The energy-speed-accuracy tradeoff in sensory adaptation

E. coli do not count single molecules

Organisms must perform sensory-motor behaviors to survive. What bounds or constraints limit behavioral performance? Previously, we found that the gradient-climbing speed of a chemotaxing Escherichia coli is near a bound set by the limited information they acquire from their chemical environments (1). Here we ask what limits their sensory accuracy. Past theoretical analyses have shown that the stochasticity of single molecule arrivals sets a fundamental limit on the precision of chemical sensing (2). Although it has been argued that bacteria approach this limit, direct evidence is lacking. Here, using information theory and quantitative experiments, we find that E. coli's chemosensing is not limited by the physics of particle counting. First, we derive the physical limit on the behaviorally-relevant information that any sensor can get about a changing chemical concentration, assuming that every molecule arriving at the sensor is recorded. Then, we derive and measure how much information E. coli's signaling pathway encodes during chemotaxis. We find that E. coli encode two orders of magnitude less information than an ideal sensor limited only by shot noise in particle arrivals. These results strongly suggest that constraints other than particle arrival noise limit E. coli's sensory fidelity.

A differentiable Gillespie algorithm for simulating chemical kinetics, parameter estimation, and designing synthetic biological circuits

The Gillespie algorithm is commonly used to simulate and analyze complex chemical reaction networks. Here, we leverage recent breakthroughs in deep learning to develop a fully differentiable variant of the Gillespie algorithm. The differentiable Gillespie algorithm (DGA) approximates discontinuous operations in the exact Gillespie algorithm using smooth functions, allowing for the calculation of gradients using backpropagation. The DGA can be used to quickly and accurately learn kinetic parameters using gradient descent and design biochemical networks with desired properties. As an illustration, we apply the DGA to study stochastic models of gene promoters. We show that the DGA can be used to: (i) successfully learn kinetic parameters from experimental measurements of mRNA expression levels from two distinct E. coli promoters and (ii) design nonequilibrium promoter architectures with desired input-output relationships. These examples illustrate the utility of the DGA for analyzing stochastic chemical kinetics, including a wide variety of problems of interest to synthetic and systems biology.

Governing principles of transcriptional logic out of equilibrium

To survive, adapt, and develop, cells respond to external and internal stimuli by tightly regulating transcription. Transcriptional regulation involves the combinatorial binding of a repertoire of transcription factors to DNA, which often results in switch-like binary outputs akin to Boolean logic gates. Recent experimental studies have demonstrated that in eukaryotes, transcription factor binding to DNA often involves energy expenditure, thereby driving the system out of equilibrium. The governing principles of transcriptional logic operations out of equilibrium remain unexplored. Here, we employ a simple two-input, single-locus model of transcription that can accommodate both equilibrium and nonequilibrium mechanisms. Using this model, we find that nonequilibrium regimes can give rise to all the logic operations accessible in equilibrium. Strikingly, energy expenditure alters the regulatory function of the two transcription factors in a mutually exclusive manner. This allows for the emergence of new logic operations that are inaccessible in equilibrium. Overall, our results show that energy expenditure can expand the range of cellular decision-making without the need for more complex promoter architectures.

Resource allocation in mammalian systems

Cells execute biological functions to support phenotypes such as growth, migration, and secretion. Complementarily, each function of a cell has resource costs that constrain phenotype. Resource allocation by a cell allows it to manage these costs and optimize their phenotypes. In fact, the management of resource constraints (e.g., nutrient availability, bioenergetic capacity, and macromolecular machinery production) shape activity and ultimately impact phenotype. In mammalian systems, quantification of resource allocation provides important insights into higher-order multicellular functions; it shapes intercellular interactions and relays environmental cues for tissues to coordinate individual cells to overcome resource constraints and achieve population-level behavior. Furthermore, these constraints, objectives, and phenotypes are context-dependent, with cells adapting their behavior according to their microenvironment, resulting in distinct steady-states. This review will highlight the biological insights gained from probing resource allocation in mammalian cells and tissues.

Role of Topology in Relaxation of One-Dimensional Stochastic Processes

Stochastic processes are commonly used models to describe dynamics of a wide variety of nonequilibrium phenomena ranging from electrical transport to biological motion. The transition matrix describing a stochastic process can be regarded as a non-Hermitian Hamiltonian. Unlike general non-Hermitian systems, the conservation of probability imposes additional constraints on the transition matrix, which can induce unique topological phenomena. Here, we reveal the role of topology in relaxation phenomena of classical stochastic processes. Specifically, we define a winding number that is related to topology of stochastic processes and show that it predicts the existence of a spectral gap that characterizes the relaxation time. Then, we numerically confirm that the winding number corresponds to the system-size dependence of the relaxation time and the characteristic transient behavior. One can experimentally realize such topological phenomena in magnetotactic bacteria and cell adhesions.

Machine learning at the mesoscale: A computation-dissipation bottleneck

The cost of information processing in physical systems calls for a trade-off between performance and energetic expenditure. Here we formulate and study a computation-dissipation bottleneck in mesoscopic systems used as input-output devices. Using both real data sets and synthetic tasks, we show how nonequilibrium leads to enhanced performance. Our framework sheds light on a crucial compromise between information compression, input-output computation and dynamic irreversibility induced by nonreciprocal interactions.

Violations of the fluctuation-dissipation theorem reveal distinct nonequilibrium dynamics of brain states

The brain is a nonequilibrium system whose dynamics change in different brain states, such as wakefulness and deep sleep. Thermodynamics provides the tools for revealing these nonequilibrium dynamics. We used violations of the fluctuation-dissipation theorem to describe the hierarchy of nonequilibrium dynamics associated with different brain states. Together with a whole-brain model fitted to empirical human neuroimaging data, and deriving the appropriate analytical expressions, we were able to capture the deviation from equilibrium in different brain states that arises from asymmetric interactions and hierarchical organization.

Nonequilibrium Effects on Information Recoverability of the Noisy Channels

We investigated the impact of nonequilibrium conditions on the transmission and recovery of information through noisy channels. By measuring the recoverability of messages from an information source, we demonstrate that the ability to recover information is connected to the nonequilibrium behavior of the information flow, particularly in terms of sequential information transfer. We discovered that the mathematical equivalence of information recoverability and entropy production characterizes the dissipative nature of information transfer. Our findings show that both entropy production (or recoverability) and mutual information increase monotonically with the nonequilibrium strength of information dynamics. These results suggest that the nonequilibrium dissipation cost can enhance the recoverability of noise messages and improve the quality of information transfer. Finally, we propose a simple model to test our conclusions and found that the numerical results support our findings.

A nonequilibrium allosteric model for receptor-kinase complexes: The role of energy dissipation in chemotaxis signaling

The Escherichia coli chemotaxis signaling pathway has served as a model system for the adaptive sensing of environmental signals by large protein complexes. The chemoreceptors control the kinase activity of CheA in response to the extracellular ligand concentration and adapt across a wide concentration range by undergoing methylation and demethylation. Methylation shifts the kinase response curve by orders of magnitude in ligand concentration while incurring a much smaller change in the ligand binding curve. Here, we show that the disproportionate shift in binding and kinase response is inconsistent with equilibrium allosteric models. To resolve this inconsistency, we present a nonequilibrium allosteric model that explicitly includes the dissipative reaction cycles driven by adenosine triphosphate (ATP) hydrolysis. The model successfully explains all existing joint measurements of ligand binding, receptor conformation, and kinase activity for both aspartate and serine receptors. Our results suggest that the receptor complex acts as an enzyme: Receptor methylation modulates the ON-state kinetics of the kinase (e.g., phosphorylation rate), while ligand binding controls the equilibrium balance between kinase ON/OFF states. Furthermore, sufficient energy dissipation is responsible for maintaining and enhancing the sensitivity range and amplitude of the kinase response. We demonstrate that the nonequilibrium allosteric model is broadly applicable to other sensor-kinase systems by successfully fitting previously unexplained data from the DosP bacterial oxygen-sensing system. Overall, this work provides a nonequilibrium physics perspective on cooperative sensing by large protein complexes and opens up research directions for understanding their microscopic mechanisms through simultaneous measurements and modeling of ligand binding and downstream responses.

Trade-offs between cost and information in cellular prediction

Living cells can leverage correlations in environmental fluctuations to predict the future environment and mount a response ahead of time. To this end, cells need to encode the past signal into the output of the intracellular network from which the future input is predicted. Yet, storing information is costly while not all features of the past signal are equally informative on the future input signal. Here, we show for two classes of input signals that cellular networks can reach the fundamental bound on the predictive information as set by the information extracted from the past signal: Push-pull networks can reach this information bound for Markovian signals, while networks that take a temporal derivative can reach the bound for predicting the future derivative of non-Markovian signals. However, the bits of past information that are most informative about the future signal are also prohibitively costly. As a result, the optimal system that maximizes the predictive information for a given resource cost is, in general, not at the information bound. Applying our theory to the chemotaxis network of Escherichia coli reveals that its adaptive kernel is optimal for predicting future concentration changes over a broad range of background concentrations, and that the system has been tailored to predicting these changes in shallow gradients.

Topologically constrained fluctuations and thermodynamics regulate nonequilibrium response

The limits on a system's response to external perturbations inform our understanding of how physical properties can be shaped by microscopic characteristics. Here, we derive constraints on the steady-state nonequilibrium response of physical observables in terms of the topology of the microscopic state space and the strength of thermodynamic driving. Notably, evaluation of these limits requires no kinetic information beyond the state-space structure. When applied to models of receptor binding, we find that sensitivity is bounded by the steepness of a Hill function with a Hill coefficient enhanced by the chemical driving beyond the structural equilibrium limit.

Noise properties of adaptation-conferring biochemical control modules

A key goal of synthetic biology is to develop functional biochemical modules with network-independent properties. Antithetic integral feedback (AIF) is a recently developed control module in which two control species perfectly annihilate each other's biological activity. The AIF module confers robust perfect adaptation to the steady-state average level of a controlled intracellular component when subjected to sustained perturbations. Recent work has suggested that such robustness comes at the unavoidable price of increased stochastic fluctuations around average levels. We present theoretical results that support and quantify this trade-off for the commonly analyzed AIF variant in the idealized limit with perfect annihilation. However, we also show that this trade-off is a singular limit of the control module: Even minute deviations from perfect adaptation allow systems to achieve effective noise suppression as long as cells can pay the corresponding energetic cost. We further show that a variant of the AIF control module can achieve significant noise suppression even in the idealized limit with perfect adaptation. This atypical configuration may thus be preferable in synthetic biology applications.

Quantitative modeling of EGF receptor ligand discrimination via internalization proofreading

The epidermal growth factor receptor (EGFR) is a central regulator of cell physiology that is stimulated by multiple distinct ligands. Although ligands bind to EGFR while the receptor is exposed on the plasma membrane, EGFR incorporation into endosomes following receptor internalization is an important aspect of EGFR signaling, with EGFR internalization behavior dependent upon the type of ligand bound. We develop quantitative modeling for EGFR recruitment to and internalization from clathrin domains, focusing on how internalization competes with ligand unbinding from EGFR. We develop two model versions: a kinetic model with EGFR behavior described as transitions between discrete states and a spatial model with EGFR diffusion to circular clathrin domains. We find that a combination of spatial and kinetic proofreading leads to enhanced EGFR internalization ratios in comparison to unbinding differences between ligand types. Various stages of the EGFR internalization process, including recruitment to and internalization from clathrin domains, modulate the internalization differences between receptors bound to different ligands. Our results indicate that following ligand binding, EGFR may encounter multiple clathrin domains before successful recruitment and internalization. The quantitative modeling we have developed describes competition between EGFR internalization and ligand unbinding and the resulting proofreading.

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.

Universal energy-accuracy tradeoffs in nonequilibrium cellular sensing

We combine stochastic thermodynamics, large deviation theory, and information theory to derive fundamental limits on the accuracy with which single cell receptors can estimate external concentrations. As expected, if the estimation is performed by an ideal observer of the entire trajectory of receptor states, then no energy consuming nonequilibrium receptor that can be divided into bound and unbound states can outperform an equilibrium two-state receptor. However, when the estimation is performed by a simple observer that measures the fraction of time the receptor is bound, we derive a fundamental limit on the accuracy of general nonequilibrium receptors as a function of energy consumption. We further derive and exploit explicit formulas to numerically estimate a Pareto-optimal tradeoff between accuracy and energy. We find this tradeoff can be achieved by nonuniform ring receptors with a number of states that necessarily increases with energy. Our results yield a thermodynamic uncertainty relation for the time a physical system spends in a pool of states and generalize the classic Berg-Purcell limit [H. C. Berg and E. M. Purcell, Biophys. J. 20, 193 (1977)0006-349510.1016/S0006-3495(77)85544-6] on cellular sensing along multiple dimensions.

Perspectives on adaptive dynamical systems

Adaptivity is a dynamical feature that is omnipresent in nature, socio-economics, and technology. For example, adaptive couplings appear in various real-world systems, such as the power grid, social, and neural networks, and they form the backbone of closed-loop control strategies and machine learning algorithms. In this article, we provide an interdisciplinary perspective on adaptive systems. We reflect on the notion and terminology of adaptivity in different disciplines and discuss which role adaptivity plays for various fields. We highlight common open challenges and give perspectives on future research directions, looking to inspire interdisciplinary approaches.

Nearly maximal information gain due to time integration in central dogma reactions

Living cells process information about their environment through the central dogma processes of transcription and translation, which drive the cellular response to stimuli. Here, we study the transfer of information from environmental input to the transcript and protein expression levels. Evaluation of both experimental and analogous simulation data reveals that transcription and translation are not two simple information channels connected in series. Instead, we demonstrate that the central dogma reactions often create a time-integrating information channel, where the translation channel receives and integrates multiple outputs from the transcription channel. This information channel model of the central dogma provides new information-theoretic selection criteria for the central dogma rate constants. Using the data for four well-studied species we show that their central dogma rate constants achieve information gain because of time integration while also keeping the loss because of stochasticity in translation relatively low (<0.5 bits).

Cascade-enhanced transport efficiency of biochemical systems

Recent developments in nonequilibrium thermodynamics, known as thermodynamic uncertainty relations, limit the system's accuracy by the amount of free-energy consumption. A transport efficiency, which can be used to characterize the capacity to control the fluctuation by means of energy cost, is a direct result of the thermodynamic uncertainty relation. According to our previous research, biochemical systems consume much lower energy cost by noise-induced oscillations to keep almost equal efficiency to maintain precise processes than that by normal oscillations. Here, we demonstrate that the performance of noise-induced oscillations propagating can be further improved through a cascade reaction mechanism. It has been discovered that it is possible to considerably enhance the transport efficiency of the biochemical reactions attained at the terminal cell, allowing the cell to use the cascade reaction mechanism to operate more precisely and efficiently. Moreover, an optimal reaction coupling strength has been predicted to maximize the transport efficiency of the terminal cell, uncovering a concrete design strategy for biochemical systems. By using the local mean field approximation, we have presented an analytical framework by extending the stochastic normal form equation to the system perturbed by external signals, providing an explanation of the optimal coupling strength.

Type-IV pili tune an adhesion-migration trade-off during surface colonization of Pseudomonas aeruginosa

Bacterial pathogenicity relies on both firm surface adhesion and cell dissemination. How twitching bacteria resolve the fundamental contradiction between adhesion and migration is unknown. To address this question, we employ live-cell imaging of type-IV pili (T4P) and therewith construct a comprehensive mathematical model of Pseudomonas aeruginosa migration. The data show that only 10% to 50% of T4P bind to substrates and contribute to migration through random extension and retraction. Individual T4P do not display a measurable sensory response to surfaces, but their number increases on cellular surface contact. Attachment to surfaces is mediated, besides T4P, by passive adhesive forces acting on the cell body. Passive adhesions slow down cell migration and result in local random motion on short time scales, which is followed by directionally persistent, superdiffusive motion on longer time scales. Moreover, passive adhesions strongly enhance surface attachment under shear flow. Δ pilA mutants, which produce no T4P, robustly stick to surfaces under shear flow. In contrast, rapidly migrating Δ pilH cells, which produce an excessive number of T4P, are easily detached by shear. Wild-type cells sacrifice migration speed for robust surface attachment by maintaining a low number of active pili. The different cell strains pertain to disjunct regimes in a generic adhesion-migration trait space. Depending on the nature of the adhesion structures, adhesion and migration are either compatible or a trade-off is required for efficient bacterial surface colonization under different conditions.

Phenotypic heterogeneity in the bacterial oxidative stress response is driven by cell-cell interactions

Genetically identical bacterial cells commonly display different phenotypes. This phenotypic heterogeneity is well known for stress responses, where it is often explained as bet hedging against unpredictable environmental threats. Here, we explore phenotypic heterogeneity in a major stress response of Escherichia coli and find it has a fundamentally different basis. We characterize the response of cells exposed to hydrogen peroxide (H2O2) stress in a microfluidic device under constant growth conditions. A machine-learning model reveals that phenotypic heterogeneity arises from a precise and rapid feedback between each cell and its immediate environment. Moreover, we find that the heterogeneity rests upon cell-cell interaction, whereby cells shield each other from H2O2 via their individual stress responses. Our work shows how phenotypic heterogeneity in bacterial stress responses can emerge from short-range cell-cell interactions and result in a collective phenotype that protects a large proportion of the population.

Kinetic Asymmetry versus Dissipation in the Evolution of Chemical Systems as Exemplified by Single Enzyme Chemotaxis

Single enzyme chemotaxis is a phenomenon by which a nonequilibrium spatial distribution of an enzyme is created and maintained by concentration gradients of the substrate and product of the catalyzed reaction. These gradients can arise either naturally through metabolism or experimentally, e.g., by flow of materials through microfluidic channels or by use of diffusion chambers with semipermeable membranes. Numerous hypotheses regarding the mechanism of this phenomenon have been proposed. Here, we discuss a mechanism based solely on diffusion and chemical reaction and show that kinetic asymmetry, a difference in the transition state energies for dissociation/association of substrate and product, and diffusion asymmetry, a difference in the diffusivities of the bound and free forms of the enzyme, are the determinates of the direction of chemotaxis and can result in either positive or negative chemotaxis, both of which have been demonstrated experimentally. Exploration of these fundamental symmetries that govern nonequilibrium behavior helps to distinguish between possible mechanisms for the evolution of a chemical system from initial to the steady state and whether the principle that determines the direction a system shifts when exposed to an external energy source is based on thermodynamics or on kinetics with the latter being supported by the results of the present paper. Our results show that, while dissipation ineluctably accompanies nonequilibrium phenomena, including chemotaxis, systems do not evolve to maximize or minimize dissipation but rather to attain greater kinetic stability and accumulate in regions where their effective diffusion coefficient is as small as possible. The chemotactic response to the chemical gradients formed by other enzymes participating in a catalytic cascade provides a mechanism for forming loose associations known as metabolons. Significantly, the direction of the effective force due to these gradients depends on the kinetic asymmetry of the enzyme and so can be nonreciprocal, where one enzyme is attracted to another enzyme, but the other enzyme is repelled by the one, in seeming contradiction to Newtons third law. This nonreciprocity is an important ingredient in the behavior of active matter.

Kinetic modeling of the chemotactic process in run-and-tumble bacteria

The chemotactic process of run-and-tumble bacteria results from modulating the tumbling rate in response to changes in chemoattractant gradients felt by the bacteria. The response has a characteristic memory time and is subject to important fluctuations. These ingredients are considered in a kinetic description of chemotaxis, allowing the computation of the stationary mobility and the relaxation times needed to reach the steady state. For large memory times, these relaxation times become large, implying that finite-time measurements give rise to nonmonotonic currents as a function of the imposed chemoattractant gradient, contrary to the stationary regime where the response is monotonic. The case of an inhomogeneous signal is analyzed. Contrary to the usual Keller-Segel model, the response is nonlocal, and the bacterial profile is smoothed with a characteristic length that grows with the memory time. Finally, the case of traveling signals is considered, where appreciable differences appear compared to memoryless chemotactic descriptions.

An allosteric switch ensures efficient unidirectional information transmission by the histidine kinase DesK from Bacillus subtilis

Phosphorylation carries chemical information in biological systems. In two-component systems (TCSs), the sensor histidine kinase and the response regulator are connected through phosphoryl transfer reactions that may be uni- or bidirectional. Directionality enables the construction of complex regulatory networks that optimize signal propagation and ensure the forward flow of information. We combined x-ray crystallography, hybrid quantum mechanics/molecular mechanics (QM/MM) simulations, and systems-integrative kinetic modeling approaches to study phosphoryl flow through the Bacillus subtilis thermosensing TCS DesK-DesR. The allosteric regulation of the histidine kinase DesK was critical to avoid back transfer of phosphoryl groups and futile phosphorylation-dephosphorylation cycles by isolating phosphatase, autokinase, and phosphotransferase activities. Interactions between the kinase's ATP-binding domain and the regulator's receiver domain placed the regulator in two distinct positions in the phosphotransferase and phosphatase complexes, thereby determining whether a key glutamine residue in DesK was properly situated to assist in the dephosphorylation reaction. Moreover, an energetically unfavorable phosphotransferase conformation when DesK was not phosphorylated minimized reverse phosphoryl transfer. DesR dimerization and a dissociative phosphoryl transfer reaction also enforced the direction of phosphoryl flow. Shorter or longer distances between the phosphoryl acceptor and donor residues shifted the phosphoryl transfer equilibrium by modulating the stabilizing effect of the Mg²⁺ cofactor. These mechanisms control the directionality of signal transmission and show how structure-encoded allostery stores and transmits information in signaling systems.

Fast Functionalization with High Performance in the Autonomous Information Engine

Mandal and Jarzynski have proposed a fully autonomous information heat engine, consisting of a demon, a mass, and a memory register interacting with a thermal reservoir. This device converts thermal energy into mechanical work by writing information to a memory register or, conversely, erasing information by consuming mechanical work. Here, we derive a speed limit inequality between the relaxation time of state transformation and the distance between the initial and final distributions, where the combination of the dynamical activity and entropy production plays an important role. Such inequality provides a hint that a speed-performance trade-off relation exists between the relaxation time to a functional state and the average production. To obtain fast functionalization while maintaining the performance, we show that the relaxation dynamics of the information heat engine can be accelerated significantly by devising an optimal initial state of the demon. Our design principle is inspired by the so-called Mpemba effect, where water freezes faster when initially heated.

OxPhos defects cause hypermetabolism and reduce lifespan in cells and in patients with mitochondrial diseases

Patients with primary mitochondrial oxidative phosphorylation (OxPhos) defects present with fatigue and multi-system disorders, are often lean, and die prematurely, but the mechanistic basis for this clinical picture remains unclear. By integrating data from 17 cohorts of patients with mitochondrial diseases (n = 690) we find evidence that these disorders increase resting energy expenditure, a state termed hypermetabolism. We examine this phenomenon longitudinally in patient-derived fibroblasts from multiple donors. Genetically or pharmacologically disrupting OxPhos approximately doubles cellular energy expenditure. This cell-autonomous state of hypermetabolism occurs despite near-normal OxPhos coupling efficiency, excluding uncoupling as a general mechanism. Instead, hypermetabolism is associated with mitochondrial DNA instability, activation of the integrated stress response (ISR), and increased extracellular secretion of age-related cytokines and metabokines including GDF15. In parallel, OxPhos defects accelerate telomere erosion and epigenetic aging per cell division, consistent with evidence that excess energy expenditure accelerates biological aging. To explore potential mechanisms for these effects, we generate a longitudinal RNASeq and DNA methylation resource dataset, which reveals conserved, energetically demanding, genome-wide recalibrations. Taken together, these findings highlight the need to understand how OxPhos defects influence the energetic cost of living, and the link between hypermetabolism and aging in cells and patients with mitochondrial diseases.

After-image formation by adaptation to dynamic color gradients

The eye's retinotopic exposure to an adapter typically produces an after-image. For example, an observer who fixates a red adapter on a gray background will see an illusory cyan after-image after removing the adapter. The after-image's content, like its color or intensity, gives insight into mechanisms responsible for adaptation and processing of a specific feature. To facilitate adaptation, vision scientists traditionally present stable, unchanging adapters for prolonged durations. How adaptation affects perception when features (e.g., color) dynamically change over time is not understood. To investigate adaptation to a dynamically changing feature, participants viewed a colored patch that changed from a color to gray, following either a direct or curved path through the (roughly) equiluminant color plane of CIE LAB space. We varied the speed and curvature of color changes across trials and experiments. Results showed that dynamic adapters produce after-images, vivid enough to be reported by the majority of participants. An after-image consisted of a color complementary to the average of the adapter's colors with a small bias towards more recent rather than initial adapter colors. The modelling of the reported after-image colors further confirmed that adaptation rapidly instigates and gradually dissipates. A second experiment replicated these results and further showed that the probability of observing an after-image diminishes only slightly when the adapter displays transient (stepwise, abrupt) color transitions. We conclude from the results that the visual system can adapt to dynamic colors, to a degree that is robust to the potential interference of transient changes in adapter content.

Energy Cost for Flocking of Active Spins: The Cusped Dissipation Maximum at the Flocking Transition

We study the energy cost of flocking in the active Ising model (AIM) and show that, besides the energy cost for self-propelled motion, an additional energy dissipation is required to power the alignment of spins. We find that this additional alignment dissipation reaches its maximum at the flocking transition point in the form of a cusp with a discontinuous first derivative with respect to the control parameter. To understand this singular behavior, we analytically solve the two- and three-site AIM models and obtain the exact dependence of the alignment dissipation on the flocking order parameter and control parameter, which explains the cusped dissipation maximum at the flocking transition. Our results reveal a trade-off between the energy cost of the system and its performance measured by the flocking speed and sensitivity to external perturbations. This trade-off relationship provides a new perspective for understanding the dynamics of natural flocks and designing optimal artificial flocking systems.

Calibrating Vision: Concepts and Questions

The idea that visual coding and perception are shaped by experience and adjust to changes in the environment or the observer is universally recognized as a cornerstone of visual processing, yet the functions and processes mediating these calibrations remain in many ways poorly understood. In this article we review a number of facets and issues surrounding the general notion of calibration, with a focus on plasticity within the encoding and representational stages of visual processing. These include how many types of calibrations there are - and how we decide; how plasticity for encoding is intertwined with other principles of sensory coding; how it is instantiated at the level of the dynamic networks mediating vision; how it varies with development or between individuals; and the factors that may limit the form or degree of the adjustments. Our goal is to give a small glimpse of an enormous and fundamental dimension of vision, and to point to some of the unresolved questions in our understanding of how and why ongoing calibrations are a pervasive and essential element of vision.

Harvesting information to control nonequilibrium states of active matter

We propose to use a correlated noise bath to drive an optically trapped Brownian particle that mimics active biological matter. Due to the flexibility and precision of our setup, we are able to control the different parameters that drive the stochastic motion of the particle with unprecedented accuracy, thus reaching strongly correlated regimes that are not easily accessible with real active matter. In particular, by using the correlation time (i.e., the "color") of the noise as a control parameter, we can trigger transitions between two nonequilibrium steady states with no expended work, but only a calorific cost. Remarkably, the measured heat production is directly proportional to the spectral entropy of the correlated noise, in a fashion that is reminiscent of Landauer's principle. Our procedure can be viewed as a method for harvesting information from the active fluctuations.

Nonequilibrium dynamics of adaptation in sensory systems

Adaptation is used by biological sensory systems to respond to a wide range of environmental signals, by adapting their response properties to the statistics of the stimulus in order to maximize information transmission. We derive rules of optimal adaptation to changes in the mean and variance of a continuous stimulus in terms of Bayesian filters and map them onto stochastic equations that couple the state of the environment to an internal variable controlling the response function. We calculate numerical and exact results for the speed and accuracy of adaptation and its impact on information transmission. We find that, in the regime of efficient adaptation, the speed of adaptation scales sublinearly with the rate of change of the environment. Finally, we exploit the mathematical equivalence between adaptation and stochastic thermodynamics to quantitatively relate adaptation to the irreversibility of the adaptation time course, defined by the rate of entropy production. Our results suggest a means to empirically quantify adaptation in a model-free and nonparametric way.

Epidermal growth factor receptor cascade prioritizes the maximization of signal transduction

Many studies have been performed to quantify cell signaling. Cell signaling molecules are phosphorylated in response to extracellular stimuli, with the phosphorylation sequence forming a signal cascade. The information gain during a signal event is given by the logarithm of the phosphorylation molecule ratio. The average information gain can be regarded as the signal transduction quantity (ST), which is identical to the Kullback-Leibler divergence (KLD), a relative entropy. We previously reported that if the total ST value in a given signal cascade is maximized, the ST rate (STR) of each signaling molecule per signal duration (min) approaches a constant value. To experimentally verify this theoretical conclusion, we measured the STR of the epidermal growth factor (EGF)-related cascade in A431 skin cancer cells following stimulation with EGF using antibody microarrays against phosphorylated signal molecules. The results were consistent with those from the theoretical analysis. Thus, signaling transduction systems may adopt a strategy that prioritizes the maximization of ST. Furthermore, signal molecules with similar STRs may form a signal cascade. In conclusion, ST and STR are promising properties for quantitative analysis of signal transduction.

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.

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.

Energy Cost of Dynamical Stabilization: Stored versus Dissipated Energy

Dynamical stabilization processes (homeostasis) are ubiquitous in nature, but the needed energetic resources for their existence have not been studied systematically. Here, we undertake such a study using the famous model of Kapitza’s pendulum, which has attracted attention in the context of classical and quantum control. This model is generalized and rendered autonomous, and we show that friction and stored energy stabilize the upper (normally unstable) state of the pendulum. The upper state can be rendered asymptotically stable, yet it does not cost any constant dissipation of energy, and only a transient energy dissipation is needed. Asymptotic stability under a single perturbation does not imply stability with respect to multiple perturbations. For a range of pendulum–controller interactions, there is also a regime where constant energy dissipation is needed for stabilization. Several mechanisms are studied for the decay of dynamically stabilized states.

The optimal strategy balancing risk and speed predicts DNA damage checkpoint override times

Checkpoints arrest biological processes allowing time for error correction. The phenomenon of checkpoint override (also known as checkpoint adaptation, slippage, or leakage), during cellular self-replication is biologically critical but currently lacks a quantitative, functional, or system-level understanding. To uncover fundamental laws governing error-correction systems, we derived a general theory of optimal checkpoint strategies, balancing the trade-off between risk and self-replication speed. Mathematically, the problem maps onto the optimization of an absorbing boundary for a random walk. We applied the theory to the DNA damage checkpoint (DDC) in budding yeast, an intensively researched model checkpoint. Using novel reporters for double-strand DNA breaks (DSBs), we first quantified the probability distribution of DSB repair in time including rare events and, secondly, the survival probability after override. With these inputs, the optimal theory predicted remarkably accurately override times as a function of DSB numbers, which we measured precisely for the first time. Thus, a first-principles calculation revealed undiscovered patterns underlying highly noisy override processes. Our multi-DSB measurements revise well-known past results and show that override is more general than previously thought.

Universal minimal cost of coherent biochemical oscillations

Biochemical clocks are essential for virtually all living systems. A biochemical clock that is isolated from an external periodic signal and subjected to fluctuations can oscillate coherently only for a finite number of oscillations. Furthermore, such an autonomous clock can oscillate only if it consumes free energy. What is the minimum amount of free-energy consumption required for a certain number of coherent oscillations? We conjecture a universal bound that answers this question. A system that oscillates coherently for N oscillations has a minimal free-energy cost per oscillation of 4π^{2}Nk_{B}T. Our bound is valid for general finite Markov processes, is conjectured based on extensive numerical evidence, is illustrated with numerical simulations of a known model for a biochemical oscillator, and applies to existing experimental data.

Improved estimation for energy dissipation in biochemical oscillations

Biochemical oscillations, regulating the timing of life processes, need consume energy to achieve good performance on crucial functions, such as high accuracy of phase period and high sensitivity to external signals. However, it is a great challenge to precisely estimate the energy dissipation in such systems. Here, based on the stochastic normal form theory (SNFT), we calculate the Pearson correlation coefficient between the oscillatory amplitude and phase, and a trade-off relation between transport efficiency and phase sensitivity can then be derived, which serves as a tighter form than the estimator resulting from the conventional thermodynamic uncertainty relation (TUR). Our findings demonstrate that a more precise energy dissipation estimation can be obtained by enhancing the sensitivity of the biochemical oscillations. Moreover, the internal noise and amplitude power effects have also been discovered.

The nonequilibrium mechanism of noise-enhanced drug synergy in HIV latency reactivation

Noise-modulating chemicals can synergize with transcriptional activators in reactivating latent HIV to eliminate latent HIV reservoirs. To understand the underlying biomolecular mechanism, we investigate a previous two-gene-state model and identify two necessary conditions for the synergy: an assumption of the inhibition effect of transcription activators on noise enhancers; and frequent transitions to the gene non-transcription-permissive state. We then develop a loop-four-gene-state model with Tat transcription/translation and find that drug synergy is mainly determined by the magnitude and direction of energy input into the genetic regulatory kinetics of the HIV promoter. The inhibition effect of transcription activators is actually a phenomenon of energy dissipation in the nonequilibrium gene transition system. Overall, the loop-four-state model demonstrates that energy dissipation plays a crucial role in HIV latency reactivation, which might be useful for improving drug effects and identifying other synergies on lentivirus latency reactivation.

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The inhibition of Activator on Noise enhancer is necessary for their synergy in reactivating HIV

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The drug synergy is a nonequilibrium phenomenon in the gene regulatory system

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The magnitude and direction of energy input determine the drug synergy

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This nonequilibrium mechanism is general without regarding molecular details

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.

Developmental energetics: Energy expenditure, budgets and metabolism during animal embryogenesis

Developing embryos are metabolically active, open systems that constantly exchange matter and energy with their environment. They function out of thermodynamic equilibrium and continuously use metabolic pathways to obtain energy from maternal nutrients, in order to fulfill the energetic requirements of growth and development. While an increasing number of studies highlight the role of metabolism in different developmental contexts, the physicochemical basis of embryogenesis, or how cellular processes use energy and matter to act together and transform a zygote into an adult organism, remains unknown. As we obtain a better understanding of metabolism, and benefit from current technology development, it is a promising time to revisit the energetic cost of development and how energetic principles may govern embryogenesis. Here, we review recent advances in methodology to measure and infer energetic parameters in developing embryos. We highlight a potential common pattern in embryonic energy expenditure and metabolic strategy across animal embryogenesis, and discuss challenges and open questions in developmental energetics.

The ecological roles of bacterial chemotaxis

How bacterial chemotaxis is performed is much better understood than why. Traditionally, chemotaxis has been understood as a foraging strategy by which bacteria enhance their uptake of nutrients and energy, yet it has remained puzzling why certain less nutritious compounds are strong chemoattractants and vice versa. Recently, we have gained increased understanding of alternative ecological roles of chemotaxis, such as navigational guidance in colony expansion, localization of hosts or symbiotic partners and contribution to microbial diversity by the generation of spatial segregation in bacterial communities. Although bacterial chemotaxis has been observed in a wide range of environmental settings, insights into the phenomenon are mostly based on laboratory studies of model organisms. In this Review, we highlight how observing individual and collective migratory behaviour of bacteria in different settings informs the quantification of trade-offs, including between chemotaxis and growth. We argue that systematically mapping when and where bacteria are motile, in particular by transgenerational bacterial tracking in dynamic environments and in situ approaches from guts to oceans, will open the door to understanding the rich interplay between metabolism and growth and the contribution of chemotaxis to microbial life.

Intuitive view of entropy production of ideal run-and-tumble particles

This work investigates the entropy production rate, Π, of the run-and-tumble model with a focus on scaling of Π as a function of the persistence time τ. It is determined that (i) Π vanishes in the limit τ→∞, marking it as an equilibrium. Stationary distributions in this limit are represented by a superposition of Boltzmann functions in analogy to a system with quenched disorder. (ii) Optimal Π is attained in the limit τ→0, marking it as a system maximally removed from equilibrium. Paradoxically, the stationary distributions in this limit have the Boltzmann form. The value of Π in this limit is that of an unconfined run-and-tumble particle and is related to the dissipation energy of a sedimenting particle. In addition to these general conclusions, this work derives an exact expression of Π for the run-and-tumble particles in a harmonic trap.

Metastable dynamics of neural circuits and networks

Cortical neurons emit seemingly erratic trains of action potentials or "spikes," and neural network dynamics emerge from the coordinated spiking activity within neural circuits. These rich dynamics manifest themselves in a variety of patterns, which emerge spontaneously or in response to incoming activity produced by sensory inputs. In this Review, we focus on neural dynamics that is best understood as a sequence of repeated activations of a number of discrete hidden states. These transiently occupied states are termed "metastable" and have been linked to important sensory and cognitive functions. In the rodent gustatory cortex, for instance, metastable dynamics have been associated with stimulus coding, with states of expectation, and with decision making. In frontal, parietal, and motor areas of macaques, metastable activity has been related to behavioral performance, choice behavior, task difficulty, and attention. In this article, we review the experimental evidence for neural metastable dynamics together with theoretical approaches to the study of metastable activity in neural circuits. These approaches include (i) a theoretical framework based on non-equilibrium statistical physics for network dynamics; (ii) statistical approaches to extract information about metastable states from a variety of neural signals; and (iii) recent neural network approaches, informed by experimental results, to model the emergence of metastable dynamics. By discussing these topics, we aim to provide a cohesive view of how transitions between different states of activity may provide the neural underpinnings for essential functions such as perception, memory, expectation, or decision making, and more generally, how the study of metastable neural activity may advance our understanding of neural circuit function in health and disease.

The voltage-depolarization performance vs the free energy cost of a single nACh receptor

The depolarization of the postsynaptic membrane potential in neuromuscular junction, which is conducted by the (stochastic) opening of nicotinic acetylcholine (nACh) receptors distributed in postsynaptic folds, is an inevitable stage in transmitting the electric signals from neural ends to muscle fibers. To perform this work, free energy must be dissipated. Being the first step in understanding the speech-accuracy-energy trade-off, we calculate the energy dissipation of a tiny membrane system that contains a single nACh receptor, i which firstly the stationary distribution of voltage is calculated based on our new method on solving the high-dimensional ODEs with variable coefficients. The ratio of the extent of depolarized-potential to the dissipated power is used to characterize the energy consuming efficiency of the system.

Escherichia coli chemotaxis is information limited

Organisms acquire and use information from their environment to guide their behaviour. However, it is unclear whether this information quantitatively limits their behavioural performance. Here, we relate information to the ability of Escherichia coli to navigate up chemical gradients, the behaviour known as chemotaxis. First, we derive a theoretical limit on the speed with which cells climb gradients, given the rate at which they acquire information. Next, we measure cells' gradient-climbing speeds and the rate of information acquisition by their chemotaxis signaling pathway. We find that E. coli make behavioural decisions with much less than the one bit required to determine whether they are swimming up-gradient. Some of this information is irrelevant to gradient climbing, and some is lost in communication to behaviour. Despite these limitations, E. coli climb gradients at speeds within a factor of two of the theoretical bound. Thus, information can limit the performance of an organism, and sensory-motor pathways may have evolved to efficiently use information acquired from the environment.

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.

Multiple functions of flagellar motility and chemotaxis in bacterial physiology

Most swimming bacteria are capable of following gradients of nutrients, signaling molecules and other environmental factors that affect bacterial physiology. This tactic behavior became one of the most-studied model systems for signal transduction and quantitative biology, and underlying molecular mechanisms are well characterized in Escherichia coli and several other model bacteria. In this review, we focus primarily on less understood aspect of bacterial chemotaxis, namely its physiological relevance for individual bacterial cells and for bacterial populations. As evident from multiple recent studies, even for the same bacterial species flagellar motility and chemotaxis might serve multiple roles, depending on the physiological and environmental conditions. Among these, finding sources of nutrients and more generally locating niches that are optimal for growth appear to be one of the major functions of bacterial chemotaxis, which could explain many chemoeffector preferences as well as flagellar gene regulation. Chemotaxis might also generally enhance efficiency of environmental colonization by motile bacteria, which involves intricate interplay between individual and collective behaviors and trade-offs between growth and motility. Finally, motility and chemotaxis play multiple roles in collective behaviors of bacteria including swarming, biofilm formation and autoaggregation, as well as in their interactions with animal and plant hosts.

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.