Conformational Entropy from Restricted Bond-Vector Motion in Proteins: The Symmetry of the Local Restrictions and Relation to NMR Relaxation

Locally mobile bond-vectors contribute to the conformational entropy of the protein, given by Sk ≡ S/k = −∫(Peq ln Peq)dΩ – ln∫dΩ. The quantity Peq = exp(−u)/Z is the orientational probability density, where Z is the partition function and u is the spatially restricting potential exerted by the immediate internal protein surroundings at the site of the motion of the bond-vector. It is appropriate to expand the potential, u, which restricts local rotational reorientation, in the basis set of the real combinations of the Wigner rotation matrix elements, D0KL. For small molecules dissolved in anisotropic media, one typically keeps the lowest even L, L = 2, nonpolar potential in axial or rhombic form. For bond-vectors anchored at the protein, the lowest odd L, L = 1, polar potential is to be used in axial or rhombic form. Here, we investigate the effect of the symmetry and polarity of these potentials onSk. For L = 1 (L = 2), Sk is the same (differs) for parallel and perpendicular ordering. The plots of Sk as a function of the coefficients of the rhombic L = 1 (L = 2) potential exhibit high-symmetry (specific low-symmetry) patterns with parameter-range-dependent sensitivity. Similar statements apply to analogous plots of the potential minima. Sk is also examined as a function of the order parameters defined in terms of u. Graphs displaying these correlations, and applications illustrating their usage, are provided. The features delineated above are generally useful for devising orienting potentials that best suit given physical circumstances. They are particularly useful for bond-vectors acting as NMR relaxation probes in proteins, when their restricted local motion is analyzed with stochastic models featuring Wigner-function-made potentials. The relaxation probes could also be molecules adsorbed at surfaces, inserted into membranes, or interlocked within metal–organic frameworks.


INTRODUCTION
Typically, proteins exhibit internal mobility. Within their scope, various structural moieties, notably bond-vectors, move locally in the presence of spatial restrictions exerted by the immediate (internal) protein surroundings. These restrictions result from the anisotropic nature of the local structure. In their presence, the bond-vector orientation is distributed nonuniformly even in cases where the local motion is undetectable, while the local ordering can be measured. The pertinent probability density functions yield conformational entropy, S k , defined (in units of the Boltzmann constant, k) as S k = −∫ (P eq ln P eq )dΩ − ln∫ dΩ. 1 P eq = exp(−u)/Z is the normalized probability density, where Z is the partition function and u is the restricting local potential. The quantity of interest is the change in conformational entropy, ΔS k , between two protein states, entailed by a physical process. 1−3 Usually, the second term in the expression for S k cancels out in calculating ΔS k .
In some cases, the restricted local bond-vector motion is observable (following appropriate isotope labeling) with NMR relaxation. Stochastic models for NMR relaxation analysis feature explicit potentials, 4,5,7,8 which straightforwardly yield S k (refs 6, 10). While this study refers to bond-vectors in proteins in general, it connects with NMR relaxation through the subset of NMR-active bond-vectors. 11,12 Let us examine this connection.
The traditional method for the analysis of NMR relaxation in proteins is model-free (MF). 13 In the MF formalism, the local spatial restrictions are expressed in terms of the squared generalized order parameter, S 2 , rather than a potential function. Analytical expressions connecting S 2 with S k for very simple axial potentials were developed; 1,12 an empirical relation which does so for several simple axial potentials was also devised. 1 A dictionary for protein side-chain entropies derived from S 2 was established in ref 14. In that study, empirical relations connecting S 2 with S k were developed utilizing reference configurational entropies and order parameters determined with molecular dynamics (MD) simulations. Actual cases using this method appear in the literature. 15 In the context of ligand binding, Wand et al. devised the empirical relation called "model-independent entropy meter" that features adjustable coefficients, which project the experimentally measured methyl-related changes in motion across the entire protein and ligand. 12 Thus, within the scope of the MF conceptualizations, the S 2to-S k conversion features in some cases simple axial potentials.
In recent years, we developed the two-body coupled-rotator slowly relaxing local structure (SRLS) approach 16−18 for the analysis of NMR relaxation in proteins. 19−22 In SRLS, the local potential is expanded in the basis set of the real linear combinations of the Wigner rotation matrix elements, D 0K L (in brief, real Wigner functions). 16,19 In accordance with typical experimental data, the lowest even L terms, and in some cases the lowest odd L terms, have been kept, yielding u = − c 0 − D 01 1 ) for polar ordering. 23−26 Nonpolar (L = 2) ordering prevails when there is inversion symmetry with respect to the origin of the director (preferred probe orientation) frame as for rigid molecules dissolved in anisotropic media. 4,5 Polar (L = 1) ordering prevails when there is no inversion symmetry with respect to the origin of the director frame. This is the case for bond-vectors anchored at the protein 23,24 (or any other relaxation probe anchored at the entity that represents the local director). In view of the various approximations, admixtures are most appropriate. The fact that MF, and in many cases SRLS, have treated local ordering from the nonpolar (L = 2) perspective is due to the fact that the theories for proteins originate in theories for small molecules.
In principle, the two-term potentials depicted above can be enhanced in SRLS by adding terms to its expression. In practice, this is often hindered by limited experimental data. We pursued the idea of potential enhancement outside of the scope of SRLS 25,26 as follows. Linear combinations of real Wigner functions with L = 1−4 were created and optimized against the corresponding potential of mean force (POMF) obtained with MD simulations. Comparison between the bestfit Wigner function and the POMF indicated that the set of terms with L = 1−4 suffices for obtaining good agreement. Moreover, using such optimized potentials, new insights into the dimerization of the Rho GTPase binding domain of plexin-B1 (in brief, plexin-B1 RBD) were gained. 26 In future work, we plan to incorporate these potentials unchanged into SRLS datafitting schemes. This will improve the picture of structural dynamics to be obtained due to better potentials, and better characterization of this picture as additional parameters can now be determined with data fitting.
The POMFs are themselves restricting potentials that can yield conformational entropy. However, they are statistical functions and as such cannot be utilized in the development promoted in this study, which is based on explicit potentials. Thus, we have at hand explicit axial and rhombic, polar and nonpolar, fairly accurate Wigner-function-made local potentials. This constitutes a rich source for conformational entropy derivation. To optimize this process in terms of the suitability of the local potential and the accuracy of the pertinent conformational entropy, it is important to determine the relation between potential form, symmetry, parity, etc. and conformational entropy. We do this here for the L = 1 and 2 potentials depicted above. Correlation graphs are provided, and their utilization is illustrated with several applications. NMR relaxation analysis using SRLS directly can benefit from this study.
A theoretical summary is given in Section 2. Our results and their discussion are described in Section 3, and our conclusions appear in Section 4.

THEORETICAL BACKGROUND
2.1. Restricting Potentials. The orienting potential, U(Ω), associated with restricted rotational reorientation, is typically given by the expansion in Wigner rotation matrix elements, D 0K where the Euler angles, Ω, describe the orientation of the probe relative to the director, which is the direction of preferential orientation in the restricting surroundings. 4,5 Note that u and the coefficients featured by eq 1 are dimensionless. It is usually assumed that the director is uniaxial (see ref 17 for the introduction of biaxiality in a manner involving one additional variable angle). Consequently, the "quantum number" M in eq 1 is zero and Ω = (0, θ, φ). One has to ensure that the potential is real; this is achieved by expanding u(Ω) in the basis set of the real combinations of the Wigner rotation matrix elements (the real Wigner functions). Finally, the infinite expansion in eq 1 has to be truncated. Keeping only the lowest even L terms, one has 5 For c 0 2 > 0, the main ordering axis orients preferentially parallel to the director; this is termed parallel ordering. For c 0 2 < 0, the main ordering axis orients preferentially perpendicular to the director; this is termed perpendicular ordering. 4,5 Keeping only the lowest odd L terms, one has: 23,24 For c 0 1 > 0 (c 0 1 < 0), the primary polar axis is parallel to the +z (−z) axis of the local ordering frame. For c 1 1 > 0 (c 1 1 < 0), the primary polar axis is tilted in the +zx (−zx) plane of the local ordering frame. 4,23 Exploring the relationship between these (and further enhanced) potentials and the conformational entropy, S k , is very broad in scope. The connection with SRLS, 19−22 which applies to proteins in solution, has been delineated above. The SRLS limit where the protein motion is frozen is the microscopic-order-macroscopic-disorder (MOMD) approach, 27 which we developed for proteins in the solid state. 28−32 SRLS and MOMD were originally developed for electron spin resonance (ESR) applications in complex fluids and proteins. [16][17][18]27 In all of these theoretical approaches, the local potential is expressed in terms of real Wigner functions. The present study is relevant to all of them and, in general, to any established stochastic model for spin relaxation analysis. 4 The Journal of Physical Chemistry B pubs.acs.org/JPCB Article The normalized probability density function required to calculate these ensemble averages is given by where u(θ,φ) is the restricting potential.

Conformational
Entropy. The entropy divided by k, S k , is defined as 1 where P eq = exp(−u)/Z is the normalized probability density, Z is the partition function, and u is the restricting potential. The change in conformational entropy is given by 1 In this study, u is given by eqs 2 or 3 and ΔS k is calculated using eqs 6 and 7. The coefficients c 0 2 , c 2 2 , c 0 1 , and c 1 1 may be positive or negative.
Let us relate to the model-free treatment of eq 7. The local spatial restrictions are implicit in the squared generalized order parameter, S 2 , defined as 13 where ⟨···⟩ denotes ensemble average. The functions in eq 8 are the (complex) Wigner rotation matrix elements with L = 2 and K = −L,...,L. We have shown that 10 For wobble-in-a-cone and one-dimensional (1D) harmonic oscillator, analytical expressions connecting S 2 with S k were developed. 1,12 For several simple axial potentials, the following empirical expression was developed 1 The parameter A is adjusted to suit the individual potentials.  Figure 1a−c shows S k as a function of the coefficient c 0 2 > 0, the order parameter S 0 2 = ⟨D 00 2 ⟩, and the squared order parameter (S 0 2 ) 2 , respectively. A comparison with MF may be conducted given that both S and S 0 2 range from 0 to 1. Note that S 0 2 = 0 corresponds to c 0 2 = 0 and S 0 2 = 1 corresponds to c 0 2 → ∞. We use 0 ≤ c 0 2 ≤ 50; any number greater than approximately 20 is virtually infinity. Figure 1c shows S k as a function of (S 0 2 ) 2 . The conformational entropy, S k , was calculated according to eq 6; S 0 2 was calculated in terms of the potential u = −c 0 2 D 00 2 according to eq 4a. Figure 1 of ref 1 shows S k as a function of S 2 obtained using eq 10 for several simple potentials. All of the curves in As expected, in all three panels of Figure 1, the entropy decreases with increasing potential or ordering strength. In the region of interest for proteins, where 1 ≲ c 0 2 ≲ 10, the S k versus c 0 2 curve is substantially steeper, hence more sensitive, than the other two curves.   . As indicated, nonpolar (L = 2) local ordering may be parallel or perpendicular. We 19−22 and others 33−35 found that typically the main ordering axis at N−H sites in proteins is C α −C α . Recalling that the director is given by the equilibrium orientation of the N−H bond, it may be deduced that, within a good approximation, perpendicular ordering prevails at these sites. 20−22 In general, the sign of the coefficient, c 0 2 , in eq 2, obtained with data f itting, determines whether the local ordering is parallel or perpendicular. This information enters the expression for S k straightforwardly (eq 6). When MF is used, data fitting determines S 2 . For perpendicular ordering to enter S k , one has to use −S in expressions such as eq 10. This has not been done, with implications illustrated below. . For parallel ordering, one has 0 ≤ c 0 2 < ∞ and 0 < S 0 2 < 1; for perpendicular ordering, one has 0 > c 0 2 > −∞ and 0 ≥ S 0 2 ≥ −0.5. The S k patterns for c 0 2 < 0 and c 0 2 > 0 differ; so do the S k patterns for S 0 2 < 0 and S 0 2 > 0. Figure 2c shows S k as a function of (S 0 2 ) 2 . A second branch yielded by negative S 0 2 values is featured in the 0−1 range; this branch is missing in MF.
Let us focus on the simplest axial polar case. L = 1 potentials are treated in refs 23, 26. In ref 24, we analyzed the 15 N−H relaxation data from the third immunoglobulin binding domain of streptococcal protein G (GB3) with rhombic L = 1 or 2 potentials and found that the results differ. Importantly, we found that the process by which GB3 binds to its cognate Fab fragment has polar character. 24 Thus, potential parity is both influential and important. Figure 3 refers to the potential u = −c 0 1 D 00 1 = −c 0 1 cos θ. Figure 3a shows S k as a function of c 0 1 for −30 ≤ c 0 1 ≤ 30, Figure  3b shows S k as a function of S 0 1 for −1 ≤ S 0 1 ≤ 1, and Figure 3c shows S k as a function of (S 0 1 ) 2 . In Figure 3a,b S k is the same for positive and negative c 0 1 (S 0 1 ), i.e., for the primary polar axis pointing along +z and −z. 23 A single branch is featured by (S 0 1 ) 2 (Figure 3c). We extend the analysis by including in it order parameters and the minima of the L = 1 and 2 potentials. For L = 1, S 0 1 is the same for +c 0 1 (primary polar axis pointing along +z) and −c 0 1 (primary polar axis pointing along −z); for L = 2, S 0 2 is not the same for +c 0 2 (parallel ordering) and −c 0 2 (perpendicular ordering). Figure 4c shows S k as a function of c 0 1 (blue), superposed on the minimum of the u = −c 0 1 cos θ potential, denoted u min , as a function of c 0 1 (red). The corresponding scales are depicted on the left and right ordinates. As one would expect, primary polar axes pointing along +z or −z yield the same patterns. Figure 4d shows S k as a function of c 0 2 (blue), superposed on u min of u = −c 0 2 (1.5 cos θ 2 −0.5) as a function of c 0 2 (red). The corresponding scales are depicted on the left and right ordinates. Both patterns differ for parallel (c 0 2 > 0) and perpendicular (c 0 2 < 0) ordering. In particular, if the potential  The Journal of Physical Chemistry B pubs.acs.org/JPCB Article minimum is −a (at θ = 0°) for parallel ordering, it will be −a/2 (at θ = 90°) for perpendicular ordering. ) that yield the same conformational entropy. We call these curves S k isolines. Figure  5c,d shows the u min isolines for the L = 1 and 2 potentials, respectively. The color codes for the values of S k and u min are given on the right of each figure. In Figure 5a,b, intense orange corresponds to large entropy and intense blue corresponds to small entropy. In Figure 5c Figure 5a) and more monotonic for u min (Figure 5c). The S k patterns are more sensitive in the middle, and the u min patterns are more sensitive in the outer region. The situation is more complicated for L = 2, which is associated with asymmetric shapes of the S k and u min isoline patterns (Figure 5b,d). While +c 2 2 and −c 2 2 yield the same isoline patterns, +c 0 2 and −c 0 2 yield different isoline patterns. The S k patterns are more sensitive in the middle, and the u min patterns are more sensitive in the outer region, in a distinctive manner. Note that the high sensitivity of the S k isoline patterns ensures good certainty in S k . ) are defined in terms of (c 0 2 , c 2 2 ) (cf. Equations 2, 3 and 4a− c). The isolines of Figure 6 are much more dispersed in conformation space than the isolines of Figure 5. Good certainty in S k is expected for potentials of relatively great, and intermediate, strength, and relatively great, and intermediate, rhombicity, as the S k isolines vary most in these regions.
N−H bonds in well-structured regions of the protein conformation feature relatively strong and highly rhombic potentials. 20, 21 In this case, it is preferable to use the correlation graphs of Figure 6. C−CH 3 bonds in proteins feature relatively weak potentials. 20,21 In that case, it is preferable to use the correlation graphs of Figure 5 (see examples below). Figure 7a shows superposed S k and u min isolines as a function of the coefficients c 0 1 and c 1 1 of the L = 1 potential. The objective is to examine the correlation between S k and u min . One can recognize a one-to-one correspondence; its precise form is revealed by Figure 7c, where S k is depicted as a function of u min . Figure 7b shows superposed S k and u min isolines as a function of the coefficients c 0 2 and c 2 2 of the L = 2 potential. The relation between S k and u min is intricate. Indeed, Figure 7d shows that, in general, multiple S k values correspond to a given value of u min .
The utilization of the correlation graphs of Figures 1−6 is illustrated below.
3.3. Applications. 3.3.1. Example 1. Statistical potentials of mean force (POMFs) can be derived directly from MD trajectories. 25,26,36,37 Figure 8a,b shows images of two POMFs representing two protein states before and after a physical event. They belong to residue G73 of plexin-B1 RBD in monomer and dimer forms, 26 but we consider them representative of a general situation where the only information available consists of POMFs.  The estimated minima are (in units of kT) 8.4 and 7.8. One could use eq 7 to determine ΔS k . However, at this stage, it is not known whether the local ordering is parallel or perpendicular. Taking u = −c 0 2 D 00 2 as a reasonable approximation, and using conjointly the graphs of Figure 2a−c, it might be possible to distinguish between these two situations. Figure 2c is likely to be particularly useful in this context. With this information in hand, one could proceed effectively with detailed analysis, where rhombic symmetry is allowed for.
3.3.2. Example 2. 15 N relaxation of the major urinary protein I (MUP-I) and its complex with the pheromone 2-secbutyl-4,5-dihydrothiazol were studied with MF at 300 K in early work. 38 The authors of ref 38 found that pheromone binding brings about increase in conformational entropy. We studied this system with SRLS in the 283−308 K range using u = −c 0 2 D 00 2 and assuming parallel ordering, to find that below approximately 300 K, S k indeed increases, but above that temperature, it decreases, upon pheromone binding. 9 At 308 K, c , hence ΔS k . Thereby, the change in conformational entropy upon pheromone binding will be determined with enhanced certainty in the interesting temperature range.
3.3.3. Example 3. Table 1  ⟩ of the alanine and methionine methyl groups at 295 K (rows 3 and 4). 39 The coefficients ⟨c 0 2 ⟩ and ⟨c 2 2 ⟩ in Table 1 represent rhombic L = 2 potentials. SRLS calculations where the potentials have rhombic symmetry are considerably more time-consuming than SRLS calculations, where the potentials have axial symmetry. One can permute the axes of the local ordering frame so that, in the new frame, the symmetry of the potential is different. 5,40 The permuted coefficients, The following situation is envisioned for the methyl groups of a given protein designated for SRLS analysis. One selects representative methyl moieties and determines c 0 2 and c 2 2 with SRLS data fitting. Using Figure 5b, the corresponding isolines are identified. For small c 0 2 and small c 2 2 , typical of methyl moieties in proteins, 39 every S k isoline has two points with c 2 2 ≈ 0. In some cases, c 0 2 of such points will be similar to the c0 2 data of the representative residues. For residues with data similar to those of the representative residues (appropriate criteria will have to be specified), it will be useful to use in SRLS calculations ĉ0 2 and ĉ2 2 ≈ 0. The geometric information will have to be updated accordingly.
3.3.4. Example 4. Table 2 shows S 0 2 and S 2 2 obtained with SRLS analysis of the N−H bonds of residues Q2 and A26 of the third immunoglobulin binding domain of streptococcal protein G (GB3) using the rhombic L = 2 potential. 41 The   Figure 6a shows better sensitivity in the region under consideration than Figure 6b. This indicates that analyzing 15 N relaxation in compact proteins such as GB3 using the rhombic L = 1 potentials is likely to yield local potentials, hence pertinent order parameters and conformational entropy, which are determined with enhanced certainty. This is useful information for future work.
3.3.5. Future Prospects. It is of interest to compare for a given NMR relaxation probe dynamic structures associated with the same value of S k . This can be accomplished by the following strategy: 1 Analyze NMR relaxation data of a given probe with SRLS and determine the "experimental" c 0 2 and c 2 2 values; 2 calculate S k (eq 6) and use Figure 5b to determine the corresponding isoline; 3 select representative pairs of c 0 2 and c 2 2 belonging to this isoline; 4 use these c 0 2 and c 2 2 pairs unchanged in SRLS data fitting and determine the corresponding local motional rates and local geometry; 5 and compare the results of steps 1 and 4.
3.3.6. Comments. (1) Lately, pressure-dependent 42 and temperature-dependent 43 studies have been performed in the context of conformational entropy derivation. The results of such studies might be useful in a project where explicit SRLS potentials and statistical MD-derived POMFs improve one another within the scope of an iterative scheme. We contemplate devising such a scheme in future work. (2) We derive conformational entropy from restricted local motions. In  The Journal of Physical Chemistry B pubs.acs.org/JPCB Article the context of NMR relaxation, the pertinent restrictions are "observed" sources. 2 There exists a different approach pursued, e.g., in ref 44, where the entropy changes are derived using an "entropy meter". The latter is an expression comprising S 2 from observed sources as well as adjustable coefficients that "project the experimentally measured changes in motion across the entire protein and ligand". The projected changes are "unobserved" sources. Such contributions are outside the scope of our study.

CONCLUSIONS
The local potentials, u, at the site of mobile bond-vectors in proteins have been expressed in terms of the real linear combinations of the Wigner rotation matrix elements, D 0K L (in brief, real Wigner functions), with L = 1 or 2. From them, the conformational entropy, S k , has been derived. To determine the effect of the symmetry (axial or rhombic) and L-parity of the local potential on the associated conformational entropy, correlation graphs between S k and the coefficients of u, as well as between S k and the order parameters defined in terms of u, have been created. The S k patterns obtained are highly specific and exhibit distinctive parameter-range-dependent sensitivity. This lays the groundwork for devising potentials for the determination of S k that best suit given physical circumstances.
NMR relaxation analysis has been invoked as a physical method that can profit substantially from these results. So can any physical method where the local restrictions are expressed in terms of real Wigner functions. We