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.
Citation: Hensen U, Lange OF, Grubmüller H (2010) Estimating Absolute Configurational Entropies of Macromolecules: The Minimally Coupled Subspace Approach. PLoS ONE 5(2): e9179. https://doi.org/10.1371/journal.pone.0009179
Editor: Jörg Langowski, German Cancer Research Center, Germany
Received: May 10, 2009; Accepted: January 25, 2010; Published: February 23, 2010
Copyright: © 2010 Hensen et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Funding: U.H. was supported by the Deutsche Forschungsgemeinschaft (research training group 782). O.F.L. was supported by the Human Frontiers of Science Program and by the Volkswagen Foundation, Grant I/80436. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.
Competing interests: The authors have declared that no competing interests exist.
Entropies are key quantities in physics, chemistry, and biology. While free energy changes govern the direction of all chemical processes including reaction equilibria, entropy changes are the underlying driving forces of ligand binding, protein folding and other phenomena driven by hydrophobic effect. Traditionally calculating entropies from atomistic ensembles of configurations of a macromolecule of atoms remains notoriously difficult.
We here propose and apply a method for calculating configurational entropies(1)where denotes the configurational probability density in the dimensional configurational space governed by the potential energy of the system. The fact that is usually on the order of several hundreds or thousands renders the evaluation of this integral quite challenging despite a number of successful attempts. – These broadly fall into three classes, (i) special-purpose perturbation type approaches, also known as thermodynamic integration , (ii) step-by-step reconstruction methods, in particular the scanning procedures introduced by Meirovitch , , (iii) direct approaches which analyse information readily available in standard equilibrium simulation trajectories –.
While perturbation approaches provide relatively accurate free energy differences also for larger systems, accurate entropies are obtained only for smaller molecules. The main obstacle, which aggravates with system size, is the sampling problem, which severely limits the accuracy, in particular for explicit solvent models , .
The most widely used direct method is the quasi-harmonic approximation  (QH), which provides an upper limit to the configurational entropy in terms of independent classical or quantum mechanical harmonic oscillators , , which is equivalent to approximating the configurational density by a multi-variate Gaussian function,with derived from the covariance matrix ,  . However, for macromolecules undergoing large conformational motions the entropy is likely to be considerably smaller than this QH upper limit due to coupling and anharmonicities and, in particular, due to the existence of multiple conformational states –. Indeed, for smaller systems such as di-saccharides  or lipids , or small subsets of larger proteins  significantly lower entropies than with QH were obtained by inclusion of anharmonicities –, ,  and pairwise correlation of QH modes .
The MCSA Scheme
Here we develop a direct method consisting of three building blocks. Results for small test systems will be presented during this introduction of the methodology to illustrate the effect of each building block. Figure 1 shows that indeed for various small test systems (alkanes, dialanine and a complete 14-residue -turn) the quasi-harmonic approximation severely overestimates the reference entropy. The reference values were obtained by thermodynamic integration (TI) gradually perturbing the systems towards an analytically tractable reference state consisting of non-interacting particles in harmonic wells, as described in methods and Refs. , . Entropy estimates obtained for all test systems are also summarized in Table 1.
Five selected alkane systems, dialanine (left), and the C-terminal turn of Protein G (right, please note that here the units are kJ/(mol K)). Thermodynamic integration (TI), density estimates over the whole configurational space (dir), full correlation analyis with subsequent clustering and kernel density estimation (FCA), quasi-harmonic (QH) and mutual information expansion estimates of 2nd (MIE2) and 3rd (MIE3) order were obtained as described in the text.
Non-Parametric Density Estimation
As the first of the three building blocks of the methodology we recently introduced a non-parametric density estimation resting on adaptive anisotropic ellipsoidal kernels  that captures the configurational density in sufficient detail. Briefly, the configurational part of the entropy in a -dimensional space is estimated from configurations according to(2)where denotes the ensemble average of an adaptive anisotropic kernel function , whose anisotropy and scaling depends on the local density at point , and whose -measure is denoted by . This formula simplifies to the well-known -nearest neighbour entropy (-NN) by fixing the kernel function to an (isotropic) sphere whose radius is chosen such that exactly configurations are within the sphere centered at configuration . In this limiting case, is the volume of the -dimensional unit sphere. NN estimators in general are entirely non-parametric and, at a finite sample size , have minimal bias  in any given number of dimensions . A major drawback, however, is the fact that due to the so-called ‘curse of dimensionality’  simple -NN estimators are applicable for up to ten dimensional configurational spaces only . In contrast, as can be seen in Fig. 1 (left, “dir”-bar), adaptive anisotropic kernels yield accurate results even for the 45-dimensional configurational space of dialanine. For the more than 500-dimensional configurational space of the 14-residue -turn, however, the ‘curse of dimensionality’  renders it impossible to improve on the quasi-harmonic approximation with direct density estimation alone (Fig. 1 right). Convergence properties and full technical details of this first MCSA module are discussed in Ref. .
Generation of Minimally Coupled Subspaces
As the second building block of our method, we apply an entropy invariant transformation such that the usually highly coupled degrees of freedom separate into optimally uncoupled subspaces, each of which being sufficiently low-dimensional to render non-parametric density estimation applicable. As the most straightforward class of entropy invariant transformations, we consider here linear orthonormal transformations of the form with . More general transformations are currently explored . We apply Full Correlation Analysis (FCA)  which minimizes mutual information by consideringwhere denote the components of and the 1-dimensional marginal density along . This procedure minimizes non-linear correlations of second and higher order  and therefore generalizes the principal component analysis (PCA) which only considers linear correlations of second order. For complex macromolecules, however, even for the optimal linear FCA transformation , considerable non-linear correlations between several degrees of freedom will remain and cannot be neglected. To address this issue, the FCA modes are subsequently clustered according to the generalized correlation coefficient , with the mutual informationbetween components and . This is achieved by assigning mode indices to clusters such that all modes with correlation coefficients larger than a certain threshold are assigned to the same cluster. This disjoint clustering defines an approximate factorization where denotes the generalized -dimensional marginal density along . This factorization is approximate in the sense that for the entropy(3)the residual entropy is small.
Such approximate factorization, of course, neglects all inter-cluster correlations. These can be pairwise correlations, and thus are small by construction, or higher-order correlations. For the latter we have to assume that they are also effectively eliminated by our threshold criterion. This assumption is supported by the observation that for the alkanes and for dialanine, with , (cf. Fig. 1). Thus, our factorization yields accurate entropies and is indeed small.
Mutual Information Expansions for Oversized Clusters
However, for the larger molecules considered here, the necessarily small threshold typically results in at least one cluster being too large for a sufficiently accurate density estimate (e.g., for the -turn ). Accordingly, while our factorization still improves the entropy estimate (cf. Fig. 1), cannot be neglected anymore. The third building block of our method addresses this issue by subdividing each oversized cluster into disjoint subclusters of sizes , , irrespective of the necessarily remaining strong correlations between these. The residual entropy contributions to the configurational entropywill be drastically increased due to non-neglegible intra-cluster contributions from all subdivided clusters , where we have omitted the argument in the rightmost two terms for brevity. We here propose to compute each via the mutual information expansion (MIE) as(4)where . Expanding the mutual information terms(5)up to second or third order, respectively, with the right-hand sum running over all possible permutations , has proven sufficiently accurate in liquid state theory  and information theory , . Indeed, for the -turn, inclusion of the remaining correlations via this expansion improved the entropy estimate (Fig. 1). For the other test systems . In contrast, for some of the test systems , such that from our observations, 3rd order MIE provides a better estimate and an upper bound to the true entropy.
Applications of MIE to macro-molecular systems can be hampered by the curse of dimensionality and combinatorial explosion of the number of terms , . In this work, the problem is circumvented by clustering into sufficiently high-dimensional () subspaces which minimizes residual inter- correlations and delays the onset of the combinatorial explosion. At the same time the subspaces are sufficiently small that even for the 3rd-order MIE no direct density estimates beyond the critical dimensionality of are required.
TATA Box Binding Protein: Protein Test Case and Error Estimate
Together, these three building blocks enable one to calculate configurational entropies even for larger biomolecules. We considered the 67-residue TATA box binding protein (TBP, pdb code 1TBA) inhibitor in two different configurations; complexed (Fig. 2 top left) and free (Fig. 2 top right). To estimate the statistical error of MCSA and QH configurational entropy estimates, for both states five independent molecular dynamics (MD) simulations were carried out using the OPLS force-field  and the TIP4P explicit solvent model  (see methods section for full simulation details). Fig. 2 shows the results obtained by the five entropy estimation methods for both complexed (left) and free (right) inhibitor. All methods estimate the free cofactor's entropy to be significantly higher than that of the bound cofactor. As can be seen, for both complexed and free cofactor, QH yields the largest estimate. The first two MCSA modules combined (kernel density estimation on little correlated configurational subspaces obtained from FCA) already yield remarkably smaller estimates, irrespective of whether a high or a low clustering threshold was chosen (hi thresh and low thresh in Fig. 2), i.e., chosing small but higher correlated subspaces or larger but lowly correlated subspaces provides similar estimates. Finally, employing all the three MCSA modules including MIE of 2nd (MIE2) and 3rd (MIE3) lowered the estimate again with, as before, the 2nd-order estimate being lower than the 3rd-order estimate.
The following techniques are used: quasi-harmonic approximation (QH); FCA with subsequent density estimation using a high clustering threshold (hi thresh) or, respectively, a low threshold (lo thresh); mutual information expansion of order 2 (MIE2) or, respectively, of order 3 (MIE3). The displayed entropy estimates are averages over five independent simulations of 100 ns each, the error bars indicate standard deviations of the mean.
The fact that the QH estimate is the largest in all cases corroborates the observations for the small test cases, and generally shows that MCSA yields improved estimates also for large macromolecules. Already the first two MCSA modules provide lower entropy estimates, even though relatively large configurational subspaces (, see Table 1) were obtained from FCA, which illustrates that indeed our kernel density estimator works accurately also for the complex high-dimensional configurational spaces spanned by proteins. Further, the fact that the clustering threshold did not affect the final estimate very much naturally reflects the fact that clustering with a high threshold yields small subspaces which are correlated, such that in Eq. 3 is large, increasing our estimate . On the other hand, clustering with a small threshold gives rise to a small but sparse sampling due to large then entails higher , such that is also increased in this case. As expected, the third MCSA module, MIE, circumvents this problem and lowers the MCSA estimate further by 404 or 397 for the free and the complexed cofactor, respectively. The 2nd-order estimate is lower than the 3rd-order estimate in all cases, which shows that also for proteins the pair correlations are generally overestimated, and inclusion of 3rd-order correlations is indeed crucial.
The statistical errors are relatively small in all cases, but generally twice as large for the free than for the complexed cofactor. We attribute this observation to the larger inherent flexibility of the free state, and hence to insufficient molecular dynamics sampling. Consequently, the MIE error for the free cofactor is over three times larger than that of the the complex. Interestingly, the MIE estimate is slightly more affected with the error for the free cofactor being three- to fourfold as high as for the complex. Due to the high number of terms to be evaluated for the MIEs (Eq. 5), already small errors of each result in relatively large errors in .
We have developed a minimally coupled subspace approach (MCSA) to estimate absolute macromolecular configurational entropies from structure ensembles which takes anharmonicities and higher-order correlations into account. The approach combines three building blocks which together allow one to calculate absolute entropies even for the highly complex configurational densities generated by the dynamics of biological macromolecules such as proteins. MCSA shares the versatility of the quasi-harmonic approach as it can be applied to unperturbed equilibrium trajectories while achieving the accuracy of special-purpose perturbation type methods. The effective dimension reduction provided by the Full Correlation Analysis allows for the application of mutual information expansions to large macromolecules. Further, the adaptive kernel non-parametric density estimation method developed for MCSA requires much weaker a-priori assumptions about the properties of the configurational densities than (quasi-)harmonic approaches. The method is applicable also to large macromolecules such as proteins. In this study, we showed that MCSA applied to the TATA box binding protein yielded significantly smaller and thus improved entropy estimates.
We note that here we focus at configurational entropies of the solute only, thus missing both the solvent as well as the solvent/solute parts. Using permutation reduction techniques , our method should be capable of capturing also these important contributions, which however lies outside the scope of the present work.
Thermodynamic Integration Reference Entropy
Absolute free energies for the test systems butane to decane, dialanine, and the ProteinG -turn were calculated by thermodynamic integration (TI). Simulation parameters cf. below. The TI scheme we have chosen to obtain the Helmholtz free energy of the fully interacting particles consists of two phases. Harmonic position restraints with a force constant were slowly switched on for each atom in the first phase, and in the second phase all force-field components were gradually switched off. Within the second phase, the charges were switched off prior to the rest of the force field. After the second phase, the system consisted of non-interacting dummy particles with mass oscillating in their respective harmonic position restraint potentials, i.e.,
The free energy of this harmonic system can be obtained analytically,where denotes the mass-weighted force constant. Hence, the thermodynamic integration yields the absolute free energyand the entropy by , where denotes the ensemble average of the potential energy.
For the TI between the systems given by (start) and (end), 21 intermediate steps were used, and the intermediate values of , 1e-6, 5e-6, 1e-5, 5e-4, 1e-4, 1e-3, 1e-2, 2e-2, 3e-2, 5e-2, 7e-2, 9e-2, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1 were distributed unevenly to obtain approximately balanced values. For each value of a trajectory of (alkanes and dialanine) or (-turn), respectively, was generated.
The error estimates of the TI reference entropies detailed in Table 1 were obtained via two ways for the alkane test systems and dialanine. First, by averaging over five independent simulations and, second, by performing blockwise averaging as derived in Ref.  over each of the 23 of each of these five trajectories. We found that the error estimates obtained by these two methods agree very well. Accordingly, for the -turn only the block averaging method was applied and the resulting error estimates are also given in Table 1.
Molecular/Stochastic Dynamics Simulations
The test systems that were compared with a thermodynamic integration reference (butane to decane, dialanine, and the ProteinG -turn) were set up as follows. Force-field parameterizations were obtained from the Dundee Prodrug server  based on the GROMOS united-atom force field . Stochastic Dynamics simulations were performed using the molecular simulations package GROMACS  in vacuo at with friction constant set to 10, dielectric constant , integration step size of and no bond constraints. Positional restraints were applied to three adjacent terminal heavy atoms. To obtain MCSA error estimates, each of the simulations was carried out five times using different starting velocities. MCSA and QH entropy estimates were obtained from trajectories of lengths (alkanes and dialanine) or (-turn), respectively, i.e. the TI entropy references required times as much computing time as MCSA and QH estimates.
The TATA box binding protein (TBP) complex (protein database entry 1TBA) was simulated using the OPLS all atom force field  in explicit TIP4P solvent  and periodic boundary conditions. NpT ensembles were simulated, with the protein and solvent coupled separately to a 300-K heat bath ().  The systems were isotropically coupled to a pressure bath at 1 bar () . Application of the Lincs  and Settle  algorithms allowed for an integration time step of . Short-range electrostatics and Lennard–Jones interactions were calculated within a cut-off of , and the neighbour list was updated every 10 steps. The particle mesh Ewald (PME) method was used for the long-range electrostatic interactions , with a grid spacing of . The free cofactor was simulated using the same parameters as above. The starting structure was obtained by removing the TBP from the X-ray structure of the complex and equilibrating for 2 ns. Entropy estimates and corresponding errors for both complexed and free cofactor were obtained from five trajectories of 200 ns length each.
Mutual Information Expansions Implementation Details
Due to the moderate regularization assumptions, our adaptive kernel density estimator is sensitive to the sparse sampling problem whose effect is highly dependent on the dimensionality. To guarantee the same accuracy of all density estimates required for the computation of the correlation terms of Eq. 5 despite different dimensionality it is, thus, necessary to ensure the same local densities around points in different terms. This is normally not provided. The mutual information between two modes and ,(6)contains differently well sampled terms in denominator and numerator, because the number of sampling points available to estimate is only half the number of sampling points available for estimating the marginal densities and (see Fig. 3). The accuracy for the estimation of the marginal densities is, consequently, possibly higher than the joint estimate yielding an inaccurate correlation estimate. To overcome this problem, we devised the concept of fill modes. Accordingly, artificially decorrelated modes are created by permuting its components , with . The marginal densities and , yielding a new expression for Eq. 6,(7)where the product of the marginal densities and is now computed from the synthetically decorrelated joint distribution , such that the same accuracy for the joint estimate is guaranteed as for the marginal estimates. Conducting this scheme on the 3rd order correlation function of three modes , and ,yields(8)where the pairwise joint distributions have been ‘filled up’ with permuted ‘fill modes’, as described above, e.g. .
a) Two arbitrarily correlated modes and marginally distributed on the axes. Correlation is clearly visible from the -distributed . The joint distribution is more sparsely sampled than both marginal distributions. b) The -distributed is decorrelated and has exactly as many sample points as the joint distribution in a), allowing precise computation of .
The sensitivity of the nearest-neigbour estimates, Eq. 2, towards the sparse sampling problem also affects the different terms of Eq. 5, which inevitably suffer from different sparse sampling problems if computed separately. Furthermore, a huge number of probability density distributions is computed more than once for the many instances of identical correlation terms appearing in that equation. Expanding over entropy terms rather than correlation terms, in contrast, yields(9)where the first summation runs over different orders until truncation order . designates how many times a certain order appears and whether it needs to be added or subtracted, and the second sum over all possible combinations . To guarantee the same estimation accuracy for all of Eq. 9, each term is filled up to truncation order yielding . Under this modification, Eq. 9 reads(10)with the number of marginal entropies,which depends on the fill mode weighting indexwhere, like above, primes indicate permuted entries.
Conceived and designed the experiments: UH OFL HG. Performed the experiments: UH OFL. Analyzed the data: UH OFL. Contributed reagents/materials/analysis tools: UH OFL. Wrote the paper: UH OFL HG.
- 1. Beveridge DL, DiCapua FM (1989) Free energy via molecular simulation: Applications to chemical and biomolecular systems. Annual Review of Biophysics and Biophysical Chemistry 18: 431–492.DL BeveridgeFM DiCapua1989Free energy via molecular simulation: Applications to chemical and biomolecular systems.Annual Review of Biophysics and Biophysical Chemistry18431492
- 2. Straatsma TP, McCammon JA (1992) Computational alchemy. Annual Review of Physical Chemistry 43: 407–435.TP StraatsmaJA McCammon1992Computational alchemy.Annual Review of Physical Chemistry43407435
- 3. Kollman P (1993) Free energy calculations: Applications to chemical and biochemical phenomena. Chem Rev 93: 2395–2417.P. Kollman1993Free energy calculations: Applications to chemical and biochemical phenomena.Chem Rev9323952417
- 4. Meirovitch H (2007) Recent developments in methodologies for calculating the entropy and free energy of biological systems by computer simulation. Curr Opin Struct Biol 17: 181–186.H. Meirovitch2007Recent developments in methodologies for calculating the entropy and free energy of biological systems by computer simulation.Curr Opin Struct Biol17181186
- 5. Peter C, Oostenbrink C, van Dorp A, van Gunsteren WF (2004) Estimating entropies from molecular dynamics simulations. J Chem Phys 120: 2652–2661.C. PeterC. OostenbrinkA. van DorpWF van Gunsteren2004Estimating entropies from molecular dynamics simulations.J Chem Phys12026522661
- 6. Cheluvaraja S, Meirovitch H (2004) Simulation method for calculating the entropy and free energy of peptides and proteins. PNAS 101: 9241–9246.S. CheluvarajaH. Meirovitch2004Simulation method for calculating the entropy and free energy of peptides and proteins.PNAS10192419246
- 7. Cheluvaraja S, Meirovitch H (2006) Calculation of the entropy and free energy of peptides by molecular dynamics simulations using the hypothetical scanning molecular dynamics method. J Chem Phys 125: 024905.S. CheluvarajaH. Meirovitch2006Calculation of the entropy and free energy of peptides by molecular dynamics simulations using the hypothetical scanning molecular dynamics method.J Chem Phys125024905
- 8. Karplus M, Kushick JN (1981) Method for estimating the configurational entropy of macromolecules. Macromolecules 14: 325–332.M. KarplusJN Kushick1981Method for estimating the configurational entropy of macromolecules.Macromolecules14325332
- 9. Schlitter J (1993) Estimation of absolute and relative entropies of macromolecules using the covariance matrix. Chemical Physics Letters 215: 617–621.J. Schlitter1993Estimation of absolute and relative entropies of macromolecules using the covariance matrix.Chemical Physics Letters215617621
- 10. Karplus M, McCammon JA (2002) Molecular dynamics simulations of biomolecules. Nat Struct Mol Biol 9: 646–652.M. KarplusJA McCammon2002Molecular dynamics simulations of biomolecules.Nat Struct Mol Biol9646652
- 11. Chang C, Chen W, Gilson M (2005) Evaluating the accuracy of the quasiharmonic approximation. J Chem Theory Comput 1: 1017–1028.C. ChangW. ChenM. Gilson2005Evaluating the accuracy of the quasiharmonic approximation.J Chem Theory Comput110171028
- 12. Chang C, Chen W, Gilson MK (2007) Ligand configurational entropy and protein binding. PNAS 104: 1534–1539.C. ChangW. ChenMK Gilson2007Ligand configurational entropy and protein binding.PNAS10415341539
- 13. Gilson MK, Zhou HX (2007) Calculation of protein-ligand binding affinities. Ann Rev Biophys Biomol Struct 36: 21–42.MK GilsonHX Zhou2007Calculation of protein-ligand binding affinities.Ann Rev Biophys Biomol Struct362142
- 14. Minh DDL, Bui JM, Chang C, Jain T, Swanson JMJ, et al. (2005) The entropic cost of protein-protein association: A case study on acetylcholinesterase binding to fasciculin-2. Biophys J 89: 25–27.DDL MinhJM BuiC. ChangT. JainJMJ Swanson2005The entropic cost of protein-protein association: A case study on acetylcholinesterase binding to fasciculin-2.Biophys J892527
- 15. Pereira CS, Kony D, Baron R, Müller M, van Gunsteren WF, et al. (2006) Conformational and dynamical properties of disaccharides in water: a molecular dynamics study. Biophysical Journal 90: 4337–4344.CS PereiraD. KonyR. BaronM. MüllerWF van Gunsteren2006Conformational and dynamical properties of disaccharides in water: a molecular dynamics study.Biophysical Journal9043374344
- 16. Baron R, deVries A, Hünenberger P, van Gunsteren W (2006) Comparison of atomic-level and coarse-grained models for liquid hydrocarbons from molecular dynamics configurational entropy estimates. J Phys Chem B 110: 8464–8473.R. BaronA. deVriesP. HünenbergerW. van Gunsteren2006Comparison of atomic-level and coarse-grained models for liquid hydrocarbons from molecular dynamics configurational entropy estimates.J Phys Chem B11084648473
- 17. Baron R, McCammon JA (2008) (thermo)dynamic role of receptor flexibility, entropy, and motional correlation in protein-ligand binding. ChemPhysChem 9: 983–988.R. BaronJA McCammon2008(thermo)dynamic role of receptor flexibility, entropy, and motional correlation in protein-ligand binding.ChemPhysChem9983988
- 18. Kolossvary I (1997) Evaluation of the molecular configuration integral in all degrees of freedom for the direct calculation of conformational free energies: Prediction of the anomeric free energy of monosaccharides. J Phys Chem A 101: 9900–9905.I. Kolossvary1997Evaluation of the molecular configuration integral in all degrees of freedom for the direct calculation of conformational free energies: Prediction of the anomeric free energy of monosaccharides.J Phys Chem A10199009905
- 19. Chang C, Potter M, Gilson M (2003) Calculation of molecular configuration integrals. J Phys Chem B 107: 1048–1055.C. ChangM. PotterM. Gilson2003Calculation of molecular configuration integrals.J Phys Chem B10710481055
- 20. Baron R, van Gunsteren W, Hünenberger P (2006) Estimating the configurational entropy from molecular dynamics simulations: anharmonicity and correlation corrections to the quasi-harmonic approximation. Trends Phys Chem 11: 87–122.R. BaronW. van GunsterenP. Hünenberger2006Estimating the configurational entropy from molecular dynamics simulations: anharmonicity and correlation corrections to the quasi-harmonic approximation.Trends Phys Chem1187122
- 21. Hensen U, Grubmüller H, Lange OF (2009) Adaptive anisotropic kernels for nonparametric estimation of absolute configurational entropies in high-dimensional configuration spaces. Phys Rev E 80: 011913.U. HensenH. GrubmüllerOF Lange2009Adaptive anisotropic kernels for nonparametric estimation of absolute configurational entropies in high-dimensional configuration spaces.Phys Rev E80011913
- 22. Tyka M, Clarke A, Sessions R (2006) An efficient, path-independent method for free-energy calculations. J Phys Chem B 110: 17212–17220.M. TykaA. ClarkeR. Sessions2006An efficient, path-independent method for free-energy calculations.J Phys Chem B1101721217220
- 23. Kraskov A, Stögbauer H, Grassberger P (2004) Estimating mutual information. Phys Rev E 69: 066138.A. KraskovH. StögbauerP. Grassberger2004Estimating mutual information.Phys Rev E69066138
- 24. Bellman RE (1961) Adaptive Control Processes. Princeton University Press. RE Bellman1961Adaptive Control ProcessesPrinceton University Press
- 25. Hnizdo V, Darian E, Fedorowicz A, Demchuk E, Li S, et al. (2007) Nearest-neighbor nonparametric method for estimating the configurational entropy of complex molecules. J Comp Chem 28: 655–668.V. HnizdoE. DarianA. FedorowiczE. DemchukS. Li2007Nearest-neighbor nonparametric method for estimating the configurational entropy of complex molecules.J Comp Chem28655668
- 26. Hennig M (2007) Entropy invariant transformations. M. Hennig2007Entropy invariant transformations.Master's thesis, Universität Jena. Master's thesis, Universität Jena.
- 27. Lange OF, Grubmüller H (2008) Full correlation analysis of conformational protein dynamics. Proteins 70: 1294–1312.OF LangeH. Grubmüller2008Full correlation analysis of conformational protein dynamics.Proteins7012941312
- 28. Lange OF, Grubmüller H (2006) Generalized correlation for biomolecular dynamics. Proteins 62: 1053–1061.OF LangeH. Grubmüller2006Generalized correlation for biomolecular dynamics.Proteins6210531061
- 29. Baranyai A, Evans DJ (1989) Direct entropy calculation from computer simulation of liquids. Phys Rev A 40: 3817–3822.A. BaranyaiDJ Evans1989Direct entropy calculation from computer simulation of liquids.Phys Rev A4038173822
- 30. Attard P, Jepps OG, Marčelja S (1997) Information content of signals using correlation function expansions of the entropy. Phys Rev E 56: 4052–4067.P. AttardOG JeppsS. Marčelja1997Information content of signals using correlation function expansions of the entropy.Phys Rev E5640524067
- 31. Attard P (1999) P. Attard1999Statistical Physics on the Eve of the Twenty-First Century, World Scientific, chapter Markov Superposition Expansion for the Entropy and Correlation Functions in Two and Three Dimensions. Statistical Physics on the Eve of the Twenty-First Century, World Scientific, chapter Markov Superposition Expansion for the Entropy and Correlation Functions in Two and Three Dimensions.
- 32. Killian BJ, Kravitz JY, Gilson MK (2007) Extraction of configurational entropy from molecular simulations via an expansion approximation. J Chem Phys 127: 024107.BJ KillianJY KravitzMK Gilson2007Extraction of configurational entropy from molecular simulations via an expansion approximation.J Chem Phys127024107
- 33. Hnizdo V, Tan J, Killian BJ, Gilson MK (2008) Efficient calculation of configurational entropy from molecular simulations by combining the mutual-information expansion and nearest-neighbor methods. J Comp Chem 29: 1605–1614.V. HnizdoJ. TanBJ KillianMK Gilson2008Efficient calculation of configurational entropy from molecular simulations by combining the mutual-information expansion and nearest-neighbor methods.J Comp Chem2916051614
- 34. Kaminski G, Friesner R, Tirado-Rives J, Jorgensen W (2001) Evaluation and reparametrization of the OPLS-AA force field for proteins via comparison with accurate quantum chemical calculations on peptides. J Phys Chem B 105: 6474–6487.G. KaminskiR. FriesnerJ. Tirado-RivesW. Jorgensen2001Evaluation and reparametrization of the OPLS-AA force field for proteins via comparison with accurate quantum chemical calculations on peptides.J Phys Chem B10564746487
- 35. Jorgensen WL, Chandrasekhar J, Madura JD, Impey RW, Klein ML (1983) Comparison of simple potential functions for simulating liquid water. J Chem Phys 79: 926–935.WL JorgensenJ. ChandrasekharJD MaduraRW ImpeyML Klein1983Comparison of simple potential functions for simulating liquid water.J Chem Phys79926935
- 36. Reinhard F, Grubmüller H (2007) Estimation of absolute solvent and solvation shell entropies via permutation reduction. J Chem Phys 126: 014102.F. ReinhardH. Grubmüller2007Estimation of absolute solvent and solvation shell entropies via permutation reduction.J Chem Phys126014102
- 37. Hess B (2002) Determining the shear viscosity of model liquids from molecular dynamics simulations. J Chem Phys 116: 209–217.B. Hess2002Determining the shear viscosity of model liquids from molecular dynamics simulations.J Chem Phys116209217
- 38. Schüttelkopf AW, van Aalten DMF (2004) PRODRG - a tool for high-throughput crystallography of protein-ligand complexes. Acta Crystallographica D 60: 1355–1363.AW SchüttelkopfDMF van Aalten2004PRODRG - a tool for high-throughput crystallography of protein-ligand complexes.Acta Crystallographica D6013551363
- 39. van Gunsteren WF, Daura X, Mark AE (1998) GROMOS force field. pp. 1211–1216.WF van GunsterenX. DauraAE Mark1998GROMOS force field.12111216Encyclopaedia of computational chemistry edition. Encyclopaedia of computational chemistry edition.
- 40. van der Spoel D, Lindahl E, Hess B, Groenhof G, Mark AE, et al. (2005) Gromacs: Fast, flexible, and free. J Comp Chem 26: 1701–1718.D. van der SpoelE. LindahlB. HessG. GroenhofAE Mark2005Gromacs: Fast, flexible, and free.J Comp Chem2617011718
- 41. Berendsen HJC, Postma JPM, van Gunsteren WF, DiNola A, Haak JR (1984) Molecular dynamics with coupling to an external bath. J Chem Phys 81: 3684–3690.HJC BerendsenJPM PostmaWF van GunsterenA. DiNolaJR Haak1984Molecular dynamics with coupling to an external bath.J Chem Phys8136843690
- 42. Hess B, Bekker H, Berendsen HJC, Fraaije JGEM (1997) Lincs: A linear constraint solver for molecular simulations. Journal of Computational Chemistry 18: 1463–1472.B. HessH. BekkerHJC BerendsenJGEM Fraaije1997Lincs: A linear constraint solver for molecular simulations.Journal of Computational Chemistry1814631472
- 43. Miyamoto S, Kollman PA (1992) Settle: An analytical version of the shake and rattle algorithm for rigid water models. J Comp Chem 13: 952–962.S. MiyamotoPA Kollman1992Settle: An analytical version of the shake and rattle algorithm for rigid water models.J Comp Chem13952962
- 44. Darden T, York D, Pedersen L (1993) Particle mesh Ewald: An N log(N) method for Ewald sums in large systems. J Chem Phys 98: 10089–10092.T. DardenD. YorkL. Pedersen1993Particle mesh Ewald: An N log(N) method for Ewald sums in large systems.J Chem Phys981008910092