Broken detailed balance at mesoscopic scales in active biological systems

Phenotyping single-cell motility in microfluidic confinement

The movement trajectories of organisms serve as dynamic read-outs of their behaviour and physiology. For microorganisms this can be difficult to resolve due to their small size and fast movement. Here, we devise a novel droplet microfluidics assay to encapsulate single micron-sized algae inside closed arenas, enabling ultralong high-speed tracking of the same cell. Comparing two model species - Chlamydomonas reinhardtii (freshwater, 2 cilia), and Pyramimonas octopus (marine, 8 cilia), we detail their highly-stereotyped yet contrasting swimming behaviours and environmental interactions. By measuring the rates and probabilities with which cells transition between a trio of motility states (smooth-forward swimming, quiescence, tumbling or excitable backward swimming), we reconstruct the control network that underlies this gait switching dynamics. A simplified model of cell-roaming in circular confinement reproduces the observed long-term behaviours and spatial fluxes, including novel boundary circulation behaviour. Finally, we establish an assay in which pairs of droplets are fused on demand, one containing a trapped cell with another containing a chemical that perturbs cellular excitability, to reveal how aneural microorganisms adapt their locomotor patterns in real-time.

Time to reach the maximum for a stationary stochastic process

We consider a one-dimensional stationary time series of fixed duration T. We investigate the time t_{m} at which the process reaches the global maximum within the time interval [0,T]. By using a path-decomposition technique, we compute the probability density function P(t_{m}|T) of t_{m} for several processes, that are either at equilibrium (such as the Ornstein-Uhlenbeck process) or out of equilibrium (such as Brownian motion with stochastic resetting). We show that for equilibrium processes the distribution of P(t_{m}|T) is always symmetric around the midpoint t_{m}=T/2, as a consequence of the time-reversal symmetry. This property can be used to detect nonequilibrium fluctuations in stationary time series. Moreover, for a diffusive particle in a confining potential, we show that the scaled distribution P(t_{m}|T) becomes universal, i.e., independent of the details of the potential, at late times. This distribution P(t_{m}|T) becomes uniform in the "bulk" 1≪t_{m}≪T and has a nontrivial universal shape in the "edge regimes" t_{m}→0 and t_{m}→T. Some of these results have been announced in a recent letter [Europhys. Lett. 135, 30003 (2021)0295-507510.1209/0295-5075/ac19ee].

Inferring entropy production rate from partially observed Langevin dynamics under coarse-graining

The entropy production rate (EPR) measures time-irreversibility in systems operating far from equilibrium. The challenge in estimating the EPR for a continuous variable system is the finite spatiotemporal resolution and the limited accessibility to all of the nonequilibrium degrees of freedom. Here, we estimate the irreversibility in partially observed systems following oscillatory dynamics governed by coupled overdamped Langevin equations. We coarse-grain an observed variable of a nonequilibrium driven system into a few discrete states and estimate a lower bound on the total EPR. As a model system, we use hair-cell bundle oscillations driven by molecular motors, such that the bundle tip position is observed, but the positions of the motors are hidden. In the observed variable space, the underlying driven process exhibits second-order semi-Markov statistics. The waiting time distributions (WTD), associated with transitions among the coarse-grained states, are non-exponential and convey the information on the broken time-reversal symmetry. By invoking the underlying time-irreversibility, we calculate a lower bound on the total EPR from the Kullback-Leibler divergence (KLD) between WTD. We show that the mean dwell-time asymmetry factor - the ratio between the mean dwell-times along the forward direction and the backward direction, can qualitatively measure the degree of broken time reversal symmetry and increases with finer spatial resolution. Finally, we apply our methodology to a continuous-time discrete Markov chain model, coarse-grained into a linear system exhibiting second-order semi-Markovian statistics, and demonstrate the estimation of a lower bound on the total EPR from irreversibility manifested only in the WTD.

Mathematical, Thermodynamical, and Experimental Necessity for Coarse Graining Empirical Densities and Currents in Continuous Space

We present general results on fluctuations and spatial correlations of the coarse-grained empirical density and current of Markovian diffusion in equilibrium or nonequilibrium steady states on all timescales. We unravel a deep connection between current fluctuations and generalized time-reversal symmetry, providing new insight into time-averaged observables. We highlight the essential role of coarse graining in space from mathematical, thermodynamical, and experimental points of view. Spatial coarse graining is required to uncover salient features of currents that break detailed balance, and a thermodynamically "optimal" coarse graining ensures the most precise inference of dissipation. Defined without coarse graining, the fluctuations of empirical density and current are proven to diverge on all timescales in dimensions higher than one, which has far-reaching consequences for the central-limit regime in continuous space. We apply the results to examples of irreversible diffusion. Our findings provide new intuition about time-averaged observables and allow for a more efficient analysis of single-molecule experiments.

Islet primary cilia motility controls insulin secretion

Primary cilia are specialized cell-surface organelles that mediate sensory perception and, in contrast to motile cilia and flagella, are thought to lack motility function. Here, we show that primary cilia in human and mouse pancreatic islets exhibit movement that is required for glucose-dependent insulin secretion. Islet primary cilia contain motor proteins conserved from those found in classic motile cilia, and their three-dimensional motion is dynein-driven and dependent on adenosine 5'-triphosphate and glucose metabolism. Inhibition of cilia motion blocks beta cell calcium influx and insulin secretion. Human beta cells have enriched ciliary gene expression, and motile cilia genes are altered in type 2 diabetes. Our findings redefine primary cilia as dynamic structures having both sensory and motile function and establish that pancreatic islet cilia movement plays a regulatory role in insulin secretion.

A unicellular walker controlled by a microtubule-based finite-state machine

Cells are complex biochemical systems whose behaviors emerge from interactions among myriad molecular components. Computation is often invoked as a general framework for navigating this cellular complexity. However, it is unclear how cells might embody computational processes such that the theories of computation, including finite-state machine models, could be productively applied. Here, we demonstrate finite-state-machine-like processing embodied in cells using the walking behavior of Euplotes eurystomus, a ciliate that walks across surfaces using fourteen motile appendages (cirri). We found that cellular walking entails regulated transitions among a discrete set of gait states. The set of observed transitions decomposes into a small group of high-probability, temporally irreversible transitions and a large group of low-probability, time-symmetric transitions, thus revealing stereotypy in the sequential patterns of state transitions. Simulations and experiments suggest that the sequential logic of the gait is functionally important. Taken together, these findings implicate a finite-state-machine-like process. Cirri are connected by microtubule bundles (fibers), and we found that the dynamics of cirri involved in different state transitions are associated with the structure of the fiber system. Perturbative experiments revealed that the fibers mediate gait coordination, suggesting a mechanical basis of gait control.

Decomposing the Local Arrow of Time in Interacting Systems

We show that the evidence for a local arrow of time, which is equivalent to the entropy production in thermodynamic systems, can be decomposed. In a system with many degrees of freedom, there is a term that arises from the irreversible dynamics of the individual variables, and then a series of non-negative terms contributed by correlations among pairs, triplets, and higher-order combinations of variables. We illustrate this decomposition on simple models of noisy logical computations, and then apply it to the analysis of patterns of neural activity in the retina as it responds to complex dynamic visual scenes. We find that neural activity breaks detailed balance even when the visual inputs do not, and that this irreversibility arises primarily from interactions between pairs of neurons.

Signatures of irreversibility in microscopic models of flocking

Flocking in d=2 is a genuine nonequilibrium phenomenon for which irreversibility is an essential ingredient. We study a class of minimal flocking models whose only source of irreversibility is self-propulsion and use the entropy production rate (EPR) to quantify the departure from equilibrium across their phase diagrams. The EPR is maximal in the vicinity of the order-disorder transition, where reshuffling of the interaction network is fast. We show that signatures of irreversibility come in the form of asymmetries in the steady-state distribution of the flock's microstates. These asymmetries occur as consequences of the time-reversal symmetry breaking in the considered self-propelled systems, independently of the interaction details. In the case of metric pairwise forces, they reduce to local asymmetries in the distribution of pairs of particles. This study suggests a possible use of pair asymmetries both to quantify the departure from equilibrium and to learn relevant information about aligning interaction potentials from data.

Eukaryotic gene regulation at equilibrium, or non?

Models of transcriptional regulation that assume equilibrium binding of transcription factors have been less successful at predicting gene expression from sequence in eukaryotes than in bacteria. This could be due to the non-equilibrium nature of eukaryotic regulation. Unfortunately, the space of possible non-equilibrium mechanisms is vast and predominantly uninteresting. The key question is therefore how this space can be navigated efficiently, to focus on mechanisms and models that are biologically relevant. In this review, we advocate for the normative role of theory-theory that prescribes rather than just describes-in providing such a focus. Theory should expand its remit beyond inferring mechanistic models from data, towards identifying non-equilibrium gene regulatory schemes that may have been evolutionarily selected, despite their energy consumption, because they are precise, reliable, fast, or otherwise outperform regulation at equilibrium. We illustrate our reasoning by toy examples for which we provide simulation code.

Emergence of local irreversibility in complex interacting systems

Living systems are fundamentally irreversible, breaking detailed balance and establishing an arrow of time. But how does the evident arrow of time for a whole system arise from the interactions among its multiple elements? We show that the local evidence for the arrow of time, which is the entropy production for thermodynamic systems, can be decomposed. First, it can be split into two components: an independent term reflecting the dynamics of individual elements and an interaction term driven by the dependencies among elements. Adapting tools from nonequilibrium physics, we further decompose the interaction term into contributions from pairs of elements, triplets, and higher-order terms. We illustrate our methods on models of cellular sensing and logical computations, as well as on patterns of neural activity in the retina as it responds to visual inputs. We find that neural activity can define the arrow of time even when the visual inputs do not, and that the dominant contribution to this breaking of detailed balance comes from interactions among pairs of neurons.

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.

Lack of Correlation between Landscape Geometry and Transition Rates

Biological cells can exist in a variety of distinct phenotypes, determined by the steady-state solutions of genetic networks governing their cell fate. A popular way of representing these states relies on the creation of landscape related to the relative occupation of these states. It is often assumed that this landscape offers direct information regarding the state-to-state transition rates, suggesting that these are related to barrier heights separating landscape minima. Here, we study a toggle triad network exhibiting multistability and directly demonstrate the lack of any direct correlation between properties of the landscape and corresponding transition rates.

Catapulting of topological defects through elasticity bands in active nematics

Active materials are those in which individual, uncoordinated local stresses drive the material out of equilibrium on a global scale. Examples of such assemblies can be seen across scales from schools of fish to the cellular cytoskeleton and underpin many important biological processes. Synthetic experiments that recapitulate the essential features of such active systems have been the object of study for decades as their simple rules allow us to elucidate the physical underpinnings of collective motion. One system of particular interest has been active nematic liquid crystals (LCs). Because of their well understood passive physics, LCs provide a rich platform to interrogate the effects of active stress. The flows and steady state structures that emerge in an active LCs have been understood to result from a competition between nematic elasticity and the local activity. However most investigations of such phenomena consider only the magnitude of the elastic resistance and not its peculiarities. Here we investigate a nematic liquid crystal and selectively change the ratio of the material's splay and bend elasticities. We show that increases in the nematic's bend elasticity specifically drives the material into an exotic steady state where elongated regions of acute bend distortion or "elasticity bands" dominate the structure and dynamics. We show that these bands strongly influence defect dynamics, including the rapid motion or "catapulting" along the disintegration of one of these bands thus converting bend distortion into defect transport. Thus, we report a novel dynamical state resultant from the competition between nematic elasticity and active stress.

Odd dynamics of living chiral crystals

Active crystals are highly ordered structures that emerge from the self-organization of motile objects, and have been widely studied in synthetic^1,2 and bacterial^3,4 active matter. Whether persistent  crystalline order can emerge  in groups of autonomously developing multicellular organisms is currently unknown. Here we show that swimming starfish embryos spontaneously assemble into chiral crystals that span thousands of spinning organisms and persist for tens of hours. Combining experiments, theory and simulations, we demonstrate that the formation, dynamics and dissolution of these living crystals are controlled by the hydrodynamic properties and the natural development of embryos. Remarkably, living chiral crystals exhibit self-sustained chiral oscillations as well as various unconventional deformation response behaviours recently predicted for odd elastic materials^5,6. Our results provide direct experimental evidence for how non-reciprocal interactions between autonomous multicellular components may facilitate non-equilibrium phases of chiral active matter.

Spectral fingerprints of nonequilibrium dynamics: The case of a Brownian gyrator

The same system can exhibit a completely different dynamical behavior when it evolves in equilibrium conditions or when it is driven out-of-equilibrium by, e.g., connecting some of its components to heat baths kept at different temperatures. Here we concentrate on an analytically solvable and experimentally relevant model of such a system-the so-called Brownian gyrator-a two-dimensional nanomachine that performs a systematic, on average, rotation around the origin under nonequilibrium conditions, while no net rotation takes place under equilibrium ones. On this example, we discuss a question whether it is possible to distinguish between two types of a behavior judging not upon the statistical properties of the trajectories of components but rather upon their respective spectral densities. The latter are widely used to characterize diverse dynamical systems and are routinely calculated from the data using standard built-in packages. From such a perspective, we inquire whether the power spectral densities possess some "fingerprint" properties specific to the behavior in nonequilibrium. We show that indeed one can conclusively distinguish between equilibrium and nonequilibrium dynamics by analyzing the cross-correlations between the spectral densities of both components in the short frequency limit, or from the spectral densities of both components evaluated at zero frequency. Our analytical predictions, corroborated by experimental and numerical results, open a new direction for the analysis of a nonequilibrium dynamics.

Observation of broken detailed balance in polymorphic transformation of bacterial flagellar filament

Living systems operate far from thermodynamic equilibrium, which usually manifests as broken detailed balance at the molecular scale. At larger scales with collective function of many molecules, the presence of non-equilibrium thermodynamics may not be evident. In bacterial motility, the switching dynamics of the flagellar rotary motor was recently discovered to be operating in non-equilibrium. However, the resulting motility pattern at the mesoscale, the run-and-tumble behavior, was normally considered to be a Poisson process that can be described by a two-state equilibrium model. Here, we studied the details of the run-and-tumble behavior by following the polymorphic transformation of the flagellar filaments, observing broken detailed balance that reveals its non-equilibrium nature. Evaluation of entropy production provided a direct measure of the lack of detailed balance, and a quantification of the rate of energy dissipation for bacterial run-and-tumble regulation.

Rectification and confinement of photokinetic bacteria in an optical feedback loop

Active particles can self-propel by exploiting locally available energy resources. When powered by light, these resources can be distributed with high resolution allowing spatio-temporal modulation of motility. Here we show that the random walks of light-driven bacteria are rectified when they swim in a structured light field that is obtained by a simple geometric transformation of a previous system snapshot. The obtained currents achieve an optimal value that we establish by general theoretical arguments. This optical feedback is used to gather and confine bacteria in high-density and high-activity regions that can be dynamically relocated and reconfigured. Moving away from the boundaries of these optically confined states, the density decays to zero in a few tens of micrometers, exhibiting steep exponential tails that suppress cell escape and ensure long-term stability. Our method is general and scalable, providing a versatile tool to produce localized and tunable active baths for microengineering applications and systematic studies of non-equilibrium phenomena in active systems.

 Light can be used to precisely modulate the speed of active particles in space and time. Here, the authors rectify and confine bacteria using an optical feedback loop that couples bacteria topast configurations.

State-space renormalization group theory of nonequilibrium reaction networks: Exact solutions for hypercubic lattices in arbitrary dimensions

Nonequilibrium reaction networks (NRNs) underlie most biological functions. Despite their diverse dynamic properties, NRNs share the signature characteristics of persistent probability fluxes and continuous energy dissipation even in the steady state. Dynamics of NRNs can be described at different coarse-grained levels. Our previous work showed that the apparent energy dissipation rate at a coarse-grained level follows an inverse power-law dependence on the scale of coarse-graining. The scaling exponent is determined by the network structure and correlation of stationary probability fluxes. However, it remains unclear whether and how the (renormalized) flux correlation varies with coarse-graining. Following Kadanoff's real space renormalization group (RG) approach for critical phenomena, we address this question by developing a state-space renormalization group theory for NRNs, which leads to an iterative RG equation for the flux correlation function. In square and hypercubic lattices, we solve the RG equation exactly and find two types of fixed point solutions. There is a family of nontrivial fixed points where the correlation exhibits power-law decay, characterized by a power exponent that can take any value within a continuous range. There is also a trivial fixed point where the correlation vanishes beyond the nearest neighbors. The power-law fixed point is stable if and only if the power exponent is less than the lattice dimension n. Consequently, the correlation function converges to the power-law fixed point only when the correlation in the fine-grained network decays slower than r^{-n} and to the trivial fixed point otherwise. If the flux correlation in the fine-grained network contains multiple stable solutions with different exponents, the RG iteration dynamics select the fixed point solution with the smallest exponent. The analytical results are supported by numerical simulations. We also discuss a possible connection between the RG flows of flux correlation with those of the Kosterlitz-Thouless transition.

Perspective: a stirring role for metabolism in cells

Based on recent findings indicating that metabolism might be governed by a limit on the rate at which cells can dissipate Gibbs energy, in this Perspective, we propose a new mechanism of how metabolic activity could globally regulate biomolecular processes in a cell. Specifically, we postulate that Gibbs energy released in metabolic reactions is used to perform work, allowing enzymes to self-propel or to break free from supramolecular structures. This catalysis-induced enzyme movement will result in increased intracellular motion, which in turn can compromise biomolecular functions. Once the increased intracellular motion has a detrimental effect on regulatory mechanisms, this will establish a feedback mechanism on metabolic activity, and result in the observed thermodynamic limit. While this proposed explanation for the identified upper rate limit on cellular Gibbs energy dissipation rate awaits experimental validation, it offers an intriguing perspective of how metabolic activity can globally affect biomolecular functions and will hopefully spark new research.

Materials science and mechanosensitivity of living matter

Living systems are composed of molecules that are synthesized by cells that use energy sources within their surroundings to create fascinating materials that have mechanical properties optimized for their biological function. Their functionality is a ubiquitous aspect of our lives. We use wood to construct furniture, bacterial colonies to modify the texture of dairy products and other foods, intestines as violin strings, bladders in bagpipes, and so on. The mechanical properties of these biological materials differ from those of other simpler synthetic elastomers, glasses, and crystals. Reproducing their mechanical properties synthetically or from first principles is still often unattainable. The challenge is that biomaterials often exist far from equilibrium, either in a kinetically arrested state or in an energy consuming active state that is not yet possible to reproduce de novo. Also, the design principles that form biological materials often result in nonlinear responses of stress to strain, or force to displacement, and theoretical models to explain these nonlinear effects are in relatively early stages of development compared to the predictive models for rubberlike elastomers or metals. In this Review, we summarize some of the most common and striking mechanical features of biological materials and make comparisons among animal, plant, fungal, and bacterial systems. We also summarize some of the mechanisms by which living systems develop forces that shape biological matter and examine newly discovered mechanisms by which cells sense and respond to the forces they generate themselves, which are resisted by their environment, or that are exerted upon them by their environment. Within this framework, we discuss examples of how physical methods are being applied to cell biology and bioengineering.

Microscopic Swarms: From Active Matter Physics to Biomedical and Environmental Applications

Microscopic swarms consisting of, e.g., active colloidal particles or microorganisms, display emergent behaviors not seen in equilibrium systems. They represent an emerging field of research that generates both fundamental scientific interest and practical technological value. This review seeks to unite the perspective of fundamental active matter physics and the perspective of practical applications of microscopic swarms. We first summarize experimental and theoretical results related to a few key aspects unique to active matter systems: the existence of long-range order, the prediction and observation of giant number fluctuations and motility-induced phase separation, and the exploration of the relations between information and order in the self-organizing patterns. Then we discuss microscopic swarms, particularly microrobotic swarms, from the perspective of applications. We introduce common methods to control and manipulate microrobotic swarms and summarize their potential applications in fields such as targeted delivery, in vivo imaging, biofilm removal, and wastewater treatment. We aim at bridging the gap between the community of active matter physics and the community of micromachines or microrobotics, and in doing so, we seek to inspire fruitful collaborations between the two communities.

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.

Quantifying brain state transition cost via Schrödinger Bridge

Quantifying brain state transition cost is a fundamental problem in systems neuroscience. Previous studies utilized network control theory to measure the cost by considering a neural system as a deterministic dynamical system. However, this approach does not capture the stochasticity of neural systems, which is important for accurately quantifying brain state transition cost. Here, we propose a novel framework based on optimal control in stochastic systems. In our framework, we quantify the transition cost as the Kullback-Leibler divergence from an uncontrolled transition path to the optimally controlled path, which is known as Schrödinger Bridge. To test its utility, we applied this framework to functional magnetic resonance imaging data from the Human Connectome Project and computed the brain state transition cost in cognitive tasks. We demonstrate correspondence between brain state transition cost and the difficulty of tasks. The results suggest that our framework provides a general theoretical tool for investigating cognitive functions from the viewpoint of transition cost.

Ultrarapid cryo-arrest of living cells on a microscope enables multiscale imaging of out-of-equilibrium molecular patterns

Imaging molecular patterns in cells by fluorescence micro- or nanoscopy has the potential to relate collective molecular behavior to cellular function. However, spatial and spectroscopic resolution is fundamentally limited by motional blur caused by finite photon fluxes and photobleaching. At physiological temperatures, photochemical reactivity does not only limit imaging at multiple scales but is also toxic to biochemical reactions that maintain cellular organization. Here, we present cryoprotectant-free ultrarapid cryo-arrest directly on a multimodal fluorescence microscope that preserves the out-of-equilibrium molecular organization of living cells. This allows the imaging of dynamic processes before cryo-arrest in combination with precise molecular pattern determination at multiple scales within the same cells under cryo-arrest. We both experimentally and theoretically show that ultrarapid cryo-arrest overcomes the fundamental resolution barrier imposed by motional blur and photochemical reactivity, enabling observation of native molecular distributions and reaction patterns that are not resolvable at physiological temperatures.

Label-Free Imaging of Nanoscale Displacements and Free-Energy Profiles of Focal Adhesions with Plasmonic Scattering Microscopy

Cell adhesion plays a critical role in cell communication, cell migration, cell proliferation, and integration of medical implants with tissues. Focal adhesions physically link the cell cytoskeleton to the extracellular matrix, but it remains challenging to image single focal adhesions directly. Here, we show that plasmonic scattering microscopy (PSM) can directly image the single focal adhesions in a label-free, real-time, and non-invasive manner with sub-micrometer spatial resolution. PSM is developed based on surface plasmon resonance (SPR) microscopy, and the evanescent illumination makes it immune to the interference of intracellular structures. Unlike the conventional SPR microscopy, PSM can provide a high signal-to-noise ratio and sub-micrometer spatial resolution for imaging the analytes with size down to a single-molecule level, thus allowing both the super-resolution lateral localization for measuring the nanoscale displacement and precise tracking of vertical distances between the analyte centroid and the sensor surface for analysis of free-energy profiles. PSM imaging of the RBL-2H3 cell with temporal resolution down to microseconds shows that the focal adhesions have random diffusion behaviors in addition to their directional movements during the antibody-mediated activation process. The free-energy mapping also shows a similar movement tendency, indicating that the cell may change its morphology upon varying the binding conditions of adhesive structures. PSM provides insights into the individual focal adhesion activities and can also serve as a promising tool for investigating the cell/surface interactions, such as cell capture and detection and tissue adhesive materials screening.

Broken detailed balance and entropy production in the human brain

To perform biological functions, living systems must break detailed balance by consuming energy and producing entropy. At microscopic scales, broken detailed balance enables a suite of molecular and cellular functions, including computations, kinetic proofreading, sensing, adaptation, and transportation. But do macroscopic violations of detailed balance enable higher-order biological functions, such as cognition and movement? To answer this question, we adapt tools from nonequilibrium statistical mechanics to quantify broken detailed balance in complex living systems. Analyzing neural recordings from hundreds of human subjects, we find that the brain violates detailed balance at large scales and that these violations increase with physical and cognitive exertion. Generally, we provide a flexible framework for investigating broken detailed balance at large scales in complex systems.

Living systems break detailed balance at small scales, consuming energy and producing entropy in the environment to perform molecular and cellular functions. However, it remains unclear how broken detailed balance manifests at macroscopic scales and how such dynamics support higher-order biological functions. Here we present a framework to quantify broken detailed balance by measuring entropy production in macroscopic systems. We apply our method to the human brain, an organ whose immense metabolic consumption drives a diverse range of cognitive functions. Using whole-brain imaging data, we demonstrate that the brain nearly obeys detailed balance when at rest, but strongly breaks detailed balance when performing physically and cognitively demanding tasks. Using a dynamic Ising model, we show that these large-scale violations of detailed balance can emerge from fine-scale asymmetries in the interactions between elements, a known feature of neural systems. Together, these results suggest that violations of detailed balance are vital for cognition and provide a general tool for quantifying entropy production in macroscopic systems.

Reversibility in nonequilibrium steady states as a measure of distance from equilibrium

From the detailed balance-like relation, we propose a measure, K^{*}, of a nonequilibrium steady-state (NESS) distance from equilibrium. We investigate in particular the NESS of a particle confined in a time-dependent harmonic potential of constant stiffness but with an ON-OFF state following a telegraph process. Experimental results coupled with simulations show that K^{*} increases at slow switching rates (far from equilibrium) and approaches to zero at equilibrium conditions. Thus, the steady-state distribution together with K^{*} fully characterizes a NESS.

Energy harvesting from anisotropic fluctuations

We consider a rudimentary model for a heat engine, known as the Brownian gyrator, that consists of an overdamped system with two degrees of freedom in an anisotropic temperature field. Whereas the hallmark of the gyrator is a nonequilibrium steady-state curl-carrying probability current that can generate torque, we explore the coupling of this natural gyrating motion with a periodic actuation potential for the purpose of extracting work. We show that path lengths traversed in the manifold of thermodynamic states, measured in a suitable Riemannian metric, represent dissipative losses, while area integrals of a work density quantify work being extracted. Thus, the maximal amount of work that can be extracted relates to an isoperimetric problem, trading off area against length of an encircling path. We derive an isoperimetric inequality that provides a universal bound on the efficiency of all cyclic operating protocols, and a bound on how fast a closed path can be traversed before it becomes impossible to extract positive work. The analysis presented provides guiding principles for building autonomous engines that extract work from anisotropic fluctuations.

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

The assessment of time irreversibility, i.e., of the lack of invariance of the statistical properties of a system under the operation of time reversal, is a topic steadily gaining attention within the research community. Irreversible dynamics have been found in many real-world systems, with alterations being connected to, for instance, pathologies in the human brain, heart and gait, or to inefficiencies in financial markets. Assessing irreversibility in time series is not an easy task, due to its many aetiologies and to the different ways it manifests in data. It is thus not surprising that several numerical methods have been proposed in the last decades, based on different principles and with different applications in mind. In this contribution we review the most important algorithmic solutions that have been proposed to test the irreversibility of time series, their underlying hypotheses, computational and practical limitations, and their comparative performance. We further provide an open-source software library that includes all tests here considered. As a final point, we show that "one size does not fit all", as tests yield complementary, and sometimes conflicting views to the problem; and discuss some future research avenues.

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.

Quantifying the Landscape of Decision Making From Spiking Neural Networks

The decision making function is governed by the complex coupled neural circuit in the brain. The underlying energy landscape provides a global picture for the dynamics of the neural decision making system and has been described extensively in the literature, but often as illustrations. In this work, we explicitly quantified the landscape for perceptual decision making based on biophysically-realistic cortical network with spiking neurons to mimic a two-alternative visual motion discrimination task. Under certain parameter regions, the underlying landscape displays bistable or tristable attractor states, which quantify the transition dynamics between different decision states. We identified two intermediate states: the spontaneous state which increases the plasticity and robustness of changes of minds and the "double-up" state which facilitates the state transitions. The irreversibility of the bistable and tristable switches due to the probabilistic curl flux demonstrates the inherent non-equilibrium characteristics of the neural decision system. The results of global stability of decision-making quantified by barrier height inferred from landscape topography and mean first passage time are in line with experimental observations. These results advance our understanding of the stochastic and dynamical transition mechanism of decision-making function, and the landscape and kinetic path approach can be applied to other cognitive function related problems (such as working memory) in brain networks.

Morphodynamics facilitate cancer cells to navigate 3D extracellular matrix

Cell shape is linked to cell function. The significance of cell morphodynamics, namely the temporal fluctuation of cell shape, is much less understood. Here we study the morphodynamics of MDA-MB-231 cells in type I collagen extracellular matrix (ECM). We systematically vary ECM physical properties by tuning collagen concentrations, alignment, and gelation temperatures. We find that morphodynamics of 3D migrating cells are externally controlled by ECM mechanics and internally modulated by Rho/ROCK-signaling. We employ machine learning to classify cell shape into four different morphological phenotypes, each corresponding to a distinct migration mode. As a result, we map cell morphodynamics at mesoscale into the temporal evolution of morphological phenotypes. We characterize the mesoscale dynamics including occurrence probability, dwell time and transition matrix at varying ECM conditions, which demonstrate the complex phenotype landscape and optimal pathways for phenotype transitions. In light of the mesoscale dynamics, we show that 3D cancer cell motility is a hidden Markov process whereby the step size distributions of cell migration are coupled with simultaneous cell morphodynamics. Morphological phenotype transitions also facilitate cancer cells to navigate non-uniform ECM such as traversing the interface between matrices of two distinct microstructures. In conclusion, we demonstrate that 3D migrating cancer cells exhibit rich morphodynamics that is controlled by ECM mechanics, Rho/ROCK-signaling, and regulate cell motility. Our results pave the way to the functional understanding and mechanical programming of cell morphodynamics as a route to predict and control 3D cell motility.

Emergent probability fluxes in confined microbial navigation

When the motion of a motile cell is observed closely, it appears erratic, and yet the combination of nonequilibrium forces and surfaces can produce striking examples of organization in microbial systems. While most of our current understanding is based on bulk systems or idealized geometries, it remains elusive how and at which length scale self-organization emerges in complex geometries. Here, using experiments and analytical and numerical calculations, we study the motion of motile cells under controlled microfluidic conditions and demonstrate that probability flux loops organize active motion, even at the level of a single cell exploring an isolated compartment of nontrivial geometry. By accounting for the interplay of activity and interfacial forces, we find that the boundary's curvature determines the nonequilibrium probability fluxes of the motion. We theoretically predict a universal relation between fluxes and global geometric properties that is directly confirmed by experiments. Our findings open the possibility to decipher the most probable trajectories of motile cells and may enable the design of geometries guiding their time-averaged motion.

Out-of-Equilibrium Biophysical Chemistry: The Case for Multidimensional, Integrated Single-Molecule Approaches

Out-of-equilibrium processes are ubiquitous across living organisms and all structural hierarchies of life. At the molecular scale, out-of-equilibrium processes (for example, enzyme catalysis, gene regulation, and motor protein functions) cause biological macromolecules to sample an ensemble of conformations over a wide range of time scales. Quantifying and conceptualizing the structure–dynamics to function relationship is challenging because continuously evolving multidimensional energy landscapes are necessary to describe nonequilibrium biological processes in biological macromolecules. In this perspective, we explore the challenges associated with state-of-the-art experimental techniques to understanding biological macromolecular function. We argue that it is time to revisit how we probe and model functional out-of-equilibrium biomolecular dynamics. We suggest that developing integrated single-molecule multiparametric force–fluorescence instruments and using advanced molecular dynamics simulations to study out-of-equilibrium biomolecules will provide a path towards understanding the principles of and mechanisms behind the structure–dynamics to function paradigm in biological macromolecules.

RNA polymerase II clusters form in line with surface condensation on regulatory chromatin

It is essential for cells to control which genes are transcribed into RNA. In eukaryotes, two major control points are recruitment of RNA polymerase II (Pol II) into a paused state, and subsequent pause release toward transcription. Pol II recruitment and pause release occur in association with macromolecular clusters, which were proposed to be formed by a liquid-liquid phase separation mechanism. How such a phase separation mechanism relates to the interaction of Pol II with DNA during recruitment and transcription, however, remains poorly understood. Here, we use live and super-resolution microscopy in zebrafish embryos to reveal Pol II clusters with a large variety of shapes, which can be explained by a theoretical model in which regulatory chromatin regions provide surfaces for liquid-phase condensation at concentrations that are too low for canonical liquid-liquid phase separation. Model simulations and chemical perturbation experiments indicate that recruited Pol II contributes to the formation of these surface-associated condensates, whereas elongating Pol II is excluded from these condensates and thereby drives their unfolding.

Nonequilibrium brain dynamics as a signature of consciousness

The cognitive functions of human and nonhuman primates rely on the dynamic interplay of distributed neural assemblies. As such, it seems unlikely that cognition can be supported by macroscopic brain dynamics at the proximity of equilibrium. We confirmed this hypothesis by investigating electrocorticography data from nonhuman primates undergoing different states of unconsciousness (sleep, and anesthesia with propofol, ketamine, and ketamine plus medetomidine), and functional magnetic resonance imaging data from humans, both during deep sleep and under propofol anesthesia. Systematically, all states of reduced consciousness unfolded at higher proximity to equilibrium compared to conscious wakefulness, as demonstrated by the computation of entropy production and the curl of probability flux in phase space. Our results establish nonequilibrium macroscopic brain dynamics as a robust signature of consciousness, opening the way for the characterization of cognition and awareness using tools from statistical mechanics.

Long range correlations and slow time scales in a boundary driven granular model

We consider a velocity field with linear viscous interactions defined on a one dimensional lattice. Brownian baths with different parameters can be coupled to the boundary sites and to the bulk sites, determining different kinds of non-equilibrium steady states or free-cooling dynamics. Analytical results for spatial and temporal correlations are provided by analytical diagonalisation of the system's equations in the infinite size limit. We demonstrate that spatial correlations are scale-free and time-scales become exceedingly long when the system is driven only at the boundaries. On the contrary, in the case a bath is coupled to the bulk sites too, an exponential correlation decay is found with a finite characteristic length. This is also true in the free cooling regime, but in this case the correlation length grows diffusively in time. We discuss the crucial role of boundary driving for long-range correlations and slow time-scales, proposing an analogy between this simplified dynamical model and dense vibro-fluidized granular materials. Several generalizations and connections with the statistical physics of active matter are also suggested.

Phase behavior and surface tension of soft active Brownian particles

We study quasi two-dimensional, monodisperse systems of active Brownian particles (ABPs) for a range of activities, stiffnesses, and densities. We develop a microscopic, analytical method for predicting the dense phase structure formed after motility-induced phase separation (MIPS) has occurred, including the dense cluster's area fraction, interparticle pressure, and radius. Our predictions are in good agreement with our Brownian dynamics simulations. We, then, derive a continuum model to investigate the relationship between the predicted interparticle pressure, the swim pressure, and the macroscopic pressure in the momentum equation. We find that formulating the point-wise macroscopic pressure as the interparticle pressure and modeling the particle activity through a spatially variant body force - as opposed to a volume-averaged swim pressure - results in consistent predictions of pressure in both the continuum model and the microscopic theory. This formulation of pressure also results in nearly zero surface tension for the phase separated domains, irrespective of activity, stiffness, and area fraction. Furthermore, using Brownian dynamics simulations and our continuum model, we showed that both the interface width and surface tension, are intrinsic characteristics of the system. On the other hand, if we were to exclude the body force induced by activity, we find that the resulting surface tension values are linearly dependent on the size of the simulation, in contrast to the statistical mechanical definition of surface tension.

Physical bioenergetics: Energy fluxes, budgets, and constraints in cells

Cells are the basic units of all living matter which harness the flow of energy to drive the processes of life. While the biochemical networks involved in energy transduction are well-characterized, the energetic costs and constraints for specific cellular processes remain largely unknown. In particular, what are the energy budgets of cells? What are the constraints and limits energy flows impose on cellular processes? Do cells operate near these limits, and if so how do energetic constraints impact cellular functions? Physics has provided many tools to study nonequilibrium systems and to define physical limits, but applying these tools to cell biology remains a challenge. Physical bioenergetics, which resides at the interface of nonequilibrium physics, energy metabolism, and cell biology, seeks to understand how much energy cells are using, how they partition this energy between different cellular processes, and the associated energetic constraints. Here we review recent advances and discuss open questions and challenges in physical bioenergetics.

Effect of boundaries on noninteracting weakly active particles in different geometries

We study analytically how noninteracting weakly active particles, for which passive Brownian diffusion cannot be neglected and activity can be treated perturbatively, distribute and behave near boundaries in various geometries. In particular, we develop a perturbative approach for the model of active particles driven by an exponentially correlated random force (active Ornstein-Uhlenbeck particles). This approach involves a relatively simple expansion of the distribution in powers of the Péclet number and in terms of Hermite polynomials. We use this approach to cleanly formulate boundary conditions, which allows us to study weakly active particles in several geometries: confinement by a single wall or between two walls in 1D, confinement in a circular or wedge-shaped region in 2D, motion near a corrugated boundary, and, finally, absorption onto a sphere. We consider how quantities such as the density, pressure, and flow of the active particles change as we gradually increase the activity away from a purely passive system. These results for the limit of weak activity help us gain insight into how active particles behave in the presence of various types of boundaries.

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.

Inertial effects on the Brownian gyrator

The recent interest into the Brownian gyrator has been confined chiefly to the analysis of Brownian dynamics both in theory and experiment despite the applicability of general cases with definite mass. Considering mass explicitly in the solution of the Fokker-Planck equation and Langevin dynamics simulations, we investigate how inertia can change the dynamics and energetics of the Brownian gyrator. In the Langevin model, the inertia reduces the nonequilibrium effects by diminishing the declination of the probability density function and the mean of a specific angular momentum, j_{θ}, as a measure of rotation. Another unique feature of the Langevin description is that rotation is maximized at a particular anisotropy while the stability of the rotation is minimized at a particular anisotropy or mass. Our results suggest that the Langevin dynamics description of the Brownian gyrator is intrinsically different from that with Brownian dynamics. In addition, j_{θ} is proven to be essential and convenient for estimating stochastic energetics such as heat currents and entropy production even in the underdamped regime.

Changes in geometrical aspects of a simple model of cilia synchronization control the dynamical state, a possible mechanism for switching of swimming gaits in microswimmers

Active oscillators, with purely hydrodynamic coupling, are useful simple models to understand various aspects of motile cilia synchronization. Motile cilia are used by microorganisms to swim and to control the flow fields in their surroundings; the patterns observed in cilia carpets can be remarkably complex, and can be changed over time by the organism. It is often not known to what extent the coupling between cilia is due to just hydrodynamic forces, and neither is it known if it is biological or physical triggers that can change the dynamical collective state. Here we treat this question from a very simplified point of view. We describe three possible mechanisms that enable a switch in the dynamical state, in a simple scenario of a chain of oscillators. We find that shape-change provides the most consistent strategy to control collective dynamics, but also imposing small changes in frequency produces some unique stable states. Demonstrating these effects in the abstract minimal model proves that these could be possible explanations for gait switching seen in ciliated micro organisms like Paramecium and others. Microorganisms with many cilia could in principle be taking advantage of hydrodynamic coupling, to switch their swimming gait through either a shape change that manifests in decreased coupling between groups of cilia, or alterations to the beat style of a small subset of the cilia.

Spatial distributions of nonconservatively interacting particles

Certain types of active systems can be treated as an equilibrium system with excess nonconservative forces driving some of the microscopic degrees of freedom. We derive results for how many particles having both conservative and nonconservative forces will behave. Treating nonconservative forces perturbatively, we show how the probability distribution of the microscopic degrees of freedom is modified from the Boltzmann distribution. We then derive approximate forms of this distribution through analyzing the nature of our perturbations. We compare the perturbative expansion for the microscopic probability distribution to an exactly solvable active system. Finally, we consider how the approximate forms for the microscopic distributions we have derived lead to different macroscopic states when coarse grained for two different kinds of systems, a collection of motile particles, and a system where nonconservative forces are applied in space. In the former, we are able to show that nonconservative forces lead to an effective attractive interaction between motile particles, and in the latter we note that by introducing nonconservative interactions between particles we modify densities through extra terms which couple to surfaces. In this way, we are able to recast certain active problems as the statistical mechanics of nonconservative forces.

Origins of eukaryotic excitability

All living cells interact dynamically with a constantly changing world. Eukaryotes, in particular, evolved radically new ways to sense and react to their environment. These advances enabled new and more complex forms of cellular behaviour in eukaryotes, including directional movement, active feeding, mating, and responses to predation. But what are the key events and innovations during eukaryogenesis that made all of this possible? Here we describe the ancestral repertoire of eukaryotic excitability and discuss five major cellular innovations that enabled its evolutionary origin. The innovations include a vastly expanded repertoire of ion channels, the emergence of cilia and pseudopodia, endomembranes as intracellular capacitors, a flexible plasma membrane and the relocation of chemiosmotic ATP synthesis to mitochondria, which liberated the plasma membrane for more complex electrical signalling involved in sensing and reacting. We conjecture that together with an increase in cell size, these new forms of excitability greatly amplified the degrees of freedom associated with cellular responses, allowing eukaryotes to vastly outperform prokaryotes in terms of both speed and accuracy. This comprehensive new perspective on the evolution of excitability enriches our view of eukaryogenesis and emphasizes behaviour and sensing as major contributors to the success of eukaryotes. This article is part of the theme issue 'Basal cognition: conceptual tools and the view from the single cell'.

Dynamics of a collection of active particles on a two-dimensional periodic undulated surface

We study the dynamics of circular disk-shaped active particles on a two-dimensional periodic undulated surface. Each particle has an internal energy mechanism which is modeled by an active friction force and it is controlled by an activity parameter [Formula: see text]. It acts as negative friction if the speed of the particle is smaller than [Formula: see text] and normal friction otherwise. Surface undulation is modeled by the periodic undulation of fixed amplitude and wavelength. The dynamics of the particle is studied for different activities and surface undulations (SU). Three types of particle dynamic is observed on varying activity and SU: confined, early time subdiffusion to diffusion and super diffusion to late time diffusion. An effective equilibrium is established by showing the Green-Kubo relation between the effective diffusivity and the velocity auto-correlation function for all activities and small SU.

Inverse Power Law Scaling of Energy Dissipation Rate in Nonequilibrium Reaction Networks

The energy dissipation rate in a nonequilibrium reaction system can be determined by the reaction rates in the underlying reaction network. By developing a coarse-graining process in state space and a corresponding renormalization procedure for reaction rates, we find that energy dissipation rate has an inverse power-law dependence on the number of microscopic states in a coarse-grained state. The dissipation scaling law requires self-similarity of the underlying network, and the scaling exponent depends on the network structure and the probability flux correlation. Existence of the inverse dissipation scaling law is shown in realistic biochemical systems such as biochemical oscillators and microtubule-kinesin active flow systems.

Modeling Cell Energy Metabolism as Weighted Networks of Non-autonomous Oscillators

Networks of oscillating processes are a common occurrence in living systems. This is as true as anywhere in the energy metabolism of individual cells. Exchanges of molecules and common regulation operate throughout the metabolic processes of glycolysis and oxidative phosphorylation, making the consideration of each of these as a network a natural step. Oscillations are similarly ubiquitous within these processes, and the frequencies of these oscillations are never truly constant. These features make this system an ideal example with which to discuss an alternative approach to modeling living systems, which focuses on their thermodynamically open, oscillating, non-linear and non-autonomous nature. We implement this approach in developing a model of non-autonomous Kuramoto oscillators in two all-to-all weighted networks coupled to one another, and themselves driven by non-autonomous oscillators. Each component represents a metabolic process, the networks acting as the glycolytic and oxidative phosphorylative processes, and the drivers as glucose and oxygen supply. We analyse the effect of these features on the synchronization dynamics within the model, and present a comparison between this model, experimental data on the glycolysis of HeLa cells, and a comparatively mainstream model of this experiment. In the former, we find that the introduction of oscillator networks significantly increases the proportion of the model's parameter space that features some form of synchronization, indicating a greater ability of the processes to resist external perturbations, a crucial behavior in biological settings. For the latter, we analyse the oscillations of the experiment, finding a characteristic frequency of 0.01–0.02 Hz. We further demonstrate that an output of the model comparable to the measurements of the experiment oscillates in a manner similar to the measured data, achieving this with fewer parameters and greater flexibility than the comparable model.

Non-Markovian data-driven modeling of single-cell motility

Trajectories of human breast cancer cells moving on one-dimensional circular tracks are modeled by the non-Markovian version of the Langevin equation that includes an arbitrary memory function. When averaged over cells, the velocity distribution exhibits spurious non-Gaussian behavior, while single cells are characterized by Gaussian velocity distributions. Accordingly, the data are described by a linear memory model which includes different random walk models that were previously used to account for various aspects of cell motility such as migratory persistence, non-Markovian effects, colored noise, and anomalous diffusion. The memory function is extracted from the trajectory data without restrictions or assumptions, thus making our approach truly data driven, and is used for unbiased single-cell comparison. The cell memory displays time-delayed single-exponential negative friction, which clearly distinguishes cell motion from the simple persistent random walk model and suggests a regulatory feedback mechanism that controls cell migration. Based on the extracted memory function we formulate a generalized exactly solvable cell migration model which indicates that negative friction generates cell persistence over long timescales. The nonequilibrium character of cell motion is investigated by mapping the non-Markovian Langevin equation with memory onto a Markovian model that involves a hidden degree of freedom and is equivalent to the underdamped active Ornstein-Uhlenbeck process.