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.

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.

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.

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.

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.

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.

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.

Even with nonnative interactions, the updated folding transition states of the homologs Proteins G & L are extensive and similar.

Experimental and computational folding studies of Proteins L & G and NuG2 typically find that sequence differences determine which of the two hairpins is formed in the transition state ensemble (TSE). However, our recent work on Protein L finds that its TSE contains both hairpins, compelling a reassessment of the influence of sequence on the folding behavior of the other two homologs. We characterize the TSEs for Protein G and NuG2b, a triple mutant of NuG2, using ψ analysis, a method for identifying contacts in the TSE. All three homologs are found to share a common and near-native TSE topology with interactions between all four strands. However, the helical content varies in the TSE, being largely absent in Proteins G & L but partially present in NuG2b. The variability likely arises from competing propensities for the formation of nonnative ß turns in the naturally occurring proteins, as observed in our TerItFix folding algorithm. All-atom folding simulations of NuG2b recapitulate the observed TSEs with four strands for 5 of 27 transition paths [Lindorff-Larsen K, Piana S, Dror RO, Shaw DE (2011) Science 334(6055):517-520]. Our data support the view that homologous proteins have similar folding mechanisms, even when nonnative interactions are present in the transition state. These findings emphasize the ongoing challenge of accurately characterizing and predicting TSEs, even for relatively simple proteins.

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.

Loss of conformational entropy in protein folding calculated using realistic ensembles and its implications for NMR-based calculations.

The loss of conformational entropy is a major contribution in the thermodynamics of protein folding. However, accurate determination of the quantity has proven challenging. We calculate this loss using molecular dynamic simulations of both the native protein and a realistic denatured state ensemble. For ubiquitin, the total change in entropy is TΔSTotal = 1.4 kcal⋅mol(-1) per residue at 300 K with only 20% from the loss of side-chain entropy. Our analysis exhibits mixed agreement with prior studies because of the use of more accurate ensembles and contributions from correlated motions. Buried side chains lose only a factor of 1.4 in the number of conformations available per rotamer upon folding (ΩU/ΩN). The entropy loss for helical and sheet residues differs due to the smaller motions of helical residues (TΔShelix-sheet = 0.5 kcal⋅mol(-1)), a property not fully reflected in the amide N-H and carbonyl C=O bond NMR order parameters. The results have implications for the thermodynamics of folding and binding, including estimates of solvent ordering and microscopic entropies obtained from NMR.

Benchmarking all-atom simulations using hydrogen exchange.

Long-time molecular dynamics (MD) simulations are now able to fold small proteins reversibly to their native structures [Lindorff-Larsen K, Piana S, Dror RO, Shaw DE (2011) Science 334(6055):517-520]. These results indicate that modern force fields can reproduce the energy surface near the native structure. To test how well the force fields recapitulate the other regions of the energy surface, MD trajectories for a variant of protein G are compared with data from site-resolved hydrogen exchange (HX) and other biophysical measurements. Because HX monitors the breaking of individual H-bonds, this experimental technique identifies the stability and H-bond content of excited states, thus enabling quantitative comparison with the simulations. Contrary to experimental findings of a cooperative, all-or-none unfolding process, the simulated denatured state ensemble, on average, is highly collapsed with some transient or persistent native 2° structure. The MD trajectories of this protein G variant and other small proteins exhibit excessive intramolecular H-bonding even for the most expanded conformations, suggesting that the force fields require improvements in describing H-bonding and backbone hydration. Moreover, these comparisons provide a general protocol for validating the ability of simulations to accurately capture rare structural fluctuations.

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.

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.

Communication: The simplified generalized entropy theory of glass-formation in polymer melts.

While a wide range of non-trivial predictions of the generalized entropy theory (GET) of glass-formation in polymer melts agree with a large number of observed universal and non-universal properties of these glass-formers and even for the dependence of these properties on monomer molecular structure, the huge mathematical complexity of the theory precludes its extension to describe, for instance, the perplexing, complex behavior observed for technologically important polymer films with thickness below ∼100 nm and for which a fundamental molecular theory is lacking for the structural relaxation. The present communication describes a hugely simplified version of the theory, called the simplified generalized entropy theory (SGET) that provides one component necessary for devising a theory for the structural relaxation of thin polymer films and thereby supplements the first required ingredient, the recently developed Flory-Huggins level theory for the thermodynamic properties of thin polymer films, before the concluding third step of combining all the components into the SGET for thin polymer films. Comparisons between the predictions of the SGET and the full GET for the four characteristic temperatures of glass-formation provide good agreement for a highly non-trivial model system of polymer melts with chains of the structure of poly(n-α olefins) systems where the GET has produced good agreement with experiment. The comparisons consider values of the relative backbone and side group stiffnesses such that the glass transition temperature decreases as the amount of excess free volume diminishes, contrary to general expectations but in accord with observations for poly(n-alkyl methacrylates). Moreover, the SGET is sufficiently concise to enable its discussion in a standard course on statistical mechanics or polymer physics.

Lattice cluster theory for dense, thin polymer films.

While the application of the lattice cluster theory (LCT) to study the miscibility of polymer blends has greatly expanded our understanding of the monomer scale molecular details influencing miscibility, the corresponding theory for inhomogeneous systems has not yet emerged because of considerable technical difficulties and much greater complexity. Here, we present a general formulation enabling the extension of the LCT to describe the thermodynamic properties of dense, thin polymer films using a high dimension, high temperature expansion. Whereas the leading order of the LCT for bulk polymer systems is essentially simple Flory-Huggins theory, the highly non-trivial leading order inhomogeneous LCT (ILCT) for a film with L layers already involves the numerical solution of 3(L - 1) coupled, highly nonlinear equations for the various density profiles in the film. The new theory incorporates the essential "transport" constraints of Helfand and focuses on the strict imposition of excluded volume constraints, appropriate to dense polymer systems, rather than the maintenance of chain connectivity as appropriate for lower densities and as implemented in self-consistent theories of polymer adsorption at interfaces. The ILCT is illustrated by presenting examples of the computed profiles of the density, the parallel and perpendicular bonds, and the chain ends for free standing and supported films as a function of average film density, chain length, temperature, interaction with support, and chain stiffness. The results generally agree with expected general trends.

Lattice model of linear telechelic polymer melts. I. Inclusion of chain semiflexibility in the lattice cluster theory.

The lattice cluster theory (LCT) for the thermodynamics of polymer systems has recently been reformulated to treat strongly interacting self-assembling polymers composed of fully flexible linear telechelic chains [J. Dudowicz and K. F. Freed, J. Chem. Phys. 136, 064902 (2012)]. Here, we further extend the LCT for linear telechelic polymer melts to include a description of chain semiflexibility, which is treated by introducing a bending energy penalty whenever a pair of consecutive bonds from a single chain lies along orthogonal directions. An analytical expression for the Helmholtz free energy is derived for the model of semiflexible linear telechelic polymer melts. The extension provides a theoretical tool for investigating the influence of chain stiffness on the thermodynamics of self-assembling telechelic polymers, and for further exploring the influence of self-assembly on glass formation in such systems.