Coexistence of two coacervate phases of polyglycine in water suggested by polymer reference interaction site model theory.

Mixing Gibbs energy and phase equilibria of aqueous solutions of polyglycine were studied theoretically by means of polymer reference interaction site model integral equation theory combined with the Gibbs-Duhem method. In addition to the ordinary liquid-liquid phase separation between dilute and concentrated solutions, the theoretical calculation predicted the coexistence of two coacervate phases, namely, the lower- and higher-density coacervates. The relative thermodynamic stabilities of these two phases change with the polymerization degree of polyglycine. The higher-density coacervate phase was rapidly stabilized by increasing the polymer length, and the lower-density phase became metastable at large polymers. The hydrogen bonds between the peptide chains were strengthened, and water was thermodynamically destabilized in the higher-density coacervate. A possible relation with the formation of amyloid fibril within a liquid droplet is also discussed.

Effects of intramolecular chain conformation on the hydration and miscibility of polyethylene glycol in water studied by means of polymer reference interaction site model theory.

To examine the conventional idea that the gauche conformation of the OCCO dihedral angle promotes the dissolution of polyethylene glycol (PEG) in water through strong hydration, the thermodynamic properties of liquid mixtures of PEG and water were studied by means of polymer reference interaction site model (PRISM) theory. The intramolecular correlation functions required as input for PRISM theory were calculated by the generator matrix method, accompanied by changes in the distribution of dihedral angles. In the infinite dilution limit, the increased probability of gauche conformation of the OCCO dihedral angles stabilizes the hydration of PEG through enhanced hydrogen bonding between the ether oxygen of PEG and water. The mixing Gibbs energies of the liquid mixtures were also calculated in the whole concentration range based on the Gibbs-Duhem equation, as per our recent proposal. A liquid-liquid phase separation was observed when all the dihedral angles of PEG were in the trans conformation; for the liquid mixture to be miscible in the whole concentration range, the introduction of the OCCO gauche conformation was found to be indispensable. The above theoretical results support the conventional idea that the OCCO gauche conformation is important for the high miscibility of PEG and water.

Comparing the influence of explicit and implicit solvation models on site-specific thermodynamic stability of proteins.

Understanding the molecular basis for protein stability requires a thermodynamic analysis of protein folding. Thermodynamic analysis is often performed by sampling many atomistic conformations using molecular simulations that employ either explicit or implicit water models. However, it remains unclear to what extent thermodynamic results from different solvation models are reliable at the molecular level. In this study, we quantify the influence of both solvation models on folding stability at the individual backbone and side chain resolutions. We assess the residue-specific folding free energy components of a ß-sheet protein and a helical protein using trajectories resulting from TIP3P explicit and generalized Born/surface area implicit solvent simulations of model proteins. We found that the thermodynamic discrepancy due to the implicit solvent mostly originates from charged side chains, followed by the under-stabilized hydrophobic ones. In contrast, the contributions of backbone residue in both proteins were comparable for explicit and implicit water models. Our study lays out the foundation for detailed thermodynamic assessment of solvation models in the context of protein simulation.

Phase equilibrium of three-component liquid systems composed of water, alcohol, and sodium chloride studied by the reference interaction-site model integral equation theory.

A theoretical method for calculating the thermodynamic properties and phase equilibria of a binary liquid mixture using the reference interaction-site model (RISM) integral equation theory, which we had proposed recently, was extended to ternary liquid systems containing salt. A novel dielectric correction of the RISM theory for a mixture of solvents was also proposed. The theory was applied to mixtures composed of water, alcohol, and NaCl, where the alcohol was either methanol or ethanol. The decrease in NaCl solubility with increasing alcohol molar fractions in the solvent was calculated. In the ethanol system, the theory yielded salt-induced liquid-liquid phase separation, which was observed experimentally in a ternary mixture of water, 1-propanol, and NaCl. The phase diagram of the ternary system was determined theoretically.

Evolutionary conservation of amino acids contributing to the protein folding transition state.

The question of whether amino acids critical to protein folding kinetics are evolutionarily conserved has been investigated intensively in the past, but no consensus has yet been reached. Recently, we have demonstrated that the transition state, dictating folding kinetics, is characterized as the state of maximum dynamic cooperativity, i.e., the state of maximum correlations between amino acid contact formations. Here, we investigate the evolutionary conservation of those amino acids contributing significantly to the dynamic cooperativity. We find a strong indication of a new kind of relationship-necessary but not sufficient causality-between the evolutionary conservation and the dynamic cooperativity: larger contributions to the dynamic cooperativity arise from more conserved residues, but not vice versa. This holds for all the protein systems for which long folding simulation trajectories are available. To our knowledge, this is the first systematic demonstration of any kind of evolutionary conservation of amino acids relevant to folding kinetics.

Atomic-level thermodynamics analysis of the binding free energy of SARS-CoV-2 neutralizing antibodies.

Understanding how protein-protein binding affinity is determined from molecular interactions at the interface is essential in developing protein therapeutics such as antibodies, but this has not yet been fully achieved. Among the major difficulties are the facts that it is generally difficult to decompose thermodynamic quantities into contributions from individual molecular interactions and that the solvent effect-dehydration penalty-must also be taken into consideration for every contact formation at the binding interface. Here, we present an atomic-level thermodynamics analysis that overcomes these difficulties and illustrate its utility through application to SARS-CoV-2 neutralizing antibodies. Our analysis is based on the direct interaction energy computed from simulated antibody-protein complex structures and on the decomposition of solvation free energy change upon complex formation. We find that the formation of a single contact such as a hydrogen bond at the interface barely contributes to binding free energy due to the dehydration penalty. On the other hand, the simultaneous formation of multiple contacts between two interface residues favorably contributes to binding affinity. This is because the dehydration penalty is significantly alleviated: the total penalty for multiple contacts is smaller than a sum of what would be expected for individual dehydrations of those contacts. Our results thus provide a new perspective for designing protein therapeutics of improved binding affinity.

Study of phase equilibria and thermodynamic properties of liquid mixtures using the integral equation theory: Application to water and alcohol mixtures.

A theoretical method for calculating the thermodynamic properties and phase equilibria of liquid-liquid mixtures using the integral equation theory is proposed. The solvation chemical potentials of the two components are evaluated by the integral equation theory and the isothermal-isobaric variation of the total density with composition is determined to satisfy the Gibbs-Duhem relation. Given the density of a pure component, the method can calculate the densities of the mixture at any composition. Furthermore, it can treat the phase equilibrium without thermodynamic inconsistency with respect to the Gibbs-Duhem relation. This method was combined with the reference interaction-site model integral equation theory and applied to mixtures of water + 1-alcohol by changing the alcohol from methanol to 1-butanol. The destabilization of the mixing Gibbs energy by increasing the hydrophobicity of the alcohol and demixing of the water-butanol mixture were reproduced. However, quantitative agreement with experiments is not satisfactory, and further improvements of the integral equation theory and the molecular models are required.

Mutation effects on FAS1 domain 4 based on structure and solubility.

Mutations in the fasciclin 1 domain 4 (FAS1-4) of transforming growth factor ß-induced protein (TGFBIp) are associated with insoluble extracellular deposits and corneal dystrophies (CDs). The decrease in solubility upon mutation has been implicated in CD; however, the exact molecular mechanisms are not well understood. Here, we performed molecular dynamics simulations followed by solvation thermodynamic analyses of the FAS1-4 domain and its three mutants-R555W, R555Q, and A546T-linked to granular corneal dystrophy type 1, Thiel-Behnke corneal dystrophy and lattice corneal dystrophy, respectively. We found that both R555W and R555Q mutants have less affinity toward solvent water relative to the wild-type protein. In the R555W mutant, a remarkable increase in solvation free energy was observed because of the structural changes near the mutation site. The mutation site W555 is buried in other hydrophobic residues, and R557 simultaneously forms salt bridges with E554 and D561. In the R555Q mutant, the increase in solvation free energy is caused by structural rearrangements far from the mutation site. R558 separately forms salt bridges with D575, E576, and E598. Thus, we thus identified the relationship between the decrease in solubility and conformational changes caused by mutations, which may be useful in designing potential therapeutics and in blocking FAS1 aggregation related to CD.

Site-Specific Backbone and Side-Chain Contributions to Thermodynamic Stabilizing Forces of the WW Domain.

The native structure of a protein is stabilized by a number of interactions such as main-chain hydrogen bonds and side-chain hydrophobic contacts. However, it has been challenging to determine how these interactions contribute to protein stability at single amino acid resolution. Here, we quantified site-specific thermodynamic stability at the molecular level to extend our understanding of the stabilizing forces in protein folding. We derived the free energy components of individual amino acid residues separately for the folding of the human Pin WW domain based on simulated structures. A further decomposition of the thermodynamic properties into contributions from backbone and side-chain groups enabled us to identify the critical residues in the secondary structure and hydrophobic core formation, without introducing physical modifications to the system as in site-directed mutagenesis methods. By relating the structural and thermodynamic changes upon folding for each residue, we find that the simultaneous formation of the backbone hydrogen bonds and side-chain contacts cooperatively stabilizes the folded structure. The identification of stabilizing interactions in a folding protein at atomic resolution will provide molecular insights into understanding the origin of the protein structure and into engineering a more stable protein.

Time-dependent communication between multiple amino acids during protein folding.

Cooperativity is considered to be a key organizing principle behind biomolecular assembly, recognition and folding. However, it has remained very challenging to quantitatively characterize how cooperative processes occur on a concerted, multiple-interaction basis. Here, we address how and when the folding process is cooperative on a molecular scale. To this end, we analyze multipoint time-correlation functions probing time-dependent communication between multiple amino acids, which were computed from long folding simulation trajectories. We find that the simultaneous multiple amino-acid contact formation, which is absent in the unfolded state, starts to develop only upon entering the folding transition path. Interestingly, the transition state, whose presence is connected to the macrostate cooperative behavior known as the two-state folding, can be identified as the state in which the amino-acid cooperativity is maximal. Thus, our work not only provides a new mechanistic view on how protein folding proceeds on a multiple-interaction basis, but also offers a conceptually novel characterization of the folding transition state and the molecular origin of the phenomenological cooperative folding behavior. Moreover, the multipoint correlation function approach adopted here is general and can be used to expand the understanding of cooperative processes in complex chemical and biomolecular systems.

Local environment effects on charged mutations for developing aggregation-resistant monoclonal antibodies.

Protein aggregation is a major concern in biotherapeutic applications of monoclonal antibodies. Introducing charged mutations is among the promising strategies to improve aggregation resistance. However, the impact of such mutations on solubilizing activity depends largely on the inserting location, whose mechanism is still not well understood. Here, we address this issue from a solvation viewpoint, and this is done by analyzing how the change in solvation free energy upon charged mutation is composed of individual contributions from constituent residues. To this end, we perform molecular dynamics simulations for a number of antibody mutants and carry out the residue-wise decomposition of the solvation free energy. We find that, in addition to the previously identified "global" principle emphasizing the key role played by the protein total net charge, a local net charge within [Formula: see text]15 Å from the mutation site exerts significant effects. For example, when the net charge of an antibody is positive, the global principle states that introducing a positively charged mutation will lead to more favorable solvation. Our finding further adds that an even more optimal mutation can be done at the site around which more positively charged residues and fewer negatively charged residues are present. Such a "local" design principle accounts for the location dependence of charged mutations, and will be useful in producing aggregation-resistant antibodies.

Folding Free Energy Landscape of Ordered and Intrinsically Disordered Proteins.

Folding funnel is the essential concept of the free energy landscape for ordered proteins. How does this concept apply to intrinsically disordered proteins (IDPs)? Here, we address this fundamental question through the explicit characterization of the free energy landscapes of the representative α-helical (HP-35) and ß-sheet (WW domain) proteins and of an IDP (pKID) that folds upon binding to its partner (KIX). We demonstrate that HP-35 and WW domain indeed exhibit the steep folding funnel: the landscape slope for these proteins is ca. -50 kcal/mol, meaning that the free energy decreases by ~5 kcal/mol upon the formation of 10% native contacts. On the other hand, the landscape of pKID is funneled but considerably shallower (slope of -24 kcal/mol), which explains why pKID is disordered in free environments. Upon binding to KIX, the landscape of pKID now becomes significantly steep (slope of -54 kcal/mol), which enables otherwise disordered pKID to fold. We also show that it is the pKID-KIX intermolecular interactions originating from hydrophobic residues that mainly confer the steep folding funnel. The present work not only provides the quantitative characterization of the protein folding free energy landscape, but also establishes the usefulness of the folding funnel concept to IDPs.

Explicit Characterization of the Free Energy Landscape of pKID-KIX Coupled Folding and Binding.

The most fundamental aspect of the free energy landscape of proteins is that it is globally funneled such that protein folding is energetically biased. Then, what are the distinctive characteristics of the landscape of intrinsically disordered proteins, apparently lacking such energetic bias, that nevertheless fold upon binding? Here, we address this fundamental issue through the explicit characterization of the free energy landscape of the paradigmatic pKID-KIX system (pKID, phosphorylated kinase-inducible domain; KIX, kinase interacting domain). This is done based on unguided, fully atomistic, explicit-water molecular dynamics simulations with an aggregated simulation time of >30 µs and on the computation of the free energy that defines the landscape. We find that, while the landscape of pKID before binding is considerably shallower than the one for a protein that autonomously folds, it becomes progressively more funneled as the binding of pKID with KIX proceeds. This explains why pKID is disordered in a free state, and the binding of pKID with KIX is a prerequisite for pKID's folding. In addition, we observe that the key event in completing the pKID-KIX coupled folding and binding is the directed self-assembly where pKID is docked upon the KIX surface to maximize the surface electrostatic complementarity, which, in turn, require pKID to adopt the correct folded structure. This key process shows up as the free energy barrier in the pKID landscape separating the intermediate nonspecific complex state and the specific complex state. The present work not only provides a detailed molecular picture of the coupled folding and binding of pKID but also expands the funneled landscape perspective to intrinsically disordered proteins.

Examining a Thermodynamic Order Parameter of Protein Folding.

Dimensionality reduction with a suitable choice of order parameters or reaction coordinates is commonly used for analyzing high-dimensional time-series data generated by atomistic biomolecular simulations. So far, geometric order parameters, such as the root mean square deviation, fraction of native amino acid contacts, and collective coordinates that best characterize rare or large conformational transitions, have been prevailing in protein folding studies. Here, we show that the solvent-averaged effective energy, which is a thermodynamic quantity but unambiguously defined for individual protein conformations, serves as a good order parameter of protein folding. This is illustrated through the application to the folding-unfolding simulation trajectory of villin headpiece subdomain. We rationalize the suitability of the effective energy as an order parameter by the funneledness of the underlying protein free energy landscape. We also demonstrate that an improved conformational space discretization is achieved by incorporating the effective energy. The most distinctive feature of this thermodynamic order parameter is that it works in pointing to near-native folded structures even when the knowledge of the native structure is lacking, and the use of the effective energy will also find applications in combination with methods of protein structure prediction.

Publisher Correction: Examining a Thermodynamic Order Parameter of Protein Folding.

A correction to this article has been published and is linked from the HTML and PDF versions of this paper. The error has been fixed in the paper.

Dynamics of Hydration Water Plays a Key Role in Determining the Binding Thermodynamics of Protein Complexes.

Interfacial waters are considered to play a crucial role in protein-protein interactions, but in what sense and why are they important? Here, using molecular dynamics simulations and statistical thermodynamic analyses, we demonstrate distinctive dynamic characteristics of the interfacial water and investigate their implications for the binding thermodynamics. We identify the presence of extraordinarily slow (~1,000 times slower than in bulk water) hydrogen-bond rearrangements in interfacial water. We rationalize the slow rearrangements by introducing the "trapping" free energies, characterizing how strongly individual hydration waters are captured by the biomolecular surface, whose magnitude is then traced back to the number of water-protein hydrogen bonds and the strong electrostatic field produced at the binding interface. We also discuss the impact of the slow interfacial waters on the binding thermodynamics. We find that, as expected from their slow dynamics, the conventional approach to the water-mediated interaction, which assumes rapid equilibration of the waters' degrees of freedom, is inadequate. We show instead that an explicit treatment of the extremely slow interfacial waters is critical. Our results shed new light on the role of water in protein-protein interactions, highlighting the need to consider its dynamics to improve our understanding of biomolecular bindings.

Computer Simulations of Intrinsically Disordered Proteins.

The investigation of intrinsically disordered proteins (IDPs) is a new frontier in structural and molecular biology that requires a new paradigm to connect structural disorder to function. Molecular dynamics simulations and statistical thermodynamics potentially offer ideal tools for atomic-level characterizations and thermodynamic descriptions of this fascinating class of proteins that will complement experimental studies. However, IDPs display sensitivity to inaccuracies in the underlying molecular mechanics force fields. Thus, achieving an accurate structural characterization of IDPs via simulations is a challenge. It is also daunting to perform a configuration-space integration over heterogeneous structural ensembles sampled by IDPs to extract, in particular, protein configurational entropy. In this review, we summarize recent efforts devoted to the development of force fields and the critical evaluations of their performance when applied to IDPs. We also survey recent advances in computational methods for protein configurational entropy that aim to provide a thermodynamic link between structural disorder and protein activity.

Anomalous Dynamics of Water Confined in Protein-Protein and Protein-DNA Interfaces.

Confined water often exhibits anomalous properties not observable in the bulk phase. Although water in hydrophobic confinement has been the focus of intense investigation, the behavior of water confined between hydrophilic surfaces, which are more frequently found in biological systems, has not been fully explored. Here, we investigate using molecular dynamics simulations dynamical properties of the water confined in hydrophilic protein-protein and protein-DNA interfaces. We find that the interfacial water exhibits glassy slow relaxations even at 300 K. In particular, the rotational dynamics show a logarithmic decay that was observed in glass-forming liquids at deeply supercooled states. We argue that such slow water dynamics are indeed induced by the hydrophilic binding surfaces, which is in opposition to the picture that the hydration water slaves protein motions. Our results will significantly impact the view on the role of water in biomolecular interactions.

New Computational Approach for External Entropy in Protein-Protein Binding.

Molecular recognition through the noncovalent association of biomolecules is of central importance in biology and pharmacology. Developing reliable computational methods for estimating binding thermodynamic parameters is therefore of great practical value. However, considerable uncertainty remains regarding the external entropy that is associated with the reduction in the external (positional and orientational) degrees of freedom upon complex formation. Here, we present a novel statistical mechanical method for computing the external entropy by extending the energetic approach we have developed for unimolecular processes to association processes. We find that, in contrary to what is postulated in most of the previous methods, intrinsic couplings between the internal and external degrees of freedom of bound complex cannot in general be neglected in the determination of the external entropy. Nevertheless, there exists the best choice of the external coordinates with which those couplings are minimized. With such a choice of the external coordinates, the lowest upper bound of the external entropy is obtained from a tractable expression, which serves as an estimate of the external entropy. Our method can be implemented in a straightforward manner with molecular dynamics simulations, and its applicability is demonstrated through the application to the barnase-barstar complex.

Thermodynamic-ensemble independence of solvation free energy.

Solvation free energy is the fundamental thermodynamic quantity in solution chemistry. Recently, it has been suggested that the partial molar volume correction is necessary to convert the solvation free energy determined in different thermodynamic ensembles. Here, we demonstrate ensemble-independence of the solvation free energy on general thermodynamic grounds. Theoretical estimates of the solvation free energy based on the canonical or grand-canonical ensemble are pertinent to experiments carried out under constant pressure without any conversion.