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Improving the Efficiency of Variationally Enhanced Sampling with Wavelet-Based Bias Potentials

Cite this: J. Chem. Theory Comput. 2022, 18, 7, 4127–4141
Publication Date (Web):June 28, 2022
https://doi.org/10.1021/acs.jctc.2c00197

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Abstract

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Collective variable-based enhanced sampling methods are routinely used on systems with metastable states, where high free energy barriers impede the proper sampling of the free energy landscapes when using conventional molecular dynamics simulations. One such method is variationally enhanced sampling (VES), which is based on a variational principle where a bias potential in the space of some chosen slow degrees of freedom, or collective variables, is constructed by minimizing a convex functional. In practice, the bias potential is taken as a linear expansion in some basis function set. So far, primarily basis functions delocalized in the collective variable space, like plane waves, Chebyshev, or Legendre polynomials, have been used. However, there has not been an extensive study of how the convergence behavior is affected by the choice of the basis functions. In particular, it remains an open question if localized basis functions might perform better. In this work, we implement, tune, and validate Daubechies wavelets as basis functions for VES. The wavelets construct orthogonal and localized bases that exhibit an attractive multiresolution property. We evaluate the performance of wavelet and other basis functions on various systems, going from model potentials to the calcium carbonate association process in water. We observe that wavelets exhibit excellent performance and much more robust convergence behavior than all other basis functions, as well as better performance than metadynamics. In particular, using wavelet bases yields far smaller fluctuations of the bias potential within individual runs and smaller differences between independent runs. Based on our overall results, we can recommend wavelets as basis functions for VES.

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1. Introduction

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A major problem impeding conventional molecular dynamics (MD) simulations is the so-called time scale or rare event problem. Often, the molecular process of interest occurs on a much longer time scale than one can simulate in practice; in other words, it is a rare event. Thus, the system stays in a metastable state during the simulation, and one does not observe transitions to other metastable states. Despite impressive developments in specialized hardware (1,2) and MD codes (3,4) that make very efficient usage of modern graphics processing units, it is unlikely that accessible time scales will increase significantly in the near future. The speedup of individual processing units has come to an end and high-performance computing relies on the usage of massive parallelization, (5) and time is not easily parallelizable. Thus, there has been considerable interest in developing advanced methods that enhance phase space sampling and overcome this time scale problem. (6−12)
A popular class of such advanced sampling methods is the so-called collective variable (CV) based enhanced sampling methods. In these methods, we identify a few relevant coarse-grained order parameters, that is, CVs, that correspond to essential slow degrees of freedom. Typically, the selection of CV is made manually by using physical and chemical intuition (13−15) and sometimes requires a bit of trial and error, while methods based on machine learning are also showing great promise in automating this task. (16−19) The slow molecular process of interest is then associated with free energy barriers separating metastable states on the free energy surface (FES) as a function of the chosen CVs. We then enhance the sampling of the FES by introducing an external bias potential that is adaptively constructed on the fly during the simulation to reduce or even wholly flatten free energy barriers. We can trace the idea of biased sampling to the original umbrella sampling method introduced in 1977. (20) The main difference between CV-based enhanced sampling methods lies in how they construct the bias potential and which kind of biased sampling is obtained. Some examples of methods that fall into the category of CV-based enhanced sampling techniques are local elevation, (21) adaptive biasing force, (22−24) energy landscape paving, (25) multiple windows umbrella sampling, (26) Gaussian-mixture umbrella sampling, (27) nonequilibrium umbrella sampling, (6,28) metadynamics, (29−31) metabasin metadynamics, (32) parallel-bias metadynamics, (33) basis function sampling, (34) Green’s function sampling, (35) artificial neural network sampling, (36) reweighted autoencoded variational Bayes for enhanced sampling, (37) on-the-fly probability-enhanced sampling, (38,39) adaptive topography of landscapes for accelerated sampling, (40) and reweighted Jarzynski sampling. (41)
Variationally enhanced sampling (VES) (42) is a recently developed CV-based enhanced sampling method based on a variational principle. It introduces a convex functional of the bias potential that is related to the relative entropy and the Kullback–Leibler divergence. (43) To minimize the functional, we generally take the bias potential as a linear expansion in some basis function set. Bias potentials based on a neural network (44) or free energy models (45−48) have also been considered in the literature. VES not only allows for obtaining FESs but can also be used to obtain kinetic properties. (49)
The focus of this paper is the choice of basis set in the linear expansion of the bias potential within VES. So far, the basis functions employed have been primarily global functions such as plane waves, Chebyshev, or Legendre polynomials that are orthogonal but delocalized in the CV space. Gaussian basis functions have also been used. (50,51) However, there has not been an extensive study of how the choice of the basis functions affects the convergence behavior. In particular, it remains an open question if basis functions that are localized in the collective variable space might perform better. While Gaussian basis functions might be the type of localized basis functions that first comes to mind, they have the disadvantage of not forming orthogonal basis sets. Instead, a more appealing option might be Daubechies wavelet-based basis sets, (52) as they are orthogonal and exhibit an attractive multiresolution property. Daubechies wavelets have recently been used as basis functions for other applications within molecular simulations, such as density functional theory (53,54) or coarse-grained potentials. (55)
In this work, we introduce the Daubechies wavelets as basis functions for the variationally enhanced sampling method. We implement the wavelets into the PLUMED 2 code, (56) tune their parameters, and evaluate their performance on various systems, going from model potentials to the calcium carbonate association process in water. (57) We also test Gaussians and cubic B-splines as other types of localized basis functions. Section 2 presents the theory of the VES method and introduces the new basis functions. Besides the theoretical properties, we also provide details on the implementation of the new functionality into the VES module of PLUMED 2. (56) In Section 3, we present the computational details of the benchmark systems. We discuss the results of the simulations in Section 4, and in Section 5, we end with some concluding remarks.

2. Theory and Methodology

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2.1. CV-Based Enhanced Sampling

We consider a molecular system described by the set of atomic coordinates r→ and a potential energy function U(r→). Without the loss of generality, we limit our discussion to the canonical (NVT) ensemble in the following. The Boltzmann distribution, which we want to sample by molecular dynamics (MD) or Monte Carlo simulations, is defined as
P(r→)=e−βU(r→)∫dr→e−βU(r→)
(1)
where β = (kBT)−1 is the inverse of the thermal energy. In collective variable (CV) based enhanced sampling methods, we identify a few relevant CVs that correspond to critical slow degrees of freedom. The equilibrium probability distribution corresponding to a set of CVs, s(r→) = {s1(r→), s2(r→), ..., sN(r→)}, is given by
P(s)=∫dr→δ(s−s(r→))P(r→)=⟨δ(s−s(r→))⟩
(2)
while the free energy surface (FES) is defined as
F(s)=−β−1log⁡P(s)+C
(3)
where C is an additive constant.
We are generally interested in systems where the FES (or, equivalently, the equilibrium probability distribution P(s)) is hard to sample by unbiased molecular dynamics simulations. For example, the FES might be characterized by many metastable basins separated by high free energy barriers such that barrier crossings occur on far greater time scales than we can afford in simulations, that is, they are rare events.
To overcome this time scale or rare event problem, we can enhance the sampling by introducing a bias potential V(s(r→)) that acts in the space of the CVs. The introduction of this bias potential will lead to a biased (i.e., non-Boltzmann) distribution given by
PV(r→)=e−β[U(r→)+V(s(r→))]∫dr→e−β[U(r→)+V(s(r→))]
(4)
Consequently, this leads to a biased CV distribution given by
PV(s)=∫dr→δ(s−s(r→))PV(r→)∝e−β[F(s)+V(s)]
(5)
that is chosen such that the sampling is easier and free energy barriers are reduced or even completely flattened.
From the biased simulation, we can obtain an ensemble average of an observable O(r→) for the unbiased simulation through reweighting
⟨O(r→)⟩=⟨O(r→)w(r→)⟩V⟨w(r→)⟩V
(6)
where w(r→) = eβV(s(r→)) is the weight of configuration r→ and the averages on the right side are obtained in the biased ensemble. In particular, we can obtain the FES for some CV set s′ by using O(r→) = δ(s′ – s′(r→))
F(s)=−β−1log⟨δ(s′−s′(r→))w(r→)⟩V+C′
(7)
where we can ignore the denominator in eq 6 as it only gives a constant shift of the FES (i.e., we can include it in the constant C′). In practice, the reweighted FES is obtained using a reweighted histogram or kernel density estimation where each sample is weighted by the bias acting on it, w(r→) = eβV(s(r→)). The reweighting procedure of eq 6 assumes a fixed bias potential, but often, it can be used for adaptively constructed bias potentials under the assumption that the bias potential is quasi-stationary, as we discuss below.

2.2. Variationally Enhanced Sampling

In the VES method introduced by Valsson and Parinello, (42) the bias potential is constructed by minimizing a convex functional given by
Ω[V]=1βlog⁡∫dse−β[F(s)+V(s)]∫dse−βF(s)+∫dsp(s)V(s)
(8)
where p(s) is a normalized probability distribution. The stationary point of this functional is given up to a constant by
V(s)=−F(s)−1βlog⁡p(s)
(9)
which, due to the convexity of Ω[V], is the global minimum. At this minimum, the CVs are distributed according to p(s), which is consequently called a “target distribution”. It can be shown that the Ω[V] functional is related to the Kullback–Leibler divergence (or relative entropy) and the cross entropy. (43)
Thus, by minimizing Ω[V], we can construct a bias potential that leads to a sampling of the CVs according to the target distribution p(s). The most straightforward choice of the target distribution is a uniform target distribution, leading to completely flat sampling in the CV space. However, we have found it better to employ a so-called well-tempered target distribution (30,58) given by p(s) = [P(s)]1/γ/ ∫ ds [P(s)]1/γ, where γ is a parameter, named bias factor, that determines how much the sampling is enhanced as compared to the equilibrium distribution P(s).
We can determine the FES directly from the bias potential through eq 9. Alternatively, we can obtain the FES, both for the biased CVs and also for any other set of CVs, by using the reweighting procedure shown in eq 6. While the VES bias potential is time-dependent, it quickly becomes quasi-stationary. Therefore, this reweighting procedure is valid after a short initial transient in the time series that is ignored. Note that, differently from metadynamics, (31,59) we generally do not need to account for time-dependent constants when performing reweighting with VES. Furthermore, under certain conditions, the VES method can also be used to obtain kinetic properties. (49)
In practice, we perform the minimization of the Ω[V] functional by assuming a functional form of the bias potential V(s; α) that depends on a set of variational parameters α = {α1, α2, ..., αM}. Thus, we go from an abstract functional minimization to a minimization of the multidimensional function Ω(α).
The most general strategy is to take the bias potential to be a linear expansion in some set of basis functions f = {f1, f2, ..., fM}:
V(s;α)=∑iαifi(s)
(10)
We can then obtain the gradient ∇Ω(α) and the Hessian HΩ(α) as
∇Ω(α)i=∂Ω(α)∂αi=−⟨fi(s)⟩V(α)+⟨fi(s)⟩p
(11)
HΩ(α)i,j=∂2Ω(α)∂αiαj=βCov[fj(s),fi(s)]V(α)
(12)
where angular brackets denote expectation values and Cov[···] is the covariance, obtained either over the bias potential or over the target distribution.
Due to statistical sampling, the estimates of the gradient and Hessian are generally noisy. Therefore, we perform the minimization of Ω(α) using stochastic optimization algorithms. In particular, the averaged stochastic gradient descent algorithm from ref (60) has proven to be a convenient choice. In this algorithm, the instantaneous parameters are updated according to the following recursion equation:
α(n+1)=α(n)−μ[∇Ω(α¯(n))+HΩ(α¯(n))(α(n)−α¯(n))]
(13)
where μ is a constant step size and the gradient and Hessian are obtained using the averaged parameters α¯(n)=1n+1∑i=0nα(i) (i.e., the bias potential depends on the averaged parameters). The parameters are updated with a relatively small stride, on the order of 1000 MD steps. Here, we only employ the diagonal part of the Hessian matrix, as generally done in VES. (42,43)

2.3. Linear Basis Functions for VES

The focus of this paper is the basis functions used in the linear expansion of the bias potential (eq 10). So far, the basis functions employed have been global functions such as plane waves (i.e., Fourier series), (42) Chebyshev polynomials, (58) or Legendre polynomials. The usage of global functions is closely related to the idea of using spectral methods for function approximation. (61) Favorable for their usage within VES, these basis functions form complete and orthogonal basis sets. However, they are delocalized in the CV space. In other words, they are non-zero over their full domain except on isolated points.
Using global or delocalized basis functions means that, during the optimization process, the bias potential will change even in parts of CV space where the MD simulation is not currently exploring. While this has not proven to be a significant issue, it is clear that delocalized basis functions might not be the optimal choice.
In this work, we consider the performance of using VES with localized basis functions, that is, functions that are non-zero on only some part of the domain of the bias potential. Therefore, they should not suffer from the issue of the bias potential changing in parts of CV space that the simulation is not currently exploring.
Examples of such localized basis functions that come to mind would be Gaussians or splines. In fact, in refs (50) and (51), the authors employed VES with Gaussian basis functions. The results obtained with this VES setup were found to be inferior to some of the results obtained with other enhanced sampling methods used by the authors (such as umbrella sampling (20)), but as no other basis functions were used with VES, it is hard to judge the performance of the Gaussian basis from their results. However, one disadvantage with using Gaussians or splines as basis functions is that they do not form orthogonal basis sets, which might affect the optimization process.
We have thus been motivated to explore the usage of wavelets as basis functions. In particular, we consider Daubechies wavelets, (52,62) which are localized functions that form orthogonal and complete basis sets. Furthermore, they have an intrinsic multiresolution property that makes it possible to iteratively add more basis functions on smaller scales in a way that preserves the orthogonality of the basis.
In the following sections, we briefly describe the new localized basis functions─Daubechies wavelets, Gaussians, and cubic B-splines─as well as Legendre and Chebyshev polynomials that we consider for comparison. These basis functions are shown in Figure 1. We give descriptions of one-dimensional basis functions only, as basis sets for higher dimensions can be obtained by considering a tensor product. For example, in two dimensions, we obtain
V(s1,s2;α)=∑i,jαi,jgi(s1)hj(s2)
(14)
where gi(s1) and hj(s2) are some one-dimensional basis functions. All the one-dimensional basis sets described in the following are defined on some given interval [a, b] and include an additional constant basis function. In practice, for MD simulations, we also need the derivatives of the basis functions to obtain the biasing force due to the external bias potential, but this is a straightforward task for all of the basis functions considered here.

Figure 1

Figure 1. Visualization of different VES basis functions used in this paper. The Sym8 wavelets, Gaussians, and cubic B-splines are localized basis functions. Here, we only show two adjacent functions, while a full basis set would include all shifted functions in the given interval (that is, [−3, 3] here). On the contrary, Legendre polynomials are delocalized functions supported on the full interval of the bias. The Legendre basis set consists of all polynomials up to a certain order; the figure shows the functions up to the quartic polynomial.

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2.4. Daubechies Wavelet Basis Functions

Daubechies developed a theory for special types of wavelets that can be used to construct complete and orthogonal basis functions. (52) These wavelets are based on using a pair of functions: the scaling function (or father wavelet) ϕ and the wavelet function (or mother wavelet) ψ. They are defined by
ϕkj(x)=2−j/2ϕ(2−jx−k)
(15)
ψkj(x)=2−j/2ψ(2−jx−k)
(16)
for a given scale j∈Z and shift k∈Z. The exact properties are set by choosing the filter coefficients hk and gk in the refinement relations given by
ϕ(x)=2∑khkϕ(2x−k)
(17)
ψ(x)=2∑kgkϕ(2x−k)
(18)
Daubechies proved that certain finite sets of filter coefficients result in orthonormal bases. Using these wavelet functions, any square-integrable function g(x) can be approximated up to an arbitrary precision by a linear combination with coefficients α
g(x)=∑kαkϕkj(x)+∑l≥j∑kαl,kψkl(x)
(19)
where the wavelet functions satisfy orthogonality relations: (63)
∫dxϕkj(x)ϕk′j(x)=δkk′
(20)
∫dxϕkj(x)ψk′j′(x)=0⁣forj≤j′
(21)
∫dxψkj(x)ψk′j′(x)=δjj′δkk′
(22)
We can see the multiresolution property of the wavelet basis functions in eq 19. Starting with the father wavelets ϕ at some scale j, an increasingly more accurate approximation is obtained by adding mother wavelets ψ at finer scales.
In this paper, we will focus on the coarsest approximation only, which corresponds to a single level of father wavelets at some scale j
g(x)=∑kαkϕkj(x)
(23)
The exact wavelet type and the scale are left for us to choose.
The wavelet type is determined by the set of filter coefficients hk and gk. Desirable properties for our application are small support of the individual function, at least C1 regularity (one continuous derivative), and the reproduction of polynomials up to a desired order.
The wavelets developed by Daubechies satisfy these properties and in fact result in the minimally supported functions for a given polynomial order. In this paper, we consider filter coefficients that result in the least asymmetric variant of these wavelets or so-called symlets. (52) The reduced asymmetry of the symlets comes at the cost of slightly reduced regularity as compared to the conventional maximum phase Daubechies wavelets. However, this does not cause problems as we only require one continuous derivative. In practice, we found the symlets to perform better than the maximum phase Daubechies wavelets. The symlets are also used in wavelet-based density functional theory calculations. (54) We will denote the symlets by SymN, where N is equal to half the number of coefficients used for construction.
The chosen number N determines the properties of the symlets, including the number of vanishing moments of the mother wavelet. Having N vanishing moments means that all polynomial functions up to the order N – 1 are orthogonal to the mother wavelet. Consequently, any polynomial of order up to N − 1 can be represented exactly by a single level of the father wavelet ϕ (i.e., the scaling function). Employing a wavelet basis with a larger N can thus helps to construct a bias potential with less regularity and steeper slopes. On the other hand, the range over which the wavelet functions are non-zero is proportional to 2N – 1. Because the basis consists of integer-shifted functions, a larger support (i.e., non-zero range) results in more overlap between functions. This makes it necessary to use more basis functions at the same scale and thus results in more expansion coefficients to optimize. After some testing, we found that using Sym8 or Sym10 yields the best results for the systems considered in this paper. Further discussion and a comparison of symlets with different numbers of vanishing moments can be found in Section S1 of the Supporting Information (SI).
The scale j of the wavelet basis can be chosen freely. Instead of selecting the scale directly, we set the desired number of basis functions. In principle, there is an infinite number of shifted wavelet functions in the basis. However, only a few of them are supported inside the range [a, b] on which the bias potential is defined. Furthermore, they are non-zero only on a small part of their domain. Thus, we choose to only include the ones with any (absolute) function value inside the bias range that is at least 1% of the maximal function value. We then calculate the required scaling to arrive at the desired number of basis functions. We did not observe disadvantages from excluding wavelets with minor contributions, while it allows us to reduce the number of coefficients to be optimized.
Generally, using a smaller scale and, consequently, more basis functions allows us to represent finer features better at the cost of needing to optimize more variational parameters. In Section S1 of the SI, we show results where we change the number of the basis functions for a fixed N value.

2.5. Gaussian Basis Functions

Gaussian basis functions are given by the mathematical expression
fi(x)=exp[−(x−μi)22σ2]
(24)
where μi is the center of the individual Gaussian and σ is a constant width parameter. The full basis set is then given by Gaussian functions with centers distributed evenly on the interval [a, b]. We add the first center at μ0 = a and define the shift between centers as d = μi – μi – 1 = (b – a)/N, where N is a user-specified integer fixing the number of basis functions.
To mitigate systematic errors at the boundaries, we add one function on each side outside the range, resulting in a total of N + 3 basis functions including the constant. As the force from the VES bias is zero outside the chosen interval by design, these additional functions will only contribute inside the bias range, similar to the boundary correction approach for metadynamics in ref (64). Although more complicated boundary correction algorithms have been developed, (65,66) we found our simple approach to work well.
The width σ of the Gaussians is set by the user. For a fixed number of Gaussians, the possible resolution of the basis can be increased by choosing Gaussians with a smaller width. However, reducing the width will reduce the overlap between Gaussians, and a too-small width will result in an ill-behaving basis set. Thus, the optimal width, which very likely is system dependent, is the smallest one that still results in good convergence. In refs (50) and (51), the width σ was set to be equal to the distance d between the centers of the Gaussians. However, as shown in Section S2 in the SI, we found improved performance when using a smaller width of σ = 0.75d. Because this yielded better results for the model systems considered here, we will show only Gaussian results obtained with this optimal width in the rest of the paper, while we refer the reader to the SI for results obtained with other σ values.

2.6. Cubic B-Spline Basis Functions

We consider the cubic B-spline basis functions from ref (67) that are given by the mathematical expression
fi(x)=h(x−μiσ)
(25)
where
h(t)={(2−|t|)3,1≤|t|≤24−6|t|2+3|t|3,|t|≤10,elsewhere
(26)
Here μi is the center of the cubic B-spline basis function, and σ is the width. The full basis set is then given by spline functions with centers distributed evenly on the interval [a, b]. The first center is set on the left boundary μ0 = a, and we define the shift between centers as d = μi – μi – 1 = (b – a)/N, where N is a user-specified integer fixing the number of basis functions. Similar to the Gaussian basis functions, to avoid boundary effects, we add functions on each side outside the range, resulting in a total of N + 3 basis functions including the constant. Different from the Gaussians, the width σ is fixed and taken as equal to the distance between centers, σ = d.

2.7. Legendre and Chebyshev Polynomial Basis Functions

Legendre and Chebyshev polynomials form sets of orthogonal basis functions on a closed interval that is matched to the range of the bias potential. Contrary to the previously described bases, the basis functions are not localized in a specific part of the interval but are non-zero except on isolated points. Chebyshev polynomials of the first kind are given by the recursion relations
C0(x)=1
(27)
C1(x)=x
(28)
Cn+1(x)=2xCn(x)−Cn−1(x)
(29)
while the recursion relations of the Legendre polynomials are
L0(x)=1
(30)
L1(x)=x
(31)
Ln+1(x)=2n+1n+1xLn(x)−nn+1Ln−1(x)
(32)
Both Chebyshev and Legendre polynomials are defined intrinsically on the interval [−1, 1] and need to be scaled and shifted when employed on different intervals. For a given interval [a, b], we use the following function to transform t ∈ [a, b] to x ∈ [−1, 1]:
x(t)=2t−(a+b)b−a
(33)

2.8. Implementation of New Basis Functions

We have implemented the new basis functions into the VES module of the PLUMED 2 code. (56,68) Our implementation is publicly available in the official PLUMED 2 GitHub repository, and it is released in version 2.8 of PLUMED.
While it was straightforward to implement Gaussians and splines, wavelets pose the problem of not having an analytic mathematical expression. Instead, in the beginning of the simulation, we generate the wavelet values and derivatives on a grid through an iterative scheme. We then use the grid as a lookup table during the simulation. This means that the computational overhead of using the wavelets is minimal. To generate the wavelet grid, both for the values and for the derivatives, we employ a vector cascade algorithm (69) that relies on finding eigenvectors of a characteristic matrix and subsequent vector–matrix multiplications to iteratively get values on an increasingly finer spaced grid. We calculate the exact values on a grid of at least 1000 points and use linear interpolation to obtain in-between values.
As localized functions are non-zero only in a small region of the total CV space, we have to modify the optimization scheme slightly. If there is no sampling in the non-zero region of a basis function during one iteration of the bias potential, the elements of gradient and Hessian corresponding to that basis function are set to zero before updating the variational parameters. This is needed because the gradient elements for these basis functions might still be non-zero due to the average over the target distribution (the second term in eq 11). Setting them to zero prevents erroneous updates of variational parameters if no sampling of the non-zero region occurred. Note that this procedure is done only for individual elements, so the total gradient vector and Hessian matrix still include non-zero elements.
We note that our implementation of the wavelet, Gaussian, and spline basis functions also supports periodic CVs. Furthermore, in addition to the least asymmetric wavelets (i.e., symlets) that we use in this work, the wavelet implementation also supports conventional maximum phase Daubechies wavelets. However, we found the latter to perform worse when compared to the symlets.

3. Computational Details

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To evaluate the performance of the different basis functions, we perform simulations on different systems, going from model potentials in one and two dimensions to a realistic system modeling the association process of calcium with carbonate in water.

3.1. Double-Well Potential

We start by considering a single particle moving in a one-dimensional model potential given by
U(x)=x4−4x2+0.7x
(34)
that has two states separated by a barrier of around 5 energy units. The form of this potential can be seen in Figure 2a. We take the x-coordinate as the CV such that the reference FES will be given by the potential above, F(x) = U(x) (up to an additive constant). We employ the ves_md_linearexpansion command line tool from the VES code for the simulations. The ves_md_linearexpansion tool implements a simple molecular dynamics integrator with a Langevin thermostat. (70) We use a time step of 0.005 and a friction coefficient of 10 for the Langevin thermostat. We set the temperature to T = 0.5/kB such that the barrier height is about 10 kBT (kB = 1). We choose to run simulations with four different basis sets: Sym8 wavelets, Gaussians, cubic B-splines, and Legendre polynomials. We expand the bias potential in the interval from −3 to 3 and fix the number of basis functions to 22 for each basis set to allow for a fair comparison. We employ a uniform target distribution and update the coefficients of the bias potential every 500 steps. The step size μ in the averaged stochastic gradient descent optimization algorithm (eq 13) was adjusted to yield the fastest convergence for each basis set. We set it to μ = 0.5 for simulations using localized basis functions and decrease it to μ = 0.1 for the simulations with Legendre polynomials. Each simulation is run for 5 × 106 steps, while the FES was determined every 5 × 104 steps via eq 9. For each basis set, we run 20 independent simulations that are started in the global minimum with different random seeds for the initial velocities and random forces.

Figure 2

Figure 2. Results for the one-dimensional double-well potential described in Section 3.1. (a) The reference FES along with the FES obtained using the wavelet basis functions at different numbers of bias iterations for one of the runs. (b) The RMS error measure (Section 3.5, eq 36) for the different basis functions as a function of the number of bias iterations. The lines denote the average over 20 independent runs, and the shaded areas denote the corresponding standard error. (c, d) The RMS error of the individual runs for (c) Sym8 wavelets and (d) Legendre polynomials. The thick lines are the same as in panel b, and the dashed lines each resemble one of the runs.

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3.2. Wolfe–Quapp Potential

The second model potential is the two-dimensional Wolfe–Quapp potential: (71,72)
U(x,y)=x4+y4−2x2−4y2+xy+0.3x+0.1y
(35)
that has two states separated by a high barrier along the y-coordinate, while along the x-coordinate, the mobility is high. The potential can be seen in Figure 3 along with projections on the x- and y-coordinates. We take both the x-coordinate and the y-coordinate as CVs such that the reference FES will be given by the potential F(x, y) = U(x, y) (up to an additive constant). We bias both CVs in the interval from −3 to 3 using 22 basis functions per CV (484 two-dimensional basis functions in total). We set the temperature to T = 1/kB. We set the step size for all simulations to μ = 0.5. We run 20 independent simulations for each basis set. Otherwise, we employ the same basis functions and simulation parameters as for the one-dimensional potential in the previous section.

Figure 3

Figure 3. Results for the two-dimensional Wolfe–Quapp potential described in Section 3.2. (a) The reference FES along with free energy projections on the x- and y-coordinates. (b, c) The free energy difference ΔF (Section 3.5, eq 38) between the two states obtained using (b) Sym8 wavelets and (c) Legendre polynomials as a function of the number of bias iterations. We show results from 20 independent simulations with dashed lines. We use solid lines for the averages and shaded areas to denote the standard errors. We denote the reference value with solid black lines. To define the areas corresponding to the two different states, we use the y = 0 line.

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3.3. Rotated Wolfe–Quapp Potential

To test the behavior when biasing only a suboptimal CV, we consider a rotated and scaled version of the Wolfe–Quapp potential. As in ref (48), the potential is rotated by an angle of θ = −0.15π. The potential energy surface is given in Figure 4 together with projections on the x- and y-coordinates. We take only the x-coordinate as a biased CV, which results in missing orthogonal slow degrees of freedom (the y-coordinate). The reference FES for the x-coordinate can be obtained by integrating over the y-coordinate, F(x) = −β–1 log ∫ dy e–βU(x, y). We use a temperature of T = 1/kB. We expand the bias potential in the interval from −3 to 3 and fix the number of basis functions to 22 for each basis set. We employ a uniform target distribution and update the coefficients of the bias potential every 500 steps. Otherwise, we employ the same basis functions and simulation parameters as for the previous two model potentials.

Figure 4

Figure 4. Results for the rotated two-dimensional Wolfe–Quapp potential described in Section 3.3. (a) The reference FES along with free energy projections on the x- and y-coordinates. Only the x-coordinate is biased. (b, c) The free energy difference ΔF (Section 3.5, eq 38) between the two states obtained using (b) Sym8 wavelets and (c) Legendre polynomials as a function of the number of bias iterations. We show results from 20 independent simulations with dashed lines. We use solid lines for the averages and shaded areas to denote the standard errors. We denote the reference value with solid black lines. To define the areas corresponding to the two different states, we use the x = 0 line.

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For this system, we observe that using the averaged stochastic gradient descent optimization algorithm does not yield good convergence for the localized basis functions. Therefore, we use the Adam stochastic gradient descent algorithm, (73) which has been used previously for VES in combination with neural networks. (44) Details of the Adam algorithm can be found in Section S3 in the SI. We notice a high sensitivity of the convergence to the step size η of the Adam algorithm. Although the standard value of η = 0.001 works in most cases, the convergence of the bias is slow, especially for simulations with Sym8 wavelets. Increasing it to η = 0.005 provides much better behavior, whereas increasing it even further results in nonconverging simulations with Legendre polynomials. We use η = 0.005 for all simulations with the Adam algorithm but note explicitly that the choice of parameters seems crucial for good convergence.
While the usage of the Adam algorithm helps improve the convergence for this system, we find a worse performance in comparison to the averaged stochastic gradient descent algorithm when testing it on the other systems considered in this paper. Therefore, further investigation is needed to understand the optimal choice for stochastic optimization. The choice very likely depends on the form of the bias potential (e.g., a linear expansion versus a neural network (44) or a bespoke model (45−48)) and the basis functions used. An interesting idea might be to combine ideas from different algorithms, similar to what was done in ref (48) where the authors introduced a combination between AdaGrad and Bach’s algorithms. However, a detailed investigation of the stochastic optimization algorithm used within VES is beyond the scope of the current work.

3.4. Calcium Carbonate Association

To study the performance of wavelet basis functions for a realistic system, we consider the association process of a calcium carbonate ion-pair in water. We use the LAMMPS code (74) (5 June 2019 release) interfaced with the PLUMED 2 code for the simulations. We employ the calcium carbonate force field developed in refs (75) and (76) and the SPC/Fw (77) water model. We follow the computational setup used in a previous metadynamics study of the association process (57) using this force field. We set up a system that contains a single Ca2+–CO32– ion-pair and 2448 water molecules in a periodic cubic box. We equilibrate the system in the NPT ensemble at a constant temperature of 300 K and a constant pressure of 1 bar for 500 ps. All subsequent simulations are performed in the NVT ensemble using a constant temperature of 300 K and a cubic box with side lengths of 41.69 Å. We run 5 ns of unbiased MD simulations from which we select in total 75 snapshots that we use as initial configurations for the biased simulations. We employ a time step of 0.001 ps. All simulations are performed at a constant temperature of 300 K using a Nosé–Hoover thermostat (78−80) with a chain length of 5 and a relaxation time of 0.1 ps. For the NPT equilibration, we employ a Nosé–Hoover barostat with a relaxation time of 1 ps to keep a constant pressure of 1 bar. Electrostatic interactions are calculated according to the PPPM method (81) with an accuracy of 10–5.
We use the same CVs as in ref (57), namely, the distance between the Ca and C atoms and the coordination number of Ca with water (see Section S5 in the SI for further details). As in the original work, (57) we use the technique of multiple walkers (82) with 25 walkers running in parallel to improve convergence, where each walker starts from a different initial configuration. We employ Sym10 wavelets or Chebyshev polynomials as basis functions. For the CV corresponding to the distance between the Ca ion and C atom of the carbonate ion, we use 60 basis functions in the range from 2 to 12 Å. For the CV corresponding to the coordination number, we use 30 basis functions in the range 5 to 9. The total number of two-dimensional basis functions is then 1200. Due to the usage of multiple walkers, we update the coefficients of the bias potential more frequently, that is, every 10 MD steps (the total number of data points for each iteration is then 250). We use the averaged stochastic gradient descent optimization algorithm with a step size of μ = 0.001 for the Sym10 wavelets. For simulations with Chebyshev polynomials, this does not always result in stable simulations, and we use a lower step size of μ = 0.0005 for these. We employ a well-tempered target distribution (58) with a bias factor of 5, where the target distribution is iteratively updated every 100 bias potential updates (1000 MD steps). We run each walkers for 3 ns, resulting in a cumulative simulation time of 75 ns.
For comparison, we also perform a well-tempered metadynamics (WTMetad) (30) simulation using the same setup as in ref (57). The bias factor is set to 5. For the Gaussians, we use an initial height of 1 kBT and widths of 0.2 Å and 0.1 for the distance and coordination number, respectively. We deposit Gaussians every 1 ps (1000 MD steps). For the metadynamics simulations, we also run each walker for 3 ns, resulting in a cumulative simulation time of 75 ns.
To focus the sampling in the part of the configuration space of interest for the association process, we add an artificial repulsive wall at a Ca–C distance of 11 Å in all simulations to prevent the ions from moving further apart. In practice, this is implemented by a harmonic bias of the form κ(x – x0)2 where we set the parameters to κ = 12 eV and x0 = 11 Å.
To obtain the reweighted FESs, we employ a reweighted kernel density estimation as implemented in PLUMED 2. We use Gaussian kernels with bandwidths of 0.05 Å and 0.05 for the Ca–C distance and coordination number CV, respectively. We ignore the first 200 ps of each walker and use samples obtained every 0.1 ps. For the metadynamics simulations, we use the c(t) reweighting scheme described in refs (31) and (59). During the metadynamics simulations, we calculate the time-dependent constant c(t) needed for the biasing weights every time a Gaussian is added using a grid of 275 × 300 over the domain [2,13] × [3,10].
To assess the stability of the simulations, we perform three independent runs using different initial configurations for each of the three biasing setups (VES with wavelets, VES with Chebyshev polynomials, and WTMetaD).

3.5. Performance Measures

To evaluate and compare the performance of the basis functions, we consider two different performance measures: the root mean square error with respect to a reference and the free energy difference between some two metastable states.
To measure the quality of the FES F(s) obtained directly from the bias through eq 9, we calculate the root mean square (RMS) error of the FES with respect to a reference as done in refs (58) and (83). Given some reference FES Fref(s), the RMS error is given by
ϵ=∫ds[F(s)−Fref(s)]2θ(ν−Fref(s))∫dsθ(ν−Fref(s))
(36)
where we perform the integration over the full CV space and θ is a Heaviside step function such that only regions with a free energy lower than a threshold value ν are considered. Since the FESs are only determined up to a constant, we shift them by their average value in the region of interest, that is, we use
F~(s)=F(s)−∫ΓdsF(s)+∫ΓdsFref(s)
(37)
to calculate the error metric in eq 36, where Γ is taken as the region of CV space where Fref(s) ≤ 4 kBT. We set the parameter ν = 8 kBT. We consider always an ensemble of multiple independent runs that are initiated with different initial conditions because a single simulation might not be representive. (84,85) We then compare the mean RMS error as well as the associated standard error of the mean.
Another performance measure we can employ is to calculate the free energy difference ΔFA, B between two different states: (31)
ΔFA,B=FA−FB=−1β⁡log⁡PAPB=−1β⁡log⁡∫Adsexp[−βF(s)]∫Bdsexp[−βF(s)]
(38)
where the domains of integration are the regions in CV space associated with the states A and B, respectively.

3.6. Data Availability

The data supporting the results reported in this paper are openly available at Zenodo (86) (DOI: 10.5281/zenodo.5851773). All LAMMPS and PLUMED 2 input files and analysis scripts required to reproduce the results reported in this paper are available on PLUMED-NEST (www.plumed-nest.org), the public repository of the PLUMED consortium, (68) as plumID:22.001 at https://www.plumed-nest.org/eggs/22/001.

4. Results and Discussion

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4.1. Model Potentials

A common way to test the performance of methodological developments of enhanced sampling methods is to consider the dynamics of a single particle on model potentials that emulate prototypical free energy landscapes. We, therefore, start by considering three model potentials, where we compare the performance of the localized basis functions (Sym8 wavelets, Gaussians, and cubic B-splines) to the delocalized Legendre polynomials that have been used as basis functions within VES so far. For these simulations, we always perform 20 independent runs for each set of basis functions and use the performance measures that we have described in Section 3.5 to compare the FESs obtained from the bias potential via eq 9.
We start by considering the one-dimensional double-well potential shown in Figure 2a that has a high free energy barrier of around 10 kBT when going from the left to right side. In panel a of Figure 2, we show an example of the FES obtained using wavelet basis functions at different bias iterations. In the SI, we present a movie showing the time evolution of the FES of exemplary simulations for all different basis sets. In panel b, we show the RMS error metric (eq 36) for the different basis functions. We can observe that, on average, the FES (or equivalently the bias) converges considerably faster with the localized basis functions than with the delocalized Legendre polynomials. Furthermore, the localized basis functions converge to a better estimate of the FES as indicated by the smaller RMS error. We can observe that the wavelets perform the best of the three localized basis functions.
In Figure 2b, we can also observe considerably larger fluctuations in the average RMS error and a larger standard error for the Legendre polynomials. The reason for this is twofold, as we can see from looking at the RMS error for the individual runs, shown in panels c and d for the wavelets and the Legendre polynomials, respectively. First, within each individual simulation, the bias potential is fluctuating more for the Legendre polynomials. Second, there is a more significant difference between runs for the Legendre polynomials. In comparison, the wavelets show a much more robust behavior with considerably smaller fluctuations within individual runs and more minor differences between runs. We can see a similar effect for the Gaussians and cubic B-splines, although they do not behave as well as the wavelets (see Figure S5 in the SI). Therefore, for this simple system, we can already see the benefits of using localized basis functions.
In the following, we will focus on the wavelets and the Legendre polynomials, while we refer the reader to the SI for results for the Gaussians and cubic B-splines. Furthermore, we will only use the free energy difference to compare the basis functions while presenting the results for the RMS error metric in the SI.
The next system that we consider is the two-dimensional Wolfe–Quapp potential (71,72) that is a commonly used model potential for testing methods. (72,87−89) We show its free energy surface, along with the free energy projections on the x- and y-coordinates, in Figure 3a. The potential has two states separated by a barrier along the y-coordinate, while the system is relatively mobile along the x-coordinate. Still, due to a strong coupling between the x- and y-coordinate, it is essential to consider both coordinates as biased CVs to get a good sampling. We thus expand the two-dimensional bias potential in a tensor product basis set of one-dimensional basis functions.
In panels b and c of Figure 3, we show the free energy difference between the two states for the wavelets and the Legendre polynomials, respectively. We can see a rather similar behavior as for the one-dimensional model potential. The wavelets exhibit far smaller fluctuations within individual runs and considerably smaller differences between runs than the Legendre polynomials. Looking at the averaged free energy difference, we can see that the wavelet simulations converge substantially better and faster than the Legendre polynomials. We can draw similar conclusions by considering the RMS error measure shown in Figure S7 in the SI.
We show the estimates of the free energy difference from the simulations with Gaussians and cubic B-spline basis functions in Figure S7 in the SI. We can observe that the Gaussians perform better than the Legendre polynomials but worse than the wavelets. However, we find that cubic B-splines perform the worst of all the basis functions and do not yield usable results for this system.
Finally, we consider a rotated version of the Wolfe–Quapp potential shown in Figure 4a that has been used as a test case for biasing suboptimal CVs. (44,48,90) We only take the x-coordinate as a CV for biasing, so we are missing the y-coordinate that is an orthogonal slow degree of freedom. We show the free energy difference between the two states in panels b and c of Figure 4. As expected, due to the usage of a suboptimal CV, the convergence behavior is slightly worse than for the previous two systems, and we need longer simulation times to obtain adequate convergence. Nevertheless, the wavelets exhibit good convergence behavior that, as before, is more robust than for the Legendre polynomials. As shown in Figure S8 in the SI, the Gaussians and the cubic B-splines perform worse than both wavelets and Legendre polynomials.
As discussed in Section 3.3, for this system, we have used a different optimization algorithm, the Adam optimizater, (73) instead of the averaged stochastic gradient descent. (60) This choice might explain a slightly different behavior in the time evolution of individual runs as compared to the previous two systems.
Having tested the localized basis functions on three different model systems, we can draw certain conclusions. The wavelet basis functions exhibit much more robust convergence behavior than the Legendre polynomials. For the wavelets, the fluctuations of the bias potential within individual runs are smaller. Additionally, the difference between independent runs is considerably smaller. The Gaussian and the cubic B-spline basis functions perform worse than the wavelets for all considered systems and do not yield usable results for some systems. Therefore, we recommend against their usage. Having established the excellent performance of the wavelets in model systems, we now move on to their use in a more realistic system.

4.2. Calcium Carbonate Association

For a more realistic system, we consider the association process of calcium carbonate in water that has previously been investigated in ref (57) using metadynamics simulations. In that work, the authors used the technique of multiple walkers (82) with 25 walkers to improve the convergence. Here, we will follow the same procedure for the wavelet and Chebyshev polynomial simulations. For comparison, we also run well-tempered metadynamics simulations using the same computational setup as used in ref (57). For each of the three biasing setups (VES with Sym10 wavelets, VES with Chebyshev polynomials, and WTMetaD), we run three independent simulations.
In Figure 5b, we show the free energy surface as a function of the two biased CVs: the distance between the calcium and the carbon atom of the carbonate and the coordination number of the calcium to the oxygens of the water molecules. We can see that to fully understand the association process, it is necessary to consider both CVs as the solvation state of the calcium, as measured by the coordination number CV, is closely coupled to the calcium–carbon distance. The minima of the FES with a Ca–C distance smaller than 4 Å correspond to the states with a contact ion-pair. The lowest state of the FES is the monodentate associated state at around 3.5 Å. At a lower coordination number and smaller distance, a second minimum corresponding to the bidentate state can be seen. For larger Ca–C distance, the ions are no longer in direct contact but are separated by the solvent. The states with a distance of around 5 Å correspond to the solvent-shared ion-pair, while the states around 7 Å denote where the solvation shells of the two ions barely touch. For even larger distances, the two ions are fully solvated.

Figure 5

Figure 5. Free energy surfaces for the calcium carbonate system described in Section 3.4. (a) Projections on the distance CV for the FESs obtained directly from the bias via eq 9 (VES) or by summing over the deposited Gaussians (WTMetad). We only show one of the runs for each biasing setup. The reference data are obtained from ref (57). (b) FES as a function of both biased CVs obtained by reweighting one of the wavelet simulations.

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To compare the different simulations, we look at the projections of the FES on the distance CV that is shown in Figure 5a. These free energy profiles are obtained at the end of simulations directly from the bias potential, that is, via eq 9 for the VES simulations or by summing up the deposited Gaussians for the WTMetad simulations. For each of the three biasing setups, we only show one representative free energy profile, while the profiles for the other runs are shown in Figure S9 in the SI. We also show three reference profiles from ref (57). All the free energy profiles are aligned such that their minimum is at zero.
We can observe in Figure 5a that all the free energy profiles obtained from our simulations are in a decent agreement with each other and the reference results from ref (57). All the simulations capture reasonably well the small barrier between the mono- and bidentate states at about 3 Å, though we should mention that this barrier in the one-dimensional profile does not represent the true barrier of the physical process due to integration over the solvent degree of freedom (i.e., the coordination number CV in the FES shown in panel b). For the dissociated state above 4 Å, we can observe that there are some differences between runs. However, we can observe similar variance between the three reference runs from ref (57) as shown in Figure S9 in the SI. Therefore, it is difficult for us to say what the correct free energy profile is. Furthermore, our results in panel b are obtained at the end of the simulations and do not reflect that the bias─and thus the obtained FES─fluctuates during the simulation. Indeed, one of the main conclusions from Section 4.1 was that the fluctuations of the bias potential within individual runs were considerably smaller for the wavelets as compared to the polynomial basis functions.
To gauge the time evolution of the bias potential and FES, we consider the free energy difference between the contact ion-pair and loosely associated states of calcium carbonate. We select the region in CV space with a distance smaller than 4 Å as the contact ion-pair state and the region with distances between 4 and 8 Å as the loosely associated state and calculate the free energy difference according to eq 38. We note that this selection of the two regions does not necessarily coincide with the chemical definitions of ion association states. (57) Here, we employ the free energy difference to monitor the stability of the bias potential and the obtained FES. In Figure 6a, we show the free energy difference obtained every 10 ps (simulation time per walker). For each of the biasing setups, we show the results from three independent runs.

Figure 6

Figure 6. Results for the calcium carbonate system described in Section 3.4. (a, c) Time evolution of the free energy difference between the region with a Ca–C distance smaller 4 Å and the region with a Ca–C distance between 4 and 8 Å. For each biasing setup, we show three independent runs where the different color shades represent the individual runs. (b, d) The average of the free energy differences obtained over the last nanosecond by using 100 samples taken every 10 ps for each simulation. The error bars show the standard deviation to signify the quality of the individual measurements. We also show the results from ref (57) as black dotted lines. In panels a and b, we use the FES obtained directly from the bias via eq 9 (VES) or by summing over the deposited Gaussians (WTMetad). In panels c and d, we use the FES obtained through reweighting where we ignore the first 200 ps of each simulation.

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In Figure 6a, we can see that the free energy differences obtained from the wavelet simulations converge faster and show less fluctuations than in the Chebyshev polynomial simulations. In particular, there are considerably larger fluctuations in the Chebyshev polynomial simulations. Furthermore, there is less difference between independent runs for the wavelets as compared to the Chebyshev polynomials. Therefore, when comparing the wavelets and the Chebyshev polynomials, we obtain the same conclusions as for the model system in Section 4.1: the wavelets exhibit less fluctuations of the bias potential within individual runs and less difference between different independent runs. The metadynamics simulations show a convergence behavior that is slightly worse than the wavelet simulations but still better than the Chebyshev polynomial simulations.
To further quantify the behavior of the simulations, we calculate the average and the standard deviation over the last nanosecond of each simulation and show the results in Figure 6b (numerical values are given in Table S1 in the SI). We chose the standard deviation because the time series from a single simulation is highly correlated and does not correspond to independent measurements. The standard deviation is thus shown as a measure of how much the free energy difference and thus also the bias fluctuate even at the end of the simulation. We can see that there is some spread in the averaged values, though all simulations agree with each other within 1 kJ/mol. We note that there is a similar spread in the three reference metadynamics simulations from ref (57) that are shown as black dotted lines in Figure 6b. Therefore, we cannot determine a reference value of the free energy difference. Noticeably, and consistent with the free energy differences time evolution in panel a, the wavelet simulations have the smallest standard deviation values, while the values are three to six times larger for the Chebyshev polynomial and metadynamics simulations.
From the results in panels a and b of Figure 6, we can conclude that the wavelets perform the best when considering the difference between independent simulations and fluctuations within runs.
So far, we have estimated the FES directly from the bias potential. An alternative way to obtain the FES is through reweighting. In fact, it is always a good practice to estimate the FES both directly from the bias potential and via reweighting and compare the results. The reweighting procedure assumes that the bias potential (i.e., the weights) is quasi-stationary. Therefore, we can expect the wavelets to perform better in this respect.
In panel c of Figure 6, we show the free energy difference values obtained from reweighted FESs every 10 ps. As before, we calculate the average and the standard deviation over the last nanosecond and present it in panel d of Figure 6, while numerical values are given in Table S1 in the SI. We can see that there are much smaller fluctuations in the free energy difference for all of the simulations as compared to panel b. All of the wavelet results agree well with each other and, when combined, yield a numerical estimate of 9.13 ± 0.04 kJ/mol (see Table S1 in the SI). There is more spread for the Chebyshev polynomial and the metadynamics simulations, but as before, all simulations agree within 1 kJ/mol. The reweighted metadynamics values tend to be lower than values obtained directly from the bias potential in panel b and closer to the wavelet results. As for the results obtained directly from the bias potential, we can conclude from the reweighted results that the wavelets perform the best when considering the difference between independent simulations and fluctuations within runs.
Overall, for the calcium carbonate association, we find that the wavelet basis functions exhibit an excellent performance. The wavelets result in a considerably better convergence behavior than the Chebyshev polynomials. The wavelet simulations also show a better convergence behavior than the metadynamics simulations.

5. Conclusions

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In this work, we have introduced the usage of Daubechies wavelets as basis functions for variationally enhanced sampling. We have implemented the wavelets into the VES module of the PLUMED 2 code, (56) tuned their parameters, and evaluated their performance on model systems and the calcium carbonate association process. Overall, the localized wavelet basis functions exhibit excellent performance and much more robust convergence behavior than the delocalized Chebyshev and Legendre polynomials used as basis functions within VES so far. In particular, the wavelet bases exhibit far smaller fluctuations of the bias potential within individual runs and smaller differences between independent runs. Less fluctuation of the bias potential is important when obtaining FESs and other equilibrium properties through reweighting as the reweighting procedure assumes a quasi-stationary bias potential. Based on our overall results, we can recommend wavelets as basis functions for variationally enhanced sampling.
We have also tested Gaussians and cubic B-splines as other types of localized basis functions. However, the Gaussian and the cubic B-spline basis functions perform worse than the wavelets for all the model systems in Section 4.1 and do not yield usable results for some systems. Therefore, we recommend against the usage of Gaussians and cubic B-splines as basis functions for VES.
One attractive feature of the wavelets basis functions is the multiresolution property displayed in eq 19. Starting with the father wavelets at some given scale, we can obtain a more accurate approximation of the FES by adding mother wavelets at finer scales. Here, we only employ a single level of father wavelets to expand the bias potential. An interesting future work would be to go beyond this and implement a multiresolution bias potential where we can increase the resolution on the fly during the simulation. Coupling this with a method to evaluate the quality of the current bias potential on the fly (for example, by using the effective sample size (38,85,91,92)) could allow us to automatically construct the VES bias potential with a predefined accuracy without the need to adapt the parameters manually.
Also, in the present work, we focused on a single type of wavelets, the family of Daubechies wavelets in their least asymmetric form. We also initially tested the Daubechies wavelets with an extremal phase. However, due to their noticeably worse performance than the symlets, we did not include them in this work’s extensive study. Nevertheless, other wavelet families could yield better performance for specific systems. For example, the boundary wavelets (93) or the multiwavelets developed by Donovan et al. (94,95) might be worthwhile to consider.

Supporting Information

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The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acs.jctc.2c00197.

  • (S1) The effect of the type and scaling parameters for the Daubechies wavelet basis functions, (S2) the effect of the width parameter for the Gaussian basis functions, (S3) the Adam stochastic gradient descent algorithm, (S4) additional figures for the model potentials, (S5) the collective variables for the calcium carbonate system, (S6) additional figures for the calcium carbonate system, and (S7) numerical results for the calcium carbonate system (PDF)

  • Video illustrating the time evolution of the FES estimates of the VES method for different basis sets (MP4)

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Most electronic Supporting Information files are available without a subscription to ACS Web Editions. Such files may be downloaded by article for research use (if there is a public use license linked to the relevant article, that license may permit other uses). Permission may be obtained from ACS for other uses through requests via the RightsLink permission system: http://pubs.acs.org/page/copyright/permissions.html.

Author Information

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  • Corresponding Author
  • Author
    • Benjamin Pampel - Max Planck Institute for Polymer Research, Ackermannweg 10, D-55128 Mainz, Germany
  • Funding

    Open access funded by Max Planck Society.

  • Notes
    The authors declare no competing financial interest.

Acknowledgments

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We thank Paolo Raiteri (Curtin University) for providing the force field for the calcium carbonate system from refs (75) and (76) and the reference data from ref (57). We also thank Stephan Goedecker (University of Basel) for valuable discussions and Robinson Cortes-Huerto and Martin Girard (Max Planck Institute for Polymer Research) for careful reading of the manuscript. We acknowledge support from the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation), Project 233630050, TRR 146 “Multiscale Simulation Methods for Soft Matter Systems”.

References

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    Shaw, D. E.; Adams, P. J.; Azaria, A.; Bank, J. A.; Batson, B.; Bell, A.; Bergdorf, M.; Bhatt, J.; Butts, J. A.; Correia, T.; Dirks, R. M.; Dror, R. O.; Eastwood, M. P.; Edwards, B.; Even, A.; Feldmann, P.; Fenn, M.; Fenton, C. H.; Forte, A.; Gagliardo, J.; Gill, G.; Gorlatova, M.; Greskamp, B.; Grossman, J. P.; Gullingsrud, J.; Harper, A.; Hasenplaugh, W.; Heily, M.; Heshmat, B. C.; Hunt, J.; Ierardi, D. J.; Iserovich, L.; Jackson, B. L.; Johnson, N. P.; Kirk, M. M.; Klepeis, J. L.; Kuskin, J. S.; Mackenzie, K. M.; Mader, R. J.; McGowen, R.; McLaughlin, A.; Moraes, M. A.; Nasr, M. H.; Nociolo, L. J.; O’Donnell, L.; Parker, A.; Peticolas, J. L.; Pocina, G.; Predescu, C.; Quan, T.; Salmon, J. K.; Schwink, C.; Shim, K. S.; Siddique, N.; Spengler, J.; Szalay, T.; Tabladillo, R.; Tartler, R.; Taube, A. G.; Theobald, M.; Towles, B.; Vick, W.; Wang, S. C.; Wazlowski, M.; Weingarten, M. J.; Williams, J. M.; Yuh, K. A. Anton 3: Twenty Microseconds of Molecular Dynamics Simulation before Lunch. In Proceedings of the International Conference for High Performance Computing, Networking, Storage and Analysis; Association for Computing Machinery, 2021.
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    Phillips, J. C.; Hardy, D. J.; Maia, J. D. C.; Stone, J. E.; Ribeiro, J. V.; Bernardi, R. C.; Buch, R.; Fiorin, G.; Hénin, J.; Jiang, W.; McGreevy, R.; Melo, M. C. R.; Radak, B. K.; Skeel, R. D.; Singharoy, A.; Wang, Y.; Roux, B.; Aksimentiev, A.; Luthey-Schulten, Z.; Kalé, L. V.; Schulten, K.; Chipot, C.; Tajkhorshid, E. Scalable Molecular Dynamics on CPU and GPU Architectures with NAMD. J. Chem. Phys. 2020, 153, 044130  DOI: 10.1063/5.0014475
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    Scalable molecular dynamics on CPU and GPU architectures with NAMD
    Phillips, James C.; Hardy, David J.; Maia, Julio D. C.; Stone, John E.; Ribeiro, Joao V.; Bernardi, Rafael C.; Buch, Ronak; Fiorin, Giacomo; Henin, Jerome; Jiang, Wei; McGreevy, Ryan; Melo, Marcelo C. R.; Radak, Brian K.; Skeel, Robert D.; Singharoy, Abhishek; Wang, Yi; Roux, Benoit; Aksimentiev, Aleksei; Luthey-Schulten, Zaida; Kale, Laxmikant V.; Schulten, Klaus; Chipot, Christophe; Tajkhorshid, Emad
    Journal of Chemical Physics (2020), 153 (4), 044130CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
    A review. NAMD is a mol. dynamics program designed for high-performance simulations of very large biol. objects on CPU- and GPU-based architectures. NAMD offers scalable performance on petascale parallel supercomputers consisting of hundreds of thousands of cores, as well as on inexpensive commodity clusters commonly found in academic environments. It is written in C++ and leans on Charm++ parallel objects for optimal performance on low-latency architectures. NAMD is a versatile, multipurpose code that gathers state-of-the-art algorithms to carry out simulations in apt thermodn. ensembles, using the widely popular CHARMM, AMBER, OPLS, and GROMOS biomol. force fields. Here, the authors review the main features of NAMD that allow both equil. and enhanced-sampling mol. dynamics simulations with numerical efficiency. The authors describe the underlying concepts used by NAMD and their implementation, most notably for handling long-range electrostatics; controlling the temp., pressure, and pH; applying external potentials on tailored grids; leveraging massively parallel resources in multiple-copy simulations; and hybrid quantum-mech./mol.-mech. descriptions. The authors detail the variety of options offered by NAMD for enhanced-sampling simulations aimed at detg. free-energy differences of either alchem. or geometrical transformations and outline their applicability to specific problems. Last, the roadmap for the development of NAMD and the authors' current efforts toward achieving optimal performance on GPU-based architectures, for pushing back the limitations that have prevented biol. realistic billion-atom objects to be fruitfully simulated, and for making large-scale simulations less expensive and easier to set up, run, and analyze are discussed. NAMD is distributed free of charge with its source code at www.ks.uiuc.edu. (c) 2020 American Institute of Physics.
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    Páll, S.; Zhmurov, A.; Bauer, P.; Abraham, M.; Lundborg, M.; Gray, A.; Hess, B.; Lindahl, E. Heterogeneous Parallelization and Acceleration of Molecular Dynamics Simulations in GROMACS. J. Chem. Phys. 2020, 153, 134110  DOI: 10.1063/5.0018516
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    Heterogeneous parallelization and acceleration of molecular dynamics simulations in GROMACS
    Pall, Szilard; Zhmurov, Artem; Bauer, Paul; Abraham, Mark; Lundborg, Magnus; Gray, Alan; Hess, Berk; Lindahl, Erik
    Journal of Chemical Physics (2020), 153 (13), 134110CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
    The introduction of accelerator devices such as graphics processing units (GPUs) has had profound impact on mol. dynamics simulations and has enabled order-of-magnitude performance advances using commodity hardware. To fully reap these benefits, it has been necessary to reformulate some of the most fundamental algorithms, including the Verlet list, pair searching, and cutoffs. Here, we present the heterogeneous parallelization and acceleration design of mol. dynamics implemented in the GROMACS codebase over the last decade. The setup involves a general cluster-based approach to pair lists and non-bonded pair interactions that utilizes both GPU and central processing unit (CPU) single instruction, multiple data acceleration efficiently, including the ability to load-balance tasks between CPUs and GPUs. The algorithm work efficiency is tuned for each type of hardware, and to use accelerators more efficiently, we introduce dual pair lists with rolling pruning updates. Combined with new direct GPU-GPU communication and GPU integration, this enables excellent performance from single GPU simulations through strong scaling across multiple GPUs and efficient multi-node parallelization. (c) 2020 American Institute of Physics.
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    Khan, H. N.; Hounshell, D. A.; Fuchs, E. R. H. Science and Research Policy at the End of Moore’s Law. Nat. Electron. 2018, 1, 14– 21,  DOI: 10.1038/s41928-017-0005-9
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    Dickson, A.; Dinner, A. R. Enhanced Sampling of Nonequilibrium Steady States. Annu. Rev. Phys. Chem. 2010, 61, 441– 459,  DOI: 10.1146/annurev.physchem.012809.103433
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    Enhanced sampling of nonequilibrium steady states
    Dickson, Alex; Dinner, Aaron R.
    Annual Review of Physical Chemistry (2010), 61 (), 441-460CODEN: ARPLAP; ISSN:0066-426X. (Annual Reviews Inc.)
    We review recent progress in methods for accelerating the convergence of simulations of nonequil. systems, specifically nonequil. umbrella sampling (NEUS) and forward flux sampling (FFS). These methods account for statistics of dynamical paths between interfaces to enforce sampling of low probability regions of phase space for computing steady-state avs., including transition rates, for systems driven arbitrarily far from equil. Recent advances in NEUS allow for efficient sampling of complex systems by focusing sampling in the vicinity of a one-dimensional manifold (string) that connects regions of interest in phase space; this procedure can be extended to the case of two strings that describe the forward and backward transition ensembles sep., which is useful, as they do not, in general, coincide. We recast FFS in the framework of NEUS to facilitate comparison of the two methods. We conclude by discussing selected applications of interest.
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    Chong, L. T.; Saglam, A. S.; Zuckerman, D. M. Path-Sampling Strategies for Simulating Rare Events in Biomolecular Systems. Curr. Opin. Struct. Biol. 2017, 43, 88– 94,  DOI: 10.1016/j.sbi.2016.11.019
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    Path-sampling strategies for simulating rare events in biomolecular systems
    Chong, Lillian T.; Saglam, Ali S.; Zuckerman, Daniel M.
    Current Opinion in Structural Biology (2017), 43 (), 88-94CODEN: COSBEF; ISSN:0959-440X. (Elsevier Ltd.)
    Despite more than three decades of effort with mol. dynamics simulations, long-timescale (ms and beyond) biol. relevant phenomena remain out of reach in most systems of interest. This is largely because important transitions, such as conformational changes and (un)binding events, tend to be rare for conventional simulations (<10 μs). That is, conventional simulations will predominantly dwell in metastable states instead of making large transitions in complex biomol. energy landscapes. In contrast, path sampling approaches focus computing effort specifically on transitions of interest. Such approaches have been in use for nearly 20 years in biomol. systems and enabled the generation of pathways and calcn. of rate consts. for ms processes, including large protein conformational changes, protein folding, and protein (un)binding.
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    Zuckerman, D. M.; Chong, L. T. Weighted Ensemble Simulation: Review of Methodology, Applications, and Software. Annu. Rev. Biophys. 2017, 46, 43– 57,  DOI: 10.1146/annurev-biophys-070816-033834
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    Weighted Ensemble Simulation: Review of Methodology, Applications, and Software
    Zuckerman, Daniel M.; Chong, Lillian T.
    Annual Review of Biophysics (2017), 46 (), 43-57CODEN: ARBNCV; ISSN:1936-122X. (Annual Reviews)
    The weighted ensemble (WE) methodol. orchestrates quasi-independent parallel simulations run with intermittent communication that can enhance sampling of rare events such as protein conformational changes, folding, and binding. The WE strategy can achieve superlinear scaling-the unbiased estn. of key observables such as rate consts. and equil. state populations to greater precision than would be possible with ordinary parallel simulation. WE software can be used to control any dynamics engine, such as std. mol. dynamics and cell-modeling packages. This article reviews the theor. basis of WE and goes on to describe successful applications to a no. of complex biol. processes-protein conformational transitions, (un)binding, and assembly processes, as well as cell-scale processes in systems biol. We furthermore discuss the challenges that need to be overcome in the next phase of WE methodol. development. Overall, the combined advances in WE methodol. and software have enabled the simulation of long-timescale processes that would otherwise not be practical on typical computing resources using std. simulation.
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    Husic, B. E.; Pande, V. S. Markov State Models: From an Art to a Science. J. Am. Chem. Soc. 2018, 140, 2386– 2396,  DOI: 10.1021/jacs.7b12191
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    Markov State Models: From an Art to a Science
    Husic, Brooke E.; Pande, Vijay S.
    Journal of the American Chemical Society (2018), 140 (7), 2386-2396CODEN: JACSAT; ISSN:0002-7863. (American Chemical Society)
    Markov state models (MSMs) are a powerful framework for analyzing dynamical systems, such as mol. dynamics (MD) simulations, that have gained widespread use over the past several decades. This perspective offers an overview of the MSM field to date, presented for a general audience as a timeline of key developments in the field. We sequentially address early studies that motivated the method, canonical papers that established the use of MSMs for MD anal., and subsequent advances in software and anal. protocols. The derivation of a variational principle for MSMs in 2013 signified a turning point from expertise-driving MSM building to a systematic, objective protocol. The variational approach, combined with best practices for model selection and open-source software, enabled a wide range of MSM anal. for applications such as protein folding and allostery, ligand binding, and protein-protein assocn. To conclude, the current frontiers of methods development are highlighted, as well as exciting applications in exptl. design and drug discovery.
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    Allison, J. R. Computational Methods for Exploring Protein Conformations. Biochem. Soc. Trans. 2020, 48, 1707– 1724,  DOI: 10.1042/BST20200193
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    Computational methods for exploring protein conformations
    Allison, Jane R.
    Biochemical Society Transactions (2020), 48 (4), 1707-1724CODEN: BCSTB5; ISSN:0300-5127. (Portland Press Ltd.)
    A review. Proteins are dynamic mols. that can transition between a potentially wide range of structures comprising their conformational ensemble. The nature of these conformations and their relative probabilities are described by a high-dimensional free energy landscape. While computer simulation techniques such as mol. dynamics simulations allow characterization of the metastable conformational states and the transitions between them, and thus free energy landscapes, to be characterised, the barriers between states can be high, precluding efficient sampling without substantial computational resources. Over the past decades, a dizzying array of methods have emerged for enhancing conformational sampling, and for projecting the free energy landscape onto a reduced set of dimensions that allow conformational states to be distinguished, known as collective variables (CVs), along which sampling may be directed. Here, a brief description of what biomol. simulation entails is followed by a more detailed exposition of the nature of CVs and methods for detg. these, and, lastly, an overview of the myriad different approaches for enhancing conformational sampling, most of which rely upon CVs, including new advances in both CV detn. and conformational sampling due to machine learning.
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    Kamenik, A. S.; Linker, S. M.; Riniker, S. Enhanced Sampling without Borders: On Global Biasing Functions and How to Reweight Them. Phys. Chem. Chem. Phys. 2021, 24, 1225– 1236,  DOI: 10.1039/D1CP04809K
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    Hénin, J.; Lelièvre, T.; Shirts, M. R.; Valsson, O.; Delemotte, L. Enhanced Sampling Methods for Molecular Dynamics Simulations. arXiv (Condensed Matter.Statistical Mechanics) , February 8, 2022, 2202.04164, ver. 1. https://arxiv.org/abs/2202.04164 (accessed 2022-02-27).
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    Fiorin, G.; Klein, M. L.; Hénin, J. Using Collective Variables to Drive Molecular Dynamics Simulations. Mol. Phys. 2013, 111, 3345– 3362,  DOI: 10.1080/00268976.2013.813594
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    Using collective variables to drive molecular dynamics simulations
    Fiorin, Giacomo; Klein, Michael L.; Henin, Jerome
    Molecular Physics (2013), 111 (22-23), 3345-3362CODEN: MOPHAM; ISSN:0026-8976. (Taylor & Francis Ltd.)
    A software framework is introduced that facilitates the application of biasing algorithms to collective variables of the type commonly employed to drive massively parallel mol. dynamics (MD) simulations. The modular framework that is presented enables one to combine existing collective variables into new ones, and combine any chosen collective variable with available biasing methods. The latter include the classic time-dependent biases referred to as steered MD and targeted MD, the temp.-accelerated MD algorithm, as well as the adaptive free-energy biases called metadynamics and adaptive biasing force. The present modular software is extensible, and portable between commonly used MD simulation engines.
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    Giberti, F.; Salvalaglio, M.; Parrinello, M. Metadynamics Studies of Crystal Nucleation. IUCrJ 2015, 2, 256– 266,  DOI: 10.1107/S2052252514027626
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    Metadynamics studies of crystal nucleation
    Giberti, Federico; Salvalaglio, Matteo; Parrinello, Michele
    IUCrJ (2015), 2 (2), 256-266CODEN: IUCRAJ; ISSN:2052-2525. (International Union of Crystallography)
    Crystn. processes are characterized by activated events and long timescales. These characteristics prevent std. mol. dynamics techniques from being efficiently used for the direct investigation of processes such as nucleation. This short review provides an overview on the use of metadynamics, a state-of-the-art enhanced sampling technique, for the simulation of phase transitions involving the prodn. of a cryst. solid. In particular the principles of metadynamics are outlined, several order parameters are described that have been or could be used in conjunction with metadynamics to sample nucleation events and then an overview is given of recent metadynamics results in the field of crystal nucleation.
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    Pietrucci, F. Strategies for the Exploration of Free Energy Landscapes: Unity in Diversity and Challenges Ahead. Rev. Phys. 2017, 2, 32– 45,  DOI: 10.1016/j.revip.2017.05.001
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    Wang, Y.; Lamim Ribeiro, J. M.; Tiwary, P. Machine Learning Approaches for Analyzing and Enhancing Molecular Dynamics Simulations. Curr. Opin. Struct. Biol. 2020, 61, 139– 145,  DOI: 10.1016/j.sbi.2019.12.016
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    Machine learning approaches for analyzing and enhancing molecular dynamics simulations
    Wang, Yihang; Lamim Ribeiro, Joao Marcelo; Tiwary, Pratyush
    Current Opinion in Structural Biology (2020), 61 (), 139-145CODEN: COSBEF; ISSN:0959-440X. (Elsevier Ltd.)
    Mol. dynamics (MD) has become a powerful tool for studying biophys. systems, due to increasing computational power and availability of software. Although MD has made many contributions to better understanding these complex biophys. systems, there remain methodol. difficulties to be surmounted. First, how to make the deluge of data generated in running even a microsecond long MD simulation human comprehensible. Second, how to efficiently sample the underlying free energy surface and kinetics. In this short perspective, we summarize machine learning based ideas that are solving both of these limitations, with a focus on their key theor. underpinnings and remaining challenges.
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    Noé, F.; Tkatchenko, A.; Müller, K.-R.; Clementi, C. Machine Learning for Molecular Simulation. Annu. Rev. Phys. Chem. 2020, 71, 361– 390,  DOI: 10.1146/annurev-physchem-042018-052331
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    Machine Learning for Molecular Simulation
    Noe, Frank; Tkatchenko, Alexandre; Mueller, Klaus-Robert; Clementi, Cecilia
    Annual Review of Physical Chemistry (2020), 71 (), 361-390CODEN: ARPLAP; ISSN:0066-426X. (Annual Reviews)
    Machine learning (ML) is transforming all areas of science. The complex and time-consuming calcns. in mol. simulations are particularly suitable for an ML revolution and have already been profoundly affected by the application of existing ML methods. Here we review recent ML methods for mol. simulation, with particular focus on (deep) neural networks for the prediction of quantum-mech. energies and forces, on coarse-grained mol. dynamics, on the extn. of free energy surfaces and kinetics, and on generative network approaches to sample mol. equil. structures and compute thermodn. To explain these methods and illustrate open methodol. problems, we review some important principles of mol. physics and describe how they can be incorporated into ML structures. Finally, we identify and describe a list of open challenges for the interface between ML and mol. simulation.
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    Gkeka, P.; Stoltz, G.; Barati Farimani, A.; Belkacemi, Z.; Ceriotti, M.; Chodera, J. D.; Dinner, A. R.; Ferguson, A. L.; Maillet, J.-B.; Minoux, H.; Peter, C.; Pietrucci, F.; Silveira, A.; Tkatchenko, A.; Trstanova, Z.; Wiewiora, R.; Lelièvre, T. Machine Learning Force Fields and Coarse-Grained Variables in Molecular Dynamics: Application to Materials and Biological Systems. J. Chem. Theory Comput. 2020, 16, 4757– 4775,  DOI: 10.1021/acs.jctc.0c00355
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    Machine Learning Force Fields and Coarse-Grained Variables in Molecular Dynamics: Application to Materials and Biological Systems
    Gkeka, Paraskevi; Stoltz, Gabriel; Barati Farimani, Amir; Belkacemi, Zineb; Ceriotti, Michele; Chodera, John D.; Dinner, Aaron R.; Ferguson, Andrew L.; Maillet, Jean-Bernard; Minoux, Herve; Peter, Christine; Pietrucci, Fabio; Silveira, Ana; Tkatchenko, Alexandre; Trstanova, Zofia; Wiewiora, Rafal; Lelievre, Tony
    Journal of Chemical Theory and Computation (2020), 16 (8), 4757-4775CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)
    A review. Machine learning encompasses tools and algorithms that are now becoming popular in almost all scientific and technol. fields. This is true for mol. dynamics as well, where machine learning offers promises of extg. valuable information from the enormous amts. of data generated by simulation of complex systems. The authors provide here a review of the authors' current understanding of goals, benefits, and limitations of machine learning techniques for computational studies on atomistic systems, focusing on the construction of empirical force fields from ab initio databases and the detn. of reaction coordinates for free energy computation and enhanced sampling.
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    Sidky, H.; Chen, W.; Ferguson, A. L. Machine Learning for Collective Variable Discovery and Enhanced Sampling in Biomolecular Simulation. Mol. Phys. 2020, 118, e1737742  DOI: 10.1080/00268976.2020.1737742
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    Machine learning for collective variable discovery and enhanced sampling in biomolecular simulation
    Sidky, Hythem; Chen, Wei; Ferguson, Andrew L.
    Molecular Physics (2020), 118 (5), e1737742/1-e1737742/21CODEN: MOPHAM; ISSN:0026-8976. (Taylor & Francis Ltd.)
    Classical mol. dynamics simulates the time evolution of mol. systems through the phase space spanned by the positions and velocities of the constituent atoms. Mol.-level thermodn., kinetic, and structural data extd. from the resulting trajectories provide valuable information for the understanding, engineering, and design of biol. and mol. materials. The cost of simulating many-body at. systems makes simulations of large mols. prohibitively expensive, and the high-dimensionality of the resulting trajectories presents a challenge for anal. Driven by advances in algorithms, hardware, and data availability, there has been a flare of interest in recent years in the applications of machine learning - esp. deep learning - to mol. simulation. These techniques have demonstrated great power and flexibility in both extg. mechanistic understanding of the important nonlinear collective variables governing the dynamics of a mol. system, and in furnishing good low-dimensional system representations with which to perform enhanced sampling or develop long-timescale dynamical models. It is the purpose of this article to introduce the key machine learning approaches, describe how they are married with statistical mech. theory into domain-specific tools, and detail applications of these approaches in understanding and accelerating biomol. simulation.
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    Torrie, G. M.; Valleau, J. P. Nonphysical Sampling Distributions in Monte Carlo Free-Energy Estimation: Umbrella Sampling. J. Comput. Phys. 1977, 23, 187– 199,  DOI: 10.1016/0021-9991(77)90121-8
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    Huber, T.; Torda, A. E.; van Gunsteren, W. F. Local Elevation: A Method for Improving the Searching Properties of Molecular Dynamics Simulation. J. Comput.-Aided Mol. Des. 1994, 8, 695– 708,  DOI: 10.1007/BF00124016
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    Local elevation: a method for improving the searching properties of molecular dynamics simulation
    Huber, Thomas; Torda, Andrew E.; van Gunsteren, Wilfred F.
    Journal of Computer-Aided Molecular Design (1994), 8 (6), 695-708CODEN: JCADEQ; ISSN:0920-654X. (ESCOM)
    The concept of memory has been introduced into a mol. dynamics algorithm. This was done so as to persuade a mol. system to visit new areas of conformational space rather than be confined to a small no. of low-energy regions. The method is demonstrated on a simple model system and the 11-residue cyclic peptide cyclosporin A. For comparison, calcns. were also performed using simulated temp. annealing and a potential energy annealing scheme. Although the method can only be applied to systems with a small no. of degrees of freedom, it offers the chance to generate a multitude of different low-energy structures, where other methods only give a single one or few. This is clearly important in problems such as drug design, where one is interested in the conformational spread of a system.
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    Calculating free energies using average force
    Darve, Eric; Pohorille, Andrew
    Journal of Chemical Physics (2001), 115 (20), 9169-9183CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
    A new, general formula that connects the derivs. of the free energy along the selected, generalized coordinates of the system with the instantaneous force acting on these coordinates is derived. The instantaneous force is defined as the force acting on the coordinate of interest so that when it is subtracted from the equations of motion the acceleration along this coordinate is zero. The formula applies to simulations in which the selected coordinates are either unconstrained or constrained to fixed values. It is shown that in the latter case the formula reduces to the expression previously derived by den Otter and Briels [Mol. Phys. 98, 773 (2000)]. If simulations are carried out without constraining the coordinates of interest, the formula leads to a new method for calcg. the free energy changes along these coordinates. This method is tested in two examples - rotation around the C-C bond of 1,2-dichloroethane immersed in water and transfer of fluoromethane across the water-hexane interface. The calcd. free energies are compared with those obtained by two commonly used methods. One of them relies on detg. the probability d. function of finding the system at different values of the selected coordinate and the other requires calcg. the av. force at discrete locations along this coordinate in a series of constrained simulations. The free energies calcd. by these three methods are in excellent agreement. The relative advantages of each method are discussed.
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    Comer, J.; Gumbart, J. C.; Hénin, J.; Lelièvre, T.; Pohorille, A.; Chipot, C. The Adaptive Biasing Force Method: Everything You Always Wanted to Know but Were Afraid to Ask. J. Phys. Chem. B 2015, 119, 1129– 1151,  DOI: 10.1021/jp506633n
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    The Adaptive Biasing Force Method: Everything You Always Wanted To Know but Were Afraid To Ask
    Comer, Jeffrey; Gumbart, James C.; Henin, Jerome; Lelievre, Tony; Pohorille, Andrew; Chipot, Christophe
    Journal of Physical Chemistry B (2015), 119 (3), 1129-1151CODEN: JPCBFK; ISSN:1520-5207. (American Chemical Society)
    In the host of numerical schemes devised to calc. free energy differences by way of geometric transformations, the adaptive biasing force algorithm has emerged as a promising route to map complex free-energy landscapes. It relies upon the simple concept that as a simulation progresses, a continuously updated biasing force is added to the equations of motion, such that in the long-time limit it yields a Hamiltonian devoid of an av. force acting along the transition coordinate of interest. This means that sampling proceeds uniformly on a flat free-energy surface, thus providing reliable free-energy ests. Much of the appeal of the algorithm to the practitioner is in its phys. intuitive underlying ideas and the absence of any requirements for prior knowledge about free-energy landscapes. Since its inception in 2001, the adaptive biasing force scheme has been the subject of considerable attention, from in-depth math. anal. of convergence properties to novel developments and extensions. The method has also been successfully applied to many challenging problems in chem. and biol. In this contribution, the method is presented in a comprehensive, self-contained fashion, discussing with a crit. eye its properties, applicability, and inherent limitations, as well as introducing novel extensions. Through free-energy calcns. of prototypical mol. systems, many methodol. aspects are examd., from stratification strategies to overcoming the so-called hidden barriers in orthogonal space, relevant not only to the adaptive biasing force algorithm but also to other importance-sampling schemes. On the basis of the discussions in this paper, a no. of good practices for improving the efficiency and reliability of the computed free-energy differences are proposed.
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    Lesage, A.; Lelièvre, T.; Stoltz, G.; Hénin, J. Smoothed Biasing Forces Yield Unbiased Free Energies with the Extended-System Adaptive Biasing Force Method. J. Phys. Chem. B 2017, 121, 3676– 3685,  DOI: 10.1021/acs.jpcb.6b10055
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    Smoothed Biasing Forces Yield Unbiased Free Energies with the Extended-System Adaptive Biasing Force Method
    Lesage, Adrien; Lelievre, Tony; Stoltz, Gabriel; Henin, Jerome
    Journal of Physical Chemistry B (2017), 121 (15), 3676-3685CODEN: JPCBFK; ISSN:1520-5207. (American Chemical Society)
    We report a theor. description and numerical tests of the extended-system adaptive biasing force method (eABF), together with an unbiased estimator of the free energy surface from eABF dynamics. Whereas the original ABF approach uses its running est. of the free energy gradient as the adaptive biasing force, eABF is built on the idea that the exact free energy gradient is not necessary for efficient exploration, and that it is still possible to recover the exact free energy sep. with an appropriate estimator. EABF does not directly bias the collective coordinates of interest, but rather fictitious variables that are harmonically coupled to them; therefore is does not require second deriv. ests., making it easily applicable to a wider range of problems than ABF. Furthermore, the extended variables present a smoother, coarse-grain-like sampling problem on a mollified free energy surface, leading to faster exploration and convergence. We also introduce CZAR, a simple, unbiased free energy estimator from eABF trajectories. EABF/CZAR converges to the phys. free energy surface faster than std. ABF for a wide range of parameters.
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    Global Optimization by Energy Landscape Paving
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    Physical Review Letters (2002), 88 (6), 068105/1-068105/4CODEN: PRLTAO; ISSN:0031-9007. (American Physical Society)
    We introduce a novel heuristic global optimization method, energy landscape paving (ELP), which combines core ideas from energy surface deformation and tabu search. In appropriate limits, ELP reduces to existing techniques. The approach is very general and flexible and is illustrated here on two protein folding problems. For these examples, the technique gives faster convergence to the global min. than previous approaches.
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    Gaussian-Mixture Umbrella Sampling
    Maragakis, Paul; van der Vaart, Arjan; Karplus, Martin
    Journal of Physical Chemistry B (2009), 113 (14), 4664-4673CODEN: JPCBFK; ISSN:1520-6106. (American Chemical Society)
    We introduce the Gaussian-mixt. umbrella sampling method (GAMUS), a biased mol. dynamics technique based on adaptive umbrella sampling that efficiently escapes free energy min. in multidimensional problems. The prior simulation data are reweighted with a max. likelihood formulation, and the new approx. probability d. is fit to a Gaussian-mixt. model, augmented by information about the unsampled areas. The method can be used to identify free energy min. in multidimensional reaction coordinates. To illustrate GAMUS, we apply it to the alanine dipeptide (2D reaction coordinate) and tripeptide (4D reaction coordinate).
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    Warmflash, A.; Bhimalapuram, P.; Dinner, A. R. Umbrella sampling for nonequilibrium processes. J. Chem. Phys. 2007, 127, 154112  DOI: 10.1063/1.2784118
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    Umbrella sampling for nonequilibrium processes
    Warmflash, Aryeh; Bhimalapuram, Prabhakar; Dinner, Aaron R.
    Journal of Chemical Physics (2007), 127 (15), 154112/1-154112/8CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
    The authors introduce an algorithm for detg. the steady-state probability distribution of an ergodic system arbitrarily far from equil. By enforcing equal sampling of different regions of phase space, as in umbrella sampling simulations of systems at equil., low probability regions are explored to a much greater extent than in phys. weighted simulations. The algorithm can be used to accumulate joint statistics for an arbitrary no. of order parameters for a system governed by any stochastic dynamics. They demonstrate the efficiency of the algorithm by applying it to a model of a genetic toggle switch which evolves irreversibly according to a continuous time Monte Carlo procedure.
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    Laio, A.; Parrinello, M. Escaping Free-Energy Minima. Proc. Natl. Acad. Sci. U. S. A. 2002, 99, 12562– 12566,  DOI: 10.1073/pnas.202427399
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    Escaping free-energy minima
    Laio, Alessandro; Parrinello, Michele
    Proceedings of the National Academy of Sciences of the United States of America (2002), 99 (20), 12562-12566CODEN: PNASA6; ISSN:0027-8424. (National Academy of Sciences)
    We introduce a powerful method for exploring the properties of the multidimensional free energy surfaces (FESs) of complex many-body systems by means of coarse-grained non-Markovian dynamics in the space defined by a few collective coordinates. A characteristic feature of these dynamics is the presence of a history-dependent potential term that, in time, fills the min. in the FES, allowing the efficient exploration and accurate detn. of the FES as a function of the collective coordinates. We demonstrate the usefulness of this approach in the case of the dissocn. of a NaCl mol. in water and in the study of the conformational changes of a dialanine in soln.
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    Barducci, A.; Bussi, G.; Parrinello, M. Well-Tempered Metadynamics: A Smoothly Converging and Tunable Free-Energy Method. Phys. Rev. Lett. 2008, 100, 020603  DOI: 10.1103/PhysRevLett.100.020603
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    Well-Tempered Metadynamics: A Smoothly Converging and Tunable Free-Energy Method
    Barducci, Alessandro; Bussi, Giovanni; Parrinello, Michele
    Physical Review Letters (2008), 100 (2), 020603/1-020603/4CODEN: PRLTAO; ISSN:0031-9007. (American Physical Society)
    We present a method for detg. the free-energy dependence on a selected no. of collective variables using an adaptive bias. The formalism provides a unified description which has metadynamics and canonical sampling as limiting cases. Convergence and errors can be rigorously and easily controlled. The parameters of the simulation can be tuned so as to focus the computational effort only on the phys. relevant regions of the order parameter space. The algorithm is tested on the reconstruction of an alanine dipeptide free-energy landscape.
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    Valsson, O.; Tiwary, P.; Parrinello, M. Enhancing Important Fluctuations: Rare Events and Metadynamics from a Conceptual Viewpoint. Annu. Rev. Phys. Chem. 2016, 67, 159– 184,  DOI: 10.1146/annurev-physchem-040215-112229
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    Enhancing Important Fluctuations: Rare Events and Metadynamics from a Conceptual Viewpoint
    Valsson, Omar; Tiwary, Pratyush; Parrinello, Michele
    Annual Review of Physical Chemistry (2016), 67 (), 159-184CODEN: ARPLAP; ISSN:0066-426X. (Annual Reviews)
    Atomistic simulations play a central role in many fields of science. However, their usefulness is often limited by the fact that many systems are characterized by several metastable states sepd. by high barriers, leading to kinetic bottlenecks. Transitions between metastable states are thus rare events that occur on significantly longer timescales than one can simulate in practice. Numerous enhanced sampling methods have been introduced to alleviate this timescale problem, including methods based on identifying a few crucial order parameters or collective variables and enhancing the sampling of these variables. Metadynamics is one such method that has proven successful in a great variety of fields. Here we review the conceptual and theor. foundations of metadynamics. As demonstrated, metadynamics is not just a practical tool but can also be considered an important development in the theory of statistical mechanics.
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    Dama, J. F.; Hocky, G. M.; Sun, R.; Voth, G. A. Exploring Valleys without Climbing Every Peak: More Efficient and Forgiving Metabasin Metadynamics via Robust On-the-Fly Bias Domain Restriction. J. Chem. Theory Comput. 2015, 11, 5638– 5650,  DOI: 10.1021/acs.jctc.5b00907
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    Exploring Valleys without Climbing Every Peak: More Efficient and Forgiving Metabasin Metadynamics via Robust On-the-Fly Bias Domain Restriction
    Dama, James F.; Hocky, Glen M.; Sun, Rui; Voth, Gregory A.
    Journal of Chemical Theory and Computation (2015), 11 (12), 5638-5650CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)
    Metadynamics is an enhanced sampling method designed to flatten free energy surfaces uniformly. However, the highest-energy regions are often irrelevant to study and dangerous to explore because systems often change irreversibly in unforeseen ways in response to driving forces in these regions, spoiling the sampling. Introducing an on-the-fly domain restriction allows metadynamics to flatten only up to a specified energy level and no further, improving efficiency and safety while decreasing the pressure on practitioners to design collective variables that are robust to otherwise irrelevant high energy driving. This paper describes a new method that achieves this using sequential on-the-fly estn. of energy wells and redefinition of the metadynamics hill shape, termed metabasin metadynamics. The energy level may be defined a priori or relative to unknown barrier energies estd. on-the-fly. Altering only the hill ensures that the method is compatible with many other advances in metadynamics methodol. The hill shape has a natural interpretation in terms of multiscale dynamics, and the computational overhead in simulation is minimal when studying systems of any reasonable size, for instance proteins or other macromols. Three example applications show that the formula is accurate and robust to complex dynamics, making metadynamics significantly more forgiving with respect to CV quality and thus more feasible to apply to the most challenging biomol. systems.
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    Pfaendtner, J.; Bonomi, M. Efficient Sampling of High-Dimensional Free-Energy Landscapes with Parallel Bias Metadynamics. J. Chem. Theory Comput. 2015, 11, 5062– 5067,  DOI: 10.1021/acs.jctc.5b00846
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    Efficient Sampling of High-Dimensional Free-Energy Landscapes with Parallel Bias Metadynamics
    Pfaendtner, Jim; Bonomi, Massimiliano
    Journal of Chemical Theory and Computation (2015), 11 (11), 5062-5067CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)
    Metadynamics accelerates sampling of mol. dynamics while reconstructing thermodn. properties of selected descriptors of the system. Its main practical difficulty originates from the compromise between keeping the no. of descriptors small for efficiently exploring their multidimensional free-energy landscape and biasing all of the slow motions of a process. Here we illustrate on a model system and on the tryptophan-cage miniprotein parallel bias metadynamics, a method that overcomes this issue by simultaneously applying multiple low-dimensional bias potentials.
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    Whitmer, J. K.; Chiu, C.-c.; Joshi, A. A.; de Pablo, J. J. Basis Function Sampling: A New Paradigm for Material Property Computation. Phys. Rev. Lett. 2014, 113, 190602  DOI: 10.1103/PhysRevLett.113.190602
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    Basis function sampling: a new paradigm for material property computation
    Whitmer, Jonathan K.; Chiu, Chi-cheng; Joshi, Abhijeet A.; de Pablo, Juan J.
    Physical Review Letters (2014), 113 (19), 190602/1-190602/5, 5 pp.CODEN: PRLTAO; ISSN:0031-9007. (American Physical Society)
    Wang-Landau sampling, and the assocd. class of flat histogram simulation methods have been remarkably helpful for calcns. of the free energy in a wide variety of phys. systems. Practically, convergence of these calcns. to a target free energy surface is hampered by reliance on parameters which are unknown a priori. Here, we derive and implement a method built upon orthogonal functions which is fast, parameter-free, and (importantly) geometrically robust. The method is shown to be highly effective in achieving convergence. An important feature of this method is its ability to attain arbitrary levels of description for the free energy. It is thus ideally suited to in silico measurement of elastic moduli and other material properties related to free energy perturbations. We demonstrate the utility of such applications by applying our method to calc. the Frank elastic consts. of the Lebwohl-Lasher model of liq. crystals.
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    Whitmer, J. K.; Fluitt, A. M.; Antony, L.; Qin, J.; McGovern, M.; de Pablo, J. J. Sculpting Bespoke Mountains: Determining Free Energies with Basis Expansions. J. Chem. Phys. 2015, 143, 044101  DOI: 10.1063/1.4927147
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    Sculpting bespoke mountains: Determining free energies with basis expansions
    Whitmer, Jonathan K.; Fluitt, Aaron M.; Antony, Lucas; Qin, Jian; McGovern, Michael; de Pablo, Juan J.
    Journal of Chemical Physics (2015), 143 (4), 044101/1-044101/6CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
    The intriguing behavior of a wide variety of phys. systems, ranging from amorphous solids or glasses to proteins, is a direct manifestation of underlying free energy landscapes riddled with local min. sepd. by large barriers. Exploring such landscapes has arguably become one of statistical physics's great challenges. A new method is proposed here for uniform sampling of rugged free energy surfaces. The method, which relies on special Green's functions to approx. the Dirac delta function, improves significantly on existing simulation techniques by providing a boundary-agnostic approach that is capable of mapping complex features in multidimensional free energy surfaces. The usefulness of the proposed approach is established in the context of a simple model glass former and model proteins, demonstrating improved convergence and accuracy over existing methods. (c) 2015 American Institute of Physics.
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    Sidky, H.; Whitmer, J. K. Learning Free Energy Landscapes Using Artificial Neural Networks. J. Chem. Phys. 2018, 148, 104111  DOI: 10.1063/1.5018708
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    Learning free energy landscapes using artificial neural networks
    Sidky, Hythem; Whitmer, Jonathan K.
    Journal of Chemical Physics (2018), 148 (10), 104111/1-104111/9CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
    Existing adaptive bias techniques, which seek to est. free energies and phys. properties from mol. simulations, are limited by their reliance on fixed kernels or basis sets which hinder their ability to efficiently conform to varied free energy landscapes. Further, user-specified parameters are in general non-intuitive yet significantly affect the convergence rate and accuracy of the free energy est. Here we propose a novel method, wherein artificial neural networks (ANNs) are used to develop an adaptive biasing potential which learns free energy landscapes. We demonstrate that this method is capable of rapidly adapting to complex free energy landscapes and is not prone to boundary or oscillation problems. The method is made robust to hyperparameters and over-fitting through Bayesian regularization which penalizes network wts. and auto-regulates the no. of effective parameters in the network. ANN sampling represents a promising innovative approach which can resolve complex free energy landscapes in less time than conventional approaches while requiring minimal user input. (c) 2018 American Institute of Physics.
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    Ribeiro, J. M. L.; Bravo, P.; Wang, Y.; Tiwary, P. Reweighted autoencoded variational Bayes for enhanced sampling (RAVE). J. Chem. Phys. 2018, 149, 072301  DOI: 10.1063/1.5025487
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    Reweighted autoencoded variational Bayes for enhanced sampling (RAVE)
    Ribeiro, Joao Marcelo Lamim; Bravo, Pablo; Wang, Yihang; Tiwary, Pratyush
    Journal of Chemical Physics (2018), 149 (7), 072301/1-072301/9CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
    Here we propose the reweighted autoencoded variational Bayes for enhanced sampling (RAVE) method, a new iterative scheme that uses the deep learning framework of variational autoencoders to enhance sampling in mol. simulations. RAVE involves iterations between mol. simulations and deep learning in order to produce an increasingly accurate probability distribution along a low-dimensional latent space that captures the key features of the mol. simulation trajectory. Using the Kullback-Leibler divergence between this latent space distribution and the distribution of various trial reaction coordinates sampled from the mol. simulation, RAVE dets. an optimum, yet nonetheless phys. interpretable, reaction coordinate and optimum probability distribution. Both then directly serve as the biasing protocol for a new biased simulation, which is once again fed into the deep learning module with appropriate wts. accounting for the bias, the procedure continuing until ests. of desirable thermodn. observables are converged. Unlike recent methods using deep learning for enhanced sampling purposes, RAVE stands out in that (a) it naturally produces a phys. interpretable reaction coordinate, (b) is independent of existing enhanced sampling protocols to enhance the fluctuations along the latent space identified via deep learning, and (c) it provides the ability to easily filter out spurious solns. learned by the deep learning procedure. The usefulness and reliability of RAVE is demonstrated by applying it to model potentials of increasing complexity, including computation of the binding free energy profile for a hydrophobic ligand-substrate system in explicit water with dissocn. time of more than 3 min, in computer time at least twenty times less than that needed for umbrella sampling or metadynamics. (c) 2018 American Institute of Physics.
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    Invernizzi, M.; Parrinello, M. Rethinking Metadynamics: From Bias Potentials to Probability Distributions. J. Phys. Chem. Lett. 2020, 11, 2731– 2736,  DOI: 10.1021/acs.jpclett.0c00497
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    Rethinking Metadynamics: From Bias Potentials to Probability Distributions
    Invernizzi, Michele; Parrinello, Michele
    Journal of Physical Chemistry Letters (2020), 11 (7), 2731-2736CODEN: JPCLCD; ISSN:1948-7185. (American Chemical Society)
    Metadynamics is an enhanced sampling method of great popularity, based on the on-the-fly construction of a bias potential that is a function of a selected no. of collective variables. We propose here a change in perspective that shifts the focus from the bias to the probability distribution reconstruction while retaining some of the key characteristics of metadynamics, such as flexible on-the-fly adjustments to the free energy est. The result is an enhanced sampling method that presents a drastic improvement in convergence speed, esp. when dealing with suboptimal and/or multidimensional sets of collective variables. The method is esp. robust and easy to use and in fact requires only a few simple parameters to be set, and it has a straightforward reweighting scheme to recover the statistics of the unbiased ensemble. Furthermore, it gives more control of the desired exploration of the phase space since the deposited bias is not allowed to grow indefinitely and it does not push the simulation to uninteresting high free energy regions. We demonstrate the performance of the method in a no. of representative examples.
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    Invernizzi, M.; Piaggi, P. M.; Parrinello, M. Unified Approach to Enhanced Sampling. Phys. Rev. X 2020, 10, 041034  DOI: 10.1103/PhysRevX.10.041034
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    Unified Approach to Enhanced Sampling
    Invernizzi, Michele; Piaggi, Pablo M.; Parrinello, Michele
    Physical Review X (2020), 10 (4), 041034CODEN: PRXHAE; ISSN:2160-3308. (American Physical Society)
    The sampling problem lies at the heart of atomistic simulations and over the years many different enhanced sampling methods have been suggested toward its soln. These methods are often grouped into two broad families. On the one hand, are methods such as umbrella sampling and metadynamics that build a bias potential based on few order parameters or collective variables. On the other hand, are tempering methods such as replica exchange that combine different thermodn. ensembles in one single expanded ensemble. We instead adopt a unifying perspective, focusing on the target probability distribution sampled by the different methods. This allows us to introduce a new class of collective-variables-based bias potentials that can be used to sample any of the expanded ensembles normally sampled via replica exchange. We also provide a practical implementation by properly adapting the iterative scheme of the recently developed on-the-fly probability enhanced sampling method [M. Invernizzi and M. Parrinello, J. Lett.11, 2731 (2020)JPCLCD1948-718510.1021/acs.jpclett.0c00497], which was originally introduced for metadynamicslike sampling. The resulting method is very general and can be used to achieve different types of enhanced sampling. It is also reliable and simple to use, since it presents only few and robust external parameters and has a straightforward reweighting scheme. Furthermore, it can be used with any no. of parallel replicas. We show the versatility of our approach with applications to multicanonical and multithermal-multibaric simulations, thermodn. integration, umbrella sampling, and combinations thereof.
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    Giberti, F.; Tribello, G. A.; Ceriotti, M. Global Free-Energy Landscapes as a Smoothly Joined Collection of Local Maps. J. Chem. Theory Comput. 2021, 17, 3292– 3308,  DOI: 10.1021/acs.jctc.0c01177
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    Global Free-Energy Landscapes as a Smoothly Joined Collection of Local Maps
    Giberti, F.; Tribello, G. A.; Ceriotti, M.
    Journal of Chemical Theory and Computation (2021), 17 (6), 3292-3308CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)
    Enhanced sampling techniques have become an essential tool in computational chem. and physics, where they are applied to sample activated processes that occur on a time scale that is inaccessible to conventional simulations. Despite their popularity, it is well known that they have constraints that hinder their applications to complex problems. The core issue lies in the need to describe the system using a small no. of collective variables (CVs). Any slow degree of freedom that is not properly described by the chosen CVs will hinder sampling efficiency. However, the exploration of configuration space is also hampered by including variables that are not relevant to describe the activated process under study. This paper presents the Adaptive Topog. of Landscape for Accelerated Sampling (ATLAS), a new biasing method capable of working with many CVs. The root idea of ATLAS is to apply a divide-and-conquer strategy, where the high-dimensional CVs space is divided into basins, each of which is described by an automatically detd., low-dimensional set of variables. A well-tempered metadynamics-like bias is constructed as a function of these local variables. Indicator functions assocd. with the basins switch on and off the local biases so that the sampling is performed on a collection of low-dimensional CV spaces that are smoothly combined to generate an effectively high-dimensional bias. The unbiased Boltzmann distribution is recovered through reweighing, making the evaluation of conformational and thermodn. properties straightforward. The decompn. of the free-energy landscape in local basins can be updated iteratively as the simulation discovers new (meta)stable states.
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  41. 41
    Bal, K. M. Reweighted Jarzynski Sampling: Acceleration of Rare Events and Free Energy Calculation with a Bias Potential Learned from Nonequilibrium Work. J. Chem. Theory Comput. 2021, 17, 6766– 6774,  DOI: 10.1021/acs.jctc.1c00574
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    41
    Reweighted Jarzynski Sampling: Acceleration of Rare Events and Free Energy Calculation with a Bias Potential Learned from Nonequilibrium Work
    Bal, Kristof M.
    Journal of Chemical Theory and Computation (2021), 17 (11), 6766-6774CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)
    We introduce a simple enhanced sampling approach for the calcn. of free energy differences and barriers along a one-dimensional reaction coordinate. First, a small no. of short nonequil. simulations are carried out along the reaction coordinate, and the Jarzynski equality is used to learn an approx. free energy surface from the nonequil. work distribution. This free energy est. is represented in a compact form as an artificial neural network and used as an external bias potential to accelerate rare events in a subsequent mol. dynamics simulation. The final free energy est. is then obtained by reweighting the equil. probability distribution of the reaction coordinate sampled under the influence of the external bias. We apply our reweighted Jarzynski sampling recipe to four processes of varying scales and complexities-spanning chem. reaction in the gas phase, pair assocn. in soln., and droplet nucleation in supersatd. vapor. In all cases, we find reweighted Jarzynski sampling to be a very efficient strategy, resulting in rapid convergence of the free energy to high precision.
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  42. 42
    Valsson, O.; Parrinello, M. Variational Approach to Enhanced Sampling and Free Energy Calculations. Phys. Rev. Lett. 2014, 113, 090601  DOI: 10.1103/PhysRevLett.113.090601
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    Variational approach to enhanced sampling and free energy calculations
    Valsson, Omar; Parrinello, Michele
    Physical Review Letters (2014), 113 (9), 090601CODEN: PRLTAO; ISSN:0031-9007. (American Physical Society)
    The ability of widely used sampling methods, such as mol. dynamics or Monte Carlo simulations, to explore complex free energy landscapes is severely hampered by the presence of kinetic bottlenecks. A large no. of solns. have been proposed to alleviate this problem. Many are based on the introduction of a bias potential which is a function of a small no. of collective variables. However constructing such a bias is not simple. Here we introduce a functional of the bias potential and an assocd. variational principle. The bias that minimizes the functional relates in a simple way to the free energy surface. This variational principle can be turned into a practical, efficient, and flexible sampling method. A no. of numerical examples are presented which include the detn. of a three-dimensional free energy surface. We argue that, beside being numerically advantageous, our variational approach provides a convenient and novel standpoint for looking at the sampling problem.
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    Valsson, O.; Parrinello, M. Variationally Enhanced Sampling. In Handbook of Materials Modeling: Methods: Theory and Modeling; Andreoni, W., Yip, S., Eds.; Springer: Cham, Switzerland, 2020, 621– 634.
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    Bonati, L.; Zhang, Y.-Y.; Parrinello, M. Neural Networks-Based Variationally Enhanced Sampling. Proc. Natl. Acad. Sci. U. S. A. 2019, 116, 17641– 17647,  DOI: 10.1073/pnas.1907975116
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    Neural networks-based variationally enhanced sampling
    Bonati, Luigi; Zhang, Yue-Yu; Parrinello, Michele
    Proceedings of the National Academy of Sciences of the United States of America (2019), 116 (36), 17641-17647CODEN: PNASA6; ISSN:0027-8424. (National Academy of Sciences)
    Sampling complex free-energy surfaces is one of the main challenges of modern atomistic simulation methods. The presence of kinetic bottlenecks in such surfaces often renders a direct approach useless. A popular strategy is to identify a small no. of key collective variables and to introduce a bias potential that is able to favor their fluctuations in order to accelerate sampling. Here, we propose to use machine-learning techniques in conjunction with the recent variationally enhanced sampling method [O. Valsson, M. Parrinello, Phys. Rev. Lett. 113, 090601 (2014)] in order to det. such potential. This is achieved by expressing the bias as a neural network. The parameters are detd. in a variational learning scheme aimed at minimizing an appropriate functional. This required the development of a more efficient minimization technique. The expressivity of neural networks allows representing rapidly varying free-energy surfaces, removes boundary effects artifacts, and allows several collective variables to be handled.
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    Piaggi, P. M.; Valsson, O.; Parrinello, M. A Variational Approach to Nucleation Simulation. Faraday Discuss. 2016, 195, 557– 568,  DOI: 10.1039/C6FD00127K
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    A variational approach to nucleation simulation
    Piaggi, Pablo M.; Valsson, Omar; Parrinello, Michele
    Faraday Discussions (2016), 195 (Reaction Rate Theory), 557-568CODEN: FDISE6; ISSN:1359-6640. (Royal Society of Chemistry)
    We study by computer simulation the nucleation of a supersatd. Lennard-Jones vapor into the liq. phase. The large free energy barriers to transition make the time scale of this process impossible to study by ordinary mol. dynamics simulations. Therefore we use a recently developed enhanced sampling method [Valsson and Parrinello, Phys. Rev. Lett.113, 090601 (2014)] based on the variational detn. of a bias potential. We differ from previous applications of this method in that the bias is constructed on the basis of the phys. model provided by the classical theory of nucleation. We examine the tech. problems assocd. with this approach. Our results are very satisfactory and will pave the way for calcg. the nucleation rates in many systems.
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    McCarty, J.; Valsson, O.; Parrinello, M. Bespoke Bias for Obtaining Free Energy Differences within Variationally Enhanced Sampling. J. Chem. Theory Comput. 2016, 12, 2162– 2169,  DOI: 10.1021/acs.jctc.6b00125
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    Bespoke Bias for Obtaining Free Energy Differences within Variationally Enhanced Sampling
    McCarty, James; Valsson, Omar; Parrinello, Michele
    Journal of Chemical Theory and Computation (2016), 12 (5), 2162-2169CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)
    Obtaining efficient sampling of multiple metastable states through mol. dynamics and hence detg. free energy differences is central for understanding many important phenomena. Here we present a new biasing strategy, which employs the recent variationally enhanced sampling approach Valsson and Parrinello Phys. Rev. Lett.2014, 113, 090601. The bias is constructed from an intuitive model of the local free energy surface describing fluctuations around metastable min. and depends on only a few parameters which are detd. variationally such that efficient sampling between states is obtained. The bias constructed in this manner largely reduces the need of finding a set of collective variables that completely spans the conformational space of interest, as they only need to be a locally valid descriptor of the system about its local min. We introduce the method and demonstrate its power on two representative examples.
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    Invernizzi, M.; Valsson, O.; Parrinello, M. Coarse Graining from Variationally Enhanced Sampling Applied to the Ginzburg–Landau Model. Proc. Natl. Acad. Sci. U. S. A. 2017, 114, 3370– 3374,  DOI: 10.1073/pnas.1618455114
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    Coarse graining from variationally enhanced sampling applied to the Ginzburg-Landau model
    Invernizzi, Michele; Valsson, Omar; Parrinello, Michele
    Proceedings of the National Academy of Sciences of the United States of America (2017), 114 (13), 3370-3374CODEN: PNASA6; ISSN:0027-8424. (National Academy of Sciences)
    A powerful way to deal with a complex system is to build a coarse-grained model capable of catching its main phys. features, while being computationally affordable. Inevitably, such coarse-grained models introduce a set of phenomenol. parameters, which are often not easily deducible from the underlying atomistic system. We present a unique approach to the calcn. of these parameters, based on the recently introduced variationally enhanced sampling method. It allows us to obtain the parameters from atomistic simulations, providing thus a direct connection between the microscopic and the mesoscopic scale. The coarse-grained model we consider is that of Ginzburg-Landau, valid around a second-order crit. point. In particular, we use it to describe a Lennard-Jones fluid in the region close to the liq.-vapor crit. point. The procedure is general and can be adapted to other coarse-grained models.
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    Invernizzi, M.; Parrinello, M. Making the Best of a Bad Situation: A Multiscale Approach to Free Energy Calculation. J. Chem. Theory Comput. 2019, 15, 2187– 2194,  DOI: 10.1021/acs.jctc.9b00032
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    Making the Best of a Bad Situation: A Multiscale Approach to Free Energy Calculation
    Invernizzi, Michele; Parrinello, Michele
    Journal of Chemical Theory and Computation (2019), 15 (4), 2187-2194CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)
    Many enhanced sampling techniques rely on the identification of a no. of collective variables that describe all the slow modes of the system. By constructing a bias potential in this reduced space, one is then able to sample efficiently and reconstruct the free energy landscape. In methods such as metadynamics, the quality of these collective variables plays a key role in convergence efficiency. Unfortunately in many systems of interest it is not possible to identify an optimal collective variable, and one must deal with the nonideal situation of a system in which some slow modes are not accelerated. The authors propose a two-step approach in which, by taking into account the residual multiscale nature of the problem, one is able to significantly speed up convergence. To do so, the authors combine an exploratory metadynamics run with an optimization of the free energy difference between metastable states, based on the recently proposed variationally enhanced sampling method. This new method is esp. suited for complex systems because of its simplicity and clear underlying phys. picture.
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    McCarty, J.; Valsson, O.; Tiwary, P.; Parrinello, M. Variationally Optimized Free-Energy Flooding for Rate Calculation. Phys. Rev. Lett. 2015, 115, 070601  DOI: 10.1103/PhysRevLett.115.070601
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    Variationally optimized free-energy flooding for rate calculation
    McCarty, James; Valsson, Omar; Tiwary, Pratyush; Parrinello, Michele
    Physical Review Letters (2015), 115 (7), 070601/1-070601/5CODEN: PRLTAO; ISSN:0031-9007. (American Physical Society)
    We propose a new method to obtain kinetic properties of infrequent events from mol. dynamics simulation. The procedure employs a recently introduced variational approach to construct a bias potential as a function of several collective variables that is designed to flood the assocd. free energy surface up to a predefined level. The resulting bias potential effectively accelerates transitions between metastable free energy min. while ensuring bias-free transition states, thus allowing accurate kinetic rates to be obtained. We test the method on a few illustrative systems for which we obtain an order of magnitude improvement in efficiency relative to previous approaches and several orders of magnitude relative to unbiased mol. dynamics. We expect an even larger improvement in more complex systems. This and the ability of the variational approach to deal efficiently with a large no. of collective variables will greatly enhance the scope of these calcns. This work is a vindication of the potential that the variational principle has if applied in innovative ways.
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    Demuynck, R.; Rogge, S. M. J.; Vanduyfhuys, L.; Wieme, J.; Waroquier, M.; Van Speybroeck, V. Efficient Construction of Free Energy Profiles of Breathing Metal─Organic Frameworks Using Advanced Molecular Dynamics Simulations. J. Chem. Theory Comput. 2017, 13, 5861– 5873,  DOI: 10.1021/acs.jctc.7b01014
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    Efficient Construction of Free Energy Profiles of Breathing Metal-Organic Frameworks Using Advanced Molecular Dynamics Simulations
    Demuynck, Ruben; Rogge, Sven M. J.; Vanduyfhuys, Louis; Wieme, Jelle; Waroquier, Michel; Van Speybroeck, Veronique
    Journal of Chemical Theory and Computation (2017), 13 (12), 5861-5873CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)
    In order to reliably predict and understand the breathing behavior of highly flexible metal-org. frameworks from thermodn. considerations, an accurate estn. of the free energy difference between their different metastable states is a prerequisite. Herein, a variety of free energy estn. methods are thoroughly tested for their ability to construct the free energy profile as a function of the unit cell vol. of MIL-53(Al). The methods comprise free energy perturbation, thermodn. integration, umbrella sampling, metadynamics, and variationally enhanced sampling. A series of mol. dynamics simulations have been performed in the frame of each of the five methods to describe structural transformations in flexible materials with the vol. as the collective variable, which offers a unique opportunity to assess their computational efficiency. Subsequently, the most efficient method, umbrella sampling, is used to construct an accurate free energy profile at different temps. for MIL-53(Al) from first principles at the PBE + D3(BJ) level of theory. This study yields insight into the importance of the different aspects such as entropy contributions and anharmonic contributions on the resulting free energy profile. As such, this thorough study provides unparalleled insight in the thermodn. of the large structural deformations of flexible materials.
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    Demuynck, R.; Wieme, J.; Rogge, S. M. J.; Dedecker, K. D.; Vanduyfhuys, L.; Waroquier, M.; Van Speybroeck, V. Protocol for Identifying Accurate Collective Variables in Enhanced Molecular Dynamics Simulations for the Description of Structural Transformations in Flexible Metal–Organic Frameworks. J. Chem. Theory Comput. 2018, 14, 5511– 5526,  DOI: 10.1021/acs.jctc.8b00725
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    Protocol for Identifying Accurate Collective Variables in Enhanced Molecular Dynamics Simulations for the Description of Structural Transformations in Flexible Metal-Organic Frameworks
    Demuynck, Ruben; Wieme, Jelle; Rogge, Sven M. J.; Dedecker, Karen D.; Vanduyfhuys, Louis; Waroquier, Michel; Van Speybroeck, Veronique
    Journal of Chemical Theory and Computation (2018), 14 (11), 5511-5526CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)
    Various kinds of flexibility have been obsd. in metal-org. frameworks, which may originate from the topol. of the material or the presence of flexible ligands. The construction of free energy profiles describing the full dynamical behavior along the phase transition path is challenging since it is not trivial to identify collective variables able to identify all metastable states along the reaction path. In this work, a systematic three-step protocol to uniquely identify the dominant order parameters for structural transformations in flexible metal-org. frameworks and subsequently construct accurate free energy profiles is presented. Methodol., this protocol is rooted in the time-structure based independent component anal. (tICA), a well-established statistical modeling technique embedded in the Markov state model methodol. and often employed to study protein folding, that allows for the identification of the slowest order parameters characterizing the structural transformation. To ensure an unbiased and systematic identification of these order parameters, the tICA decompn. is performed based on information from a prior replica exchange (RE) simulation, as this technique enhances the sampling along all degrees of freedom of the system simultaneously. From this simulation, the tICA procedure exts. the order parameters - often structural parameters - that characterize the slowest transformations in the material. Subsequently, these order parameters are adopted in traditional enhanced sampling methods such as umbrella sampling, thermodn. integration, and variationally enhanced sampling to construct accurate free energy profiles capturing the flexibility in these nanoporous materials. In this work, the applicability of this tICA-RE protocol is demonstrated by detg. the slowest order parameters in both MIL-53(Al) and CAU-13, which exhibit a strongly different type of flexibility. The obtained free energy profiles as a function of this extd. order parameter are furthermore compared to the profiles obtained when adopting less-suited collective variables, indicating the importance of systematically selecting the relevant order parameters to construct accurate free energy profiles for flexible metal-org. frameworks, which is in correspondence with exptl. findings. The method succeeds in mapping the full free energy surface in terms of appropriate collective variables for MOFs exhibiting linker flexibility. For CAU-13, we show the decreased stability of the closed pore phase by systematically adding adsorbed xylene mols. in the framework.
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    Daubechies, I. Orthonormal Bases of Compactly Supported Wavelets. Commun. Pure Appl. Math. 1988, 41, 909– 996,  DOI: 10.1002/cpa.3160410705
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    Mohr, S.; Ratcliff, L. E.; Boulanger, P.; Genovese, L.; Caliste, D.; Deutsch, T.; Goedecker, S. Daubechies Wavelets for Linear Scaling Density Functional Theory. J. Chem. Phys. 2014, 140, 204110  DOI: 10.1063/1.4871876
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    Daubechies wavelets for linear scaling density functional theory
    Mohr, Stephan; Ratcliff, Laura E.; Boulanger, Paul; Genovese, Luigi; Caliste, Damien; Deutsch, Thierry; Goedecker, Stefan
    Journal of Chemical Physics (2014), 140 (20), 204110/1-204110/16CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
    We demonstrate that Daubechies wavelets can be used to construct a minimal set of optimized localized adaptively contracted basis functions in which the Kohn-Sham orbitals can be represented with an arbitrarily high, controllable precision. Ground state energies and the forces acting on the ions can be calcd. in this basis with the same accuracy as if they were calcd. directly in a Daubechies wavelets basis, provided that the amplitude of these adaptively contracted basis functions is sufficiently small on the surface of the localization region, which is guaranteed by the optimization procedure described in this work. This approach reduces the computational costs of d. functional theory calcns., and can be combined with sparse matrix algebra to obtain linear scaling with respect to the no. of electrons in the system. Calcns. on systems of 10 000 atoms or more thus become feasible in a systematic basis set with moderate computational resources. Further computational savings can be achieved by exploiting the similarity of the adaptively contracted basis functions for closely related environments, e.g., in geometry optimizations or combined calcns. of neutral and charged systems. (c) 2014 American Institute of Physics.
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    Ratcliff, L. E.; Dawson, W.; Fisicaro, G.; Caliste, D.; Mohr, S.; Degomme, A.; Videau, B.; Cristiglio, V.; Stella, M.; D’Alessandro, M.; Goedecker, S.; Nakajima, T.; Deutsch, T.; Genovese, L. Flexibilities of Wavelets as a Computational Basis Set for Large-Scale Electronic Structure Calculations. J. Chem. Phys. 2020, 152, 194110  DOI: 10.1063/5.0004792
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    Flexibilities of wavelets as a computational basis set for large-scale electronic structure calculations
    Ratcliff, Laura E.; Dawson, William; Fisicaro, Giuseppe; Caliste, Damien; Mohr, Stephan; Degomme, Augustin; Videau, Brice; Cristiglio, Viviana; Stella, Martina; D'Alessandro, Marco; Goedecker, Stefan; Nakajima, Takahito; Deutsch, Thierry; Genovese, Luigi
    Journal of Chemical Physics (2020), 152 (19), 194110CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
    The BigDFT project was started in 2005 with the aim of testing the advantages of using a Daubechies wavelet basis set for Kohn-Sham (KS) d. functional theory (DFT) with pseudopotentials. This project led to the creation of the BigDFT code, which employs a computational approach with optimal features of flexibility, performance, and precision of the results. In particular, the employed formalism has enabled the implementation of an algorithm able to tackle DFT calcns. of large systems, up to many thousands of atoms, with a computational effort that scales linearly with the no. of atoms. In this work, we recall some of the features that have been made possible by the peculiar properties of Daubechies wavelets. In particular, we focus our attention on the usage of DFT for large-scale systems. We show how the localized description of the KS problem, emerging from the features of the basis set, is helpful in providing a simplified description of large-scale electronic structure calcns. We provide some examples on how such a simplified description can be employed, and we consider, among the case-studies, the SARS-CoV-2 main protease. (c) 2020 American Institute of Physics.
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    Maiolo, M.; Vancheri, A.; Krause, R.; Danani, A. Wavelets as Basis Functions to Represent the Coarse-Graining Potential in Multiscale Coarse Graining Approach. J. Comput. Phys. 2015, 300, 592– 604,  DOI: 10.1016/j.jcp.2015.07.039
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    Tribello, G. A.; Bonomi, M.; Branduardi, D.; Camilloni, C.; Bussi, G. PLUMED 2: New Feathers for an Old Bird. Comput. Phys. Commun. 2014, 185, 604– 613,  DOI: 10.1016/j.cpc.2013.09.018
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    PLUMED 2: New feathers for an old bird
    Tribello, Gareth A.; Bonomi, Massimiliano; Branduardi, Davide; Camilloni, Carlo; Bussi, Giovanni
    Computer Physics Communications (2014), 185 (2), 604-613CODEN: CPHCBZ; ISSN:0010-4655. (Elsevier B.V.)
    Enhancing sampling and analyzing simulations are central issues in mol. simulation. Recently, we introduced PLUMED, an open-source plug-in that provides some of the most popular mol. dynamics (MD) codes with implementations of a variety of different enhanced sampling algorithms and collective variables (CVs). The rapid changes in this field, in particular new directions in enhanced sampling and dimensionality redn. together with new hardware, require a code that is more flexible and more efficient. We therefore present PLUMED 2 here-a complete rewrite of the code in an object-oriented programming language (C++). This new version introduces greater flexibility and greater modularity, which both extends its core capabilities and makes it far easier to add new methods and CVs. It also has a simpler interface with the MD engines and provides a single software library contg. both tools and core facilities. Ultimately, the new code better serves the ever-growing community of users and contributors in coping with the new challenges arising in the field.
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    Kellermeier, M.; Raiteri, P.; Berg, J. K.; Kempter, A.; Gale, J. D.; Gebauer, D. Entropy Drives Calcium Carbonate Ion Association. ChemPhysChem 2016, 17, 3535– 3541,  DOI: 10.1002/cphc.201600653
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    Entropy drives calcium carbonate ion association
    Kellermeier, Matthias; Raiteri, Paolo; Berg, John; Kempter, Andreas; Gale, Julian; Gebauer, Denis
    ChemPhysChem (2016), 17 (21), 3535-3541CODEN: CPCHFT; ISSN:1439-4235. (Wiley-VCH Verlag GmbH & Co. KGaA)
    The understanding of the mol. mechanisms underlying the early stages of crystn. is still incomplete. In the case of calcium carbonate, exptl. and computational evidence suggests that phase sepn. relies on so-called pre-nucleation clusters (PNCs). A thorough thermodn. anal. of the enthalpic and entropic contributions to the overall free energy of PNC formation derived from three independent methods demonstrates that solute clustering is driven by entropy. This can be quant. rationalized by the release of water mols. from ion hydration layers, explaining why ion assocn. is not limited to simple ion pairing. The key role of water release in this process suggests that PNC formation should be a common phenomenon in aq. solns.
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    Valsson, O.; Parrinello, M. Well-Tempered Variational Approach to Enhanced Sampling. J. Chem. Theory Comput. 2015, 11, 1996– 2002,  DOI: 10.1021/acs.jctc.5b00076
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    Well-Tempered Variational Approach to Enhanced Sampling
    Valsson, Omar; Parrinello, Michele
    Journal of Chemical Theory and Computation (2015), 11 (5), 1996-2002CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)
    We propose a simple yet effective iterative scheme that allows us to employ the well-tempered distribution as a target distribution for the collective variables in our recently introduced variational approach to enhanced sampling and free energy calcns. The performance of the scheme is evaluated for the three-dimensional free energy surface of alanine tetrapeptide where the convergence can be rather poor when employing the uniform target distribution. Using the well-tempered target distribution on the other hand results in a significant improvement in convergence. The results obsd. in this paper indicate that the well-tempered distribution is in most cases the preferred and recommended choice for the target distribution in the variational approach.
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    Tiwary, P.; Parrinello, M. A Time-Independent Free Energy Estimator for Metadynamics. J. Phys. Chem. B 2015, 119, 736– 742,  DOI: 10.1021/jp504920s
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    A Time-Independent Free Energy Estimator for Metadynamics
    Tiwary, Pratyush; Parrinello, Michele
    Journal of Physical Chemistry B (2015), 119 (3), 736-742CODEN: JPCBFK; ISSN:1520-5207. (American Chemical Society)
    Metadynamics is a powerful and well-established enhanced sampling method for exploring and quantifying free energy surfaces of complex systems as a function of appropriately chosen variables. In the limit of long simulation time, metadynamics converges to the exact free energy surface plus a time-dependent const. The authors analyze in detail this time-dependent const. The authors show an easy way to calc. it, and by explicitly calcg. the time dependence of this const., they are able to derive a time-independent and locally convergent free energy estimator for metadynamics. The authors also derive an alternate procedure for obtaining the full unbiased distributions of generic operators from biased metadynamics simulations and explicitly test its usefulness.
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    Baftizadeh, F.; Cossio, P.; Pietrucci, F.; Laio, A. Protein Folding and Ligand-Enzyme Binding from Bias-Exchange Metadynamics Simulations. Curr. Phys. Chem. 2012, 2, 79– 91,  DOI: 10.2174/1877946811202010079
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    Protein folding and ligand-enzyme binding from bias-exchange metadynamics simulations
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    Current Physical Chemistry (2012), 2 (1), 79-91CODEN: CPCUBU; ISSN:1877-9468. (Bentham Science Publishers Ltd.)
    A review. Bias-Exchange Metadynamics is a powerful technique that can be used for reconstructing the free energy and for enhancing the conformational search in complex biol. systems. In this method, a large set of collective variables (CVs) is chosen and several metadynamics simulations are performed on different replicas of the system, each replica biasing a different CV. Exchanges between the bias potentials are periodically attempted according to a replica exchange scheme, and this process is repeated until convergence of the free energy profiles is obtained. Bias-Exchange Metadynamics has been used to understand several different biol. phenomena. In particular, due to the efficaciously multidimensional nature of the bias, it is useful to study the folding process of small-to-medium size proteins, and ligand-enzyme binding. This review intends to provide a comprehensive description of the algorithm and the approach used to analyze its output. We focus on the practical aspects that need to be addressed when one attempts to apply the method to study protein systems: choice of the appropriate set of parameters and CVs, proper treatment of boundary conditions, convergence criteria, and derivation of a thermodn. and kinetic model of the system from the simulation results.
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    Crespo, Y.; Marinelli, F.; Pietrucci, F.; Laio, A. Metadynamics Convergence Law in a Multidimensional System. Phys. Rev. E 2010, 81, 055701(R)  DOI: 10.1103/PhysRevE.81.055701
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    Metadynamics is an efficient method for simulation of the free energy of many-particle systems. Over the last several years it has been applied to study a wide variety of systems, ranging from simple fluids to biol. macromols. The method relies on uniform sampling along specified collective variables or order parameters. Such order parameters, however, are often bounded, and metadynamics algorithms as originally developed suffer from systematic errors at the corresponding boundaries. While several approaches have been proposed in the past to correct these errors for unidimensional systems, no method exists to fully correct these errors in multi-dimensional systems at points where multiple boundaries meet. Here we present a correction scheme that circumvents this limitation. (c) 2013 American Institute of Physics.
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    We show how to derive a simple integrator for the Langevin equation and illustrate how it is possible to check the accuracy of the obtained distribution on the fly, using the concept of effective energy introduced in a recent paper [J. Chem. Phys. 126, 014101 (2007)]. Our integrator leads to correct sampling also in the difficult high-friction limit. We also show how these ideas can be applied in practical simulations, using a Lennard-Jones crystal as a paradigmatic case.
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    Wolfe, S.; Schlegel, H. B.; Csizmadia, I. G.; Bernardi, F. Chemical Dynamics of Symmetric and Asymmetric Reaction Coordinates. J. Am. Chem. Soc. 1975, 97, 2020– 2024,  DOI: 10.1021/ja00841a005
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    Journal of the American Chemical Society (1975), 97 (8), 2020-4CODEN: JACSAT; ISSN:0002-7863.
    The question of whether a rotation-inversion process that results in the interconversion of two enantiomeric or identical species can be described by a unique asymmetric reaction coordinate was subjected to a quantum mechanical analysis, employing the symmetry properties of the reaction surface, and the results applied to two cases: case I, in which one transition state separates reagents and products; and case II, in which a stable intermediate appears on the reaction coordinate. No unique path exists for case I, i.e., all asymmetric reaction coordinates are indistinguishable. This conclusion holds under equilibrium conditions, nonequilibrium conditions, and photochem. excitation. For case II, distinguishable asymmetric paths can exist under nonequilibrium conditions. These are related to each other in a diastereomeric sense, in contrast to case I, in which various reaction paths differ in an enantiomeric sense.
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    A growing string method for the reaction pathway defined by a Newton trajectory
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    Journal of Chemical Physics (2005), 122 (17), 174106/1-174106/11CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
    The reaction path is an important concept of theor. chem. We use a projection operator for the following of the Newton trajectory (NT) along the reaction valley of the potential energy surface. We describe the numerical scheme for the string method, adapting the proposal of a growing string (GS) by [Peters et al.,J. Chem. Phys. 120, 7877 (2004)]. The combination of the Newton projector and the growing string idea is an improvement of both methods, and a great saving of the no. of iterations needed to find the pathway over the saddle point. This combination GS-NT is at the best of our knowledge new. We employ two different corrector methods: first, the use of projected gradient steps, and second a conjugated gradient method, the CG+ method of Liu, Nocedal, and Waltz, generalized by projectors. The executed examples are Lennard-Jones clusters, LJ7 and LJ22, and an N-methyl-alanyl-acetamide (alanine dipeptide) rearrangement between the min. C7ax and C5. For the latter, the growing string calcn. is interfaced with the GASSIAN03 quantum chem. software package.
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    Fast parallel algorithms for short-range molecular dynamics
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    Three parallel algorithms for classical mol. dynamics are presented. The first assigns each processor a fixed subset of atoms; the second assigns each a fixed subset of inter-at. forces to compute; the third assigns each a fixed spatial region. The algorithms are suitable for mol. dynamics models which can be difficult to parallelize efficiently - those with short-range forces where the neighbors of each atom change rapidly. They can be implemented on any distributed-memory parallel machine which allows for message-passing of data between independently executing processors. The algorithms are tested on a std. Lennard-Jones benchmark problem for system sizes ranging from 500 to 100,000,000 atoms on several parallel supercomputers - the nCUBE 2, Intel iPSC/860 and Paragon, and Cray T3D. Comparing the results to the fastest reported vectorized Cray Y-MP and C90 algorithm shows that the current generation of parallel machines is competitive with conventional vector supercomputers even for small problems. For large problems, the spatial algorithm achieves parallel efficiencies of 90% and a 1840-node Intel Paragon performs up to 165 faster than a single Cray C90 processor. Trade-offs between the three algorithms and guidelines for adapting them to more complex mol. dynamics simulations are also discussed.
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    Demichelis, R.; Raiteri, P.; Gale, J. D.; Quigley, D.; Gebauer, D. Stable Prenucleation Mineral Clusters Are Liquid-like Ionic Polymers. Nat. Commun. 2011, 2, 590,  DOI: 10.1038/ncomms1604
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    Stable prenucleation mineral clusters are liquid-like ionic polymers
    Demichelis Raffaella; Raiteri Paolo; Gale Julian D; Quigley David; Gebauer Denis
    Nature communications (2011), 2 (), 590 ISSN:.
    Calcium carbonate is an abundant substance that can be created in several mineral forms by the reaction of dissolved carbon dioxide in water with calcium ions. Through biomineralization, organisms can harness and control this process to form various functional materials that can act as anything from shells through to lenses. The early stages of calcium carbonate formation have recently attracted attention as stable prenucleation clusters have been observed, contrary to classical models. Here we show, using computer simulations combined with the analysis of experimental data, that these mineral clusters are made of an ionic polymer, composed of alternating calcium and carbonate ions, with a dynamic topology consisting of chains, branches and rings. The existence of a disordered, flexible and strongly hydrated precursor provides a basis for explaining the formation of other liquid-like amorphous states of calcium carbonate, in addition to the non-classical behaviour during growth of amorphous calcium carbonate.
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    Raiteri, P.; Demichelis, R.; Gale, J. D. Thermodynamically Consistent Force Field for Molecular Dynamics Simulations of Alkaline-Earth Carbonates and Their Aqueous Speciation. J. Phys. Chem. C 2015, 119, 24447– 24458,  DOI: 10.1021/acs.jpcc.5b07532
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    Thermodynamically Consistent Force Field for Molecular Dynamics Simulations of Alkaline-Earth Carbonates and Their Aqueous Speciation
    Raiteri, Paolo; Demichelis, Raffaella; Gale, Julian D.
    Journal of Physical Chemistry C (2015), 119 (43), 24447-24458CODEN: JPCCCK; ISSN:1932-7447. (American Chemical Society)
    In recent years atomistic simulations have become increasingly important in providing mol. insight to complement expts. Even for the seemingly simple case of ion-pair formation a detailed atomistic picture of the structure and relative stability of the contact, solvent-shared and solvent-sepd. ion pairs can only be readily achieved by computer simulation. Here a new force field parametrization for the alk.-earth carbonate interactions in water has been developed by fitting against exptl. thermodn. and structural data. We demonstrate that the present force field can accurately reproduce the dynamics and thermodn. of the ions in soln., which is the key to producing quant. accurate data that can be compared against expt.
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    Wu, Y.; Tepper, H. L.; Voth, G. A. Flexible Simple Point-Charge Water Model with Improved Liquid-State Properties. J. Chem. Phys. 2006, 124, 024503  DOI: 10.1063/1.2136877
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    Flexible simple point-charge water model with improved liquid-state properties
    Wu, Yujie; Tepper, Harald L.; Voth, Gregory A.
    Journal of Chemical Physics (2006), 124 (2), 024503/1-024503/12CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
    In order to introduce flexibility into the simple point-charge (SPC) water model, the impact of the intramol. degrees of freedom on liq. properties was systematically studied in this work as a function of many possible parameter sets. It was found that the diffusion const. is extremely sensitive to the equil. bond length and that this effect is mainly due to the strength of intermol. hydrogen bonds. The static dielec. const. was found to be very sensitive to the equil. bond angle via the distribution of intermol. angles in the liq.: A larger bond angle will increase the angle formed by two mol. dipoles, which is particularly significant for the first solvation shell. This result is in agreement with the work of Hochtl et al. [J. Chem. Phys. 109, 4927 (1998)]. A new flexible simple point-charge water model was derived by optimizing bulk diffusion and dielec. consts. to the exptl. values via the equil. bond length and angle. Due to the large sensitivities, the parametrization only slightly perturbs the mol. geometry of the base SPC model. Extensive comparisons of thermodn., structural, and kinetic properties indicate that the new model is much improved over the std. SPC model and its overall performance is comparable to or even better than the extended SPC model.
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    A unified formulation of the constant-temperature molecular-dynamics methods
    Nose, Shuichi
    Journal of Chemical Physics (1984), 81 (1), 511-19CODEN: JCPSA6; ISSN:0021-9606.
    Three recently proposed const. temp. mol. dynamics methods [N., (1984) (1); W. G. Hoover et al., (1982) (2); D. J. Evans and G. P. Morris, (1983) (2); and J. M. Haile and S. Gupta, 1983) (3)] are examd. anal. via calcg. the equil. distribution functions and comparing them with that of the canonical ensemble. Except for effects due to momentum and angular momentum conservation, method (1) yields the rigorous canonical distribution in both momentum and coordinate space. Method (2) can be made rigorous in coordinate space, and can be derived from method (1) by imposing a specific constraint. Method (3) is not rigorous and gives a deviation of order N-1/2 from the canonical distribution (N the no. of particles). The results for the const. temp.-const. pressure ensemble are similar to the canonical ensemble case.
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    Canonical dynamics: Equilibrium phase-space distributions
    Hoover
    Physical review. A, General physics (1985), 31 (3), 1695-1697 ISSN:0556-2791.
    There is no expanded citation for this reference.
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    Tuckerman, M. E.; Alejandre, J.; López-Rendón, R.; Jochim, A. L.; Martyna, G. J. A Liouville-operator Derived Measure-Preserving Integrator for Molecular Dynamics Simulations in the Isothermal–Isobaric Ensemble. J. Phys. A: Math. Gen. 2006, 39, 5629– 5651,  DOI: 10.1088/0305-4470/39/19/S18
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    A Liouville-operator derived measure-preserving integrator for molecular dynamics simulations in the isothermal-isobaric ensemble
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    A review. The const.-pressure, const.-temp. (NPT) mol. dynamics approach is re-examd. from the viewpoint of deriving a new measure-preserving reversible geometric integrator for the equations of motion. The underlying concepts of non-Hamiltonian phase-space anal., measure-preserving integrators and the symplectic property for Hamiltonian systems are briefly reviewed. In addn., current measure-preserving schemes for the const.-vol., const.-temp. ensemble are also reviewed. A new geometric integrator for the NPT method is presented, is shown to preserve the correct phase-space vol. element and is demonstrated to perform well in realistic examples. Finally, a multiple time-step version of the integrator is presented for treating systems with motion on several time scales.
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    Hockney, R. W.; Eastwood, J. W. Computer Simulation Using Particles; CRC Press: Boca Raton, FL, 1988.
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    Raiteri, P.; Laio, A.; Gervasio, F. L.; Micheletti, C.; Parrinello, M. Efficient Reconstruction of Complex Free Energy Landscapes by Multiple Walkers Metadynamics. J. Phys. Chem. B 2006, 110, 3533– 3539,  DOI: 10.1021/jp054359r
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    Efficient Reconstruction of Complex Free Energy Landscapes by Multiple Walkers Metadynamics
    Raiteri, Paolo; Laio, Alessandro; Gervasio, Francesco Luigi; Micheletti, Cristian; Parrinello, Michele
    Journal of Physical Chemistry B (2006), 110 (8), 3533-3539CODEN: JPCBFK; ISSN:1520-6106. (American Chemical Society)
    Recently, we have introduced a new method, metadynamics, which is able to sample rarely occurring transitions and to reconstruct the free energy as a function of several variables with a controlled accuracy. This method has been successfully applied in many different fields, ranging from chem. to biophysics and ligand docking and from material science to crystal structure prediction. We present an important development that speeds up metadynamics calcns. by orders of magnitude and renders the algorithm much more robust. We use multiple interacting simulations, walkers, for exploring and reconstructing the same free energy surface. Each walker contributes to the history-dependent potential that, in metadynamics, is an est. of the free energy. We show that the error on the reconstructed free energy does not depend on the no. of walkers, leading to a fully linear scaling algorithm even on inexpensive loosely coupled clusters of PCs. In addn., we show that the accuracy and stability of the method are much improved by combining it with a weighted histogram anal. We check the validity of our new method on a realistic application.
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    Branduardi, D.; Bussi, G.; Parrinello, M. Metadynamics with Adaptive Gaussians. J. Chem. Theory Comput. 2012, 8, 2247– 2254,  DOI: 10.1021/ct3002464
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    Metadynamics with Adaptive Gaussians
    Branduardi, Davide; Bussi, Giovanni; Parrinello, Michele
    Journal of Chemical Theory and Computation (2012), 8 (7), 2247-2254CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)
    Metadynamics is an established sampling method aimed at reconstructing the free-energy surface relative to a set of appropriately chosen collective variables. In std. metadynamics, the free-energy surface is filled by the addn. of Gaussian potentials of preassigned and typically diagonal covariance. Asymptotically the free-energy surface is proportional to the bias deposited. Here, we consider the possibility of using Gaussians whose variance is adjusted on the fly to the local properties of the free-energy surface. We suggest two different prescriptions: one is based on the local diffusivity and the other on the local geometrical properties. We further examine the problem of extg. the free-energy surface when using adaptive Gaussians. We show that the std. relation between the bias and the free energy does not hold. In the limit of narrow Gaussians an explicit correction can be evaluated. In the general case, we propose to use instead a relation between bias and free energy borrowed from umbrella sampling. This relation holds for all kinds of incrementally deposited bias. We illustrate on the case of alanine dipeptide the advantage of using adaptive Gaussians in conjunction with the new free-energy estimator both in terms of accuracy and speed of convergence.
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    Coveney, P. V.; Wan, S. On the Calculation of Equilibrium Thermodynamic Properties from Molecular Dynamics. Phys. Chem. Chem. Phys. 2016, 18, 30236– 30240,  DOI: 10.1039/C6CP02349E
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    On the calculation of equilibrium thermodynamic properties from molecular dynamics
    Coveney, Peter V.; Wan, Shunzhou
    Physical Chemistry Chemical Physics (2016), 18 (44), 30236-30240CODEN: PPCPFQ; ISSN:1463-9076. (Royal Society of Chemistry)
    The purpose of statistical mechanics is to provide a route to the calcn. of macroscopic properties of matter from their constituent microscopic components. It is well known that the macrostates emerge as ensemble avs. of microstates. However, this is more often stated than implemented in computer simulation studies. Here we consider foundational aspects of statistical mechanics which are overlooked in most textbooks and research articles that purport to compute macroscopic behavior from microscopic descriptions based on classical mechanics and show how due attention to these issues leads in directions which have not been widely appreciated in the field of mol. dynamics simulation.
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    Grossfield, A.; Patrone, P. N.; Roe, D. R.; Schultz, A. J.; Siderius, D.; Zuckerman, D. M. Best Practices for Quantification of Uncertainty and Sampling Quality in Molecular Simulations. LiveCoMS 2019, 1, 5067,  DOI: 10.33011/livecoms.1.1.5067
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    Pampel, B.; Valsson, O. Improving the Efficiency of Variationally Enhanced Sampling with Wavelet-Based Bias Potentials (v1.0) [Data set]. Zenodo , 2022. DOI: 10.5281/zenodo.5851773 .
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    Aguilar-Mogas, A.; Giménez, X.; Bofill, J. M. Implementation of an Algorithm Based on the Runge-Kutta-Fehlberg Technique and the Potential Energy as a Reaction Coordinate to Locate Intrinsic Reaction Paths. J. Comput. Chem. 2010, 2510– 2525,  DOI: 10.1002/jcc.21539
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  • Abstract

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    Figure 1

    Figure 1. Visualization of different VES basis functions used in this paper. The Sym8 wavelets, Gaussians, and cubic B-splines are localized basis functions. Here, we only show two adjacent functions, while a full basis set would include all shifted functions in the given interval (that is, [−3, 3] here). On the contrary, Legendre polynomials are delocalized functions supported on the full interval of the bias. The Legendre basis set consists of all polynomials up to a certain order; the figure shows the functions up to the quartic polynomial.

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    Figure 2

    Figure 2. Results for the one-dimensional double-well potential described in Section 3.1. (a) The reference FES along with the FES obtained using the wavelet basis functions at different numbers of bias iterations for one of the runs. (b) The RMS error measure (Section 3.5, eq 36) for the different basis functions as a function of the number of bias iterations. The lines denote the average over 20 independent runs, and the shaded areas denote the corresponding standard error. (c, d) The RMS error of the individual runs for (c) Sym8 wavelets and (d) Legendre polynomials. The thick lines are the same as in panel b, and the dashed lines each resemble one of the runs.

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    Figure 3

    Figure 3. Results for the two-dimensional Wolfe–Quapp potential described in Section 3.2. (a) The reference FES along with free energy projections on the x- and y-coordinates. (b, c) The free energy difference ΔF (Section 3.5, eq 38) between the two states obtained using (b) Sym8 wavelets and (c) Legendre polynomials as a function of the number of bias iterations. We show results from 20 independent simulations with dashed lines. We use solid lines for the averages and shaded areas to denote the standard errors. We denote the reference value with solid black lines. To define the areas corresponding to the two different states, we use the y = 0 line.

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    Figure 4

    Figure 4. Results for the rotated two-dimensional Wolfe–Quapp potential described in Section 3.3. (a) The reference FES along with free energy projections on the x- and y-coordinates. Only the x-coordinate is biased. (b, c) The free energy difference ΔF (Section 3.5, eq 38) between the two states obtained using (b) Sym8 wavelets and (c) Legendre polynomials as a function of the number of bias iterations. We show results from 20 independent simulations with dashed lines. We use solid lines for the averages and shaded areas to denote the standard errors. We denote the reference value with solid black lines. To define the areas corresponding to the two different states, we use the x = 0 line.

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    Figure 5

    Figure 5. Free energy surfaces for the calcium carbonate system described in Section 3.4. (a) Projections on the distance CV for the FESs obtained directly from the bias via eq 9 (VES) or by summing over the deposited Gaussians (WTMetad). We only show one of the runs for each biasing setup. The reference data are obtained from ref (57). (b) FES as a function of both biased CVs obtained by reweighting one of the wavelet simulations.

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    Figure 6

    Figure 6. Results for the calcium carbonate system described in Section 3.4. (a, c) Time evolution of the free energy difference between the region with a Ca–C distance smaller 4 Å and the region with a Ca–C distance between 4 and 8 Å. For each biasing setup, we show three independent runs where the different color shades represent the individual runs. (b, d) The average of the free energy differences obtained over the last nanosecond by using 100 samples taken every 10 ps for each simulation. The error bars show the standard deviation to signify the quality of the individual measurements. We also show the results from ref (57) as black dotted lines. In panels a and b, we use the FES obtained directly from the bias via eq 9 (VES) or by summing over the deposited Gaussians (WTMetad). In panels c and d, we use the FES obtained through reweighting where we ignore the first 200 ps of each simulation.

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      A review. Proteins are dynamic mols. that can transition between a potentially wide range of structures comprising their conformational ensemble. The nature of these conformations and their relative probabilities are described by a high-dimensional free energy landscape. While computer simulation techniques such as mol. dynamics simulations allow characterization of the metastable conformational states and the transitions between them, and thus free energy landscapes, to be characterised, the barriers between states can be high, precluding efficient sampling without substantial computational resources. Over the past decades, a dizzying array of methods have emerged for enhancing conformational sampling, and for projecting the free energy landscape onto a reduced set of dimensions that allow conformational states to be distinguished, known as collective variables (CVs), along which sampling may be directed. Here, a brief description of what biomol. simulation entails is followed by a more detailed exposition of the nature of CVs and methods for detg. these, and, lastly, an overview of the myriad different approaches for enhancing conformational sampling, most of which rely upon CVs, including new advances in both CV detn. and conformational sampling due to machine learning.
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      Kamenik, A. S.; Linker, S. M.; Riniker, S. Enhanced Sampling without Borders: On Global Biasing Functions and How to Reweight Them. Phys. Chem. Chem. Phys. 2021, 24, 1225– 1236,  DOI: 10.1039/D1CP04809K
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      Hénin, J.; Lelièvre, T.; Shirts, M. R.; Valsson, O.; Delemotte, L. Enhanced Sampling Methods for Molecular Dynamics Simulations. arXiv (Condensed Matter.Statistical Mechanics) , February 8, 2022, 2202.04164, ver. 1. https://arxiv.org/abs/2202.04164 (accessed 2022-02-27).
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      Fiorin, G.; Klein, M. L.; Hénin, J. Using Collective Variables to Drive Molecular Dynamics Simulations. Mol. Phys. 2013, 111, 3345– 3362,  DOI: 10.1080/00268976.2013.813594
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      Using collective variables to drive molecular dynamics simulations
      Fiorin, Giacomo; Klein, Michael L.; Henin, Jerome
      Molecular Physics (2013), 111 (22-23), 3345-3362CODEN: MOPHAM; ISSN:0026-8976. (Taylor & Francis Ltd.)
      A software framework is introduced that facilitates the application of biasing algorithms to collective variables of the type commonly employed to drive massively parallel mol. dynamics (MD) simulations. The modular framework that is presented enables one to combine existing collective variables into new ones, and combine any chosen collective variable with available biasing methods. The latter include the classic time-dependent biases referred to as steered MD and targeted MD, the temp.-accelerated MD algorithm, as well as the adaptive free-energy biases called metadynamics and adaptive biasing force. The present modular software is extensible, and portable between commonly used MD simulation engines.
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      Giberti, F.; Salvalaglio, M.; Parrinello, M. Metadynamics Studies of Crystal Nucleation. IUCrJ 2015, 2, 256– 266,  DOI: 10.1107/S2052252514027626
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      Metadynamics studies of crystal nucleation
      Giberti, Federico; Salvalaglio, Matteo; Parrinello, Michele
      IUCrJ (2015), 2 (2), 256-266CODEN: IUCRAJ; ISSN:2052-2525. (International Union of Crystallography)
      Crystn. processes are characterized by activated events and long timescales. These characteristics prevent std. mol. dynamics techniques from being efficiently used for the direct investigation of processes such as nucleation. This short review provides an overview on the use of metadynamics, a state-of-the-art enhanced sampling technique, for the simulation of phase transitions involving the prodn. of a cryst. solid. In particular the principles of metadynamics are outlined, several order parameters are described that have been or could be used in conjunction with metadynamics to sample nucleation events and then an overview is given of recent metadynamics results in the field of crystal nucleation.
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      Pietrucci, F. Strategies for the Exploration of Free Energy Landscapes: Unity in Diversity and Challenges Ahead. Rev. Phys. 2017, 2, 32– 45,  DOI: 10.1016/j.revip.2017.05.001
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      Wang, Y.; Lamim Ribeiro, J. M.; Tiwary, P. Machine Learning Approaches for Analyzing and Enhancing Molecular Dynamics Simulations. Curr. Opin. Struct. Biol. 2020, 61, 139– 145,  DOI: 10.1016/j.sbi.2019.12.016
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      Machine learning approaches for analyzing and enhancing molecular dynamics simulations
      Wang, Yihang; Lamim Ribeiro, Joao Marcelo; Tiwary, Pratyush
      Current Opinion in Structural Biology (2020), 61 (), 139-145CODEN: COSBEF; ISSN:0959-440X. (Elsevier Ltd.)
      Mol. dynamics (MD) has become a powerful tool for studying biophys. systems, due to increasing computational power and availability of software. Although MD has made many contributions to better understanding these complex biophys. systems, there remain methodol. difficulties to be surmounted. First, how to make the deluge of data generated in running even a microsecond long MD simulation human comprehensible. Second, how to efficiently sample the underlying free energy surface and kinetics. In this short perspective, we summarize machine learning based ideas that are solving both of these limitations, with a focus on their key theor. underpinnings and remaining challenges.
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      Noé, F.; Tkatchenko, A.; Müller, K.-R.; Clementi, C. Machine Learning for Molecular Simulation. Annu. Rev. Phys. Chem. 2020, 71, 361– 390,  DOI: 10.1146/annurev-physchem-042018-052331
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      Machine Learning for Molecular Simulation
      Noe, Frank; Tkatchenko, Alexandre; Mueller, Klaus-Robert; Clementi, Cecilia
      Annual Review of Physical Chemistry (2020), 71 (), 361-390CODEN: ARPLAP; ISSN:0066-426X. (Annual Reviews)
      Machine learning (ML) is transforming all areas of science. The complex and time-consuming calcns. in mol. simulations are particularly suitable for an ML revolution and have already been profoundly affected by the application of existing ML methods. Here we review recent ML methods for mol. simulation, with particular focus on (deep) neural networks for the prediction of quantum-mech. energies and forces, on coarse-grained mol. dynamics, on the extn. of free energy surfaces and kinetics, and on generative network approaches to sample mol. equil. structures and compute thermodn. To explain these methods and illustrate open methodol. problems, we review some important principles of mol. physics and describe how they can be incorporated into ML structures. Finally, we identify and describe a list of open challenges for the interface between ML and mol. simulation.
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      Gkeka, P.; Stoltz, G.; Barati Farimani, A.; Belkacemi, Z.; Ceriotti, M.; Chodera, J. D.; Dinner, A. R.; Ferguson, A. L.; Maillet, J.-B.; Minoux, H.; Peter, C.; Pietrucci, F.; Silveira, A.; Tkatchenko, A.; Trstanova, Z.; Wiewiora, R.; Lelièvre, T. Machine Learning Force Fields and Coarse-Grained Variables in Molecular Dynamics: Application to Materials and Biological Systems. J. Chem. Theory Comput. 2020, 16, 4757– 4775,  DOI: 10.1021/acs.jctc.0c00355
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      Machine Learning Force Fields and Coarse-Grained Variables in Molecular Dynamics: Application to Materials and Biological Systems
      Gkeka, Paraskevi; Stoltz, Gabriel; Barati Farimani, Amir; Belkacemi, Zineb; Ceriotti, Michele; Chodera, John D.; Dinner, Aaron R.; Ferguson, Andrew L.; Maillet, Jean-Bernard; Minoux, Herve; Peter, Christine; Pietrucci, Fabio; Silveira, Ana; Tkatchenko, Alexandre; Trstanova, Zofia; Wiewiora, Rafal; Lelievre, Tony
      Journal of Chemical Theory and Computation (2020), 16 (8), 4757-4775CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)
      A review. Machine learning encompasses tools and algorithms that are now becoming popular in almost all scientific and technol. fields. This is true for mol. dynamics as well, where machine learning offers promises of extg. valuable information from the enormous amts. of data generated by simulation of complex systems. The authors provide here a review of the authors' current understanding of goals, benefits, and limitations of machine learning techniques for computational studies on atomistic systems, focusing on the construction of empirical force fields from ab initio databases and the detn. of reaction coordinates for free energy computation and enhanced sampling.
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      Sidky, H.; Chen, W.; Ferguson, A. L. Machine Learning for Collective Variable Discovery and Enhanced Sampling in Biomolecular Simulation. Mol. Phys. 2020, 118, e1737742  DOI: 10.1080/00268976.2020.1737742
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      Machine learning for collective variable discovery and enhanced sampling in biomolecular simulation
      Sidky, Hythem; Chen, Wei; Ferguson, Andrew L.
      Molecular Physics (2020), 118 (5), e1737742/1-e1737742/21CODEN: MOPHAM; ISSN:0026-8976. (Taylor & Francis Ltd.)
      Classical mol. dynamics simulates the time evolution of mol. systems through the phase space spanned by the positions and velocities of the constituent atoms. Mol.-level thermodn., kinetic, and structural data extd. from the resulting trajectories provide valuable information for the understanding, engineering, and design of biol. and mol. materials. The cost of simulating many-body at. systems makes simulations of large mols. prohibitively expensive, and the high-dimensionality of the resulting trajectories presents a challenge for anal. Driven by advances in algorithms, hardware, and data availability, there has been a flare of interest in recent years in the applications of machine learning - esp. deep learning - to mol. simulation. These techniques have demonstrated great power and flexibility in both extg. mechanistic understanding of the important nonlinear collective variables governing the dynamics of a mol. system, and in furnishing good low-dimensional system representations with which to perform enhanced sampling or develop long-timescale dynamical models. It is the purpose of this article to introduce the key machine learning approaches, describe how they are married with statistical mech. theory into domain-specific tools, and detail applications of these approaches in understanding and accelerating biomol. simulation.
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      Torrie, G. M.; Valleau, J. P. Nonphysical Sampling Distributions in Monte Carlo Free-Energy Estimation: Umbrella Sampling. J. Comput. Phys. 1977, 23, 187– 199,  DOI: 10.1016/0021-9991(77)90121-8
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      Huber, T.; Torda, A. E.; van Gunsteren, W. F. Local Elevation: A Method for Improving the Searching Properties of Molecular Dynamics Simulation. J. Comput.-Aided Mol. Des. 1994, 8, 695– 708,  DOI: 10.1007/BF00124016
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      Local elevation: a method for improving the searching properties of molecular dynamics simulation
      Huber, Thomas; Torda, Andrew E.; van Gunsteren, Wilfred F.
      Journal of Computer-Aided Molecular Design (1994), 8 (6), 695-708CODEN: JCADEQ; ISSN:0920-654X. (ESCOM)
      The concept of memory has been introduced into a mol. dynamics algorithm. This was done so as to persuade a mol. system to visit new areas of conformational space rather than be confined to a small no. of low-energy regions. The method is demonstrated on a simple model system and the 11-residue cyclic peptide cyclosporin A. For comparison, calcns. were also performed using simulated temp. annealing and a potential energy annealing scheme. Although the method can only be applied to systems with a small no. of degrees of freedom, it offers the chance to generate a multitude of different low-energy structures, where other methods only give a single one or few. This is clearly important in problems such as drug design, where one is interested in the conformational spread of a system.
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      Darve, E.; Pohorille, A. Calculating Free Energies Using Average Force. J. Chem. Phys. 2001, 115, 9169– 9183,  DOI: 10.1063/1.1410978
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      Calculating free energies using average force
      Darve, Eric; Pohorille, Andrew
      Journal of Chemical Physics (2001), 115 (20), 9169-9183CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
      A new, general formula that connects the derivs. of the free energy along the selected, generalized coordinates of the system with the instantaneous force acting on these coordinates is derived. The instantaneous force is defined as the force acting on the coordinate of interest so that when it is subtracted from the equations of motion the acceleration along this coordinate is zero. The formula applies to simulations in which the selected coordinates are either unconstrained or constrained to fixed values. It is shown that in the latter case the formula reduces to the expression previously derived by den Otter and Briels [Mol. Phys. 98, 773 (2000)]. If simulations are carried out without constraining the coordinates of interest, the formula leads to a new method for calcg. the free energy changes along these coordinates. This method is tested in two examples - rotation around the C-C bond of 1,2-dichloroethane immersed in water and transfer of fluoromethane across the water-hexane interface. The calcd. free energies are compared with those obtained by two commonly used methods. One of them relies on detg. the probability d. function of finding the system at different values of the selected coordinate and the other requires calcg. the av. force at discrete locations along this coordinate in a series of constrained simulations. The free energies calcd. by these three methods are in excellent agreement. The relative advantages of each method are discussed.
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      Comer, J.; Gumbart, J. C.; Hénin, J.; Lelièvre, T.; Pohorille, A.; Chipot, C. The Adaptive Biasing Force Method: Everything You Always Wanted to Know but Were Afraid to Ask. J. Phys. Chem. B 2015, 119, 1129– 1151,  DOI: 10.1021/jp506633n
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      The Adaptive Biasing Force Method: Everything You Always Wanted To Know but Were Afraid To Ask
      Comer, Jeffrey; Gumbart, James C.; Henin, Jerome; Lelievre, Tony; Pohorille, Andrew; Chipot, Christophe
      Journal of Physical Chemistry B (2015), 119 (3), 1129-1151CODEN: JPCBFK; ISSN:1520-5207. (American Chemical Society)
      In the host of numerical schemes devised to calc. free energy differences by way of geometric transformations, the adaptive biasing force algorithm has emerged as a promising route to map complex free-energy landscapes. It relies upon the simple concept that as a simulation progresses, a continuously updated biasing force is added to the equations of motion, such that in the long-time limit it yields a Hamiltonian devoid of an av. force acting along the transition coordinate of interest. This means that sampling proceeds uniformly on a flat free-energy surface, thus providing reliable free-energy ests. Much of the appeal of the algorithm to the practitioner is in its phys. intuitive underlying ideas and the absence of any requirements for prior knowledge about free-energy landscapes. Since its inception in 2001, the adaptive biasing force scheme has been the subject of considerable attention, from in-depth math. anal. of convergence properties to novel developments and extensions. The method has also been successfully applied to many challenging problems in chem. and biol. In this contribution, the method is presented in a comprehensive, self-contained fashion, discussing with a crit. eye its properties, applicability, and inherent limitations, as well as introducing novel extensions. Through free-energy calcns. of prototypical mol. systems, many methodol. aspects are examd., from stratification strategies to overcoming the so-called hidden barriers in orthogonal space, relevant not only to the adaptive biasing force algorithm but also to other importance-sampling schemes. On the basis of the discussions in this paper, a no. of good practices for improving the efficiency and reliability of the computed free-energy differences are proposed.
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      Lesage, A.; Lelièvre, T.; Stoltz, G.; Hénin, J. Smoothed Biasing Forces Yield Unbiased Free Energies with the Extended-System Adaptive Biasing Force Method. J. Phys. Chem. B 2017, 121, 3676– 3685,  DOI: 10.1021/acs.jpcb.6b10055
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      Smoothed Biasing Forces Yield Unbiased Free Energies with the Extended-System Adaptive Biasing Force Method
      Lesage, Adrien; Lelievre, Tony; Stoltz, Gabriel; Henin, Jerome
      Journal of Physical Chemistry B (2017), 121 (15), 3676-3685CODEN: JPCBFK; ISSN:1520-5207. (American Chemical Society)
      We report a theor. description and numerical tests of the extended-system adaptive biasing force method (eABF), together with an unbiased estimator of the free energy surface from eABF dynamics. Whereas the original ABF approach uses its running est. of the free energy gradient as the adaptive biasing force, eABF is built on the idea that the exact free energy gradient is not necessary for efficient exploration, and that it is still possible to recover the exact free energy sep. with an appropriate estimator. EABF does not directly bias the collective coordinates of interest, but rather fictitious variables that are harmonically coupled to them; therefore is does not require second deriv. ests., making it easily applicable to a wider range of problems than ABF. Furthermore, the extended variables present a smoother, coarse-grain-like sampling problem on a mollified free energy surface, leading to faster exploration and convergence. We also introduce CZAR, a simple, unbiased free energy estimator from eABF trajectories. EABF/CZAR converges to the phys. free energy surface faster than std. ABF for a wide range of parameters.
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      Hansmann, U. H. E.; Wille, L. T. Global Optimization by Energy Landscape Paving. Phys. Rev. Lett. 2002, 88, 068105  DOI: 10.1103/PhysRevLett.88.068105
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      Global Optimization by Energy Landscape Paving
      Hansmann, Ulrich H. E.; Wille, Luc T.
      Physical Review Letters (2002), 88 (6), 068105/1-068105/4CODEN: PRLTAO; ISSN:0031-9007. (American Physical Society)
      We introduce a novel heuristic global optimization method, energy landscape paving (ELP), which combines core ideas from energy surface deformation and tabu search. In appropriate limits, ELP reduces to existing techniques. The approach is very general and flexible and is illustrated here on two protein folding problems. For these examples, the technique gives faster convergence to the global min. than previous approaches.
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      Kästner, J. Umbrella Sampling. Wiley Interdiscip. Rev.: Comput. Mol. Sci. 2011, 1, 932– 942,  DOI: 10.1002/wcms.66
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      Maragakis, P.; van der Vaart, A.; Karplus, M. Gaussian-Mixture Umbrella Sampling. J. Phys. Chem. B 2009, 113, 4664– 4673,  DOI: 10.1021/jp808381s
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      Gaussian-Mixture Umbrella Sampling
      Maragakis, Paul; van der Vaart, Arjan; Karplus, Martin
      Journal of Physical Chemistry B (2009), 113 (14), 4664-4673CODEN: JPCBFK; ISSN:1520-6106. (American Chemical Society)
      We introduce the Gaussian-mixt. umbrella sampling method (GAMUS), a biased mol. dynamics technique based on adaptive umbrella sampling that efficiently escapes free energy min. in multidimensional problems. The prior simulation data are reweighted with a max. likelihood formulation, and the new approx. probability d. is fit to a Gaussian-mixt. model, augmented by information about the unsampled areas. The method can be used to identify free energy min. in multidimensional reaction coordinates. To illustrate GAMUS, we apply it to the alanine dipeptide (2D reaction coordinate) and tripeptide (4D reaction coordinate).
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      Umbrella sampling for nonequilibrium processes
      Warmflash, Aryeh; Bhimalapuram, Prabhakar; Dinner, Aaron R.
      Journal of Chemical Physics (2007), 127 (15), 154112/1-154112/8CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
      The authors introduce an algorithm for detg. the steady-state probability distribution of an ergodic system arbitrarily far from equil. By enforcing equal sampling of different regions of phase space, as in umbrella sampling simulations of systems at equil., low probability regions are explored to a much greater extent than in phys. weighted simulations. The algorithm can be used to accumulate joint statistics for an arbitrary no. of order parameters for a system governed by any stochastic dynamics. They demonstrate the efficiency of the algorithm by applying it to a model of a genetic toggle switch which evolves irreversibly according to a continuous time Monte Carlo procedure.
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      Laio, A.; Parrinello, M. Escaping Free-Energy Minima. Proc. Natl. Acad. Sci. U. S. A. 2002, 99, 12562– 12566,  DOI: 10.1073/pnas.202427399
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      Escaping free-energy minima
      Laio, Alessandro; Parrinello, Michele
      Proceedings of the National Academy of Sciences of the United States of America (2002), 99 (20), 12562-12566CODEN: PNASA6; ISSN:0027-8424. (National Academy of Sciences)
      We introduce a powerful method for exploring the properties of the multidimensional free energy surfaces (FESs) of complex many-body systems by means of coarse-grained non-Markovian dynamics in the space defined by a few collective coordinates. A characteristic feature of these dynamics is the presence of a history-dependent potential term that, in time, fills the min. in the FES, allowing the efficient exploration and accurate detn. of the FES as a function of the collective coordinates. We demonstrate the usefulness of this approach in the case of the dissocn. of a NaCl mol. in water and in the study of the conformational changes of a dialanine in soln.
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      Barducci, A.; Bussi, G.; Parrinello, M. Well-Tempered Metadynamics: A Smoothly Converging and Tunable Free-Energy Method. Phys. Rev. Lett. 2008, 100, 020603  DOI: 10.1103/PhysRevLett.100.020603
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      Well-Tempered Metadynamics: A Smoothly Converging and Tunable Free-Energy Method
      Barducci, Alessandro; Bussi, Giovanni; Parrinello, Michele
      Physical Review Letters (2008), 100 (2), 020603/1-020603/4CODEN: PRLTAO; ISSN:0031-9007. (American Physical Society)
      We present a method for detg. the free-energy dependence on a selected no. of collective variables using an adaptive bias. The formalism provides a unified description which has metadynamics and canonical sampling as limiting cases. Convergence and errors can be rigorously and easily controlled. The parameters of the simulation can be tuned so as to focus the computational effort only on the phys. relevant regions of the order parameter space. The algorithm is tested on the reconstruction of an alanine dipeptide free-energy landscape.
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      Valsson, O.; Tiwary, P.; Parrinello, M. Enhancing Important Fluctuations: Rare Events and Metadynamics from a Conceptual Viewpoint. Annu. Rev. Phys. Chem. 2016, 67, 159– 184,  DOI: 10.1146/annurev-physchem-040215-112229
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      Enhancing Important Fluctuations: Rare Events and Metadynamics from a Conceptual Viewpoint
      Valsson, Omar; Tiwary, Pratyush; Parrinello, Michele
      Annual Review of Physical Chemistry (2016), 67 (), 159-184CODEN: ARPLAP; ISSN:0066-426X. (Annual Reviews)
      Atomistic simulations play a central role in many fields of science. However, their usefulness is often limited by the fact that many systems are characterized by several metastable states sepd. by high barriers, leading to kinetic bottlenecks. Transitions between metastable states are thus rare events that occur on significantly longer timescales than one can simulate in practice. Numerous enhanced sampling methods have been introduced to alleviate this timescale problem, including methods based on identifying a few crucial order parameters or collective variables and enhancing the sampling of these variables. Metadynamics is one such method that has proven successful in a great variety of fields. Here we review the conceptual and theor. foundations of metadynamics. As demonstrated, metadynamics is not just a practical tool but can also be considered an important development in the theory of statistical mechanics.
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      Dama, J. F.; Hocky, G. M.; Sun, R.; Voth, G. A. Exploring Valleys without Climbing Every Peak: More Efficient and Forgiving Metabasin Metadynamics via Robust On-the-Fly Bias Domain Restriction. J. Chem. Theory Comput. 2015, 11, 5638– 5650,  DOI: 10.1021/acs.jctc.5b00907
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      Exploring Valleys without Climbing Every Peak: More Efficient and Forgiving Metabasin Metadynamics via Robust On-the-Fly Bias Domain Restriction
      Dama, James F.; Hocky, Glen M.; Sun, Rui; Voth, Gregory A.
      Journal of Chemical Theory and Computation (2015), 11 (12), 5638-5650CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)
      Metadynamics is an enhanced sampling method designed to flatten free energy surfaces uniformly. However, the highest-energy regions are often irrelevant to study and dangerous to explore because systems often change irreversibly in unforeseen ways in response to driving forces in these regions, spoiling the sampling. Introducing an on-the-fly domain restriction allows metadynamics to flatten only up to a specified energy level and no further, improving efficiency and safety while decreasing the pressure on practitioners to design collective variables that are robust to otherwise irrelevant high energy driving. This paper describes a new method that achieves this using sequential on-the-fly estn. of energy wells and redefinition of the metadynamics hill shape, termed metabasin metadynamics. The energy level may be defined a priori or relative to unknown barrier energies estd. on-the-fly. Altering only the hill ensures that the method is compatible with many other advances in metadynamics methodol. The hill shape has a natural interpretation in terms of multiscale dynamics, and the computational overhead in simulation is minimal when studying systems of any reasonable size, for instance proteins or other macromols. Three example applications show that the formula is accurate and robust to complex dynamics, making metadynamics significantly more forgiving with respect to CV quality and thus more feasible to apply to the most challenging biomol. systems.
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      Pfaendtner, J.; Bonomi, M. Efficient Sampling of High-Dimensional Free-Energy Landscapes with Parallel Bias Metadynamics. J. Chem. Theory Comput. 2015, 11, 5062– 5067,  DOI: 10.1021/acs.jctc.5b00846
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      Efficient Sampling of High-Dimensional Free-Energy Landscapes with Parallel Bias Metadynamics
      Pfaendtner, Jim; Bonomi, Massimiliano
      Journal of Chemical Theory and Computation (2015), 11 (11), 5062-5067CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)
      Metadynamics accelerates sampling of mol. dynamics while reconstructing thermodn. properties of selected descriptors of the system. Its main practical difficulty originates from the compromise between keeping the no. of descriptors small for efficiently exploring their multidimensional free-energy landscape and biasing all of the slow motions of a process. Here we illustrate on a model system and on the tryptophan-cage miniprotein parallel bias metadynamics, a method that overcomes this issue by simultaneously applying multiple low-dimensional bias potentials.
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      Whitmer, J. K.; Chiu, C.-c.; Joshi, A. A.; de Pablo, J. J. Basis Function Sampling: A New Paradigm for Material Property Computation. Phys. Rev. Lett. 2014, 113, 190602  DOI: 10.1103/PhysRevLett.113.190602
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      Basis function sampling: a new paradigm for material property computation
      Whitmer, Jonathan K.; Chiu, Chi-cheng; Joshi, Abhijeet A.; de Pablo, Juan J.
      Physical Review Letters (2014), 113 (19), 190602/1-190602/5, 5 pp.CODEN: PRLTAO; ISSN:0031-9007. (American Physical Society)
      Wang-Landau sampling, and the assocd. class of flat histogram simulation methods have been remarkably helpful for calcns. of the free energy in a wide variety of phys. systems. Practically, convergence of these calcns. to a target free energy surface is hampered by reliance on parameters which are unknown a priori. Here, we derive and implement a method built upon orthogonal functions which is fast, parameter-free, and (importantly) geometrically robust. The method is shown to be highly effective in achieving convergence. An important feature of this method is its ability to attain arbitrary levels of description for the free energy. It is thus ideally suited to in silico measurement of elastic moduli and other material properties related to free energy perturbations. We demonstrate the utility of such applications by applying our method to calc. the Frank elastic consts. of the Lebwohl-Lasher model of liq. crystals.
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      Whitmer, J. K.; Fluitt, A. M.; Antony, L.; Qin, J.; McGovern, M.; de Pablo, J. J. Sculpting Bespoke Mountains: Determining Free Energies with Basis Expansions. J. Chem. Phys. 2015, 143, 044101  DOI: 10.1063/1.4927147
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      Sculpting bespoke mountains: Determining free energies with basis expansions
      Whitmer, Jonathan K.; Fluitt, Aaron M.; Antony, Lucas; Qin, Jian; McGovern, Michael; de Pablo, Juan J.
      Journal of Chemical Physics (2015), 143 (4), 044101/1-044101/6CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
      The intriguing behavior of a wide variety of phys. systems, ranging from amorphous solids or glasses to proteins, is a direct manifestation of underlying free energy landscapes riddled with local min. sepd. by large barriers. Exploring such landscapes has arguably become one of statistical physics's great challenges. A new method is proposed here for uniform sampling of rugged free energy surfaces. The method, which relies on special Green's functions to approx. the Dirac delta function, improves significantly on existing simulation techniques by providing a boundary-agnostic approach that is capable of mapping complex features in multidimensional free energy surfaces. The usefulness of the proposed approach is established in the context of a simple model glass former and model proteins, demonstrating improved convergence and accuracy over existing methods. (c) 2015 American Institute of Physics.
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      Sidky, H.; Whitmer, J. K. Learning Free Energy Landscapes Using Artificial Neural Networks. J. Chem. Phys. 2018, 148, 104111  DOI: 10.1063/1.5018708
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      Learning free energy landscapes using artificial neural networks
      Sidky, Hythem; Whitmer, Jonathan K.
      Journal of Chemical Physics (2018), 148 (10), 104111/1-104111/9CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
      Existing adaptive bias techniques, which seek to est. free energies and phys. properties from mol. simulations, are limited by their reliance on fixed kernels or basis sets which hinder their ability to efficiently conform to varied free energy landscapes. Further, user-specified parameters are in general non-intuitive yet significantly affect the convergence rate and accuracy of the free energy est. Here we propose a novel method, wherein artificial neural networks (ANNs) are used to develop an adaptive biasing potential which learns free energy landscapes. We demonstrate that this method is capable of rapidly adapting to complex free energy landscapes and is not prone to boundary or oscillation problems. The method is made robust to hyperparameters and over-fitting through Bayesian regularization which penalizes network wts. and auto-regulates the no. of effective parameters in the network. ANN sampling represents a promising innovative approach which can resolve complex free energy landscapes in less time than conventional approaches while requiring minimal user input. (c) 2018 American Institute of Physics.
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      Ribeiro, J. M. L.; Bravo, P.; Wang, Y.; Tiwary, P. Reweighted autoencoded variational Bayes for enhanced sampling (RAVE). J. Chem. Phys. 2018, 149, 072301  DOI: 10.1063/1.5025487
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      Reweighted autoencoded variational Bayes for enhanced sampling (RAVE)
      Ribeiro, Joao Marcelo Lamim; Bravo, Pablo; Wang, Yihang; Tiwary, Pratyush
      Journal of Chemical Physics (2018), 149 (7), 072301/1-072301/9CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
      Here we propose the reweighted autoencoded variational Bayes for enhanced sampling (RAVE) method, a new iterative scheme that uses the deep learning framework of variational autoencoders to enhance sampling in mol. simulations. RAVE involves iterations between mol. simulations and deep learning in order to produce an increasingly accurate probability distribution along a low-dimensional latent space that captures the key features of the mol. simulation trajectory. Using the Kullback-Leibler divergence between this latent space distribution and the distribution of various trial reaction coordinates sampled from the mol. simulation, RAVE dets. an optimum, yet nonetheless phys. interpretable, reaction coordinate and optimum probability distribution. Both then directly serve as the biasing protocol for a new biased simulation, which is once again fed into the deep learning module with appropriate wts. accounting for the bias, the procedure continuing until ests. of desirable thermodn. observables are converged. Unlike recent methods using deep learning for enhanced sampling purposes, RAVE stands out in that (a) it naturally produces a phys. interpretable reaction coordinate, (b) is independent of existing enhanced sampling protocols to enhance the fluctuations along the latent space identified via deep learning, and (c) it provides the ability to easily filter out spurious solns. learned by the deep learning procedure. The usefulness and reliability of RAVE is demonstrated by applying it to model potentials of increasing complexity, including computation of the binding free energy profile for a hydrophobic ligand-substrate system in explicit water with dissocn. time of more than 3 min, in computer time at least twenty times less than that needed for umbrella sampling or metadynamics. (c) 2018 American Institute of Physics.
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      Invernizzi, M.; Parrinello, M. Rethinking Metadynamics: From Bias Potentials to Probability Distributions. J. Phys. Chem. Lett. 2020, 11, 2731– 2736,  DOI: 10.1021/acs.jpclett.0c00497
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      Rethinking Metadynamics: From Bias Potentials to Probability Distributions
      Invernizzi, Michele; Parrinello, Michele
      Journal of Physical Chemistry Letters (2020), 11 (7), 2731-2736CODEN: JPCLCD; ISSN:1948-7185. (American Chemical Society)
      Metadynamics is an enhanced sampling method of great popularity, based on the on-the-fly construction of a bias potential that is a function of a selected no. of collective variables. We propose here a change in perspective that shifts the focus from the bias to the probability distribution reconstruction while retaining some of the key characteristics of metadynamics, such as flexible on-the-fly adjustments to the free energy est. The result is an enhanced sampling method that presents a drastic improvement in convergence speed, esp. when dealing with suboptimal and/or multidimensional sets of collective variables. The method is esp. robust and easy to use and in fact requires only a few simple parameters to be set, and it has a straightforward reweighting scheme to recover the statistics of the unbiased ensemble. Furthermore, it gives more control of the desired exploration of the phase space since the deposited bias is not allowed to grow indefinitely and it does not push the simulation to uninteresting high free energy regions. We demonstrate the performance of the method in a no. of representative examples.
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      Invernizzi, M.; Piaggi, P. M.; Parrinello, M. Unified Approach to Enhanced Sampling. Phys. Rev. X 2020, 10, 041034  DOI: 10.1103/PhysRevX.10.041034
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      Unified Approach to Enhanced Sampling
      Invernizzi, Michele; Piaggi, Pablo M.; Parrinello, Michele
      Physical Review X (2020), 10 (4), 041034CODEN: PRXHAE; ISSN:2160-3308. (American Physical Society)
      The sampling problem lies at the heart of atomistic simulations and over the years many different enhanced sampling methods have been suggested toward its soln. These methods are often grouped into two broad families. On the one hand, are methods such as umbrella sampling and metadynamics that build a bias potential based on few order parameters or collective variables. On the other hand, are tempering methods such as replica exchange that combine different thermodn. ensembles in one single expanded ensemble. We instead adopt a unifying perspective, focusing on the target probability distribution sampled by the different methods. This allows us to introduce a new class of collective-variables-based bias potentials that can be used to sample any of the expanded ensembles normally sampled via replica exchange. We also provide a practical implementation by properly adapting the iterative scheme of the recently developed on-the-fly probability enhanced sampling method [M. Invernizzi and M. Parrinello, J. Lett.11, 2731 (2020)JPCLCD1948-718510.1021/acs.jpclett.0c00497], which was originally introduced for metadynamicslike sampling. The resulting method is very general and can be used to achieve different types of enhanced sampling. It is also reliable and simple to use, since it presents only few and robust external parameters and has a straightforward reweighting scheme. Furthermore, it can be used with any no. of parallel replicas. We show the versatility of our approach with applications to multicanonical and multithermal-multibaric simulations, thermodn. integration, umbrella sampling, and combinations thereof.
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      Giberti, F.; Tribello, G. A.; Ceriotti, M. Global Free-Energy Landscapes as a Smoothly Joined Collection of Local Maps. J. Chem. Theory Comput. 2021, 17, 3292– 3308,  DOI: 10.1021/acs.jctc.0c01177
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      Global Free-Energy Landscapes as a Smoothly Joined Collection of Local Maps
      Giberti, F.; Tribello, G. A.; Ceriotti, M.
      Journal of Chemical Theory and Computation (2021), 17 (6), 3292-3308CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)
      Enhanced sampling techniques have become an essential tool in computational chem. and physics, where they are applied to sample activated processes that occur on a time scale that is inaccessible to conventional simulations. Despite their popularity, it is well known that they have constraints that hinder their applications to complex problems. The core issue lies in the need to describe the system using a small no. of collective variables (CVs). Any slow degree of freedom that is not properly described by the chosen CVs will hinder sampling efficiency. However, the exploration of configuration space is also hampered by including variables that are not relevant to describe the activated process under study. This paper presents the Adaptive Topog. of Landscape for Accelerated Sampling (ATLAS), a new biasing method capable of working with many CVs. The root idea of ATLAS is to apply a divide-and-conquer strategy, where the high-dimensional CVs space is divided into basins, each of which is described by an automatically detd., low-dimensional set of variables. A well-tempered metadynamics-like bias is constructed as a function of these local variables. Indicator functions assocd. with the basins switch on and off the local biases so that the sampling is performed on a collection of low-dimensional CV spaces that are smoothly combined to generate an effectively high-dimensional bias. The unbiased Boltzmann distribution is recovered through reweighing, making the evaluation of conformational and thermodn. properties straightforward. The decompn. of the free-energy landscape in local basins can be updated iteratively as the simulation discovers new (meta)stable states.
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      Bal, K. M. Reweighted Jarzynski Sampling: Acceleration of Rare Events and Free Energy Calculation with a Bias Potential Learned from Nonequilibrium Work. J. Chem. Theory Comput. 2021, 17, 6766– 6774,  DOI: 10.1021/acs.jctc.1c00574
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      Reweighted Jarzynski Sampling: Acceleration of Rare Events and Free Energy Calculation with a Bias Potential Learned from Nonequilibrium Work
      Bal, Kristof M.
      Journal of Chemical Theory and Computation (2021), 17 (11), 6766-6774CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)
      We introduce a simple enhanced sampling approach for the calcn. of free energy differences and barriers along a one-dimensional reaction coordinate. First, a small no. of short nonequil. simulations are carried out along the reaction coordinate, and the Jarzynski equality is used to learn an approx. free energy surface from the nonequil. work distribution. This free energy est. is represented in a compact form as an artificial neural network and used as an external bias potential to accelerate rare events in a subsequent mol. dynamics simulation. The final free energy est. is then obtained by reweighting the equil. probability distribution of the reaction coordinate sampled under the influence of the external bias. We apply our reweighted Jarzynski sampling recipe to four processes of varying scales and complexities-spanning chem. reaction in the gas phase, pair assocn. in soln., and droplet nucleation in supersatd. vapor. In all cases, we find reweighted Jarzynski sampling to be a very efficient strategy, resulting in rapid convergence of the free energy to high precision.
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      Valsson, O.; Parrinello, M. Variational Approach to Enhanced Sampling and Free Energy Calculations. Phys. Rev. Lett. 2014, 113, 090601  DOI: 10.1103/PhysRevLett.113.090601
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      Variational approach to enhanced sampling and free energy calculations
      Valsson, Omar; Parrinello, Michele
      Physical Review Letters (2014), 113 (9), 090601CODEN: PRLTAO; ISSN:0031-9007. (American Physical Society)
      The ability of widely used sampling methods, such as mol. dynamics or Monte Carlo simulations, to explore complex free energy landscapes is severely hampered by the presence of kinetic bottlenecks. A large no. of solns. have been proposed to alleviate this problem. Many are based on the introduction of a bias potential which is a function of a small no. of collective variables. However constructing such a bias is not simple. Here we introduce a functional of the bias potential and an assocd. variational principle. The bias that minimizes the functional relates in a simple way to the free energy surface. This variational principle can be turned into a practical, efficient, and flexible sampling method. A no. of numerical examples are presented which include the detn. of a three-dimensional free energy surface. We argue that, beside being numerically advantageous, our variational approach provides a convenient and novel standpoint for looking at the sampling problem.
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      Valsson, O.; Parrinello, M. Variationally Enhanced Sampling. In Handbook of Materials Modeling: Methods: Theory and Modeling; Andreoni, W., Yip, S., Eds.; Springer: Cham, Switzerland, 2020, 621– 634.
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      Bonati, L.; Zhang, Y.-Y.; Parrinello, M. Neural Networks-Based Variationally Enhanced Sampling. Proc. Natl. Acad. Sci. U. S. A. 2019, 116, 17641– 17647,  DOI: 10.1073/pnas.1907975116
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      Neural networks-based variationally enhanced sampling
      Bonati, Luigi; Zhang, Yue-Yu; Parrinello, Michele
      Proceedings of the National Academy of Sciences of the United States of America (2019), 116 (36), 17641-17647CODEN: PNASA6; ISSN:0027-8424. (National Academy of Sciences)
      Sampling complex free-energy surfaces is one of the main challenges of modern atomistic simulation methods. The presence of kinetic bottlenecks in such surfaces often renders a direct approach useless. A popular strategy is to identify a small no. of key collective variables and to introduce a bias potential that is able to favor their fluctuations in order to accelerate sampling. Here, we propose to use machine-learning techniques in conjunction with the recent variationally enhanced sampling method [O. Valsson, M. Parrinello, Phys. Rev. Lett. 113, 090601 (2014)] in order to det. such potential. This is achieved by expressing the bias as a neural network. The parameters are detd. in a variational learning scheme aimed at minimizing an appropriate functional. This required the development of a more efficient minimization technique. The expressivity of neural networks allows representing rapidly varying free-energy surfaces, removes boundary effects artifacts, and allows several collective variables to be handled.
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      Piaggi, P. M.; Valsson, O.; Parrinello, M. A Variational Approach to Nucleation Simulation. Faraday Discuss. 2016, 195, 557– 568,  DOI: 10.1039/C6FD00127K
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      A variational approach to nucleation simulation
      Piaggi, Pablo M.; Valsson, Omar; Parrinello, Michele
      Faraday Discussions (2016), 195 (Reaction Rate Theory), 557-568CODEN: FDISE6; ISSN:1359-6640. (Royal Society of Chemistry)
      We study by computer simulation the nucleation of a supersatd. Lennard-Jones vapor into the liq. phase. The large free energy barriers to transition make the time scale of this process impossible to study by ordinary mol. dynamics simulations. Therefore we use a recently developed enhanced sampling method [Valsson and Parrinello, Phys. Rev. Lett.113, 090601 (2014)] based on the variational detn. of a bias potential. We differ from previous applications of this method in that the bias is constructed on the basis of the phys. model provided by the classical theory of nucleation. We examine the tech. problems assocd. with this approach. Our results are very satisfactory and will pave the way for calcg. the nucleation rates in many systems.
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      McCarty, J.; Valsson, O.; Parrinello, M. Bespoke Bias for Obtaining Free Energy Differences within Variationally Enhanced Sampling. J. Chem. Theory Comput. 2016, 12, 2162– 2169,  DOI: 10.1021/acs.jctc.6b00125
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      Bespoke Bias for Obtaining Free Energy Differences within Variationally Enhanced Sampling
      McCarty, James; Valsson, Omar; Parrinello, Michele
      Journal of Chemical Theory and Computation (2016), 12 (5), 2162-2169CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)
      Obtaining efficient sampling of multiple metastable states through mol. dynamics and hence detg. free energy differences is central for understanding many important phenomena. Here we present a new biasing strategy, which employs the recent variationally enhanced sampling approach Valsson and Parrinello Phys. Rev. Lett.2014, 113, 090601. The bias is constructed from an intuitive model of the local free energy surface describing fluctuations around metastable min. and depends on only a few parameters which are detd. variationally such that efficient sampling between states is obtained. The bias constructed in this manner largely reduces the need of finding a set of collective variables that completely spans the conformational space of interest, as they only need to be a locally valid descriptor of the system about its local min. We introduce the method and demonstrate its power on two representative examples.
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      Invernizzi, M.; Valsson, O.; Parrinello, M. Coarse Graining from Variationally Enhanced Sampling Applied to the Ginzburg–Landau Model. Proc. Natl. Acad. Sci. U. S. A. 2017, 114, 3370– 3374,  DOI: 10.1073/pnas.1618455114
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      Coarse graining from variationally enhanced sampling applied to the Ginzburg-Landau model
      Invernizzi, Michele; Valsson, Omar; Parrinello, Michele
      Proceedings of the National Academy of Sciences of the United States of America (2017), 114 (13), 3370-3374CODEN: PNASA6; ISSN:0027-8424. (National Academy of Sciences)
      A powerful way to deal with a complex system is to build a coarse-grained model capable of catching its main phys. features, while being computationally affordable. Inevitably, such coarse-grained models introduce a set of phenomenol. parameters, which are often not easily deducible from the underlying atomistic system. We present a unique approach to the calcn. of these parameters, based on the recently introduced variationally enhanced sampling method. It allows us to obtain the parameters from atomistic simulations, providing thus a direct connection between the microscopic and the mesoscopic scale. The coarse-grained model we consider is that of Ginzburg-Landau, valid around a second-order crit. point. In particular, we use it to describe a Lennard-Jones fluid in the region close to the liq.-vapor crit. point. The procedure is general and can be adapted to other coarse-grained models.
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      Invernizzi, M.; Parrinello, M. Making the Best of a Bad Situation: A Multiscale Approach to Free Energy Calculation. J. Chem. Theory Comput. 2019, 15, 2187– 2194,  DOI: 10.1021/acs.jctc.9b00032
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      Making the Best of a Bad Situation: A Multiscale Approach to Free Energy Calculation
      Invernizzi, Michele; Parrinello, Michele
      Journal of Chemical Theory and Computation (2019), 15 (4), 2187-2194CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)
      Many enhanced sampling techniques rely on the identification of a no. of collective variables that describe all the slow modes of the system. By constructing a bias potential in this reduced space, one is then able to sample efficiently and reconstruct the free energy landscape. In methods such as metadynamics, the quality of these collective variables plays a key role in convergence efficiency. Unfortunately in many systems of interest it is not possible to identify an optimal collective variable, and one must deal with the nonideal situation of a system in which some slow modes are not accelerated. The authors propose a two-step approach in which, by taking into account the residual multiscale nature of the problem, one is able to significantly speed up convergence. To do so, the authors combine an exploratory metadynamics run with an optimization of the free energy difference between metastable states, based on the recently proposed variationally enhanced sampling method. This new method is esp. suited for complex systems because of its simplicity and clear underlying phys. picture.
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      McCarty, J.; Valsson, O.; Tiwary, P.; Parrinello, M. Variationally Optimized Free-Energy Flooding for Rate Calculation. Phys. Rev. Lett. 2015, 115, 070601  DOI: 10.1103/PhysRevLett.115.070601
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      Variationally optimized free-energy flooding for rate calculation
      McCarty, James; Valsson, Omar; Tiwary, Pratyush; Parrinello, Michele
      Physical Review Letters (2015), 115 (7), 070601/1-070601/5CODEN: PRLTAO; ISSN:0031-9007. (American Physical Society)
      We propose a new method to obtain kinetic properties of infrequent events from mol. dynamics simulation. The procedure employs a recently introduced variational approach to construct a bias potential as a function of several collective variables that is designed to flood the assocd. free energy surface up to a predefined level. The resulting bias potential effectively accelerates transitions between metastable free energy min. while ensuring bias-free transition states, thus allowing accurate kinetic rates to be obtained. We test the method on a few illustrative systems for which we obtain an order of magnitude improvement in efficiency relative to previous approaches and several orders of magnitude relative to unbiased mol. dynamics. We expect an even larger improvement in more complex systems. This and the ability of the variational approach to deal efficiently with a large no. of collective variables will greatly enhance the scope of these calcns. This work is a vindication of the potential that the variational principle has if applied in innovative ways.
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    50. 50
      Demuynck, R.; Rogge, S. M. J.; Vanduyfhuys, L.; Wieme, J.; Waroquier, M.; Van Speybroeck, V. Efficient Construction of Free Energy Profiles of Breathing Metal─Organic Frameworks Using Advanced Molecular Dynamics Simulations. J. Chem. Theory Comput. 2017, 13, 5861– 5873,  DOI: 10.1021/acs.jctc.7b01014
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      50
      Efficient Construction of Free Energy Profiles of Breathing Metal-Organic Frameworks Using Advanced Molecular Dynamics Simulations
      Demuynck, Ruben; Rogge, Sven M. J.; Vanduyfhuys, Louis; Wieme, Jelle; Waroquier, Michel; Van Speybroeck, Veronique
      Journal of Chemical Theory and Computation (2017), 13 (12), 5861-5873CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)
      In order to reliably predict and understand the breathing behavior of highly flexible metal-org. frameworks from thermodn. considerations, an accurate estn. of the free energy difference between their different metastable states is a prerequisite. Herein, a variety of free energy estn. methods are thoroughly tested for their ability to construct the free energy profile as a function of the unit cell vol. of MIL-53(Al). The methods comprise free energy perturbation, thermodn. integration, umbrella sampling, metadynamics, and variationally enhanced sampling. A series of mol. dynamics simulations have been performed in the frame of each of the five methods to describe structural transformations in flexible materials with the vol. as the collective variable, which offers a unique opportunity to assess their computational efficiency. Subsequently, the most efficient method, umbrella sampling, is used to construct an accurate free energy profile at different temps. for MIL-53(Al) from first principles at the PBE + D3(BJ) level of theory. This study yields insight into the importance of the different aspects such as entropy contributions and anharmonic contributions on the resulting free energy profile. As such, this thorough study provides unparalleled insight in the thermodn. of the large structural deformations of flexible materials.
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    51. 51
      Demuynck, R.; Wieme, J.; Rogge, S. M. J.; Dedecker, K. D.; Vanduyfhuys, L.; Waroquier, M.; Van Speybroeck, V. Protocol for Identifying Accurate Collective Variables in Enhanced Molecular Dynamics Simulations for the Description of Structural Transformations in Flexible Metal–Organic Frameworks. J. Chem. Theory Comput. 2018, 14, 5511– 5526,  DOI: 10.1021/acs.jctc.8b00725
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      Protocol for Identifying Accurate Collective Variables in Enhanced Molecular Dynamics Simulations for the Description of Structural Transformations in Flexible Metal-Organic Frameworks
      Demuynck, Ruben; Wieme, Jelle; Rogge, Sven M. J.; Dedecker, Karen D.; Vanduyfhuys, Louis; Waroquier, Michel; Van Speybroeck, Veronique
      Journal of Chemical Theory and Computation (2018), 14 (11), 5511-5526CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)
      Various kinds of flexibility have been obsd. in metal-org. frameworks, which may originate from the topol. of the material or the presence of flexible ligands. The construction of free energy profiles describing the full dynamical behavior along the phase transition path is challenging since it is not trivial to identify collective variables able to identify all metastable states along the reaction path. In this work, a systematic three-step protocol to uniquely identify the dominant order parameters for structural transformations in flexible metal-org. frameworks and subsequently construct accurate free energy profiles is presented. Methodol., this protocol is rooted in the time-structure based independent component anal. (tICA), a well-established statistical modeling technique embedded in the Markov state model methodol. and often employed to study protein folding, that allows for the identification of the slowest order parameters characterizing the structural transformation. To ensure an unbiased and systematic identification of these order parameters, the tICA decompn. is performed based on information from a prior replica exchange (RE) simulation, as this technique enhances the sampling along all degrees of freedom of the system simultaneously. From this simulation, the tICA procedure exts. the order parameters - often structural parameters - that characterize the slowest transformations in the material. Subsequently, these order parameters are adopted in traditional enhanced sampling methods such as umbrella sampling, thermodn. integration, and variationally enhanced sampling to construct accurate free energy profiles capturing the flexibility in these nanoporous materials. In this work, the applicability of this tICA-RE protocol is demonstrated by detg. the slowest order parameters in both MIL-53(Al) and CAU-13, which exhibit a strongly different type of flexibility. The obtained free energy profiles as a function of this extd. order parameter are furthermore compared to the profiles obtained when adopting less-suited collective variables, indicating the importance of systematically selecting the relevant order parameters to construct accurate free energy profiles for flexible metal-org. frameworks, which is in correspondence with exptl. findings. The method succeeds in mapping the full free energy surface in terms of appropriate collective variables for MOFs exhibiting linker flexibility. For CAU-13, we show the decreased stability of the closed pore phase by systematically adding adsorbed xylene mols. in the framework.
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      Daubechies, I. Orthonormal Bases of Compactly Supported Wavelets. Commun. Pure Appl. Math. 1988, 41, 909– 996,  DOI: 10.1002/cpa.3160410705
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      Mohr, S.; Ratcliff, L. E.; Boulanger, P.; Genovese, L.; Caliste, D.; Deutsch, T.; Goedecker, S. Daubechies Wavelets for Linear Scaling Density Functional Theory. J. Chem. Phys. 2014, 140, 204110  DOI: 10.1063/1.4871876
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      Daubechies wavelets for linear scaling density functional theory
      Mohr, Stephan; Ratcliff, Laura E.; Boulanger, Paul; Genovese, Luigi; Caliste, Damien; Deutsch, Thierry; Goedecker, Stefan
      Journal of Chemical Physics (2014), 140 (20), 204110/1-204110/16CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
      We demonstrate that Daubechies wavelets can be used to construct a minimal set of optimized localized adaptively contracted basis functions in which the Kohn-Sham orbitals can be represented with an arbitrarily high, controllable precision. Ground state energies and the forces acting on the ions can be calcd. in this basis with the same accuracy as if they were calcd. directly in a Daubechies wavelets basis, provided that the amplitude of these adaptively contracted basis functions is sufficiently small on the surface of the localization region, which is guaranteed by the optimization procedure described in this work. This approach reduces the computational costs of d. functional theory calcns., and can be combined with sparse matrix algebra to obtain linear scaling with respect to the no. of electrons in the system. Calcns. on systems of 10 000 atoms or more thus become feasible in a systematic basis set with moderate computational resources. Further computational savings can be achieved by exploiting the similarity of the adaptively contracted basis functions for closely related environments, e.g., in geometry optimizations or combined calcns. of neutral and charged systems. (c) 2014 American Institute of Physics.
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      Ratcliff, L. E.; Dawson, W.; Fisicaro, G.; Caliste, D.; Mohr, S.; Degomme, A.; Videau, B.; Cristiglio, V.; Stella, M.; D’Alessandro, M.; Goedecker, S.; Nakajima, T.; Deutsch, T.; Genovese, L. Flexibilities of Wavelets as a Computational Basis Set for Large-Scale Electronic Structure Calculations. J. Chem. Phys. 2020, 152, 194110  DOI: 10.1063/5.0004792
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      Flexibilities of wavelets as a computational basis set for large-scale electronic structure calculations
      Ratcliff, Laura E.; Dawson, William; Fisicaro, Giuseppe; Caliste, Damien; Mohr, Stephan; Degomme, Augustin; Videau, Brice; Cristiglio, Viviana; Stella, Martina; D'Alessandro, Marco; Goedecker, Stefan; Nakajima, Takahito; Deutsch, Thierry; Genovese, Luigi
      Journal of Chemical Physics (2020), 152 (19), 194110CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
      The BigDFT project was started in 2005 with the aim of testing the advantages of using a Daubechies wavelet basis set for Kohn-Sham (KS) d. functional theory (DFT) with pseudopotentials. This project led to the creation of the BigDFT code, which employs a computational approach with optimal features of flexibility, performance, and precision of the results. In particular, the employed formalism has enabled the implementation of an algorithm able to tackle DFT calcns. of large systems, up to many thousands of atoms, with a computational effort that scales linearly with the no. of atoms. In this work, we recall some of the features that have been made possible by the peculiar properties of Daubechies wavelets. In particular, we focus our attention on the usage of DFT for large-scale systems. We show how the localized description of the KS problem, emerging from the features of the basis set, is helpful in providing a simplified description of large-scale electronic structure calcns. We provide some examples on how such a simplified description can be employed, and we consider, among the case-studies, the SARS-CoV-2 main protease. (c) 2020 American Institute of Physics.
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      Maiolo, M.; Vancheri, A.; Krause, R.; Danani, A. Wavelets as Basis Functions to Represent the Coarse-Graining Potential in Multiscale Coarse Graining Approach. J. Comput. Phys. 2015, 300, 592– 604,  DOI: 10.1016/j.jcp.2015.07.039
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      Tribello, G. A.; Bonomi, M.; Branduardi, D.; Camilloni, C.; Bussi, G. PLUMED 2: New Feathers for an Old Bird. Comput. Phys. Commun. 2014, 185, 604– 613,  DOI: 10.1016/j.cpc.2013.09.018
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      PLUMED 2: New feathers for an old bird
      Tribello, Gareth A.; Bonomi, Massimiliano; Branduardi, Davide; Camilloni, Carlo; Bussi, Giovanni
      Computer Physics Communications (2014), 185 (2), 604-613CODEN: CPHCBZ; ISSN:0010-4655. (Elsevier B.V.)
      Enhancing sampling and analyzing simulations are central issues in mol. simulation. Recently, we introduced PLUMED, an open-source plug-in that provides some of the most popular mol. dynamics (MD) codes with implementations of a variety of different enhanced sampling algorithms and collective variables (CVs). The rapid changes in this field, in particular new directions in enhanced sampling and dimensionality redn. together with new hardware, require a code that is more flexible and more efficient. We therefore present PLUMED 2 here-a complete rewrite of the code in an object-oriented programming language (C++). This new version introduces greater flexibility and greater modularity, which both extends its core capabilities and makes it far easier to add new methods and CVs. It also has a simpler interface with the MD engines and provides a single software library contg. both tools and core facilities. Ultimately, the new code better serves the ever-growing community of users and contributors in coping with the new challenges arising in the field.
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      Kellermeier, M.; Raiteri, P.; Berg, J. K.; Kempter, A.; Gale, J. D.; Gebauer, D. Entropy Drives Calcium Carbonate Ion Association. ChemPhysChem 2016, 17, 3535– 3541,  DOI: 10.1002/cphc.201600653
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      Entropy drives calcium carbonate ion association
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      ChemPhysChem (2016), 17 (21), 3535-3541CODEN: CPCHFT; ISSN:1439-4235. (Wiley-VCH Verlag GmbH & Co. KGaA)
      The understanding of the mol. mechanisms underlying the early stages of crystn. is still incomplete. In the case of calcium carbonate, exptl. and computational evidence suggests that phase sepn. relies on so-called pre-nucleation clusters (PNCs). A thorough thermodn. anal. of the enthalpic and entropic contributions to the overall free energy of PNC formation derived from three independent methods demonstrates that solute clustering is driven by entropy. This can be quant. rationalized by the release of water mols. from ion hydration layers, explaining why ion assocn. is not limited to simple ion pairing. The key role of water release in this process suggests that PNC formation should be a common phenomenon in aq. solns.
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      Well-Tempered Variational Approach to Enhanced Sampling
      Valsson, Omar; Parrinello, Michele
      Journal of Chemical Theory and Computation (2015), 11 (5), 1996-2002CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)
      We propose a simple yet effective iterative scheme that allows us to employ the well-tempered distribution as a target distribution for the collective variables in our recently introduced variational approach to enhanced sampling and free energy calcns. The performance of the scheme is evaluated for the three-dimensional free energy surface of alanine tetrapeptide where the convergence can be rather poor when employing the uniform target distribution. Using the well-tempered target distribution on the other hand results in a significant improvement in convergence. The results obsd. in this paper indicate that the well-tempered distribution is in most cases the preferred and recommended choice for the target distribution in the variational approach.
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      A Time-Independent Free Energy Estimator for Metadynamics
      Tiwary, Pratyush; Parrinello, Michele
      Journal of Physical Chemistry B (2015), 119 (3), 736-742CODEN: JPCBFK; ISSN:1520-5207. (American Chemical Society)
      Metadynamics is a powerful and well-established enhanced sampling method for exploring and quantifying free energy surfaces of complex systems as a function of appropriately chosen variables. In the limit of long simulation time, metadynamics converges to the exact free energy surface plus a time-dependent const. The authors analyze in detail this time-dependent const. The authors show an easy way to calc. it, and by explicitly calcg. the time dependence of this const., they are able to derive a time-independent and locally convergent free energy estimator for metadynamics. The authors also derive an alternate procedure for obtaining the full unbiased distributions of generic operators from biased metadynamics simulations and explicitly test its usefulness.
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      Baftizadeh, F.; Cossio, P.; Pietrucci, F.; Laio, A. Protein Folding and Ligand-Enzyme Binding from Bias-Exchange Metadynamics Simulations. Curr. Phys. Chem. 2012, 2, 79– 91,  DOI: 10.2174/1877946811202010079
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      Protein folding and ligand-enzyme binding from bias-exchange metadynamics simulations
      Baftizadeh, Fahimeh; Cossio, Pilar; Pietrucci, Fabio; Laio, Alessandro
      Current Physical Chemistry (2012), 2 (1), 79-91CODEN: CPCUBU; ISSN:1877-9468. (Bentham Science Publishers Ltd.)
      A review. Bias-Exchange Metadynamics is a powerful technique that can be used for reconstructing the free energy and for enhancing the conformational search in complex biol. systems. In this method, a large set of collective variables (CVs) is chosen and several metadynamics simulations are performed on different replicas of the system, each replica biasing a different CV. Exchanges between the bias potentials are periodically attempted according to a replica exchange scheme, and this process is repeated until convergence of the free energy profiles is obtained. Bias-Exchange Metadynamics has been used to understand several different biol. phenomena. In particular, due to the efficaciously multidimensional nature of the bias, it is useful to study the folding process of small-to-medium size proteins, and ligand-enzyme binding. This review intends to provide a comprehensive description of the algorithm and the approach used to analyze its output. We focus on the practical aspects that need to be addressed when one attempts to apply the method to study protein systems: choice of the appropriate set of parameters and CVs, proper treatment of boundary conditions, convergence criteria, and derivation of a thermodn. and kinetic model of the system from the simulation results.
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      Crespo, Y.; Marinelli, F.; Pietrucci, F.; Laio, A. Metadynamics Convergence Law in a Multidimensional System. Phys. Rev. E 2010, 81, 055701(R)  DOI: 10.1103/PhysRevE.81.055701
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      McGovern, Michael; de Pablo, Juan
      Journal of Chemical Physics (2013), 139 (8), 084102/1-084102/5CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
      Metadynamics is an efficient method for simulation of the free energy of many-particle systems. Over the last several years it has been applied to study a wide variety of systems, ranging from simple fluids to biol. macromols. The method relies on uniform sampling along specified collective variables or order parameters. Such order parameters, however, are often bounded, and metadynamics algorithms as originally developed suffer from systematic errors at the corresponding boundaries. While several approaches have been proposed in the past to correct these errors for unidimensional systems, no method exists to fully correct these errors in multi-dimensional systems at points where multiple boundaries meet. Here we present a correction scheme that circumvents this limitation. (c) 2013 American Institute of Physics.
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      The PLUMED consortium Promoting Transparency and Reproducibility in Enhanced Molecular Simulations. Nat. Methods 2019, 16, 670– 673,  DOI: 10.1038/s41592-019-0506-8
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      Physical Review E: Statistical, Nonlinear, and Soft Matter Physics (2007), 75 (5-2), 056707/1-056707/7CODEN: PRESCM; ISSN:1539-3755. (American Physical Society)
      We show how to derive a simple integrator for the Langevin equation and illustrate how it is possible to check the accuracy of the obtained distribution on the fly, using the concept of effective energy introduced in a recent paper [J. Chem. Phys. 126, 014101 (2007)]. Our integrator leads to correct sampling also in the difficult high-friction limit. We also show how these ideas can be applied in practical simulations, using a Lennard-Jones crystal as a paradigmatic case.
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      Journal of the American Chemical Society (1975), 97 (8), 2020-4CODEN: JACSAT; ISSN:0002-7863.
      The question of whether a rotation-inversion process that results in the interconversion of two enantiomeric or identical species can be described by a unique asymmetric reaction coordinate was subjected to a quantum mechanical analysis, employing the symmetry properties of the reaction surface, and the results applied to two cases: case I, in which one transition state separates reagents and products; and case II, in which a stable intermediate appears on the reaction coordinate. No unique path exists for case I, i.e., all asymmetric reaction coordinates are indistinguishable. This conclusion holds under equilibrium conditions, nonequilibrium conditions, and photochem. excitation. For case II, distinguishable asymmetric paths can exist under nonequilibrium conditions. These are related to each other in a diastereomeric sense, in contrast to case I, in which various reaction paths differ in an enantiomeric sense.
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      A growing string method for the reaction pathway defined by a Newton trajectory
      Quapp, Wolfgang
      Journal of Chemical Physics (2005), 122 (17), 174106/1-174106/11CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
      The reaction path is an important concept of theor. chem. We use a projection operator for the following of the Newton trajectory (NT) along the reaction valley of the potential energy surface. We describe the numerical scheme for the string method, adapting the proposal of a growing string (GS) by [Peters et al.,J. Chem. Phys. 120, 7877 (2004)]. The combination of the Newton projector and the growing string idea is an improvement of both methods, and a great saving of the no. of iterations needed to find the pathway over the saddle point. This combination GS-NT is at the best of our knowledge new. We employ two different corrector methods: first, the use of projected gradient steps, and second a conjugated gradient method, the CG+ method of Liu, Nocedal, and Waltz, generalized by projectors. The executed examples are Lennard-Jones clusters, LJ7 and LJ22, and an N-methyl-alanyl-acetamide (alanine dipeptide) rearrangement between the min. C7ax and C5. For the latter, the growing string calcn. is interfaced with the GASSIAN03 quantum chem. software package.
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      Three parallel algorithms for classical mol. dynamics are presented. The first assigns each processor a fixed subset of atoms; the second assigns each a fixed subset of inter-at. forces to compute; the third assigns each a fixed spatial region. The algorithms are suitable for mol. dynamics models which can be difficult to parallelize efficiently - those with short-range forces where the neighbors of each atom change rapidly. They can be implemented on any distributed-memory parallel machine which allows for message-passing of data between independently executing processors. The algorithms are tested on a std. Lennard-Jones benchmark problem for system sizes ranging from 500 to 100,000,000 atoms on several parallel supercomputers - the nCUBE 2, Intel iPSC/860 and Paragon, and Cray T3D. Comparing the results to the fastest reported vectorized Cray Y-MP and C90 algorithm shows that the current generation of parallel machines is competitive with conventional vector supercomputers even for small problems. For large problems, the spatial algorithm achieves parallel efficiencies of 90% and a 1840-node Intel Paragon performs up to 165 faster than a single Cray C90 processor. Trade-offs between the three algorithms and guidelines for adapting them to more complex mol. dynamics simulations are also discussed.
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      Demichelis, R.; Raiteri, P.; Gale, J. D.; Quigley, D.; Gebauer, D. Stable Prenucleation Mineral Clusters Are Liquid-like Ionic Polymers. Nat. Commun. 2011, 2, 590,  DOI: 10.1038/ncomms1604
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      Stable prenucleation mineral clusters are liquid-like ionic polymers
      Demichelis Raffaella; Raiteri Paolo; Gale Julian D; Quigley David; Gebauer Denis
      Nature communications (2011), 2 (), 590 ISSN:.
      Calcium carbonate is an abundant substance that can be created in several mineral forms by the reaction of dissolved carbon dioxide in water with calcium ions. Through biomineralization, organisms can harness and control this process to form various functional materials that can act as anything from shells through to lenses. The early stages of calcium carbonate formation have recently attracted attention as stable prenucleation clusters have been observed, contrary to classical models. Here we show, using computer simulations combined with the analysis of experimental data, that these mineral clusters are made of an ionic polymer, composed of alternating calcium and carbonate ions, with a dynamic topology consisting of chains, branches and rings. The existence of a disordered, flexible and strongly hydrated precursor provides a basis for explaining the formation of other liquid-like amorphous states of calcium carbonate, in addition to the non-classical behaviour during growth of amorphous calcium carbonate.
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      Raiteri, P.; Demichelis, R.; Gale, J. D. Thermodynamically Consistent Force Field for Molecular Dynamics Simulations of Alkaline-Earth Carbonates and Their Aqueous Speciation. J. Phys. Chem. C 2015, 119, 24447– 24458,  DOI: 10.1021/acs.jpcc.5b07532
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      Thermodynamically Consistent Force Field for Molecular Dynamics Simulations of Alkaline-Earth Carbonates and Their Aqueous Speciation
      Raiteri, Paolo; Demichelis, Raffaella; Gale, Julian D.
      Journal of Physical Chemistry C (2015), 119 (43), 24447-24458CODEN: JPCCCK; ISSN:1932-7447. (American Chemical Society)
      In recent years atomistic simulations have become increasingly important in providing mol. insight to complement expts. Even for the seemingly simple case of ion-pair formation a detailed atomistic picture of the structure and relative stability of the contact, solvent-shared and solvent-sepd. ion pairs can only be readily achieved by computer simulation. Here a new force field parametrization for the alk.-earth carbonate interactions in water has been developed by fitting against exptl. thermodn. and structural data. We demonstrate that the present force field can accurately reproduce the dynamics and thermodn. of the ions in soln., which is the key to producing quant. accurate data that can be compared against expt.
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      Wu, Y.; Tepper, H. L.; Voth, G. A. Flexible Simple Point-Charge Water Model with Improved Liquid-State Properties. J. Chem. Phys. 2006, 124, 024503  DOI: 10.1063/1.2136877
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      Flexible simple point-charge water model with improved liquid-state properties
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      In order to introduce flexibility into the simple point-charge (SPC) water model, the impact of the intramol. degrees of freedom on liq. properties was systematically studied in this work as a function of many possible parameter sets. It was found that the diffusion const. is extremely sensitive to the equil. bond length and that this effect is mainly due to the strength of intermol. hydrogen bonds. The static dielec. const. was found to be very sensitive to the equil. bond angle via the distribution of intermol. angles in the liq.: A larger bond angle will increase the angle formed by two mol. dipoles, which is particularly significant for the first solvation shell. This result is in agreement with the work of Hochtl et al. [J. Chem. Phys. 109, 4927 (1998)]. A new flexible simple point-charge water model was derived by optimizing bulk diffusion and dielec. consts. to the exptl. values via the equil. bond length and angle. Due to the large sensitivities, the parametrization only slightly perturbs the mol. geometry of the base SPC model. Extensive comparisons of thermodn., structural, and kinetic properties indicate that the new model is much improved over the std. SPC model and its overall performance is comparable to or even better than the extended SPC model.
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      Tuckerman, M. E.; Alejandre, J.; López-Rendón, R.; Jochim, A. L.; Martyna, G. J. A Liouville-operator Derived Measure-Preserving Integrator for Molecular Dynamics Simulations in the Isothermal–Isobaric Ensemble. J. Phys. A: Math. Gen. 2006, 39, 5629– 5651,  DOI: 10.1088/0305-4470/39/19/S18
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      Efficient Reconstruction of Complex Free Energy Landscapes by Multiple Walkers Metadynamics
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      Journal of Physical Chemistry B (2006), 110 (8), 3533-3539CODEN: JPCBFK; ISSN:1520-6106. (American Chemical Society)
      Recently, we have introduced a new method, metadynamics, which is able to sample rarely occurring transitions and to reconstruct the free energy as a function of several variables with a controlled accuracy. This method has been successfully applied in many different fields, ranging from chem. to biophysics and ligand docking and from material science to crystal structure prediction. We present an important development that speeds up metadynamics calcns. by orders of magnitude and renders the algorithm much more robust. We use multiple interacting simulations, walkers, for exploring and reconstructing the same free energy surface. Each walker contributes to the history-dependent potential that, in metadynamics, is an est. of the free energy. We show that the error on the reconstructed free energy does not depend on the no. of walkers, leading to a fully linear scaling algorithm even on inexpensive loosely coupled clusters of PCs. In addn., we show that the accuracy and stability of the method are much improved by combining it with a weighted histogram anal. We check the validity of our new method on a realistic application.
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      Branduardi, D.; Bussi, G.; Parrinello, M. Metadynamics with Adaptive Gaussians. J. Chem. Theory Comput. 2012, 8, 2247– 2254,  DOI: 10.1021/ct3002464
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      Metadynamics with Adaptive Gaussians
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      Journal of Chemical Theory and Computation (2012), 8 (7), 2247-2254CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)
      Metadynamics is an established sampling method aimed at reconstructing the free-energy surface relative to a set of appropriately chosen collective variables. In std. metadynamics, the free-energy surface is filled by the addn. of Gaussian potentials of preassigned and typically diagonal covariance. Asymptotically the free-energy surface is proportional to the bias deposited. Here, we consider the possibility of using Gaussians whose variance is adjusted on the fly to the local properties of the free-energy surface. We suggest two different prescriptions: one is based on the local diffusivity and the other on the local geometrical properties. We further examine the problem of extg. the free-energy surface when using adaptive Gaussians. We show that the std. relation between the bias and the free energy does not hold. In the limit of narrow Gaussians an explicit correction can be evaluated. In the general case, we propose to use instead a relation between bias and free energy borrowed from umbrella sampling. This relation holds for all kinds of incrementally deposited bias. We illustrate on the case of alanine dipeptide the advantage of using adaptive Gaussians in conjunction with the new free-energy estimator both in terms of accuracy and speed of convergence.
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      Coveney, P. V.; Wan, S. On the Calculation of Equilibrium Thermodynamic Properties from Molecular Dynamics. Phys. Chem. Chem. Phys. 2016, 18, 30236– 30240,  DOI: 10.1039/C6CP02349E
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      On the calculation of equilibrium thermodynamic properties from molecular dynamics
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      The purpose of statistical mechanics is to provide a route to the calcn. of macroscopic properties of matter from their constituent microscopic components. It is well known that the macrostates emerge as ensemble avs. of microstates. However, this is more often stated than implemented in computer simulation studies. Here we consider foundational aspects of statistical mechanics which are overlooked in most textbooks and research articles that purport to compute macroscopic behavior from microscopic descriptions based on classical mechanics and show how due attention to these issues leads in directions which have not been widely appreciated in the field of mol. dynamics simulation.
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      Grossfield, A.; Patrone, P. N.; Roe, D. R.; Schultz, A. J.; Siderius, D.; Zuckerman, D. M. Best Practices for Quantification of Uncertainty and Sampling Quality in Molecular Simulations. LiveCoMS 2019, 1, 5067,  DOI: 10.33011/livecoms.1.1.5067
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      Aguilar-Mogas, A.; Giménez, X.; Bofill, J. M. Implementation of an Algorithm Based on the Runge-Kutta-Fehlberg Technique and the Potential Energy as a Reaction Coordinate to Locate Intrinsic Reaction Paths. J. Comput. Chem. 2010, 2510– 2525,  DOI: 10.1002/jcc.21539
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      Implementation of an algorithm based on the Runge-Kutta-Fehlberg technique and the potential energy as a reaction coordinate to locate intrinsic reaction paths
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      The intrinsic reaction coordinate (IRC) curve is used widely as a representation of the Reaction Path and can be parameterized taking the potential energy as a reaction coordinate (Aguilar-Mogas et al., J Chem Phys 2008, 128, 104102). Taking this parameterization and its variational nature, an algorithm is proposed that permits to locate this type of curve joining two points from an arbitrary curve that joints the same initial and final points. The initial and final points are min. of the potential energy surface assocd. with the geometry of reactants and products of the reaction whose mechanism is under study. The arbitrary curves are moved toward the IRC curve by a Runge-Kutta-Fehlberg technique. This technique integrates a set of differential equations resulting from the minimization until value zero of the line integral over the Weierstrass E-function. The Weierstrass E-function is related with the second variation in the theory of calculus of variations. The algorithm has been proved in real chem. systems. © 2010 Wiley Periodicals, Inc. J Comput Chem, 2010.
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      Bofill, J. M.; Quapp, W.; Caballero, M. Locating Transition States on Potential Energy Surfaces by the Gentlest Ascent Dynamics. Chem. Phys. Lett. 2013, 583, 203– 208,  DOI: 10.1016/j.cplett.2013.07.074
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      Chemical Physics Letters (2013), 583 (), 203-208CODEN: CHPLBC; ISSN:0009-2614. (Elsevier B.V.)
      The system of ordinary differential equations for the method of the gentlest ascent dynamics (GAD) has been derived which was previously proposed [W. E and X. Zhou, Nonlinearity 24, 1831 (2011)]. For this purpose we use diverse projection operators to a given initial direction. Using simple examples we explain the two possibilities of a GAD curve: it can directly find the transition state by a gentlest ascent, or it can go the roundabout way over a turning point and then find the transition state going downhill along its ridge. An outlook to generalised formulas for higher order saddle-points is added.
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      Zhang, X.-J.; Shang, C.; Liu, Z.-P. Double-Ended Surface Walking Method for Pathway Building and Transition State Location of Complex Reactions. J. Chem. Theory Comput. 2013, 9, 5745– 5753,  DOI: 10.1021/ct4008475
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      Double-Ended Surface Walking Method for Pathway Building and Transition State Location of Complex Reactions
      Zhang, Xiao-Jie; Shang, Cheng; Liu, Zhi-Pan
      Journal of Chemical Theory and Computation (2013), 9 (12), 5745-5753CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)
      Toward the activity prediction with large-scale computations, here a double-ended surface walking (DESW) method is developed for connecting two min. on a potential energy surface (PES) and locating the assocd. transition state (TS) using only the first derivs. The method operates two images starting from the initial and the final states, resp., to walk in a stepwise manner toward each other. The surface walking involves repeated bias potential addn. and local relaxation with the constrained Broyden dimer method to correct the walking direction. We apply the method to a model PES, a large set of gas phase Baker reactions, and complex surface catalytic reactions, which demonstrates that the DESW method can establish a low energy pathway linking two min. even without iterative optimization of the pathway, from which the TS can be located readily. By comparing the efficiency of the new method with the existing methods, we show that the DESW method is much less computationally demanding and is applicable for reactions with complex PESs. We hope that the DESW method may be integrated with the PES sampling methods for automated reaction prediction.
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      We introduce an enhanced sampling method that is based on constructing a model probability d. from which a bias potential is derived. The model relies on the fact that in a phys. system most of the configurations visited can be grouped into isolated metastable islands. With each island we assoc. a distribution that is fitted to a Gaussian mixt. The different distributions are linearly combined together with coeffs. that are computed self-consistently. This leads to an integrated procedure for discovering new metastable states, exploring reaction pathways, computing free energy differences, and estg. reaction rates.
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      To quantify the progress in the development of algorithms and force fields used in mol. simulations, a general method for the assessment of the sampling quality is needed. Statistical mechanics principles suggest the populations of phys. states characterize equil. sampling in a fundamental way. We therefore develop an approach for analyzing the variances in state populations, which quantifies the degree of sampling in terms of the effective sample size (ESS). The ESS ests. the no. of statistically independent configurations contained in a simulated ensemble. The method is applicable to both traditional dynamics simulations as well as more modern (e.g., multicanonical) approaches. Our procedure is tested in a variety of systems from toy models to atomistic protein simulations. We also introduce a simple automated procedure to obtain approx. phys. states from dynamic trajectories: this allows sample-size estn. in systems for which phys. states are not known in advance.
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  • Supporting Information

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    The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acs.jctc.2c00197.

    • (S1) The effect of the type and scaling parameters for the Daubechies wavelet basis functions, (S2) the effect of the width parameter for the Gaussian basis functions, (S3) the Adam stochastic gradient descent algorithm, (S4) additional figures for the model potentials, (S5) the collective variables for the calcium carbonate system, (S6) additional figures for the calcium carbonate system, and (S7) numerical results for the calcium carbonate system (PDF)

    • Video illustrating the time evolution of the FES estimates of the VES method for different basis sets (MP4)


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