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We illustrate three methods for calculating the binodals of phase-separated condensates from molecular simulations. Because molecular simulations can only be carried out for small system sizes, correction for finite sizes may be required for making direct comparison between calculated results and experimental data. We first summarize the three methods and then present detailed implementation of each method on a Lennard-Jones fluid. In the first method, chemical potentials are calculated over a range of particle densities in canonical-ensemble simulations; the densities of the dilute and dense phases at the given temperature are then found by a Maxwell equal-area construction. In Gibbs-ensemble Monte Carlo, the exchange between separated dilute and dense phases is simulated to obtain their densities. Lastly, slab-geometry molecular dynamics simulations model the dilute and dense phases in coexistence and yield not only their densities but also their interfacial tension. The three types of simulations are carried out for a range of system sizes, and the results are scaled to generate the binodals corrected for finite system sizes. Size-corrected interfacial tension is also produced from slab-geometry molecular dynamics simulations.

Dynamics is a crucial link between sequence and function for intrinsically disordered proteins (IDPs). NMR spin relaxation is a powerful technique for characterizing the sequence-dependent backbone dynamics of IDPs. Of particular interest is the15N transverse relaxation rate (R2), which reports on slower dynamics (10s of ns up to 1 μs and beyond). NMR and molecular dynamics (MD) simulations have shown that local interactions and secondary structure formation slow down backbone dynamics and raiseR2. ElevatedR2has been suggested to be indicators of propensities of membrane association, liquid-liquid phase separation, and other functional processes. Here we present a sequence-based method, SeqDYN, for predictingR2of IDPs. TheR2value of a residue is expressed as the product of contributing factors from all residues, which attenuate with increasing sequence distance from the central residue. The mathematical model has 21 parameters, representing the correlation length (where the attenuation is at 50%) and the amplitudes of the contributing factors of the 20 types of amino acids. Training on a set of 45 IDPs reveals a correlation length of 5.6 residues, aromatic and long branched aliphatic amino acids and Arg asR2promotors whereas Gly and short polar amino acids asR2suppressors. The prediction accuracy of SeqDYN is competitive against that of recent MD simulations using IDP-specific force fields. For a structured protein, SeqDYN prediction representsR2in the unfolded state. SeqDYN is available as a web server athttps://zhougroup-uic.github.io/SeqDYNidp/for rapidR2prediction.Significance StatementHow the sequences of intrinsically disordered proteins (IDPs) code for functions is still an enigma. Dynamics, in particular residue-specific dynamics, holds crucial clues. Enormous efforts have been spent to characterize residue-specific dynamics of IDPs, mainly through NMR spin relaxation experiments. Here we present a sequence-based method, SeqDYN, for predicting residue-specific backbone dynamics of IDPs. SeqDYN employs a mathematical model with 21 parameters and is trained on 45 IDPs. It provides not only rapid, accurate prediction but also insightful physical interpretation of sequence-dependent IDP dynamics.

Nonspecific binding of crowder proteins with functional proteins is likely prevalent in vivo, yet direct quantitative evidence, let alone residue-specific information, is scarce. Here we present NMR characterization showing that bovine serum albumin (BSA) weakly but preferentially interacts with HPr. Notably, the binding interface overlaps with that for HPr’s specific partner protein, EIN, leading to competition. The crowder protein thus reduces the EIN-HPr binding affinity and accelerates the dissociation of the native complex. In contrast, Ficoll-70 stabilizes the native complex and slows its dissociation, as one would expect from excluded-volume and microviscosity effects. Our atomistic modeling of macromolecular crowding rationalizes the experimental data, and provides quantitative insights into the energetics of protein-crowder interactions. The integrated NMR and modeling study yields benchmarks for the effects of crowded cellular environments on protein-protein specific interactions, with implications for evolution regarding how nonspecific binding can be minimized or exploited.

The crowded cellular environments provide ample opportunities for proteins to interact with bystander macromolecules, yet direct evidence, let alone residue-specific information, for such nonspecific binding is rare. Here, by combining NMR spectroscopy and atomistic modeling, we investigated how crowders influence the association equilibrium and kinetics of two protein partners, EIN and HPr. Ficoll-70 increases the EIN-HPr binding affinity whereas bovine serum albumin (BSA) decreases the affinity. The opposite effects of the two crowders are quantitatively explained by atomistic modeling, which shows that the stabilizing effect of Ficoll-70 arises from volume exclusion favoring the bound state. In contrast, the destabilizing effect of BSA arises from preferential soft interactions with the free state; notably, BSA has favorable electrostatic interactions with positively charged HPr residues within the EIN-binding site. Some of the residues from this site indeed experience significant chemical shift perturbation when titrated with BSA, while the relaxation rates of HPr backbone amides exhibit overall elevation. Furthermore, relaxation dispersion data indicate that Ficoll-70 and BSA both slow down the EIN-HPr association rate, but change the dissociate rate in opposite directions. The observations on kinetics are accounted for by two effects of the crowders: increasing the solution microviscosity and reshaping the EIN-HPr interaction energy surface. The kind of preferential interactions between BSA and HPr that leads to competition with EIN should be prevalent in cellular environments. Our NMR results and atomistic modeling provide benchmarks, at both qualitative and quantitative levels, for the effects of crowded cellular environments on protein-protein specific interactions.Significance StatementAlthough nonspecific binding of crowder macromolecules with functional proteins is likely prevalentin vivo, direct evidence is rare. Here we present NMR characterizations showing that bovine serum albumin preferentially interacts with a specific binding site on HPr, leading to competition with the latter’s partner EIN. The preferential interactions result in destabilization of the EIN-HPr native complex and speedup of its dissociation, contrary to expectations from excluded-volume and viscosity effects. Atomistic modeling of macromolecular crowding rationalizes the experimental observations, and provides qualitative and quantitative insight into the influences of the crowded cellular environment on protein-protein specific interactions. Our work also has implications for evolution, regarding how nonspecific binding can be either minimized or exploited for gaining new functions.

The second virial coefficient, B(2), measures a protein solution’s deviation from ideal behavior. It is widely used to predict or explain solubility, crystallization condition, aggregation propensity, and critical temperature for liquid-liquid phase separation. B(2) is determined by the interaction energy between two protein molecules, specifically, by the integration of the Mayer f-function in the relative configurational space (translation and rotation) of the two molecules. Simple theoretical models, such as one attributed to Derjaguin, Landau, Verwey, and Overbeek (DLVO), can fit the dependence of B(2) on salt concentrations. However, model parameters derived often are physically unrealistic and hardly transferable from protein to protein. Previous B(2) calculations incorporating atomistic details were done with limited sampling in the configurational space, due to enormous computational cost. Our FMAP method, based on fast Fourier transform, can considerably accelerate such calculations, and here we adapt it to calculate B(2) values for proteins represented at the atomic level in implicit solvent. After tuning of a single parameter in the energy function, FMAPB2 predicts well the B(2) values for lysozyme and other proteins over wide ranges of solvent conditions (salt concentration, pH, and temperature). The method is available as a web server at http://pipe.rcc.fsu.edu/fmapb2.

Intracellular membraneless organelles and their myriad cellular functions have garnered tremendous recent interest. It is becoming well accepted that they form via liquid-liquid phase separation (LLPS) of protein mixtures (often including RNA), where the organelles correspond to a protein-rich droplet phase coexisting with a protein-poor bulk phase. The major protein components contain disordered regions and often also RNA-binding domains, and the disordered fragments on their own easily undergo LLPS. By contrast, LLPS for structured proteins has been observed infrequently. The contrasting phase behaviors can be explained by modeling disordered and structured proteins, respectively, as polymers and colloids. These physical models also provide a better understanding of the regulation of droplet formation by cellular signals and its dysregulation leading to diseases.

Conformational malleability allows intrinsically disordered proteins (IDPs) to respond agilely to their environments, such as nonspecifically interacting with in vivo bystander macromolecules (or crowders). Previous studies have emphasized conformational compaction of IDPs due to steric repulsion by macromolecular crowders, but effects of soft attraction are largely unexplored. Here we studied the conformational ensembles of the IDP FlgM in both polymer and protein crowders by small-angle neutron scattering. As crowder concentrations increased, the mean radius of gyration of FlgM first decreased but then exhibited an uptick. Ensemble optimization modeling indicated that FlgM conformations under protein crowding segregated into two distinct populations, one compacted and one extended. Coarse-grained simulations showed that compacted conformers fit into an interstitial void and occasionally bind to a surrounding crowder, whereas extended conformers snake through interstitial crevices and bind multiple crowders simultaneously. Crowder-induced conformational segregation may facilitate various cellular functions of IDPs.

The many bystander macromolecules in the crowded cellular environments present both steric repulsion and weak attraction to proteins undergoing folding or binding and hence impact the thermodynamic and kinetic properties of these processes. The weak not nonrandom binding with bystander macromolecules may facilitate subcellular localization and biological function. Weak binding also leads to the emergence of a protein-rich, droplet phase, which has been implicated in regulating a variety of cellular functions. All these important problems can now be addressed by realistic modeling of intermolecular interactions. Configurational sampling of concentrated protein solutions is an ongoing challenge.

In cellular environments, proteins not only interact with their specific partners but also encounter a high concentration of bystander macromolecules, or crowders. Nonspecific interactions with macromolecular crowders modulate the activities of proteins, but our knowledge about the rules of nonspecific interactions is still very limited. In previous work, we presented experimental evidence that macromolecular crowders acted competitively in inhibiting the binding of maltose binding protein (MBP) with its ligand maltose. Competition between a ligand and an inhibitor may result from binding to either the same site or different conformations of the protein. Maltose binds to the cleft between two lobes of MBP, and in a series of mutants, the affinities increased with an increase in the extent of lobe closure. Here we investigated whether macromolecular crowders also have a conformational or site preference when binding to MBP. The affinities of a polymer crowder, Ficoll70, measured by monitoring tryptophan fluorescence were 3-6-fold higher for closure mutants than for wild-type MBP. Competition between the ligand and crowder, as indicated by fitting of titration data and directly by nuclear magnetic resonance spectroscopy, and their similar preferences for closed MBP conformations further suggest the scenario in which the crowder, like maltose, preferentially binds to the interlobe cleft of MBP. Similar observations were made for bovine serum albumin as a protein crowder. Conformational and site preferences in MBP-crowder binding allude to the paradigm that nonspecific interactions can possess hallmarks of molecular recognition, which may be essential for intracellular organizations including colocalization of proteins and liquid-liquid phase separation.

Recently, we (Qin, S.; Zhou, H. X. J. Chem. Theory Comput.2013, 9, 4633–4643) developed the FFT-based method for Modeling Atomistic Proteins–crowder interactions, henceforth FMAP. Given its potential wide use for calculating effects of crowding on protein folding and binding free energies, here we aimed to optimize the accuracy and speed of FMAP. FMAP is based on expressing protein–crowder interactions as correlation functions and evaluating the latter via fast Fourier transform (FFT). The numerical accuracy of FFT improves as the grid spacing for discretizing space is reduced, but at increasing computational cost. We sought to speed up FMAP calculations by using a relatively coarse grid spacing of 0.6 Å and then correcting for discretization errors. This strategy was tested for different types of interactions (hard-core repulsion, nonpolar attraction, and electrostatic interaction) and over a wide range of protein–crowder systems. We were able to correct for the numerical errors on hard-core repulsion and nonpolar attraction by an 8% inflation of atomic hard-core radii and on electrostatic interaction by a 5% inflation of the magnitudes of protein atomic charges. The corrected results have higher accuracy and enjoy a speedup of more than 100-fold over those obtained using a fine grid spacing of 0.15 Å. With this optimization of accuracy and speed, FMAP may become a practical tool for realistic modeling of protein folding and binding in cell-like environments.

We report the first assessment of blind predictions of water positions at protein-protein interfaces, performed as part of the critical assessment of predicted interactions (CAPRI) community-wide experiment. Groups submitting docking predictions for the complex of the DNase domain of colicin E2 and Im2 immunity protein (CAPRI Target 47), were invited to predict the positions of interfacial water molecules using the method of their choice. The predictions-20 groups submitted a total of 195 models-were assessed by measuring the recall fraction of water-mediated protein contacts. Of the 176 high- or medium-quality docking models-a very good docking performance per se-only 44% had a recall fraction above 0.3, and a mere 6% above 0.5. The actual water positions were in general predicted to an accuracy level no better than 1.5 A, and even in good models about half of the contacts represented false positives. This notwithstanding, three hotspot interface water positions were quite well predicted, and so was one of the water positions that is believed to stabilize the loop that confers specificity in these complexes. Overall the best interface water predictions was achieved by groups that also produced high-quality docking models, indicating that accurate modelling of the protein portion is a determinant factor. The use of established molecular mechanics force fields, coupled to sampling and optimization procedures also seemed to confer an advantage. Insights gained from this analysis should help improve the prediction of protein-water interactions and their role in stabilizing protein complexes.

Predictions of protein-protein binders and binding affinities have traditionally focused on features pertaining to the native complexes. In developing a computational method for predicting protein-protein association rate constants, we introduced the concept of transient complex after mapping the interaction energy surface. The transient complex is located at the outer boundary of the bound-state energy well, having near-native separation and relative orientation between the subunits but not yet formed most of the short-range native interactions. We found that the width of the binding funnel and the electrostatic interaction energy of the transient complex are among the features predictive of binders and binding affinities. These ideas were very promising for the five affinity-related targets (T43–45, 55, and 56) of CAPRI rounds 20–27. For T43, we ranked the single crystallographic complex as number 1 and were one of only two groups that clearly identified that complex as a true binder; for T44, we ranked the only design with measurable binding affinity as number 4. For the nine docking targets, continuing on our success in previous CAPRI rounds, we produced 10 medium-quality models for T47 and acceptable models for T48 and T49. We conclude that the interaction energy landscape and the transient complex in particular will complement existing features in leading to better prediction of binding affinities.

Community-wide blind prediction experiments such as CAPRI and CASP provide an objective measure of the current state of predictive methodology. Here we describe a community-wide assessment of methods to predict the effects of mutations on protein-protein interactions. Twenty-two groups predicted the effects of comprehensive saturation mutagenesis for two designed influenza hemagglutinin binders and the results were compared with experimental yeast display enrichment data obtained using deep sequencing. The most successful methods explicitly considered the effects of mutation on monomer stability in addition to binding affinity, carried out explicit side-chain sampling and backbone relaxation, evaluated packing, electrostatic, and solvation effects, and correctly identified around a third of the beneficial mutations. Much room for improvement remains for even the best techniques, and large-scale fitness landscapes should continue to provide an excellent test bed for continued evaluation of both existing and new prediction methodologies.

Recent experimental studies of protein folding and binding under crowded solutions suggest that crowding agents exert subtle influences on the thermodynamic and kinetic properties of the proteins. While some of the crowding effects can be understood qualitatively from simple models of the proteins, quantitative rationalization of these effects requires an atomistic representation of the protein molecules in modeling their interactions with crowders. A computational approach, known as postprocessing, has opened the door for atomistic modeling of crowding effects. This review summarizes the applications of the postprocessing approach for studying crowding effects on the thermodynamics and kinetics of protein folding, conformational transition, and binding. The integration of atomistic modeling with experiments in crowded solutions promises new insight into biochemical processes in cellular environments.

We present the results for CAPRI Round 30, the first joint CASP-CAPRI experiment, which brought together experts from the protein structure prediction and protein–protein docking communities. The Round comprised 25 targets from amongst those submitted for the CASP11 prediction experiment of 2014. The targets included mostly homodimers, a few homotetramers, and two heterodimers, and comprised protein chains that could readily be modeled using templates from the Protein Data Bank. On average 24 CAPRI groups and 7 CASP groups submitted docking predictions for each target, and 12 CAPRI groups per target participated in the CAPRI scoring experiment. In total more than 9500 models were assessed against the 3D structures of the corresponding target complexes. Results show that the prediction of homodimer assemblies by homology modeling techniques and docking calculations is quite successful for targets featuring large enough subunit interfaces to represent stable associations. Targets with ambiguous or inaccurate oligomeric state assignments, often featuring crystal contact-sized interfaces, represented a confounding factor. For those, a much poorer prediction performance was achieved, while nonetheless often providing helpful clues on the correct oligomeric state of the protein. The prediction performance was very poor for genuine tetrameric targets, where the inaccuracy of the homology-built subunit models and the smaller pair-wise interfaces severely limited the ability to derive the correct assembly mode. Our analysis also shows that docking procedures tend to perform better than standard homology modeling techniques and that highly accurate models of the protein components are not always required to identify their association modes with acceptable accuracy. We are most grateful to the PDBe at the European Bioinformatics Institute in Hinxton, UK, for hosting the CAPRI website. Our deepest thanks go to all the structural biologists and to the following structural genomics initiatives: Northeast Structural Genomics Consortium, Joint Center for Structural Genomics, NatPro PSI:Biology, New York Structural Genomics Research Center, Midwest Center for Structural Genomics, Structural Genomics Consortium, for contributing the targets for this joint CASP-CAPRI experiment. MFL acknowledges support from the FRABio FR3688 Research Federation “Structural & Functional Biochemistry of Biomolecular Assemblies.”

Chemical potential is a fundamental property for determining thermodynamic equilibria involving exchange of molecules, such as between two phases of molecular systems. Previously, we developed the fast Fourier transform (FFT)-based method for Modeling Atomistic Protein-crowder interactions (FMAP) to calculate excess chemical potentials according to the Widom insertion. Intermolecular interaction energies were expressed as correlation functions and evaluated via FFT. Here, we extend this method to calculate liquid-liquid phase equilibria of macromolecular solutions. Chemical potentials are calculated by FMAP over a wide range of molecular densities, and the condition for coexistence of low- and high-density phases is determined by the Maxwell equal-area rule. When benchmarked on Lennard-Jones fluids, our method produces an accurate phase diagram at 18% of the computational cost of the current best method. Importantly, the gain in computational speed increases dramatically as the molecules become more complex, leading to many orders of magnitude in speed up for atomistically represented proteins. We demonstrate the power of FMAP by reporting the first results for the liquid-liquid coexistence curve of γII-crystallin represented at the all-atom level. Our method may thus open the door to accurate determination of phase equilibria for macromolecular mixtures such as protein-protein mixtures and protein-RNA mixtures, that are known to undergo liquid-liquid phase separation, both in vitro and in vivo.

The crowded environment of cells poses a challenge for rapid protein-protein association. Yet, it has been established that the rates of association are similar in crowded and in dilute solutions. Here we probe the pathway leading to fast association between TEM1 β-lactamase and its inhibitor protein BLIP in crowded solutions. We show that the affinity of the encounter complex, the rate of final complex formation, and the structure of the transition state are similar in crowded solutions and in buffer. The experimental results were reproduced by calculations based on the transient-complex theory for protein association. Both experiments and calculations suggest that while crowding agents decrease the diffusion constant of the associating proteins, they also induce an effective excluded-volume attraction between them. The combination of the two opposing effects thus results in nearly identical overall association rates in diluted and crowded solutions.

PI2PE (http://pipe.sc.fsu.edu) is a suite of four web servers for predicting a variety of folding- and binding-related properties of proteins. These include the solvent accessibility of amino acids upon protein folding, the amino acids forming the interfaces of protein–protein and protein–nucleic acid complexes, and the binding rate constants of these complexes. Three of the servers debuted in 2007, and have garnered ∼2,500 unique users and finished over 30,000 jobs. The functionalities of these servers are now enhanced, and a new sever, for predicting the binding rate constants, has been added. Together, these web servers form a pipeline from protein sequence to tertiary structure, then to quaternary structure, and finally to binding kinetics.

In CAPRI rounds 13-19, we submitted models that are of acceptable or higher quality for 6 of the total of 13 targets. This success builds on our record in previous CAPRI rounds. The docking problem can be divided into two steps. In the first, translational/rotational and conformational space is searched to generate a pool of docked poses; the success of this search step is measured by whether near-native poses are included in the pool. In the second step, the pool is selected for near-native poses. In our previous assessment of CAPRI results, we suggested that the search problem is largely solved; a remaining problem is to select near-native poses. Our work in these new rounds of CAPRI was guided by this assessment. To solve the selection problem, we used an assortment of criteria on the interfaces of candidate poses. In one extreme, represented by T29, with very little known interface information, our criterion for top models was based on interface prediction. Poses in which the predicted interface residues occurred in interfaces were selected. Our model 1 for T29 was of medium quality. In the other extreme, represented by T40, with reliably known interface information, our selection was solely based on such information. Nine of the ten models submitted for T40 were of high (3 models), medium (4 models), and acceptable (2 models) quality. Our strategy of mixing predicted and known interface information appears to be widely applicable for the selection of near-native poses.

Macromolecular crowding has long been known to significantly affect protein oligomerization, and yet no direct quantitative measurements appear to have been made of its effects on the binding free energy of the elemental step of adding a single subunit. Here, we report the effects of two crowding agents on the binding free energy of two subunits in the Escherichia coli polymerase III holoenzyme. The crowding agents are found, paradoxically, to have only a modest stabilizing effect, of the order of 1 kcal/mol, on the binding of the two subunits. Systematic variations in the level of stabilization with crowder size are nevertheless observed. The data are consistent with theoretical predictions based on atomistic modeling of excluded-volume interactions with crowders. We reconcile the apparent paradox presented by our data by noting that the modest effects of crowding on elemental binding steps are cumulative, and thus lead to substantial stabilization of higher oligomers. Correspondingly, the effects of small variations in the level of crowding during the lifetime of a cell may be magnified, suggesting that crowding may play a role in increased susceptibility to protein aggregation-related diseases with aging.

We have developed a computational approach for predicting protein-protein association rates (Alsallaq and Zhou, Structure 2007, 15, 215). Here we expand the range of applicability of this approach to protein-RNA binding and report the first results for protein-RNA binding rates predicted from atomistic modeling. The system studied is the U1A protein and stem/loop II of the U1 small nuclear RNA. Experimentally it was observed that the binding rate is significantly reduced by increasing salt concentration while the dissociation changes little with salt concentration, and charges distant from the binding site make marginal contribution to the binding rate. These observations are rationalized. Moreover, predicted effects of salt and charge mutations are found to be in quantitative agreement with experimental results.

Docking of unbound protein structures into a complex has gained significant progress in recent years, but nonetheless still poses a great challenge. We have pursued a holistic approach to docking which brings together effective methods at different stages. First, protein-protein interaction sites are predicted or obtained from experimental studies in the literature. Interface prediction/experimental data are then used to guide the generation of docked poses or to rank docked poses generated from an unbiased search. Finally, selected models are refined by lengthy molecular dynamics (MD) simulations in explicit water. For CAPRI target T27, we used information on interaction sites as input to drive docking and as a filter to rank docked poses. Lead candidates were then clustered according to RMSD among them. From the clustering, 10 models were selected and subject to refinement by MD simulations. Our Model 7 is rated number one among all submissions according to L_rmsd. Six of our other submissions are rated acceptable. As scorer, eight of our submissions are rated acceptable. Proteins 2007. © 2007 Wiley-Liss, Inc.

The negatively charged phosphates of nucleic acids are often paired with positively charged residues upon binding proteins. It was thus counter-intuitive when previous Poisson-Boltzmann (PB) calculations gave positive energies from electrostatic interactions, meaning that they destabilize protein-nucleic acid binding. Our own PB calculations on protein-protein binding have shown that the sign and the magnitude of the electrostatic component are sensitive to the specification of the dielectric boundary in PB calculations. A popular choice for the boundary between the solute low dielectric and the solvent high dielectric is the molecular surface; an alternative is the van der Waals (vdW) surface. In line with results for protein-protein binding, in this article, we found that PB calculations with the molecular surface gave positive electrostatic interaction energies for two protein-RNA complexes, but the signs are reversed when the vdW surface was used. Therefore, whether destabilizing or stabilizing effects are predicted depends on the choice of the dielectric boundary. The two calculation protocols, however, yielded similar salt effects on the binding affinity. Effects of charge mutations differentiated the two calculation protocols; PB calculations with the vdW surface had smaller deviations overall from experimental data.

We have improved the multiple linear regression (MLR) algorithm for protein secondary structure prediction by combining it with the evolutionary information provided by multiple sequence alignment of PSI-BLAST. On the CB513 dataset, the three states average overall per-residue accuracy, Q(3), reached 76.4%, while segment overlap accuracy, SOV99, reached 73.2%, using a rigorous jackknife procedure and the strictest reduction of eight states DSSP definition to three states. This represents an improvement of approximately 5% on overall per-residue accuracy compared with previous work. The relative solvent accessibility prediction also benefited from this combination of methods. The system achieved 77.7% average jackknifed accuracy for two states prediction based on a 25% relative solvent accessibility mode, with a Mathews' correlation coefficient of 0.548. The improved MLR secondary structure and relative solvent accessibility prediction server is available at http://spg.biosci.tsinghua.edu.cn/.

Chemical potential is a fundamental property for determining thermodynamic equilibria involving exchanges of molecules, such as between two phases of molecular systems. Previously we developed the FMAP method for calculating excess chemical potentials according to Widom insertion [1, 2]. Intermolecular interaction energies were expressed as correlation times and evaluated via fast Fourier transform. Here we extend this method to calculate phase equilibria. Chemical potentials are calculated by FMAP at a wide range of molecular densities and the boundary between low- and high-density phases is identified by the Maxwell equal-area rule. When benchmarked on a Lennard-Jones fluid, our method produces an accurate phase diagram at 10% of the computational cost of the present best method. Importantly, the gain in computational speed increases dramatically as the molecules become more complex, potentially reaching many orders of magnitude in speedup for atomistically represented proteins [2]. Our method thus opens the door to accurate determination of phase equilibria for macromolecular mixtures. Of particular interest are protein-protein mixtures and protein-RNA mixtures that involve poly-valent interactions. These mixtures are known to undergo a liquid-liquid phase separation, both in vitro and in vivo. Our phase calculations for these systems will lead to much need physical understanding of the liquid-liquid phase separation and how it is used for many biological functions.1. Qin, S., Zhou, H. X. J. Chem. Theory Comput. 2013, 9, 4633−4643.2. Qin, S., Zhou, H. X. J. Chem. Theory Comput. 2014, 10, 2824-2835.

Due to their conformational malleability, intrinsically disordered proteins (IDPs) are particularly susceptible to influences of crowded cellular environments. Here we report a computational study of the effects of macromolecular crowding on the conformational ensemble of a coarse-grained IDP model, by using two approaches. In one, the IDP is simulated along with the crowders; in the other, crowder-free simulations are postprocessed to predict the conformational ensembles under crowding. We found significant decreases in the radius of gyration of the IDP under crowding, and suggest repulsive interactions with crowders as a common cause for chain compaction in a number of experimental studies. The postprocessing approach accurately reproduced the conformational ensembles of the IDP in the direct simulations here, and holds enormous potential for realistic modeling of IDPs under crowding, by permitting thorough conformation sampling for the proteins even when they and the crowders are both represented at the all-atom level.

It is now well recognized that macromolecular crowding can exert significant effects on protein folding and binding stability. In order to calculate such effects in direct simulations of proteins mixed with bystander macromolecules, the latter (referred to as crowders) are usually modeled as spheres and the proteins represented at a coarse-grained level. Our recently developed postprocessing approach allows the proteins to be represented at the all-atom level but, for computational efficiency, has only been implemented for spherical crowders. Modeling crowder molecules in cellular environments and in vitro experiments as spheres may distort their effects on protein stability. Here we present a new method that is capable for treating aspherical crowders. The idea, borrowed from protein-protein docking, is to calculate the excess chemical potential of the proteins in crowded solution by fast Fourier transform (FFT). As the first application, we studied the effects of ellipsoidal crowders on the folding and binding free energies of all-atom proteins, and found, in agreement with previous direct simulations with coarse-grained protein models, that the aspherical crowders exert greater stabilization effects than spherical crowders of the same volume. Moreover, as demonstrated here, the FFT-based method has the important property that its computational cost does not increase strongly even when the level of details in representing the crowders is increased all the way to all-atom, thus significantly accelerating realistic modeling of protein folding and binding in cell-like environments.

PI2PE (http://pipe.sc.fsu.edu) is a suite of four web servers for predicting a variety of folding- and binding-related properties of proteins. These include the solvent accessibility of amino acids upon protein folding, the amino acids forming the interfaces of protein–protein and protein–nucleic acid complexes, and the binding rate constants of these complexes. Three of the servers debuted in 2007, and have garnered ∼2,500 unique users and finished over 30,000 jobs. The functionalities of these servers are now enhanced, and a new sever, for predicting the binding rate constants, has been added. Together, these web servers form a pipeline from protein sequence to tertiary structure, then to quaternary structure, and finally to binding kinetics.

In cellular environments, two protein molecules on their way to form a specific complex encounter many bystander macromolecules. The latter molecules, or crowders, affect both the energetics of the interaction between the test molecules and the dynamics of their relative motion. In earlier work (Zhou and Szabo 1991 J. Chem. Phys. 95 5948-52), it has been shown that, in modeling the association kinetics of the test molecules, the presence of crowders can be accounted for by their energetic and dynamic effects. The recent development of the transient-complex theory for protein association in dilute solutions makes it possible to easily incorporate the energetic and dynamic effects of crowders. The transient complex refers to a late on-pathway intermediate, in which the two protein molecules have near-native relative separation and orientation, but have yet to form the many short-range specific interactions of the native complex. The transient-complex theory predicts the association rate constant as k(a) = k(a0)exp(-ΔG*(el)/k(B)T), where k(a0) is the 'basal' rate constant for reaching the transient complex by unbiased diffusion, and the Boltzmann factors captures the influence of long-range electrostatic interactions between the protein molecules. Crowders slow down the diffusion, therefore reducing the basal rate constant (to k(ac0)), and induce an effective interaction energy ΔG(c). We show that the latter interaction energy for atomistic proteins in the presence of spherical crowders is 'long'-ranged, allowing the association rate constant under crowding to be computed as k(ac) = k(ac0)exp[-(ΔG*(el) + ΔG*(c))/k(B)T]. Applications demonstrate that this computational method allows for realistic modeling of protein association kinetics under crowding.

In cellular environments the total macromolecule concentrations can exceed 300 g/l. Such crowded conditions can significantly affect stability, kinetics, and function of proteins. Previous studies on the effects of crowding have focused on the excluded-volume interaction. Recent studies with protein crowders and cell lysates suggested that, in addition to excluded-volume interaction, these crowding agents also have a soft interaction with the test protein chymotrypsin inhibitor 2 (CI2). The latter interaction has a comparable, or even greater effect on CI2 folding stability than the former interaction. We have developed a postprocessing method for modeling crowding [1, 2]. In this method we simulate the end states (e.g., folded and unfolded states) of a protein in a dilute solution and then calculate the free energies of transferring these end-state conformations to a crowded solution. We thus bypass the slow transitions between end states, which have necessitated coarse-grained representations by others. The postprocessing method was applied previously [1-3] to treat excluded-volume interaction, by either an insertion algorithm [1] or a theory involving numerically calculated radius, area, and volume [4], with the test proteins represented at the all-atom level. Here we treat the additional soft interaction, by using a fast Fourier transform based algorithm. We use the new method to study the total effects of protein crowders on CI2 folding stability and compare the results to experimental data. The success in modeling both excluded-volume and soft interactions of protein crowders is an important step toward understanding the total impact of cellular environments.[1] Qin & Zhou (2009) Biophys. J. 97, 12.[2] Qin, Minh, McCammon & Zhou (2010) J. Phys. Chem. Lett. 1, 107.[3] Dong, Qin & Zhou (2010) PLoS Comput. Biol. 6, e1000833.[4] Qin & Zhou (2010) Phys. Rev. E 81, 031919.

Actin is an abundant protein that constitutes a main component of the eukaryotic cytoskeleton. Its polymerization and depolymerization are regulated by a variety of actin-binding proteins. Their functions range from nucleation of actin polymerization to sequestering G-actin in 1∶1 complexes. The kinetics of forming these complexes, with rate constants varying at least three orders of magnitude, is critical to the distinct regulatory functions. Previously we have developed a transient-complex theory for computing protein association mechanisms and association rate constants. The transient complex refers to an intermediate in which the two associating proteins have near-native separation and relative orientation but have yet to form short-range specific interactions of the native complex. The association rate constant is predicted as k a = k a0 , where k a0 is the basal rate constant for reaching the transient complex by free diffusion, and the Boltzmann factor captures the bias of long-range electrostatic interactions. Here we applied the transient-complex theory to study the association kinetics of seven actin-binding proteins with G-actin. These proteins exhibit three classes of association mechanisms, due to their different molecular shapes and flexibility. The 1000-fold k a variations among them can mostly be attributed to disparate electrostatic contributions. The basal rate constants also showed variations, resulting from the different shapes and sizes of the interfaces formed by the seven actin-binding proteins with G-actin. This study demonstrates the various ways that actin-binding proteins use physical properties to tune their association mechanisms and rate constants to suit distinct regulatory functions., Author Summary Actin polymerization and depolymerization drive cell motility and are regulated by a variety of actin-binding proteins. The widely-varying rate constants (k a) of the actin-binding proteins associating with G-actin, spanning at least three orders of magnitude, appear to be tuned for their distinct regulatory functions. Here we applied our previously developed transient-complex theory to study the association kinetics of seven actin-binding proteins with G-actin. These proteins exhibit three classes of association mechanisms, due to their different molecular shapes and flexibility. The 1000-fold k a variations among them can mostly be attributed to disparate inter-protein electrostatic interactions. By computing the association mechanisms and quantifying the physical determinants of association rate constants, the present study reveals critical links between the structure and function of the actin-binding proteins.

International audience; The CAPRI (Critical Assessment of Predicted Interactions) and CASP (Critical Assessment of protein Structure Prediction) experiments have demonstrated the power of community-wide tests of methodology in assessing the current state of the art and spurring progress in the very challenging areas of protein docking and structure prediction. We sought to bring the power of community-wide experiments to bear on a very challenging protein design problem that provides a complementary but equally fundamental test of current understanding of protein-binding thermodynamics. We have generated a number of designed protein-protein interfaces with very favorable computed binding energies but which do not appear to be formed in experiments, suggesting that there may be important physical chemistry missing in the energy calculations. A total of 28 research groups took up the challenge of determining what is missing: we provided structures of 87 designed complexes and 120 naturally occurring complexes and asked participants to identify energetic contributions and/or structural features that distinguish between the two sets. The community found that electrostatics and solvation terms partially distinguish the designs from the natural complexes, largely due to the nonpolar character of the designed interactions. Beyond this polarity difference, the community found that the designed binding surfaces were, on average, structurally less embedded in the designed monomers, suggesting that backbone conformational rigidity at the designed surface is important for realization of the designed function. These results can be used to improve computational design strategies, but there is still much to be learned; for example, one designed complex, which does form in experiments, was classified by all metrics as a nonbinder.

The four cytokines erythropoietin (EPO), interleukin-4 (IL4), human growth hormone (hGH), and prolactin (PRL) all form four-helix bundles and bind to type I cytokine receptors. However, their receptor-binding rate constants span a 5000-fold range. Here, we quantitatively rationalize these vast differences in rate constants by our transient-complex theory for protein-protein association. In the transient complex, the two proteins have near-native separation and relative orientation, but have yet to form the short-range specific interactions of the native complex. The theory predicts the association rate constant as k(a)=k(a0)exp(-ΔG(el)(∗)/k(B)T) where k(a0) is the basal rate constant for reaching the transient complex by random diffusion, and the Boltzmann factor captures the rate enhancement due to electrostatic attraction. We found that the vast differences in receptor-binding rate constants of the four cytokines arise mostly from the differences in charge complementarity among the four cytokine-receptor complexes. The basal rate constants (k(a0)) of EPO, IL4, hGH, and PRL were similar (5.2 × 10(5) M(-1)s(-1), 2.4 × 10(5) M(-1)s(-1), 1.7 × 10(5) M(-1)s(-1), and 1.7 × 10(5) M(-1)s(-1), respectively). However, the average electrostatic free energies (ΔG(e1)(∗)) were very different (-4.2 kcal/mol, -2.4 kcal/mol, -0.1 kcal/mol, and -0.5 kcal/mol, respectively, at ionic strength=160 mM). The receptor-binding rate constants predicted without adjusting any parameters, 6.2 × 10(8) M(-1)s(-1), 1.3 × 10(7) M(-1)s(-1), 2.0 × 10(5) M(-1)s(-1), and 7.6 × 10(4) M(-1)s(-1), respectively, for EPO, IL4, hGH, and PRL agree well with experimental results. We uncover that these diverse rate constants are anticorrelated with the circulation concentrations of the cytokines, with the resulting cytokine-receptor binding rates very close to the limits set by the half-lives of the receptors, suggesting that these binding rates are functionally relevant and perhaps evolutionarily tuned. Our calculations also reproduced well-observed effects of mutations and ionic strength on the rate constants and produced a set of mutations on the complex of hGH with its receptor that putatively enhances the rate constant by nearly 100-fold through increasing charge complementarity. To quantify charge complementarity, we propose a simple index based on the charge distribution within the binding interface, which shows good correlation with ΔG(e1)(∗). Together these results suggest that protein charges can be manipulated to tune k(a) and control biological function.

The association rate constants (k(a)) of proteins with other proteins or other macromolecular targets are a fundamental biophysical property. Observed rate constants span over ten orders of magnitude, from 1 to 10(10) M(-1)s(-1). Protein association can be rate limited either by the diffusional approach of the subunits to form a transient complex, with near-native separation and orientation but without short-range native interactions, or by the subsequent conformational rearrangement to form the native complex. Our transient-complex theory showed promise in predicting k(a) in the diffusion-limited regime. Here, we develop it into a web server called TransComp (http://pipe.sc.fsu.edu/transcomp/) and report on the server's accuracy and robustness based on applications to over 100 protein complexes. We expect this server to be a valuable tool for systems biology applications and for kinetic characterization of protein-protein and protein-nucleic acid association in general.

In CAPRI rounds 13-19, we submitted models that are of acceptable or higher quality for 6 of the total of 13 targets. This success builds on our record in previous CAPRI rounds. The docking problem can be divided into two steps. In the first, translational/rotational and conformational space is searched to generate a pool of docked poses; the success of this search step is measured by whether near-native poses are included in the pool. In the second step, the pool is selected for near-native poses. In our previous assessment of CAPRI results, we suggested that the search problem is largely solved; a remaining problem is to select near-native poses. Our work in these new rounds of CAPRI was guided by this assessment. To solve the selection problem, we used an assortment of criteria on the interfaces of candidate poses. In one extreme, represented by T29, with very little known interface information, our criterion for top models was based on interface prediction. Poses in which the predicted interface residues occurred in interfaces were selected. Our model 1 for T29 was of medium quality. In the other extreme, represented by T40, with reliably known interface information, our selection was solely based on such information. Nine of the ten models submitted for T40 were of high (3 models), medium (4 models), and acceptable (2 models) quality. Our strategy of mixing predicted and known interface information appears to be widely applicable for the selection of near-native poses.

Macromolecular crowding inside cells affects the thermodynamic and kinetic properties of proteins. The scaled particle theory (SPT) has played an important role toward establishing a qualitative picture for the effects of crowding. However, SPT-based modeling lacks molecular details. Molecular dynamics simulations overcome this limitation, but at great computational cost. Here, we present a theoretical method for modeling crowding at the atomic level. The method makes it possible to achieve exhaustive conformational sampling in modeling crowding effects and to tackle challenges posed by large protein oligomers and by complex mixtures of crowders.

Conformational changes are directly linked to protein functions. Inside cells, the equilibria and transition rates between different conformations may be affected by macromolecular crowding. Recently we have developed a “postprocessing” approach for modeling crowding effects [1]. With this approach, proteins can now be represented at the atomic level when modeling crowding effects. Here this approach is used to study how crowding affects the equilibria between open and closed conformations of 8 proteins: Trp repressor, adenylate kinase (AK), orotidine phosphate decarboxylase (ODCase), yeast protein-disulfide isomerase (PDI), hemoglobin, DNA β-glucosltransferase, cyanovirin-N, and Ap4A hydrolase. For each protein, molecular dynamics simulations of the open and closed states are separately run. Representative open and closed conformations are then used to calculate the crowding-induced changes in chemical potential for the two states. The difference in chemical-potential change between the two states finally predicts the effects of crowding on the population ratio of the two states. Crowding is found to reduce the open population to various extents. In the presence of crowders with a 20-A radius and occupying 35% of volume, the open-to-closed population ratios of Trp repressor, ODCase, AK, and PDI are reduced by 40%, 46%, 64%, and 80%, respectively. The reductions for the remaining four proteins are 15-30%. As expected, the four proteins experiencing the stronger crowding effects are those with larger conformational changes between open and closed states (e.g., as measured by the change in radius of gyration). Larger proteins also tend to experience stronger crowding effects than smaller ones [e.g., comparing PDI (504 residues) and Trp repressor (107 residues)]. These quantitative results and qualitative trends will serve as valuable guide for expected crowding effects on protein conformation changes inside cells.[1] S. Qin, and H.-X. Zhou, Biophys J 97, 12 (2009).

The internal dynamics of proteins inside of cells may be affected by the crowded intracellular environments. Here, we test a novel approach to simulations of crowding, in which simulations in the absence of crowders are postprocessed to predict crowding effects, against the direct approach of simulations in the presence of crowders. The effects of crowding on the flap dynamics of HIV-1 protease predicted by the postprocessing approach are found to agree well with those calculated by the direct approach. The postprocessing approach presents distinct advantages over the direct approach in terms of accuracy and speed and is expected to have broad impact on atomistic simulations of macromolecular crowding.