We have recently suggested an approximate computer simulation technique, based on the scanning method, which enables one to extract the entropy (and hence the free energy) of polymer chains from a relatively small sample. So far this technique has been discussed in terms of chains with excluded volume (EV); in the present work we extend its scope to chains, which also have finite interactions. In order to obtain better approximations for the entropy we utilize the concepts of the generalized Monte Carlo procedure suggested by Schmidt [Phys. Rev. Lett. 51, 2175 (1983)] (and discussed in the preceding paper). We analyze our results from the preceding paper for self-attracting random walks (without EV) on square and simple-cubic lattices, and show that our techniques lead to accuracy which is better than 0.3% for both the free energy and the entropy. For chain models which include EV, we use, in addition to Schmidt's procedure, an alternative one based on ``importance sampling.'' We test these two procedures as applied to self-avoiding walks (SAW's) on a square lattice, where the SAW's are unbounded in space or bounded in relatively small ``boxes.'' Very accurate results for the entropy are also obtained here. It turns out, however, that importance sampling is slightly more efficient than the Schmidt procedure.

Computer simulation of the free energy of polymer chains with excluded volume and with finite interactions

Fluctuations in backbone dihedral angles in the α-helical conformation of homopolypeptides are studied based on an assumption that the conformational energy function of a polypeptide consisting of n amino-acid residues can be approximated by a 2n-dimensional parabola around the minimum point in the range of fluctuations. A formula is derived that relates 〈ΔθiΔθj〉, the mean value of the product of deviations of dihedral angles ϕi and ψi (collectively designated by θi) from their energy minimum values, with a matrix inverse to the second derivative matrix F,n of the conformational energy function at the minimum point. A method of calculating the inverse matrix Fn−1 explicitly is given. The method is applied to calculating 〈ΔθiΔθj〉 for the α-helices of poly(L-alanine) and polyglycine. The autocorrelations 〈(Δϕi)2〉 and 〈(Δψi)2〉 at 300°K are found to be about 66 deg2 and 49 deg2, respectively, for poly(L-alanine), and 84 deg2 and 116 deg2, respectively, for polyglycine. The length of correlations of fluctuations along the chain is found for both polypeptides to be about eight residues long.

Fluctuations of an α‐helix

Predicting Three‐Dimensional Structures of Oligopeptides

A method is developed to extract the entropy of polypeptides and proteins from samples of conformations. It is based on techniques suggested previously by Meirovitch, and has the advantage that it can be applied not only to states in which the molecule undergoes harmonic or quasiharmonic conformational fluctuations, but also to the random coil, as well as to mixtures of these extreme states. In order to confine the search to a region of conformational space corresponding to a stable state, the transition probabilities are determined not by “looking to the future,” as in the previous method [H. Meirovitch and H. A. Scheraga (1986) J. Chem. Phys.84, 6369–6375], but by analyzing the previous steps in the generation of the chain. The method is applied to a model of decaglycine with rigid geometry, using the potential energy function ECEPP (Empirical Conformational Energy Program for Peptides). The model is simulated with the Metropolis Monte Carlo method to generate samples of conformations in the α-helical and hairpin regions, respectively, at T = 100 K. For the α-helix, the four dihedral angles of the N- and C-terminal residues are found to undergo full rotational variation. The results show that the α-helix is a more stable structure than the hairpin. Both its Helmholtz free energy F and energy E are lower than those of the hairpin by ΔF ∼ 0.4 and ΔE ∼ 0.3 kcal/mole/residue, respectively. It should be noted that the contribution of the entropy ΔS to ΔF is significant (TΔS ∼ 0.1 kcal/mole/residue). Also, the entropy of the α-helix is found to be larger than that of the hairpin. This is a result of the extra entropy arising from the rotational freedom about the four terminal single bonds of the α-helix.

Stability of polypeptide conformational states as determined by computer simulation of the free energy.

Even though poly(L-valine) and poly(L-isoleucine) both contain residues that are branched at their β-carbon atoms, they exhibit a different behavior of their Zimm-Bragg helix-growth parameter s in aqueous solution. This quantity increases with temperature for poly(L-valine) but decreases for poly(L-isoleucine). The origin of this behavioral difference was examined by computing theoretical values of s versus temperature from interatomic interaction energies, taking solvent (hydrophobic and hydrophilic) effects into account. The calculated s versus temperature curves for both homopolymers are consistent with the observed experimental behavior. The two homopolymers behave differently because of differences in the change in the number of hydration-shell water molecules accompanying their helix–coil transitions. The larger isoleucine side chains are more crowded together in both the α-helical and coil forms than are those of valine. Therefore, there is a smaller change in hydration of the isoleucine side chains compared to that of the valine side chains in the helix–coil transition. By analyzing the effects of hydration on the s versus temperature curves, it is possible to account also for the experimental curve for poly(L-leucine), which exhibits an intermediate behavior between those for poly(L-valine) and poly(L-isoleucine).

Molecular theory of the helix-coil transition in polyamino acids. V. Explanation of the different conformational behavior of valine, isoleucine, and leucine in aqueous solution

New Method for Calculating the Conformational Entropy of a Regular Helix

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New Method for Calculating the Conformational Entropy of a Regular Helix