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 on Sk. 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, defined (in units of the Boltzmann constant, k) as S = −∫(Peq ln Peq)dΩ − ln∫dΩ.[1]Peq = 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, between two protein states, entailed by a physical process.[1−3] Usually, the second term in the expression for S cancels out in calculating ΔS.

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 (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, S2, rather than a potential function. Analytical expressions connecting S2 with S 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 S2 was established in ref (14). In that study, empirical relations connecting S2 with S 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 S2-to-S 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, D0 (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 = – c02D002 – c22 (D022 + D0–22) for nonpolar ordering[17,18,20−22] and u = −c01D001 – c11 (D0–11 – D011) 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 best-fit 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 data-fitting 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 . Our results and their discussion are described in Section , and our conclusions appear in Section .

Theoretical Background

Restricting Potentials

The orienting potential, U(Ω), associated with restricted rotational reorientation, is typically given by the expansion in Wigner rotation matrix elements, D0(Ω)[4,5]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 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 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 has to be truncated. Keeping only the lowest even L terms, one has[5]For c02 > 0, the main ordering axis orients preferentially parallel to the director; this is termed parallel ordering. For c02 < 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 c01 > 0 (c01 < 0), the primary polar axis is parallel to the +z (−z) axis of the local ordering frame. For c11 > 0 (c11 < 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, 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.[16−18,27] 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.[4,5]

Parameters of Interest Defined in Terms of the Restricting Potentials

Order Parameters

Order parameters are ensemble averages of real Wigner functions defined in terms of restricting potentials. Here, we are using the order parameters[4,23]The normalized probability density function required to calculate these ensemble averages is given bywhere u(θ,φ) is the restricting potential.

Conformational Entropy

The entropy divided by k, S, is defined as[1]where Peq = 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 or 3 and ΔS is calculated using eqs and 7. The coefficients c02, c22, c01, and c11 may be positive or negative.

Let us relate to the model-free treatment of eq . The local spatial restrictions are implicit in the squared generalized order parameter, S2, defined as[13]where ⟨···⟩ denotes ensemble average. The functions in eq 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 S2 with S 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. This equation was obtained with a different parameterization in ref (14). Only positive values of S (which, as pointed out above, correspond to parallel ordering) are considered. Empirical relations for methyl groups were developed within the scope of a dictionary-type framework in ref (14) (see above). A comprehensive empirical relation was developed in ref (12) (see above).

Results and Discussion

Axial Potentials

The first axial (K = 0) even L term in the Wigner function expansion yields the potential u = −c02D002 = −c02 (1.5 cos2 θ – 0.5). Figure a–c shows S as a function of the coefficient c02 > 0, the order parameter S02 = ⟨D002⟩, and the squared order parameter (S02)2, respectively. A comparison with MF may be conducted given that both S and S02 range from 0 to 1. Note that S02 = 0 corresponds to c02 = 0 and S02 = 1 corresponds to c02 → ∞. We use 0 ≤ c02 ≤ 50; any number greater than approximately 20 is virtually infinity.

Figure 1

Conformational entropy, S, as a function of (a) the coefficient, c02, of the potential u = −c02D002, (b) the order parameter, S02 = ⟨D002⟩, and (c) the squared order parameter, (S02)2. c02 ranges from 0 to 50; S02 and (S02)2 range from 0 to 1.

Figure c shows S as a function of (S02)2. The conformational entropy, S, was calculated according to eq ; S02 was calculated in terms of the potential u = −c02D002 according to eq . Figure 1 of ref (1) shows S as a function of S2 obtained using eq for several simple potentials. All of the curves in Figure 1 of ref (1) agree qualitatively with the curve in Figure c; none agrees with it quantitatively, although many of the simple potentials considered in ref (1) are limiting cases of u = −c02D002.

As expected, in all three panels of Figure , the entropy decreases with increasing potential or ordering strength. In the region of interest for proteins, where 1 ≲ c02 ≲ 10, the S versus c02 curve is substantially steeper, hence more sensitive, than the other two curves.

Let us focus on c02. 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 Cα–Cα. 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, c02, in eq , obtained with data fitting, determines whether the local ordering is parallel or perpendicular. This information enters the expression for S straightforwardly (eq ). When MF is used, data fitting determines S2. For perpendicular ordering to enter S, one has to use −S in expressions such as eq . This has not been done, with implications illustrated below.

Figure a–c is analogous to Figure a–c, except that now both parallel and perpendicular orderings are featured by considering the full parameter range of c02. For parallel ordering, one has 0 ≤ c02 < ∞ and 0 < S02 < 1; for perpendicular ordering, one has 0 > c02 > −∞ and 0 ≥ S02 ≥ −0.5. The S patterns for c02 < 0 and c02 > 0 differ; so do the S patterns for S02 < 0 and S02 > 0. Figure c shows S as a function of (S02)2. A second branch yielded by negative S02 values is featured in the 0–1 range; this branch is missing in MF.

Figure 2

Conformational entropy, S, as a function of (a) the coefficient, c02, of the potential u = −c02D002, (b) the order parameter, S02 = ⟨D002⟩, and (c) the squared order parameter, (S02)2. c02 ranges from −20 to 20, S02 ranges from −0.5 to 1.0, and (S02)2 ranges from 0 to 1. The long branch in Figure c is associated with c02 > 0 and S02 > 0; the short branch is associated with c02 < 0 and S02 < 0.

Let us focus on the simplest axial polar case. L = 1 potentials are treated in refs (23, 26). In ref (24), we analyzed the 15N–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 refers to the potential u = −c01D001 = −c01 cos θ. Figure a shows S as a function of c01 for −30 ≤ c01 ≤ 30, Figure b shows S as a function of S01 for −1 ≤ S01 ≤ 1, and Figure c shows S as a function of (S01)2. In Figure a,b S is the same for positive and negative c01 (S01), i.e., for the primary polar axis pointing along +z and −z.[23] A single branch is featured by (S01)2 (Figure c).

Figure 3

Conformational entropy, S, as a function of (a) the coefficient, c01, of the potential u = −c01D001, (b) the order parameter, S01 = ⟨D001⟩, and (c) the squared order parameter, (S01)2. c01 ranges from −30 to 30, S01 from −1.0 to 1.0, and (S02)2 from 0 to 1.

We extend the analysis by including in it order parameters and the minima of the L = 1 and 2 potentials. Figure a shows S01= ⟨D001⟩ as a function of c01 for −30 ≤ c01 ≤ 30, and Figure b shows S02 = ⟨D002⟩ as a function of c02 for −30 ≤ c02 ≤ 30. For L = 1, S01 is the same for +c01 (primary polar axis pointing along +z) and −c01 (primary polar axis pointing along −z); for L = 2, S02 is not the same for +c02 (parallel ordering) and −c02 (perpendicular ordering).

Figure 4

Order parameter, S01, as a function of the coefficient, c01, of the potential u = −c01D001 (a). Order parameter, S02, as a function of the potential coefficient, c02, of the potential u = −c02D002 (b). Conformational entropy, S (blue), and the minimum, umin, of u = −c01D001 (red), as a function of c01 (c). Conformational entropy, S (blue), and the minimum, umin, of u = −c02D002 (red), as a function of c02 (d).

Figure c shows S as a function of c01 (blue), superposed on the minimum of the u = −c01 cos θ potential, denoted umin, as a function of c01 (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 d shows S as a function of c02 (blue), superposed on umin of u = −c02 (1.5 cos θ2–0.5) as a function of c02 (red). The corresponding scales are depicted on the left and right ordinates. Both patterns differ for parallel (c02 > 0) and perpendicular (c02 < 0) ordering. In particular, if the potential minimum is −a (at θ = 0°) for parallel ordering, it will be −a/2 (at θ = 90°) for perpendicular ordering.

Rhombic Potentials

Equations and 3 show the functional forms of the rhombic L = 2 and 1 potentials, respectively. Figure a,b shows S as a function of the potential coefficients c01 and c11 (c02 and c22). Figure c,d shows umin as a function of the coefficients c01 and c11 (c02 and c22). In all of the Figure simulations, and in the simulations of Figures and 7, 1681 data points were used; the same results were obtained with a larger number of data points.

Figure 5

Conformational entropy, S, as a function of the coefficients, c01 and c11, of the potential u = −c01D001 – c11 (D0-11 – D011) (a). Conformational entropy, S, as a function of the coefficients, c02 and c22, of the potential u = −c02D002 – c22 (D022 + D0-22) (b). The minimum, umin, of u = −c01D001 – c11 (D0-11 – D011) as a function of c01 and c11 (c). The minimum, umin, of u = −c02D002 – c22 (D022 + D0-22) as a function of c02 and c22 (d). Color codes are on the right of each panel. A total of 1681 data points were used in generating each panel of this figure.

Figure 6

Conformational entropy, S, as a function of the order parameters, S01 and S11, defined in terms of the potential u = −c01D001 – c11 (D0-11 – D011) (a). Conformational entropy, S, as a function of the order parameters, S02 and S22, defined in terms of the potential u = −c02D002 – c22 (D022 + D0-22) (b). umin of u = −c01D001 – c11 (D0-11 – D011) as a function of S01 and S11 (c). umin of u = −c02D002 – c22 (D022 + D0-22) as a function of S02 and S22 (d). Color codes and number of data points as in Figure .

Figure 7

(a) Superposed conformational entropy, S (blue), and potential minimum, umin (red), as a function of the coefficients, c01 and c11, of the potential u = −c01D001 – c11 (D0-11 – D011). (b) S of (a) as a function of umin of (a). (c) Superposed conformational entropy, S (blue), and potential minimum, umin (red), as a function of the coefficients, c02 and c22, of the potential u = −c02D002 – c22 (D022 + D0-22). (d) S of (c) as a function of umin of (c). Number of data points as in Figure .

The curves depicted in Figure a represent the group of points with coordinates (c01, c11) that yield the same conformational entropy, S; those depicted in Figure b represent the group of points with coordinates (c02, c22) that yield the same conformational entropy. We call these curves S isolines. Figure c,d shows the umin isolines for the L = 1 and 2 potentials, respectively. The color codes for the values of S and umin are given on the right of each figure. In Figure a,b, intense orange corresponds to large entropy and intense blue corresponds to small entropy. In Figure c,d, intense orange corresponds to shallow potentials and intense blue corresponds to deep potentials.

L = 1 potentials (Figure a,c), which are shallow and nearly axial (small |c01| and |c11|), yield large entropy; those which are deep and highly rhombic (large |c01| and |c11|) yield small entropy. In-between the changes are less monotonic for S (Figure a) and more monotonic for umin (Figure c). The S patterns are more sensitive in the middle, and the umin 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 and umin isoline patterns (Figure b,d). While +c22 and −c22 yield the same isoline patterns, +c02 and −c02 yield different isoline patterns. The S patterns are more sensitive in the middle, and the umin patterns are more sensitive in the outer region, in a distinctive manner. Note that the high sensitivity of the S isoline patterns ensures good certainty in S.

Figure a–d is analogous to Figure a–d, with the coordinates being order parameters instead of potential coefficients. (S01, S11) are defined in terms of (c01, c11), and (S02, S22) are defined in terms of (c02, c22) (cf. Equations , 3 and 4a–c). The isolines of Figure are much more dispersed in conformation space than the isolines of Figure . Good certainty in S is expected for potentials of relatively great, and intermediate, strength, and relatively great, and intermediate, rhombicity, as the S 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 . C–CH3 bonds in proteins feature relatively weak potentials.[20,21] In that case, it is preferable to use the correlation graphs of Figure (see examples below).

Figure a shows superposed S and umin isolines as a function of the coefficients c01 and c11 of the L = 1 potential. The objective is to examine the correlation between S and umin. One can recognize a one-to-one correspondence; its precise form is revealed by Figure c, where S is depicted as a function of umin. Figure b shows superposed S and umin isolines as a function of the coefficients c02 and c22 of the L = 2 potential. The relation between S and umin is intricate. Indeed, Figure d shows that, in general, multiple S values correspond to a given value of umin.

The utilization of the correlation graphs of Figures –6 is illustrated below.

Applications

Example 1

Statistical potentials of mean force (POMFs) can be derived directly from MD trajectories.[25,26,36,37]Figure a,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.

Figure 8

MD-derived potentials of mean force for the N–H bonds of residue G73 of plexin-B1 RBD in monomer (Figure a) and dimer (Figure b) forms.[9,36,37] The minima of these potentials (in units of kT) are 8.4 and 7.8, respectively.

The estimated minima are (in units of kT) 8.4 and 7.8. One could use eq to determine ΔS. However, at this stage, it is not known whether the local ordering is parallel or perpendicular. Taking u = −c02D002 as a reasonable approximation, and using conjointly the graphs of Figure a–c, it might be possible to distinguish between these two situations. Figure c 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.

Example 2

15N relaxation of the major urinary protein I (MUP-I) and its complex with the pheromone 2-sec-butyl-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 = −c02D002 and assuming parallel ordering, to find that below approximately 300 K, S indeed increases, but above that temperature, it decreases, upon pheromone binding.[9]

At 308 K, c02 is on the order of 15–17 in both forms of MUP-I. At 283 K, the majority of the c02 values are on the order of 25 in the free form and between 15 and 20 in the bound form (Figure 7 of ref (9)). ΔS is small at both temperatures. The S versus positive c02 curve in Figure a shows that the dependence of S on c02 is nearly linear for large c02 and much steeper for smaller c02. Therefore decreasing the pressure at 283 K would lower c02 differentially increasing Δc02, hence ΔS. Thereby, the change in conformational entropy upon pheromone binding will be determined with enhanced certainty in the interesting temperature range.

Example 3

Table shows the average values of the potential coefficients ⟨c02⟩ and ⟨c22⟩ for all of the methyl moieties of the complex of Ca2+–calmodulin with the peptide smMLCKp at the temperatures depicted (rows 1 and 2). It also shows ⟨c02⟩ and ⟨c22⟩ of the alanine and methionine methyl groups at 295 K (rows 3 and 4).[39]

Table 1

Average Potential Coefficients, ⟨c02⟩ and ⟨c22⟩, of All of the Methyl Groups of the Complex of Ca2+–Calmodulin with the Peptide smMLCLp at 288 and 308 K (Rows 1 and 2), and ⟨c02⟩ and ⟨c22⟩ of Alanine (A) and Methionine (M) at 295 K (Rows 3 and 4)a

	T, K	⟨c02⟩	⟨c22⟩	Sk
1	288	0.92	–0.68	1.69
2	308	0.39	–0.74	1.77
3	295	0.22	–0.98	1.72
4	295	0.65	–0.50	1.76

Reproduced with permission from ref (39). Copyright 2011 of the American Chemical Society.

The coefficients ⟨c02⟩ and ⟨c22⟩ in Table 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, ĉ02 and ĉ22, are given by[40]The following situation is envisioned for the methyl groups of a given protein designated for SRLS analysis. One selects representative methyl moieties and determines c02 and c22 with SRLS data fitting. Using Figure b, the corresponding isolines are identified. For small c02 and small c22, typical of methyl moieties in proteins,[39] every S isoline has two points with c22 ≈ 0. In some cases, c02 of such points will be similar to the ĉ02 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 ĉ02 and ĉ22 ≈ 0. The geometric information will have to be updated accordingly.

Example 4

Table shows S02 and S22 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]

Table 2

S02 and S22 of Residues Q2 and A26 of GB3 Obtained with SRLS Analysis of 15N Relaxation.[41]S Derived from the c02 and c22 Values That Yielded These S02 and S22 Values (eqs , 4a, and 4c)a

1	2	3	4	5
residue	structural element	S02	S22	Sk
Q2	β1 strand	–0.49	1.08	–0.37
A26	α helix	–0.42	1.13	–0.70

Reproduced with permission from ref (41). Copyright 2012 of the American Chemical Society.

The points (S02, S22) with values (−0.49, 1.08) and (−0.42, 1.13) are located in the upper left corner of Figure b, as they represent strong perpendicular ordering and large rhombicity. Figure b refers to the rhombic L = 2 potential, whereas Figure a refers to the rhombic L = 1 potential. Figure a shows better sensitivity in the region under consideration than Figure b. This indicates that analyzing 15N 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.

Future Prospects

It is of interest to compare for a given NMR relaxation probe dynamic structures associated with the same value of S. This can be accomplished by the following strategy:[1] Analyze NMR relaxation data of a given probe with SRLS and determine the “experimental” c02 and c22 values;[2] calculate S (eq ) and use Figure b to determine the corresponding isoline;[3] select representative pairs of c02 and c22 belonging to this isoline;[4] use these c02 and c22 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.

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 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 S2 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, D0 (in brief, real Wigner functions), with L = 1 or 2. From them, the conformational entropy, S, 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 and the coefficients of u, as well as between Sk and the order parameters defined in terms of u, have been created. The S patterns obtained are highly specific and exhibit distinctive parameter-range-dependent sensitivity. This lays the groundwork for devising potentials for the determination of S 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 use here the amide bond and the methyl moiety of proteins as examples of NMR relaxation probes. Additional examples are molecular moieties adsorbed as surfaces, embedded in membranes, or interlocked in metal–organic frameworks.