Fundamental Aspects of Phase-Separated Biomolecular Condensates.

Biomolecular condensates, formed through phase separation, are upending our understanding in much of molecular, cell, and developmental biology. There is an urgent need to elucidate the physicochemical foundations of the behaviors and properties of biomolecular condensates. Here we aim to fill this need by writing a comprehensive, critical, and accessible review on the fundamental aspects of phase-separated biomolecular condensates. We introduce the relevant theoretical background, present the theoretical basis for the computation and experimental measurement of condensate properties, and give mechanistic interpretations of condensate behaviors and properties in terms of interactions at the molecular and residue levels.

Calculating Structure Factors of Protein Solutions by Atomistic Modeling of Protein-Protein Interactions.

We present a method, FMAPS(q), for calculating the structure factor, S(q), of a protein solution, by extending our fast Fourier transform-based modeling of atomistic protein-protein interactions (FMAP) approach. The interaction energy consists of steric, nonpolar attractive, and electrostatic terms that are additive among all pairs of atoms between two protein molecules. In the present version, we invoke the free-rotation approximation, such that the structure factor is given by the Fourier transform of the protein center-center distribution function gC(R). At low protein concentrations, gC(R) can be approximated as e-ßW(R), where W(R) is the potential of mean force along the center-center distance R. We calculate W(R) using FMAPB2, a member of the FMAP class of methods that is specialized for the second virial coefficient [Qin and Zhou, J Phys Chem B 123 (2019) 8203-8215]. For higher protein concentrations, we obtain S(q) by a modified random-phase approximation, which is a perturbation around the steric-only energy function. Without adjusting any parameters, the calculated structure factors for lysozyme and bovine serum albumin at various ionic strengths, temperatures, and protein concentrations are all in reasonable agreement with those measured by small-angle X-ray or neutron scattering. This initial success motivates further developments, including removing approximations and parameterizing the interaction energy function.

Surface Charge Can Modulate Phase Separation of Multidomain Proteins.

Phase separation has emerged as an important mechanism explaining the formation of certain biomolecular condensates. Biological phase separation is often driven by the multivalent interactions of modular protein domains. Beyond valency, the physical features of folded domains that promote phase separation are poorly understood. We used a model system─the small ubiquitin modifier (SUMO) and its peptide ligand, the SUMO interaction motif (SIM)─to examine how domain surface charge influences multivalency-driven phase separation. Phase separation of polySUMO and polySIM was altered by pH via a change in the protonation state of SUMO surface histidines. These effects were recapitulated by histidine mutations, which modulated SUMO solubility and polySUMO-polySIM phase separation in parallel and were quantitatively explained by atomistic modeling of weak interactions among proteins in the system. Thus, surface charge can tune the phase separation of multivalent proteins, suggesting a means of controlling phase separation biologically, evolutionarily, and therapeutically.

Atomistic modeling of liquid-liquid phase equilibrium explains dependence of critical temperature on γ-crystallin sequence.

Liquid-liquid phase separation of protein solutions has regained heightened attention for its biological importance and pathogenic relevance. Coarse-grained models are limited when explaining residue-level effects on phase equilibrium. Here we report phase diagrams for γ-crystallins using atomistic modeling. The calculations were made possible by combining our FMAP method for computing chemical potentials and Brownian dynamics simulations for configurational sampling of dense protein solutions, yielding the binodal and critic temperature (Tc). We obtain a higher Tc for a known high-Tc γ-crystallin, γF, than for a low-Tc paralog, γB. The difference in Tc is corroborated by a gap in second virial coefficient. Decomposition of inter-protein interactions reveals one amino-acid substitution between γB and γF, from Ser to Trp at position 130, as the major contributor to the difference in Tc. This type of analysis enables us to link phase equilibrium to amino-acid sequence and to design mutations for altering phase equilibrium.

Atomistic Modeling of Liquid-Liquid Phase Equilibrium Explains Dependence of Critical Temperature on γ-Crystallin Sequence.

Liquid-liquid phase separation of protein solutions has regained heightened attention for its biological importance and pathogenic relevance. Coarse-grained models are limited when explaining residue-level effects on phase equilibrium. Here we report phase diagrams for γ-crystallins using atomistic modeling. The calculations were made possible by combining our FMAP method for computing chemical potentials and Brownian dynamics simulations for configurational sampling of dense protein solutions, yielding the binodal and critic temperature ( T c ). We obtain a higher T c for a known high- T c γ-crystallin, γF, than for a low- T c paralog, γB. The difference in T c is corroborated by a gap in second virial coefficient. Decomposition of inter-protein interactions reveals one amino-acid substitution between γB and γF, from Ser to Trp at position 130, as the major contributor to the difference in T c . This type of analysis enables us to link phase equilibrium to amino-acid sequence and to design mutations for altering phase equilibrium.

Predicting the Sequence-Dependent Backbone Dynamics of Intrinsically Disordered Proteins.

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 the 15N 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 raise R2. Elevated R2 has 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 predicting R2 of IDPs. The R2 value 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 as R2 promotors whereas Gly and short polar amino acids as R2 suppressors. The prediction accuracy of SeqDYN is competitive against that of recent MD simulations using IDP-specific force fields. For a structured protein, SeqDYN prediction represents R2 in the unfolded state. SeqDYN is available as a web server at https://zhougroup-uic.github.io/SeqDYNidp/ for rapid R2 prediction.

Calculating Binodals and Interfacial Tension of Phase-Separated Condensates from Molecular Simulations with Finite-Size Corrections.

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.

ReSMAP: Web Server for Predicting Residue-Specific Membrane-Association Propensities of Intrinsically Disordered Proteins.

The functional processes of many proteins involve the association of their intrinsically disordered regions (IDRs) with acidic membranes. We have identified the membrane-association characteristics of IDRs using extensive molecular dynamics (MD) simulations and validated them with NMR spectroscopy. These studies have led to not only deep insight into functional mechanisms of IDRs but also to intimate knowledge regarding the sequence determinants of membrane-association propensities. Here we turned this knowledge into a web server called ReSMAP, for predicting the residue-specific membrane-association propensities from IDR sequences. The membrane-association propensities are calculated from a sequence-based partition function, trained on the MD simulation results of seven IDRs. Robustness of the prediction is demonstrated by leaving one IDR out of the training set. We anticipate there will be many applications for the ReSMAP web server, including rapid screening of IDR sequences for membrane association.

Preferential Interactions of a Crowder Protein with the Specific Binding Site of a Native Protein Complex.

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 nuclear magnetic resonance (NMR) characterization showing that bovine serum albumin weakly but preferentially interacts with the histidine carrier protein (HPr). Notably, the binding interface overlaps with that for HPr's specific partner protein, EIN, leading to competition. The crowder protein thus decreases 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.

Calculation of Second Virial Coefficients of Atomistic Proteins Using Fast Fourier Transform.

The second virial coefficient, B2, 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. B2 is determined by the interaction energy between two protein molecules and, 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 B2 on salt concentrations. However, model parameters derived often are physically unrealistic and hardly transferable from protein to protein. Previous B2 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 B2 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 B2 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 .

Transfer Free Energies of Test Proteins Into Crowded Protein Solutions Have Simple Dependence on Crowder Concentration.

The effects of macromolecular crowding on the thermodynamic properties of test proteins are determined by the latter's transfer free energies from a dilute solution to a crowded solution. The transfer free energies in turn are determined by effective protein-crowder interactions. When these interactions are modeled at the all-atom level, the transfer free energies may defy simple predictions. Here we investigated the dependence of the transfer free energy (Δµ) on crowder concentration. We represented both the test protein and the crowder proteins atomistically, and used a general interaction potential consisting of hard-core repulsion, non-polar attraction, and solvent-screened electrostatic terms. The chemical potential was rigorously calculated by FMAP (Qin and Zhou, 2014), which entails expressing the protein-crowder interaction terms as correlation functions and evaluating them via fast Fourier transform (FFT). To high accuracy, the transfer free energy can be decomposed into an excluded-volume component (Δµe-v), arising from the hard-core repulsion, and a soft-attraction component (Δµs-a), arising from non-polar and electrostatic interactions. The decomposition provides physical insight into crowding effects, in particular why such effects are very modest on protein folding stability. Further decomposition of Δµs-a into non-polar and electrostatic components does not work, because these two types of interactions are highly correlated in contributing to Δµs-a. We found that Δµe-v fits well to the generalized fundamental measure theory (Qin and Zhou, 2010), which accounts for atomic details of the test protein but approximates the crowder proteins as spherical particles. Most interestingly, Δµs-a has a nearly linear dependence on crowder concentration. The latter result can be understood within a perturbed virial expansion of Δµ (in powers of crowder concentration), with Δµe-v as reference. Whereas the second virial coefficient deviates strongly from that of the reference system, higher virial coefficients are close to their reference counterparts, thus leaving the linear term to make the dominant contribution to Δµs-a.

Both Ligands and Macromolecular Crowders Preferentially Bind to Closed Conformations of Maltose Binding Protein.

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.

Atomistic Modeling of Intrinsically Disordered Proteins Under Polyethylene Glycol Crowding: Quantitative Comparison with Experimental Data and Implication of Protein-Crowder Attraction.

The malleability of intrinsically disordered proteins (IDPs) has generated great interest in understanding how their conformations respond to crowded cellular environments. Experiments can report gross properties such as fluorescence resonance energy transfer (FRET) efficiency but cannot resolve the conformational ensembles of IDPs and their interactions with macromolecular crowders. Computation can in principle provide the latter information but in practice has been hampered by the enormous expense for realistic modeling of IDPs and crowders and for sufficient conformational sampling. Here, taking advantage of a powerful method called FMAP (fast Fourier transform-based modeling of atomistic protein-crowder interactions), we computed how the conformational ensembles of three IDPs are modified in concentrated polyethylene glycol (PEG) 6000 solutions. We represented the IDPs at the all-atom level and the PEG molecules at a coarse-grained level and calculated the experimental observable, i.e., FRET efficiency. Whereas accounting for only steric repulsion of PEG led to overestimation of crowding effects, quantitative agreement with experimental data was obtained upon including mild IDP-PEG attraction. The present work demonstrates that realistic modeling of IDPs under crowded conditions for direct comparison with experiments is now achievable.

Why Do Disordered and Structured Proteins Behave Differently in Phase Separation?

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.

Intrinsically Disordered Protein Exhibits Both Compaction and Expansion under Macromolecular Crowding.

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.

Protein folding, binding, and droplet formation in cell-like conditions.

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 but 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.

Fast Method for Computing Chemical Potentials and Liquid-Liquid Phase Equilibria of Macromolecular Solutions.

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.

Prediction of homoprotein and heteroprotein complexes by protein docking and template-based modeling: A CASP-CAPRI experiment.

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. Proteins 2016; 84(Suppl 1):323-348. © 2016 Wiley Periodicals, Inc.

Further Development of the FFT-based Method for Atomistic Modeling of Protein Folding and Binding under Crowding: Optimization of Accuracy and Speed.

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.