The thermodynamic potential of a solvated molecule

In this subsection we recall the basic equations for the thermodynamic potential of a solvated molecule, following the book of McQuarrie [31] and the review of Gilson et al. [8] and Roux and Simonson [32]. We will use in the following molecular instead of molar quantities.

The standard chemical potential is expressed in terms of configurational integrals: (1) with , where k_B is the Boltzmann constant and T is the temperature, h is Planck’s constant, the mass of atom i of the solute and solute (and solvent, in the following) coordinates are indicated by (and , respectively). P⁰ is the standard pressure, the partial molar volume of the solute, C⁰ is the standard 1 M concentration and N_Av is Avogadro’s number.

The solvation potential of mean force is defined by the configuration integral: (2)

In what follows we will not consider the kinetic energy terms in the partition functions because they will cancel in processes involving isolated proteins, and the term will be neglected because it is typically small for proteins and its variations upon conformational changes are negligible. For instance upon folding protein volume changes by less than 0.5 percent at standard pressure [33], which amounts to a mean free energy variation of just about 20 J/mol for a protein of 50 kDa. The picture would obviously change completely at high pressures.

We may use the probability distribution (3) to express the entropy as: (4) The first term is a classical conformational entropy term for the solute.

The second term is the entropy due to the solute-solvent interactions.

Note that the latter term cannot be simply estimated from the known or modeled dependence of on the temperature because it depends also on the conformational space available which depends in turn on the temperature.

In the present work we are interested in the characterization of the first term, i.e. the entropy term arising from the conformational freedom of the protein.

Calculation of entropy from samples in conformational space

We follow here the approach of Singh et al. [13] developed and applied by Hnizdo, Gilson and coworkers [14, 15, 34, 35]. The approach considers n configurational samples for a random s-dimensional variable distributed according to the probability density p(x₁, x₂, …, x_s) (or for short). Given an s-dimensional sphere of radius r centered at a reasonable estimate for the local probability density () may be expressed by counting how many samples are found inside the volume of the sphere V_r. If this number is k, then is reasonably defined by the equation: (5) Note that in general V_r depends on the metric defined on the space which is considered. For euclidean distances the volume is given by: (6) where Γ() is the Γ function.

With an estimate for the density the entropy can be estimated (we omit here the k_B factor for generality) as the average of : (7) For each sample i we take a sphere of radius r_i and count the number of samples k_i found in the sphere. By substituting Eqs 5 and 6 into the above equation we can reasonably approximate entropy by: (8) In the approach described by Singh et al. [13] the radius of the sphere centered about each conformational sample is taken as the distance to the k-th nearest neighbor and k_i is therefore the same for all samples. The above heuristic equations are corrected as to provide an unbiased estimator of the entropy for the probability distribution : (9) In the above equation R_i,k is the distance of the i-th sample from its k-th nearest neighbor, Γ() is, as before, the Γ function, γ is Euler’s constant (0.5772…) and L_k−1 is defined as: L₀ = 0, .

It is easy to recognize the similarity of the heuristic and exact equation.

Note that R_i,k has a unit of measure and therefore the numerical value of depends on the latter unit. This is consistent with the fact that in the equation for the entropy S = −k_B∫plog(p) the probability density p has units of measure.

To illustrate the symbols in the above equation consider a long molecular dynamics trajectory for a small molecule possessing only two rotatable bonds. The i-th (out of 1000) sample of the conformational space could be represented by the 2 torsional angles describing its conformation (assuming rigid bonds and angles) taken from the i-th snapshot. In this case the dimensionality of the system s would be 2 and the number of samples n would be 1000.

The distance d of each sample from all other samples would be computed according to the metric chosen (for torsional angles this is described below), and ranked, e.g. for the i-th snapshot R_i,k would be then the k-th shortest computed distance. For each choice k we could use Eq (9) to estimate the configurational entropy of the molecule. In the following, unless otherwise stated, we will use k = 1, i.e. for each sample we consider the distance from its first nearest neighbor.

Limitations in the sampling capabilities of molecular dynamics simulations

A caution word is due about the possibility of sampling the conformational space of a molecule by molecular dynamics simulations. Indeed typical simulation times extend up to a hundred nanoseconds. It is likely that on this time scale sampling of the conformational space will not be complete. This consideration will apply to the approach used in this work as well as to other approaches found in the literature.

When comparing the entropy of different macrostates of the same molecule the effect of neglected accessible conformational space could be greatly reduced if the missing portion of sampled conformations is similar in the two simulations.

We will not address the issue here in detail as it affects similarly all methods that estimate conformational entropy from molecular dynamics simulations snapshots. For the shortest simulations we will only check convergence by considering the first and the last half of the simulations and compare the results obtained from the two subsets of snapshots. Even such test does not guarantee that other conformations will not be accessible in longer simulation times.

The main interest here however is a proof of principle and to show that even from standard simulations it is possible to get valuable information about conformational entropy.

Calculation of the conformational entropy from molecular dynamics simulations

A widespread approach to compute the conformational entropy of a protein uses semi-classical quasi-harmonic analysis [36–41] or, in a refined approach, fitting probability distributions (see e.g. [42, 43]). Here we assume that the largest changes in the conformational entropy of a protein in most interesting biological processes arise from energetic restraints in the torsional freedom of backbone and sidechain groups. For this reason we consider torsional angles as variables and we neglect bond lengths and bond angles. The latter approximation amounts to assuming that the torsional degrees of freedom are decoupled from bond and angle degrees of freedom, which is mostly a good approximation for proteins, and that the vibrational partition function of bonds and angles is not significantly changed by conformational changes [38].

The latter observation might be system dependent. To give an estimate of the quality of this approximation, we considered the particular case of the residue undergoing the largest changes in conformational freedom between the holo/active and apo/inactive EPAC simulations described later. We computed the entropy using the method of Schlitter [40] as implemented in the software Carma [44] from molecular dynamics trajectories fixing or not the bond lengths and angles to their equilibrium values. In this way we were able to assess the bond and angle relative contributions to the entropy. The change in entropy between holo/active and apo/inactive, after fitting the residue to the first snapshot, was 0.264 kJ/(mol K) for free covalent geometry and 0.279 kJ/(mol K) for fixed covalent geometry. The effect appears therefore rather limited, compared to the overall computed entropy or the effect of the fitting reference chosen. This is in line with the observed changes in bond length and angle variances (about 6% with different signs). Note that the computed entropy change appears artifactually very large, possibly as a consequence of the adopted harmonic oscillator model.

We consider only the torsional degrees of freedom involving heavy atoms or polar hydrogens of serine, threonine, tyrosine, whereas proline, aromatic rings, amide and guanidinium moieties are taken as rigid. Other terminal hydrogens that could define torsional angles, but possess ternary symmetry, i.e. methyl and amino groups are considered free to rotate (or more generally we assume that their conformational distributions are not affected by the processes considered here) and therefore the contribution to the entropy of any conformational change will be zero for these moieties. For torsional angles distances we use the distance d₁ (d for simplicity here) studied in the work of Hnizdo and coworkers [20], which amounts to the circular distance in the torsional angle space. For two angular vectors and of n angles each, and with periodicities Θ_i the distance is defined as: (10) with θ_1i and θ_2i reported in the interval It must be noted that on changing from cartesian to BAT (bond, angle, torsion) coordinates the Jacobian of the transformation enters the configurational integral [8, 45]. Under the assumption of rigid bond and angle geometry this extra term cancels out between reference (or initial) and final state of the system. The proof of independence of the Jacobian from the configuration has been given by Go and Scheraga [46] cited in ref. [45] for a linear chain. The proof can be extended to a branched chain and for an arbitray atom taken as the reference point for global translation. For this reason it will not be further considered here.

Nearest neighbor estimate of entropy is superior to classical harmonic analysis

It might be questioned whether the approach used here is needed at all when other methods are widely used. Most widely used methods are based on the quasi-harmonic approximation. The limitations and remedies to the quasi-harmonic approximation have been thoroughly described by McCammon and coworkers [42, 43] and Numata et al. [18]. We will not deal with this subject here but just provide a simple example demonstrating the issue.

If a system freely moving in one-dimension is considered and the range of movement is restricted, the quasi-harmonic approximation will give the correct change of entropy upon restriction. Indeed the square root of the variance for a uniform distribution is proportional to the range of the distribution itself. However it is sufficient to consider a toy model where we have a single freely rotatable bond (torsion angle θ) with three atoms fixed and the end atom moving, to see that the quasi-harmonic approximation breaks down.

We consider the unrestricted rotation as the reference state (probability density and a restricted rotational macrostate where the rotational angle is freely varying between θ₁ and θ₂ with Δθ = θ₂ − θ₁. The entropies of the two probability distributions are: and with the difference in entropy from the free to the restricted macrostate being When we compute (classical) entropies using the harmonic approximation we must compute the variance-covariance matrix for two coordinates. The analysis will result in two harmonic oscillators whose classical entropy change from free to restricted macrostate is overestimating the true entropy for all choices of Δθ. In particular for a reasonable restriction to an interval of Δθ = π/3 the entropy is overestimated roughly by a factor 2, whereas it is estimated with an average root mean square error of 5 percent by the nearest neighbor method with just one hundred samples.

Calculation of unfolded proteins conformational entropy from database structures

Crystallographic structures of proteins are generally the most representative conformations adopted by proteins in their range of stability, and for this reason are considered a good representation of the native protein ensemble. However, the conformational preferences of amino acids observed experimentally in the folded structure are not excluded from the conformational diversity of the set of partially folded and unfolded conformations that proteins can adopt. Indeed it is reasonable to assume that a reversible process like folding (at least for small proteins) simply freezes a set of amino acid conformations which can be adopted in the unfolded protein ensemble.

This principle has driven the refinement of available forcefields [47–49] and received an important validation from long time molecular dynamics simulations where the superiority of forcefield refined against experimental torsion angles distributions, and further refined afterwards, were able not only to fold small protein domains, but also to reproduce their folding rates [49, 50].

Based on the above considerations, we collected all possible conformations of all residues not directly involved in secondary structures of proteins, thus avoiding artifacts due to correlations, in the culled pdb dataset (3600 proteins) [51], as they provide samples of the conformational space available to aminoacids in their unfolded state, averaged over sequence details. This procedure assumes the presence of an identical interaction between the neighboring residues (i.e. the rest of the chain) and the unfolded and/or the database samples.

Therefore additional correlations between aminoacids in the unfolded state are not considered.

It should be noted that obtaining the same information from molecular dynamics simulations would not be easy because of the presumably largely asymmetric simulation box.

Obviously the accessible conformational space depends on the temperature, but it is reasonable to expect that this dependence is limited for stable proteins in their stability range. Indeed we can consider that for each torsional angle there is a number of preferred values (rotamers) and that the range of values explored about the most frequent ones increases with temperature. This situation is very similar to that of the harmonic oscillator. For a harmonic oscillator with frequency ν the vibrational partition function q_vib, the corresponding entropy S_vib and its derivative with respect to temperature, in the high temperature limit are given by: (11) (12) (13) If we approximate the effect of temperature on the conformational landscape of unfolded proteins with that of a collection of N harmonic oscillators, where N is the number of torsional degrees of freedom we obtain: (14) We may therefore expect that the dependence of the computed entropy on the temperature would be rather mild. We consider a typical range of temperatures of 30 K around 300 K, where most proteins are stable. To avoid misunderstanding we remark that here we are considering the effect of temperature on the variability of each single rotamer, i.e. about a well defined torsional energy minimum, and not different rotamers which are already sampled in the dataset.

The expected change in the entropy (per oscillator) corresponding to changes in distribution about the minimum energy rotamers due to temperature are limited by which appears negligible compared to the overall entropy associated with the loss of conformational freedom.

All database proteins have been processed adding hydrogens by the program Reduce [52]. Torsional angles have been computed using the program Molmol [53]. Only rotatable torsional angles involving four heavy atoms or three heavy atoms and a polar hydrogen have been considered.

For polar hydrogens only residues involved in hydrogen bonds have been considered in order to avoid large artifactual entropy contributions from the preference in trans conformation for the program Reduce in the absence of favorable interactions. This choice was adopted for tyrosines, serines, threonines and cysteines. Cysteines involved or not in disulfide bridges were considered separately for analysis.

Distances in the torsional angles space are computed by taking euclidean distances in the periodic boundary n-dimensional angular space. In order to reduce the large conformational space, torsion angles involving different aminoacids have been considered as independent and therefore entropies are computed on a residue basis. The entropy corresponding to each aminoacid type, whose conformation is defined by the n–dimensional vector of torsion angles θ, is calculated with reference to the uniform distribution in the angular space, i.e.: (15) (16) where Θ_i is the periodicity of the i-th torsional angle, typically 2π.

Note that the volume of a n-dimensional sphere enters Eq (9) [34] and therefore suitable corrections to the equation should be applied when considering distances larger than half of the minimum periodicity of the torsional angles. This was not the case for our analysis because of a sufficiently large number of samples, leading to short nearest neighbor distances. Exact formulae have been reported by Hnizdo and coworkers [20].

The conformational entropy of folding

Wildtype β2-microglobulin was simulated for 30 ns, after equilibration at 320 K as previously described [54].

The entropy for each set of conformations was computed using Eq (9) with reference to the uniform torsion angle distribution exactly as in the previous section, taking 300 snapshots from the 30 ns molecular dynamics simulation. Then, the reference entropy computed from residues in the database representing unfolded amino acids was subtracted from the computed entropy, and the difference was taken as the conformational entropy of folding.

The conformational entropy of binding

The calculation of the binding free energy of association of biomolecules is very complex due to the large number of interacting bodies. In order to understand and possibly design novel interactions the free energy of binding is often approximated by the sum of different contributions, under the assumption that they can be decoupled. A derivation of this approach and an exhaustive discussion of its limitations has been given by Gilson et al. [8]. Typically the attention is focused on solute degrees of freedom and solvent degrees of freedom are taken into account using the potential of mean force [32]. Evaluation of the entropy loss upon interaction is not straightfoward because the potential of mean force depends implicitly on the temperature.

We take the derivative of the Gibbs’ free energy: (17) where: (18) The choice of coordinates must allow easy calculation of the probability distribution p(). In this respect, following Gilson et al. [8], we choose the cartesian coordinates and of a reference atom on molecules A and B, respectively and other two atoms, assumed to be rigidly positioned with respect to the formers, on each molecule, to define the overall rotation state of the molecule.

We use as a criterion that the choice of the atoms must be the most insensitive with respect to dynamics in the bound state. This is done in order to minimize the correlation of these coordinates with other degrees of freedom, and therefore to decouple them from internal degrees of freedom.

For the complex we consider the coordinate system defined for molecule A as the reference one and we define the position of molecule B relative to , i. e. the position of molecule B is defined by . The entropic contribution due to loss in rototranslational freedom is computed considering that all positions and orientations are still possible for the complex whereas the position and the orientation of molecule B is restricted, with respect to the position and orientation of the reference system of coordinates of molecule A.

Let us now denote by ξ_A and ξ_B the overall rotational degrees of freedom and by and the internal degrees of freedom.

With this notation the entropy of complex formation may be then written as: (19) (20) (21) (22) (23) (24) (25) where the integrals over the bound state require a definition of the bound state, which is typically easy to attain if the bound state has a deep energy minimimum. Otherwise it may be chosen as to adhere to experimental signal distinguishing bound from free molecules.

The first term of the above equation is difficult to treat and it has been traditionally estimated assuming that rotational and translational degrees of freedom are independent. The latter assumption seems however poorly grounded and mostly motivated by the difficulty of treating the two degrees of freedom together. We will follow however the same approach here for simplicity.

Note that however, in order to use a distance based estimation of the entropy, a distance between rotation states must be defined. Rotation states are denoted equivalently by quaternions or 3 × 3 rotation matrices. There are several metrics which can be defined in the space of rotations [55]. It must be noted that Eq (9) assumes an euclidean distance in a cartesian space and therefore the number of points of randomly distributed n-dimensional variables within an n-dimensional distance d increases as .

Once a metric is defined we should compute the volume in rotation space corresponding to each distance and substitute it in the original Eq (9).

If we consider the representation of rotations which specifies the polar and azimuthal angles ϕ, ψ of the rotation axis and the rotation angle θ, the probability measure for a uniform distribution in rotation space is that with the axis of rotation uniformly distributed over the solid angle 4π and the rotation angle θ distributed in the range [0, π] with probability density [56]: (26) The metric we have used defines the distance d between two rotations described by 3 × 3 matrices R₁ and R₂ as: (27) where Tr is the trace operator, is the angle of rotation about the rotation axis for the composite rotation .

With this metric the volume in rotation space is: (28) Note that here we considered rotations in a range of π and not 2π to avoid double counting of rotations, because a rotation by an angle θ about the vector is the same as the rotation by an angle −θ about the vector .

Comparison with other methods

Entropies were also calculated according to other methods to provide comparison. A simple histogram method was applied where the range of the variables is divided in bins and the probability distribution is approximated as a piece-wise constant function on the bins. The constant value of the approximated probability distribution within bin i is the ratio of the number of counts n_i in the bin over the total number of counts n_c, i. e. . The entropy is computed as with reference to the uniform distribution of the same number of counts on a specified interval.

For translational entropies the set of vectors was histogrammed in 3D and the entropy computed with respect to the standard 1 M reference state. The width of the 3D bins was 0.133 Å ×0.133 Å ×0.133 Å for the results reported here.

For rotational entropies the rotation matrices were converted into a set of three angles: two (ϕ, ψ) to specify the rotation axis, and one (θ) to specify the rotation around that axis. θ was allowed to vary only between and . Since the random distribution of rotation is not uniform over the three angular ranges, the range was divided in such a way that the integral over each bin was uniform. With this prescription the three angular ranges were divided in 10 × 10 × 100 bins, respectively. The reference state was the uniform distribution of rotations.

As mentioned above, the method due to Schlitter [40] estimating entropies from the covariance matrix was used as implemented in the software Carma [44]. The source code was modified as to plot the contributions of each eigenvector to the entropy.

Molecular dynamics simulations

Conformational entropy of unfolded proteins

In order to compute conformational entropy changes upon folding it is necessary to have a reference for the unfolded state. An explicit simulation of the unfolded state of a polypeptide chain would be unpractical because of both equilibrium constant, dictating the relative population, and kinetic constants coupled with free energy landscape, dictating the length of the simulation time.

Reference entropy values computed here may be used for computing conformational entropy differences from simulations of folded proteins, assuming, as usual, that aminoacid conformations in the unfolded state are essentially the same for all unfolded proteins. The assumption is in line with the typical lack of any secondary chemical shift observed in NMR spectra of unfolded proteins [71]. Conformational entropies for unfolded proteins (with reference to uniform torsional angle distribution) have been estimated from the conformation of aminoacids in public structural databases, as motivated in the Materials and methods section. Residues from the culled pdb dataset [51] not involved in secondary structures were taken as representative of unfolded conformations. The number of samples for the different aminoacids within protein sequence is reported in Table 1, together with entropies estimated from the nearest neighbor distances. Backbone angles ϕ or ψ were removed from the set of torsional angles to simulate residues at N- and C-termini, respectively. The entropy computed taking different k-nearest neighbors was reasonably consistent with the adopted choice of 1st-nearest neighbors. We report in Fig 1 the computed entropy for the alanine residue (only two torsional angles, ϕ and ψ, both with 2π periodicity) and those of valine, isoleucine, tryptophane and lysine (3, 4, 5 and 6 torsional angles, respectively) versus the average distance of the nearest-neighbors which increases as k increases. The entropy versus distance is reported instead of the entropy versus k because the average distance of the k-nearest-neighbors represents the resolution of the corresponding histogram approach to probability and entropy computation. As expected with increasing average distances details of the distribution are lost and the entropy increases. As the average distance approaches half of the minimum periodicity of the torsional angles, corrections should be made to the reference hyperspherical volume implied in Eq (9). In the absence of corrections artifactual curve behaviour is apparent. In the figure the onset of such effects takes place at 180 degrees. The entropy computed by the present method cannot be compared straightforwardly with the many estimates of sidechain entropy reported in the literature [72] because here the distribution of all residue torsional angles are considered together. In an approximate analysis we subtracted the entropy corresponding to alanine (as a model for all aminoacids backbone) from all other entropies and compared the resulting values with the values calculated by Koehl and Delarue [73]. The results correlate well (r = 0.86), although the estimated entropy is on average 1.8 times larger, most likely because of finer resolution due to larger database and different method. The trend of computed entropy with average nearest neighbor distance is observed in Fig 1.

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Fig 1. Entropy for residues in unfolded proteins calculated according to the k^th nearest neighbor vs. the average distance of the k^th nearest neighbor.

Distances are increasing with k. The curves are relative to alanine (continuous line), valine (dotted line), isoleucine (short dashed line), tryptophane (long dashed line) and lysine (dot dashed line).

https://doi.org/10.1371/journal.pone.0132356.g001

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Table 1. Entropy and average nearest neighbor distance of unfolded aminoacids in k_B units.

The row CYS (ϕ, ψ, χ₁) refers to cysteines involved in disulfide bridges. For N- and C-terminal residues see text.

https://doi.org/10.1371/journal.pone.0132356.t001

Conformational entropy of a folded protein: β2-microglobulin

The conformational entropy change upon folding is obtained in this section as the difference in the conformational entropy computed from molecular dynamics simulation between the folded state of the protein and the reference entropies for the unfolded state computed from database samples in the previous section.

The system chosen here is β2-microglobulin which is a small, well characterized protein.

300 snapshots from 30 ns molecular dynamics simulations of β2-microglobulin have been converted in torsional angles and entropy was computed, residue by residue. It should be noted that in the approach only correlations among degrees of freedom of the same residues are considered, whereas those among torsional angles of different residues would lead to very high dimensional space. This issue will be addressed in this subsection.

The entropies computed for the model residues in the unfolded protein (see previous subsection) have been subtracted from the computed entropies. The plot of computed entropy vs. residue number is sensitive to the nature of each residue, however, it is worth noting that the first and last residue display small but positive differential entropy. It is likely that this is due to the fact that these are really terminal residues, with more conformational freedom than the model ones, which were taken from residues in the sequence with the terminal torsional angles ignored. For all other residues the individual entropies range from -6.1 k_B for R81 to -0.4 and -0.7 k_B for R89 and K15 in the mobile FG and AB loop, respectively. It is reassuring that most mobile residues show entropies close to those of the unfolded residues.

The residues displaying the largest differential entropy are mostly located in stable regions of the protein. For instance, R81 (-6.1 k_B) is close to C80 involved in a disulfide bridge in the most stable region of the protein. R81 is partly exposed to the solvent but is involved in a salt bridge with residue D38. The picture is however somehow obscured by the individual nature of residues. In order to overcome this difficulty we consider the sum of entropies of residues clustering together. To this end we list for each residue the contacting residues and sum their computed entropies as representative of the entropy corresponding to a local opening reaction involving that residue. By local opening we mean the process of partial or global unfolding, where an amide hydrogen becomes exposed to the solvent, whose global entropy is measured e.g. by the Bluu-Tramp hydrogen-exchange NMR experiment [3, 4].

The results are reported in Fig 2 where the single residue and the cluster (i.e. the sum over neighboring residues) entropies are reported, together with experimental results. The latter include not only the conformational but also solvation entropy, and serve only the purpose to show the scale of the phenomena under consideration. It is interesting to note that the experimental entropy values which are most largely different from the conformational ones are those for the only three leucines for which experimental information is available, L39, L40 and L87 and the two aromatic residues F22 and Y66. For these residues hydrophobic solvation contributions to entropy are expected to be relevant. For other hydrophobic or aromatic residues data (Y10, Y26, Y67, V82, V93, I46, W95) the same large difference is however not observed, pointing out the importance of specific details of local opening rather than a general hydrophobic effect, which are not captured by our simple model for local openings.

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Fig 2. Entropy versus residue number from 30 ns β2-microglobulin molecular dynamics simulation.

Residue computed entropy (dashed line), summed entropy corresponding to residue neighborhood (continuous line). In the low part of the figure residues with defined secondary structure are indicated by black boxes. Black dots with error bars represent experimentally determined entropy from Bluu-Tramp hydrogen-deuterium exchange NMR experiments [3]. The latter experiments measure the enthalpy and entropy of the process (partial or global unfolding) which exposes protein amide hydrogens to solvent.

https://doi.org/10.1371/journal.pone.0132356.g002

It should be noted that the approach followed here neglects correlations between degrees of freedom belonging to different amino acids. For a complete treatment of such correlations, approaches have been developed by Gilson and coworkers [15, 74]. The correlation within the same amino acid are expected to be large and actually many studies of protein conformations have demonstrated the correlation between secondary structure and side chain torsional angles [75]. It is however true that backbone angles of adjacent residues show often correlated motions which preserve the overall direction of the chain [76]. Such correlations have been observed very early in molecular dynamics simulations [77]. In order to assess the magnitude of this effect we computed the entropies S_ϕi and S_ψi−1 corresponding to the 1-dimensional distribution of angles ϕ_i and ψ_i−1 separated by the peptide bond and the entropy S_{ϕi, ψi−1} corresponding to the 2-dimensional joint distribution. The difference (S_ϕi+S_ψi−1−S_{ϕi, ψi−1}), i.e. the mutual information of ϕ_i and ψ_i−1 [74] was found to be positive, as expected, and on average of small entity (0.13 ± 0.05 k_B). An extreme test for contacting side chains is provided by the degrees of freedom of cysteines involved in a disulfide bridge, where a strong correlation is expected. Also here the explicit calculation of the entropy corresponding to the two cysteines compared to the two isolated residues gave a positive difference (1.5 k_B) which however amounts to ca. 10% of the overall entropy of the two residues, before subtraction of free residues one, and 30% of the entropy after subtraction. We expect that in the absence of restraints on chain direction in the unfolded state, the correlation should be rather small. A similar test was performed for all torsional angles of R81 with all other torsional angles of β₂-microglobulin. The average mutual information is about 0.06 k_B. Overall these data suggest that correlations, other than those within the same aminoacids may give corrections to the conformational entropies which could be in the range of one order of magnitude smaller than the single amino acids conformational entropies of folding.

For this rather short simulation we checked also convergence. The trajectory was divided in two and the results for the separated halves were plotted together with the results for the total simulation time. As it can be seen in Fig 3 the results are rather stable with few exceptions. The largest difference is observed for K58 which loses close interaction with D59 in the second half of the simulation becoming more flexible.

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Fig 3. Entropy versus residue number from 30 ns β2-microglobulin molecular dynamics simulation (thick line) and from the first and second half of the simulation (thin lines).

https://doi.org/10.1371/journal.pone.0132356.g003

Conformational entropy and order parameters

In this section we address the relationship between conformational entropies and order parameters, which are measured by NMR typically on HN vectors and are taken as a measure of conformational dynamics of each aminoacid.

The system chosen for this purpose is the exchange protein directly activated by cAMP (EPAC) which is a receptor undergoing functionally critical conformational and dynamics changes upon binding cAMP.

Changes in dynamics have been monitored by changes in HN order parameters, as measured by NMR, used, in turn, to measure changes in conformational entropy.

Four simulations, starting from the active and inactive conformations in the apo and holo forms, were previously performed, analysed and compared with experimental data by Melacini and coworkers [28]. We use the distance-based estimates of entropies for the four trajectories, which compared well with experimental data, and use this example to address the following issues:

1. how well the order parameters S², typically measured on HN vectors using NMR, correlate with the entropy associated with backbone degrees of freedom;
2. how well the the entropy associated with backbone degrees of freedom correlates with the entropy of all torsional degrees of freedom.

For the first question, a straightforward correlation of the order parameters of the HN vector with the conformational freedom of backbone torsion angles is not expected because the HN vector could be kept in place e.g. by a hydrogen bond even with floppy flanking peptide moieties. Rather, an agreement between order parameters and entropies averaged over chain segments should be expected. Indeed the raw correlation between backbone entropy and the order parameter computed on MD trajectories is poor, besides a necessary transformation between unbounded entropy and bounded order parameter. However, when the backbone entropy is smoothed using the lowess algorithm [78] implemented in the software R [79], with the parameter f set to 0.02, the (anti)correlation increases to -0.61 (Fig 4). The plot of order parameter versus the computed backbone entropy shows a semiquantitative relationship between the two. Notwithstanding the very different assumptions underlying the present calculation and the harmonic oscillation model used by Wand and coworkers [80], the points are scattered about a sigmoidal curve, thus confirming the insight provided by the original link between order parameters and associated entropy.

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Fig 4. HN vector order parameter versus the computed backbone entropy on all four EPAC MD trajectories.

The single simlations show some differences, but deviations from a sigmoidal behaviour is found for all four simulations.

https://doi.org/10.1371/journal.pone.0132356.g004

The changes in the order parameter S² occurring upon conformational transitions may be associated, based on the concept that they reflect overall conformational freedom changes, to changes in conformational entropy. For this reason we examine how backbone entropy, which is semiquantitatively measured by the order parameter S², is correlated with overall conformational entropy. In other words we wish to estimate the latter from the measured HN order parameter S².

We computed the entropy associated with backbone and all torsional angles for all four simulations and fitted the overall conformational entropy as a linear combination of backbone conformational entropy (coefficient 1.120) and the number of sidechain rotational degrees of freedom (coefficient -1.425), assuming implicitly that each degree of freedom gives a similar entropic contribution. Alternatively the difference between the absolute entropy of all degrees of freedom and backbone degrees of freedom could be fit as linearly dependent on the number of sidechain degrees of freedom (coefficient -1.633) with similar results. Residues alanine, glycine and proline that do not possess torsions other than backbone ones were removed from the comparison. The fit shows a strong correlation with fitted data (correlation coefficient 0.859). This analysis does not take into account whether the residue is exposed or not. A slight modification of the approach considers also the exposed area of the residue. In this case the estimated absolute entropy of the residue is a weighted sum of the backbone entropy (weight: 1.08), of the number of degrees of freedom of the sidechain (weight: -1.78) and the exposed surface area in Ų (weight: 0.012). With this simple linear model the correlation coefficient increases to 0.906 and the root mean square deviation of the predicted versus computed absolute entropy is 0.76 k_B (Fig 5).

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Fig 5. Residue entropy computed for all four EPAC MD simulations.

On the x-axis the entropy is computed using all residue torsion angles, on the y-axis the entropy is fitted as a linear combination of backbone entropy, the number of sidechain degrees of freedom and the solvent accessible surface area of the residue. Glycine, Alanine and Proline residues that do not possess sidechain torsion angles, are not reported in the figure.

https://doi.org/10.1371/journal.pone.0132356.g005

Rototranslational entropy loss upon binding: OppA-tripeptide complexes and β2-microglobulin association with a citrate coated gold surface

The free energy of binding of tripeptides with the E. coli peptide protein binding has been extensively studied by computational methods [30]. We address here the rototranslational entropy loss upon binding. We first select an almost rigid three-atom fragment which is most decoupled from internal degrees of freedom of the peptide. To this end we consider the N-CA-C fragment of each of the three aminoacids and we compute the root mean square fluctuations of the three atoms upon superposition of the OppA regions involved in binding. All three aminoacids gave similar results, depending on the specific tripeptide, and therefore we kept the central aminoacid backbone atoms as the reference triad atoms.

Overall complex rototranslation and the movements internal to OppA protein were removed by superimposing backbone atoms of residues 32–37, 245–246, 269–271, 397–404, 414–419 of the OppA protein to the starting structure. The superimposed structure was then used for superposition of the three reference ligand backbone atoms and the resulting rotation matrix and translation vector were written. This procedure is illustrated in Fig 6.

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Fig 6. Graphical representation of how the rotation matrix and translation vector describing the rototranslational state of the tripeptide ligand with respect to OppA is obtained.

First, overall rototranslation of the complex is removed by superposition of frame residues. Second, the central residue N, CA and C atoms are superimposed on the reference structure. The resulting rotation matrix and translation vector describe the rototranslational state of the tripeptide ligand.

https://doi.org/10.1371/journal.pone.0132356.g006

The resulting computed entropies are reported in Table 2. The reference states are the randomly oriented rigid triad of backbone atoms (S = −k_Blog(4π²)), for the rotational state, and the 1M concentration state, for the translational state (−k_Blog(1/1660)), with angles measured in radians and distances in Å. The resulting rototranslational entropy of association at room temperature is ca. 10 kcal/mol. A similar value was predicted earlier based on semiquantitive considerations by others (e.g. [39, 81]).

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Table 2. Rotational and translational entropies in k_B units for the binding of 20 tripeptides to OppA protein.

In parentheses values computed using the histogram method are reported. The average nearest-neighbor distance (in Å for translations and in rad for rotational distances) is reported in Av. dist. columns.

https://doi.org/10.1371/journal.pone.0132356.t002

Rotational and translational entropies have been computed using also the histogram method for the same set of 1000 snapshots. The results are dependent on the number of bins used. The latter was chosen as to limit the number of bins containing a single sample. With this choice the agreement is remarkable with correlation coefficients over 0.96 and mean square root difference of less than 0.2 k_B (Table 2).

In order to study a less tightly bound system we consider β2-microglobulin non-covalently bound to a gold nanoparticle coated with citrate. Here the protein is limited in its movements along the axis perpendicular to the surface plane and in its orientations. We have considered the distance of the center of geometry of the most structured regions from the plane through the oxygen atoms of the citrate bound to the gold surface. The reference rotation state is computed through superposition of the backbone atoms of the residues that contact most the surface, i.e. I1, K3, K58, W60 with the corresponding atoms in the first structure. After setting a common reference frame for the bound state, the rotation matrix is found by superposition of secondary structure elements. Because the bound state is defined irrespective of the orientation and position over the plane, the computed rotational entropy must be computed with a modified formula taking into account the reduction in dimensionality. The translational entropy is computed by Eq (9) with dimension one and taking the distance between two sample points as the difference in the distance from the surface plane.

Similarly for the rotational entropy of the molecule in contact with the surface it should be considered that it should be corrected by an additional term k_Blog(2π) to take into account any rotation about the axis perpendicular to the surface plane. With these corrections the overall translational and rotational entropy change from the 1M reference state to the surface bound molecule amounts to -2.1 k_B and -4.8 k_B, respectively. These figures, which are considerably smaller than the corresponding ones for tight binding, could be underestimated because of the expected correlation between translational and rotational degrees of freedom for the bound state.