Physical Review Research

A survey of the literature reveals notable discrepancies among the purported exact results for the spectra of stochastic gene expression models. For self-repressing gene circuits, previous studies ([Phys. Rev. Lett. 99, 108103 (2007)], [Phys. Rev. E 83,062902 (2011)], [J. Chem. Phys. 160, 074105 (2024)], and [bioRxiv 2025.02.05.635946 (2025)]) have provided different exact solutions for the eigenvalues of the generator matrix. In this work, we propose a unified Hilbert space framework for the spectral theory of stochastic gene expression. Based on this framework, we analytically derive the spectra for models of constitutive, bursty, and autoregulated gene expression. The eigenvalues and eigenvectors obtained are then used to construct an exact spectral representation of the time-dependent distribution of gene product numbers. The spectral gap between the zero eigenvalue and the first nonzero eigenvalue, which reflects the relaxation rate of the system towards its steady state, is then compared with the prediction of the deterministic model, and we find that deterministic modeling fails to capture the relaxation rate when autoregulation is strong. In particular, our results demonstrate that for infinite-dimensional operators such as in stochastic gene expression models, many conclusions in linear algebra do not apply, and one must rely on the modern theory of functional analysis.

Our eyes are never still. Even when fixating, they exhibit small, jittery motions. While it has long been argued that these fixational eye movements (FEMs) aid the acquisition of visual information, a complete theoretical description of their impact on the information available in the early visual system has been lacking. Here we build FEMs into a minimal theoretical model of the early visual response, including the critical process of adaptation (a fading response to a fixed image). We establish the effect of FEMs on the mutual information between a visual stimulus and this response. Our approach identifies the key dimensionless parameters that characterise the effect of fixational eye movements, and reveals the regimes in which this effect can be beneficial, detrimental or negligible. Taking parameter values appropriate for human vision, we show that the spatio-temporal couplings induced by fixational eye movements can explain the qualitative features of the human contrast sensitivity function as well as classic experiments on temporal integration. To our knowledge, the consideration of fixational eye movements in this latter context is novel, and our results suggest the need for future experiments to determine the mechanisms by which spatial and temporal responses are coupled in the human visual system.

Although many animals rely on visual information to navigate, optic flow is inherently ambiguous as it confounds information about motion speed and object distance. As a result, the visual feedback produced by a given motor command is context-dependent and requires an appropriately adapted response. Recent experiments have investigated how the fish Danionella cerebrum use visual cues to stabilize their position against simulated external currents. Logarithmic sensorimotor transformations have been proposed to enable adaptive responses to perturbations while preventing delay-induced instabilities. Here, we develop the theoretical framework introduced for continuous locomotion to show how logarithmic coding naturally gives rise to this adaptive behavior. The system is modeled by a nonlinear delay differential equation, which is analyzed using dynamical systems theory. We further analyze experimental data to uncover the mechanisms underlying swimming initiation and positional drift correction. Finally, we extend our framework to intermittent locomotion, resulting in a nonlinear difference equation, and show that it still produces robust adaptive behavior. This is motivated by the literature on zebrafish, where visuomotor stabilization has been extensively studied, but burst-and-coast swimming obscures the underlying adaptation mechanism. We show that our theory can reproduce the experimental results reported for motor adaptation in zebrafish without invoking internal models. Overall, our results highlight logarithmic coding as a unifying principle for visuomotor stability across continuous and intermittent locomotor regimes.

"Einstein from noise" (EfN) is a prominent example of the model bias phenomenon, where systematic errors in the statistical model lead to spurious but consistent estimates. In the EfN experiment, one falsely believes that a set of observations contains noisy, shifted copies of a template signal (e.g., an Einstein image), whereas in reality, it contains only pure noise observations. To estimate the signal, the observations are first aligned with the template using cross-correlation and then averaged. Although the observations contain nothing but noise, it was recognized early on that this process produces a signal that resembles the template signal! This model bias pitfall was at the heart of a central scientific controversy about validation techniques in structural biology. This paper provides a comprehensive statistical analysis of the EfN phenomenon above. We show that the Fourier phases of the EfN estimator (namely, the average of the aligned noise observations) converge to the Fourier phases of the template signal, thereby explaining the observed structural similarity. Additionally, we prove that the convergence rate of Fourier phases is inversely proportional to the number of noise observations and, in the high-dimensional regime, to the Fourier magnitudes of the template signal. Moreover, in the high-dimensional regime, the EfN estimator converges to a scaled version of the template signal. This work not only deepens the theoretical understanding of the EfN phenomenon but also highlights potential pitfalls in template matching techniques and emphasizes the need for careful interpretation of noisy observations across disciplines in engineering, statistics, physics, and biology.

Sperm modulate their flagellar symmetry to navigate through complex physico-chemical environments and achieve reproductive function. Yet it remains elusive how sperm swim forwards despite the inherent asymmetry of several components that constitutes the flagellar engine. Despite the critical importance of symmetry, or the lack of it, on sperm navigation and its physiological state, there is no methodology to date that can robustly detect the symmetry state of the beat in free-swimming sperm in 3D. How does symmetric progressive swimming emerge even for asymmetric beating, and how can beating (a)symmetry be inferred experimentally? Here, we numerically resolve the fluid mechanics of swimming around asymmetrically beating spermatozoa. This reveals that sperm spinning critically regularizes swimming into persistently symmetric paths in 3D, allowing sperm to swim forwards despite any imperfections on the beat. The sperm orientation in three-dimensions, and not the swimming path, can inform the symmetry state of the beat, eliminating the need of tracking the flagellum in 3D. We report a surprising correspondence between the movement of sperm and spinning-top experiments, indicating that the flagellum drives "spinning-top" type rotations during sperm swimming, and that this parallel is not a mere analogy. These results may prove essential in future studies on the role of (a)symmetry in spinning and swimming microorganisms and micro-robots, as body orientation detection has been vastly overlooked in favour of swimming path detection. Altogether, sperm rotation may provide a foolproof mechanism for forward propulsion and navigation in nature that would otherwise not be possible for flagella with broken symmetry.

We investigate the dynamics and hydrodynamics of a human spermatozoa swimming freely in 3D. We simultaneously track the sperm flagellum and the sperm head orientation in the laboratory frame of reference via high-speed high-resolution 4D (3D+t) microscopy, and extract the flagellar waveform relative to the body frame of reference, as seen from a frame of reference that translates and rotates with the sperm in 3D. Numerical fluid flow reconstructions of sperm motility are performed utilizing the experimental 3D waveforms, with excellent accordance between predicted and observed 3D sperm kinematics. The reconstruction accuracy is validated by directly comparing the three linear and three angular sperm velocities with experimental measurements. Our microhydrodynamic analysis reveals a novel fluid flow pattern, characterized by a pair of vortices that circulate in opposition to each other along the sperm cell. Finally, we show that the observed sperm counter-vortices are not unique to the experimental beat, and can be reproduced by idealised waveform models, thus suggesting a fundamental flow structure for free-swimming sperm propelled by a 3D beating flagellum.

A key goal of synthetic biology is to establish 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 others 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.

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: pushpull 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.

Foraging is a central decision-making behavior performed by all animals, essential to garnishing enough energy for an organism to survive. Similarly, mating is crucial for evolutionary continuity and offspring production. Mate choice is one of the central tenets of sexual selection, driving major evolutionary processes, and can be regarded as a decision-making process between potential mating partners. Often researchers have used coarse-grained models to describe macroscopic phenomenology pertaining to mate choice without detailed quantitative mechanisms of how animals use individual and environmental signals to guide their mating decisions. In this letter, we show that mate choice can be cast as a foraging problem, and we present an analytically tractable optimal foraging-inspired mechanistic theory of decision-making underlying mate choice. We begin from the premise that deciding upon which partner with which to mate is at its core a stochastic decision-making process. Agents adopt a variety of decision strategies, tuned by decision thresholds for leaving or committing to a mate. We find that sensitive leaving thresholds are favored independently of signal availability in the population. By contrast, optimal thresholds for committing to a mate depend upon signal availability in the population, with signal-rich populations generally favoring less eager strategies compared to signal-poor populations.

In Nature there are significant relationships known between microorganisms from two kingdoms of life, as in the supply of vitamin B12 by bacteria to algae. Such interactions motivate general investigations into the spatio-temporal dynamics of metabolite exchanges. Here we study by experiment and theory a model system: a coculture of the bacterium B. subtilis, an obligate aerobe that is chemotactic to oxygen, and a nonmotile mutant of the alga C. reinhardtii, which photosynthetically produces oxygen when illuminated. Strikingly, when a shaft of light illuminates a thin, initially uniform suspension of the two, the chemotactic influx of bacteria to the photosyn-thetically active region leads to expulsion of the algae from that area. This effect arises from algal transport due to spatially-varying collective behavior of bacteria, and is mathematically related to the "turbulent diamagnetism" associated with magnetic flux expulsion in stars.

Micro-calorimetry offers significant potential as a quantitative method for studying the structure and function of biological systems, for instance, by probing the excess heat released by cellular or sub-cellular structures, isothermal or not, when external parameters change. We present the conceptual framework of nonequilibrium calorimetry, and as illustrations, we compute the heat capacity of biophysical models with few degrees of freedom related to ciliar motion (rowing model) and molecular motor motion (flashing ratchets). Our quantitative predictions reveal intriguing dependencies of the (nonequilibrium) heat capacity as a function of relevant biophysical parameters, which can even take negative values as a result of biological activity.

Bacteria navigate their environment by biasing their swimming direction toward beneficial chemicals and away from harmful ones. Out of all the chemicals bacteria respond to, oxygen stands out due to its ubiquitous presence, distinct influence on bacterial metabolism and motility, and historical role in chemotaxis research. However, a coherent understanding of bacterial motility in oxygen gradients, known as aerotaxis, remains elusive, as evidenced by conflicting reports on the migration direction of the model organism Escherichia coli in self-generated oxygen gradients. Here, by combining experiments, simulations, and theory, we provide a unified framework elucidating the fundamental biophysical principle governing bacterial aerotaxis. We track the migration of bacteria in a capillary channel under self-generated oxygen gradients and show that the migration direction depends on the overall bacterial density. At high densities, bacteria migrate toward regions of higher oxygen concentration, whereas at low densities, they move in the opposite direction. We identify a critical bacterial density at which collective migration ceases, despite the presence of oxygen gradients. A kinetic theory, based on the assumption that bacteria seek an optimal oxygen concentration, is then developed to quantitatively explain our experimental findings. We validate this hypothesis by demonstrating the biased movement of individual bacteria in a dense suspension and proposing a signaling pathway that enables this behavior. Thus, by bridging the molecular level understanding of the signaling pathway, the motility of single bacteria in oxygen gradients, and the collective population dynamics shaped by oxygen diffusion and consumption, our study provides a comprehensive understanding of aerotaxis, addressing the long-standing controversy over how bacteria response to non-uniform oxygen distributions pervasive in microbial habitats.

It is known that mechanical interactions couple a cell to its neighbors, enabling a feedback loop to regulate tissue growth. However, the interplay between cell-cell adhesion strength, local cell density and force fluctuations in regulating cell proliferation is poorly understood. Here, we show that spatial variations in the tumor growth rates, which depend on the location of cells within tissue spheroids, are strongly influenced by cell-cell adhesion. As the strength of the cell-cell adhesion increases, intercellular pressure initially decreases, enabling dormant cells to more readily enter into a proliferative state. We identify an optimal cell-cell adhesion regime where pressure on a cell is a minimum, allowing for maximum proliferation. We use a theoretical model to validate this novel collective feedback mechanism coupling adhesion strength, local stress fluctuations and proliferation. Our results predict the existence of a non-monotonic proliferation behavior as a function of adhesion strength, consistent with experimental results. Several experimental implications of the proposed role of cell-cell adhesion in proliferation are quantified, making our model predictions amenable to further experimental scrutiny. We show that the mechanism of contact inhibition of proliferation, based on a pressure-adhesion feedback loop, serves as a unifying mechanism to understand the role of cell-cell adhesion in proliferation.

The functional properties of the outermost retinal circuits involved in color discrimination are not well understood. Recent experimental work on zebrafish has elucidated the in-vivo activity of photoreceptors and horizontal cells as a function of the stimulus spectrum, highlighting the appearance of chromatic-opponent signals at the first synaptic connection between cones and horizontal cells. These findings, together with the observed lack of gap junctions, suggest that the mechanism yielding early color-opponency in zebrafish is dominated by inhibitory feedback. We discuss the observed neuronal activity in the context of efficient codification of chromatic information, hypothesizing that opponent chromatic signals provide optimal codification, minimizing signal redundancy. We examine whether these functional properties are general across species by studying the dynamic properties of dichromatic and trichromatic outer retinal networks. Our findings show that dominant inhibitory feedback mechanisms provide an unambiguous codification of chromatic stimuli, whereas this property is not guaranteed in networks with strong excitatory inter-cone connections, for example via gap junctions. This provides a plausible explanation for the absence of gap junctions observed in the outermost zebrafish retinal layers. In addition, our study suggests that the simplest zebrafish-like network with dominant inhibitory feedback capable of optimally codifying chromatic information requires at least two successive inhibitory feedback layers. Finally, we contrast the chromatic codification performance of zebrafish-inspired retinal networks with networks having different opsin combinations. We find that optimal combinations lead to a chromatic codification improvement of only 13% compared with zebrafish opsins, suggesting that the zebrafish retina performs nearly optimal codification of chromatic information in its habitat. 2 Author summaryRecent experimental work has evidenced that outer neuronal circuits in the zebrafish retina use color-opponent mechanisms to codify and transmit chromatic information at the first synaptic contact between cones and horizontal cells. Inspired by these findings, we propose a data-driven model to study physiological and dynamical properties of outer retinal networks and their implications for color codification across vertebrate retinal circuits. We first study our model in a large parameter space, finding that the primary biological mechanism leading to color-opponent signals is mediated by dominant inhibitory feedback, e.g., via horizontal cell synaptic connections. In contrast, strong coupling among cones leads to ambiguous chromatic codification, undesirable in the outer retina. Then, we parameterize the model using zebrafish experimental data and quantify its chromatic codification performance. Our results suggest that trichromatic retinas with inhibitory feedback are highly efficient and capture most of the chromatic information variance typical from zebrafish environments. More specifically, a comparison among zebrafish-inspired retinal networks suggests that zebrafish retinal circuits are near-optimal chromatic codification of their natural chromatic information.

Sensory information propagates through successive processing stages in the brain, where synaptic weight patterns between stations determine how downstream neurons decode information from upstream populations. Although optimized synaptic connectivity can enhance information transmission, it requires precise weight tuning. Recent evidence depicting substantial synaptic volatility raises two fundamental questions: How does coarse-tuning of synaptic connectivity affect information transmission? What strategies could the nervous system employ to maintain reliable communication despite synaptic fluctuations? We addressed these questions by analyzing the signal-to-noise ratio (SNR) for binary stimulus discrimination under two decoding schemes: a naive population average and an optimized linear decoder. For the naive decoder, we found that SNR remains largely insensitive to synaptic imprecision, since performance is already limited by correlated noise in neuronal responses. For the optimal decoder, we identified three distinct regimes, that is, weak, moderate and strong coarse-tuning. Under weak coarse-tuning, SNR2 scales linearly with population size N. Under moderate coarse-tuning, scaling becomes sublinear. Strikingly, under strong coarse-tuning, the regime most consistent with observed neuronal heterogeneity, SNR saturates and can not be improved by recruiting larger populations. This limitation persists even when incorporating feedforward or recurrent network architectures. These findings suggest that in the biologically relevant regime of strong coarse-tuning, naive and optimal decoders can achieve qualitatively similar performance. The analysis shows that effective readout under synaptic volatility is constrained to an invariant low-dimensional manifold aligned with the naive decoder, potentially pointing to a fundamental principle for robust neural computation in the face of ongoing synaptic remodeling.

Cell migration in vivo is often guided by chemical signals. Such chemotaxis, such as performed by immune cells migrating to a wound site, is complicated by the complex geometry inside living tissues. In this study, we extend our theoretical model of branched-cell migration on a network by introducing chemokine sources to explore the cellular response. The model predicts a speed-accuracy tradeoff, whereby slow cells are significantly more accurate and able to follow efficiently a weak chemoattractant signal. We then compare the models predictions with experimental observations of neutrophils migrating to the site of laser-inflicted wound in a zebrafish larva fin, and migrating in-vitro inside a regular lattice of pillars. We find that the model captures the details of the sub-cellular response to the chemokine gradient, as well as the large-scale migration response. This comparison suggests that the neutrophils behave as fast cells, compromising their chemotaxis accuracy, which explains the functionality of these immune cells.

Hydrodynamic coordination of cilia is ubiquitous in biology. It is commonly modeled using the steady Stokes equations. The flow around ciliated cells, however, exhibits finite time vorticity diffusion, requiring a dynamical description. We present a model of elastic cilia coupled by transient viscous flow in the bulk fluid. Therein, vorticity diffusion impacts cilia coordination qualitatively and quantitatively. In particular, pairs of cilia synchronize in antiphase for long diffusion times. Moreover, metachronal waves occur in cilia chains larger than the viscous penetration depth, whereas global synchronization occurs in Stokes flow.

Understanding how cells sense mechanical cues and regulate migration is crucial in the development, fibrosis, and oncogenesis processes. However, a comprehensive physical picture of cell migration remains lacking, given the diverse environmental properties and cell physiologies. Here, we generalize the motor-clutch model to the whole-cell level and systematically investigate the effects of substrate stiffness, friction, and actin polymerization speed on cell migration. We unveil three distinct migration modes: gripping, slipping, and stick-slip. Notably, stiffness sensing occurs exclusively in the stick-slip mode, which requires a low substrate stiffness and a minimum actin polymerization speed as necessary conditions. Intriguingly, the optimal substrate stiffness that maximizes the migration speed is inversely proportional to the actin polymerization speed. Moreover, the maximal speed only depends on the nature of the clutch molecules, independent of substrate properties. We reveal the boundary criteria between the three migration modes and demonstrate that fast- and slow-migrating cells can coexist in an isogenic cell population without the need for biochemical feedback loops.

Receptor-mediated ligand endocytosis is traditionally viewed as a negative feedback mechanism for signal attenuation. Here we show that ligand removal can paradoxically enhance directional information in autonomous cell-cell attraction. Many cell systems migrate toward one another in the absence of externally imposed gradients, implying that secretion, diffusion, and uptake must themselves generate usable directional cues. We develop a surface-resolved theory of a finite-sized detector exposed to a nearby source and derive analytical expressions for the steady-state ligand field. The resulting concentration profiles are governed by a single dimensionless Damkohler number that compares receptor-mediated endocytosis to diffusive ligand transport. Increasing ligand removal lowers extracellular ligand concentrations and reduces absolute concentration differences across the detector surface, but preferentially enhances relative surface anisotropy. Thus, destroying the signal can increase the usable information encoded in relative gradients. Incorporating nonlinear downstream processing reveals a tradeoff between contrast enhancement and signal depletion that yields a well-defined optimal endocytosis rate, in a regime consistent with experimentally measured receptor internalization kinetics. These results recast receptor-mediated endocytosis as an extracellular information-processing mechanism that reshapes self-generated gradients to enhance directional information.

Cell adhesion is a fundamental biological process that governs cell proliferation, differentiation, migration, and tissue development. Cells adhere to the extracellular matrix through specialized transmembrane proteins, whose structures and functions are well studied. However, how mechanical, chemical, and biological factors interact to regulate these proteins and hence to shape cross-scale adhesion dynamics from molecular clustering to cellular migration remains unclear. Here, we propose a multiscale mechano-biochemical coupling framework to investigate the dynamics of cell-substrate adhesions, incorporating comprehensive molecular steps in the integrin life cycle, including activation, clustering, signal transduction and internalization. Our model elucidates the roles of caveolin transport and actin flow in modulating integrin dynamics and FA morphology. We identify the antagonistic interplay between integrin internalization and clustering that governs cross-scale adhesion dynamics. Furthermore, our model quantitatively demonstrates how the substrate stiffness regulates the integrin clustering size and internalization rate. These findings provide mechanistic insights into the regulation of cell migration, particularly the transition between durotaxis and negative durotaxis, driven by intracellular and extracellular microenvironmental factors. Our model offers an effective framework for understanding the cross-scale regulation process of cell adhesion involved in physiological and pathological activities, such as stem cell differentiation and cancer metastasis.