Learning the non-equilibrium dynamics of Brownian movies

Learning dynamical models of single and collective cell migration: a review

Single and collective cell migration are fundamental processes critical for physiological phenomena ranging from embryonic development and immune response to wound healing and cancer metastasis. To understand cell migration from a physical perspective, a broad variety of models for the underlying physical mechanisms that govern cell motility have been developed. A key challenge in the development of such models is how to connect them to experimental observations, which often exhibit complex stochastic behaviours. In this review, we discuss recent advances in data-driven theoretical approaches that directly connect with experimental data to infer dynamical models of stochastic cell migration. Leveraging advances in nanofabrication, image analysis, and tracking technology, experimental studies now provide unprecedented large datasets on cellular dynamics. In parallel, theoretical efforts have been directed towards integrating such datasets into physical models from the single cell to the tissue scale with the aim of conceptualising the emergent behaviour of cells. We first review how this inference problem has been addressed in both freely migrating and confined cells. Next, we discuss why these dynamics typically take the form of underdamped stochastic equations of motion, and how such equations can be inferred from data. We then review applications of data-driven inference and machine learning approaches to heterogeneity in cell behaviour, subcellular degrees of freedom, and to the collective dynamics of multicellular systems. Across these applications, we emphasise how data-driven methods can be integrated with physical active matter models of migrating cells, and help reveal how underlying molecular mechanisms control cell behaviour. Together, these data-driven approaches are a promising avenue for building physical models of cell migration directly from experimental data, and for providing conceptual links between different length-scales of description.

Nonadditivity in interactions between three membrane-wrapped colloidal spheres

Many cell functions require a concerted effort from multiple membrane proteins, for example, for signaling, cell division, and endocytosis. One contribution to their successful self-organization stems from the membrane deformations that these proteins induce. While the pairwise interaction potential of two membrane-deforming spheres has recently been measured, membrane-deformation-induced interactions have been predicted to be nonadditive, and hence their collective behavior cannot be deduced from this measurement. We here employ a colloidal model system consisting of adhesive spheres and giant unilamellar vesicles to test these predictions by measuring the interaction potential of the simplest case of three membrane-deforming, spherical particles. We quantify their interactions and arrangements and, for the first time, experimentally confirm and quantify the nonadditive nature of membrane-deformation-induced interactions. We furthermore conclude that there exist two favorable configurations on the membrane: (1) a linear and (2) a triangular arrangement of the three spheres. Using Monte Carlo simulations, we corroborate the experimentally observed energy minima and identify a lowering of the membrane deformation as the cause for the observed configurations. The high symmetry of the preferred arrangements for three particles suggests that arrangements of many membrane-deforming objects might follow simple rules.

Quantifying spatiotemporal patterns in classical and quantum systems out of equilibrium

A rich variety of nonequilibrium dynamical phenomena and processes unambiguously calls for the development of general numerical techniques to probe and estimate a complex interplay between spatial and temporal degrees of freedom in many-body systems of completely different nature. In this work we provide a solution to this problem by adopting a structural complexity measure to quantify spatiotemporal patterns in the time-dependent digital representation of a system. On the basis of very limited amount of data our approach allows us to distinguish different dynamical regimes and define critical parameters in both classical and quantum systems. By the example of the discrete time crystal realized in nonequilibrium quantum systems we provide a complete low-level characterization of this nontrivial dynamical phase with only processing bitstrings, which can be considered as a valuable alternative to previous studies based on the calculations of qubit correlation functions.

The lack of temporal brain dynamics asymmetry as a signature of impaired consciousness states

Life is a constant battle against equilibrium. From the cellular level to the macroscopic scale, living organisms as dissipative systems require the violation of their detailed balance, i.e. metabolic enzymatic reactions, in order to survive. We present a framework based on temporal asymmetry as a measure of non-equilibrium. By means of statistical physics, it was discovered that temporal asymmetries establish an arrow of time useful for assessing the reversibility in human brain time series. Previous studies in human and non-human primates have shown that decreased consciousness states such as sleep and anaesthesia result in brain dynamics closer to the equilibrium. Furthermore, there is growing interest in the analysis of brain symmetry based on neuroimaging recordings and since it is a non-invasive technique, it can be extended to different brain imaging modalities and applied at different temporo-spatial scales. In the present study, we provide a detailed description of our methodological approach, paying special attention to the theories that motivated this work. We test, for the first time, the reversibility analysis in human functional magnetic resonance imaging data in patients suffering from disorder of consciousness. We verify that the tendency of a decrease in the asymmetry of the brain signal together with the decrease in non-stationarity are key characteristics of impaired consciousness states. We expect that this work will open the way for assessing biomarkers for patients' improvement and classification, as well as motivating further research on the mechanistic understanding underlying states of impaired consciousness.

Stochastic line integrals and stream functions as metrics of irreversibility and heat transfer

Stochastic line integrals are presented as a useful metric for quantitatively characterizing irreversibility and detailed balance violation in noise-driven dynamical systems. A particular realization is the stochastic area, recently studied in coupled electrical circuits. Here we provide a general framework for understanding properties of stochastic line integrals and clarify their implementation for experiments and simulations. For two-dimensional systems, stochastic line integrals can be expressed in terms of a stream function, the sign of which determines the orientation of nonequilibrium steady-state probability currents. Theoretical results are supported by numerical studies of an overdamped two-dimensional mass-spring system driven out of equilibrium. Additionally, the stream function permits analytical understanding of the scaling dependence of stochastic area growth rate on key parameters such as the noise strength for both linear and nonlinear springs.

Information-driven transitions in projections of underdamped dynamics

Low-dimensional representations of underdamped systems often provide useful insights and analytical tractability. Here, we build such representations via information projections, obtaining an optimal model that captures the most information on observed spatial trajectories. We show that, in paradigmatic systems, the minimization of the information loss drives the appearance of a discontinuous transition in the optimal model parameters. Our results raise serious warnings for general inference approaches, and they unravel fundamental properties of effective dynamical representations impacting several fields, from biophysics to dimensionality reduction.

From predicting to learning dissipation from pair correlations of active liquids

Active systems, which are driven out of equilibrium by local non-conservative forces, can adopt unique behaviors and configurations. Towards designing such materials, an important challenge is to precisely connect the static structure of active systems to the dissipation of energy induced by the local driving. Here, we use tools from liquid-state theories and machine learning to take on these challenges. We first demonstrate analytically for an isotropic active matter system that dissipation and pair correlations are closely related when driving forces behave like an active temperature. We then extend a nonequilibrium mean-field framework for predicting these pair correlations which, unlike most existing approaches, is applicable even for strongly interacting particles and far from equilibrium, to predict dissipation in these systems. Based on this theory, we reveal analytically a robust relation between dissipation and structure which holds even as the system approaches a nonequilibrium phase transition. Finally, we construct a neural network which maps static configurations of particles to their dissipation rate without any prior knowledge of the underlying dynamics. Our results open novel perspectives on the interplay between dissipation and organization out-of-equilibrium.

Information Thermodynamics of the Transition-Path Ensemble

The reaction coordinate describing a transition between reactant and product is a fundamental concept in the theory of chemical reactions. Within transition-path theory, a quantitative definition of the reaction coordinate is found in the committor, which is the probability that a trajectory initiated from a given microstate first reaches the product before the reactant. Here we develop an information-theoretic origin for the committor and show how selecting transition paths from a long ergodic equilibrium trajectory induces entropy production which exactly equals the information that system dynamics provide about the reactivity of trajectories. This equality of entropy production and dynamical information generation also holds at the level of arbitrary individual coordinates, providing parallel measures of the coordinate's relevance to the reaction, each of which is maximized by the committor.

Irreversibility in linear systems with colored noise

Time irreversibility is a distinctive feature of nonequilibrium dynamics and several measures of irreversibility have been introduced to assess the distance from thermal equilibrium of a stochastically driven system. While the dynamical noise is often approximated as white, in many real applications the time correlations of the random forces can actually be significantly long-lived compared to the relaxation times of the driven system. We analyze the effects of temporal correlations in the noise on commonly used measures of irreversibility and demonstrate how the theoretical framework for white-noise-driven systems naturally generalizes to the case of colored noise. Specifically, we express the autocorrelation function, the area enclosing rates, and mean phase space velocity in terms of solutions of a Lyapunov equation and in terms of their white-noise limit values.

Estimating Entropy Production from Waiting Time Distributions

Living systems operate far from thermal equilibrium by converting the chemical potential of ATP into mechanical work to achieve growth, replication, or locomotion. Given time series observations of intra-, inter-, or multicellular processes, a key challenge is to detect nonequilibrium behavior and quantify the rate of free energy consumption. Obtaining reliable bounds on energy consumption and entropy production directly from experimental data remains difficult in practice, as many degrees of freedom typically are hidden to the observer, so that the accessible coarse-grained dynamics may not obviously violate detailed balance. Here, we introduce a novel method for bounding the entropy production of physical and living systems which uses only the waiting time statistics of hidden Markov processes and, hence, can be directly applied to experimental data. By determining a universal limiting curve, we infer entropy production bounds from experimental data for gene regulatory networks, mammalian behavioral dynamics, and numerous other biological processes. Further considering the asymptotic limit of increasingly precise biological timers, we estimate the necessary entropic cost of heartbeat regulation in humans, dogs, and mice.

Improved bounds on entropy production in living systems

Living systems maintain or increase local order by working against the second law of thermodynamics. Thermodynamic consistency is restored as they consume free energy, thereby increasing the net entropy of their environment. Recently introduced estimators for the entropy production rate have provided major insights into the efficiency of important cellular processes. In experiments, however, many degrees of freedom typically remain hidden to the observer, and, in these cases, existing methods are not optimal. Here, by reformulating the problem within an optimization framework, we are able to infer improved bounds on the rate of entropy production from partial measurements of biological systems. Our approach yields provably optimal estimates given certain measurable transition statistics. In contrast to prevailing methods, the improved estimator reveals nonzero entropy production rates even when nonequilibrium processes appear time symmetric and therefore may pretend to obey detailed balance. We demonstrate the broad applicability of this framework by providing improved bounds on the energy consumption rates in a diverse range of biological systems including bacterial flagella motors, growing microtubules, and calcium oscillations within human embryonic kidney cells.