Transition Path Theory Research Articles

Statistical Mechanical Theories of Membrane Permeability.

A popular theoretical framework to compute the permeability coefficient of a molecule is provided by the classic Smoluchowski-Kramers treatment of the steady-state diffusive flux across a free-energy barrier. Within this framework, commonly termed "inhomogeneous solubility-diffusion" (ISD), the permeability, P, is expressed in closed form in terms of the potential of mean force and position-dependent diffusivity of the molecule of interest along the membrane normal. In principle, both quantities can be calculated from all-atom MD simulations. Although several methods exist for calculating the position-dependent diffusivity, each of these is at best an estimate. In addition, the ISD model does not account for memory effects along the chosen reaction coordinate. For these reasons, it is important to seek alternative theoretical formulations to determine the permeability coefficient that are able to account for the factors ignored by the ISD approximation. Using Green-Kubo linear response theory, we establish the familiar constitutive relation between the flux density across the membrane and the difference in the concentration of a permeant molecule, j = PΔC. On this basis, we derive a time-correlation function expression for the nonequilibrium flux across a membrane that is reminiscent of the transmission coefficient in the reactive flux formalism treatment of transition rates. An analysis based on the transition path theory framework is exploited to derive alternative expressions for the permeability coefficient. The different strategies are illustrated with stochastic simulations based on the generalized Langevin equation in addition to unbiased molecular dynamics simulations of water permeation of a lipid bilayer.

Kinetic Network in Milestoning: Clustering, Reduction, and Transition Path Analysis.

We present a reduction of the Milestoning (ReM) algorithm to analyze the high-dimensional Milestoning kinetic network. The algorithm reduces the Milestoning network to low dimensions but preserves essential kinetic information, such as local residence time, exit time, and mean first passage time between any two states. This is achieved in three steps. First, nodes (milestones) in the high-dimensional Milestoning network are grouped into clusters based on the metastability identified by an auxiliary continuous-time Markov chain. Our clustering method is applicable not only to time-reversible networks but also to nonreversible networks generated from practical simulations with statistical fluctuations. Second, a reduced network is established via network transformation, containing only the core sets of clusters as nodes. Finally, transition pathways are analyzed in the reduced network based on the transition path theory. The algorithm is illustrated using a toy model and a solvated alanine dipeptide in two and four dihedral angles.

Conformational Dynamics in Corynebacterium glutamicum Diaminopimelate Epimerase: Insights from Ligand Parameterization, Atomistic Simulation, and Markov State Modeling.

The microbial enzyme diaminopimelate epimerase (DapF), a vital enzyme in the lysine biosynthetic pathway, catalyzes the conversion of L, L-diaminopimelate (L, L-DAP) to D, L-diaminopimelate (D, L-DAP) using a catalytic cysteine dyad with one cysteine in thiol state and another in thiolate. Under oxidizing conditions, the catalytic cysteines of apo DapF form a disulfide bond that alters the structure and function of DapF. Given its potential as a target for antimicrobial resistance treatments, understanding DapF's functional dynamics is imperative. In the present work, we employ microsecond-scale all-atom molecular dynamics simulations of product-bound DapF and apo-DapF under oxidized and reduced conditions. We employ a polarized charge model for the ligand and the active site residues, which was necessary to preserve the electrostatic environment in the active site and retain the ligand in the active site. The product-bound DapF and apo-DapF in oxidized and reduced conditions exhibit a closed, semi-open, and open conformation, respectively, as identified using the internal coordinates of the dimeric enzyme and the principal component analysis. The conformational switch is guided by the dynamic catalytic (DC) loop, loop II, and loop III movements in the active site. The time scale of the close-to-open conformational transition is estimated to be 0.8 μs through Markov state modeling (MSM) and transition path theory (TPT). The present study explains the role of various active site residues and loops in ligand binding and protein dynamics in the DapF enzyme under different redox conditions. Such information will be helpful in future inhibitor design studies targeting the DapF enzyme.

Clustering molecular energy landscapes by adaptive network embedding

In order to efficiently explore the chemical space of all possible small molecules, a common approach is to compress the dimension of the system to facilitate downstream machine learning tasks. Towards this end, we present a data-driven approach for clustering potential energy landscapes of molecular structures by applying recently developed Network Embedding techniques to obtain latent variables defined through the embedding function. To scale up the method, we also incorporate an entropy sensitive adaptive scheme for hierarchical sampling of the energy landscape, based on Metadynamics and Transition Path Theory. Taking into account the kinetic information implied by the energy landscape of a system, we can interpret dynamical node-node relationships in reduced dimensions. We demonstrate the framework through Lennard-Jones clusters and a human DNA sequence.

Transition path theory for diffusive search with stochastic resetting

Many chemical reactions can be formulated in terms of particle diffusion in a complex energy landscape. Transition path theory (TPT) is a theoretical framework for describing the direct (reaction) pathways from reactant to product states within this energy landscape, and calculating the effective reaction rate. It is now the standard method for analyzing rare events between long lived states. In this paper, we consider a completely different application of TPT, namely, a dual-aspect diffusive search process in which a particle alternates between collecting cargo from a source domain A and then delivering it to a target domain B. The rate of resource accumulation at the target, k AB , is determined by the statistics of direct (reactive or transport) paths from A to B. Rather than considering diffusion in a complex energy landscape, we focus on pure diffusion with stochastic resetting. Resetting introduces two non-trivial problems in the application of TPT. First, the process is not time-reversal invariant, which is reflected by the fact that there exists a unique non-equilibrium stationary state (NESS). Second, calculating k AB involves determining the total probability flux of direct transport paths across a dividing surface S between A and B. This requires taking into account discontinuous jumps across S due to resetting. We derive a general expression for k AB and show that it is independent of the choice of dividing surface. Finally, using the example of diffusion in a finite interval, we show that there exists an optimal resetting rate at which k AB is maximized. We explore how this feature depends on model parameters.

On Inactivation of the Coronavirus Main Protease.

A deeper understanding of the inactive conformations of the coronavirus main protease (MPro) could inform the design of allosteric drugs. Based on extensive molecular dynamics simulations, we built a Markov State Model to investigate structural changes that can inactivate the SARS-CoV-2 MPro. In a subset of structures, one subunit of the homodimer assumes an inactive conformation that resembles an inactive crystal structure. However, contradicting the widely held half-of-sites activity hypothesis, the most populated enzyme structures have two active subunits. We then used transition path theory (TPT) and the Jensen-Shannon Divergence (JSD) to pinpoint residues involved in the inactivation process. A π stack between Phe140 and His163 is a key feature that can distinguish active and inactive conformations of MPro. Each subunit has unique inactive conformations stabilized by π stacking interactions involving residues Phe140, Tyr118, His163, and His172, a hydrogen bonding network centered around His163 and His172, and a modified network of interactions in the dimer interface. The importance of these residues in maintaining an active structure explains the sensitivity of enzymatic activity to site-directed mutagenesis.

A Kinetic Transition Network Model Reveals the Diversity of Protein Dimer Formation Mechanisms.

Protein homodimers have been classified as three-state or two-state dimers depending on whether a folded monomer forms before association, but the details of the folding-binding mechanisms are poorly understood. Kinetic transition networks of conformational states have provided insight into the folding mechanisms of monomeric proteins, but extending such a network to two protein chains is challenging as all the relative positions and orientations of the chains need to be included, greatly increasing the number of degrees of freedom. Here, we present a simplification of the problem by grouping all states of the two chains into two layers: a dissociated and an associated layer. We combined our two-layer approach with the Wako-Saito-Muñoz-Eaton method and used Transition Path Theory to investigate the dimer formation kinetics of eight homodimers. The analysis reveals a remarkable diversity of dimer formation mechanisms. Induced folding, conformational selection, and rigid docking are often simultaneously at work, and their contribution depends on the protein concentration. Pre-folded structural elements are always present at the moment of association, and asymmetric binding mechanisms are common. Our two-layer network approach can be combined with various methods that generate discrete states, yielding new insights into the kinetics and pathways of flexible binding processes.

Optimal control for sampling the transition path process and estimating rates

Many processes in nature such as conformal changes in biomolecules and clusters of interacting particles, genetic switches, mechanical or electromechanical oscillators with added noise, and many others are modeled using stochastic differential equations with small white noise. The study of rare transitions between metastable states in such systems is of great interest and importance. The direct simulation of rare transitions is difficult due to long waiting times. Transition path theory is a mathematical framework for the quantitative description of rare events. Its crucial component is the committor function, the solution to a boundary value problem for the backward Kolmogorov equation. The key fact exploited in this work is that the optimal controller constructed from the committor leads to the generation of transition trajectories exclusively. We prove this fact for a broad class of stochastic differential equations. Moreover, we demonstrate that the committor computed for a dimensionally reduced system and then lifted to the original phase space still allows us to construct an effective controller and estimate the transition rate with reasonable accuracy. Furthermore, we propose an all-the-way-through scheme for computing the committor via neural networks, sampling the transition trajectories, and estimating the transition rate without meshing the space. We apply the proposed methodology to four test problems: the overdamped Langevin dynamics with Mueller’s potential and the rugged Mueller potential in 10D, the noisy bistable Duffing oscillator, and Lennard-Jones-7 in 2D.

Macrophage phenotype transitions in a stochastic gene-regulatory network model

Polarization is the process by which a macrophage cell commits to a phenotype based on external signal stimulation. To know how this process is affected by random fluctuations and events within a cell is of utmost importance to better understand the underlying dynamics and predict possible phenotype transitions. For this purpose, we develop a stochastic modeling approach for the macrophage polarization process. We classify phenotype states using the Robust Perron Cluster Analysis and quantify transition pathways and probabilities by applying Transition Path Theory. Depending on the model parameters, we identify four bistable and one tristable phenotype configuration. We find that bistable transitions are fast but their states less robust. In contrast, phenotype transitions in the tristable situation have a comparatively long time duration, which reflects the robustness of the states. The results indicate parallels in the overall transition behavior of macrophage cells with other heterogeneous and plastic cell types, such as cancer cells. Our approach allows for a probabilistic interpretation of macrophage phenotype transitions and biological inference on phenotype robustness. In general, the methodology can easily be adapted to other systems where random state switches are known to occur.

Pathways of hLL-3717-29 Aggregation Give Insight into the Mechanism of α-Amyloid Formation.

α-amyloids present a novel self-assembly principle that can be utilized to prepare functional biomaterials. Evidence of α-amyloid formation in the active core of the human LL-37 protein (comprising residues 17 to 29) was associated with this peptide's membranolytic property. Though mechanistic pathways of β-amyloid formation are known, such studies are scarce in α-amyloids. Modern computational techniques allow such mechanistic studies in molecular detail. Here, we propose aggregation pathways in hLL-3717-29 through molecular dynamics simulations. We first identified oligomers among peptides based on a distance criterion. The distribution of oligomers was then used to build Markov state models from which pathways were obtained using the framework of transition path theory. We checked the structural stability of the peptides during oligomerization, which is crucial from their functional point of view. We also investigated the key residues that participate in oligomer formation, the interactions between them, and the effect of residue mutations on the binding free energy of the peptides. Our findings suggest that larger oligomers are produced from the association of smaller and intermediate oligomers. The peptides retain their helical structure during aggregation with transient occurrences of 3-10 helix and turns. Hydrophobic interactions are vital in the aggregation of these peptides with Ile24 playing a crucial role. Mutation of this residue to alanine decreases the peptides' binding free energy, resulting in reduced aggregation tendency.

On the Mechanism of Polaritonic Rate Suppression from Quantum Transition Paths.

Polariton chemistry holds promise for facilitating mode-selective chemical reactions, but the underlying mechanism behind the rate modifications observed under strong vibrational coupling is not well understood. Using the recently developed quantum transition path theory, we have uncovered a mechanism of resonant suppression of a thermal reaction rate in a simple model polaritonic system consisting of a reactive mode in a bath confined to a lossless microcavity with a single photon mode. We observed the formation of a polariton during rate-limiting transitions on reactive pathways and identified the concomitant rate suppression as being due to hybridization between the reactive mode and the cavity mode, which inhibits bath-mediated tunneling. The transition probabilities that define the quantum master equation can be directly translated into a visualization of the corresponding polariton energy landscape. This landscape exhibits a double funnel structure with a large barrier between the initial and final states.

An Efficient Path Classification Algorithm Based on Variational Autoencoder to Identify Metastable Path Channels for Complex Conformational Changes.

Conformational changes (i.e., dynamic transitions between pairs of conformational states) play important roles in many chemical and biological processes. Constructing the Markov state model (MSM) from extensive molecular dynamics (MD) simulations is an effective approach to dissect the mechanism of conformational changes. When combined with transition path theory (TPT), MSM can be applied to elucidate the ensemble of kinetic pathways connecting pairs of conformational states. However, the application of TPT to analyze complex conformational changes often results in a vast number of kinetic pathways with comparable fluxes. This obstacle is particularly pronounced in heterogeneous self-assembly and aggregation processes. The large number of kinetic pathways makes it challenging to comprehend the molecular mechanisms underlying conformational changes of interest. To address this challenge, we have developed a path classification algorithm named latent-space path clustering (LPC) that efficiently lumps parallel kinetic pathways into distinct metastable path channels, making them easier to comprehend. In our algorithm, MD conformations are first projected onto a low-dimensional space containing a small set of collective variables (CVs) by time-structure-based independent component analysis (tICA) with kinetic mapping. Then, MSM and TPT are constructed to obtain the ensemble of pathways, and a deep learning architecture named the variational autoencoder (VAE) is used to learn the spatial distributions of kinetic pathways in the continuous CV space. Based on the trained VAE model, the TPT-generated ensemble of kinetic pathways can be embedded into a latent space, where the classification becomes clear. We show that LPC can efficiently and accurately identify the metastable path channels in three systems: a 2D potential, the aggregation of two hydrophobic particles in water, and the folding of the Fip35 WW domain. Using the 2D potential, we further demonstrate that our LPC algorithm outperforms the previous path-lumping algorithms by making substantially fewer incorrect assignments of individual pathways to four path channels. We expect that LPC can be widely applied to identify the dominant kinetic pathways underlying complex conformational changes.

Improving the stability of temporal statistics in transition path theory with sparse data

Ulam’s method is a popular discretization scheme for stochastic operators that involves the construction of a transition probability matrix controlling a Markov chain on a set of cells covering some domain. We consider an application to satellite-tracked undrogued surface-ocean drifting buoy trajectories obtained from the National Oceanic and Atmospheric Administration Global Drifter Program dataset. Motivated by the motion of Sargassum in the tropical Atlantic, we apply Transition Path Theory (TPT) to drifters originating off the west coast of Africa to the Gulf of Mexico. We find that the most common case of a regular covering by equal longitude–latitude side cells can lead to a large instability in the computed transition times as a function of the number of cells used. We propose a different covering based on a clustering of the trajectory data that is stable against the number of cells in the covering. We also propose a generalization of the standard transition time statistic of TPT that can be used to construct a partition of the domain of interest into weakly dynamically connected regions.

Discovering Reaction Pathways, Slow Variables, and Committor Probabilities with Machine Learning.

A significant challenge faced by atomistic simulations is the difficulty, and often impossibility, to sample the transitions between metastable states of the free-energy landscape associated with slow molecular processes. Importance-sampling schemes represent an appealing option to accelerate the underlying dynamics by smoothing out the relevant free-energy barriers, but require the definition of suitable reaction-coordinate (RC) models expressed in terms of compact low-dimensional sets of collective variables (CVs). While most computational studies of slow molecular processes have traditionally relied on educated guesses based on human intuition to reduce the dimensionality of the problem at hand, a variety of machine-learning (ML) algorithms have recently emerged as powerful alternatives to discover meaningful CVs capable of capturing the dynamics of the slowest degrees of freedom. Considering a simple paradigmatic situation in which the long-time dynamics is dominated by the transition between two known metastable states, we compare two variational data-driven ML methods based on Siamese neural networks aimed at discovering a meaningful RC model─the slowest decorrelating CV of the molecular process, and the committor probability to first reach one of the two metastable states. One method is the state-free reversible variational approach for Markov processes networks (VAMPnets), or SRVs─the other, inspired by the transition path theory framework, is the variational committor-based neural networks, or VCNs. The relationship and the ability of these methodologies to discover the relevant descriptors of the slow molecular process of interest are illustrated with a series of simple model systems. We also show that both strategies are amenable to importance-sampling schemes through an appropriate reweighting algorithm that approximates the kinetic properties of the transition.

Survival strategies of artificial active agents

Artificial cells can be engineered to display dynamics sharing remarkable features in common with the survival behavior of living organisms. In particular, such active systems can respond to stimuli provided by the environment and undertake specific displacements to remain out of equilibrium, e.g. by moving towards regions with higher fuel concentration. In spite of the intense experimental activity aiming at investigating this fascinating behavior, a rigorous definition and characterization of such “survival strategies” from a statistical physics perspective is still missing. In this work, we take a first step in this direction by adapting and applying to active systems the theoretical framework of Transition Path Theory, which was originally introduced to investigate rare thermally activated transitions in passive systems. We perform experiments on camphor disks navigating Petri dishes and perform simulations in the paradigmatic active Brownian particle model to show how the notions of transition probability density and committor function provide the pivotal concepts to identify survival strategies, improve modeling, and obtain and validate experimentally testable predictions. The definition of survival in these artificial systems paves the way to move beyond simple observation and to formally characterize, design and predict complex life-like behaviors.

Sampling-Dependent Transition Paths of Iceland–Scotland Overflow Water

Abstract In this note, we apply transition path theory (TPT) from Markov chains to shed light on the problem of Iceland–Scotland Overflow Water (ISOW) equatorward export. A recent analysis of observed trajectories of submerged floats demanded revision of the traditional abyssal circulation theory, which postulates that ISOW should steadily flow along a deep boundary current (DBC) around the subpolar North Atlantic prior to exiting it. The TPT analyses carried out here allow attention to be focused on the portions of flow from the origin of ISOW to the region where ISOW exits the subpolar North Atlantic and suggest that insufficient sampling may be biasing the aforementioned demand. The analyses, appropriately adapted to represent a continuous input of ISOW, are carried out on three time-homogeneous Markov chains modeling the ISOW flow. One is constructed using a high number of simulated trajectories homogeneously covering the flow domain. The other two use much fewer trajectories which heterogeneously cover the domain. The trajectories in the latter two chains are observed trajectories or simulated trajectories subsampled at the observed frequency. While the densely sampled chain supports a well-defined DBC, whether this is a peculiarity of the simulation considered or not, the more heterogeneously sampled chains do not, irrespective of the nature of the trajectories used, i.e., observed or simulated. Studying the sampling sensitivity of the Markov chains, we can give recommendations for enlarging the existing float dataset to improve the significance of conclusions about long-time-asymptotic aspects of the ISOW circulation.

Machine-guided path sampling to discover mechanisms of molecular self-organization

Molecular self-organization driven by concerted many-body interactions produces the ordered structures that define both inanimate and living matter. Here we present an autonomous path sampling algorithm that integrates deep learning and transition path theory to discover the mechanism of molecular self-organization phenomena. The algorithm uses the outcome of newly initiated trajectories to construct, validate and—if needed—update quantitative mechanistic models. Closing the learning cycle, the models guide the sampling to enhance the sampling of rare assembly events. Symbolic regression condenses the learned mechanism into a human-interpretable form in terms of relevant physical observables. Applied to ion association in solution, gas-hydrate crystal formation, polymer folding and membrane-protein assembly, we capture the many-body solvent motions governing the assembly process, identify the variables of classical nucleation theory, uncover the folding mechanism at different levels of resolution and reveal competing assembly pathways. The mechanistic descriptions are transferable across thermodynamic states and chemical space.

Transition Path Theory for Langevin Dynamics on Manifolds: Optimal Control and Data-Driven Solver

We present a data-driven point of view for rare events, which represent conformational transitions in biochemical reactions modeled by overdamped Langevin dynamics on manifolds in high dimensions. We first reinterpret the transition state theory and the transition path theory from the optimal control viewpoint. Given a point cloud probing the manifold, we construct a discrete Markov chain with a -matrix computed from an approximated Voronoi tesselation via the point cloud. We use this -matrix to compute a discrete committor function whose level set automatically orders the point cloud. Then based on the committor function, an optimally controlled random walk on point clouds is constructed and utilized to efficiently sample transition paths, which become an almost sure event in time instead of a rare event in the original reaction dynamics. To compute the mean transition path efficiently, a local averaging algorithm based on the optimally controlled random walk is developed, which adapts the finite temperature string method to the controlled Monte Carlo samples. Numerical examples on sphere/torus including a conformational transition for the alanine dipeptide in vacuum are conducted to illustrate the data-driven solver for the transition path theory on point clouds. The mean transition path obtained via the controlled Monte Carlo simulations highly coincides with the computed dominant transition path in the transition path theory.

Exploring the use of Transition Path Theory in building an oil spill prediction scheme

The Transition Path Theory (TPT) of complex systems has proven to be a robust means to statistically characterize the ensemble of trajectories that connect any two preset flow regions, say 𝒜 and ℬ, directly. More specifically, transition paths are such that they start in 𝒜 and then go to ℬ without detouring back to 𝒜 or ℬ. This way, they make an effective contribution to the transport from 𝒜 to ℬ. Here, we explore its use for building a scheme that enables predicting the evolution of an oil spill in the ocean. This involves appropriately adapting TPT such that it includes a reservoir that pumps oil into a typically open domain. Additionally, we lift up the restriction of the oil not to return to the spill site en route to a region that is targeted to be protected. TPT is applied on oil trajectories available up to the present, e.g., as integrated using velocities produced by a data assimilative system or as inferred from high-frequency radars, to make a prediction of transition oil paths beyond, without relying on forecasted oil trajectories. As a proof of concept, we consider a hypothetical oil spill in the Trion oil field, under development within the Perdido Foldbelt in the northwestern Gulf of Mexico, and the Deepwater Horizon oil spill. This is done using trajectories integrated from climatological and hindcast surface velocity and winds as well as produced by satellite-tracked surface drifting buoys, in each case discretized into a Markov chain that provides a framework for the TPT-based prediction.

Data-Driven Transition Path Analysis Yields a Statistical Understanding of Sudden Stratospheric Warming Events in an Idealized Model

Abstract Atmospheric regime transitions are highly impactful as drivers of extreme weather events, but pose two formidable modeling challenges: predicting the next event (weather forecasting) and characterizing the statistics of events of a given severity (the risk climatology). Each event has a different duration and spatial structure, making it hard to define an objective “average event.” We argue here that transition path theory (TPT), a stochastic process framework, is an appropriate tool for the task. We demonstrate TPT’s capacities on a wave–mean flow model of sudden stratospheric warmings (SSWs) developed by Holton and Mass, which is idealized enough for transparent TPT analysis but complex enough to demonstrate computational scalability. Whereas a recent article (Finkel et al. 2021) studied near-term SSW predictability, the present article uses TPT to link predictability to long-term SSW frequency. This requires not only forecasting forward in time from an initial condition, but also backward in time to assess the probability of the initial conditions themselves. TPT enables one to condition the dynamics on the regime transition occurring, and thus visualize its physical drivers with a vector field called the reactive current. The reactive current shows that before an SSW, dissipation and stochastic forcing drive a slow decay of vortex strength at lower altitudes. The response of upper-level winds is late and sudden, occurring only after the transition is almost complete from a probabilistic point of view. This case study demonstrates that TPT quantities, visualized in a space of physically meaningful variables, can help one understand the dynamics of regime transitions.