A probabilistic method for the calculation of energy of hydrophobic interactions

A theoretical approach to the quantitative estimation of the energy of hydrophobic interactions on the molecular level has been developed. The model is based on the fundamental relationship between the probability of shaping of a cavity of the excluded volume in liquid water that results from fluctuations of density and free energy of prime hydrophobic solvates, hard spheres. This probability was estimated with the use of the probabilistic method in combination with experimentally observed data on density and radial distribution function of allocation. Free energy of hydrophobic interactions for a complex consisting of several hard spheres was determined. The critical value of the particles included in the complex was estimated.     

Free-energy contributions to complex formation between botulinum neurotoxin type B and synaptobrevin fragment

Free-energy terms that contribute to complex formation between the catalytic domain of botulinum neurotoxin type B (BoNT/B-L(C)) and a 36-residue synaptobrevin fragment were estimated by using a combination of microscopic simulations and continuum methods. The complex for a non-hydrolyzed substrate was calculated by optimizing an energy function applied to the X-ray co-crystal structure of BoNT/B-L(C) bound with reaction products from a cleaved synaptobrevin peptide, refined to high crystallographic thermal factors. The estimated absolute binding affinity of the simulation structure is in good qualitative agreement with the experimental free energy of Michaelis complex formation, given the approximations of the model calculations. The simulation structure revealed significant complex stabilization from the hydrophobic effect, while the electrostatic cost of releasing water molecules from the interface determined to be highly unfavorable. By partitioning the total electrostatic and hydrophobic terms into residue free-energy contributions, a binding-affinity 'signature' for synaptobrevin was developed from the optimized conformation. The results demonstrate the effect of substrate length on complex formation and identify a peripheral high-affinity binding site near the N-terminal region that might initiate cooperative activation responsible for the large minimal substrate length requirement. The so-called SNARE motif is observed to contribute negligible free energy of binding.     

Energetics of hydrophobic matching in lipid-protein interactions

Lipid chain length modulates the activity of transmembrane proteins by mismatch between the hydrophobic span of the protein and that of the lipid membrane. Relative binding affinities of lipids with different chain lengths are used to estimate the excess free energy of lipid-protein interaction that arises from hydrophobic mismatch. For a wide range of integral proteins and peptides, the energy cost is much less than the elastic penalty of fully stretching or compressing the lipid chains to achieve complete hydrophobic matching. The chain length dependences of the free energies of lipid association are described by a model that combines elastic chain extension with a free energy term that depends linearly on the extent of residual mismatch. The excess free energy densities involved lie in the region of 0.5-2.0 k_BT.nm⁻². Values of this size could arise from exposure of hydrophobic groups to polar portions of the lipid or protein, but not directly to water, or alternatively from changes in tilt of the transmembrane helices that are energetically comparable to those activating mechanosensitive channels. The influence of hydrophobic mismatch on dimerization of transmembrane helices and their transfer between lipid vesicles, and on shifts in chain-melting transitions of lipid bilayers by incorporated proteins, is analyzed by using the same thermodynamic model. Segmental order parameters confirm that elastic lipid chain distortions are insufficient to compensate fully for the mismatch, but the dependence on chain length with tryptophan-anchored peptides requires that the free energy density of hydrophobic mismatch should increase with increasing extent of mismatch.     

Calculating the free energy of association of transmembrane helices

A large number of experimental studies have been devoted to quantifying the interaction between transmembrane (TM) helices in detergent micelles and, more recently, in bilayers. Theoretical calculation of association free energy of TM helices would be useful for predicting the propensity of given sequences to oligomerize and for understanding the difference between association in micelles and in bilayers. In this article, the theoretical foundation for calculating the standard association free energy of TM helices is laid out and is applied to glycophorin A in both micelles and bilayers. The standard association free energy is decomposed into the effective energy, translational, rotational, and conformational entropy terms. The effective energy of association is obtained by molecular dynamics simulations in an implicit membrane model. The translational and rotational entropy of association is calculated from the probability distribution of the translational and rotational degrees of freedom obtained from the molecular dynamics simulations. The side-chain conformational entropy of association is estimated from the probability distribution obtained by rigid rotation of all side-chain dihedral angles. The calculated standard association free energy of glycophorin A in N-dodecylphosphocholine micelles is in good agreement with the experimental value. The translational entropy cost is larger, whereas the rotational entropy cost is smaller in bilayers than in micelles. The standard association free energy in 1,2-dimyristoyl-sn-glycero-3-phosphocholine bilayers is calculated to be approximately 1.3 kcal/mol more favorable than in N-dodecylphosphocholine micelles, consistent with available experimental data.     

Kinetic analysis of local structure formations in protein folding

The protein folding process is described by a cluster model based on the assumption that local structures or clusters are formed at an early stage in different regions of the polypeptide chain. Possible local structural elements in a globular protein are helices, bends, and hydrophobic cores whose formation is presumably determined by the interaction with the environment. Thus the tendency of local structure formation is expressed by a surface free energy of the cluster, which is assigned to the interface between the cluster and its environment. The probability of finding the chain of N residues with k clusters and m residues in the cluster is represented by a cluster distribution map. The cluster model exhibits a distinct two-state-like equilibrium transition, which can be seen on this map as well-separated native and denatured populations at the midpoint of the transition. The native population is localized at k ≈ 1 and m ≈ N, while the position of the denatured population can vary significantly depending on the surface free energy of the cluster. If the surface free energy is strong, the denatured population is localized near k = 0 and m = 0. On the other hand, if the surface free energy is weak, the denatured population is localized at high k and m values. The dynamics of the cluster model are treated as a stochastic process involving the transition from a state (k,m) to one of its six neighbors. The transition probability for each transition is determined by the free energy difference between two states; thus no activation process is assumed. However, the conversion of the two macrostates, native and denatured populations, involves the free energy activation due to the cooperative interaction of the macrosystem. The dynamics are analyzed by following the time evolution of the population profile on the cluster distribution map. Kinetic schemes are proposed to describe the multistep mechanism of protein folding and unfolding.     

Calculation of the Affinity of the λ Repressor-Operator Complex Based on Free Energy Component Analysis

A calculation of the binding free energy of the u repressor-operator complex is described based on free energy component analysis. The calculations are based on a thermodynamic cycle of seven steps decomposed into a total of 24 individual components. The values of these terms are estimated using a combination of empirical potential functions from AMBER, generalized Born - solvent accessibility calculations, elementary statistical mechanics and semiempirical physicochemical properties. Two alternative approaches are compared, one based on the crystal structure of the complex and the other based on the molecular dynamics simulation of the u repressor-operator complex. The calculated affinity is m 19.7 kcal/mol from the crystal structure calculation and m 17.9 kcal/mol from the MD method. The corresponding experimental affinity of the complex is about m 12.6 kcal/mol, indicating reasonable agreement between theory and experiment, considering the approximations involved in the computational methodology. The results are analyzed in terms of contributions from electrostatics, van der Waal interactions, the hydrophobic effect and solvent release. The capabilities and limitations of free energy component methodology are assessed and discussed on the basis of these results.     

Probing the Disordered Domain of the Nuclear Pore Complex through Coarse-Grained Molecular Dynamics Simulations

The distribution of disordered proteins (FG-nups) that line the transport channel of the nuclear pore complex (NPC) is investigated by means of coarse-grained molecular dynamics simulations. A one-bead-per-amino-acid model is presented that accounts for the hydrophobic/hydrophilic and electrostatic interactions between different amino acids, polarity of the solvent, and screening of free ions. The results indicate that the interaction of the FG-nups forms a high-density, doughnut-like distribution inside the NPC, which is rich in FG-repeats. We show that the obtained distribution is encoded in the amino-acid sequence of the FG-nups and is driven by both electrostatic and hydrophobic interactions. To explore the relation between structure and function, we have systematically removed different combinations of FG-nups from the pore to simulate inviable and viable NPCs that were previously studied experimentally. The obtained density distributions show that the maximum density of the FG-nups inside the pore does not exceed 185 mg/mL in the inviable NPCs, whereas for the wild-type and viable NPCs, this value increases to 300 mg/mL. Interestingly, this maximum density is not correlated to the total mass of the FG-nups, but depends sensitively on the specific combination of essential Nups located in the central plane of the NPC.     

Overall charge and local charge density of pectin determines the enthalpic and entropic contributions to complexation with β-lactoglobulin

The complex formation between β-lactoglobulin and pectins of varying overall charge and local charge density were investigated. Isothermal titration calorimetry experiments were carried out to determine the enthalpic contribution to the complex formation at pH 4.25 and various ionic strengths. Complex formation was found to be an exothermic process for all conditions. Combination with previously published binding constants by Sperber et al. (Sperber, B. L. H. M.; Cohen Stuart, M. A.; Schols, H. A.; Voragen, A. G. J.; Norde, W. Biomacromolecules 2009, 10, 3246-3252) allows for the determination of the changes in the Gibbs energy and the change in entropy of the system upon complex formation between β-lactoglobulin and pectin. The local charge density of pectin is found to determine the balance between enthalpic and entropic contributions. For a high local charge density pectin, the main contribution to the Gibbs energy is of an enthalpic nature, supported by a favorable entropy effect due to the release of small counterions. A pectin with a low local charge density has a more even distribution of the enthalpic and entropic part to the change of the Gibbs energy. The enthalpic part is reduced due to the lower charge density, while the relative increase of the entropic contribution is thought to be caused by a change in the location of the binding place for pectin on the β-lactoglobulin molecule. The association of the hydrophobic methyl esters on pectin with an exposed hydrophobic region on β-lg results in the release of water molecules from the hydrophobic region and surrounding the methyl esters of the pectin molecule. An increase in the ionic strength decreases the enthalpic contribution due to the shielding of electrostatic attraction in favor of the entropic contribution, supporting the idea that the release of water molecules from hydrophobic areas plays a part in the complex formation.     

Probing the Disordered Domain of the Nuclear Pore Complex through Coarse-Grained Molecular Dynamics Simulations

The distribution of disordered proteins (FG-nups) that line the transport channel of the nuclear pore complex (NPC) is investigated by means of coarse-grained molecular dynamics simulations. A one-bead-per-amino-acid model is presented that accounts for the hydrophobic/hydrophilic and electrostatic interactions between different amino acids, polarity of the solvent, and screening of free ions. The results indicate that the interaction of the FG-nups forms a high-density, doughnut-like distribution inside the NPC, which is rich in FG-repeats. We show that the obtained distribution is encoded in the amino-acid sequence of the FG-nups and is driven by both electrostatic and hydrophobic interactions. To explore the relation between structure and function, we have systematically removed different combinations of FG-nups from the pore to simulate inviable and viable NPCs that were previously studied experimentally. The obtained density distributions show that the maximum density of the FG-nups inside the pore does not exceed 185 mg/mL in the inviable NPCs, whereas for the wild-type and viable NPCs, this value increases to 300 mg/mL. Interestingly, this maximum density is not correlated to the total mass of the FG-nups, but depends sensitively on the specific combination of essential Nups located in the central plane of the NPC.     

Physico-chemical mechanisms of transport by micropinocytosis

Blood plasma proteins adsorption on a capillar endoteliocyte membrane surface can be accompanied by the formation of protein clusters. Their superficial protein density is about 10(16) protein globules per m2. The surface density of free energy of such protein layer in a cluster is estimated according to total energy of individual protein globules hydrophobic contribution, and its value is approximately 5 mJ X m-2. It is identical to local variation quantity of the membrane free energy. An alternation of the free surface energy must lead to the appearance of chemically induced and bending moments and to the membrane distortion accompanied by the caveole forming. The blood hydrostatic pressure in capillar lumen (approximately 33 X 10(2)N X m-2) creates the membrane isotropic tension which is proportional to its value and to caveole radius. The latter according to Laplas' equation closes the caveole into the vesicula with the radius approximately 40 nm. The transport of free vesicula by cytoplasmic currents to the basal surface of an endoteliocyte results in exocytosis (which proceeds approximately 0.1 s) with a release of the vesicula surface free energy (approximately 10(-16) J). The capillar endotelium vesicular transport is the indivisible endocytosis-exocytosis process characterized by "the turnover" of the cell plasmic membrane matter without its loss.     

Simulations of kinetically irreversible protein aggregate structure.

We have simulated the structure of kinetically irreversible protein aggregates in two-dimensional space using a lattice-based Monte-Carlo routine. Our model specifically accounts for the intermolecular interactions between hydrophobic and hydrophilic protein surfaces and a polar solvent. The simulations provide information about the aggregate density, the types of inter-monomer contacts and solvent content within the aggregates, the type and extent of solvent exposed perimeter, and the short- and long-range order all as a function of (i) the extent of monomer hydrophobic surface area and its distribution on the model protein surface and (ii) the magnitude of the hydrophobic-hydrophobic contact energy. An increase in the extent of monomer hydrophobic surface area resulted in increased aggregate densities with concomitant decreased system free energies. These effects are accompanied by increases in the number of hydrophobic-hydrophobic contacts and decreases in the solvent-exposed hydrophobic surface area of the aggregates. Grouping monomer hydrophobic surfaces in a single contiguous stretch resulted in lower aggregate densities and lower short range order. More favorable hydrophobic-hydrophobic contact energies produced structures with higher densities but the number of unfavorable protein-protein contacts was also observed to increase; greater configurational entropy produced the opposite effect. Properties predicted by our model are in good qualitative agreement with available experimental observations.     

Hydration of krypton and consideration of clathrate models of hydrophobic effects from the perspective of quasi-chemical theory

Ab initio molecular dynamics (AIMD) results on a krypton-water liquid solution are presented and compared to recent XAFS results for the radial hydration structure for a Kr atom in liquid water solution. Though these AIMD calculations have important limitations of scale, the comparisons with the liquid solution results are satisfactory and significantly different from the radial distributions extracted from the data on the solid Kr/H(2)O clathrate hydrate phase. The calculations also produce the coordination number distribution that can be examined for metastable coordination structures suggesting possibilities for clathrate-like organization; none are seen in these results. Clathrate pictures of hydrophobic hydration are discussed, as is the quasi-chemical theory that should provide a basis for clathrate pictures. Outer shell contributions are discussed and estimated; they are positive and larger than the positive experimental hydration free energy of Kr(aq), implying that inner shell contributions must be negative and of comparable size. Clathrate-like inner shell hydration structures on a Kr atom solute are obtained for some, but not all, of the coordination number cases observed in the simulation. The structures found have a delicate stability. Inner shell coordination structures extracted from the simulation of the liquid, and then subjected to quantum chemical optimization, always decomposed. Interactions with the outer shell material are decisive in stabilizing coordination structures observed in liquid solution and in clathrate phases. The primitive quasi-chemical estimate that uses a dielectric model for the influence of the outer shell material on the inner shell equilibria gives a contribution to hydration free energy that is positive and larger than the experimental hydration free energy. The 'what are we to tell students' question about hydrophobic hydration, often answered with structural clathrate pictures, is then considered; we propose an alternative answer that is consistent with successful molecular theories of hydrophobic effects and based upon distinctive observable properties of liquid water. Considerations of parsimony, for instance Ockham's razor, then suggest that additional structural hypotheses in response to 'what are we to tell students' are not required at this stage.     

Amphipathic helixes and plasma lipoproteins: Thermodynamic and geometric considerations

In this paper analyses are made of the thermodynamic and geometric properties of the predicted association between amphipathic helixes and phospholipid vesicles. From thermodynamic considerations it is proposed that a major driving force for such an association is the negative free energy gained by the transfer of a number of hydrophobic residues (contained within the non-polar faces of amphipathic helixes), from water to the interior of a phospholipid bilayer. The mechanism proposed is that in the aqueous state a potentially amphipathic sequence forms a non-helical hydrophobic patch on the surface of the apolipoprotein. Formation of an amphipathic helix and simultaneous burial of the hydrophobic residues in the surface of a phospholipid bilayer provides the driving force for lipid association. From this model an estimate of the upperlimit for the hydrophobically driven free energy of lipid association (?40?65 kcal/mol) is calculated for the 4 apolipoproteins with known sequences.On the basis of geometrical considerations a model for an intermediate state of high density lipoprotein (HDL) synthesis is proposed. This model consists of a cholesterol-containing phospholipid bilayer disc whose ‘naked’ hydrophobic edges are shielded from the aqueous phase by amphipathic helixes of the apolipoproteins. Exposure of these ‘bicycle tire’ micelles to the enzyme lecithin: cholesterol acyl transferase (LCAT) is postulated to result in the formation of mature spherical HDL particles with cholesteryl ester forming a neutral lipid core.     

Energetics of inclusion-induced bilayer deformations.

The material properties of lipid bilayers can affect membrane protein function whenever conformational changes in the membrane-spanning proteins perturb the structure of the surrounding bilayer. This coupling between the protein and the bilayer arises from hydrophobic interactions between the protein and the bilayer. We analyze the free energy cost associated with a hydrophobic mismatch, i.e., a difference between the length of the protein's hydrophobic exterior surface and the average thickness of the bilayer's hydrophobic core, using a (liquid-crystal) elastic model of bilayer deformations. The free energy of the deformation is described as the sum of three contributions: compression-expansion, splay-distortion, and surface tension. When evaluating the interdependence among the energy components, one modulus renormalizes the other: e.g., a change in the compression-expansion modulus affects not only the compression-expansion energy but also the splay-distortion energy. The surface tension contribution always is negligible in thin solvent-free bilayers. When evaluating the energy per unit distance (away from the inclusion), the splay-distortion component dominates close to the bilayer/inclusion boundary, whereas the compression-expansion component is more prominent further away from the boundary. Despite this complexity, the bilayer deformation energy in many cases can be described by a linear spring formalism. The results show that, for a protein embedded in a membrane with an initial hydrophobic mismatch of only 1 A, an increase in hydrophobic mismatch to 1.3 A can increase the Boltzmann factor (the equilibrium distribution for protein conformation) 10-fold due to the elastic properties of the bilayer.     

混交林测树因子概率分布模型的构建及应用

基于最大熵原理,针对目前对混交林测树因子概率分布模型研究的不足,提出了联合最大熵概率密度函数,该函数具有如下特点：1）函数的每一组成部分都是相互联系的最大熵函数,故可以综合混交林各主要组成树种测树因子的概率分布信息;2）函数是具有双权重的概率表达式,能体现混交林结构复杂的特点,在最大限度地利用混交林每一主要树种测树因子概率分布信息的同时,还能精确地全面反映混交林测树因子概率分布规律;3）函数的结构简洁、性能优良.用天目山自然保护区的混交林样地对混交林测树因子概率分布模型进行了应用与检验,结果表明：模型的拟合精度(R²=0.9655)与检验精度(R²=
0.9772)都较高.说明联合最大熵概率密度函数可以作为混交林测树因子概率分布模型,为全面了解混交林林分结构提供了一种可行的方法.     

Free energy landscape of protein folding in water: explicit vs. implicit solvent

The Generalized Born (GB) continuum solvent model is arguably the most widely used implicit solvent model in protein folding and protein structure prediction simulations; however, it still remains an open question on how well the model behaves in these large-scale simulations. The current study uses the beta-hairpin from C-terminus of protein G as an example to explore the folding free energy landscape with various GB models, and the results are compared to the explicit solvent simulations and experiments. All free energy landscapes are obtained from extensive conformation space sampling with a highly parallel replica exchange method. Because solvation model parameters are strongly coupled with force fields, five different force field/solvation model combinations are examined and compared in this study, namely the explicit solvent model: OPLSAA/SPC model, and the implicit solvent models: OPLSAA/SGB (Surface GB), AMBER94/GBSA (GB with Solvent Accessible Surface Area), AMBER96/GBSA, and AMBER99/GBSA. Surprisingly, we find that the free energy landscapes from implicit solvent models are quite different from that of the explicit solvent model. Except for AMBER96/GBSA, all other implicit solvent models find the lowest free energy state not the native state. All implicit solvent models show erroneous salt-bridge effects between charged residues, particularly in OPLSAA/SGB model, where the overly strong salt-bridge effect results in an overweighting of a non-native structure with one hydrophobic residue F52 expelled from the hydrophobic core in order to make better salt bridges. On the other hand, both AMBER94/GBSA and AMBER99/GBSA models turn the beta-hairpin in to an alpha-helix, and the alpha-helical content is much higher than the previously reported alpha-helices in an explicit solvent simulation with AMBER94 (AMBER94/TIP3P). Only AMBER96/GBSA shows a reasonable free energy landscape with the lowest free energy structure the native one despite an erroneous salt-bridge between D47 and K50. Detailed results on free energy contour maps, lowest free energy structures, distribution of native contacts, alpha-helical content during the folding process, NOE comparison with NMR, and temperature dependences are reported and discussed for all five models.     

Association of protein-DNA recognition complexes: electrostatic and nonelectrostatic effects

In this study the electrostatic and nonelectrostatic contributions to the binding free energy of a number of different protein-DNA recognition complexes are investigated. To determine the electrostatic effects in the protein-DNA association the Poisson-Boltzmann approach was applied. Overall the salt-dependent electrostatic free energy opposed binding in all protein-DNA complexes except one, and the salt-independent electrostatic contribution favored binding in more than half of the complexes. Further the salt-dependent electrostatic free energy increased with higher ionic concentrations and therefore complex association is stronger opposed at higher ionic concentrations. The hydrophobic effect in the protein-DNA complexes was determined from the buried accessible surface area and the surface tension. A majority of the complexes showed more polar than nonpolar buried accessible surface area. Interestingly the buried DNA-accessible surface area was preferentially hydrophilic, only in one complex a slightly more hydrophobic buried accessible surface area was observed. A quite sophisticated balance between several different free energy components seems to be responsible for determining the free energy of binding in protein-DNA systems.     

Sparse sampling of water density fluctuations near liquid-vapor coexistence

Abstract

The free energetics of water density fluctuations in bulk water, at interfaces, and in hydrophobic confinement inform the hydration of hydrophobic solutes as well as their interactions and assembly. The characterisation of such free energetics is typically performed using enhanced sampling techniques such as umbrella sampling. In umbrella sampling, order parameter distributions obtained from adjacent biased simulations must overlap in order to estimate free energy differences between biased ensembles. Many biased simulations are typically required to ensure such overlap, which exacts a steep computational cost. We recently introduced a sparse sampling method, which circumvents the overlap requirement by using thermodynamic integration to estimate free energy differences between biased ensembles. Here we build upon and generalise sparse sampling for characterising the free energetics of water density fluctuations in systems near liquid-vapor coexistence. We also introduce sensible heuristics for choosing the biasing potential parameters and strategies for adaptively refining them, which facilitate the estimation of such free energetics accurately and efficiently. We illustrate the method by characterising the free energetics of cavitation in a large volume in bulk water. We also use sparse sampling to characterise the free energetics of capillary evaporation for water confined between two hydrophobic plates. In both cases, sparse sampling is nearly two orders of magnitude faster than umbrella sampling. Given its efficiency, the sparse sampling method is particularly well suited for characterising free energy landscapes for systems wherein umbrella sampling is prohibitively expensive.     

Free energetics of rigid body association of ubiquitin binding domains: A biochemical model for binding mediated by hydrophobic interaction

Weak intermolecular interactions, such as hydrophobic associations, underlie numerous biomolecular recognition processes. Ubiquitin is a small protein that represents a biochemical model for exploring thermodynamic signatures of hydrophobic association as it is widely held that a major component of ubiquitin's binding to numerous partners is mediated by hydrophobic regions on both partners. Here, we use atomistic molecular dynamics simulations in conjunction with the Adaptive Biasing Force sampling method to compute potentials of mean force (the reversible work, or free energy, associated with the binding process) to investigate the thermodynamic signature of complexation in this well‐studied biochemical model of hydrophobic association. We observe that much like in the case of a purely hydrophobic solute (i.e., graphene, carbon nanotubes), association is favored by entropic contributions from release of water from the interprotein regions. Moreover, association is disfavored by loss of enthalpic interactions, but unlike in the case of purely hydrophobic solutes, in this case protein‐water interactions are lost and not compensated for by additional water‐water interactions generated upon release of interprotein and moreso, hydration, water. We further find that relative orientations of the proteins that mutually present hydrophobic regions of each protein to its partner are favored over those that do not. In fact, the free energy minimum as predicted by a force field based method recapitulates the experimental NMR solution structure of the complex. Proteins 2014; 82:1453–1468. © 2014 Wiley Periodicals, Inc.