Biased Ensembles Research Articles

Large-deviations approach to thermalization: the case of harmonic chains with conservative noise

We investigate the possibility of characterizing the different thermalization pathways through a large-deviation approach. Specifically, we consider clean, disordered and quasi-periodic harmonic chains under energy and momentum-conserving noise. For their associated master equations, describing the dynamics of normal modes energies, we compute the fluctuations of activity and dynamical entropy in the corresponding biased ensembles. First-order dynamical phase transition are found that originates from different activity regions in action space. At the transitions, the steady-state in the biased ensembles changes from extended to localized, yielding a kind of condensation in normal-modes space. For the disordered and quasi-periodic models, we argue that the phase-diagram has a critical point at a finite value of the disorder or potential strength.

Guidelines for Free-Energy Calculations Involving Charge Changes.

The Coulomb interactions in molecular simulations are inherently approximated due to the finite size of the molecular box sizes amenable to current-day compute power. Several methods exist for treating long-range electrostatic interactions, yet these approaches are subject to various finite-size-related artifacts. Lattice-sum methods are frequently used to approximate long-range interactions; however, these approaches also suffer from artifacts which become particularly pronounced for free-energy calculations that involve charge changes. The artifacts, however, also affect the sampling when plain simulations are performed, leading to a biased ensemble. Here, we investigate two previously described model systems to determine if artifacts continue to play a role when overall neutral boxes are considered, in the context of both free-energy calculations and sampling. We find that ensuring that no net-charge changes take place, while maintaining a neutral simulation box, may be sufficient provided that the simulation boxes are large enough. Addition of salt to the solution (when appropriate) can further alleviate the remaining artifacts in the sampling or the calculated free-energy differences. We provide practical guidelines to avoid finite-size artifacts.

Controlling Conditional Expectations by Zero-Determinant Strategies

Zero-determinant strategies are memory-one strategies in repeated games which unilaterally enforce linear relations between expected payoffs of players. Recently, the concept of zero-determinant strategies was extended to the class of memory-n strategies with nge 1, which enables more complicated control of payoffs by one player. However, what we can do by memory-n zero-determinant strategies is still not clear. Here, we show that memory-n zero-determinant strategies in repeated games can be used to control conditional expectations of payoffs. Equivalently, they can be used to control expected payoffs in biased ensembles, where a history of action profiles with large value of bias function is more weighted. Controlling conditional expectations of payoffs is useful for strengthening zero-determinant strategies, because players can choose conditions in such a way that only unfavorable action profiles to one player are contained in the conditions. We provide several examples of memory-n zero-determinant strategies in the repeated prisoner’s dilemma game. We also explain that a deformed version of zero-determinant strategies is easily extended to the memory-n case.

Mechanical analysis of a dynamical phase transition for particles in a channel

We analyze biased ensembles of trajectories for a two-dimensional system of particles, evolving by Langevin dynamics in a channel geometry. This bias controls the degree of particle clustering. On biasing to large clustering, we observe a dynamical phase transition where the particles break symmetry and accumulate at one of the walls. We analyze the mechanical properties of this symmetry-broken state using the Irving-Kirkwood stress tensor. The biased ensemble is characterized by body forces that originate in random thermal noises, but they have finite averages in the presence of the bias. We discuss the connection of these forces to Doob's transform and optimal control theory.

Estimation of binding rates and affinities from multiensemble Markov models and ligand decoupling.

Accurate and efficient simulation of the thermodynamics and kinetics of protein-ligand interactions is crucial for computational drug discovery. Multiensemble Markov Model (MEMM) estimators can provide estimates of both binding rates and affinities from collections of short trajectories but have not been systematically explored for situations when a ligand is decoupled through scaling of non-bonded interactions. In this work, we compare the performance of two MEMM approaches for estimating ligand binding affinities and rates: (1) the transition-based reweighting analysis method (TRAM) and (2) a Maximum Caliber (MaxCal) based method. As a test system, we construct a small host-guest system where the ligand is a single uncharged Lennard-Jones (LJ) particle, and the receptor is an 11-particle icosahedral pocket made from the same atom type. To realistically mimic a protein-ligand binding system, the LJ ϵ parameter was tuned, and the system was placed in a periodic box with 860 TIP3P water molecules. A benchmark was performed using over 80 µs of unbiased simulation, and an 18-state Markov state model was used to estimate reference binding affinities and rates. We then tested the performance of TRAM and MaxCal when challenged with limited data. Both TRAM and MaxCal approaches perform better than conventional Markov state models, with TRAM showing better convergence and accuracy. We find that subsampling of trajectories to remove time correlation improves the accuracy of both TRAM and MaxCal and that in most cases, only a single biased ensemble to enhance sampled transitions is required to make accurate estimates.

Enhanced sampling without borders: on global biasing functions and how to reweight them.

Molecular dynamics (MD) simulations are a powerful tool to follow the time evolution of biomolecular motions in atomistic resolution. However, the high computational demand of these simulations limits the timescales of motions that can be observed. To resolve this issue, so called enhanced sampling techniques are developed, which extend conventional MD algorithms to speed up the simulation process. Here, we focus on techniques that apply global biasing functions. We provide a broad overview of established enhanced sampling methods and promising new advances. As the ultimate goal is to retrieve unbiased information from biased ensembles, we also discuss benefits and limitations of common reweighting schemes. In addition to concisely summarizing critical assumptions and implications, we highlight the general application opportunities as well as uncertainties of global enhanced sampling.

Irreversibility and Biased Ensembles in Active Matter: Insights from Stochastic Thermodynamics

Active systems evade the rules of equilibrium thermodynamics by constantly dissipating energy at the level of their microscopic components. This energy flux stems from the conversion of a fuel, present in the environment, into sustained individual motion. It can lead to collective effects without any equilibrium equivalent, some of which can be rationalized by using equilibrium tools to recapitulate nonequilibrium transitions. An important challenge is then to delineate systematically to what extent the character of these active transitions is genuinely distinct from equilibrium analogs. We review recent works that use stochastic thermodynamics tools to identify, for active systems, a measure of irreversibility comprising a coarse-grained or informatic entropy production. We describe how this relates to the underlying energy dissipation or thermodynamic entropy production, and how it is influenced by collective behavior. Then, we review the possibility of constructing thermodynamic ensembles out of equilibrium, where trajectories are biased toward atypical values of nonequilibrium observables. We show that this is a generic route to discovering unexpected phase transitions in active matter systems, which can also inform their design.

Dynamical phase transition in the activity-biased fully-connected random field Ising model: connection with glass-forming systems

We analyse biased ensembles of trajectories for the random-field Ising model on a fully-connected lattice, which is described exactly by mean-field theory. By coupling the activity of the system to a dynamical biasing field, we find a range of dynamical phase transitions, including spontaneous symmetry breaking into ordered states. For weak bias, the phase behaviour is controlled by extrema of the free energy, which may be local minima or saddle points. For large bias, the system tends to states of extremal activity, which may differ strongly from free energy minima. We discuss connections of these results to random first-order transition theory of glasses, which motivates an extension of the analysis to random-field Ising models where the dynamical activity is not symmetric under magnetisation reversal.

Long-ranged correlations in large deviations of local clustering.

In systems of diffusing particles, we investigate large deviations of a time-averaged measure of clustering around one particle. We focus on biased ensembles of trajectories, which realize large-deviation events. The bias acts on a single particle, but elicits a response that spans the whole system. We analyze this effect through the lens of macroscopic fluctuation theory, focusing on the coupling of the bias to hydrodynamic modes. This explains that the dynamical free energy has nontrivial scaling relationships with the system size, in 1 and 2 spatial dimensions. We show that the long-ranged response to a bias on one particle also has consequences when biasing two particles.

Multi-Species Stochastic Model and Related Effective Site-Dependent Transition Rates

The dynamical rules in an auxiliary stochastic process that generate the biased ensemble of rare events are nonlocal. For the systems with one type of particles, it is shown that one can find special cases for which the transition rates of the auxiliary stochastic generator can be local. In this paper, we investigate this possibility for a system of classical hard-core particles with more than one type of particles moving on a one-dimensional lattice with open boundary conditions. We assume that the nature of interactions in the original process is local and we also assume that the original rates are constant. We will show that under certain conditions on transition rates, the auxiliary stochastic generator can be local but its transition rates depend on the site number. Two-species exclusion model is considered as an example. Considering the activity of particles of type 1 as a time-integrated observable, the above-mentioned conditions are investigated.

Dissipation controls transport and phase transitions in active fluids: mobility, diffusion and biased ensembles

Active fluids operate by constantly dissipating energy at the particle level to perform a directed motion, yielding dynamics and phases without any equilibrium equivalent. The emerging behaviors have been studied extensively, yet deciphering how local energy fluxes control the collective phenomena is still largely an open challenge. We provide generic relations between the activity-induced dissipation and the transport properties of an internal tracer. By exploiting a mapping between active fluctuations and disordered driving, our results reveal how the local dissipation, at the basis of self-propulsion, constrains internal transport by reducing the mobility and the diffusion of particles. Then, we employ techniques of large deviations to investigate how interactions are affected when varying dissipation. This leads us to shed light on a microscopic mechanism to promote clustering at low dissipation, and we also show the existence of collective motion at high dissipation. Overall, these results illustrate how tuning dissipation provides an alternative route to phase transitions in active fluids.

Large deviations and optimal control forces for hard particles in one dimension

We analyse large deviations of the dynamical activity in one-dimensional systems of diffusing hard particles. Using an optimal-control representation of the large-deviation problem, we analyse effective interaction forces which can be added to the system, to aid sampling of biased ensembles of trajectories. We find several distinct regimes, as a function of the activity and the system size: we present approximate analytical calculations that characterise the effective interactions in several of these regimes. For high activity the system is hyperuniform and the interactions are long-ranged and repulsive. For low activity, there is a near-equilibrium regime described by macroscopic fluctuation theory, characterised by long-ranged attractive forces. There is also a far-from-equilibrium regime in which one of the interparticle gaps becomes macroscopic and the interactions depend strongly on the size of this gap. We discuss the extent to which transition path sampling of these ensembles is improved by adding suitable control forces.

How Dissipation Constrains Fluctuations in Nonequilibrium Liquids: Diffusion, Structure, and Biased Interactions

The dynamics and structure of nonequilibrium liquids, driven by non-conservative forces which can be either external or internal, generically hold the signature of the net dissipation of energy in the thermostat. Yet, disentangling precisely how dissipation changes collective effects remains challenging in many-body systems due to the complex interplay between driving and particle interactions. First, we combine explicit coarse-graining and stochastic calculus to obtain simple relations between diffusion, density correlations and dissipation in nonequilibrium liquids. Based on these results, we consider large-deviation biased ensembles where trajectories mimic the effect of an external drive. The choice of the biasing function is informed by the connection between dissipation and structure derived in the first part. Using analytical and computational techniques, we show that biasing trajectories effectively renormalizes interactions in a controlled manner, thus providing intuition on how driving forces can lead to spatial organization and collective dynamics. Altogether, our results show how tuning dissipation provides a route to alter the structure and dynamics of liquids and soft materials.

Network-based Biased Tree Ensembles (NetBiTE) for Drug Sensitivity Prediction and Drug Sensitivity Biomarker Identification in Cancer

We present the Network-based Biased Tree Ensembles (NetBiTE) method for drug sensitivity prediction and drug sensitivity biomarker identification in cancer using a combination of prior knowledge and gene expression data. Our devised method consists of a biased tree ensemble that is built according to a probabilistic bias weight distribution. The bias weight distribution is obtained from the assignment of high weights to the drug targets and propagating the assigned weights over a protein-protein interaction network such as STRING. The propagation of weights, defines neighborhoods of influence around the drug targets and as such simulates the spread of perturbations within the cell, following drug administration. Using a synthetic dataset, we showcase how application of biased tree ensembles (BiTE) results in significant accuracy gains at a much lower computational cost compared to the unbiased random forests (RF) algorithm. We then apply NetBiTE to the Genomics of Drug Sensitivity in Cancer (GDSC) dataset and demonstrate that NetBiTE outperforms RF in predicting IC50 drug sensitivity, only for drugs that target membrane receptor pathways (MRPs): RTK, EGFR and IGFR signaling pathways. We propose based on the NetBiTE results, that for drugs that inhibit MRPs, the expression of target genes prior to drug administration is a biomarker for IC50 drug sensitivity following drug administration. We further verify and reinforce this proposition through control studies on, PI3K/MTOR signaling pathway inhibitors, a drug category that does not target MRPs, and through assignment of dummy targets to MRP inhibiting drugs and investigating the variation in NetBiTE accuracy.

Rate constants in spatially inhomogeneous systems.

We present a theory and accompanying importance sampling method for computing rate constants in spatially inhomogeneous systems. Using the relationship between rate constants and path space partition functions, we illustrate that the relative change in the rate of a rare event through space is isomorphic to the calculation of a free energy difference, albeit in a trajectory ensemble. Like equilibrium free energies, relative rate constants can be estimated by importance sampling. An extension to transition path sampling is proposed that combines biased path ensembles and weighted histogram analysis to accomplish this estimate. We show that rate constants can also be decomposed into different contributions, including relative changes in stability, barrier height, and flux. This decomposition provides a means of interpretation and insight into rare processes in complex environments. We verify these ideas with a simple model of diffusion with spatially varying diffusivity and illustrate their utility in a model of ion pair dissociation near an electrochemical interface.

Protein p Ka's from Adaptive Landscape Flattening Instead of Constant-pH Simulations.

Protein acid/base constants, or p Ka's are often computed from Monte Carlo or molecular dynamics simulations at a series of constant pH values. Instead, we propose to adaptively flatten the free energy landscape in the space of protonation states. The flattening is achieved by a Wang-Landau Monte Carlo, where a bias potential is constructed adaptively during an initial phase, such that all protonation states achieve comparable probabilities. Biased ensembles of states are then reweighted by subtracting out the bias and adding a pH-dependent free energy term. Titration curves constructed for three test proteins agreed, within the small numerical uncertainty, with those obtained earlier from the constant-pH approach.

Sparse sampling of water density fluctuations near liquid-vapor coexistence

The free energetics of water density fluctuations in bulk water, at interfaces, and in hydrophobic confinement inform the hydration of hydrophobic solutes as well as their interactions and assembly. The characterisation of such free energetics is typically performed using enhanced sampling techniques such as umbrella sampling. In umbrella sampling, order parameter distributions obtained from adjacent biased simulations must overlap in order to estimate free energy differences between biased ensembles. Many biased simulations are typically required to ensure such overlap, which exacts a steep computational cost. We recently introduced a sparse sampling method, which circumvents the overlap requirement by using thermodynamic integration to estimate free energy differences between biased ensembles. Here we build upon and generalise sparse sampling for characterising the free energetics of water density fluctuations in systems near liquid-vapor coexistence. We also introduce sensible heuristics for choosing the biasing potential parameters and strategies for adaptively refining them, which facilitate the estimation of such free energetics accurately and efficiently. We illustrate the method by characterising the free energetics of cavitation in a large volume in bulk water. We also use sparse sampling to characterise the free energetics of capillary evaporation for water confined between two hydrophobic plates. In both cases, sparse sampling is nearly two orders of magnitude faster than umbrella sampling. Given its efficiency, the sparse sampling method is particularly well suited for characterising free energy landscapes for systems wherein umbrella sampling is prohibitively expensive.

Enantiodromic effective generators of a Markov jump process with Gallavotti-Cohen symmetry.

This paper deals with the properties of the stochastic generators of the effective (driven) processes associated with atypical values of transition-dependent time-integrated currents with Gallavotti-Cohen symmetry in Markov jump processes. Exploiting the concept of biased ensemble of trajectories by introducing a biasing field s, we show that the stochastic generators of the effective processes associated with the biasing fields s and E-s are enantiodromic with respect to each other where E is the conjugated field to the current. We illustrate our findings by considering an exactly solvable creation-annihilation process of classical particles with nearest-neighbor interactions defined on a one-dimensional lattice.

Absence of dissipation in trajectory ensembles biased by currents

We consider biased ensembles of trajectories associated with large deviations of currents in equilibrium systems. The biased ensembles are characterised by non-zero currents and lack the time-reversal symmetry of the equilibrium state. In cases where the equilibrium system has an inversion symmetry which is broken by the bias, we show that the biased ensembles retain a generalised time-reversal symmetry, involving a spatial transformation that inverts the current. This means that these ensembles lack dissipation. Hence, they differ significantly from non-equilibrium steady states where currents are induced by external forces. One consequence of this result is that maximum entropy assumptions (MaxEnt/MaxCal), widely used for modelling thermal systems away from equilibrium, have quite unexpected implications, including apparent superfluid behaviour in a classical model of shear flow.

Multiensemble Markov models of molecular thermodynamics and kinetics

We introduce the general transition-based reweighting analysis method (TRAM), a statistically optimal approach to integrate both unbiased and biased molecular dynamics simulations, such as umbrella sampling or replica exchange. TRAM estimates a multiensemble Markov model (MEMM) with full thermodynamic and kinetic information at all ensembles. The approach combines the benefits of Markov state models-clustering of high-dimensional spaces and modeling of complex many-state systems-with those of the multistate Bennett acceptance ratio of exploiting biased or high-temperature ensembles to accelerate rare-event sampling. TRAM does not depend on any rate model in addition to the widely used Markov state model approximation, but uses only fundamental relations such as detailed balance and binless reweighting of configurations between ensembles. Previous methods, including the multistate Bennett acceptance ratio, discrete TRAM, and Markov state models are special cases and can be derived from the TRAM equations. TRAM is demonstrated by efficiently computing MEMMs in cases where other estimators break down, including the full thermodynamics and rare-event kinetics from high-dimensional simulation data of an all-atom protein-ligand binding model.