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Simulation method for calculating the entropy and free energy of peptides and proteins

Edited by Harold A. Scheraga, Cornell University, Ithaca, NY (received for review December 10, 2003)
Abstract
A method called complete hypothetical scanning Monte Carlo has been introduced for calculating the absolute entropy, S, and free energy, F, of fluids. Here, the method is extended to peptide chains in vacuum. Thus, S is calculated from a given sample by reconstructing each conformation stepbystep by using transition probabilities (TPs); at each step, part of the chain coordinates have already been determined (the “frozen past”), and the TP is obtained from a Monte Carlo simulation of the (future) part of the chain whose TPs as yet have not been calculated. Very accurate results for S and F are obtained for the helix, extended, and hairpin microstates of a simplified model of decaglycine (Gly)_{10} and (Gly)_{16}. These results agree well with results obtained by the quasiharmonic approximation and the local states method. The complete HSMC method can be applied to a macromolecule with any degree of flexibility, ranging from local fluctuations to a random coil. Also, the difference in stability, ΔF_{mn} = F_{m}  F_{n} between significantly different microstates m and n can be obtained from two simulations only without the need to resort to thermodynamic integration. Our longterm goal is to extend this method to any peptide and apply it to a peptide immersed in a box with explicit water.
In ref. 1, White and Meirovitch discuss the importance and difficulties of calculating the absolute free energy, F, and entropy, S; however, their role in computational structural biology should be further emphasized. The energy surface of a protein, commonly defined by a force field, is highly rugged, consisting of a tremendous number of local minima (2), where the native structure corresponds to the localized energy well with the lowest F. However, molecular dynamics simulations have shown (3, 4) that even a protein with a well defined structure fluctuates significantly within a region called wide microstate (e.g., the conformational region of an αhelix of a peptide) that typically consists of many localized energy wells. A peptide or protein, or protein segments such as surface loops, can exhibit an intermediate flexibility, where several wide microstates are populated significantly at thermodynamic equilibrium. It is essential to be able to identify these wide microstates, m, and to calculate F_{m} , which lead to their relative populations and to weighted averages of various quantities that can be compared with experimental values (5, 6). F_{m} is useful particularly if m and n differ significantly; then, calculating the difference, ΔF_{mn} = F_{m}  F_{n} is straightforward, whereas calculating it by thermodynamic integration might be prohibitive (see refs. 7–12 and references therein).
In ref. 1, the hypothetical scanning (HS) method for calculating the absolute F and S (10) has been further developed and applied to liquid argon and water. This method, named complete hypothetical scanning Monte Carlo (HSMC), is extended here to a peptide in vacuum or peptide described by an implicit solvation. As a first step, we treat a simplified model of decaglycine (Gly)_{10} simulated by Monte Carlo (MC) (13) at three wide microstates: helix, extended, and hairpin. Each sample conformation is reconstructed gradually by calculating transition probabilities (TP) for the dihedral angles and fixing the related atoms at their positions. A TP is obtained by an MC simulation of the future (yet unfixed) part of the chain and, to avoid the escape of a future sample from the corresponding microstate, we impose restrictions obtained from the local states (LS) method for calculating S (5, 6, 14–18). Therefore, the entire procedure is a hybrid of two techniques and, to test its performance for larger peptides, we also study a 16residue polyglycine (Gly)_{16} in the helix and extended wide microstates. The complete HSMC results for S and F are found to be in a very good agreement with results obtained by the LS method and the quasiharmonic (QH) approximation (19, 20). Our longterm goal is to extend the complete HSMC method to any peptide and to apply it to a peptide immersed in a box with explicit water.
Theory and Methodology
The Model and Statistical Mechanics Considerations. We have first studied decaglycine, NH_{2}(Gly)_{10}CONH_{2}, modeled by the amber96 force field in vacuum (21), where the charges of the end groups were neutralized. For simplicity, we denote the dihedral angles ϕ _{i} ,Ψ _{i} , and ω _{i} ordered along the chain by α _{k} , k = 1,3N = 30, where N is the number of residues [the extension for (Gly)_{16} is straightforward]. The partition function, Z, is an integral over the function exp(E/k _{B} T) (E is the potential energy and k _{B}, the Boltzmann constant) with respect to the Cartesian coordinates over the whole conformational space, Ω. However, for a stable wide microstate, the integration is carried out only over the limited region Ω_{0} that defines the wide microstate. To apply the HS or LS methods, one has to change the variables of integration from Cartesian to internal coordinates, which makes the integral dependent also on the Jacobian, J. For a linear chain, J has been shown to be independent of the dihedral angles and is a simple function of the bond angles and bond lengths. Thus, if the potentials of these “hard variables” are strong, their average values can be assigned to J, which to a good approximation can be taken out of the integral (see refs. 19, 22, and 23).
For the same reason, one can assume a more restrictive model (see below), where the bond angles and bond lengths are kept constant at their average values and thus the corresponding exp(E/k _{B} T) values can be taken out of the integral as well. In particular, notice that, although the contribution of bond stretching to the absolute entropy is not small, it is expected to be similar for different wide microstates of the same molecule. Therefore, to a good approximation, the contribution of bond stretching to the differences ΔS_{m,n} and ΔF_{m,n} between wide microstates m and n cancels out. Assuming that the bond lengths are not correlated with the bond and dihedral angles enables one to carry out the integration over the bond lengths; if a similar assumption is made for the bond angles, the remaining integral becomes a function of the 3N dihedral angles (22, 23) The partition function is where the prefactor D is a product of J and the integral over the bond lengths and bond angles; D depends on T and the units in which the bond lengths and bond angles are expressed. For calculating ΔS_{m,n} and ΔF_{m,n} of two wide microstates of the same molecule, lnD cancels and can be ignored (notice, however, that D contributes to the absolute F and S). The probability density corresponding to Z (Eq. 1 ) is and the exact entropy (defined up to an additive constant) is
Thus far we have described the transformation from a peptide model represented by Cartesian coordinates to a model represented by dihedral angles. This is needed for applying both the LS and HS methods. However, we have found MC simulations in Cartesian coordinates to be extremely inefficient; therefore, we have studied a relatively simple model of ployglycine based on the amber96 force field with constant bond lengths and bond angles, an option available in the program tinker (http://dasher.wustl.edu/tinker). Thus, the dihedral angles ϕ _{i} , Ψ _{i} , and ω _{i} become the variables of an MC procedure significantly more efficient than that based on Cartesian coordinates; for the present model of rigid geometry, the application of the LS and HS methods is therefore direct. Obviously, keeping the bond angles constant is a temporary restriction applied only in this initial study of the complete HSMC method. In what follows, for simplicity, the various methods will be described as applied to this model of polyglycine.
The Exact Scanning Method. The exact scanning method is a stepbystep construction procedure for polymer chains (24) and thus is equivalent to the MC and MD procedures in the sense that large samples generated by all these methods lead to the same averages and fluctuations within the statistical errors. With the exact scanning method an Nresidue conformation of polyglycine in the helical region (Ω_{0}), is built by defining the dihedral angles α _{k} step by step with TPs and adding the related atoms; for example, the angle ϕ determines the coordinates of the two hydrogens connected to C^{α}, and the position of C′ (16, 25). Thus, at step k, k1 dihedral angles α_{1,}..., α _{k1} have already been determined, they and the related structure (the past) are kept constant, and α _{k} should be defined with the exact TP density ρ(α _{k} α _{k1} ... α_{1}), where dα _{k} is a small segment centered at α _{k} , and Z_{f} (α _{k} ... α_{1}) is a future partition function defined over the helical region Ω_{0} by integrating over the future conformations defined by α _{k+1} ... dα _{3N} (within Ω_{0}), where the past angles, α_{1}... α _{k} , are held fixed,
The probability density of the entire conformation is Because of the equivalence between the MC and the scanning method mentioned above, one can assume that a given MC sample has rather been generated by the scanning method, which enables one to reconstruct each conformation and to calculate the TP densities that hypothetically were used to create it stepbystep. This idea can be implemented in two different ways, by the LS and HS methods. Because some elements of the LS method are implemented within the framework of the complete HSMC method, we describe the LS method first.
The LS Method. In the first step, the MC sample (of a given wide microstate) is visited, and the variability range Δα _{k} is calculated (5, 6, 14–18). where α _{k} (max) and α _{k} (min) are the maximum and minimum values of α _{k} found in the sample, respectively. Next, the ranges Δα _{k} are divided into l equal segments, where l is the discretization parameter. We denote these segments by ν _{k} , (ν _{k} = 1, l). Thus, an angle α _{k} is now represented by the segment ν _{k} to which it belongs, and a conformation i is expressed by the corresponding vector of segments [ν_{1}(i), ν_{2}(i),..., ν _{3N} (i)]. Under this discretization approximation, ρ(α _{k} α _{k1} ... α_{1}) can be estimated by where n(ν _{k} ,..., ν_{1}) is the number of times the LS [i.e., the partial vector (ν _{k} ,...,ν_{1}) representing (α _{k} ,...,α_{1})] appears in the sample. Because the number of local states increases exponentially with k, one has to resort to approximations based on smaller LSs that consist of ν _{k} and the b angles preceding it along the chain, i.e., the vector (ν _{k} , ν _{k1} ,..., ν _{kb} ); b is called the correlation parameter. The sample is visited for the second time, and for a given b, one calculates the number of occurrences n(ν _{k} , ν _{k1} ,..., ν _{kb} ) of all of the local states from which a set of TP ρ(ν _{k} ν _{k1} ,..., ν _{kb} ) are defined. The sample is then visited for the third time, and for each member i of the sample, one determines the 3N local states and the corresponding TP, whose product defines an approximate probability density ρ _{i} (b, l) for conformation i: The larger are b and l, the better the approximation (given enough statistics). ρ _{i} (b, l) allows defining an approximate entropy and free energy functional, S ^{A} and F ^{A}, which constitute rigorous upper and lower bounds for the correct values, respectively (25), where 〈E〉 is the Boltzmann average of the potential energy, estimated from the MC sample, and ρ^{B} is the Boltzmann probability density with which the sample was created.
S ^{A} is estimated from a Boltzmann sample of size n by S̄ ^{A}:
As discussed in ref. 1, the fluctuation ΔF of the correct free energy is zero, whereas the approximate F ^{A} has finite fluctuation, ΔF ^{A} (estimated by ), which is expected to decrease as the approximation improves (17).
The LS method can be applied to any chain flexibility, i.e., it is not limited to harmonic or QH fluctuations (19, 20, 22, 23, 26). Thus, free energy differences between wide microstates with significant structural differences can be calculated, which is a difficult task with methods based on thermodynamic integration.
Approximate HS Method. The idea of the HS method is to reconstruct each sample conformation step by step, obtaining the TP density of each α _{k} (Eq. 4 ) by calculating the future partition functions Z_{f} . However, a systematic integration of Z_{f} within the limits of Ω_{0} is difficult and becomes impractical for a large peptide where Ω_{0} is unknown; therefore, thus far, HS was applied only to selfavoiding walks (SAWs) on a lattice, where Ω_{0} is the entire space and Z_{f} is calculated approximately by enumerating only future SAWs of f steps (i.e., α _{k} ,..., α _{k+f1} ), rather than of N  k + 1 steps (27, 28).
The Complete HSMC Method. With the complete HSMC method applied to peptides (like for fluids), one calculates at each reconstruction step k of conformation i the TP density, ρ(α _{k} α _{k1} ... α_{1}), from n_{f} MC steps (trials) (13), where the entire future of the peptide can move by changing the future angles α _{k} ,..., α _{3N} , whereas the dihedral angles α_{1},..., α _{k1} (defining the past) are kept fixed at their values in conformation i. A small segment (bin) δα _{k} (see Eq. 4 ) is centered at α _{k} , and the number of MC visits to this bin during the simulation, n _{visit}, is calculated; one obtains where the relation becomes exact for a very large n_{f} . The product of these TP densities leads to the probability density of the entire chain (Eqs. 6 and 10 ). Notice that, unlike the systematic calculation of Z_{f} , where the limits of Ω_{0} are in practice unknown, with the complete HSMC procedure, the future structures generated by MC at each step k remain in general within the limits of the wide microstate Ω_{0} defined by the analyzed MC sample. In some cases, however, the future samples were found to escape from this region; therefore, before applying the complete HSMC method, the LS method is applied to the analyzed sample and the α _{k} (min) and α _{k} (max) values (Eq. 7 ) are calculated; they are then used to keep the future structures within Ω_{0} by rejecting MC moves with angle values beyond those of α _{k} (min) and α _{k} (max). Although complete HSMC considers the entire future, in practice ρ(α _{k} α _{k1} ... α_{1}) (Eq. 13 ) will be somewhat approximate due to insufficient future sampling, imperfect random number generator, etc.; therefore, the corresponding free energy, F ^{A} (Eqs. 10 and 11 ), will underestimate slightly the correct value, where its fluctuation ΔF ^{A} (Eq. 12 ) does not vanish.
Results and Discussion
We have first studied three wide microstates of (Gly)_{10}: helix, hairpin, and extended. Samples of these wide microstates were generated by the Metropolis MC procedure (13) at 100 K where a trial structure is obtained by randomly changing all of the 30 dihedral angles, each within ±1° of its current value. These simulations were started from helical, extended, and hairpin structures obtained by minimizing the energy of the corresponding structures, ϕ _{k} = Ψ _{k} = 55°, and ω _{k} = 180°, ϕ _{k} = Ψ _{k} = ω _{k} = 180°, and two extended strands of four residues connected by a type I′ turn. The first 5,000 MC steps were used for equilibration, and from the following 50,000 steps, after every 100 steps the current structure was retained for future analysis; in this way, three equal samples of 500 structures were generated. It should be pointed out that preliminary simulations at 300 K resulted in unstable samples (i.e., the structures escaped from their wide microstates); therefore, the temperature was decreased to 100 K, where the helix and extended simulations were found to be very stable, whereas the hairpin sample remained stable only up to the first 50,000 MC steps. The corresponding Δα _{k} values (Eq. 7 ) are relatively small (see Table 1), representing relatively concentrated samples. Notice, however, that due to correlations, each wide microstate is significantly smaller than the corresponding region, Δα_{1×}Δα_{2×}.... ×Δα_{30}.
To apply the complete HSMC method, each conformation was reconstructed step by step by TPs obtained from MC simulations of the future part carried out in the same way as described above with the additional restriction that a trial conformation with α _{k} larger than α _{k} (max) or smaller than α _{k} (min) is rejected. To check the convergence of the results, they were calculated for four future sample sizes, n_{f} = 20,000, 40,000, 80,000, and 160,000. For the same reason, we studied for each dihedral angle four bin sizes, δ = Δα _{k} /15, Δα _{k} /10, Δα _{k} /5, and 20° centered at α _{k} . Notice that as for the LS method, the bin size is proportional to Δα _{k} . If the counts of the smallest bin are smaller than 50, the bin size is increased to the next size and, if necessary, to the next one (δ = Δα _{k} /5); the same is applied to the second size bin. In the case of zero counts, n _{visit} is taken to be 1; notice, however, that zero counts is a very rare event.
Results for the entropy (TS) appear in Table 2 for various n_{f} values and bin sizes; the results for the largest bin (of 20°) are not provided, because they are significantly worse than those obtained for the smaller bins. All of the HSMC results are based on samples of 400 structures (of the entire samples of 500 conformations), and the statistical errors were obtained from the fluctuations and results based on partial samples. The accuracy of complete HSMC can always be improved by decreasing the bin size and increasing the future sample size, meaning that correspondingly S ^{A} (Eq. 11 ) is expected to decrease [provided that the probability density is defined on the same conformational space (i.e., the wide microstate) as the Boltzmann probability density used to generate the sample]. Indeed, for each bin, the entropy decreases (or remain constant) as n_{f} increases, where the only exception is the entropy for the helix based on the smallest sample, n_{f} = 20,000, which is smaller than the entropies of the larger samples; this probably stems from an HSMC probability density that is defined on only a partial region of the helical wide microstate due to insufficient sampling.
The entropy results for the extended microstate for a given n_{f} are the same for the different bin sizes, and the results for n_{f} = 80,000 and 160,000 are converged within the error bars. The same applies to the hairpin results for the two smallest bins. The helix results behave differently, where for each bin they are constant for the three largest n_{f} values, whereas they decrease with decreasing the bin size and probably have not yet completely converged. However, within the accuracy of the usual force fields, entropy and free energy differences smaller than 0.1 kcal/mol are in general ignored; therefore, even the helix results can be considered as converged.
It is of interest to compare the complete HSMC results with those obtained by other methods. For that, we increased the samples of the three wide microstates from 500 to 30,000 structures by imposing the restriction on the MC procedure that a trial conformation with α _{k} larger than α _{k} (max) or smaller than α _{k} (min) obtained for the initial sample (Table 1) is rejected. We applied the QH approximation (19, 20) to a subsample of 4,000 conformations, where and σ is the determinant of the covariance matrix of the 3N dihedral angles. We also applied the LS method (with correlation parameter, b = 1 and l = 10) to the entire increased sample. The QH results presented in Table 2 are very close to the complete HSMC values, probably because the three samples are approximately QH. The LS and HSMC entropies are equal for the extended microstate, because the angular correlations along the chain are short, and b = 1 already captures most of them. On the other hand, the range of these correlations increases for the helix and the hairpin, and the LS entropies, as expected, become slightly larger (upper bounds) than the HSMC values.
In Table 3, complete HSMC results are presented for the free energy, which is defined by F ^{A} (Eq. 10 ), as discussed for the entropy above. These results are given only for the smallest bin, because the free energies for the other bins can be obtained from the entropies of Table 2; as expected, the free energy increases as the approximation improves (i.e., as n_{f} is increased). Again, the QH results are close to the HSMC values, and the LS result is close to the HSMC value for the extended microstate and smaller for the other two microstates. Notice, however, that the energy components of QH and LS are calculated from the corresponding larger samples (see previous paragraph). We also provide in Table 3 results for the average energy (obtained from 400 structures) and the fluctuations of the energy and free energy. As expected, the free energy fluctuations decrease as n_{f} is increased, and for n_{f} = 160,000, they are four times smaller than the corresponding fluctuations of the energy, except for the hairpin where the ratio is ≈2.
The main interest in this study is to determine the relative stability of the three wide microstates. In the upper part of Table 4, we present results for the differences, TΔS, ΔF, and ΔE between these microstates for (Gly)_{10}. Within their uncertainty of 0.1–0.2 kcal/mol, the differences are very stable for the three bin sizes, for n_{f} = 40,000–160,000, for samples as small as 200 conformations, and for the helixextended differences also for 100 conformations. This demonstrates that, in practice, complete HSMC can be quite efficient. For the model studied, the helix is the most stable, where its free energy is lower by 14.3 and 24.1 kcal/mol than that of the hairpin and extended microstates, respectively. These differences are mostly governed by the energy differences, 11.9 and 27.95 kcal/mol, where the TΔS values are only 1.7 and 3.9 kcal/mol, respectively.
It is of interest to test the performance of complete HSMC for larger peptides, and we therefore also applied it to (Gly)_{16}. Two samples of size 600 each spanning the extended and helical wide microstates were generated by MC [as described for (Gly)_{10}], where 400 and 600 conformations of them were reconstructed by HSMC, respectively. The dihedral angle values of these samples are concentrated around their canonical values with deviations Δα _{k} (Eq. 4 ) very close to those obtained for (Gly)_{10} in Table 1, where significant differences exist only for ΔΨ of the first and last residues. Results for TS ^{A}, F ^{A}, and its fluctuation, ΔF ^{A}, appear in Tables 5 and 6, which are structured as Tables 2 and 3. The corresponding results for TS ^{A} are basically unchanged (i.e., converged) as bin size decreases, i.e., in going from Δα _{k} /5 (results not shown) to Δα _{k} /10, and to Δα _{k} /15. Within each bin size, the helix results are constant as well, meaning that already a future sample size of n_{f} = 40,000 is sufficient. On the other hand, for the extended microstate, the results for each bin size decrease as n_{f} is increased and larger n_{f} is needed to reach convergence; for n_{f} = 320,000, the expected extrapolated result is TS ^{A} = 31.90, which is used in calculating the differences for (Gly)_{16} in Table 4.
As expected, for both microstates, the LS results (b = 1, l = 10) slightly overestimate the HSMC values, whereas the QH results are equal to the HSMC values within a relatively large statistical error. The free energy fluctuations, as expected, decrease monotonically as the approximation improves, and they are smaller than the energy fluctuations by a factor of 4.7 and 2.2 for the helix and extended microstates, respectively. The LS and QH results were obtained from relatively large samples of 5·10^{4} and 25·10^{3} conformations, respectively, hence the corresponding energies are slightly different from those based on the smaller HSMC samples; thus, whereas the F ^{A}(LS) and F(QH) are close to F ^{A}(HSMC), a strict comparison is not straightforward. In Table 4 the differences, TΔS, ΔF, and ΔE for the extended and helix microstates are presented with acceptable errors of 0.2–0.3 kcal/mol (see above). It should be pointed out that the results for S ^{A} and the energy of the helix scale with increasing peptide size, whereas the energy of the extended state does not; therefore, ΔE, and ΔF, do not scale in going from (Gly)_{10} to (Gly)_{16}.
At this stage of development of the complete HSMC method, reconstructing a single conformation of (Gly)_{10} based on n_{f} = 160,000 requires ≈90min central processing unit (CPU) time on a 2.6GHz Athlon processor, meaning that a n_{f} = 40,000 run, which is sufficient for providing the 0.1–0.2 kcal/mol accuracy, requires 23min CPU time; for (Gly)_{16}, the time increases by a factor of ≈2.2. However, one can increase the efficiency further by decreasing the amount of sampling (n_{f} ) for the smaller future peptides and using importance sampling methods to enhance the number of counts.
Summary
We have introduced here the complete HSMC method for a peptide chain in vacuum. In this initial study, we sought to treat a simple model with minimal degrees of freedom and therefore chose a polyglycine model with constant bond lengths and bond angles described by the amber force field. Although ignoring the contribution of the bond lengths to differences in entropy is a valid approximation, the contribution of the bond angles is significant and should not be ignored; however, adding this contribution is straightforward and is currently being studied by us. To be able to generate stable wide microstates around helix, extended, and hairpin structures, the temperature was decreased to 100 K; the corresponding samples were approximately QH, which allowed comparing the HSMC results with those obtained by the QH method. However, unlike the QH method, the complete HSMC is general, in the sense that it can be applied to any chain flexibility, where side chains visit all of the available rotamers, for example. Complete HSMC is probably the only method for calculating the absolute entropy of peptide chains that, practically, is exact.
Acknowledgments
This work was supported by National Institutes of Health Grant R01 GM66090 and in part by National Institutes of Health Grant R01 GM61916.
Footnotes

↵ * To whom correspondence should be addressed. Email: hagaim{at}pitt.edu.

This paper was submitted directly (Track II) to the PNAS office.

Abbreviations: HSMC, hypothetical scanning Monte Carlo; TP, transition probabilities; LS, local states; QH, quasiharmonic; HS, hypothetical scanning; MC, Monte Carlo.
 Copyright © 2004, The National Academy of Sciences
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