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Allostery in proteins involves, broadly speaking, ligand-induced conformational transitions that modulate function at active sites distal to where the ligand binds. In contrast, the concept of cooperativity (in the sense used in phase transition theory) is often invoked to understand protein folding and, therefore, function. The modern view on allostery is one based on dynamics and hinges on the time-dependent interactions between key residues in a complex network, interactions that determine the free-energy profile for the reaction at the distal site. Here, we merge allostery and cooperativity, and we discuss a joint model with features of both. In our model, the active-site reaction is replaced by the reaction pathway that leads to protein folding, and the presence or absence of the effector is replaced by mutant-vs-wild type changes in key residues. To this end, we employ our recently introduced time-lagged independent component analysis (tICA) correlation approach [Ray et al. Proc. Natl. Acad. Sci. 118(43) (2021), e2100943118] to identify the allosteric role of distant residues in the folded-state dynamics of a large protein. In this work, we apply the technique to identify key residues that have a significant role in the folding of a small, fast folding-protein, chignolin. Using extensive enhanced sampling simulations, we critically evaluate the accuracy of the predictions by mutating each residue one at a time and studying how the mutations change the underlying free energy landscape of the folding process. We observe that mutations in those residues whose associated backbone torsion angles have a high correlation score can indeed lead to loss of stability of the folded configuration. We also provide a rationale based on interaction energies between individual residues with the rest of the protein to explain this effect. From these observations, we conclude that the tICA correlation score metric is a useful tool for predicting the role of individual residues in the correlated dynamics of proteins and can find application to the problem of identifying regions of protein that are either most vulnerable to mutations or— mutatis mutandis—to binding events that affect their functionality.

We consider the recently developed weighted ensemble milestoning (WEM) scheme [J. Chem. Phys. 152, 234114 (2020)], and test its capability of simulating ligand-receptor dissociation dynamics. We performed WEM simulations on the following host-guest systems: Na$^+$/Cl$^-$ ion pair and 4-hydroxy-2-butanone (BUT) ligand with FK506 binding protein (FKBP). As proof or principle, we show that the WEM formalism reproduces the Na$^+$/Cl$^-$ ion pair dissociation timescale and the free energy profile obtained from long conventional MD simulation. To increase accuracy of WEM calculations applied to kinetics and thermodynamics in protein-ligand binding, we introduced a modified WEM scheme called weighted ensemble milestoning with restraint release (WEM-RR), which can increase the number of starting points per milestone without adding additional computational cost. WEM-RR calculations obtained a ligand residence time and binding free energy in agreement with experimental and previous computational results. Moreover, using the milestoning framework, the binding time and rate constants, dissociation constant and the committor probabilities could also be calculated at a low computational cost. We also present an analytical approach for estimating the association rate constant ($k_{\text{on}}$) when binding is primarily diffusion driven. We show that the WEM method can efficiently calculate multiple experimental observables describing ligand-receptor binding/unbinding and is a promising candidate for computer-aided inhibitor design.

Contractile tails are key components of the biological nanomachinery involved in cell membrane puncturing, where they provide a means to deliver molecules and ions inside cells. Two intriguing examples of contractile tails are those from bacteriophage T4 and R2-pyocin. Although the two systems are different in terms of biological activity, they share a fascinatingly similar injection mechanism, during which the tail sheaths of both systems contract from a so-called extended state to around half of their length (the contracted state), accompanied by release of elastic energy originally stored in the sheath. Despite the great prevalence and biomedical importance of contractile delivery systems, many fundamental details of their injection machinery and dynamics are still unknown. In this work, we calculate the bending and torsional stiffness constants of a helical tail sheath strand of bacteriophage T4 and R2-pyocin, in both extended and contracted states, using molecular dynamics simulations of about one-sixth of the entire sheath. Differences in stiffness constants between the two systems are rationalized by comparing their all-atom monomer structures, changes in sheath architecture on contraction, and differences in interstrand interactions. The calculated coefficients indicate that the T4 strand is stiffer for both bending and torsion than the corresponding R2-pyocin strands in both extended and contracted conformations. The sheath strands also have greater stiffness in the contracted state for both systems. As the main application of this study, we describe how the stiffness constants can be incorporated in a model to simulate the dynamics of contractile nanoinjection machineries.

The milestoning algorithm of Elber and co-workers creates a framework for computing the time scale of processes that are too long and too complex to be studied using simply brute force simulations. The fundamental objects involved in the milestoning algorithm are the first passage time distributions KAB(τ) between adjacent conformational milestones A and B. The method proposed herein aims to further enhance milestoning (or other interface based sampling methods) by employing an artificially applied force, akin to a wind that blows the trajectories from their initial to their final states, and by subsequently applying corrective weights to the trajectories to yield the true first passage time distributions KAB(τ) in a fraction of the computation time required for unassisted calculations. The re-weighting method is rooted in the formalism of stochastic path integrals. The theoretical basis for the technique and numerical examples are presented.The milestoning algorithm of Elber and co-workers creates a framework for computing the time scale of processes that are too long and too complex to be studied using simply brute force simulations. The fundamental objects involved in the milestoning algorithm are the first passage time distributions KAB(τ) between adjacent conformational milestones A and B. The method proposed herein aims to further enhance milestoning (or other interface based sampling methods) by employing an artificially applied force, akin to a wind that blows the trajectories from their initial to their final states, and by subsequently applying corrective weights to the trajectories to yield the true first passage time distributions KAB(τ) in a fraction of the computation time required for unassisted calculations. The re-weighting method is rooted in the formalism of stochastic path integrals. The theoretical basis for the technique and numerical examples are presented.

In the milestoning framework, and more generally in related transition interface sampling schemes, one significantly enhances the calculation of relaxation rates for complex equilibrium kinetics from molecular dynamics simulations between the milestones or interfaces. The goal of the present paper is to advance milestoning applications into the realm of non-equilibrium statistical mechanics, in particular, to calculate entire time correlation functions. In order to accomplish this, we introduce a novel methodology for obtaining the flux through a given milestone configuration as a function of both time and initial configuration and build upon it with a novel formalism describing autocorrelation for Langevin motion in a discrete configuration space. The method is then applied to three different test systems: a harmonic oscillator, which we solve analytically, a two-well potential, which is solved numerically, and an atomistic molecular dynamics simulation of alanine dipeptide.

A novel aspect in the area of mechano-chemistry concerns the effect of external forces on enzyme activity, i.e., the existence of mechano-catalytic coupling. Recent experiments on enzyme-catalyzed disulphide bond reduction in proteins under the effect of a force applied on the termini of the protein substrate reveal an unexpected biphasic force dependence for the bond cleavage rate. Here, using atomistic molecular dynamics simulations combined with Smoluchowski theory, we propose a model for this behavior. For a broad range of forces and systems, the model reproduces the experimentally observed rates by solving a reaction-diffusion equation for a “protein coordinate” diffusing in a force-dependent effective potential. The atomistic simulations are used to compute, from first principles, the parameters of the model via a quasiharmonic analysis. Additionally, the simulations are also used to provide details about the microscopic degrees of freedom that are important for the underlying mechano-catalysis.

Several recent implementations of algorithms for sampling reaction pathways employ a strategy for placing interfaces or milestones across the reaction coordinate manifold. Interfaces can be introduced such that the full feature space describing the dynamics of a macromolecule is divided into Voronoi (or other) cells, and the global kinetics of the molecular motions can be calculated from the set of fluxes through the interfaces between the cells. Although some methods of this type are exact for an arbitrary set of cells, in practice, the calculations will converge fastest when the interfaces are placed in regions where they can best capture transitions between configurations corresponding to local minima. The aim of this paper is to introduce a fully automated machine-learning algorithm for defining a set of cells for use in kinetic sampling methodologies based on subdividing the dynamical feature space; the algorithm requires no intuition about the system or input from the user and scales to high-dimensional systems.

An ad hoc method for calculating the entropy of a biomolecular system from the covariance matrix of the atomic fluctuations is analyzed. It is shown that its essential assumption can be eliminated by a quasiharmonic analysis. The computer time required for use of the latter is of the same order as that of the former.

The “Smart Walking” Monte Carlo algorithm is examined. In general, due to a bias imposed by the interbasin trial move, the algorithm does not satisfy detailed balance. While it has been shown that it can provide good estimates of equilibrium averages for certain potentials, for other potentials the estimates are poor. A modified version of the algorithm, Smart Darting Monte Carlo, which obeys the detailed balance condition, is proposed. Calculations on a one-dimensional model potential, on a Lennard-Jones cluster and on the alanine dipeptide demonstrate the accuracy and promise of the method for deeply quenched systems.

Generalized Monte Carlo and molecular dynamics algorithms which provide enhanced sampling of the phase space in the calculation of equilibrium thermodynamic properties is presented. The algorithm samples trial moves from a generalized statistical distribution derived from a modification of the Gibbs–Shannon entropy proposed by Tsallis. Results for a one-dimensional model potential demonstrate that the algorithm leads to a greatly enhanced rate of barrier crossing and convergence in the calculation of equilibrium averages. Comparison is made with standard Metropolis Monte Carlo and the J-walking algorithm of Franz, Freeman and Doll. Application to a 13-atom Lennard-Jones cluster demonstrates the ease with which the algorithm may be applied to complex molecular systems.

We present a simple method for utilizing experimental data to improve the efficiency of numerical calculations of free energy profiles from molecular dynamics simulations. The method involves umbrella sampling simulations with restraining potentials based on a known approximate estimate of the free energy profile derived solely from experimental data. The use of the experimental data results in optimal restraining potentials, guides the simulation along relevant pathways, and decreases overall computational time. In demonstration of the method, two systems are showcased. First, guided, unguided (regular) umbrella sampling simulations and exhaustive sampling simulations are compared to each other in the calculation of the free energy profile for the distance between the ends of a pentapeptide. The guided simulation use restraints based on a simulated “experimental” potential of mean force of the end-to-end distance that would be measured by fluorescence resonance energy transfer (obtained from exhaustive sampling). Statistical analysis shows a dramatic improvement in efficiency for a 5 window guided umbrella sampling over 5 and 17 window unguided umbrella sampling simulations. Moreover, the form of the potential of mean force for the guided simulations evolves, as one approaches convergence, along the same milestones as the extensive simulations, but exponentially faster. Second, the method is further validated by replicating the forced unfolding pathway of the titin I27 domain using guiding umbrella sampling potentials determined from actual single molecule pulling data. Comparison with unguided umbrella sampling reveals that the use of guided sampling encourages unfolding simulations to converge faster to a forced unfolding pathway that agrees with previous results and produces a more accurate potential of mean force.

A method exploiting the properties of an artificial (nonphysical) Langevin dynamics with a negative frictional coefficient along a suitable manifold and positive friction in the perpendicular directions is presented for the enhanced calculation of time-correlation functions for rare event problems. Exact time-correlation functions that describe the kinetics of the transitions for the all-positive, physical system can be calculated by reweighting the generated trajectories according to stochastic path integral treatment involving a functional weight based on an Onsager-Machlup action functional. The method is tested on a prototypical multidimensional model system featuring the main elements of conformational space characteristic of complex condensed matter systems. Using the present method, accurate estimates of rate constants require at least three order of magnitudes fewer trajectories than regular Langevin dynamics. The method is particularly useful in calculating kinetic properties in the context of multidimensional energy landscapes that are characteristic of complex systems such as proteins and nucleic acids.

We derive expressions for the equilibrium entropy and energy changes in the context of the Jarzynski equality relating nonequilibrium work to equilibrium free energy. The derivation is based on a stochastic path integral technique that reweights paths at different temperatures. Stochastic dynamics generated by either a Langevin equation or a Metropolis Monte Carlo scheme are treated. The approach enables the entropy-energy decomposition from trajectories evolving at a single-temperature and does not require simulations or measurements at two or more temperatures. Both finite difference and analytical formulae are derived. Testing is performed on a prototypical model system and the method is compared with existing thermodynamic integration and thermodynamic perturbation approaches for entropy-energy decomposition. The new formulae are also put in the context of more general, dynamics-independent expressions that derive from either a fluctuation theorem or the Feynman-Kac theorem.

We present and analyze a general method to calculate time correlation functions from molecular dynamics on scaled potentials for complex systems for which simulation is affected by broken ergodicity. Depending on the value of the scaling factor, correlations can be calculated for times that can be orders of magnitude longer than those accessible to direct simulations. We show that the exact value of the time correlation functions of the original system (i.e., with unscaled potential) can be obtained, in principle, using an action-reweighting scheme based on a stochastic path-integral formalism. Two tests (involving a bistable potential model and a dipeptide bond-vector orientational relaxation) are exemplified to showcase the strengths, as well as the limitations of the approach, and a procedure for the estimation of the time-dependent standard deviation error is outlined.

We present a novel computational method, the skewed-momenta method (Skew'M), which applies a bias to the Maxwell distribution of initial momenta used to generate ensembles of trajectories. As a result, conformational transitions are accentuated and kinetic properties are calculated more effectively. The connection to the related puddle jumping method is discussed. A reweighting scheme permits the exact calculation of kinetic properties. Applications are presented for the rapid calculation of rate constants for molecular isomerization, and for the efficient reconstruction of free-energy profiles using a straightforward modification of the Jarzynski identity.