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Understanding dynamics in complex systems is challenging because there are many degrees of freedom, and those that are most important for describing events of interest are often not obvious. The leading eigenfunctions of the transition operator are useful for visualization, and they can provide an efficient basis for computing statistics, such as the likelihood and average time of events (predictions). Here, we develop inexact iterative linear algebra methods for computing these eigenfunctions (spectral estimation) and making predictions from a dataset of short trajectories sampled at finite intervals. We demonstrate the methods on a low-dimensional model that facilitates visualization and a high-dimensional model of a biomolecular system. Implications for the prediction problem in reinforcement learning are discussed. 

Many sampling strategies commonly used in molecular dynamics, such as umbrella sampling and alchemical free energy methods, involve sampling from multiple states. The Multistate Bennett Acceptance Ratio (MBAR) formalism is a widely used way of recombining the resulting data. However, the error of the MBAR estimator is not well-understood: previous error analyses of MBAR assumed independent samples. In this work, we derive a central limit theorem for MBAR estimates in the presence of correlated data, further justifying the use of MBAR in practical applications. Moreover, our central limit theorem yields an estimate of the error that can be decomposed into contributions from the individual Markov chains used to sample the states. This gives additional insight into how sampling in each state affects the overall error. We demonstrate our error estimator on an umbrella sampling calculation of the free energy of isomerization of the alanine dipeptide and an alchemical calculation of the hydration free energy of methane. Our numerical results demonstrate that the time required for the Markov chain to decorrelate in individual states can contribute considerably to the total MBAR error, highlighting the importance of accurately addressing the effect of sample correlation. 

We consider an immersed elastic body that is actively driven through a structured fluid by a motor or an external force. The behavior of such a system generally cannot be solved analytically, necessitating the use of numerical methods. However, current numerical methods omit important details of the microscopic structure and dynamics of the fluid, which can modulate the magnitudes and directions of viscoelastic restoring forces. To address this issue, we develop a simulation platform for modeling viscoelastic media with tensorial elasticity. We build on the lattice Boltzmann algorithm and incorporate viscoelastic forces, elastic immersed objects, a microscopic orientation field, and coupling between viscoelasticity and the orientation field. We demonstrate our method by characterizing how the viscoelastic restoring force on a driven immersed object depends on various key parameters as well as the tensorial character of the elastic response. We find that the restoring force depends non-monotonically on the rate of diffusion of the stress and the size of the object. We further show how the restoring force depends on the relative orientation of the microscopic structure and the pulling direction. These results imply that accounting for previously neglected physical features, such as stress diffusion and the microscopic orientation field, can improve the realism of viscoelastic simulations. We discuss possible applications and extensions to the method. 

In nature, several ciliated protists possess the remarkable ability to execute ultrafast motions using protein assemblies called myonemes, which contract in response to Ca 2+ ions. Existing theories, such as actomyosin contractility and macroscopic biomechanical latches, do not adequately describe these systems, necessitating development of models to understand their mechanisms. In this study, we image and quantitatively analyze the contractile kinematics observed in two ciliated protists ( Vorticella sp. and Spirostomum sp.), and, based on the mechanochemistry of these organisms, we propose a minimal mathematical model that reproduces our observations as well as those published previously. Analyzing the model reveals three distinct dynamic regimes, differentiated by the rate of chemical driving and the importance of inertia. We characterize their unique scaling behaviors and kinematic signatures. Besides providing insights into Ca 2+ -powered myoneme contraction in protists, our work may also inform the rational design of ultrafast bioengineered systems such as active synthetic cells. 

Transition path theory provides a statistical description of the dynamics of a reaction in terms of local spatial quantities. In its original formulation, it is limited to reactions that consist of trajectories flowing from a reactant set A to a product set B. We extend the basic concepts and principles of transition path theory to reactions in which trajectories exhibit a specified sequence of events and illustrate the utility of this generalization on examples. 

Transition path theory computes statistics from ensembles of reactive trajectories. A common strategy for sampling reactive trajectories is to control the branching and pruning of trajectories so as to enhance the sampling of low probability segments. However, it can be challenging to apply transition path theory to data from such methods because determining whether configurations and trajectory segments are part of reactive trajectories requires looking backward and forward in time. Here, we show how this issue can be overcome efficiently by introducing simple data structures. We illustrate the approach in the context of nonequilibrium umbrella sampling, but the strategy is general and can be used to obtain transition path theory statistics from other methods that sample segments of unbiased trajectories. 

The cyanobacterial clock presents a unique opportunity to understand the biochemical basis of circadian rhythms. The core oscillator, composed of the KaiA, KaiB, and KaiC proteins, has been extensively studied, but a complete picture of its connection to the physiology of the cell is lacking. To identify previously unknown components of the clock, we used KaiB locked in its active fold as bait in an immunoprecipitation/mass spectrometry approach. We found that the most abundant interactor, other than KaiC, was a putative diguanylate cyclase protein predicted to contain multiple Per-Arnt-Sim (PAS) domains, which we propose to name KidA. Here we show that KidA directly binds to the fold-switched active form of KaiB through its N-terminal PAS domains. We found that KidA shortens the period of the circadian clock both in vivo and in vitro and alters the ability of the clock to entrain to light-dark cycles. The dose-dependent effect of KidA on the clock period could be quantitatively recapitulated by a mathematical model in which KidA stabilizes the fold-switched form of KaiB, favoring rebinding to KaiC. Put together, our results show that the period and amplitude of the clock can be modulated by regulating the access of KaiB to the fold-switched form.