Learning Force Fields from Stochastic Trajectories

Abstract

When monitoring the dynamics of stochastic systems, such as interacting particles agitated by thermal noise, disentangling deterministic forces from Brownian motion is challenging. Indeed, we show that there is an information-theoretic bound, the capacity of the system when viewed as a communication channel, that limits the rate at which information about the force field can be extracted from a Brownian trajectory. This capacity provides an upper bound to the system's entropy production rate, and quantifies the rate at which the trajectory becomes distinguishable from pure Brownian motion. We propose a practical method, Stochastic Force Inference, that uses this information to approximate force fields and spatially variable diffusion coefficients. This technique readily permits the evaluation of out-of-equilibrium currents and entropy production with a limited amount of data.

Learning the non-equilibrium dynamics of Brownian movies

“…To this end, we employ a recently introduced method, Stochastic Force Inference 24 (SFI), for the inverse Brownian dynamics problem. Briefly, this procedure is based on a least-squares approximation of the diffusion and drift fields using a basis of known functions (such as polynomials).…”

“…DCA relies on the measurement of an intuitive trajectory-based non-equilibrium quantity: the area enclosing rate (AER) matrix associated to a general set of coordinates y . The elements of the AER matrix, in Itô convention, are defined by 24 , 38 , 60 – 62 where y i denotes the i -th coordinate centered around its mean value and 〈⋅〉 a time average. This non-equilibrium measure quantifies the average area enclosed by the trajectory in phase space per unit time.…”

“…Such dissipative currents can be quantified by the entropy production rate 39 , which is a measure of the irreversibility of the dynamics 40 . New approaches have been developed to measure this rate in real systems 22 , 24 , shedding light onto the structure of dissipative processes 19 and their impact on the dynamics of living matter 20 . However, it remains an outstanding challenge to accurately infer the entropy production rate by analyzing Brownian movies of such systems.…”

“…Traditional approaches to measure microscopic forces and analyze time-lapse microscopy data typically rely on tracking the position or shape of well-defined probes, such as tracer beads, fluorescent proteins and filaments, or simply on exploiting the natural contrast of the intracellular medium to obtain such tracks 14 – 17 , 29 , 31 , 34 , 41 – 44 . The tracer trajectories can be studied through stochastic analysis techniques to extract an effective model for their dynamics and infer quantities like the entropy production rate 19 , 20 , 22 , 24 , 45 – 48 . There are, however, many cases in which tracking is impractical 49 , 50 , due to limited resolution or simply because there are no recognizable objects to use as tracers.…”

“…This allows us to perform a dimensional reduction, leading to a representation of the movie as a stochastic trajectory in this component space. Finally, we employ a recently introduced method, Stochastic Force Inference (SFI) 24 , to analyze such trajectories. Our approach not only yields an estimate of the entropy production rate of a Brownian movie, which is a controlled lower bound to the system’s total entropy production rate, but also important dynamical information such as a time-resolved force map of the imaged system.…”