The hypothetical scanning ͑HS͒ method is a general approach for calculating the absolute entropy S and free energy F by analyzing Boltzmann samples obtained by Monte Carlo or molecular dynamics techniques. With HS applied to a fluid, each configuration i of the sample is reconstructed by gradually placing the molecules in their positions at i using transition probabilities ͑TPs͒. At each step of the process the system is divided into two parts, the already treated molecules ͑the ''past''͒, which are fixed, and the as yet unspecified ͑mobile͒ ''future'' molecules. Obtaining the TP exactly requires calculating partition functions over all positions of the future molecules in the presence of the frozen past, thus it is customary to invoke various approximations to best represent these quantities. In a recent publication ͓Proc. Natl. Acad. Sci. USA 101, 9235 ͑2004͔͒ we developed a version of HS called complete HSMC, where each TP is calculated from an MC simulation involving all of the future molecules ͑the complete future͒; the method was applied very successfully to Lennard-Jones systems ͑liquid argon͒ and a box of TIP3P water molecules. In its basic implementation the method provides lower and upper bounds for F, where the latter can be evaluated only for relatively small systems. Here we introduce a new expression for an upper bound, which can be evaluated for larger systems. We also propose a new exact expression for F and verify its effectiveness. These free energy functionals lead to significantly improved accuracy ͑as applied to the liquid systems above͒ which is comparable to our thermodynamic integration results. We formalize and discuss theoretical aspects of HSMC that have not been addressed in previous studies. Additionally, several functionals are developed and shown to provide the free energy through the analysis of a single configuration.

The scanning simulation method is applied to a model of polymer adsorption in which a single self-avoiding walk is terminally attached to an attracting impenetrable surface on a simple cubic lattice. Relatively long chains are studied, of up to 1000 steps, which enable us to obtain new estimates for the reciprocal transition temperature IEI/kBTa = Ba = 0.291 ± 0.001 (E is the interaction energy ofa monomer with the surface), the crossover exponent ~ = 0.530 ± 0.007 and the free energy exponents at Ta, r~B = 1.304 ± 0.006 and r~~ = 0.805 ± 0.015. At T = 00 we obtain, rl = 0.687 ± 0.005, rll = -0.38 ± 0.02, and the effective coordination number q = 4.6839 ± 0.0001, which are in good agreement with estimates obtained by other methods. At T> Ta we demonstrate the existence of strong correction to scaling for the perpendicular part of the mean-square end-to-end distance (R 2) 1 and for the monomer concentration profile p(z) (z is the distance from the surface). At T = 00 the leading correction to scaling term for (R 2) 1 is c/ HI', where cz -0.9 and ",zO.4 is close to 0.5 obtained for the random walk model in the preceding paper. This means that the asymptotic regime, in which these corrections become negligible, corresponds to a large polymer length that is not realized experimentally. Close enough to Ta we demonstrate for our lattice model the validity of various scaling forms predicted by Eisenriegler, Kremer, and Binder [J. Chem. Phys. 77, 6296 (1982)] for a continuum model on the basis of the n-vector model.= -dkB T and denote their critical values by Ta and Ba

Abstract:The hypothetical scanning molecular dynamics (HSMD) method is a relatively new technique for calculating the absolute entropy, S, and free energy, F, from a given sample generated by any simulation procedure. Thus, each sample conformation, i, is reconstructed by calculating transition probabilities that their product leads to the probability of i, hence to the entropy. HSMD is an exact method where all interactions are considered, and the only approximation is due to insufficient sampling. In previous studies HSMD (and HS Monte Carlo -HSMC) has been applied very successfully to liquid argon, TIP3P water, self-avoiding walks, and peptides in a R-helix, extended, and hairpin microstates. In this paper HSMD is developed further as applied to the flexible 7-residue surface loop, 304-310 (Gly-His-Gly-Ala-Gly-GlySer) of the enzyme porcine pancreatic R-amylase. We are mainly interested in entropy and free energy differences ∆S ) S free -S bound (and ∆F)F free -F bound ) between the free and bound microstates of the loop, which are obtained from two separate MD samples of these microstates without the need to carry out thermodynamic integration. As for peptides, we find that relatively large systematic errors in S free and S bound (and F free and F bound ) are cancelled in ∆S (∆F) which is thus obtained efficiently with high accuracy, i.e., with a statistical error of 0.1-0.2 kcal/mol (T)300 K) using the AMBER force field and AMBER with the implicit solvation GB/SA. We provide theoretical arguments in support of this cancellation, discuss in detail the problems involved in the computational definition of a microstate in conformational space, suggest potential ways for enhancing efficiency further, and describe the next development where explicit water will replace implicit solvation.

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Hypothetical scanning (HS) is a method for calculating the absolute entropy S and free energy F from a sample generated by any simulation technique. With this approach each sample configuration is reconstructed with the help of transition probabilities (TPs) and their product leads to the configuration's probability, hence to the entropy. Recently a new way for calculating the TPs by Monte Carlo (MC) simulations has been suggested, where all system interactions are taken into account. Therefore, this method--called HSMC--is in principle exact where the only approximation is due to insufficient sampling. HSMC has been applied very successfully to liquid argon, TIP3P water, self-avoiding walks on a lattice, and peptides. Because molecular dynamics (MD) is considered to be significantly more efficient than MC for a compact polymer chain, in this paper HSMC is extended to MD simulations as applied to peptides. Like before, we study decaglycine in vacuum but for the first time also a peptide with side chains, (Val)(2)(Gly)(6)(Val)(2). The transition from MC to MD requires implementing essential changes in the reconstruction process of HSMD. Results are calculated for three microstates, helix, extended, and hairpin. HSMD leads to very stable differences in entropy TDeltaS between these microstates with small errors of 0.1-0.2 kcal/mol (T=100 K) for a wide range of calculation parameters with extremely high efficiency. Various aspects of HSMD and plans for future work are discussed.

Estimation of the energy from a given Boltzmann sample is straightforward since one just has to average the contribution of the individual configurations. On the other hand, calculation of the absolute entropy, S (hence the absolute free energy F) is difficult because it depends on the entire (unknown) ensemble. We have developed a new method called, "the hypothetical scanning molecular dynamics" (HSMD) for calculating the absolute S from a given sample (generated by any simulation technique). In other words, S (like the energy) is "written" on the sample configurations, where HSMD provides a prescription of how to "read" it. In practice, each sample conformation, i is reconstructed with transition probabilities and their product leads to the probability of i, hence to the entropy. HSMD is an exact method where all interactions are considered and the only approximation is due to insufficient sampling. In previous studies HSMD (and HS Monte Carlo -HSMC) has been extended systematically to systems of increasing complexity, where the most recent is the 7-residue mobile loop, 304-310 (Gly-His-Gly-Ala-Gly-Gly-Ser) of the enzyme porcine pancreatic α-amylase modeled by the AMBER force field and AMBER with the implicit solvation GB/SA (paper I). In the present paper we make a step further and extend HSMD to the same loop capped with TIP3P explicit water at 300 K. As in paper I, we are mainly interested in entropy and free energy differences between the free and bound microstates of the loop, which are obtained from two separate MD samples of these microstates. The contribution of the loop to S and F is calculated by HSMD and that of water by a particular thermodynamic integration procedure. As expected, the free microstate is more stable than the bound microstate by a total free energy difference, F free − F bound = −4.8 ± 1, as compared to −25.5 kcal/mol obtained with GB/SA. We find that relatively large systematic errors in the loop entropies, S free (loop) and S bound (loop) are cancelled in their difference which is thus obtained efficiently and with high accuracy, i.e. with a statistical error of 0.1 kcal/mol. This cancellation, which has been observed in previous HSMD studies, is in accord with theoretical arguments given in paper I.

A new approach, the hypothetical scanning Monte Carlo (HSMC), for calculating the absolute entropy, S, and free energy, F, has been introduced recently and applied first to fluids (argon and water) and later to peptides. In this paper the method is further developed for peptide chains in vacuum. S is calculated from a given MC sample by reconstructing each sample conformation i step-by-step, i.e., calculating transition probabilities (TPs) for the dihedral and bond angles and fixing the related atoms at their positions. At step k of the process the chain's coordinates that have already been determined are kept fixed (the "frozen past") and TP(k) is obtained from a MC simulation of the "future" part of the chain whose TPs as yet have not been determined; when the process is completed the contribution of conformation i to the entropy is, S(i) approximately -ln Pi(k) TP(k). In a recent paper we studied polyglycine chains, modeled by the AMBER force field with constant bond lengths and bond angles (the rigid model). Decaglycine [(Gly)(10)] was studied in the helical, extended, and hairpin microstates, while (Gly)(16) was treated only in the first two microstates. In this paper the samples are increased and restudied, (Gly)(16) is also investigated in the hairpin microstate, and for (Gly)(10) approximations are tested where only part of the future is considered for calculating the TPs. We calculate upper and lower bounds for F and demonstrate that like for fluids, F can be obtained from multiple reconstructions of a single conformation. We also test a more realistic model of (Gly)(10) where the bond angles are allowed to move (the flexible model). Very accurate results for S and F are obtained which are compared to results obtained by the quasiharmonic approximation and the local states method. Thus, differences in entropy and free energy between the three microstates are obtained within errors of 0.1-0.3 kcal/mol. The HSMC method can be applied to a macromolecule with any degree of flexibility, ranging from local fluctuations to a random coil. The present results demonstrate that the difference in stability, DeltaF(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 long-term goal is to extend this method to any peptide and apply it to a peptide immersed in a box with explicit water.

Under certain experimental conditions peptide solutions can generate sufficient NMR data for structure determination. Yet, unlike proteins, peptides typically prevail as ensembles of interconverting structures, and therefore, the experimental variables (which are intensities of cross peaks in nuclear Overhauser enhancement spectroscopy (NOESY) spectra, or NOEs) are average quantities. The process of structure elucidation is complex, and many related questions are still open to debate. We have developed a new theoretical methodology for treating ensembles of interconverting conformations which is based on purely thermodynamic grounds. The peptide is described by a potential energy function, and its conformational space is viewed as a collection of microstates, which are local conformational regions around energy minima. The overall methodology enables one to identify the thermodynamically most stable microstates, determine their populations, calculate the individual microstate NOEs, and obtain the overall NOEs as averages over the individual contributions, weighted by the microstate populations. In this paper we develop for the first time theoretical methods for calculating the relative contribution of microstates to the partition function, as their minimum energy is increased above the global energy minimum (GEM). This is necessary for determining which microstates should be considered in detail in the next stage of the analysis. The new methods presented herein are applied to the pentapeptide Leu-enkephalin (H-Tyr-Gly-Gly-Phe-Leu-OH) described by the potential energy function ECEPP, with calculations carried out at 280 K. We find that the microstates contained within relatively small ranges of 2 to 3 kcal/mol above the GEM (or above the lowest harmonic free energy) constitute 0.59 to 0.75 of the partition function. This should be compared with energy ranges up to 15 kcaymol set arbitrarily in previous studies. A detailed comparison of theoretical predictions with experimental data, obtained from a cryoprotective Leu-enkephalin solution at 280 K, will be carried out in the next paper of this series.