Broken detailed balance at mesoscopic scales in active biological systems

On spontaneous oscillations of hair bundles in the amphibian papilla

Hair cells play a critical role in the auditory system, acting as key agents in active sound detection. Studying living hair cells ex vivo provides valuable insights into the mechanisms underlying sound detection. In this study, we investigated the nonlinear dynamics of hair bundle oscillations. We developed a robust ex vivo preparation of the amphibian papilla, a bullfrog's auditory organ, and observed spontaneous oscillations of hair bundles. These oscillations were classified into three distinct types: regular, bursting, and spiking. Regular oscillators demonstrated stable oscillations with a well-defined dominant frequency. Bursting oscillators alternated between periods of stable oscillatory activity and quiescence, while spiking oscillators were mostly quiescent, interrupted by brief oscillatory bursts. The oscillation frequencies ranged from 1 to 90 Hz, with approximately 95% of cells oscillating below 40 Hz.

Tissue Active Matter: Integrating Mechanics and Signaling into Dynamical Models

The importance of physical forces in the morphogenesis, homeostatic function, and pathological dysfunction of multicellular tissues is being increasingly characterized, both theoretically and experimentally. Analogies between biological systems and inert materials such as foams, gels, and liquid crystals have provided striking insights into the core design principles underlying multicellular organization. However, these connections can seem surprising given that a key feature of multicellular systems is their ability to constantly consume energy, providing an active origin for the forces that they produce. Key emerging questions are, therefore, to understand whether and how this activity grants tissues novel properties that do not have counterparts in classical materials, as well as their consequences for biological function. Here, we review recent discoveries at the intersection of active matter and tissue biology, with an emphasis on how modeling and experiments can be combined to understand the dynamics of multicellular systems. These approaches suggest that a number of key biological tissue-scale phenomena, such as morphogenetic shape changes, collective migration, or fate decisions, share unifying design principles that can be described by physical models of tissue active matter.

Perspective: Time irreversibility in systems observed at coarse resolution

A broken time-reversal symmetry, i.e., broken detailed balance, is central to non-equilibrium physics and is a prerequisite for life. However, it turns out to be quite challenging to unambiguously define and quantify time-reversal symmetry (and violations thereof) in practice, that is, from observations. Measurements on complex systems have a finite resolution and generally probe low-dimensional projections of the underlying dynamics, which are well known to introduce memory. In situations where many microscopic states become "lumped" onto the same observable "state" or when introducing "reaction coordinates" to reduce the dimensionality of data, signatures of a broken time-reversal symmetry in the microscopic dynamics become distorted or masked. In this Perspective, we highlight why, in defining and discussing time-reversal symmetry and quantifying its violations, the precise underlying assumptions on the microscopic dynamics, the coarse graining, and further reductions are not a technical detail. These assumptions decide whether the conclusions that are drawn are physically sound or inconsistent. We summarize recent findings in the field and reflect upon key challenges.

Concentration buffering and noise reduction in non-equilibrium phase-separating systems

Biomolecular condensates have been proposed to buffer intracellular concentrations and reduce noise. However, concentrations need not be buffered in multicomponent systems, leading to a non-constant saturation concentration (csat) when individual components are varied. Simplified equilibrium considerations suggest that noise reduction might be closely related to concentration buffering and that a fixed saturation concentration is required for noise reduction to be effective. Here, we present a theoretical analysis to demonstrate that these suggestions do not apply to mesoscopic fluctuating systems. We show that concentration buffering and noise reduction are distinct concepts, which cannot be used interchangeably. We further demonstrate that concentration buffering and a constant csat are neither necessary nor sufficient for noise reduction to be effective. Clarity about these concepts is important for studying the role of condensates in controlling cellular noise and for the interpretation of concentration relationships in cells. A record of this paper's transparent peer review process is included in the supplemental information.

Description of chemical systems by means of response functions

In this paper we introduce a formalism that allows to describe the response of a part of a biochemical system in terms of renewal equations. In particular, we examine under which conditions the interactions between the different parts of a chemical system, described by means of linear ODEs, can be represented in terms of renewal equations. We show also how to apply the formalism developed in this paper to some particular types of linear and non-linear ODEs, modelling some biochemical systems of interest in biology (for instance, some time-dependent versions of the classical Hopfield model of kinetic proofreading). We also analyse some of the properties of the renewal equations that we are interested in, as the long-time behaviour of their solution. Furthermore, we prove that the kernels characterising the renewal equations derived by biochemical system with reactions that satisfy the detail balance condition belong to the class of completely monotone functions.

Interphase chromatin biophysics and mechanics: new perspectives and open questions

The physical organization and properties of chromatin within the interphase nucleus are intimately linked to a wide range of functional DNA-based processes. In this context, interphase chromatin mechanics - that is, how chromatin, physically, responds to forces - is gaining increasing attention. Recent methodological advances for probing the force-response of chromatin in cellulo open new avenues for research, as well as new questions. This review discusses emerging views from these approaches and others, including recent in vitro single-molecule studies of cohesin and condensin motor activities, providing insights into physical and material aspects of chromatin, its plasticity in the context of functional processes, its nonequilibrium or 'active matter' properties, and the importance of factors such as chromatin fiber tension and stiffness. This growing field offers exciting opportunities to better understand the interplay between interphase chromosome structure, dynamics, mechanics, and functions.

Algorithmic Approaches for Assessing Multiscale Irreversibility in Time Series: Review and Comparison

Many physical and biological phenomena are characterized by time asymmetry, and are referred to as irreversible. Time-reversal symmetry breaking is in fact the hallmark of systems operating away from equilibrium and reflects the power dissipated by driving the system away from it. Time asymmetry may manifest in a wide range of time scales; quantifying irreversibility in such systems thus requires methods capable of detecting time asymmetry in a multiscale fashion. In this contribution we review the main algorithmic solutions that have been proposed to detect time irreversibility, and evaluate their performance and limitations when used in a multiscale context using several well-known synthetic dynamical systems. While a few of them have a general applicability, most tests yield conflicting results on the same data, stressing that a "one size fits all" solution is still to be achieved. We conclude presenting some guidelines for the interested practitioner, as well as general considerations on the meaning of multiscale time irreversibility.

In Situ Mechanics of the Cytoskeleton

Not only for man-made architecture but also for living cells, the relationship between force and structure is a fundamental properties that governs their mechanical behaviors. However, our knowledge of the mechanical properties of intracellular structures is very limited because of the lack of direct measurement methods. We established high-force intracellular magnetic tweezers that can generate calibrated forces up to 10 nN, enabling direct force measurements of the cytoskeleton. Using this method, we show that the strain field of the microtubule and actin meshwork follow the same scaling, suggesting that the two cytoskeletal systems behave as an integrated elastic body. Furthermore, quantification of structural response of single microtubules demonstrates that microtubules are enclosed by the elastic medium of filamentous actin. Our results defining the force-structure relationship of the cytoskeleton serve as a framework to understand cellular behaviors by direct intracellular mechanical measurement.

Irreversible dynamics of a continuum driven by active matter

We study the fluctuational behavior of overdamped elastic filaments (e.g., strings or rods) driven by active matter which induces irreversibility. The statistics of discrete normal modes are translated into the continuum of the position representation which allows discernment of the spatial structure of dissipation and fluctuational work done by the active forces. The mapping of force statistics onto filament statistics leads to a generalized fluctuation-dissipation relation which predicts the components of the stochastic area tensor and its spatial proxy, the irreversibility field. We illustrate the general theory with explicit results for a tensioned string between two fixed endpoints. Plots of the stochastic area tensor components in the discrete plane of mode pairs reveal how the active forces induce spatial correlations of displacement along the filament. The irreversibility field provides additional quantitative insight into the relative spatial distributions of fluctuational work and dissipative response.

A local-global principle for nonequilibrium steady states

The global steady state of a system in thermal equilibrium exponentially favors configurations with lesser energy. This principle is a powerful explanation of self-organization because energy is a local property of configurations. For nonequilibrium systems, there is no such property for which an analogous principle holds, hence no common explanation of the diverse forms of self-organization they exhibit. However, a flurry of recent empirical results has shown that a local property of configurations called "rattling" predicts the steady states of some nonequilibrium systems, leading to claims of a far-reaching principle of nonequilibrium self-organization. But for which nonequilibrium systems is rattling accurate, and why? We develop a theory of rattling in terms of Markov processes that gives simple and precise answers to these key questions. Our results show that rattling predicts a broader class of nonequilibrium steady states than has been claimed and for different reasons than have been suggested. Its predictions hold to an extent determined by the relative variance of, and correlation between, the local and global "parts" of a steady state. We show how these quantities characterize the local-global relationships of various random walks on random graphs, spin-glass dynamics, and models of animal collective behavior. Surprisingly, we find that the core idea of rattling is so general as to apply to equilibrium and nonequilibrium systems alike.

Deciphering the intracellular forces shaping mitochondrial motion

We propose a novel quantitative method to explore the forces affecting mitochondria within living cells in an almost non-invasive fashion. This new tool enables the detection of localized mechanical impulses on these organelles that occur amidst the stationary fluctuations caused by the thermal jittering in the cytoplasm. Recent experimental evidence shows that the action of mechanical forces has important effects on the dynamics, morphology and distribution of mitochondria in cells. In particular, their crosstalk with the cytoskeleton has been found to alter these organelles function; however, the mechanisms underlying this phenomenon are largely unknown. Our results highlight the different functions that cytoskeletal networks play in shaping mitochondrial dynamics. This work presents a novel technique to extend our knowledge of how the impact of mechanical cues can be quantified at the single organelle level. Moreover, this approach can be expanded to the study of other organelles or biopolymers.

Dynamic Tracking of Biological Processes Using Near-Infrared Fluorescent Single-Walled Carbon Nanotubes

Biological processes are characterized by dynamic and elaborate temporal patterns driven by the interplay of genes, proteins, and cellular components that are crucial for adaptation to changing environments. This complexity spans from molecular to organismal scales, necessitating their real-time monitoring and tracking to unravel the active processes that fuel living systems and enable early disease detection, personalized medicine, and drug development. Single-walled carbon nanotubes (SWCNTs), with their unique physicochemical and optical properties, have emerged as promising tools for real-time tracking of such processes. This perspective highlights the key properties of SWCNTs that make them ideal for such monitoring. Subsequently, it surveys studies utilizing SWCNTs to track dynamic biological phenomena across hierarchical levels─from molecules to cells, tissues, organs, and whole organisms─acknowledging their pivotal role in advancing this field. Finally, the review outlines challenges and future directions, aiming to expand the frontier of real-time biological monitoring using SWCNTs, contributing to deeper insights and novel applications in biomedicine.

Accessing activity and viscoelastic properties of artificial and living systems from passive measurement

Living systems are complex dynamic entities that operate far from thermodynamic equilibrium. Their active, non-equilibrium behaviour requires energy to drive cellular organization and dynamics. Unfortunately, most statistical mechanics approaches are not valid in non-equilibrium situations, forcing researchers to use intricate and often invasive methods to study living processes. Here we experimentally demonstrate that an observable termed mean back relaxation quantifies the active mechanics of living cells from passively observed particle trajectories. The mean back relaxation represents the average trajectory of a particle after a recent motion and is calculated from three-point probabilities. We show that this parameter allows the detection of broken detailed balance in confined systems. We experimentally observe that it provides access to the non-equilibrium generating energy and viscoelastic properties of artificial bulk materials and living cells. These findings suggest that the mean back relaxation can function as a marker of non-equilibrium dynamics and is a non-invasive avenue to determine viscoelastic material properties from passive measurements.

Broken detailed balance and entropy production in directed networks

The structure of a complex network plays a crucial role in determining its dynamical properties. In this paper , we show that the the degree to which a network is directed and hierarchically organized is closely associated with the degree to which its dynamics break detailed balance and produce entropy. We consider a range of dynamical processes and show how different directed network features affect their entropy production rate. We begin with an analytical treatment of a two-node network followed by numerical simulations of synthetic networks using the preferential attachment and Erdös-Renyi algorithms. Next, we analyze a collection of 97 empirical networks to determine the effect of complex real-world topologies. Finally, we present a simple method for inferring broken detailed balance and directed network structure from multivariate time series and apply our method to identify non-equilibrium dynamics and hierarchical organisation in both human neuroimaging and financial time series. Overall, our results shed light on the consequences of directed network structure on non-equilibrium dynamics and highlight the importance and ubiquity of hierarchical organisation and non-equilibrium dynamics in real-world systems.

Single-cell morphodynamical trajectories enable prediction of gene expression accompanying cell state change

Extracellular signals induce changes to molecular programs that modulate multiple cellular phenotypes, including proliferation, motility, and differentiation status. The connection between dynamically adapting phenotypic states and the molecular programs that define them is not well understood. Here we develop data-driven models of single-cell phenotypic responses to extracellular stimuli by linking gene transcription levels to "morphodynamics" - changes in cell morphology and motility observable in time-lapse image data. We adopt a dynamics-first view of cell state by grouping single-cell trajectories into states with shared morphodynamic responses. The single-cell trajectories enable development of a first-of-its-kind computational approach to map live-cell dynamics to snapshot gene transcript levels, which we term MMIST, Molecular and Morphodynamics-Integrated Single-cell Trajectories. The key conceptual advance of MMIST is that cell behavior can be quantified based on dynamically defined states and that extracellular signals alter the overall distribution of cell states by altering rates of switching between states. We find a cell state landscape that is bound by epithelial and mesenchymal endpoints, with distinct sequences of epithelial to mesenchymal transition (EMT) and mesenchymal to epithelial transition (MET) intermediates. The analysis yields predictions for gene expression changes consistent with curated EMT gene sets and provides a prediction of thousands of RNA transcripts through extracellular signal-induced EMT and MET with near-continuous time resolution. The MMIST framework leverages true single-cell dynamical behavior to generate molecular-level omics inferences and is broadly applicable to other biological domains, time-lapse imaging approaches and molecular snapshot data.

Deep learning probability flows and entropy production rates in active matter

Active matter systems, from self-propelled colloids to motile bacteria, are characterized by the conversion of free energy into useful work at the microscopic scale. They involve physics beyond the reach of equilibrium statistical mechanics, and a persistent challenge has been to understand the nature of their nonequilibrium states. The entropy production rate and the probability current provide quantitative ways to do so by measuring the breakdown of time-reversal symmetry. Yet, their efficient computation has remained elusive, as they depend on the system's unknown and high-dimensional probability density. Here, building upon recent advances in generative modeling, we develop a deep learning framework to estimate the score of this density. We show that the score, together with the microscopic equations of motion, gives access to the entropy production rate, the probability current, and their decomposition into local contributions from individual particles. To represent the score, we introduce a spatially local transformer network architecture that learns high-order interactions between particles while respecting their underlying permutation symmetry. We demonstrate the broad utility and scalability of the method by applying it to several high-dimensional systems of active particles undergoing motility-induced phase separation (MIPS). We show that a single network trained on a system of 4,096 particles at one packing fraction can generalize to other regions of the phase diagram, including to systems with as many as 32,768 particles. We use this observation to quantify the spatial structure of the departure from equilibrium in MIPS as a function of the number of particles and the packing fraction.

Entropy Production of Run-and-Tumble Particles

We analyze the entropy production in run-and-tumble models. After presenting the general formalism in the framework of the Fokker-Planck equations in one space dimension, we derive some known exact results in simple physical situations (free run-and-tumble particles and harmonic confinement). We then extend the calculation to the case of anisotropic motion (different speeds and tumbling rates for right- and left-oriented particles), obtaining exact expressions of the entropy production rate. We conclude by discussing the general case of heterogeneous run-and-tumble motion described by space-dependent parameters and extending the analysis to the case of d-dimensional motions.

Variance sum rule for entropy production

Entropy production is the hallmark of nonequilibrium physics, quantifying irreversibility, dissipation, and the efficiency of energy transduction processes. Despite many efforts, its measurement at the nanoscale remains challenging. We introduce a variance sum rule (VSR) for displacement and force variances that permits us to measure the entropy production rate σ in nonequilibrium steady states. We first illustrate it for directly measurable forces, such as an active Brownian particle in an optical trap. We then apply the VSR to flickering experiments in human red blood cells. We find that σ is spatially heterogeneous with a finite correlation length, and its average value agrees with calorimetry measurements. The VSR paves the way to derive σ using force spectroscopy and time-resolved imaging in living and active matter.

Does the brain behave like a (complex) network? I. Dynamics

Graph theory is now becoming a standard tool in system-level neuroscience. However, endowing observed brain anatomy and dynamics with a complex network structure does not entail that the brain actually works as a network. Asking whether the brain behaves as a network means asking whether network properties count. From the viewpoint of neurophysiology and, possibly, of brain physics, the most substantial issues a network structure may be instrumental in addressing relate to the influence of network properties on brain dynamics and to whether these properties ultimately explain some aspects of brain function. Here, we address the dynamical implications of complex network, examining which aspects and scales of brain activity may be understood to genuinely behave as a network. To do so, we first define the meaning of networkness, and analyse some of its implications. We then examine ways in which brain anatomy and dynamics can be endowed with a network structure and discuss possible ways in which network structure may be shown to represent a genuine organisational principle of brain activity, rather than just a convenient description of its anatomy and dynamics.

Bioelectric control of locomotor gaits in the walking ciliate Euplotes

Diverse animal species exhibit highly stereotyped behavioral actions and locomotor sequences as they explore their natural environments. In many such cases, the neural basis of behavior is well established, where dedicated neural circuitry contributes to the initiation and regulation of certain response sequences. At the microscopic scale, single-celled eukaryotes (protists) also exhibit remarkably complex behaviors and yet are completely devoid of nervous systems. Here, to address the question of how single cells control behavior, we study locomotor patterning in the exemplary hypotrich ciliate Euplotes, a highly polarized cell, which actuates a large number of leg-like appendages called cirri (each a bundle of ∼25-50 cilia) to swim in fluids or walk on surfaces. As it navigates its surroundings, a walking Euplotes cell is routinely observed to perform side-stepping reactions, one of the most sophisticated maneuvers ever observed in a single-celled organism. These are spontaneous and stereotyped reorientation events involving a transient and fast backward motion followed by a turn. Combining high-speed imaging with simultaneous time-resolved electrophysiological recordings, we show that this complex coordinated motion sequence is tightly regulated by rapid membrane depolarization events, which orchestrate the activity of different cirri on the cell. Using machine learning and computer vision methods, we map detailed measurements of cirri dynamics to the cell's membrane bioelectrical activity, revealing a differential response in the front and back cirri. We integrate these measurements with a minimal model to understand how Euplotes-a unicellular organism-manipulates its membrane potential to achieve real-time control over its motor apparatus.

Nematic Torques in Scalar Active Matter: When Fluctuations Favor Polar Order and Persistence

We study the impact of nematic alignment on scalar active matter in the disordered phase. We show that nematic torques control the emergent physics of particles interacting via pairwise forces and can either induce or prevent phase separation. The underlying mechanism is a fluctuation-induced renormalization of the mass of the polar field that generically arises from nematic torques. The correlations between the fluctuations of the polar and nematic fields indeed conspire to increase the particle persistence length, contrary to what phenomenological computations predict. This effect is generic and our theory also quantitatively accounts for how nematic torques enhance particle accumulation along confining boundaries and opposes demixing in mixtures of active and passive particles.

Machine learning interpretable models of cell mechanics from protein images

Cellular form and function emerge from complex mechanochemical systems within the cytoplasm. Currently, no systematic strategy exists to infer large-scale physical properties of a cell from its molecular components. This is an obstacle to understanding processes such as cell adhesion and migration. Here, we develop a data-driven modeling pipeline to learn the mechanical behavior of adherent cells. We first train neural networks to predict cellular forces from images of cytoskeletal proteins. Strikingly, experimental images of a single focal adhesion (FA) protein, such as zyxin, are sufficient to predict forces and can generalize to unseen biological regimes. Using this observation, we develop two approaches-one constrained by physics and the other agnostic-to construct data-driven continuum models of cellular forces. Both reveal how cellular forces are encoded by two distinct length scales. Beyond adherent cell mechanics, our work serves as a case study for integrating neural networks into predictive models for cell biology.

α-divergence improves the entropy production estimation via machine learning

Recent years have seen a surge of interest in the algorithmic estimation of stochastic entropy production (EP) from trajectory data via machine learning. A crucial element of such algorithms is the identification of a loss function whose minimization guarantees the accurate EP estimation. In this study we show that there exists a host of loss functions, namely, those implementing a variational representation of the α-divergence, which can be used for the EP estimation. By fixing α to a value between -1 and 0, the α-NEEP (Neural Estimator for Entropy Production) exhibits a much more robust performance against strong nonequilibrium driving or slow dynamics, which adversely affects the existing method based on the Kullback-Leibler divergence (α=0). In particular, the choice of α=-0.5 tends to yield the optimal results. To corroborate our findings, we present an exactly solvable simplification of the EP estimation problem, whose loss function landscape and stochastic properties give deeper intuition into the robustness of the α-NEEP.

Methods and Measures for Investigating Microscale Motility

Motility is an essential factor for an organism's survival and diversification. With the advent of novel single-cell technologies, analytical frameworks, and theoretical methods, we can begin to probe the complex lives of microscopic motile organisms and answer the intertwining biological and physical questions of how these diverse lifeforms navigate their surroundings. Herein, we summarize the main mechanisms of microscale motility and give an overview of different experimental, analytical, and mathematical methods used to study them across different scales encompassing the molecular-, individual-, to population-level. We identify transferable techniques, pressing challenges, and future directions in the field. This review can serve as a starting point for researchers who are interested in exploring and quantifying the movements of organisms in the microscale world.

Irreversible Mesoscale Fluctuations Herald the Emergence of Dynamical Phases

We study fluctuating field models with spontaneously emerging dynamical phases. We consider two typical transition scenarios associated with parity-time symmetry breaking: oscillatory instabilities and critical exceptional points. An analytical investigation of the low-noise regime reveals a drastic increase of the mesoscopic entropy production toward the transitions. For an illustrative model of two nonreciprocally coupled Cahn-Hilliard fields, we find physical interpretations in terms of actively propelled interfaces and a coupling of eigenmodes of the linearized dynamics near the critical exceptional point.

Inference of the force pattern acting on a semiflexible filament from shape analysis

The study of the active forces acting on semiflexible filaments networks such as the cytoskeleton requires noninvasive tools able to explore the deformation of single filaments in their natural environment. We propose here a practical method based on the solution of the hydrodynamic beam equation in the presence of transverse forces. We found that the derivative of the local curvature presents discontinuities that match the location of the applied forces, in contrast to the smooth curvature function obtained for the case of compressing longitudinal forces. These patterns can be easily appreciated in a kymograph of the curvature, which also reflects the temporal behavior of the forces. We assessed the method performance with numerical simulations describing the deformation of single microtubules provoked by the action of intracellular active forces.

Stationary shapes of axisymmetric vesicles beyond lowest-energy configurations

We conduct a systematic exploration of the energy landscape of vesicle morphologies within the framework of the Helfrich model. Vesicle shapes are determined by minimizing the elastic energy subject to constraints of constant area and volume. The results show that pressurized vesicles can adopt higher-energy spindle-like configurations that require the action of point forces at the poles. If the internal pressure is lower than the external one, multilobed shapes are predicted. We utilize our results to rationalize the experimentally observed spindle shapes of giant vesicles in a uniform AC field.

Effective temperature for an intermittent bistable potential

Thermodynamics of far-from-equilibrium systems often require measurement of effective parameters such as temperature. Whether such approach is valid for the general case of resetting protocols, active systems, or of confined systems under time-varying fields is still under investigation. We report on the effect of switching ON-OFF of an asymmetric bistable potential to the mean first passage time (MFPT) of a probed particle to go from one potential minima to the other. Experimental results coupled with numerical simulations shows the potential becoming more symmetric at slow switching. Moreover, the MFPT deviates from equilibrium condition with an effective temperature, Teff < T, at slow switching but approaches room temperature, T, at fast switching. For each switching rate, we quantify how far the system is from equilibrium by measuring deviation from a detailed balance like relation and the net circulation of flux present in phase-space. Both analysis suggest equilibrium condition are met at high switching.

Active Spaghetti: Collective Organization in Cyanobacteria

Filamentous cyanobacteria can show fascinating examples of nonequilibrium self-organization, which, however, are not well understood from a physical perspective. We investigate the motility and collective organization of colonies of these simple multicellular lifeforms. As their area density increases, linear chains of cells gliding on a substrate show a transition from an isotropic distribution to bundles of filaments arranged in a reticulate pattern. Based on our experimental observations of individual behavior and pairwise interactions, we introduce a nonreciprocal model accounting for the filaments' large aspect ratio, fluctuations in curvature, motility, and nematic interactions. This minimal model of active filaments recapitulates the observations, and rationalizes the appearance of a characteristic length scale in the system, based on the Péclet number of the cyanobacteria filaments.

Large deviations of the stochastic area for linear diffusions

The area enclosed by the two-dimensional Brownian motion in the plane was studied by Lévy, who found the characteristic function and probability density of this random variable. For other planar processes, in particular ergodic diffusions described by linear stochastic differential equations (SDEs), only the expected value of the stochastic area is known. Here we calculate the generating function of the stochastic area for linear SDEs, which can be related to the integral of the angular momentum, and extract from the result the large deviation functions characterizing the dominant part of its probability density in the long-time limit, as well as the effective SDE describing how large deviations arise in that limit. In addition, we obtain the asymptotic mean of the stochastic area, which is known to be related to the probability current, and the asymptotic variance, which is important for determining from observed trajectories whether or not a diffusion is reversible. Examples of reversible and irreversible linear SDEs are studied to illustrate our results.

Motor-free contractility of active biopolymer networks

Contractility in animal cells is often generated by molecular motors such as myosin, which require polar substrates for their function. Motivated by recent experimental evidence of motor-independent contractility, we propose a robust motor-free mechanism that can generate contraction in biopolymer networks without the need for substrate polarity. We show that contractility is a natural consequence of active binding-unbinding of crosslinkers that breaks the principle of detailed balance, together with the asymmetric force-extension response of semiflexible biopolymers. We have extended our earlier work to discuss the motor-free contraction of viscoelastic biopolymer networks. We calculate the resulting contractile velocity using a microscopic model and show that it can be reduced to a simple coarse-grained model under certain limits. Our model may provide an explanation of recent reports of motor-independent contractility in cells. Our results also suggest a mechanism for generating contractile forces in synthetic active materials.

Thermodynamic Bound on the Asymmetry of Cross-Correlations

The principle of microscopic reversibility says that, in equilibrium, two-time cross-correlations are symmetric under the exchange of observables. Thus, the asymmetry of cross-correlations is a fundamental, measurable, and often-used statistical signature of deviation from equilibrium. Here we find a simple and universal inequality that bounds the magnitude of asymmetry by the cycle affinity, i.e., the strength of thermodynamic driving. Our result applies to a large class of systems and all state observables, and it suggests a fundamental thermodynamic cost for various nonequilibrium functions quantified by the asymmetry. It also provides a powerful tool to infer affinity from measured cross-correlations, in a different and complementary way to the thermodynamic uncertainty relations. As an application, we prove a thermodynamic bound on the coherence of noisy oscillations, which was previously conjectured by Barato and Seifert [Phys. Rev. E 95, 062409 (2017)PRESCM2470-004510.1103/PhysRevE.95.062409]. We also derive a thermodynamic bound on directed information flow in a biochemical signal transduction model.

Efficient control protocols for an active Ornstein-Uhlenbeck particle

Designing a protocol to efficiently drive a stochastic system is an active field of research. Here we extend such control theory to an active Ornstein-Uhlenbeck particle (AOUP) in a bistable potential, driven by a harmonic trap. We find that protocols designed to minimize the excess work (up to linear response) perform better than naive protocols with constant velocity for a wide range of protocol durations.

Inferring scale-dependent non-equilibrium activity using carbon nanotubes

In living systems, irreversible, yet stochastic, molecular interactions form multiscale structures (such as cytoskeletal networks), which mediate processes (such as cytokinesis and cellular motility) in a close relationship between the structure and function. However, owing to a lack of methods to quantify non-equilibrium activity, their dynamics remain poorly characterized. Here, by measuring the time-reversal asymmetry encoded in the conformational dynamics of filamentous single-walled carbon nanotubes embedded in the actomyosin network of Xenopus egg extract, we characterize the multiscale dynamics of non-equilibrium activity encoded in bending-mode amplitudes. Our method is sensitive to distinct perturbations to the actomyosin network and the concentration ratio of adenosine triphosphate to adenosine diphosphate. Thus, our method can dissect the functional coupling of microscopic dynamics to the emergence of larger scale non-equilibrium activity. We relate the spatiotemporal scales of non-equilibrium activity to the key physical parameters of a semiflexible filament embedded in a non-equilibrium viscoelastic environment. Our analysis provides a general tool to characterize steady-state non-equilibrium activity in high-dimensional spaces.

Non-reversibility outperforms functional connectivity in characterisation of brain states in MEG data

Characterising brain states during tasks is common practice for many neuroscientific experiments using electrophysiological modalities such as electroencephalography (EEG) and magnetoencephalography (MEG). Brain states are often described in terms of oscillatory power and correlated brain activity, i.e. functional connectivity. It is, however, not unusual to observe weak task induced functional connectivity alterations in the presence of strong task induced power modulations using classical time-frequency representation of the data. Here, we propose that non-reversibility, or the temporal asymmetry in functional interactions, may be more sensitive to characterise task induced brain states than functional connectivity. As a second step, we explore causal mechanisms of non-reversibility in MEG data using whole brain computational models. We include working memory, motor, language tasks and resting-state data from participants of the Human Connectome Project (HCP). Non-reversibility is derived from the lagged amplitude envelope correlation (LAEC), and is based on asymmetry of the forward and reversed cross-correlations of the amplitude envelopes. Using random forests, we find that non-reversibility outperforms functional connectivity in the identification of task induced brain states. Non-reversibility shows especially better sensitivity to capture bottom-up gamma induced brain states across all tasks, but also alpha band associated brain states. Using whole brain computational models we find that asymmetry in the effective connectivity and axonal conduction delays play a major role in shaping non-reversibility across the brain. Our work paves the way for better sensitivity in characterising brain states during both bottom-up as well as top-down modulation in future neuroscientific experiments.

Violation of Detailed Balance in Quantum Open Systems

We consider the dynamics of a quantum system immersed in a dilute gas at thermodynamic equilibrium using a quantum Markovian master equation derived by applying the low-density limit technique. It is shown that the Gibbs state at the bath temperature is always stationary while the detailed balance condition at this state can be violated beyond the Born approximation. This violation is generically related to the absence of time-reversal symmetry for the scattering T matrix, which produces a thermalization mechanism that allows the presence of persistent probability and heat currents at thermal equilibrium. This phenomenon is illustrated by a model of an electron hopping between three quantum dots in an external magnetic field.

Entropy by Neighbor Distance as a New Measure for Characterizing Spatiotemporal Orders in Microscopic Collective Systems

Collective systems self-organize to form globally ordered spatiotemporal patterns. Finding appropriate measures to characterize the order in these patterns will contribute to our understanding of the principles of self-organization in all collective systems. Here we examine a new measure based on the entropy of the neighbor distance distributions in the characterization of collective patterns. We study three types of systems: a simulated self-propelled boid system, two active colloidal systems, and one centimeter-scale robotic swarm system. In all these systems, the new measure proves sensitive in revealing active phase transitions and in distinguishing steady states. We envision that the entropy by neighbor distance could be useful for characterizing biological swarms such as bird flocks and for designing robotic swarms.

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.

Macroscopic quorum sensing sustains differentiating embryonic stem cells

Cells can secrete molecules that help each other's replication. In cell cultures, chemical signals might diffuse only within a cell colony or between colonies. A chemical signal's interaction length-how far apart interacting cells are-is often assumed to be some value without rigorous justifications because molecules' invisible paths and complex multicellular geometries pose challenges. Here we present an approach, combining mathematical models and experiments, for determining a chemical signal's interaction length. With murine embryonic stem (ES) cells as a testbed, we found that differentiating ES cells secrete FGF4, among others, to communicate over many millimeters in cell culture dishes and, thereby, form a spatially extended, macroscopic entity that grows only if its centimeter-scale population density is above a threshold value. With this 'macroscopic quorum sensing', an isolated macroscopic, but not isolated microscopic, colony can survive differentiation. Our integrated approach can determine chemical signals' interaction lengths in generic multicellular communities.

Nonequilibrium currents in stochastic field theories: A geometric insight

We introduce a formalism to study nonequilibrium steady-state probability currents in stochastic field theories. We show that generalizing the exterior derivative to functional spaces allows identification of the subspaces in which the system undergoes local rotations. In turn, this allows prediction of the counterparts in the real, physical space of these abstract probability currents. The results are presented for the case of the Active Model B undergoing motility-induced phase separation, which is known to be out of equilibrium but whose steady-state currents have not yet been observed, as well as for the Kardar-Parisi-Zhang equation. We locate and measure these currents and show that they manifest in real space as propagating modes localized in regions with nonvanishing gradients of the fields.

Thermophoresis beyond Local Thermodynamic Equilibrium

We measure the thermophoresis of polysterene beads over a wide range of temperature gradients and find a pronounced nonlinear phoretic characteristic. The transition to the nonlinear behavior is marked by a drastic slowing down of thermophoretic motion and is characterized by a Péclet number of order unity as corroborated for different particle sizes and salt concentrations. The data follow a single master curve covering the entire nonlinear regime for all system parameters upon proper rescaling of the temperature gradients with the Péclet number. For low thermal gradients, the thermal drift velocity follows a theoretical linear model relying on the local-equilibrium assumption, while linear theoretical approaches based on hydrodynamic stresses, ignoring fluctuations, predict significantly slower thermophoretic motion for steeper thermal gradients. Our findings suggest that thermophoresis is fluctuation dominated for small gradients and crosses over to a drift-dominated regime for larger Péclet numbers in striking contrast to electrophoresis.

Dissipation and energy propagation across scales in an active cytoskeletal material

Living systems are intrinsically nonequilibrium: They use metabolically derived chemical energy to power their emergent dynamics and self-organization. A crucial driver of these dynamics is the cellular cytoskeleton, a defining example of an active material where the energy injected by molecular motors cascades across length scales, allowing the material to break the constraints of thermodynamic equilibrium and display emergent nonequilibrium dynamics only possible due to the constant influx of energy. Notwithstanding recent experimental advances in the use of local probes to quantify entropy production and the breaking of detailed balance, little is known about the energetics of active materials or how energy propagates from the molecular to emergent length scales. Here, we use a recently developed picowatt calorimeter to experimentally measure the energetics of an active microtubule gel that displays emergent large-scale flows. We find that only approximately one-billionth of the system's total energy consumption contributes to these emergent flows. We develop a chemical kinetics model that quantitatively captures how the system's total thermal dissipation varies with ATP and microtubule concentrations but that breaks down at high motor concentration, signaling an interference between motors. Finally, we estimate how energy losses accumulate across scales. Taken together, these results highlight energetic efficiency as a key consideration for the engineering of active materials and are a powerful step toward developing a nonequilibrium thermodynamics of living systems.

Targeted Computed Tomography Visualization and Healing of Inflammatory Bowel Disease by Orally Delivered Bacterial-Flagella-Inspired Polydiiododiacetylene Nanofibers

Accurate diagnosis and timely therapeutic intervention of inflammatory bowel disease (IBD) is essential in preventing the progression of the disease, although it still represents an insurmountable challenge. Here we report the design of bacterial-flagella-inspired polydiiododiacetylene (PIDA) nanofibers and its performance in targeted computed tomography (CT) imaging and on-demand therapeutic intervention of IBD. With a morphology mimicking bacterial flagella, PIDA nanofibers attach on the mucus layer of the gastrointestinal (GI) tract after oral administration, evenly distributing on the GI surface to portray the GI lining under CT scan within 2 h. PIDA can retain for a longer time in the damaged mucosa at the inflamed lesions than in normal GI tissues to enable the targeted CT visualization of IBD. PIDA also scavenges reactive oxygen species and ameliorates gut dysbiosis attributed to its iodine-substituted polydiacetylene structure, so that the enriched PIDA nanofibers at the targeted IBD lesions can alleviate the inflammation while maintaining the gut microbiota homeostasis, thus promoting the rebalance of GI microenvironment and the mucosal healing.

Challenges in Inferring the Directionality of Active Molecular Processes from Single-Molecule Fluorescence Resonance Energy Transfer Trajectories

We discuss some of the practical challenges that one faces in using stochastic thermodynamics to infer directionality of molecular machines from experimental single-molecule trajectories. Because of the limited spatiotemporal resolution of single-molecule experiments and because both forward and backward transitions between the same pairs of states cannot always be detected, differentiating between the forward and backward directions of, e.g., an ATP-consuming molecular machine that operates periodically, turns out to be a nontrivial task. Using a simple extension of a Markov-state model that is commonly employed to analyze single-molecule transition-path measurements, we illustrate how irreversibility can be hidden from such measurements but in some cases can be uncovered when non-Markov effects in low-dimensional single-molecule trajectories are considered.

Early EEG monitoring predicts clinical outcome in patients with moderate to severe traumatic brain injury

There is a need for reliable predictors in patients with moderate to severe traumatic brain injury to assist clinical decision making. We assess the ability of early continuous EEG monitoring at the intensive care unit (ICU) in patients with traumatic brain injury (TBI) to predict long term clinical outcome and evaluate its complementary value to current clinical standards. We performed continuous EEG measurements in patients with moderate to severe TBI during the first week of ICU admission. We assessed the Extended Glasgow Outcome Scale (GOSE) at 12 months, dichotomized into poor (GOSE 1-3) and good (GOSE 4-8) outcome. We extracted EEG spectral features, brain symmetry index, coherence, aperiodic exponent of the power spectrum, long range temporal correlations, and broken detailed balance. A random forest classifier using feature selection was trained to predict poor clinical outcome based on EEG features at 12, 24, 48, 72 and 96 h after trauma. We compared our predictor with the IMPACT score, the best available predictor, based on clinical, radiological and laboratory findings. In addition we created a combined model using EEG as well as the clinical, radiological and laboratory findings. We included hundred-seven patients. The best prediction model using EEG parameters was found at 72 h after trauma with an AUC of 0.82 (0.69-0.92), specificity of 0.83 (0.67-0.99) and sensitivity of 0.74 (0.63-0.93). The IMPACT score predicted poor outcome with an AUC of 0.81 (0.62-0.93), sensitivity of 0.86 (0.74-0.96) and specificity of 0.70 (0.43-0.83). A model using EEG and clinical, radiological and laboratory parameters resulted in a better prediction of poor outcome (p < 0.001) with an AUC of 0.89 (0.72-0.99), sensitivity of 0.83 (0.62-0.93) and specificity of 0.85 (0.75-1.00). EEG features have potential use for predicting clinical outcome and decision making in patients with moderate to severe TBI and provide complementary information to current clinical standards.

In vitro reconstitution of small GTPase regulation

Small GTPases play essential roles in the organization of eukaryotic cells. In recent years, it has become clear that their intracellular functions result from intricate biochemical networks of the GTPase and their regulators that dynamically bind to a membrane surface. Due to the inherent complexities of their interactions, however, revealing the underlying mechanisms of action is often difficult to achieve from in vivo studies. This review summarizes in vitro reconstitution approaches developed to obtain a better mechanistic understanding of how small GTPase activities are regulated in space and time.

Jarzyski's Equality and Crooks' Fluctuation Theorem for General Markov Chains with Application to Decision-Making Systems

We define common thermodynamic concepts purely within the framework of general Markov chains and derive Jarzynski's equality and Crooks' fluctuation theorem in this setup. In particular, we regard the discrete-time case, which leads to an asymmetry in the definition of work that appears in the usual formulation of Crooks' fluctuation theorem. We show how this asymmetry can be avoided with an additional condition regarding the energy protocol. The general formulation in terms of Markov chains allows transferring the results to other application areas outside of physics. Here, we discuss how this framework can be applied in the context of decision-making. This involves the definition of the relevant quantities, the assumptions that need to be made for the different fluctuation theorems to hold, as well as the consideration of discrete trajectories instead of the continuous trajectories, which are relevant in physics.

Model-Free Measurement of Local Entropy Production and Extractable Work in Active Matter

Time-reversal symmetry breaking and entropy production are universal features of nonequilibrium phenomena. Despite its importance in the physics of active and living systems, the entropy production of systems with many degrees of freedom has remained of little practical significance because the high dimensionality of their state space makes it difficult to measure. Here we introduce a local measure of entropy production and a numerical protocol to estimate it. We establish a connection between the entropy production and extractability of work in a given region of the system and show how this quantity depends crucially on the degrees of freedom being tracked. We validate our approach in theory, simulation, and experiments by considering systems of active Brownian particles undergoing motility-induced phase separation, as well as active Brownian particles and E.coli in a rectifying device in which the time-reversal asymmetry of the particle dynamics couples to spatial asymmetry to reveal its effects on a macroscopic scale.