Decomposition of energy and free energy changes by following the flow of work along reaction path

Trajectory Statistical Learning of the Potential Mean of Force and Diffusion Coefficient from Molecular Dynamics Simulations

Central to studying the conformational changes of a complex protein is understanding the dynamics and energetics involved. Phenomenologically, structural dynamics can be formulated using an overdamped Langevin model along an observable, e.g., the distance between two residues in the protein. The Langevin model is specified by the deterministic force (the potential of mean force, PMF) and stochastic force (characterized by the diffusion coefficient, D). It is therefore of great interest to be able to extract both PMF and D from an observable time series but under the same computational framework. Here, we approach this challenge in molecular dynamics (MD) simulations by treating it as a missing-data Bayesian estimation problem. An important distinction in our methodology is that the entire MD trajectory, as opposed to the individual data elements, is used as the statistical variable in Bayesian imputation. This idea is implemented through an eigen-decomposition procedure for a time-symmetrized Fokker-Planck equation, followed by maximizing the likelihood for parameter estimation. The mathematical expressions for the functional derivatives used in learning PMF and D also provide new physical insights for the manner by which the information on both the deterministic and stochastic forces is encoded in the dynamics data. An all-atom MD simulation of a nontrivial biomolecule case is used to illustrate the application of this approach. We show that, interestingly, the results of trajectory statistical learning can motivate new order parameters for an improved description of the kinetic bottlenecks in conformational changes. Complementing purely data-driven or black-box methods, this work underscores the advantages of physics-based machine learning in gaining chemical insights from quantitative parameter estimation.

Free energy simulations to understand the effect of Met → Ala mutations at positions 205, 206 and 213 on stability of human prion protein

Prion diseases are a family of infectious amyloid diseases affecting human and animals. Prion propagation in transmissible spongiform encephalopathies is associated with the unfolding and conversion of normal cellular prion protein into its pathogenic scrapie form. Understanding the fundamentals of prion protein aggregation caused by mutations is crucial to unravel the pathology of prion diseases. To help understand the contributions of individual residues to the stability of the human prion protein, we have carried out free energy simulations based on atomistic molecular dynamics trajectories. We focus on Met → Ala mutations at positions 205, 206 and 213, which are mostly buried residues located on helix 3 of the protein. The simulations predicted that all three mutations destabilize the prion protein. Changes in unfolding free energy upon mutation, ∆∆G, are 3.10 ± 0.79, 2.00 ± 0.26 and 3.06 ± 0.66 kcal/mol for M205A, M206A and M213A, respectively, in excellent agreement with the corresponding experimental values of 3.09 ± 0.28, 1.50 ± 0.34 and 3.12 ± 0.27 kcal/mol [T. Hart et al. (2009) PNAS 106, 5651-5656]. Component analysis indicates that the major contributions to the loss of protein stability arise from van der Waals interactions for the M205A and M206A mutations, and from van der Waals and covalent energy terms for M213A. Interestingly, while free energy contributions from a majority of residues neighboring the mutation sites tend to stabilize the wild type, there are a few residues stabilizing the mutant side chains. Our results show that this approach to free energy calculation can be very useful for understanding the detailed mechanism of human prion protein stability.

An approach to calculate the free energy changes of surface reactions using free energy decomposition on ab initio brute-force molecular dynamics trajectories

To understand the mechanisms and kinetics of catalytic reactions in heterogeneous catalysis, ab initio molecular dynamics is one of the powerful methods used to explore the free energy surface (FES) of surface elementary steps.

Revisiting the Dielectric Constant Effect on the Nucleophile and Leaving Group of Prototypical Backside SN2 Reactions: A Reaction Force and Atomic Contribution Analysis

The solvent effect on the nucleophile and leaving group atoms of the prototypical F^- + CH₃Cl → CH₃F + Cl^- backside bimolecular nucleophilic substitution reaction (S_N2) is analyzed employing the reaction force and the atomic contributions methods on the intrinsic reaction coordinate (IRC). Solvent effects were accounted for using the polarizable continuum solvent model. Calculations were performed employing 11 dielectric constants, ε, ranging from 1.0 to 78.5, to cover a wide spectrum of solvents. The reaction force data reveal that the solvent mainly influences the region of the IRC preceding the energy barrier, where the structural rearrangement to reach the transition state occurs. A detailed analysis of the atomic role in the reaction as a function of ε reveals that the nucleophile and the carbon atom are the ones that contribute the most to the energy barrier. In addition, we investigated the effect of the choice of nucleophile and leaving group on the ΔE₀ and ΔE^‡ of Y^- + CH₃X → YCH₃ + X^- (X, Y = F, Cl, Br, I) in aqueous solution. Our analysis allowed us to find relationships between the atomic contributions to the activation energy and leaving group ability and nucleophilicity.

Calculation of Enzyme Fluctuograms from All-Atom Molecular Dynamics Simulation

In this work, a computational framework is presented to compute the time evolution of force constants for a coarse grained (CG) elastic network model along an all-atom molecular dynamics trajectory of a protein system. Obtained via matching distance fluctuations, these force constants represent strengths of mechanical coupling between CG beads. Variation of coupling strengths with time is hence termed the fluctuogram of protein dynamics. In addition to the schematic procedure and implementation considerations, several ways of combining force constants and data analysis are presented to illustrate the potential application of protein fluctuograms. The unique angle provided by the fluctuogram expands the scope of atomistic simulations and is expected to impact upon fundamental understanding of protein dynamics as well as protein engineering technologies.

Computing the relative stabilities and the per-residue components in protein conformational changes

Protein molecules often undergo conformational changes. In order to gain insights into the forces that drive such changes, it would be useful to have a method that computes the per-residue contributions to the conversion free energy. Here, we describe the "confine-convert-release" (CCR) method, which is applicable to large conformational changes. We show that CCR correctly predicts the stable states of several "chameleon" sequences that have previously been challenging for molecular simulations. CCR can often discriminate better from worse predictions of native protein models in critical assessment of protein structure prediction (CASP). We show how the total conversion free energies can be parsed into per-residue free-energy components. Such parsing gives insights into which amino acids are most responsible for given transformations. For example, here we are able to "reverse-engineer" the known design principles of the chameleon proteins. This opens up the possibility for systematic improvements in structure-prediction scoring functions, in the design of protein conformational switches, and in interpreting protein mechanisms at the amino-acid level.

Expectation-maximization of the potential of mean force and diffusion coefficient in Langevin dynamics from single molecule FRET data photon by photon

The dynamics of a protein along a well-defined coordinate can be formally projected onto the form of an overdamped Lagevin equation. Here, we present a comprehensive statistical-learning framework for simultaneously quantifying the deterministic force (the potential of mean force, PMF) and the stochastic force (characterized by the diffusion coefficient, D) from single-molecule Förster-type resonance energy transfer (smFRET) experiments. The likelihood functional of the Langevin parameters, PMF and D, is expressed by a path integral of the latent smFRET distance that follows Langevin dynamics and realized by the donor and the acceptor photon emissions. The solution is made possible by an eigen decomposition of the time-symmetrized form of the corresponding Fokker-Planck equation coupled with photon statistics. To extract the Langevin parameters from photon arrival time data, we advance the expectation-maximization algorithm in statistical learning, originally developed for and mostly used in discrete-state systems, to a general form in the continuous space that allows for a variational calculus on the continuous PMF function. We also introduce the regularization of the solution space in this Bayesian inference based on a maximum trajectory-entropy principle. We use a highly nontrivial example with realistically simulated smFRET data to illustrate the application of this new method.

String method for calculation of minimum free-energy paths in Cartesian space in freely-tumbling systems

The string method is a molecular-simulation technique that aims to calculate the minimum free-energy path of a chemical reaction or conformational transition, in the space of a pre-defined set of reaction coordinates that is typically highly dimensional. Any descriptor may be used as a reaction coordinate, but arguably the Cartesian coordinates of the atoms involved are the most unprejudiced and intuitive choice. Cartesian coordinates, however, present a non-trivial problem, in that they are not invariant to rigid-body molecular rotations and translations, which ideally ought to be unrestricted in the simulations. To overcome this difficulty, we reformulate the framework of the string method to integrate an on-the-fly structural-alignment algorithm. This approach, referred to as SOMA (String method with Optimal Molecular Alignment), enables the use of Cartesian reaction coordinates in freely tumbling molecular systems. In addition, this scheme permits the dissection of the free-energy change along the most probable path into individual atomic contributions, thus revealing the dominant mechanism of the simulated process. This detailed analysis also provides a physically-meaningful criterion to coarse-grain the representation of the path. To demonstrate the accuracy of the method we analyze the isomerization of the alanine dipeptide in vacuum and the chair-to-inverted-chair transition of β-D mannose in explicit water. Notwithstanding the simplicity of these systems, the SOMA approach reveals novel insights into the atomic mechanism of these isomerizations. In both cases, we find that the dynamics and the energetics of these processes are controlled by interactions involving only a handful of atoms in each molecule. Consistent with this result, we show that a coarse-grained SOMA calculation defined in terms of these subsets of atoms yields nearidentical minimum free-energy paths and committor distributions to those obtained via a highly-dimensional string.

"Fluctuograms" reveal the intermittent intra-protein communication in subtilisin Carlsberg and correlate mechanical coupling with co-evolution

The mechanism of intra-protein communication and allosteric coupling is key to understanding the structure-property relationship of protein function. For subtilisin Carlsberg, the Ca²⁺-binding loop is distal to substrate-binding and active sites, yet the serine protease function depends on Ca²⁺ binding. The atomic molecular dynamics (MD) simulations of apo and Ca²⁺-bound subtilisin show similar structures and there is no direct evidence that subtilisin has alternative conformations. To model the intra-protein communication due to Ca²⁺ binding, we transform the sequential segments of an atomic MD trajectory into separate elastic network models to represent anharmonicity and nonlinearity effectively as the temporal and spatial variation of the mechanical coupling network. In analogy to the spectrogram of sound waves, this transformation is termed the “fluctuogram” of protein dynamics. We illustrate that the Ca²⁺-bound and apo states of subtilisin have different fluctuograms and that intra-protein communication proceeds intermittently both in space and in time. We found that residues with large mechanical coupling variation due to Ca²⁺ binding correlate with the reported mutation sites selected by directed evolution for improving the stability of subtilisin and its activity in a non-aqueous environment. Furthermore, we utilize the fluctuograms calculated from MD to capture the highly correlated residues in a multiple sequence alignment. We show that in addition to the magnitude, the variance of coupling strength is also an indicative property for the sequence correlation observed in a statistical coupling analysis. The results of this work illustrate that the mechanical coupling networks calculated from atomic details can be used to correlate with functionally important mutation sites and co-evolution.

A hallmark of protein molecules is their machine-like behaviors while carrying out biological functions. At the molecular level, molecular signals such as binding a metal ion at an action site can cause long-range effects and alter protein function. Such phenomena are often referred to as intra-protein communication or allosteric coupling. Elucidating the underlying mechanisms could lead to novel discovery of molecular modulators to regulate protein function in a more specific and effective manner. A long-standing puzzle is the roles of the anharmonicity and nonlinearity in protein dynamics. To incorporate these characters in modeling intra-protein communication, we devise a “fluctuogram” analysis to record the choreography of allosteric coupling in an atomic molecular dynamics simulation. We show that fluctuogram analysis can bridge the results of physics-based simulation and sequence alignment in bioinformatics by capturing the residues that exhibit high correlation in a multiple sequence alignment. We also show that the fluctuograms calculated from atomic details have the potential to be applied as a tool to select mutation sites for modulating protein function.