A simulation method for calculating the absolute entropy and free energy of fluids: Application to liquid argon and water

Abstract

The hypothetical scanning (HS) method is a general approach for calculating the absolute entropy and free energy by analyzing Boltzmann samples obtained by Monte Carlo (MC) or molecular dynamics techniques. With HS applied to a fluid, each configuration i of the sample is reconstructed by adding its atoms gradually to the initially empty volume, i.e., by placing them in their positions at i using transition probabilities (TPs). At each step of the process, the volume is divided into two parts, the already visited part (the ''past'') and the ''future'' part, where obtaining the TP requires calculating partition functions over the future part in the presence of the frozen past. In recent publications, the TPs were calculated approximately by taking into account only partial future. Here we present a ''complete HSMC'' procedure, where the TPs are calculated from MC simulations carried out over the complete future. The complete HSMC method is applied to systems of liquid argon and the TIP3P model of water, and very good results for the free energy are obtained, as compared with results obtained by thermodynamic integration.T he absolute entropy, S, is a measure of order and is also a fundamental component of the absolute Helmholtz free energy, F, F ϭ E Ϫ TS, where E is the energy and T is the absolute temperature. The free energy is a key quantity as it constitutes the correct criterion of stability, which is mandatory for determining the relative populations of protein structures, for example. However, calculation of S and therefore F of a complex system, such as a peptide or a protein in water by computer simulation, is an extremely difficult problem (1-4). Using any simulation technique, it is relatively easy to calculate the energy, E i , which is ''written'' on system configuration i in terms of microscopic interactions (e.g., Lennard-Jones interactions of argon). On the other hand, calculating S Ϸ Ϫln P i B requires knowledge of the value of the Boltzmann probability, which is the sampling probability. However, P i B is not provided directly by the commonly used dynamical techniques, Metropolis Monte Carlo (MC) (5) and molecular dynamics (MD) (6 -7), therefore, F is unknown as well. In most cases, calculation of F is based on reversible thermodynamic integration (TI) techniques that provide the difference in the free energy, ⌬F m,n , between two states m and n, (e.g., helical and hairpin states of a peptide) and only when the absolute entropy of one state is known can that of the other be obtained. Although TI is a robust approach (refs. 1-4, 8, and 9 and references therein), for proteins, such integration is feasible only if the structural variance between the two states is very small; otherwise, the integration path can become prohibitively lengthy and complex. Therefore, it is important to develop methods that provide ln P i B at least approximately, enabling one to calculate the absolute F m and F n from two samples of the states m and n; in this case ⌬F m,n ϭ F m Ϫ F n can be calculated even for significa...

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

“…Notice that the fluctuation of an approximate free energy (i.e., based on an approximate probability density) is finite and it is expected to decrease as the approximation improves. 8,9,[20][21][22][23][24]50 C. Exact Scanning Procedure. The HSMC method is based on the ideas of the exact scanning method, which is a step-bystep construction procedure for a peptide.…”

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

“…Free energies are given as A c /εN, where A c is the configurational free energy defined in Eq. (20), ε is the standard Lennard-Jones energy parameter, and N is the number of atoms. …”

“…All values correspond to the configurational free energy, A c [Eq. (20], per atom, and in energy units of ε (i.e., A c /εN). Shown here are the free energy estimates F A , F B , F G B , F M , F G M , and F D , and the fluctuation in F A , σ A .…”

“…Another important point is that this methodology is capable, at least in principle, of yielding the exact Boltzmann probability (and therefore the exact free energy) in the limit of infinite sampling. Recently, we have successfully applied HSMC to liquids (argon19 -21 and water 20,21 ), peptides in vacuum, 22-24 and lattice polymers. 25,26 Present efforts in our lab have been aimed at adapting this methodology for new systems, and widening its applicability in general.…”

A black-box re-weighting analysis can correct flawed simulation data

“…To provide a proof of principle for our reweighting approach, we will determine free energy differences ( F), between states for various test systems (15)(16)(17)(18)(19)(20)(21).…”