**Cite This:**

*J. Chem. Theory Comput.*2020, 16, 4, 2042-2052

# Data Reweighting in Metadynamics Simulations

- Timo M. SchäferTimo M. SchäferInstitut für Physik, Johannes Gutenberg-Universität Mainz, Mainz, GermanyGraduate School Materials Science in Mainz, Mainz, GermanyMore by Timo M. Schäfer
- and
- Giovanni Settanni
*****Giovanni SettanniInstitut für Physik, Johannes Gutenberg-Universität Mainz, Mainz, GermanyMax Planck Graduate Center mit der Johannes Gutenberg-Universität Mainz, Mainz, Germany*****Email: [email protected]More by Giovanni Settanni

## Abstract

The data collected along a metadynamics simulation can be used to recover information about the underlying unbiased system by means of a reweighting procedure. Here, we analyze the behavior of several reweighting techniques in terms of the quality of the reconstruction of the underlying unbiased free energy landscape in the early stages of the simulation and propose a simple reweighting scheme that we relate to the other techniques. We then show that the free energy landscape reconstructed from reweighted data can be more accurate than the negative bias potential depending on the reweighting technique, the stage of the simulation, and the adoption of well-tempered or standard metadynamics. While none of the tested reweighting techniques from the literature provides the most accurate results in all the analyzed situations, the one proposed here, in addition to helping simplifying the reweighting procedure, converges quickly and precisely to the underlying free energy surface in all the considered cases, thus allowing for an efficient use of limited simulation data.

## 1. Introduction

## 2. Results and Discussion

### 2.1. Reweighting Models

*V*(

**,**

*r**t*) at time

*t*and at point

**in coordinate space is defined as a function of the past trajectory of the system. In particular it is a sum of Gaussian hills defined in the**

*r**d*-dimensional space of collective variables

**with height**

*s**W*and widths

**σ**, deposited with a period Δ

*t*:

*U*(

**) +**

*r**V*(

**(**

*s***),**

*r**t*), where

*U*(

**) is the unbiased internal energy of the system.**

*r**w*(

**,**

*r**t*) is also a function of the bias potential. A right choice of

*w*(

**,**

*r**t*) insures that unbiased averages of an arbitrary operator

*A*can be calculated via

_{0}denoting the unbiased average, and ⟨···⟩

_{b}denoting the average over the biased trajectory.

*V*(

**(**

*s***),**

*r**t*) the instantaneous probability density function (pdf) is given as

**indirectly through**

*r***, the same is the case for**

*s**w*, and the expression above can be further modified: (14,16)

*F*(

**), the unbiased free energy.**

*s**F*(

**) from the bias potential (12,13,15,22)**

*s**∞*) and

*c*(

*t*) is a time-dependent offset which can be chosen so that

*F*(

*s*) becomes asymptotically time-independent. (16)

*F*(

*s*) from eq 11 into eq 9, which leads to Tiwary’s weights. (16,19) In the case of standard metadynamics, this results in

**is extended to the sampled regions of the collective variable space. Thus, the weight according to this scheme is proportional to the exponential of the bias potential normalized by the average of the exponential of the bias potential over the space of the collective variables.**

*s**F*(

*s*) from the end of the simulation, resulting in the following expression (for standard metadynamics):

*t*

_{f}is the final time of the simulation. In the later stages of the trajectory, where

*V*(

*s*,

*t*) →

*V*(

*s*,

*t*

_{f}), eq 13 converges to the Tiwary’s weights of eq 12. On the other hand, considering for simplicity the case of standard metadynamics, in the early part of the trajectory for

*t*≪

*t*

_{f},

*V*(

*s*,

*t*

_{f}) dominates in the term at the numerator, so the integrals at the numerator and denominator of eq 13 are approximately equal, and the expression simplifies to exponential weights:

*g*

_{t′}(

**) to the bias potential, a constant term would be subtracted from it, uniformly over the space of**

*s***. In this way, we will have a modified bias potential**

*s**t*. The modified bias has the property that at any time

*t*. Now, replacing the bias

*V*in eq 14, with

*V*′ from eq 15, leads to a balanced exponential reweighting scheme:

^{βV(s,t)}⟩

_{s}, while in the balanced exponential, it is normalized by

*e*

^{β⟨V(s,t)⟩s}. The average of the exponential in Tiwary’s method, although analytically more accurate, converges more slowly than the exponential of the average bias because it is very sensitive to values of

*V*at the upper tail of its distribution. For the same reason, Tiwary’s method may result in a larger run-to-run variability due to the different ways the distribution of

*V*may be progressively sampled in different runs.

*P*

_{b}(

**,**

*s**t*) = e

^{βV (s,t)/(γ–1)}. In our hands the second method provides more accurate estimates of the free energy in most of the circumstances, so we will only show those results. From the biased histogram, the unbiased histogram is obtained as

*P*

_{0}(

**) = e**

*s*^{βV (s,tf)}

*P*

_{b}(

**,**

*s**t*

_{f}). This method is based on the construction and time evolution of histograms, rather than the assignment of weights to sampled conformations. To cast it into weights, let us define

*n*

_{h}(

**) the partial histogram obtained from the data collected between the deposition of the (**

*s**h*– 1)th and

*h*th Gaussians. After deposition of the

*h*th Gaussian, the partial histogram is added to overall histogram and reweighted according to eq 17. Thus, the weight of the partial histogram on the overall histogram becomes

*n*

_{h}(

**)**

*s**e*

^{–β(V̇(s,t)+ċ(t))Δt}. At the next Gaussian deposition, a similar process occurs, and the weight of our initial partial histogram becomes

*V*(

**,**

*s**t*

_{h}) = = . Equation 19 tells us that every sample collected between the (

*h*– 1)th and the

*h*th Gaussian deposition has a weight:

*M*terms, where

*M*is the total number of Gaussians added to the bias at the final time

*t*=

*t*

_{f}. The first

*h*– 1 terms, all included in the expression

*V*(

**,**

*s**t*

_{h–1}), come from the Gaussians deposited until time

*t*

_{h–1}and are dependent on

**; the rest are uniform offsets coming from Gaussians deposited at later stages. The expression in the exponential of eq 20 is defined apart from an additive constant**

*s**C*. If we choose we obtain

_{b}indicates an average over the biased ensemble. We note that eq 21 recalls eq 16 with the difference that in eq 16 the averages of the Gaussians are simple averages while in eq 21 they are made in the biased ensemble. In the case of standard metadynamics, where the bias factor γ →

*∞*, the averages in the biased distribution e

^{βV(s,t)/(γ–1)}become simple averages in

**space and the two methods should provide approximately similar results, apart from the fact that in Bonomi’s methods all evaluations of the bias potential are done on binned data, rather than the unbinned data used in eq 16. This latter difference turns out to have important consequences in the case of estimating averages of observables that were not directly biased during the metadynamics simulations, as will be shown later. In any case, for the comparison, we will use the output of the program from ref (14) made available via PLUMED 1.3 and not eq 20.**

*s**t*

_{f}of the bias potential and are not a function of time.

### 2.2. Unidimensional Model System

*U*(

*x*) = (

*x*

^{2}– 1)

^{2}. The simplicity of the model allows us to associate precisely the effects of each parameter of the metadynamics to the dynamics of the particle. For the sake of testing the reweighting models, we do not even need to run a real molecular dynamics. Instead we built a simple Monte Carlo scheme where the particle can move randomly in the interval [

*x*– Δ

*x*:

*x*+ Δ

*x*] where

*x*is its present position and Δ

*x*is fixed to 0.05. Moves are accepted according to a Metropolis scheme where the energy of the system is

*U*(

*x*) +

*V*(

*x*,

*t*),

*V*(

*x*,

*t*) being the metadynamics bias.

*t*is the deposition period,

*v*is the volume, and σ is the width of the Gaussian hills (the height of the Gaussian is then ). We explored a wide spectrum of values of the three metadynamics parameters, while we fixed the length of the simulations to 2 × 10

^{7}steps. The temperature of the system used in the Metropolis scheme was set to

^{1}/

_{10}th of the height of the free energy barrier separating the two minima, which insures that unbiased simulations would not converge quickly. A total of 72 independent trajectories were collected for each set of parameters. The position of the particle was saved every 100 steps unless reported differently and, later, discretized into bins of size 0.005.

*F*(

*x*) =

*U*(

*x*) itself. We used the simulation data to estimate it either using the negative bias potential

*n*

_{x}is the number of discretized position bins in the sum. The sum is limited to the values of

*x*sampled during the simulation and to the interval [-1.5,1.5] to exclude the external parts of the landscape which may reach very high free energy values and provide dominant contributions to the integral. The FESs obtained with the various methods are aligned by subtracting the average value over the points in the sum to obtain the RMSD

_{FES}. The RMSD values are averaged across all runs with the same metadynamics parameters. The error of the free energy estimates measured at the end of the simulations is high at low filling/flooding rates

*v*/Δ

*t*, where the length of the simulations is not sufficient to properly sample the free energy landscape, as there is no time to cross the free energy barrier. For sufficiently large flooding rates, instead, the error becomes approximately proportional to as already reported. (25) We then fixed the metadynamics parameters to values providing a sufficiently fast flooding rate and monitored the average error as a function of simulation time (Figure 1b). In this case the balanced-exponential reweighted FES and, to a lower extent, the negative bias potential converge significantly faster to the true FES than the Tiwary reweighted FES. The balanced exponential FESs show run-to-run variability lower than Tiwary’s method. Both balanced exponential and Tiwary’s method show that the tends to a constant toward the end of the simulations (Figure 1b inset), as expected. (14) The accuracy of the negative bias reaches a plateau, related to the continuous addition of new hills of fixed height to the bias potential, as per standard metadynamics, which produces finite fluctuations around the true FES. The fluctuations of the balanced-exponential and the Tiwary reweighted FES around the correct FES decrease with time due to the time dependence of the size of the weights (see below). The behavior of the fluctuations of the two reweighted FESs can also be observed directly in the Figure 1a.

*F*|) between the two minima of the double well potential averaged across the independent runs (Figure 1c) shows that the bias potential and the balanced-exponential reweighting reach low values faster than the method proposed by Tiwary and Parrinello, although the fluctuations in this case are larger than those for the RMSD. The balanced exponential and Tiwary’s method eventually converge to similar values smaller than the negative bias and with smaller fluctuations. The time series of the height of the free energy barrier Δ

*F*

^{‡}(or, better, its deviations from the expected value of 1, Figure 1d), which measures how well the methods deal with high free energy regions like transition states that are sampled rarely, shows, on average, that the balanced-exponential reweighted FES manages to converge several times faster than the Tiwary reweighted one or the bias potential, and it does so consistently across the runs (small standard deviation). Also in this case, eventually, on longer simulation time scales, Tiwary’s method reaches a similar accuracy.

_{FES}. Compared to parts b–e of Figure 1, it seems that the convergence of RMSD

_{V}of the two reweighting techniques to within a standard deviation occurs at a time larger than required for the balanced-exponential FES to converge to the true FES. The time of convergence of the RMSD

_{V}of the two reweighting schemes may then provide an indication about the state of convergence of the simulation.

### 2.3. Alanine Dipeptide with Standard Metadynamics

*C*

_{7eq}(φ = −1.382, ψ = 1.005), the local minimum

*C*

_{ax}(φ = 1.257, ψ = −0.880), as well as the connecting transition state ‡ (φ = 1.885, ψ = −2.136).

*k*

_{b}

*T*from the global minimum and

*n*

_{φ,ψ}is the number of bins used in the sum, see Figure 3a), as well as the deviation |ΔΔ

*F*

_{C7eq–Cax}| of the estimated free energy difference between minima from the reference (Figure 3b), and the deviation |ΔΔ

*F*

_{C7eq–‡}| of the estimated height of the free energy barrier (Figure 3c).

_{FES}that is 3 times larger than the balanced exponential. Closer inspection shows that in the runs where the local minimum

*C*

_{ax}is reached early in the simulation, Tiwary’s reweighting does not converge on the reference FES due to wrong weighting of the conformations of the local minimum (Figures S1 and S2 in the Supporting Information).

^{4}), where the exponential reweight outperforms Tiwary’s reweight in terms of matching the reference free energy, the effective sample size of Tiwary’s and balanced exponential reweighting are approximately 0.19 and 0.06 of the total sample size, respectively, so less than an order of magnitude difference. This means that the lower size of the sample in balanced exponential reweighting is more than compensated for by the quality of the weights in terms of matching the reference FES.

_{V}. Also, in this case, the time of convergence of Tiwary’s and balanced exponential provides and indication of the degree of convergence of the simulation. In this regime, Branduardi’s methods shows a much smaller distance to the negative bias than the other methods. Branduardi’s method indeed is supposed to asymptotically match the negative bias.

_{FES}remains higher than the other methods, the lower parts of the FES are reproduced with relative accuracy (low error on the free energy difference between minima) while the high free energy regions are not (large error on the height of the free energy barrier). On the other hand, Bonomi’s method, which is also not specifically indicated for standard metadynamics, provides good results also in this context, with accuracies similar to those of the balanced exponential method, confirming the similarity of the methods in the case of standard metadynamics highlighted in the section Reweighting Models.

### 2.4. Alanine Dipeptide with Standard Metadynamics on a Single Collective Variable

*C*

_{ax}early in the trajectory). In this case, a major problem is the limited sampling in the ψ direction, which, as mentioned, has low correlation with the φ direction used in the bias. In particular, for a given φ “slice” of the FES corresponding to a minimum of the free energy, the values of ψ with high free energy will be sampled rarely, because of the bias pushing the system away from there based only on the value of φ. This problem appears in the balanced exponential reweighting scheme, where outliers are present (Figure 6b) with very low weights (i.e., very high free energy) located at the border of the sampled regions and corresponding to regions visited early in the simulation and never visited again. These outliers contribute substantially to increase the RMSD

_{FES}although in reality the rest of the reweighted FES is close to the reference. To limit the influence of this phenomenon, we measured the RMSD

_{FES}of the various reweighting schemes limiting the sum of the differences to the bins visited in the last 1 ns of simulation, thus excluding the few bins with unusually high free energy due to limited sampling.

_{FES}(Figure 7a), in this case, shows that balanced exponential and Tiwary’s methods are, on average, more accurate than the others, both in the early and the later stages of the simulations followed by Branduardi’s method. However, Tiwary’s and Branduardi’s method are both affected by a large run-to-run variability: the standard deviation of the RMSD

_{FES}is about twice as big as the balanced exponential (width of the error bands in Figure 7a). In the case of Tiwary’s method, this is again due to a fraction of the runs entering the local minimum

*C*

_{ax}early in the trajectory (Figure S3 in Supporting Information). Also the estimate of the free energy difference between minima in the case of the balanced exponential (Figure 7b) converges faster to more accurate values than Tiwary’s and Bonomi’s methods and shows smaller run-to-run fluctuations. The data on the estimate of the height of the free energy barrier are to be taken with care, as the TS lies close to the border of the sampled region (Figure 6) and suffers from limited sampling. In any case, the balanced exponential shows on average to estimate the barrier accurately followed by Tiwary’s and Branduardi’s methods. Here we observed that, in Bonomi’s method, using the averaging of the hills over the biased distribution (rewtype = 0) does not provide very good estimates of the free energy barrier height. Averaging by means of the histogram (rewtype = 1) showed better results, similar to Branduardi’s method (data not shown).

### 2.5. Alanine Dipeptide with Well-Tempered Metadynamics

_{FES}(which is focused on the regions around the minima), the balanced exponential and Tiwary’s estimates are only slightly more accurate than the other methods, on average (Figure 9a), although the first one shows lower run-to-run variability than the latter (i.e., standard deviation of RMSD

_{FES}is almost half) in analogy to the previously analyzed cases. The Branduardi estimate, after a transient, outperforms rescaled negative bias estimates (as already observed in ref (15)). Bonomi’s reweighting requires a slightly longer transient to reach accuracies similar to the other methods but in the later stages of the simulations there is substantial overlap among the various methods, confirming what reported in ref (21) for longer time scales. The fact that the height of the Gaussians is reduced during the course of the simulations helps to reduce the weight differences between early and late trajectory snapshots (Figure 9e,f) and insures that the for all FES estimates (Figure 9a inset and Figure S4 in the Supporting Information). One other aspect that is similar to both standard and WT-metadynamics is that, in the early stage of the simulations, the estimated FES from balanced exponential reweighting is closer to the estimate obtained from the negative bias than Tiwary’s estimate (Figures 9d and 3d), and the convergence of the two values can be used as an indication of the convergence of the simulation. As a final note, we would like to point out that the estimates obtained using the balanced exponential, Bonomi’s and Tiwary’s method on standard metadynamics runs reach (after 8 ns) accuracies comparable to those obtained using the rescaled negative bias from WT-metadynamics runs (Figure 9a–d and 3a–d).

## 3. Conclusions

## Supporting Information

The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acs.jctc.9b00867.

Figures S1–S4 (PDF)

## Terms & Conditions

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.

## Acknowledgments

We thank Giovanni Bussi for critical reading of the manuscript and Friederike Schmid and Marialore Sulpizi for fruitful discussions. T.M.S. gratefully acknowledges financial support from the Graduate School Materials Science in Mainz. G.S. gratefully acknowledges financial support from the Max-Planck Graduate Center with the University of Mainz. We gratefully acknowledge support with computing time from the HPC facility Hazelhen at the High Performance Computing Center Stuttgart and the HPC facility Mogon at the university of Mainz. This work was supported by the German Science Foundation within the SFB 1066 Project Q1.

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Chem.*2002,*53*, 291– 318, DOI: 10.1146/annurev.physchem.53.082301.113146Google Scholar8https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD38Xks1Ois7g%253D&md5=fbf79b52e751dfd87a73c345b9581898Transition path sampling: throwing ropes over rough mountain passes, in the darkBolhuis, Peter G.; Chandler, David; Dellago, Christoph; Geissler, Phillip L.Annual Review of Physical Chemistry (2002), 53 (), 291-318CODEN: ARPLAP; ISSN:0066-426X. (Annual Reviews Inc.)A review is given of the concepts and methods of transition path sampling. These methods allow computational studies of rare events without requiring prior knowledge of mechanisms, reaction coordinates, and transition states. Based upon a statistical mechanics of trajectory space, they provide a perspective with which time dependent phenomena, even for systems driven far from equil., can be examd. with the same types of importance sampling tools that in the past have been applied so successfully to static equil. properties.**9**Darve, E.; Rodríguez-Gómez, D.; Pohorille, A. Adaptive biasing force method for scalar and vector free energy calculations.*J. Chem. Phys.*2008,*128*, 144120, DOI: 10.1063/1.2829861Google Scholar9https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD1cXkvFyiu74%253D&md5=6f6eb47d685e873d1ff35ffdc9ae66cbAdaptive biasing force method for scalar and vector free energy calculationsDarve, Eric; Rodriguez-Gomez, David; Pohorille, AndrewJournal of Chemical Physics (2008), 128 (14), 144120/1-144120/13CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)In free energy calcns. based on thermodn. integration, it is necessary to compute the derivs. of the free energy as a function of one (scalar case) or several (vector case) order parameters. We derive in a compact way a general formulation for evaluating these derivs. as the av. of a mean force acting on the order parameters, which involves first derivs. with respect to both Cartesian coordinates and time. This is in contrast with the previously derived formulas, which require first and second derivs. of the order parameter with respect to Cartesian coordinates. As illustrated in a concrete example, the main advantage of this new formulation is the simplicity of its use, esp. for complicated order parameters. It is also straightforward to implement in a mol. dynamics code, as can be seen from the pseudo-code given at the end. We further discuss how the approach based on time derivs. can be combined with the adaptive biasing force method, an enhanced sampling technique that rapidly yields uniform sampling of the order parameters, and by doing so greatly improves the efficiency of free energy calcns. Using the backbone dihedral angles Φ and Ψ in N-acetylalanyl-N'-methylamide as a numerical example, we present a technique to reconstruct the free energy from its derivs., a calcn. that presents some difficulties in the vector case because of the statistical errors affecting the derivs. (c) 2008 American Institute of Physics.**10**Maragliano, L.; Vanden-Eijnden, E. A temperature accelerated method for sampling free energy and determining reaction pathways in rare events simulations.*Chem. Phys. Lett.*2006,*426*, 168– 175, DOI: 10.1016/j.cplett.2006.05.062Google Scholar10https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD28XmslGns7c%253D&md5=791e4865671fb36662b57f2342f8d95eA temperature accelerated method for sampling free energy and determining reaction pathways in rare events simulationsMaragliano, Luca; Vanden-Eijnden, EricChemical Physics Letters (2006), 426 (1-3), 168-175CODEN: CHPLBC; ISSN:0009-2614. (Elsevier B.V.)A method for sampling efficiently the free energy landscape of a complex system with respect to some given collective variables is proposed. Inspired by metadynamics [A. Laio, M. Parrinello, Proc. Nat. Acad. Sci. USA 99 (2002) 12562], we introduce an extended system where the collective variables are treated as dynamical ones and show that this allows to sample the free energy landscape of these variables directly. The sampling is accelerated by using an artificially high temp. for the collective variables. The validity of the method is established using general results for systems with multiple time-scales, and its numerical efficiency is also discussed via error anal. We also show how the method can be modified in order to sample the reactive pathways in free energy space, and thereby analyze the mechanism of a reaction. Finally, we discuss how the method can be generalized and used as an alternative to the Kirkwood generalized thermodn. integration approach for the calcn. of free energy differences.**11**Laio, A.; Parrinello, M. Escaping free-energy minima.*Proc. Natl. Acad. Sci. U. S. A.*2002,*99*, 12562– 12566, DOI: 10.1073/pnas.202427399Google Scholar11https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD38XnvFGiurc%253D&md5=48d5bc7436f3ef9d78369671e70fa608Escaping free-energy minimaLaio, Alessandro; Parrinello, MicheleProceedings 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.**12**Bussi, G.; Laio, A.; Parrinello, M. Equilibrium free energies from nonequilibrium metadynamics.*Phys. Rev. Lett.*2006,*96*, 090601 DOI: 10.1103/PhysRevLett.96.090601Google Scholar12https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD28XitlSis7o%253D&md5=b4cf2dd6399f8cd58be36692776c38e5Equilibrium Free Energies from Nonequilibrium MetadynamicsBussi, Giovanni; Laio, Alessandro; Parrinello, MichelePhysical Review Letters (2006), 96 (9), 090601/1-090601/4CODEN: PRLTAO; ISSN:0031-9007. (American Physical Society)We propose a new formalism to map history-dependent metadynamics in a Markovian process. We apply this formalism to model Langevin dynamics and det. the equil. distribution of a collection of simulations. We demonstrate that the reconstructed free energy is an unbiased est. of the underlying free energy and anal. derive an expression for the error. The present results can be applied to other history-dependent stochastic processes, such as Wang-Landau sampling.**13**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.100.020603Google Scholar13https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD1cXovFensQ%253D%253D&md5=701ccfeee476c2e9a5d1e5a6b0e82197Well-Tempered Metadynamics: A Smoothly Converging and Tunable Free-Energy MethodBarducci, Alessandro; Bussi, Giovanni; Parrinello, MichelePhysical 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.**14**Bonomi, M.; Barducci, A.; Parrinello, M. Reconstructing the equilibrium Boltzmann distribution from well-tempered metadynamics.*J. Comput. Chem.*2009,*30*, 1615– 1621, DOI: 10.1002/jcc.21305Google Scholar14https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD1MXnsFCiur8%253D&md5=da4d0b135d1edc29919698c5540671abReconstructing the equilibrium Boltzmann distribution from well-tempered metadynamicsBonomi, M.; Barducci, A.; Parrinello, M.Journal of Computational Chemistry (2009), 30 (11), 1615-1621CODEN: JCCHDD; ISSN:0192-8651. (John Wiley & Sons, Inc.)Metadynamics is a widely used and successful method for reconstructing the free-energy surface of complex systems as a function of a small no. of suitably chosen collective variables. This is achieved by biasing the dynamics of the system. The bias acting on the collective variables distorts the probability distribution of the other variables. Here we present a simple reweighting algorithm for recovering the unbiased probability distribution of any variable from a well-tempered metadynamics simulation. We show the efficiency of the reweighting procedure by reconstructing the distribution of the four backbone dihedral angles of alanine dipeptide from two and even one dimensional metadynamics simulation. © 2009 Wiley Periodicals, Inc. J Comput Chem 2009.**15**Branduardi, D.; Bussi, G.; Parrinello, M. Metadynamics with adaptive Gaussians.*J. Chem. Theory Comput.*2012,*8*, 2247, DOI: 10.1021/ct3002464Google Scholar15https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC38Xnt1WrsLc%253D&md5=abed7a6d34ff4797d7cbdc3167ad9060Metadynamics with Adaptive GaussiansBranduardi, Davide; Bussi, Giovanni; Parrinello, MicheleJournal 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.**16**Tiwary, P.; Parrinello, M. A Time-Independent Free Energy Estimator for Metadynamics.*J. Phys. Chem. B*2015,*119*, 736– 742, DOI: 10.1021/jp504920sGoogle Scholar16https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC2cXhtFChur%252FL&md5=10fd77b982ca3cde1559ee7c02361a8cA Time-Independent Free Energy Estimator for MetadynamicsTiwary, Pratyush; Parrinello, MicheleJournal 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.**17**Tiana, G. Estimation of microscopic averages from metadynamics.*Eur. Phys. J. B*2008,*63*, 235, DOI: 10.1140/epjb/e2008-00232-8Google Scholar17https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD1cXos1Kjtr8%253D&md5=075258cbee399d1c2230239c6eb76673Estimation of microscopic averages from metadynamicsTiana, G.European Physical Journal B: Condensed Matter and Complex Systems (2008), 63 (2), 235-238CODEN: EPJBFY; ISSN:1434-6028. (EDP Sciences)With the help of metadynamics it is possible to calc. efficiently the free energy of systems displaying high energy barriers as a function of few selected "collective variables". In doing this, the contribution of all the other degrees of freedom ("microscopic" variables) is averaged out and, thus, lost. In the following it is shown that it is possible to calc. the thermal av. of these microscopic degrees of freedom during the metadynamics, not loosing this piece of information. The method is tested on a two-dimensional toy system and on a small mol., that is dialanine.**18**Smiatek, J.; Heuer, A. Calculation of free energy landscapes: A histogram reweighted metadynamics approach.*J. Comput. Chem.*2011,*32*, 2084– 2096, DOI: 10.1002/jcc.21790Google Scholar18https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC3MXnvFaksbk%253D&md5=1fdd5dcaf029fcb487d719e9c1d7f49aCalculation of free energy landscapes: A histogram reweighted metadynamics approachSmiatek, Jens; Heuer, AndreasJournal of Computational Chemistry (2011), 32 (10), 2084-2096CODEN: JCCHDD; ISSN:0192-8651. (John Wiley & Sons, Inc.)We present an efficient method for the calcn. of free energy landscapes. Our approach involves a history-dependent bias potential, which is evaluated on a grid. The corresponding free energy landscape is constructed via a histogram reweighting procedure a posteriori. Because of the presence of the bias potential, it can be also used to accelerate rare events. In addn., the calcd. free energy landscape is not restricted to the actual choice of collective variables and can in principle be extended to auxiliary variables of interest without further numerical effort. The applicability is shown for several examples. We present numerical results for the alanine dipeptide and the Met-Enkephalin in explicit soln. to illustrate our approach. Furthermore, we derive an empirical formula that allows the prediction of the computational cost for the ordinary metadynamics variant in comparison with our approach, which is validated by a dimensionless representation. © 2011 Wiley Periodicals, Inc. J Comput Chem, 2011.**19**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.2013.09.018Google Scholar19https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC3sXhs1yqs7fJ&md5=292009aab558d0ef1108bb9a5f036c40PLUMED 2: New feathers for an old birdTribello, Gareth A.; Bonomi, Massimiliano; Branduardi, Davide; Camilloni, Carlo; Bussi, GiovanniComputer 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.**20**Bonomi, M.; Branduardi, D.; Bussi, G.; Camilloni, C.; Provasi, D.; Raiteri, P.; Donadio, D.; Marinelli, F.; Pietrucci, F.; Broglia, R. A.; Parrinello, M. PLUMED: A portable plugin for free-energy calculations with molecular dynamics.*Comput. Phys. Commun.*2009,*180*, 1961, DOI: 10.1016/j.cpc.2009.05.011Google Scholar20https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD1MXhtV2kt7fL&md5=49638734f589b5df1e0f3752f62ab663PLUMED: A portable plugin for free-energy calculations with molecular dynamicsBonomi, Massimiliano; Branduardi, Davide; Bussi, Giovanni; Camilloni, Carlo; Provasi, Davide; Raiteri, Paolo; Donadio, Davide; Marinelli, Fabrizio; Pietrucci, Fabio; Broglia, Ricardo A.; Parrinello, MicheleComputer Physics Communications (2009), 180 (10), 1961-1972CODEN: CPHCBZ; ISSN:0010-4655. (Elsevier B.V.)A program aimed at free-energy calcns. in mol. systems is presented. It consists of a series of routines that can be interfaced with the most popular classical mol. dynamics (MD) codes through a simple patching procedure. This leaves the possibility for the user to exploit many different MD engines depending on the system simulated and on the computational resources available. Free-energy calcns. can be performed as a function of many collective variables, with a particular focus on biol. problems, and using state-of-the-art methods such as metadynamics, umbrella sampling, and Jarzynski-equation based steered MD. The present software, written in ANSI-C language, can be easily interfaced with both Fortran and C/C++ codes.**21**Gimondi, I.; Tribello, G. A.; Salvalaglio, M. Building maps in collective variable space.*J. Chem. Phys.*2018,*149*, 104104, DOI: 10.1063/1.5027528Google Scholar21https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC1cXhslWis7jE&md5=ea69647708f72ff9e7bf95bcad815d47Building maps in collective variable spaceGimondi, Ilaria; Tribello, Gareth A.; Salvalaglio, MatteoJournal of Chemical Physics (2018), 149 (10), 104104/1-104104/13CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)Enhanced sampling techniques such as umbrella sampling and metadynamics are now routinely used to provide information on how the thermodn. potential, or free energy, depends on a small no. of collective variables (CVs). The free energy surfaces that one exts. by using these techniques provide a simplified or coarse-grained representation of the configurational ensemble. In this work, we discuss how auxiliary variables can be mapped in CV space. We show that maps of auxiliary variables allow one to analyze both the physics of the mol. system under investigation and the quality of the reduced representation of the system that is encoded in a set of CVs. We apply this approach to analyze the degeneracy of CVs and to compute entropy and enthalpy surfaces in CV space both for conformational transitions in alanine dipeptide and for phase transitions in carbon dioxide mol. crystals under pressure. (c) 2018 American Institute of Physics.**22**Dama, J. F.; Parrinello, M.; Voth, G. A. Well-tempered metadynamics converges asymptotically.*Phys. Rev. Lett.*2014,*112*, 240602, DOI: 10.1103/PhysRevLett.112.240602Google Scholar22https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC2cXhvV2jt73I&md5=c2d5a025586d883ba0aabbb6990fb680Well-tempered metadynamics converges asymptoticallyDama, James F.; Parrinello, Michele; Voth, Gregory A.Physical Review Letters (2014), 112 (24), 240602/1-240602/6, 6 pp.CODEN: PRLTAO; ISSN:0031-9007. (American Physical Society)Metadynamics is a versatile and capable enhanced sampling method for the computational study of soft matter materials and biomol. systems. However, over a decade of application and several attempts to give this adaptive umbrella sampling method a firm theor. grounding prove that a rigorous convergence anal. is elusive. This Letter describes such an anal., demonstrating that well-tempered metadynamics converges to the final state it was designed to reach and, therefore, that the simple formulas currently used to interpret the final converged state of tempered metadynamics are correct and exact. The results do not rely on any assumption that the collective variable dynamics are effectively Brownian or any idealizations of the hill deposition function; instead, they suggest new, more permissive criteria for the method to be well behaved. The results apply to tempered metadynamics with or without adaptive Gaussians or boundary corrections and whether the bias is stored approx. on a grid or exactly.**23**Laio, A.; Gervasio, F. L. Metadynamics: a method to simulate rare events and reconstruct the free energy in biophysics, chemistry and material science.*Rep. Prog. Phys.*2008,*71*, 126601, DOI: 10.1088/0034-4885/71/12/126601Google Scholar23https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD1MXhtFyntrk%253D&md5=cd84cfc103f97c7d7ccf09fc434e2478Metadynamics: a method to stimulate rare events and reconstruct the free energy in biophysics, chemistry and material scienceLaio, Alessandro; Gervasio, Francesco L.Reports on Progress in Physics (2008), 71 (12), 126601/1-126601/22CODEN: RPPHAG; ISSN:0034-4885. (Institute of Physics Publishing)A review. Metadynamics is a powerful algorithm that can be used both for reconstructing the free energy and for accelerating rare events in systems described by complex Hamiltonians, at the classical or at the quantum level. In the algorithm the normal evolution of the system is biased by a history-dependent potential constructed as a sum of Gaussians centered along the trajectory followed by a suitably chosen set of collective variables. The sum of Gaussians is exploited for reconstructing iteratively an estimator of the free energy and forcing the system to escape from local min. This review is intended to provide a comprehensive description of the algorithm, with a focus on the practical aspects that need to be addressed when one attempts to apply metadynamics to a new system: (i) the choice of the appropriate set of collective variables; (ii) the optimal choice of the metadynamics parameters and (iii) how to control the error and ensure convergence of the algorithm.**24**Marinelli, F.; Pietrucci, F.; Laio, A.; Piana, S. A Kinetic Model of Trp-Cage Folding from Multiple Biased Molecular Dynamics Simulations.*PLoS Comput. Biol.*2009,*5*, e1000452 DOI: 10.1371/journal.pcbi.1000452Google Scholar24https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A280%3ADC%252BD1MrktFSitg%253D%253D&md5=42e8c3d2eeb63936b027aac05072325eA kinetic model of trp-cage folding from multiple biased molecular dynamics simulationsMarinelli Fabrizio; Pietrucci Fabio; Laio Alessandro; Piana StefanoPLoS computational biology (2009), 5 (8), e1000452 ISSN:.Trp-cage is a designed 20-residue polypeptide that, in spite of its size, shares several features with larger globular proteins.Although the system has been intensively investigated experimentally and theoretically, its folding mechanism is not yet fully understood. Indeed, some experiments suggest a two-state behavior, while others point to the presence of intermediates. In this work we show that the results of a bias-exchange metadynamics simulation can be used for constructing a detailed thermodynamic and kinetic model of the system. The model, although constructed from a biased simulation, has a quality similar to those extracted from the analysis of long unbiased molecular dynamics trajectories. This is demonstrated by a careful benchmark of the approach on a smaller system, the solvated Ace-Ala3-Nme peptide. For theTrp-cage folding, the model predicts that the relaxation time of 3100 ns observed experimentally is due to the presence of a compact molten globule-like conformation. This state has an occupancy of only 3% at 300 K, but acts as a kinetic trap.Instead, non-compact structures relax to the folded state on the sub-microsecond timescale. The model also predicts the presence of a state at Calpha-RMSD of 4.4 A from the NMR structure in which the Trp strongly interacts with Pro12. This state can explain the abnormal temperature dependence of the Pro12-delta3 and Gly11-alpha3 chemical shifts. The structures of the two most stable misfolded intermediates are in agreement with NMR experiments on the unfolded protein. Our work shows that, using biased molecular dynamics trajectories, it is possible to construct a model describing in detail the Trp-cage folding kinetics and thermodynamics in agreement with experimental data.**25**Laio, A.; Rodriguez-Fortea, A.; Gervasio, F. L.; Ceccarelli, M.; Parrinello, M. Assessing the Accuracy of Metadynamics †.*J. Phys. Chem. B*2005,*109*, 6714– 6721, DOI: 10.1021/jp045424kGoogle Scholar25https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD2MXht1ygurs%253D&md5=f0946da733889a7e05d4f97f6608468bAssessing the Accuracy of MetadynamicsLaio, Alessandro; Rodriguez-Fortea, Antonio; Gervasio, Francesco Luigi; Ceccarelli, Matteo; Parrinello, MicheleJournal of Physical Chemistry B (2005), 109 (14), 6714-6721CODEN: JPCBFK; ISSN:1520-6106. (American Chemical Society)Metadynamics is a powerful technique that has been successfully exploited to explore the multidimensional free energy surface of complex polyat. systems and predict transition mechanisms in very different fields, ranging from chem. and solid-state physics to biophysics. We here derive an explicit expression for the accuracy of the methodol. and provide a way to choose the parameters of the method in order to optimize its performance.**26**Berendsen, H. J. C.; van der Spoel, D.; van Drunen, R. GROMACS: A message-passing parallel molecular dynamics implementation.*Comput. Phys. Commun.*1995,*91*, 43– 56, DOI: 10.1016/0010-4655(95)00042-EGoogle Scholar26https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK2MXps1Wrtr0%253D&md5=04d823aeab28ca374efb86839c705179GROMACS: A message-passing parallel molecular dynamics implementationBerendsen, H. J. C.; van der Spoel, D.; van Drunen, R.Computer Physics Communications (1995), 91 (1-3), 43-56CODEN: CPHCBZ; ISSN:0010-4655. (Elsevier)A parallel message-passing implementation of a mol. dynamics (MD) program that is useful for bio(macro)mols. in aq. environment is described. The software has been developed for a custom-designed 32-processor ring GROMACS (Groningen MAchine for Chem. Simulation) with communication to and from left and right neighbors, but can run on any parallel system onto which a a ring of processors can be mapped and which supports PVM-like block send and receive calls. The GROMACS software consists of a preprocessor, a parallel MD and energy minimization program that can use an arbitrary no. of processors (including one), an optional monitor, and several anal. tools. The programs are written in ANSI C and available by ftp (information: [email protected]). The functionality is based on the GROMOS (Groningen Mol. Simulation) package (van Gunsteren and Berendsen, 1987; BIOMOS B.V., Nijenborgh 4, 9747 AG Groningen). Conversion programs between GROMOS and GROMACS formats are included.The MD program can handle rectangular periodic boundary conditions with temp. and pressure scaling. The interactions that can be handled without modification are variable non-bonded pair interactions with Coulomb and Lennard-Jones or Buckingham potentials, using a twin-range cut-off based on charge groups, and fixed bonded interactions of either harmonic or constraint type for bonds and bond angles and either periodic or cosine power series interactions for dihedral angles. Special forces can be added to groups of particles (for non-equil. dynamics or for position restraining) or between particles (for distance restraints). The parallelism is based on particle decompn. Interprocessor communication is largely limited to position and force distribution over the ring once per time step.**27**Case, D. A.; Cheatham, T. E.; Darden, T.; Gohlke, H.; Luo, R.; Merz, K. M.; Onufriev, A.; Simmerling, C.; Wang, B.; Woods, R. J. The Amber biomolecular simulation programs.*J. Comput. Chem.*2005,*26*, 1668– 1688, DOI: 10.1002/jcc.20290Google Scholar27https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD2MXht1SlsbbM&md5=93be29ff894bab96c783d24e9886c7d0The amber biomolecular simulation programsCase, David A.; Cheatham, Thomas E., III; Darden, Tom; Gohlke, Holger; Luo, Ray; Merz, Kenneth M., Jr.; Onufriev, Alexey; Simmerling, Carlos; Wang, Bing; Woods, Robert J.Journal of Computational Chemistry (2005), 26 (16), 1668-1688CODEN: JCCHDD; ISSN:0192-8651. (John Wiley & Sons, Inc.)The authors describe the development, current features, and some directions for future development of the Amber package of computer programs. This package evolved from a program that was constructed in the late 1970s to do Assisted Model Building with Energy Refinement, and now contains a group of programs embodying a no. of powerful tools of modern computational chem., focused on mol. dynamics and free energy calcns. of proteins, nucleic acids, and carbohydrates.**28**Kish, L.*Survey Sampling*; John Wiley & Sons, Ltd.; New York, 1965.Google ScholarThere is no corresponding record for this reference.**29**Frenkel, D.; Smit, B.*Understanding Molecular Simulations*, 2nd ed.; Computational Science; Academic Press: 2002; Vol. 1.Google ScholarThere is no corresponding record for this reference.**30**Bussi, G. Personal communication.Google ScholarThere is no corresponding record for this reference.

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## References

ARTICLE SECTIONSThis article references 30 other publications.

**1**Bernardi, R. C.; Melo, M. C.; Schulten, K. Enhanced sampling techniques in molecular dynamics simulations of biological systems.*Biochim. Biophys. Acta, Gen. Subj.*2015,*1850*, 872– 877, DOI: 10.1016/j.bbagen.2014.10.0191https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC2cXhvVanurbN&md5=44e3ac1aa042c35ee08aa86a6a63e78eEnhanced sampling techniques in molecular dynamics simulations of biological systemsBernardi, Rafael C.; Melo, Marcelo C. R.; Schulten, KlausBiochimica et Biophysica Acta, General Subjects (2015), 1850 (5), 872-877CODEN: BBGSB3; ISSN:0304-4165. (Elsevier B.V.)A review. Mol. dynamics has emerged as an important research methodol. covering systems to the level of millions of atoms. However, insufficient sampling often limits its application. The limitation is due to rough energy landscapes, with many local min. sepd. by high-energy barriers, which govern the biomol. motion. In the past few decades methods have been developed that address the sampling problem, such as replica-exchange mol. dynamics, metadynamics and simulated annealing. Here the authors present an overview over theses sampling methods in an attempt to shed light on which should be selected depending on the type of system property studied. Enhanced sampling methods have been employed for a broad range of biol. systems and the choice of a suitable method is connected to biol. and phys. characteristics of the system, in particular system size. While metadynamics and replica-exchange mol. dynamics are the most adopted sampling methods to study biomol. dynamics, simulated annealing is well suited to characterize very flexible systems. The use of annealing methods for a long time was restricted to simulation of small proteins; however, a variant of the method, generalized simulated annealing, can be employed at a relatively low computational cost to large macromol. complexes. Mol. dynamics trajectories frequently do not reach all relevant conformational substates, for example those connected with biol. function, a problem that can be addressed by employing enhanced sampling algorithms. This article is part of a Special Issue entitled Recent developments of mol. dynamics.**2**Dror, R. O.; Dirks, R. M.; Grossman, J.; Xu, H.; Shaw, D. E. Biomolecular Simulation: A Computational Microscope for Molecular Biology.*Annu. Rev. Biophys.*2012,*41*, 429– 452, DOI: 10.1146/annurev-biophys-042910-1552452https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC38Xpt1yhs7s%253D&md5=3f872bcd93c1c2141ef3f020c5c6d45dBiomolecular simulation: a computational microscope for molecular biologyDror, Ron O.; Dirks, Robert M.; Grossman, J. P.; Xu, Huafeng; Shaw, David E.Annual Review of Biophysics (2012), 41 (), 429-452CODEN: ARBNCV; ISSN:1936-122X. (Annual Reviews Inc.)A review. Mol. dynamics simulations capture the behavior of biol. macromols. in full at. detail, but their computational demands, combined with the challenge of appropriately modeling the relevant physics, have historically restricted their length and accuracy. Dramatic recent improvements in achievable simulation speed and the underlying phys. models have enabled at.-level simulations on timescales as long as milliseconds that capture key biochem. processes such as protein folding, drug binding, membrane transport, and the conformational changes crit. to protein function. Such simulation may serve as a computational microscope, revealing biomol. mechanisms at spatial and temporal scales that are difficult to observe exptl. We describe the rapidly evolving state of the art for at.-level biomol. simulation, illustrate the types of biol. discoveries that can now be made through simulation, and discuss challenges motivating continued innovation in this field.**3**Abrams, C.; Bussi, G. Enhanced sampling in molecular dynamics using metadynamics, replica-exchange, and temperature-acceleration.*Entropy*2014,*16*, 163– 199, DOI: 10.3390/e160101633https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC2cXptV2htbg%253D&md5=7e27e14ffdacdd6de1e89f512b0f96ceEnhanced sampling in molecular dynamics using metadynamics, replica-exchange, and temperature-accelerationAbrams, Cameron; Bussi, GiovanniEntropy (2014), 16 (1), 163-199, 37 pp.CODEN: ENTRFG; ISSN:1099-4300. (MDPI AG)We review a selection of methods for performing enhanced sampling in mol. dynamics simulations. We consider methods based on collective variable biasing and on tempering, and offer both historical and contemporary perspectives. In collective-variable biasing, we first discuss methods stemming from thermodn. integration that use mean force biasing, including the adaptive biasing force algorithm and temp. acceleration. We then turn to methods that use bias potentials, including umbrella sampling and metadynamics. We next consider parallel tempering and replica-exchange methods. We conclude with a brief presentation of some combination methods.**4**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)90121-8There is no corresponding record for this reference.**5**Hansmann, U. H. Parallel tempering algorithm for conformational studies of biological molecules.*Chem. Phys. Lett.*1997,*281*, 140– 150, DOI: 10.1016/S0009-2614(97)01198-65https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK2sXotFWktr4%253D&md5=ddfed8309757ca88a834a1cf3c80cb08Parallel tempering algorithm for conformational studies of biological moleculesHansmann, Ulrich H. E.Chemical Physics Letters (1997), 281 (1,2,3), 140-150CODEN: CHPLBC; ISSN:0009-2614. (Elsevier Science B.V.)The effectiveness of a new algorithm, parallel tempering, is studied for numerical simulations of biol. mols. These mols. suffer from a rough energy landscape. The resulting slowing down in numerical simulations is overcome by the new method. This is demonstrated by performing simulations with high statistics for one of the simplest peptides, Met-enkephalin. The numerical effectiveness of the new technique was found to be much better than traditional methods and is comparable to sophisticated methods like generalized ensemble techniques.**6**Sugita, Y.; Okamoto, Y. Replica-exchange molecular dynamics method for protein folding.*Chem. Phys. Lett.*1999,*314*, 141– 151, DOI: 10.1016/S0009-2614(99)01123-96https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK1MXotVKrsLc%253D&md5=0fec0ff81ca7806c1e1ac29e5f50ce19Replica-exchange molecular dynamics method for protein foldingSugita, Y.; Okamoto, Y.Chemical Physics Letters (1999), 314 (1,2), 141-151CODEN: CHPLBC; ISSN:0009-2614. (Elsevier Science B.V.)We have developed a formulation for mol. dynamics algorithm for the replica-exchange method. The effectiveness of the method for the protein-folding problem is tested with the penta-peptide Met-enkephalin. The method can overcome the multiple-min. problem by exchanging non-interacting replicas of the system at several temps. From only one simulation run, one can obtain probability distributions in canonical ensemble for a wide temp. range using multiple-histogram re-weighting techniques, which allows the calcn. of any thermodn. quantity as a function of temp. in that range.**7**Sugita, Y.; Okamoto, Y. Replica-exchange multicanonical algorithm and multicanonical replica-exchange method for simulating systems with rough energy landscape.*Chem. Phys. Lett.*2000,*329*, 261– 270, DOI: 10.1016/S0009-2614(00)00999-47https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD3cXnsFWgtLg%253D&md5=0f3829688faf51d80e0efcb58ffff3e3Replica-exchange multicanonical algorithm and multicanonical replica-exchange method for simulating systems with rough energy landscapeSugita, Y.; Okamoto, Y.Chemical Physics Letters (2000), 329 (3,4), 261-270CODEN: CHPLBC; ISSN:0009-2614. (Elsevier Science B.V.)We propose two efficient algorithms for configurational sampling of systems with rough energy landscape. The first one is a new method for the detn. of the multi-canonical wt. factor. In this method, a short replica-exchange simulation is performed and the multi-canonical wt. factor is obtained by the multiple histogram reweighting techniques. The second one is a further extension of the first in which a replica-exchange multi-canonical simulation is performed with a small no. of replicas. These new algorithms are particularly useful for studying the protein folding problem.**8**Bolhuis, P. G.; Chandler, D.; Dellago, C.; Geissler, P. L. TRANSITION PATH SAMPLING: Throwing Ropes Over Rough Mountain Passes, in the Dark.*Annu. Rev. Phys. Chem.*2002,*53*, 291– 318, DOI: 10.1146/annurev.physchem.53.082301.1131468https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD38Xks1Ois7g%253D&md5=fbf79b52e751dfd87a73c345b9581898Transition path sampling: throwing ropes over rough mountain passes, in the darkBolhuis, Peter G.; Chandler, David; Dellago, Christoph; Geissler, Phillip L.Annual Review of Physical Chemistry (2002), 53 (), 291-318CODEN: ARPLAP; ISSN:0066-426X. (Annual Reviews Inc.)A review is given of the concepts and methods of transition path sampling. These methods allow computational studies of rare events without requiring prior knowledge of mechanisms, reaction coordinates, and transition states. Based upon a statistical mechanics of trajectory space, they provide a perspective with which time dependent phenomena, even for systems driven far from equil., can be examd. with the same types of importance sampling tools that in the past have been applied so successfully to static equil. properties.**9**Darve, E.; Rodríguez-Gómez, D.; Pohorille, A. Adaptive biasing force method for scalar and vector free energy calculations.*J. Chem. Phys.*2008,*128*, 144120, DOI: 10.1063/1.28298619https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD1cXkvFyiu74%253D&md5=6f6eb47d685e873d1ff35ffdc9ae66cbAdaptive biasing force method for scalar and vector free energy calculationsDarve, Eric; Rodriguez-Gomez, David; Pohorille, AndrewJournal of Chemical Physics (2008), 128 (14), 144120/1-144120/13CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)In free energy calcns. based on thermodn. integration, it is necessary to compute the derivs. of the free energy as a function of one (scalar case) or several (vector case) order parameters. We derive in a compact way a general formulation for evaluating these derivs. as the av. of a mean force acting on the order parameters, which involves first derivs. with respect to both Cartesian coordinates and time. This is in contrast with the previously derived formulas, which require first and second derivs. of the order parameter with respect to Cartesian coordinates. As illustrated in a concrete example, the main advantage of this new formulation is the simplicity of its use, esp. for complicated order parameters. It is also straightforward to implement in a mol. dynamics code, as can be seen from the pseudo-code given at the end. We further discuss how the approach based on time derivs. can be combined with the adaptive biasing force method, an enhanced sampling technique that rapidly yields uniform sampling of the order parameters, and by doing so greatly improves the efficiency of free energy calcns. Using the backbone dihedral angles Φ and Ψ in N-acetylalanyl-N'-methylamide as a numerical example, we present a technique to reconstruct the free energy from its derivs., a calcn. that presents some difficulties in the vector case because of the statistical errors affecting the derivs. (c) 2008 American Institute of Physics.**10**Maragliano, L.; Vanden-Eijnden, E. A temperature accelerated method for sampling free energy and determining reaction pathways in rare events simulations.*Chem. Phys. Lett.*2006,*426*, 168– 175, DOI: 10.1016/j.cplett.2006.05.06210https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD28XmslGns7c%253D&md5=791e4865671fb36662b57f2342f8d95eA temperature accelerated method for sampling free energy and determining reaction pathways in rare events simulationsMaragliano, Luca; Vanden-Eijnden, EricChemical Physics Letters (2006), 426 (1-3), 168-175CODEN: CHPLBC; ISSN:0009-2614. (Elsevier B.V.)A method for sampling efficiently the free energy landscape of a complex system with respect to some given collective variables is proposed. Inspired by metadynamics [A. Laio, M. Parrinello, Proc. Nat. Acad. Sci. USA 99 (2002) 12562], we introduce an extended system where the collective variables are treated as dynamical ones and show that this allows to sample the free energy landscape of these variables directly. The sampling is accelerated by using an artificially high temp. for the collective variables. The validity of the method is established using general results for systems with multiple time-scales, and its numerical efficiency is also discussed via error anal. We also show how the method can be modified in order to sample the reactive pathways in free energy space, and thereby analyze the mechanism of a reaction. Finally, we discuss how the method can be generalized and used as an alternative to the Kirkwood generalized thermodn. integration approach for the calcn. of free energy differences.**11**Laio, A.; Parrinello, M. Escaping free-energy minima.*Proc. Natl. Acad. Sci. U. S. A.*2002,*99*, 12562– 12566, DOI: 10.1073/pnas.20242739911https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD38XnvFGiurc%253D&md5=48d5bc7436f3ef9d78369671e70fa608Escaping free-energy minimaLaio, Alessandro; Parrinello, MicheleProceedings 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.**12**Bussi, G.; Laio, A.; Parrinello, M. Equilibrium free energies from nonequilibrium metadynamics.*Phys. Rev. Lett.*2006,*96*, 090601 DOI: 10.1103/PhysRevLett.96.09060112https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD28XitlSis7o%253D&md5=b4cf2dd6399f8cd58be36692776c38e5Equilibrium Free Energies from Nonequilibrium MetadynamicsBussi, Giovanni; Laio, Alessandro; Parrinello, MichelePhysical Review Letters (2006), 96 (9), 090601/1-090601/4CODEN: PRLTAO; ISSN:0031-9007. (American Physical Society)We propose a new formalism to map history-dependent metadynamics in a Markovian process. We apply this formalism to model Langevin dynamics and det. the equil. distribution of a collection of simulations. We demonstrate that the reconstructed free energy is an unbiased est. of the underlying free energy and anal. derive an expression for the error. The present results can be applied to other history-dependent stochastic processes, such as Wang-Landau sampling.**13**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.100.02060313https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD1cXovFensQ%253D%253D&md5=701ccfeee476c2e9a5d1e5a6b0e82197Well-Tempered Metadynamics: A Smoothly Converging and Tunable Free-Energy MethodBarducci, Alessandro; Bussi, Giovanni; Parrinello, MichelePhysical 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.**14**Bonomi, M.; Barducci, A.; Parrinello, M. Reconstructing the equilibrium Boltzmann distribution from well-tempered metadynamics.*J. Comput. Chem.*2009,*30*, 1615– 1621, DOI: 10.1002/jcc.2130514https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD1MXnsFCiur8%253D&md5=da4d0b135d1edc29919698c5540671abReconstructing the equilibrium Boltzmann distribution from well-tempered metadynamicsBonomi, M.; Barducci, A.; Parrinello, M.Journal of Computational Chemistry (2009), 30 (11), 1615-1621CODEN: JCCHDD; ISSN:0192-8651. (John Wiley & Sons, Inc.)Metadynamics is a widely used and successful method for reconstructing the free-energy surface of complex systems as a function of a small no. of suitably chosen collective variables. This is achieved by biasing the dynamics of the system. The bias acting on the collective variables distorts the probability distribution of the other variables. Here we present a simple reweighting algorithm for recovering the unbiased probability distribution of any variable from a well-tempered metadynamics simulation. We show the efficiency of the reweighting procedure by reconstructing the distribution of the four backbone dihedral angles of alanine dipeptide from two and even one dimensional metadynamics simulation. © 2009 Wiley Periodicals, Inc. J Comput Chem 2009.**15**Branduardi, D.; Bussi, G.; Parrinello, M. Metadynamics with adaptive Gaussians.*J. Chem. Theory Comput.*2012,*8*, 2247, DOI: 10.1021/ct300246415https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC38Xnt1WrsLc%253D&md5=abed7a6d34ff4797d7cbdc3167ad9060Metadynamics with Adaptive GaussiansBranduardi, Davide; Bussi, Giovanni; Parrinello, MicheleJournal 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.**16**Tiwary, P.; Parrinello, M. A Time-Independent Free Energy Estimator for Metadynamics.*J. Phys. Chem. B*2015,*119*, 736– 742, DOI: 10.1021/jp504920s16https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC2cXhtFChur%252FL&md5=10fd77b982ca3cde1559ee7c02361a8cA Time-Independent Free Energy Estimator for MetadynamicsTiwary, Pratyush; Parrinello, MicheleJournal 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.**17**Tiana, G. Estimation of microscopic averages from metadynamics.*Eur. Phys. J. B*2008,*63*, 235, DOI: 10.1140/epjb/e2008-00232-817https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD1cXos1Kjtr8%253D&md5=075258cbee399d1c2230239c6eb76673Estimation of microscopic averages from metadynamicsTiana, G.European Physical Journal B: Condensed Matter and Complex Systems (2008), 63 (2), 235-238CODEN: EPJBFY; ISSN:1434-6028. (EDP Sciences)With the help of metadynamics it is possible to calc. efficiently the free energy of systems displaying high energy barriers as a function of few selected "collective variables". In doing this, the contribution of all the other degrees of freedom ("microscopic" variables) is averaged out and, thus, lost. In the following it is shown that it is possible to calc. the thermal av. of these microscopic degrees of freedom during the metadynamics, not loosing this piece of information. The method is tested on a two-dimensional toy system and on a small mol., that is dialanine.**18**Smiatek, J.; Heuer, A. Calculation of free energy landscapes: A histogram reweighted metadynamics approach.*J. Comput. Chem.*2011,*32*, 2084– 2096, DOI: 10.1002/jcc.2179018https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC3MXnvFaksbk%253D&md5=1fdd5dcaf029fcb487d719e9c1d7f49aCalculation of free energy landscapes: A histogram reweighted metadynamics approachSmiatek, Jens; Heuer, AndreasJournal of Computational Chemistry (2011), 32 (10), 2084-2096CODEN: JCCHDD; ISSN:0192-8651. (John Wiley & Sons, Inc.)We present an efficient method for the calcn. of free energy landscapes. Our approach involves a history-dependent bias potential, which is evaluated on a grid. The corresponding free energy landscape is constructed via a histogram reweighting procedure a posteriori. Because of the presence of the bias potential, it can be also used to accelerate rare events. In addn., the calcd. free energy landscape is not restricted to the actual choice of collective variables and can in principle be extended to auxiliary variables of interest without further numerical effort. The applicability is shown for several examples. We present numerical results for the alanine dipeptide and the Met-Enkephalin in explicit soln. to illustrate our approach. Furthermore, we derive an empirical formula that allows the prediction of the computational cost for the ordinary metadynamics variant in comparison with our approach, which is validated by a dimensionless representation. © 2011 Wiley Periodicals, Inc. J Comput Chem, 2011.**19**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.2013.09.01819https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC3sXhs1yqs7fJ&md5=292009aab558d0ef1108bb9a5f036c40PLUMED 2: New feathers for an old birdTribello, Gareth A.; Bonomi, Massimiliano; Branduardi, Davide; Camilloni, Carlo; Bussi, GiovanniComputer 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.**20**Bonomi, M.; Branduardi, D.; Bussi, G.; Camilloni, C.; Provasi, D.; Raiteri, P.; Donadio, D.; Marinelli, F.; Pietrucci, F.; Broglia, R. A.; Parrinello, M. PLUMED: A portable plugin for free-energy calculations with molecular dynamics.*Comput. Phys. Commun.*2009,*180*, 1961, DOI: 10.1016/j.cpc.2009.05.01120https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD1MXhtV2kt7fL&md5=49638734f589b5df1e0f3752f62ab663PLUMED: A portable plugin for free-energy calculations with molecular dynamicsBonomi, Massimiliano; Branduardi, Davide; Bussi, Giovanni; Camilloni, Carlo; Provasi, Davide; Raiteri, Paolo; Donadio, Davide; Marinelli, Fabrizio; Pietrucci, Fabio; Broglia, Ricardo A.; Parrinello, MicheleComputer Physics Communications (2009), 180 (10), 1961-1972CODEN: CPHCBZ; ISSN:0010-4655. (Elsevier B.V.)A program aimed at free-energy calcns. in mol. systems is presented. It consists of a series of routines that can be interfaced with the most popular classical mol. dynamics (MD) codes through a simple patching procedure. This leaves the possibility for the user to exploit many different MD engines depending on the system simulated and on the computational resources available. Free-energy calcns. can be performed as a function of many collective variables, with a particular focus on biol. problems, and using state-of-the-art methods such as metadynamics, umbrella sampling, and Jarzynski-equation based steered MD. The present software, written in ANSI-C language, can be easily interfaced with both Fortran and C/C++ codes.**21**Gimondi, I.; Tribello, G. A.; Salvalaglio, M. Building maps in collective variable space.*J. Chem. Phys.*2018,*149*, 104104, DOI: 10.1063/1.502752821https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC1cXhslWis7jE&md5=ea69647708f72ff9e7bf95bcad815d47Building maps in collective variable spaceGimondi, Ilaria; Tribello, Gareth A.; Salvalaglio, MatteoJournal of Chemical Physics (2018), 149 (10), 104104/1-104104/13CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)Enhanced sampling techniques such as umbrella sampling and metadynamics are now routinely used to provide information on how the thermodn. potential, or free energy, depends on a small no. of collective variables (CVs). The free energy surfaces that one exts. by using these techniques provide a simplified or coarse-grained representation of the configurational ensemble. In this work, we discuss how auxiliary variables can be mapped in CV space. We show that maps of auxiliary variables allow one to analyze both the physics of the mol. system under investigation and the quality of the reduced representation of the system that is encoded in a set of CVs. We apply this approach to analyze the degeneracy of CVs and to compute entropy and enthalpy surfaces in CV space both for conformational transitions in alanine dipeptide and for phase transitions in carbon dioxide mol. crystals under pressure. (c) 2018 American Institute of Physics.**22**Dama, J. F.; Parrinello, M.; Voth, G. A. Well-tempered metadynamics converges asymptotically.*Phys. Rev. Lett.*2014,*112*, 240602, DOI: 10.1103/PhysRevLett.112.24060222https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC2cXhvV2jt73I&md5=c2d5a025586d883ba0aabbb6990fb680Well-tempered metadynamics converges asymptoticallyDama, James F.; Parrinello, Michele; Voth, Gregory A.Physical Review Letters (2014), 112 (24), 240602/1-240602/6, 6 pp.CODEN: PRLTAO; ISSN:0031-9007. (American Physical Society)Metadynamics is a versatile and capable enhanced sampling method for the computational study of soft matter materials and biomol. systems. However, over a decade of application and several attempts to give this adaptive umbrella sampling method a firm theor. grounding prove that a rigorous convergence anal. is elusive. This Letter describes such an anal., demonstrating that well-tempered metadynamics converges to the final state it was designed to reach and, therefore, that the simple formulas currently used to interpret the final converged state of tempered metadynamics are correct and exact. The results do not rely on any assumption that the collective variable dynamics are effectively Brownian or any idealizations of the hill deposition function; instead, they suggest new, more permissive criteria for the method to be well behaved. The results apply to tempered metadynamics with or without adaptive Gaussians or boundary corrections and whether the bias is stored approx. on a grid or exactly.**23**Laio, A.; Gervasio, F. L. Metadynamics: a method to simulate rare events and reconstruct the free energy in biophysics, chemistry and material science.*Rep. Prog. Phys.*2008,*71*, 126601, DOI: 10.1088/0034-4885/71/12/12660123https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD1MXhtFyntrk%253D&md5=cd84cfc103f97c7d7ccf09fc434e2478Metadynamics: a method to stimulate rare events and reconstruct the free energy in biophysics, chemistry and material scienceLaio, Alessandro; Gervasio, Francesco L.Reports on Progress in Physics (2008), 71 (12), 126601/1-126601/22CODEN: RPPHAG; ISSN:0034-4885. (Institute of Physics Publishing)A review. Metadynamics is a powerful algorithm that can be used both for reconstructing the free energy and for accelerating rare events in systems described by complex Hamiltonians, at the classical or at the quantum level. In the algorithm the normal evolution of the system is biased by a history-dependent potential constructed as a sum of Gaussians centered along the trajectory followed by a suitably chosen set of collective variables. The sum of Gaussians is exploited for reconstructing iteratively an estimator of the free energy and forcing the system to escape from local min. This review is intended to provide a comprehensive description of the algorithm, with a focus on the practical aspects that need to be addressed when one attempts to apply metadynamics to a new system: (i) the choice of the appropriate set of collective variables; (ii) the optimal choice of the metadynamics parameters and (iii) how to control the error and ensure convergence of the algorithm.**24**Marinelli, F.; Pietrucci, F.; Laio, A.; Piana, S. A Kinetic Model of Trp-Cage Folding from Multiple Biased Molecular Dynamics Simulations.*PLoS Comput. Biol.*2009,*5*, e1000452 DOI: 10.1371/journal.pcbi.100045224https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A280%3ADC%252BD1MrktFSitg%253D%253D&md5=42e8c3d2eeb63936b027aac05072325eA kinetic model of trp-cage folding from multiple biased molecular dynamics simulationsMarinelli Fabrizio; Pietrucci Fabio; Laio Alessandro; Piana StefanoPLoS computational biology (2009), 5 (8), e1000452 ISSN:.Trp-cage is a designed 20-residue polypeptide that, in spite of its size, shares several features with larger globular proteins.Although the system has been intensively investigated experimentally and theoretically, its folding mechanism is not yet fully understood. Indeed, some experiments suggest a two-state behavior, while others point to the presence of intermediates. In this work we show that the results of a bias-exchange metadynamics simulation can be used for constructing a detailed thermodynamic and kinetic model of the system. The model, although constructed from a biased simulation, has a quality similar to those extracted from the analysis of long unbiased molecular dynamics trajectories. This is demonstrated by a careful benchmark of the approach on a smaller system, the solvated Ace-Ala3-Nme peptide. For theTrp-cage folding, the model predicts that the relaxation time of 3100 ns observed experimentally is due to the presence of a compact molten globule-like conformation. This state has an occupancy of only 3% at 300 K, but acts as a kinetic trap.Instead, non-compact structures relax to the folded state on the sub-microsecond timescale. The model also predicts the presence of a state at Calpha-RMSD of 4.4 A from the NMR structure in which the Trp strongly interacts with Pro12. This state can explain the abnormal temperature dependence of the Pro12-delta3 and Gly11-alpha3 chemical shifts. The structures of the two most stable misfolded intermediates are in agreement with NMR experiments on the unfolded protein. Our work shows that, using biased molecular dynamics trajectories, it is possible to construct a model describing in detail the Trp-cage folding kinetics and thermodynamics in agreement with experimental data.**25**Laio, A.; Rodriguez-Fortea, A.; Gervasio, F. L.; Ceccarelli, M.; Parrinello, M. Assessing the Accuracy of Metadynamics †.*J. Phys. Chem. B*2005,*109*, 6714– 6721, DOI: 10.1021/jp045424k25https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD2MXht1ygurs%253D&md5=f0946da733889a7e05d4f97f6608468bAssessing the Accuracy of MetadynamicsLaio, Alessandro; Rodriguez-Fortea, Antonio; Gervasio, Francesco Luigi; Ceccarelli, Matteo; Parrinello, MicheleJournal of Physical Chemistry B (2005), 109 (14), 6714-6721CODEN: JPCBFK; ISSN:1520-6106. (American Chemical Society)Metadynamics is a powerful technique that has been successfully exploited to explore the multidimensional free energy surface of complex polyat. systems and predict transition mechanisms in very different fields, ranging from chem. and solid-state physics to biophysics. We here derive an explicit expression for the accuracy of the methodol. and provide a way to choose the parameters of the method in order to optimize its performance.**26**Berendsen, H. J. C.; van der Spoel, D.; van Drunen, R. GROMACS: A message-passing parallel molecular dynamics implementation.*Comput. Phys. Commun.*1995,*91*, 43– 56, DOI: 10.1016/0010-4655(95)00042-E26https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK2MXps1Wrtr0%253D&md5=04d823aeab28ca374efb86839c705179GROMACS: A message-passing parallel molecular dynamics implementationBerendsen, H. J. C.; van der Spoel, D.; van Drunen, R.Computer Physics Communications (1995), 91 (1-3), 43-56CODEN: CPHCBZ; ISSN:0010-4655. (Elsevier)A parallel message-passing implementation of a mol. dynamics (MD) program that is useful for bio(macro)mols. in aq. environment is described. The software has been developed for a custom-designed 32-processor ring GROMACS (Groningen MAchine for Chem. Simulation) with communication to and from left and right neighbors, but can run on any parallel system onto which a a ring of processors can be mapped and which supports PVM-like block send and receive calls. The GROMACS software consists of a preprocessor, a parallel MD and energy minimization program that can use an arbitrary no. of processors (including one), an optional monitor, and several anal. tools. The programs are written in ANSI C and available by ftp (information: [email protected]). The functionality is based on the GROMOS (Groningen Mol. Simulation) package (van Gunsteren and Berendsen, 1987; BIOMOS B.V., Nijenborgh 4, 9747 AG Groningen). Conversion programs between GROMOS and GROMACS formats are included.The MD program can handle rectangular periodic boundary conditions with temp. and pressure scaling. The interactions that can be handled without modification are variable non-bonded pair interactions with Coulomb and Lennard-Jones or Buckingham potentials, using a twin-range cut-off based on charge groups, and fixed bonded interactions of either harmonic or constraint type for bonds and bond angles and either periodic or cosine power series interactions for dihedral angles. Special forces can be added to groups of particles (for non-equil. dynamics or for position restraining) or between particles (for distance restraints). The parallelism is based on particle decompn. Interprocessor communication is largely limited to position and force distribution over the ring once per time step.**27**Case, D. A.; Cheatham, T. E.; Darden, T.; Gohlke, H.; Luo, R.; Merz, K. M.; Onufriev, A.; Simmerling, C.; Wang, B.; Woods, R. J. The Amber biomolecular simulation programs.*J. Comput. Chem.*2005,*26*, 1668– 1688, DOI: 10.1002/jcc.2029027https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD2MXht1SlsbbM&md5=93be29ff894bab96c783d24e9886c7d0The amber biomolecular simulation programsCase, David A.; Cheatham, Thomas E., III; Darden, Tom; Gohlke, Holger; Luo, Ray; Merz, Kenneth M., Jr.; Onufriev, Alexey; Simmerling, Carlos; Wang, Bing; Woods, Robert J.Journal of Computational Chemistry (2005), 26 (16), 1668-1688CODEN: JCCHDD; ISSN:0192-8651. (John Wiley & Sons, Inc.)The authors describe the development, current features, and some directions for future development of the Amber package of computer programs. This package evolved from a program that was constructed in the late 1970s to do Assisted Model Building with Energy Refinement, and now contains a group of programs embodying a no. of powerful tools of modern computational chem., focused on mol. dynamics and free energy calcns. of proteins, nucleic acids, and carbohydrates.**28**Kish, L.*Survey Sampling*; John Wiley & Sons, Ltd.; New York, 1965.There is no corresponding record for this reference.**29**Frenkel, D.; Smit, B.*Understanding Molecular Simulations*, 2nd ed.; Computational Science; Academic Press: 2002; Vol. 1.There is no corresponding record for this reference.**30**Bussi, G. Personal communication.There is no corresponding record for this reference.

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