Calculation of the entropy and free energy of peptides by molecular dynamics simulations using the hypothetical scanning molecular dynamics method

Developing end-point methods for absolute binding free energy calculation using the Boltzmann-quasiharmonic model

Understanding the physical forces underlying receptor-ligand binding requires robust methods for analyzing the binding thermodynamics. In end-point binding free energy methods the binding free energy is naturally decomposable into physically intuitive contributions such as the solvation free energy and configurational entropy that can provide insights. Here we present a new end-point method called EE-BQH (Effective Energy-Boltzmann-Quasiharmonic) which combines the Boltzmann-Quasiharmonic model for configurational entropy with different solvation free energy methods, such as the continuum solvent PBSA model and the integral equation-based 3D-RISM, to estimate the absolute binding free energy. We compare EE-BQH with other treatments of configurational entropy such as Quasiharmonic models in internal coordinates (QHIC) and in Cartesian coordinates (QHCC), and Normal Mode analysis (NMA), by testing them on the octa acids host-guest complexes from the SAMPL8 blind challenge. The accuracies in the calculated absolute binding free energies strongly depend on the configurational entropy and solvation free energy methods used. QHIC and BQH yield the best agreements with the established potential of mean force (PMF) estimates, with R² of ∼0.7 and mean unsigned error of ∼1.7 kcal mol^-1. These results from the end-point calculations are also in similar agreement with experiments. While 3D-RISM in combination with QHIC or BQH lead to reasonable correlations with the PMF results and experiments, the calculated absolute binding free energies are underestimated by ∼5 kcal mol^-1. While the binding is accompanied by a significant reduction in the ligand translational/rotational entropy, the change in the torsional entropy in these host-guest systems is slightly positive. Compared with BQH, QHIC underestimates the reduction of configurational entropy because of the non-Gaussian probability distributions in the ligand rotation and a small number of torsions. The study highlights the crucial role of configurational entropy in determining binding and demonstrates the potential of using the new end-point method to provide insights in more complex protein-ligand systems.

The origin of the cooperativity in the streptavidin-biotin system: A computational investigation through molecular dynamics simulations

Previous experimental study measuring the binding affinities of biotin to the wild type streptavidin (WT) and three mutants (S45A, D128A and S45A/D128A double mutant) has shown that the loss of binding affinity from the double mutation is larger than the direct sum of those from two single mutations. The origin of this cooperativity has been investigated in this work through molecular dynamics simulations and the end-state free energy method using the polarized protein-specific charge. The results show that this cooperativity comes from both the enthalpy and entropy contributions. The former contribution mainly comes from the alternations of solvation free energy. Decomposition analysis shows that the mutated residues nearly have no contributions to the cooperativity. Instead, N49 and S88, which are located at the entry of the binding pocket and interact with the carboxyl group of biotin, make the dominant contribution among all the residues in the first binding shell around biotin.

Macromolecular Entropy Can Be Accurately Computed from Force

A method is presented to evaluate a molecule's entropy from the atomic forces calculated in a molecular dynamics simulation. Specifically, diagonalization of the mass-weighted force covariance matrix produces eigenvalues which in the harmonic approximation can be related to vibrational frequencies. The harmonic oscillator entropies of each vibrational mode may be summed to give the total entropy. The results for a series of hydrocarbons, dialanine and a β hairpin are found to agree much better with values derived from thermodynamic integration than results calculated using quasiharmonic analysis. Forces are found to follow a harmonic distribution more closely than coordinate displacements and better capture the underlying potential energy surface. The method's accuracy, simplicity, and computational similarity to quasiharmonic analysis, requiring as input force trajectories instead of coordinate trajectories, makes it readily applicable to a wide range of problems.

Absolute free energies of biomolecules from unperturbed ensembles

Computing the absolute free energy of a macromolecule's structural state, F, is a challenging problem of high relevance. This study presents a method that computes F using only information from an unperturbed simulation of the macromolecule in the relevant conformational state, ensemble, and environment. Absolute free energies produced by this method, dubbed Valuation of Local Configuration Integral with Dynamics (VALOCIDY), enable comparison of alternative states. For example, comparing explicitly solvated and vaporous states of amino acid side-chain analogs produces solvation free energies in good agreement with experiments. Also, comparisons between alternative conformational states of model heptapeptides (including the unfolded state) produce free energy differences in agreement with data from μs molecular-dynamics simulations and experimental propensities. The potential of using VALOCIDY in computational protein design is explored via a small design problem of stabilizing a β-turn structure. When VALOCIDY-based estimation of folding free energy is used as the design metric, the resulting sequence folds into the desired structure within the atomistic force field used in design. The VALOCIDY-based approach also recognizes the distinct status of the native sequence regardless of minor details of the starting template structure, in stark contrast with a traditional fixed-backbone approach.

The F130L mutation in streptavidin reduces its binding affinity to biotin through electronic polarization effect

Recently, Baugh et al. discovered that a distal point mutation (F130L) in streptavidin causes no distinct variation to the structure of the binding pocket but a 1000-fold reduction in biotin binding affinity. In this work, we carry out molecular dynamics simulations and apply an end-state free energy method to calculate the binding free energies of biotin to wild type streptavidin and its F130L mutant. The absolute binding affinities based on AMBER charge are repulsive, and the mutation induced binding loss is underestimated. When using the polarized protein-specific charge, the absolute binding affinities are significantly enhanced. In particular, both the absolute and relative binding affinities are in line with the experimental measurements. Further investigation indicates that polarization effect is indispensable in both the generation of structural ensembles and the calculation of interaction energies. This work verifies Baugh's conjecture that electrostatic polarization effect plays an essential role in modulating the binding affinity of biotin to the streptavidin through F130L mutation.

Develop and test a solvent accessible surface area-based model in conformational entropy calculations

It is of great interest in modern drug design to accurately calculate the free energies of protein-ligand or nucleic acid-ligand binding. MM-PBSA (molecular mechanics Poisson-Boltzmann surface area) and MM-GBSA (molecular mechanics generalized Born surface area) have gained popularity in this field. For both methods, the conformational entropy, which is usually calculated through normal-mode analysis (NMA), is needed to calculate the absolute binding free energies. Unfortunately, NMA is computationally demanding and becomes a bottleneck of the MM-PB/GBSA-NMA methods. In this work, we have developed a fast approach to estimate the conformational entropy based upon solvent accessible surface area calculations. In our approach, the conformational entropy of a molecule, S, can be obtained by summing up the contributions of all atoms, no matter they are buried or exposed. Each atom has two types of surface areas, solvent accessible surface area (SAS) and buried SAS (BSAS). The two types of surface areas are weighted to estimate the contribution of an atom to S. Atoms having the same atom type share the same weight and a general parameter k is applied to balance the contributions of the two types of surface areas. This entropy model was parametrized using a large set of small molecules for which their conformational entropies were calculated at the B3LYP/6-31G* level taking the solvent effect into account. The weighted solvent accessible surface area (WSAS) model was extensively evaluated in three tests. For convenience, TS values, the product of temperature T and conformational entropy S, were calculated in those tests. T was always set to 298.15 K through the text. First of all, good correlations were achieved between WSAS TS and NMA TS for 44 protein or nucleic acid systems sampled with molecular dynamics simulations (10 snapshots were collected for postentropy calculations): the mean correlation coefficient squares (R²) was 0.56. As to the 20 complexes, the TS changes upon binding; TΔS values were also calculated, and the mean R² was 0.67 between NMA and WSAS. In the second test, TS values were calculated for 12 proteins decoy sets (each set has 31 conformations) generated by the Rosetta software package. Again, good correlations were achieved for all decoy sets: the mean, maximum, and minimum of R² were 0.73, 0.89, and 0.55, respectively. Finally, binding free energies were calculated for 6 protein systems (the numbers of inhibitors range from 4 to 18) using four scoring functions. Compared to the measured binding free energies, the mean R² of the six protein systems were 0.51, 0.47, 0.40, and 0.43 for MM-GBSA-WSAS, MM-GBSA-NMA, MM-PBSA-WSAS, and MM-PBSA-NMA, respectively. The mean rms errors of prediction were 1.19, 1.24, 1.41, 1.29 kcal/mol for the four scoring functions, correspondingly. Therefore, the two scoring functions employing WSAS achieved a comparable prediction performance to that of the scoring functions using NMA. It should be emphasized that no minimization was performed prior to the WSAS calculation in the last test. Although WSAS is not as rigorous as physical models such as quasi-harmonic analysis and thermodynamic integration (TI), it is computationally very efficient as only surface area calculation is involved and no structural minimization is required. Moreover, WSAS has achieved a comparable performance to normal-mode analysis. We expect that this model could find its applications in the fields like high throughput screening (HTS), molecular docking, and rational protein design. In those fields, efficiency is crucial since there are a large number of compounds, docking poses, or protein models to be evaluated. A list of acronyms and abbreviations used in this work is provided for quick reference.

Free energy considerations for nucleic acids with dangling ends near a surface: a coarse grained approach

A coarse grained model for the thermodynamics of nucleic acid hybridization near surfaces has been extended and parameterized to consider the contribution of unpaired dangling ends. The parameters of the model differ when representing a double stranded DNA section or a single stranded DNA section. The thermodynamic effects of the possibility of different dangling end combinations were considered in the presence of different types of surfaces. Configurational sampling was achieved by the Metropolis Monte Carlo method. To gain a more complete picture of the free energy changes, an estimation of the conformational entropy was included. We find a strong thermodynamic effect for dangling mismatches due to sequence requirements when they are nearer the surface as opposed to being held away from the surface.

Calculation of configurational entropy with a Boltzmann-quasiharmonic model: the origin of high-affinity protein-ligand binding

Accurate assessment of configurational entropy remains a large challenge in biology. While many methods exist to calculate configurational entropy, there is a balance between accuracy and computational demands. Here we calculate ligand and protein conformational entropies using the Boltzmann-quasiharmonic (BQH) method, which treats the first-order entropy term by the Boltzmann expression for entropy while determining correlations using the quasiharmonic model. This method is tested by comparison with the exact Clausius expression for entropy on a range of test molecules ranging from small ligands to a protein. Using the BQH method, we then analyze the rotational and translational (R/T) entropy change upon ligand binding for five protein complexes to explore the origins of extremely tight affinity. The results suggest that in these systems such affinity is achieved by a combination of simultaneously maintaining good protein-ligand contacts while allowing significant residual R/T motion of the ligand through suitable protein motions.

Relative stability of the open and closed conformations of the active site loop of streptavidin

The eight-residue surface loop, 45-52 (Ser, Ala, Val, Gly, Asn, Ala, Glu, Ser), of the homotetrameric protein streptavidin has a "closed" conformation in the streptavidin-biotin complex, where the corresponding binding affinity is one of the strongest found in nature (ΔG ∼ -18 kcal∕mol). However, in most of the crystal structures of apo (unbound) streptavidin, the loop conformation is "open" and typically exhibits partial disorder and high B-factors. Thus, it is plausible to assume that the loop structure is changed from open to closed upon binding of biotin, and the corresponding difference in free energy, ΔF = F(open) - F(closed) in the unbound protein, should therefore be considered in the total absolute free energy of binding. ΔF (which has generally been neglected) is calculated here using our "hypothetical scanning molecular-dynamics" (HSMD) method. We use a protein model in which only the atoms closest to the loop are considered (the "template") and they are fixed in the x-ray coordinates of the free protein; the x-ray conformation of the closed loop is attached to the same (unbound) template and both systems are capped with the same sphere of TIP3P water. Using the force field of the assisted model building with energy refinement (AMBER), we carry out two separate MD simulations (at temperature T = 300 K), starting from the open and closed conformations, where only the atoms of the loop and water are allowed to move (the template-water and template-loop interactions are considered). The absolute F(open) and F(closed) (of loop + water) are calculated from these trajectories, where the loop and water contributions are obtained by HSMD and a thermodynamic integration (TI) process, respectively. The combined HSMD-TI procedure leads to total (loop + water) ΔF = -27.1 ± 2.0 kcal∕mol, where the entropy TΔS constitutes 34% of ΔF, meaning that the effect of S is significant and should not be ignored. Also, ΔS is positive, in accord with the high flexibility of the open loop observed in crystal structures, while the energy ΔE is unexpectedly negative, thus also adding to the stability of the open loop. The loop and the 250 capped water molecules are the largest system studied thus far, which constitutes a test for the efficiency of HSMD-TI; this efficiency and technical issues related to the implementation of the method are also discussed. Finally, the result for ΔF is a prediction that will be considered in the calculation of the absolute free energy of binding of biotin to streptavidin, which constitutes our next project.

Extending fragment-based free energy calculations with library Monte Carlo simulation: annealing in interaction space

Pre-calculated libraries of molecular fragment configurations have previously been used as a basis for both equilibrium sampling (via library-based Monte Carlo) and for obtaining absolute free energies using a polymer-growth formalism. Here, we combine the two approaches to extend the size of systems for which free energies can be calculated. We study a series of all-atom poly-alanine systems in a simple dielectric solvent and find that precise free energies can be obtained rapidly. For instance, for 12 residues, less than an hour of single-processor time is required. The combined approach is formally equivalent to the annealed importance sampling algorithm; instead of annealing by decreasing temperature, however, interactions among fragments are gradually added as the molecule is grown. We discuss implications for future binding affinity calculations in which a ligand is grown into a binding site.

Entropy localization in proteins

The configurational entropy of a protein is under physiological conditions a major contributor to the free energy. Its quantitative characterization is therefore an important step toward the understanding of protein function. The configurational entropy of the oncoprotein MDM2, whose determination is a challenge by experiment alone, is studied here by means of 0.4 μs molecular dynamics computer simulations in both the presence and absence of the p53-peptide ligand. By characterizing protein motions in dihedral angle space, it is found that the motional amplitudes change considerably upon ligand binding while correlations between dihedral angle motions are remarkably well conserved. This applies for backbone and side-chain dihedral angle pairs at both short- and long-range distance to the binding site. As a direct consequence, the change of the configurational entropy can be decomposed into a sum of local contributions. This significantly facilitates the understanding of the relationship between protein dynamics and thermodynamics, which is important, for example, in the context of protein-ligand and protein-protein interactions. The findings also have implications for the direct derivation of entropy changes from site-specific dynamics measurements as afforded by NMR spectroscopy.

Entropy and Free Energy of a Mobile Loop Based on the Crystal Structures of the Free and Bound Proteins

A mobile loop changes its conformation from "open" (free enzyme) to "closed" upon ligand binding. The difference in the Helmholtz free energy, ΔF(loop) between these states sheds light on the mechanism of binding. With our "hypothetical scanning molecular dynamics" (HSMD-TI) method ΔF(loop) = F(free) - F(bound) where F(free) and F(bound) are calculated from two MD samples of the free and bound loop states; the contribution of water is obtained by a thermodynamic integration (TI) procedure. In previous work the free and bound loop structures were both attached to the same "template" which was "cut" from the crystal structure of the free protein. Our results for loop 287-290 of AcetylCholineEsterase agree with the experiment, ΔF(loop)~ -4 kcal/mol if the density of the TIP3P water molecules capping the loop is close to that of bulk water, i.e., N(water) = 140 - 180 waters in a sphere of a 18 Å radius. Here we calculate ΔF(loop) for the more realistic case, where two templates are "cut" from the crystal structures, 2dfp.pdb (bound) and 2ace.pdb (free), where N(water) = 40 - 160; this requires adding a computationally more demanding (second) TI procedure. While the results for N(water) ≤ 140 are computationally sound, ΔF(loop) is always positive (18 ± 2 kcal/mol for N(water) = 140). These (disagreeing) results are attributed to the large average B-factor, 41.6 of 2dfp (23.4 Å(2) for 2ace). While this conformational uncertainty is an inherent difficulty, the (unstable) results for N(water) = 160 suggest that it might be alleviated by applying different (initial) structural optimizations to each template.

Methods for calculating the absolute entropy and free energy of biological systems based on ideas from polymer physics

The commonly used simulation techniques, Metropolis Monte Carlo (MC) and molecular dynamics (MD) are of a dynamical type which enables one to sample system configurations i correctly with the Boltzmann probability, P(i)(B), while the value of P(i)(B) is not provided directly; therefore, it is difficult to obtain the absolute entropy, S approximately -ln P(i)(B), and the Helmholtz free energy, F. With a different simulation approach developed in polymer physics, a chain is grown step-by-step with transition probabilities (TPs), and thus their product is the value of the construction probability; therefore, the entropy is known. Because all exact simulation methods are equivalent, i.e. they lead to the same averages and fluctuations of physical properties, one can treat an MC or MD sample as if its members have rather been generated step-by-step. Thus, each configuration i of the sample can be reconstructed (from nothing) by calculating the TPs with which it could have been constructed. This idea applies also to bulk systems such as fluids or magnets. This approach has led earlier to the "local states" (LS) and the "hypothetical scanning" (HS) methods, which are approximate in nature. A recent development is the hypothetical scanning Monte Carlo (HSMC) (or molecular dynamics, HSMD) method which is based on stochastic TPs where all interactions are taken into account. In this respect, HSMC(D) can be viewed as exact and the only approximation involved is due to insufficient MC(MD) sampling for calculating the TPs. The validity of HSMC has been established by applying it first to liquid argon, TIP3P water, self-avoiding walks (SAW), and polyglycine models, where the results for F were found to agree with those obtained by other methods. Subsequently, HSMD was applied to mobile loops of the enzymes porcine pancreatic alpha-amylase and acetylcholinesterase in explicit water, where the difference in F between the bound and free states of the loop was calculated. Currently, HSMD is being extended for calculating the absolute and relative free energies of ligand-enzyme binding. We describe the whole approach and discuss future directions.

Estimating absolute configurational entropies of macromolecules: the minimally coupled subspace approach

We develop a general minimally coupled subspace approach (MCSA) to compute absolute entropies of macromolecules, such as proteins, from computer generated canonical ensembles. Our approach overcomes limitations of current estimates such as the quasi-harmonic approximation which neglects non-linear and higher-order correlations as well as multi-minima characteristics of protein energy landscapes. Here, Full Correlation Analysis, adaptive kernel density estimation, and mutual information expansions are combined and high accuracy is demonstrated for a number of test systems ranging from alkanes to a 14 residue peptide. We further computed the configurational entropy for the full 67-residue cofactor of the TATA box binding protein illustrating that MCSA yields improved results also for large macromolecular systems.

Toward accurate microscopic calculation of solvation entropies: extending the restraint release approach to studies of solvation effects

The evaluation of the solvation entropies is a major conceptual and practical challenge. On the one hand, it is interesting to quantify the factors that are responsible for the solvation entropies in solutions, whereas on the other, it is essential to be able to assess the contributions of the solvation entropies to the binding free energies and related properties. In fact, the solvation entropies are neglected in almost all of the studies of the binding entropies. The main problem is that widely used approaches, such as the quasiharmonic (QH) approximation, do not provide reliable results particularly in cases of shallow potential and multidimensional surfaces while brute force evaluations of the entropic effects by simulating temperature dependence of the free energy converges very slowly. This paper addresses the above issue by starting with an analysis of the factors that are responsible for the negative solvation entropy of ions, showing that it is not due to the change in the solvent vibration modes or to the solvent force constant but to the changes in the solvent configurational space upon change in the solute charges. We begin by clarifying that when one deals with aqueous solutions, it is easy to evaluate the corresponding entropic effect by the Langevin dipole (LD) treatment. However, in this work we are interested in developing a general microscopic tool that can be used to study similar effects in the proteins. To this end, we explore the ability of our restraint release (RR) approach to evaluate the solvation entropy. We start this analysis by reviewing the foundation of this approach and in particular, the requirements of minimizing the enthalpy contribution to the RR free energy. We then establish that our approach is not a specialized harmonic treatment but a rather powerful general approach. Moving to the main topic of this work, we demonstrate that the RR approach provides quantitative results for the solvation entropies of monovalent and divalent ions and effectively captures the physics of these entropic effects. The success of the current approach indicates that it should be applicable to the studies of the solvation entropies in the proteins and also, in examining hydrophobic effects. Thus, we believe that the RR approach provides a powerful tool for evaluating the corresponding contributions to the binding entropies and, eventually, to the binding free energies. This holds promise for extending the information theory modeling to proteins and protein-ligand complexes in aqueous solutions and consequently, facilitating computer-aided drug design.

Adaptive anisotropic kernels for nonparametric estimation of absolute configurational entropies in high-dimensional configuration spaces

The quasiharmonic approximation is the most widely used estimate for the configurational entropy of macromolecules from configurational ensembles generated from atomistic simulations. This method, however, rests on two assumptions that severely limit its applicability, (i) that a principal component analysis yields sufficiently uncorrelated modes and (ii) that configurational densities can be well approximated by Gaussian functions. In this paper we introduce a nonparametric density estimation method which rests on adaptive anisotropic kernels. It is shown that this method provides accurate configurational entropies for up to 45 dimensions thus improving on the quasiharmonic approximation. When embedded in the minimally coupled subspace framework, large macromolecules of biological interest become accessible, as demonstrated for the 67-residue coldshock protein.

Absolute free energy and entropy of a mobile loop of the enzyme acetylcholinesterase

The loop 287-290 (Ile, Phe, Arg, and Phe) of the protein acetylcholinesterase (AChE) changes its structure upon interaction of AChE with diisopropylphosphorofluoridate (DFP). Reversible dissociation measurements suggest that the free-energy (F) penalty for the loop displacement is DeltaF=Ffree-Fbound approximately -4 kcal/mol. Therefore, this loop has been the target of two studies by Olson's group for testing the efficiency of procedures for calculating F. In this paper, we test for the first time the performance of our "hypothetical scanning molecular dynamics" (HSMD) method and the validity of the related modeling for a loop with bulky side chains in explicit water. Thus, we consider only atoms of the protein that are the closest to the loop (they constitute the "template"), where the rest of the atoms are ignored. The template's atoms are fixed in the X-ray coordinates of the free protein, and the loop is capped with a sphere of TIP3P water molecules; also, the X-ray structure of the bound loop is attached to the free template. We carry out two separate MD simulations starting from the free and bound X-ray structures, where only the atoms of the loop and water are allowed to move while the template-water and template-loop (AMBER) interactions are considered. The absolute Ffree and Fbound (of the loop and water) are calculated from the corresponding trajectories. A main objective of this paper is to assess the reliability of this model, and for this several template sizes are studied capped with 80-220 water molecules. We find that consistent results for the free energy (which also agree with the experimental data above) require a template larger than a minimal size and a number of water molecules approximately equal to the experimental density of bulk water. For example, we obtain DeltaFtotal=DeltaFwater+DeltaFloop=-3.1+/-2.5 and -3.6+/-4 kcal/mol for a template consisting of 944 atoms and a sphere containing 160 and 180 waters, respectively. Our calculations demonstrate the important contribution of water to the total free energy. Namely, for water densities close to the experimental value, DeltaFwater is always negative leading thereby to a negative DeltaFtotal (while DeltaFloop is always positive). Also, the contribution of the water entropy TDeltaSwater to DeltaFtotal is significant. Various aspects related to the efficiency of HSMD are tested and improved, and plans for future studies are discussed.

Configurational entropy in protein-peptide binding: computational study of Tsg101 ubiquitin E2 variant domain with an HIV-derived PTAP nonapeptide

Configurational entropy is thought to influence biomolecular processes, but there are still many open questions about this quantity, including its magnitude, its relationship to molecular structure, and the importance of correlation. The mutual information expansion (MIE) provides a novel and systematic approach to extracting configurational entropy changes due to correlated motions from molecular simulations. We present the first application of the MIE method to protein-ligand binding using multiple molecular dynamics simulations to study the association of the ubiquitin E2 variant domain of the protein Tsg101 and an HIV-derived nonapeptide. This investigation utilizes the second-order MIE approximation, which accounts for correlations between all pairs of degrees of freedom. The computed change in configurational entropy is large and has a major contribution from changes in pairwise correlation. The results also reveal intricate structure-entropy relationships. Thus, the present analysis suggests that in order for a model of binding to be accurate, it must include a careful accounting of configurational entropy changes.

In silico relationship between configurational entropy and soft degrees of freedom in proteins and peptides

Configurational entropies from temperature-dependent free energy calculations of proteins and peptides are compared with entropies determined as the sum of the Shannon entropies of a kernel density estimate of the dihedral angle probability distributions. We find that the entropy changes can be separated into independent contributions from hard and soft degrees of freedom. Furthermore, because the effects of motional correlations between dihedral angles on entropy differences are small, the functional role of configurational entropy changes can be interpreted in a spatially resolved manner.

Entropy and free energy of a mobile protein loop in explicit water

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 CarloHSMC) has been extended systematically to systems of increasing complexity, where the most recent is the seven-residue mobile loop, 304-310 (Gly-His-Gly-Ala-Gly-Gly-Ser) of the enzyme porcine pancreatic alpha-amylase modeled by the AMBER force field and AMBER with the implicit solvation GB/SA (paper I, Cheluvaraja, S.; Meirovitch, H. J. Chem. Theory Comput. 2008, 4, 192). 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, Ffree-Fbound=-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, Sfree(loop) and Sbound(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.

Evaluation of configurational entropy methods from peptide folding-unfolding simulation

A 4-micros molecular dynamics simulation of the second beta-hairpin of the B1 domain of streptococcal protein G is used to characterize the free energy surface and to evaluate different configurational entropy estimators. From the equilibrium folding-unfolding trajectory, 200 000 conformers are clustered according to their root-mean-square deviation (RMSD). The height of the free energy barrier between pairs of clusters is found to be significantly correlated with their pairwise RMSD. Relative free energies and relative configurational entropies of the clusters are determined by explicit evaluation of the partition functions of the different clusters. These entropies are used to evaluate different entropy estimators for the largest 20 clusters as well as a subensemble comprising exclusively extended conformers. It is found that the quasi-harmonic entropy estimator operating in dihedral angle space performs better than the one using Cartesian coordinates. A recent generalization of the quasi-harmonic approach that computes Shannon entropies of probability distributions obtained by projecting the conformers along the eigenvectors of the covariance matrix performs similarly well. For the best entropy estimators, a linear correlation coefficient between 0.92 and 0.97 is found. Unexpectedly, when correlations between dihedral angles are neglected, the agreement with the reference entropies improved.

Extraction of configurational entropy from molecular simulations via an expansion approximation

A method is presented for extracting the configurational entropy of solute molecules from molecular dynamics simulations, in which the entropy is computed as an expansion of multidimensional mutual information terms, which account for correlated motions among the various internal degrees of freedom of the molecule. The mutual information expansion is demonstrated to be equivalent to estimating the full-dimensional configurational probability density function (PDF) using the generalized Kirkwood superposition approximation (GKSA). While the mutual information expansion is derived to the full dimensionality of the molecule, the current application uses a truncated form of the expansion in which all fourth- and higher-order mutual information terms are neglected. Truncation of the mutual information expansion at the nth order is shown to be equivalent to approximating the full-dimensional PDF using joint PDFs with dimensionality of n or smaller by successive application of the GKSA. The expansion method is used to compute the absolute (classical) configurational entropy in a basis of bond-angle-torsion internal coordinates for several small molecules as well as the change in entropy upon binding for a small host-guest system. Convergence properties of the computed entropy values as a function of simulation time are investigated and comparisons are made with entropy values from the second generation Mining Minima software. These comparisons demonstrate a deviation in -TS of no more than about 2 kcal/mol for all cases in which convergence has been obtained.

Recent developments in methodologies for calculating the entropy and free energy of biological systems by computer simulation

The Helmholtz free energy, F, plays an important role in proteins because of their rugged potential energy surface, which is 'decorated' with a tremendous number of local wells (denoted microstates, m). F governs protein folding, whereas differences DeltaF(mn) determine the relative populations of microstates that are visited by a flexible cyclic peptide or a flexible protein segment (e.g. a surface loop). Recently developed methodologies for calculating DeltaF(mn) (and entropy differences, DeltaS(mn)) mainly use thermodynamic integration and calculation of the absolute F; interesting new approaches in these categories are the adaptive integration method and the hypothetical scanning molecular dynamics method, respectively.