Challenges and Advantages of Accounting for Backbone Flexibility in Prediction of Protein-Protein Complexes.

Predicting protein binding is a core problem of computational biophysics. That this objective can be partly achieved with some amount of success using docking algorithms based on rigid protein models is remarkable, although going further requires allowing for protein flexibility. However, accurately capturing the conformational changes upon binding remains an enduring challenge for docking algorithms. Here, we adapt our Upside folding model, where side chains are represented as multi-position beads, to explore how flexibility may impact predictions of protein-protein complexes. Specifically, the Upside model is used to investigate where backbone flexibility helps, which types of interactions are important, and what is the impact of coarse graining. These efforts also shed light on the relative challenges posed by folding and docking. After training the Upside energy function for docking, the model is competitive with the established all-atom methods. However, allowing for backbone flexibility during docking is generally detrimental, as the presence of comparatively minor (3-5 Å) deviations relative to the docked structure has a large negative effect on performance. While this issue appears to be inherent to current forcefield-guided flexible docking methods, systems involving the co-folding of flexible loops such as antibody-antigen complexes represent an interesting exception. In this case, binding is improved when backbone flexibility is allowed using the Upside model.

Prediction and Validation of a Protein's Free Energy Surface Using Hydrogen Exchange and (Importantly) Its Denaturant Dependence.

The denaturant dependence of hydrogen-deuterium exchange (HDX) is a powerful measurement to identify the breaking of individual H-bonds and map the free energy surface (FES) of a protein including the very rare states. Molecular dynamics (MD) can identify each partial unfolding event with atomic-level resolution. Hence, their combination provides a great opportunity to test the accuracy of simulations and to verify the interpretation of HDX data. For this comparison, we use Upside, our new and extremely fast MD package that is capable of folding proteins with an accuracy comparable to that of all-atom methods. The FESs of two naturally occurring and two designed proteins are so generated and compared to our NMR/HDX data. We find that Upside's accuracy is considerably improved upon modifying the energy function using a new machine-learning procedure that trains for proper protein behavior including realistic denatured states in addition to stable native states. The resulting increase in cooperativity is critical for replicating the HDX data and protein stability, indicating that we have properly encoded the underlying physiochemical interactions into an MD package. We did observe some mismatch, however, underscoring the ongoing challenges faced by simulations in calculating accurate FESs. Nevertheless, our ensembles can identify the properties of the fluctuations that lead to HDX, whether they be small-, medium-, or large-scale openings, and can speak to the breadth of the native ensemble that has been a matter of debate.

On the Interpretation of Force-Induced Unfolding Studies of Membrane Proteins Using Fast Simulations.

Single-molecule force spectroscopy has proven extremely beneficial in elucidating folding pathways for membrane proteins. Here, we simulate these measurements, conducting hundreds of unfolding trajectories using our fast Upside algorithm for slow enough speeds to reproduce key experimental features that may be missed using all-atom methods. The speed also enables us to determine the logarithmic dependence of pulling velocities on the rupture levels to better compare to experimental values. For simulations of atomic force microscope measurements in which force is applied vertically to the C-terminus of bacteriorhodopsin, we reproduce the major experimental features including even the back-and-forth unfolding of single helical turns. When pulling laterally on GlpG to mimic the experiment, we observe quite different behavior depending on the stiffness of the spring. With a soft spring, as used in the experimental studies with magnetic tweezers, the force remains nearly constant after the initial unfolding event, and a few pathways and a high degree of cooperativity are observed in both the experiment and simulation. With a stiff spring, however, the force drops to near zero after each major unfolding event, and numerous intermediates are observed along a wide variety of pathways. Hence, the mode of force application significantly alters the perception of the folding landscape, including the number of intermediates and the degree of folding cooperativity, important issues that should be considered when designing experiments and interpreting unfolding data.

Lattice theory for binding of linear polymers to a solid substrate from polymer melts: I. Influence of chain connectivity on molecular binding and adsorption.

Most theories of the binding of molecules to surfaces or for the association between molecules treat the binding species as structureless entities and neglect their rigidity and the changes in their stiffness induced by the binding process. The binding species are also taken to be "ideal," meaning that the existence of van der Waals interactions and changes in these interactions upon molecular binding are also neglected. An understanding of the thermodynamics of these multifunctional molecular binding processes has recently come into focus in the context of the molecular binding of complex molecules, such as dendrimers and DNA grafted nanoparticles, to surfaces where the degree of binding cooperativity and selectivity, as well as the location of the binding transition, are found to be sensitive to the number of binding units constrained to a larger scale polymeric scaffold. We address the fundamental problem of molecular binding by extending classical Langmuir theory to describe the particular example of the reversible binding of semiflexible polymer chains to a solid substrate under melt conditions. The polymer chains are assumed to have a variable number N of binding units (segments) and to exhibit variable bending energies and van der Waals interactions in the bulk and on the surface, in addition to strong directional interactions with the surface. The resulting generalized Langmuir theory is applied to the examination of the influence of the chain connectivity of ideal polymers on the surface coverage Θ, transition binding temperature T1/2 at which Θ = 1/2, and on the derivative |dΘ/dT|T=T1/2 and the constant volume specific heat of binding, Cv bind, measures of the cooperativity and "sharpness" of the binding transition, respectively. Paper II is devoted to the impact of the van der Waals attractive interactions and chain stiffness on the reversible binding of nonideal polymer chains to a solid surface, including the enthalpy-entropy compensation phenomenon observed experimentally in many molecular and particle binding processes.

Lattice theory for binding of linear polymers to a solid substrate from polymer melts. II. Influence of van der Waals interactions and chain semiflexibility on molecular binding and adsorption.

The polymeric Langmuir theory, developed in Paper I [J. Dudowicz et al., J. Chem. Phys. 151, 124706 (2019)], is employed to investigate the influence of van der Waals interactions and chain rigidity on the thermodynamics of the reversible molecular binding to interfaces in one component polymer fluids (polymer melts). Both van der Waals interactions and chain stiffness are found to influence the temperature variation of the surface coverage Θ, the binding transition itself, and the cooperativity of molecular binding. Re-entrancy of the surface coverage Θ(T) is found to arise when the intermolecular interactions are sufficiently attractive to cause a liquid-vapor like phase separation in the interfacial region, a phenomenon that can occur in the binding of both small molecules and polymer chains to surfaces.

Trajectory-based training enables protein simulations with accurate folding and Boltzmann ensembles in cpu-hours.

An ongoing challenge in protein chemistry is to identify the underlying interaction energies that capture protein dynamics. The traditional trade-off in biomolecular simulation between accuracy and computational efficiency is predicated on the assumption that detailed force fields are typically well-parameterized, obtaining a significant fraction of possible accuracy. We re-examine this trade-off in the more realistic regime in which parameterization is a greater source of error than the level of detail in the force field. To address parameterization of coarse-grained force fields, we use the contrastive divergence technique from machine learning to train from simulations of 450 proteins. In our procedure, the computational efficiency of the model enables high accuracy through the precise tuning of the Boltzmann ensemble. This method is applied to our recently developed Upside model, where the free energy for side chains is rapidly calculated at every time-step, allowing for a smooth energy landscape without steric rattling of the side chains. After this contrastive divergence training, the model is able to de novo fold proteins up to 100 residues on a single core in days. This improved Upside model provides a starting point both for investigation of folding dynamics and as an inexpensive Bayesian prior for protein physics that can be integrated with additional experimental or bioinformatic data.

Accurate calculation of side chain packing and free energy with applications to protein molecular dynamics.

To address the large gap between time scales that can be easily reached by molecular simulations and those required to understand protein dynamics, we present a rapid self-consistent approximation of the side chain free energy at every integration step. In analogy with the adiabatic Born-Oppenheimer approximation for electronic structure, the protein backbone dynamics are simulated as preceding according to the dictates of the free energy of an instantaneously-equilibrated side chain potential. The side chain free energy is computed on the fly, allowing the protein backbone dynamics to traverse a greatly smoothed energetic landscape. This computation results in extremely rapid equilibration and sampling of the Boltzmann distribution. Our method, termed Upside, employs a reduced model involving the three backbone atoms, along with the carbonyl oxygen and amide proton, and a single (oriented) side chain bead having multiple locations reflecting the conformational diversity of the side chain's rotameric states. We also introduce a novel, maximum-likelihood method to parameterize the side chain interactions using protein structures. We demonstrate state-of-the-art accuracy for predicting χ1 rotamer states while consuming only milliseconds of CPU time. Our method enables rapidly equilibrating coarse-grained simulations that can nonetheless contain significant molecular detail. We also show that the resulting free energies of the side chains are sufficiently accurate for de novo folding of some proteins.

Dielectric virial expansion of polarizable dipolar spheres.

The dielectric virial expansion is developed for composite systems with embedded interacting dielectric dipolar spheres. Introducing a multiple-scattering expansion for the polarization energy in the presence of an external field enables the derivation of a virial expansion for the polarizability. Substituting the polarizability into the Clausius-Mossotti relation yields the virial series for the effective medium permittivity. When the dipole moment of the particles or inclusions vanishes, the leading-order term in the series reduces to the Maxwell-Garnett mixing rule, whereas the higher-order terms provide corrections that become important at higher densities. The dielectric virial coefficients are readily evaluated by replacing the surface charge contributions with image lines. Numerical data are presented for the second virial coefficients to illustrate the effects of polarization.

A Membrane Burial Potential with H-Bonds and Applications to Curved Membranes and Fast Simulations.

We use the statistics of a large and curated training set of transmembrane helical proteins to develop a knowledge-based potential that accounts for the dependence on both the depth of burial of the protein in the membrane and the degree of side-chain exposure. Additionally, the statistical potential includes depth-dependent energies for unsatisfied backbone hydrogen bond donors and acceptors, which are found to be relatively small, ∼2 RT. Our potential accurately places known proteins within the bilayer. The potential is applied to the mechanosensing MscL channel in membranes of varying thickness and curvature, as well as to the prediction of protein structure. The potential is incorporated into our new Upside molecular dynamics algorithm. Notably, we account for the exchange of protein-lipid interactions for protein-protein interactions as helices contact each other, thereby avoiding overestimating the energetics of helix association within the membrane. Simulations of most multimeric complexes find that isolated monomers and the oligomers retain the same orientation in the membrane, suggesting that the assembly of prepositioned monomers presents a viable mechanism of oligomerization.

Response to Comment on "Innovative scattering analysis shows that hydrophobic disordered proteins are expanded in water".

Best et al claim that we provide no convincing basis to assert that a discrepancy remains between FRET and SAXS results on the dimensions of disordered proteins under physiological conditions. We maintain that a clear discrepancy is apparent in our and other recent publications, including results shown in the Best et al comment. A plausible origin is fluorophore interactions in FRET experiments.

Lattice theory of competitive binding: Influence of van der Waals interactions on molecular binding and adsorption to a solid substrate from binary liquid mixtures.

The reversible binding of molecules to surfaces is one of the most fundamental processes in condensed fluids, with obvious applications in the molecular separation of materials, chromatographic characterization, and material processing. Motivated in particular by the ubiquitous occurrence of binding processes in molecular biology and self-assembly, we have developed a lattice type theory of competitive molecular binding to solid substrates from binary mixtures of two small molecule liquids that interact between themselves by van der Waals forces in addition to exhibiting binding interactions with the solid surface. The derived theory, in contrast to previously existing theoretical frameworks, enables us to investigate the influence of van der Waals interactions on interfacial binding and selective molecular adsorption. For reference, the classic Langmuir theory of adsorption is recovered when all van der Waals interaction energies between the molecules in the bulk liquid phase and those on the surface are formally set to zero. Illustrative calculations are performed for the binding of molecules to a solid surface from pure liquids and from their binary mixtures. The properties analyzed include the surface coverage θ, the binding transition temperature Tbind, the individual surface coverages, θA and θC, and the relative surface coverages, σAC≡θA/θC or σCA≡θC/θA. The latter two quantities coincide with the degrees of adsorption directly determined from experimental adsorption measurements. The Langmuir theory is shown to apply formally under a wide range of conditions where the original enthalpies (Δh or ΔhA and ΔhC) and entropies (Δs or ΔsA and ΔsC) of the binding reactions are simply replaced by their respective "effective" counterparts (Δheff or ΔhAeff and ΔhCeff and Δseff or ΔsAeff and ΔsCeff), whose values depend on the strength of der Waals interactions and of the "bare" free energy parameters (Δh or ΔhA and ΔhC, and Δs or ΔsA and ΔsC). Numerous instances of entropy-enthalpy compensation between these effective free energy parameters follow from our calculations, confirming previous reports on this phenomenon obtained from experimental studies of molecular binding processes in solution.

Innovative scattering analysis shows that hydrophobic disordered proteins are expanded in water.

A substantial fraction of the proteome is intrinsically disordered, and even well-folded proteins adopt non-native geometries during synthesis, folding, transport, and turnover. Characterization of intrinsically disordered proteins (IDPs) is challenging, in part because of a lack of accurate physical models and the difficulty of interpreting experimental results. We have developed a general method to extract the dimensions and solvent quality (self-interactions) of IDPs from a single small-angle x-ray scattering measurement. We applied this procedure to a variety of IDPs and found that even IDPs with low net charge and high hydrophobicity remain highly expanded in water, contrary to the general expectation that protein-like sequences collapse in water. Our results suggest that the unfolded state of most foldable sequences is expanded; we conjecture that this property was selected by evolution to minimize misfolding and aggregation.

Mixtures of two self- and mutually-associating liquids: Phase behavior, second virial coefficients, and entropy-enthalpy compensation in the free energy of mixing.

The theoretical description of the phase behavior of polymers dissolved in binary mixtures of water and other miscible solvents is greatly complicated by the self- and mutual-association of the solvent molecules. As a first step in treating these complex and widely encountered solutions, we have developed an extension of Flory-Huggins theory to describe mixtures of two self- and mutually-associating fluids comprised of small molecules. Analytic expressions are derived here for basic thermodynamic properties of these fluid mixtures, including the spinodal phase boundaries, the second osmotic virial coefficients, and the enthalpy and entropy of mixing these associating solvents. Mixtures of this kind are found to exhibit characteristic closed loop phase boundaries and entropy-enthalpy compensation for the free energy of mixing in the low temperature regime where the liquid components are miscible. As discussed by Widom et al. [Phys. Chem. Chem. Phys. 5, 3085 (2003)], these basic miscibility trends, quite distinct from those observed in non-associating solvents, are defining phenomenological characteristics of the "hydrophobic effect." We find that our theory of mixtures of associating fluids captures at least some of the thermodynamic features of real aqueous mixtures.

Image method for electrostatic energy of polarizable dipolar spheres.

The multiple-scattering theory for the electrostatics of many-body systems of monopolar spherical particles, embedded in a dielectric medium, is generalized to describe the electrostatics of these particles with embedded dipoles and multipoles. The Neumann image line construction for the electrostatic polarization produced by one particle is generalized to compute the energy, forces, and torques for the many-body system as functions of the positions of the particles. The approach is validated by comparison with direct numerical calculation, and the convergence rate is analyzed and expressed in terms of the discontinuity in dielectric contrast and particle density. As an illustration of this formalism, the stability of small particle clusters is analyzed. The theory is developed in a form that can readily be adapted to Monte Carlo and molecular dynamics simulations for polarizable particles and, more generally, to study the interactions among polarizable molecules.

Comparative Study of the Collective Dynamics of Proteins and Inorganic Nanoparticles.

Molecular dynamics simulations of ubiquitin in water/glycerol solutions are used to test the suggestion by Karplus and coworkers that proteins in their biologically active state should exhibit a dynamics similar to 'surface-melted' inorganic nanoparticles (NPs). Motivated by recent studies indicating that surface-melted inorganic NPs are in a 'glassy' state that is an intermediate dynamical state between a solid and liquid, we probe the validity and significance of this proposed analogy. In particular, atomistic simulations of ubiquitin in solution based on CHARMM36 force field and pre-melted Ni NPs (Voter-Chen Embedded Atom Method potential) indicate a common dynamic heterogeneity, along with other features of glass-forming (GF) liquids such as collective atomic motion in the form of string-like atomic displacements, potential energy fluctuations and particle displacements with long range correlations ('colored' or 'pink' noise), and particle displacement events having a power law scaling in magnitude, as found in earthquakes. On the other hand, we find the dynamics of ubiquitin to be even more like a polycrystalline material in which the α-helix and ß-sheet regions of the protein are similar to crystal grains so that the string-like collective atomic motion is concentrated in regions between the α-helix and ß-sheet domains.

Generalized entropy theory of glass-formation in fully flexible polymer melts.

The generalized entropy theory (GET) offers many insights into how molecular parameters influence polymer glass-formation. Given the fact that chain rigidity often plays a critical role in understanding the glass-formation of polymer materials, the GET was originally developed based on models of semiflexible chains. Consequently, all previous calculations within the GET considered polymers with some degree of chain rigidity. Motivated by unexpected results from computer simulations of fully flexible polymer melts concerning the dependence of thermodynamic and dynamic properties on the cohesive interaction strength (ϵ), the present paper employs the GET to explore the influence of ϵ on glass-formation in models of polymer melts with a vanishing bending rigidity, i.e., fully flexible polymer melts. In accord with simulations, the GET for fully flexible polymer melts predicts that basic dimensionless thermodynamic properties (such as the reduced thermal expansion coefficient and isothermal compressibility) are universal functions of the temperature scaled by ϵ in the regime of low pressures. Similar scaling behavior is also found for the configurational entropy density in the GET for fully flexible polymer melts. Moreover, we find that the characteristic temperatures of glass-formation increase linearly with ϵ and that the fragility is independent of ϵ in fully flexible polymer melts, predictions that are again consistent with simulations of glass-forming polymer melts composed of fully flexible chains. Beyond an explanation of these general trends observed in simulations, the GET for fully flexible polymer melts predicts the presence of a positive residual configurational entropy at low temperatures, indicating a return to Arrhenius relaxation in the low temperature glassy state.

Image method for induced surface charge from many-body system of dielectric spheres.

Charged dielectric spheres embedded in a dielectric medium provide the simplest model for many-body systems of polarizable ions and charged colloidal particles. We provide a multiple scattering formulation for the total electrostatic energy for such systems and demonstrate that the polarization energy can be rapidly evaluated by an image method that generalizes the image methods for conducting spheres. Individual contributions to the total electrostatic energy are ordered according to the number of polarized surfaces involved, and each additional surface polarization reduces the energy by a factor of (a/R)3ϵ, where a is the sphere radius, R the average inter-sphere separation, and ϵ the relevant dielectric mismatch at the interface. Explicit expressions are provided for both the energy and the forces acting on individual spheres, which can be readily implemented in Monte Carlo and molecular dynamics simulations of polarizable charged spheres, thereby avoiding costly computational techniques that introduce a surface charge distribution that requires numerical solution.

Relation Between Solvent Quality and Phase Behavior of Ternary Mixtures of Polymers and Two Solvents that Exhibit Cononsolvency.

The phase boundaries of polymer solutions in mixed solvents can be extremely complex due to the many competing van der Waals and associative interactions that can arise in these ubiquitous and technologically important complex fluids. The present paper focuses specific attention on ternary solutions of polymers (B) dissolved in a mixture of two solvents (A, C) that competitively associate with the polymer. We are particularly concerned with explaining the origin of the peculiar phenomenon of cononsolvency in mixed solvents, where a mixture of two individually good solvents behaves effectively as a poor solvent. Our computations are based on a recently developed generalization of Flory-Huggins theory that incorporates the competitive solvation of a polymer by two associating solvents. On the basis of this framework, we evaluate the limit of polymer phase stability (spinodal curves) and the second osmotic virial coefficient [Formula: see text] as a function of temperature and the composition of the pure solvent mixture that is maintained in osmotic equilibrium with the ternary solution. The calculations provide new insights into the miscibility patterns of ternary A/B/C polymer solutions exhibiting cononsolvency.

Self-assembly and glass-formation in a lattice model of telechelic polymer melts: Influence of stiffness of the sticky bonds.

Telechelic polymers are chain macromolecules that may self-assemble through the association of their two mono-functional end groups (called "stickers"). A deep understanding of the relation between microscopic molecular details and the macroscopic physical properties of telechelic polymers is important in guiding the rational design of telechelic polymer materials with desired properties. The lattice cluster theory (LCT) for strongly interacting, self-assembling telechelic polymers provides a theoretical tool that enables establishing the connections between important microscopic molecular details of self-assembling polymers and their bulk thermodynamics. The original LCT for self-assembly of telechelic polymers considers a model of fully flexible linear chains [J. Dudowicz and K. F. Freed, J. Chem. Phys. 136, 064902 (2012)], while our recent work introduces a significant improvement to the LCT by including a description of chain semiflexibility for the bonds within each individual telechelic chain [W.-S. Xu and K. F. Freed, J. Chem. Phys. 143, 024901 (2015)], but the physically associative (or called "sticky") bonds between the ends of the telechelics are left as fully flexible. Motivated by the ubiquitous presence of steric constraints on the association of real telechelic polymers that impart an additional degree of bond stiffness (or rigidity), the present paper further extends the LCT to permit the sticky bonds to be semiflexible but to have a stiffness differing from that within each telechelic chain. An analytical expression for the Helmholtz free energy is provided for this model of linear telechelic polymer melts, and illustrative calculations demonstrate the significant influence of the stiffness of the sticky bonds on the self-assembly and thermodynamics of telechelic polymers. A brief discussion is also provided for the impact of self-assembly on glass-formation by combining the LCT description for this extended model of telechelic polymers with the Adam-Gibbs relation between the structural relaxation time and the configurational entropy.

Cooperative folding near the downhill limit determined with amino acid resolution by hydrogen exchange.

The relationship between folding cooperativity and downhill, or barrier-free, folding of proteins under highly stabilizing conditions remains an unresolved topic, especially for proteins such as λ-repressor that fold on the microsecond timescale. Under aqueous conditions where downhill folding is most likely to occur, we measure the stability of multiple H bonds, using hydrogen exchange (HX) in a λYA variant that is suggested to be an incipient downhill folder having an extrapolated folding rate constant of 2 × 10(5) s(-1) and a stability of 7.4 kcal·mol(-1) at 298 K. At least one H bond on each of the three largest helices (α1, α3, and α4) breaks during a common unfolding event that reflects global denaturation. The use of HX enables us to both examine folding under highly stabilizing, native-like conditions and probe the pretransition state region for stable species without the need to initiate the folding reaction. The equivalence of the stability determined at zero and high denaturant indicates that any residual denatured state structure minimally affects the stability even under native conditions. Using our ψ analysis method along with mutational ϕ analysis, we find that the three aforementioned helices are all present in the folding transition state. Hence, the free energy surface has a sufficiently high barrier separating the denatured and native states that folding appears cooperative even under extremely stable and fast folding conditions.