Calculation of the entropy of lattice polymer models from Monte Carlo trajectories

Abstract

While lattice models are used extensively for macromolecules (synthetic polymers proteins, etc), calculation of the absolute entropy, S, and the free energy, F, from a given Monte Carlo (MC) trajectory is not straightforward. Recently we have developed the hypothetical scanning MC (HSMC) method for calculating S and F of fluids. Here we extend HSMC to self-avoiding walks on a square lattice and discuss its wide applicability to complex polymer lattice models. HSMC is independent of existing techniques and thus constitutes an independent research tool; it provides rigorous upper and lower bounds for F, which can be obtained from a very small sample and even from a single chain conformation.Lattice models have been utilized to study a wide range of phenomena in polymer physics [1][2][3][4][5] as well as in structural biology, mainly related to protein folding and stability [6-9] (Refs 1-9 constitute a very limited representation of hundreds of papers published in the last 15 years). Because of their simplicity these models have been invaluable tools for understanding global properties that do not depend strongly on molecular details. Such models vary in complexity, ranging from self-avoiding walks on a square lattice to chain models on enriched 3D lattices with a large effective coordination number.Commonly, these systems are simulated by variants of Metropolis Monte Carlo (MC) -a dynamical method that enables one to generate samples of chain configurations i distributed according to their Boltzmann probability, P i B , from which equilibrium information can be extracted [10]. Using MC it is straightforward to calculate properties that are measured directly from i, such as the potential energy E i . On the other hand, the value of P i B cannot be obtained in a straightforward manner, which makes it difficult to calculate the absolute entropy, S ~ -lnP i B directly, i.e., as a byproduct of the simulation (like E i ). There is a strong interest in S as a measure of order and as an essential ingredient of the free energy, F=E−TS, where T is the absolute temperature; F constitutes the criterion of stability, which is mandatory in structure determination of proteins, for example. Furthermore, because MC simulations constitute models for dynamical processes, one would seek to calculate changes in F and S during a relaxation process, by assuming local equilibrium in certain parts along the MC trajectory; a classic example is simulation of protein folding [11].S, and F are commonly calculated by thermodynamic integration (TI) techniques [12][13][14] that do not operate on a given MC sample but requires conducting a separate set of MC simulations. This is a robust approach that enables one to calculate differences, ΔS ab and ΔF ab , between two states a and b of a system; however, if the structural variance of such states is large (e.g., helical and hairpin states of a polypeptide) the integration from state a to b becomes difficult *Corresponding author

Calculation of the Entropy and Free Energy from Monte Carlo Simulations of a Peptide Stretched by an External Force

“…The present results demonstrate further the versatility of the HSMC method, which has been applied thus far to liquid argon and TIP3P water, 22.23 self-avoiding walks on a lattice, 26 and models of decaglycine. 24,25 To further enhance the performance of HSMC, we are extending it now to molecular dynamics (rather than MC) simulations, where our long-term goal is to develop software that enables one to apply the method to a general peptide consisting of any sequence of amino acid residues in implicit as well as explicit solvent.…”

“…26 In particular, two models of polyglycine molecules of 10 and 16 residues, described by the AMBER force field 27 in vacuum, were studied. One model is based on constant bond lengths and bond angles (the rigid model), and the other consists only of constant bond lengths (the flexible model).…”

Calculation of the entropy of random coil polymers with the hypothetical scanning Monte Carlo method

“…They are equal to the TI and series results for all N accept for N=599, where S scan is smaller due to a bias (for generating compact chains) that was not removed completely by the importance sampling procedure [Eq. (6)]. For N ≥ 249 the statistical errors of S scan are significantly larger than those of S TI .…”

Free volume hypothetical scanning molecular dynamics method for the absolute free energy of liquids

“…The probability (density) for a future atom to be located at the target position x′, is directly measured through raw counts, which is again done by centering a small counting cube at x′. The resulting transition probability density, ρ x′,FV = TP k , is calculated as, (25) This is the same expression as for ρ occ in Eq. (23), however the important difference is the different simulation conditions (FV vs. EV), therefore the TP densities are different.…”