Temperature compensation through kinetic regulation in biochemical oscillators.

Nearly all circadian clocks maintain a period that is insensitive to temperature changes, a phenomenon known as temperature compensation (TC). Yet, it is unclear whether there is any common feature among different systems that exhibit TC. From a general timescale invariance, we show that TC relies on the existence of certain period-lengthening reactions wherein the period of the system increases strongly with the rates in these reactions. By studying several generic oscillator models, we show that this counterintuitive dependence is nonetheless a common feature of oscillators in the nonlinear (far-from-onset) regime where the oscillation can be separated into fast and slow phases. The increase of the period with the period-lengthening reaction rates occurs when the amplitude of the slow phase in the oscillation increases with these rates while the progression speed in the slow phase is controlled by other rates of the system. The positive dependence of the period on the period-lengthening rates balances its inverse dependence on other kinetic rates in the system, which gives rise to robust TC in a wide range of parameters. We demonstrate the existence of such period-lengthening reactions and their relevance for TC in all four model systems we considered. Theoretical results for a model of the Kai system are supported by experimental data. A study of the energy dissipation also shows that better TC performance requires higher energy consumption. Our study unveils a general mechanism by which a biochemical oscillator achieves TC by operating in parameter regimes far from the onset where period-lengthening reactions exist.

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.

The inverse variance-flatness relation in stochastic gradient descent is critical for finding flat minima.

Despite tremendous success of the stochastic gradient descent (SGD) algorithm in deep learning, little is known about how SGD finds generalizable solutions at flat minima of the loss function in high-dimensional weight space. Here, we investigate the connection between SGD learning dynamics and the loss function landscape. A principal component analysis (PCA) shows that SGD dynamics follow a low-dimensional drift-diffusion motion in the weight space. Around a solution found by SGD, the loss function landscape can be characterized by its flatness in each PCA direction. Remarkably, our study reveals a robust inverse relation between the weight variance and the landscape flatness in all PCA directions, which is the opposite to the fluctuation-response relation (aka Einstein relation) in equilibrium statistical physics. To understand the inverse variance-flatness relation, we develop a phenomenological theory of SGD based on statistical properties of the ensemble of minibatch loss functions. We find that both the anisotropic SGD noise strength (temperature) and its correlation time depend inversely on the landscape flatness in each PCA direction. Our results suggest that SGD serves as a landscape-dependent annealing algorithm. The effective temperature decreases with the landscape flatness so the system seeks out (prefers) flat minima over sharp ones. Based on these insights, an algorithm with landscape-dependent constraints is developed to mitigate catastrophic forgetting efficiently when learning multiple tasks sequentially. In general, our work provides a theoretical framework to understand learning dynamics, which may eventually lead to better algorithms for different learning tasks.

Mechanosensitive remodeling of the bacterial flagellar motor is independent of direction of rotation.

Motility is important for the survival and dispersal of many bacteria, and it often plays a role during infections. Regulation of bacterial motility by chemical stimuli is well studied, but recent work has added a new dimension to the problem of motility control. The bidirectional flagellar motor of the bacterium Escherichia coli recruits or releases torque-generating units (stator units) in response to changes in load. Here, we show that this mechanosensitive remodeling of the flagellar motor is independent of direction of rotation. Remodeling rate constants in clockwise rotating motors and in counterclockwise rotating motors, measured previously, fall on the same curve if plotted against torque. Increased torque decreases the off rate of stator units from the motor, thereby increasing the number of active stator units at steady state. A simple mathematical model based on observed dynamics provides quantitative insight into the underlying molecular interactions. The torque-dependent remodeling mechanism represents a robust strategy to quickly regulate output (torque) in response to changes in demand (load).

Stochastic Gradient Descent Introduces an Effective Landscape-Dependent Regularization Favoring Flat Solutions.

Generalization is one of the most important problems in deep learning, where there exist many low-loss solutions due to overparametrization. Previous empirical studies showed a strong correlation between flatness of the loss landscape at a solution and its generalizability, and stochastic gradient descent (SGD) is crucial in finding the flat solutions. To understand the effects of SGD, we construct a simple model whose overall loss landscape has a continuous set of degenerate (or near-degenerate) minima and the loss landscape for a minibatch is approximated by a random shift of the overall loss function. By direct simulations of the stochastic learning dynamics and solving the underlying Fokker-Planck equation, we show that due to its strong anisotropy the SGD noise introduces an additional effective loss term that decreases with flatness and has an overall strength that increases with the learning rate and batch-to-batch variation. We find that the additional landscape-dependent SGD loss breaks the degeneracy and serves as an effective regularization for finding flat solutions. As a result, the flatness of the overall loss landscape increases during learning and reaches a higher value (flatter minimum) for a larger SGD noise strength before the noise strength reaches a critical value when the system fails to converge. These results, which are verified in realistic neural network models, elucidate the role of SGD for generalization, and they may also have important implications for hyperparameter selection for learning efficiently without divergence.

Filtering input fluctuations in intensity and in time underlies stochastic transcriptional pulses without feedback.

Stochastic pulsatile dynamics have been observed in an increasing number of biological circuits with known mechanism involving feedback control and bistability. Surprisingly, recent single-cell experiments in Escherichia coli flagellar synthesis showed that flagellar genes are activated in stochastic pulses without the means of feedback. However, the mechanism for pulse generation in these feedbackless circuits has remained unclear. Here, by developing a system-level stochastic model constrained by a large set of single-cell E. coli flagellar synthesis data from different strains and mutants, we identify the general underlying design principles for generating stochastic transcriptional pulses without feedback. Our study shows that an inhibitor (YdiV) of the transcription factor (FlhDC) creates a monotonic ultrasensitive switch that serves as a digital filter to eliminate small-amplitude FlhDC fluctuations. Furthermore, we find that the high-frequency (fast) fluctuations of FlhDC are filtered out by integration over a timescale longer than the timescale of the input fluctuations. Together, our results reveal a filter-and-integrate design for generating stochastic pulses without feedback. This filter-and-integrate mechanism enables a general strategy for cells to avoid premature activation of the expensive downstream gene expression by filtering input fluctuations in both intensity and time so that the system only responds to input signals that are both strong and persistent.

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.

Optimal compressed sensing strategies for an array of nonlinear olfactory receptor neurons with and without spontaneous activity.

There are numerous different odorant molecules in nature but only a relatively small number of olfactory receptor neurons (ORNs) in brains. This "compressed sensing" challenge is compounded by the constraint that ORNs are nonlinear sensors with a finite dynamic range. Here, we investigate possible optimal olfactory coding strategies by maximizing mutual information between odor mixtures and ORNs' responses with respect to the bipartite odor-receptor interaction network (ORIN) characterized by sensitivities between all odorant-ORN pairs. For ORNs without spontaneous (basal) activity, we find that the optimal ORIN is sparse-a finite fraction of sensitives are zero, and the nonzero sensitivities follow a broad distribution that depends on the odor statistics. We show analytically that sparsity in the optimal ORIN originates from a trade-off between the broad tuning of ORNs and possible interference. Furthermore, we show that the optimal ORIN enhances performances of downstream learning tasks (reconstruction and classification). For ORNs with a finite basal activity, we find that having inhibitory odor-receptor interactions increases the coding capacity and the fraction of inhibitory interactions increases with the ORN basal activity. We argue that basal activities in sensory receptors in different organisms are due to the trade-off between the increase in coding capacity and the cost of maintaining the spontaneous basal activity. Our theoretical findings are consistent with existing experiments and predictions are made to further test our theory. The optimal coding model provides a unifying framework to understand the peripheral olfactory systems across different organisms.

Escape band in Escherichia coli chemotaxis in opposing attractant and nutrient gradients.

It is commonly believed that bacterial chemotaxis helps cells find food. However, not all attractants are nutrients, and not all nutrients are strong attractants. Here, by using microfluidic experiments, we studied Escherichia coli chemotaxis behavior in the presence of a strong chemoattractant (e.g., aspartate or methylaspartate) gradient and an opposing gradient of diluted tryptone broth (TB) growth medium. Our experiments showed that cells initially accumulate near the strong attractant source. However, after the peak cell density (h) reaches a critical value [Formula: see text], the cells form a "escape band" (EB) that moves toward the chemotactically weaker but metabolically richer nutrient source. By using various mutant strains and varying experimental conditions, we showed that the competition between Tap and Tar receptors is the key molecular mechanism underlying the formation of the escape band. A mathematical model combining chemotaxis signaling and cell growth was developed to explain the experiments quantitatively. The model also predicted that the width w and the peak position [Formula: see text] of EB satisfy two scaling relations: [Formula: see text] and [Formula: see text], where l is the channel length. Both scaling relations were verified by experiments. Our study shows that the combination of nutrient consumption, population growth, and chemotaxis with multiple receptors allows cells to search for optimal growth condition in complex environments with conflicting sources.

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

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

Error-Speed Correlations in Biopolymer Synthesis.

Synthesis of biopolymers such as DNA, RNA, and proteins are biophysical processes aided by enzymes. The performance of these enzymes is usually characterized in terms of their average error rate and speed. However, because of thermal fluctuations in these single-molecule processes, both error and speed are inherently stochastic quantities. In this Letter, we study fluctuations of error and speed in biopolymer synthesis and show that they are in general correlated. This means that, under equal conditions, polymers that are synthesized faster due to a fluctuation tend to have either better or worse errors than the average. The error-correction mechanism implemented by the enzyme determines which of the two cases holds. For example, discrimination in the forward reaction rates tends to grant smaller errors to polymers with faster synthesis. The opposite occurs for discrimination in monomer rejection rates. Our results provide an experimentally feasible way to identify error-correction mechanisms by measuring the error-speed correlations.

A dual regulation mechanism of histidine kinase CheA identified by combining network-dynamics modeling and system-level input-output data.

It is challenging to decipher molecular mechanisms in biological systems from system-level input-output data, especially for complex processes that involve interactions among multiple components. We addressed this general problem for the bacterial histidine kinase CheA, the activity of which is regulated in chemotaxis signaling complexes by bacterial chemoreceptors. We developed a general network model to describe the dynamics of the system, treating the receptor complex with coupling protein CheW and the P3P4P5 domains of kinase CheA as a regulated enzyme with two substrates, ATP and P1, the phosphoryl-accepting domain of CheA. Our simple network model allowed us to search hypothesis space systematically. For different and progressively more complex regulation schemes, we fit our models to a large set of input-output data with the aim of identifying the simplest possible regulation mechanisms consistent with the data. Our modeling and analysis revealed novel dual regulation mechanisms in which receptor activity regulated ATP binding plus one other process, either P1 binding or phosphoryl transfer between P1 and ATP. Strikingly, in our models receptor control affected the kinetic rate constants of substrate association and dissociation equally and thus did not alter the respective equilibrium constants. We suggest experiments that could distinguish between the two dual-regulation mechanisms. This systems-biology approach of combining modeling and a large input-output dataset should be applicable for studying other complex biological processes.

Distinct signaling of Drosophila chemoreceptors in olfactory sensory neurons.

In Drosophila, olfactory sensory neurons (OSNs) rely primarily on two types of chemoreceptors, odorant receptors (Ors) and ionotropic receptors (Irs), to convert odor stimuli into neural activity. The cellular signaling of these receptors in their native OSNs remains unclear because of the difficulty of obtaining intracellular recordings from Drosophila OSNs. Here, we developed an antennal preparation that enabled the first recordings (to our knowledge) from targeted Drosophila OSNs through a patch-clamp technique. We found that brief odor pulses triggered graded inward receptor currents with distinct response kinetics and current-voltage relationships between Or- and Ir-driven responses. When stimulated with long-step odors, the receptor current of Ir-expressing OSNs did not adapt. In contrast, Or-expressing OSNs showed a strong Ca(2+)-dependent adaptation. The adaptation-induced changes in odor sensitivity obeyed the Weber-Fechner relation; however, surprisingly, the incremental sensitivity was reduced at low odor backgrounds but increased at high odor backgrounds. Our model for odor adaptation revealed two opposing effects of adaptation, desensitization and prevention of saturation, in dynamically adjusting odor sensitivity and extending the sensory operating range.

Swarming in the Dirt: Ordered Flocks with Quenched Disorder.

The effect of quenched (frozen) disorder on the collective motion of active particles is analyzed. We find that active polar systems are far more robust against quenched disorder than equilibrium ferromagnets. Long-ranged order (a nonzero average velocity ⟨v⟩) persists in the presence of quenched disorder even in spatial dimensions d=3; in d=2, quasi-long-ranged order (i.e., spatial velocity correlations that decay as a power law with distance) occurs. In equilibrium systems, only quasi-long-ranged order in d=3 and short-ranged order in d=2 are possible. Our theoretical predictions for two dimensions are borne out by simulations.

Barrier Crossing in Escherichia coli Chemotaxis.

We study cell navigation in spatiotemporally complex environments by developing a microfluidic racetrack device that creates a traveling wave with multiple peaks and a tunable wave speed. We find that while the population-averaged chemotaxis drift speed increases with wave speed for low wave speed, it decreases sharply for high wave speed. This reversed dependence of population-averaged chemotaxis drift speed on wave speed is caused by a "barrier-crossing" phenomenon, where a cell hops backwards from one peak attractant location to the peak behind by crossing an unfavorable (barrier) region with low attractant concentrations. By using a coarse-grained model of chemotaxis, we map bacterial motility in an attractant field to the random motion of an overdamped particle in an effective potential. The observed barrier-crossing phenomenon of living cells and its dependence on the spatiotemporal profile of attractant concentration are explained quantitatively by Kramers reaction rate theory.

Information processing in bacteria: memory, computation, and statistical physics: a key issues review.

Living systems have to constantly sense their external environment and adjust their internal state in order to survive and reproduce. Biological systems, from as complex as the brain to a single E. coli cell, have to process these data in order to make appropriate decisions. How do biological systems sense external signals? How do they process the information? How do they respond to signals? Through years of intense study by biologists, many key molecular players and their interactions have been identified in different biological machineries that carry out these signaling functions. However, an integrated, quantitative understanding of the whole system is still lacking for most cellular signaling pathways, not to say the more complicated neural circuits. To study signaling processes in biology, the key thing to measure is the input-output relationship. The input is the signal itself, such as chemical concentration, external temperature, light (intensity and frequency), and more complex signals such as the face of a cat. The output can be protein conformational changes and covalent modifications (phosphorylation, methylation, etc), gene expression, cell growth and motility, as well as more complex output such as neuron firing patterns and behaviors of higher animals. Due to the inherent noise in biological systems, the measured input-output dependence is often noisy. These noisy data can be analysed by using powerful tools and concepts from information theory such as mutual information, channel capacity, and the maximum entropy hypothesis. This information theory approach has been successfully used to reveal the underlying correlations between key components of biological networks, to set bounds for network performance, and to understand possible network architecture in generating observed correlations. Although the information theory approach provides a general tool in analysing noisy biological data and may be used to suggest possible network architectures in preserving information, it does not reveal the underlying mechanism that leads to the observed input-output relationship, nor does it tell us much about which information is important for the organism and how biological systems use information to carry out specific functions. To do that, we need to develop models of the biological machineries, e.g. biochemical networks and neural networks, to understand the dynamics of biological information processes. This is a much more difficult task. It requires deep knowledge of the underlying biological network-the main players (nodes) and their interactions (links)-in sufficient detail to build a model with predictive power, as well as quantitative input-output measurements of the system under different perturbations (both genetic variations and different external conditions) to test the model predictions to guide further development of the model. Due to the recent growth of biological knowledge thanks in part to high throughput methods (sequencing, gene expression microarray, etc) and development of quantitative in vivo techniques such as various florescence technology, these requirements are starting to be realized in different biological systems. The possible close interaction between quantitative experimentation and theoretical modeling has made systems biology an attractive field for physicists interested in quantitative biology. In this review, we describe some of the recent work in developing a quantitative predictive model of bacterial chemotaxis, which can be considered as the hydrogen atom of systems biology. Using statistical physics approaches, such as the Ising model and Langevin equation, we study how bacteria, such as E. coli, sense and amplify external signals, how they keep a working memory of the stimuli, and how they use these data to compute the chemical gradient. In particular, we will describe how E. coli cells avoid cross-talk in a heterogeneous receptor cluster to keep a ligand-specific memory. We will also study the thermodynamic costs of adaptation for cells to maintain an accurate memory. The statistical physics based approach described here should be useful in understanding design principles for cellular biochemical circuits in general.

Discovery of novel chemoeffectors and rational design of Escherichia coli chemoreceptor specificity.

Bacterial chemoreceptors mediate chemotactic responses to diverse stimuli. Here, by using an integrated in silico, in vitro, and in vivo approach, we screened a large compound library and found eight novel chemoeffectors for the Escherichia coli chemoreceptor Tar. Six of the eight new Tar binding compounds induce attractant responses, and two of them function as antagonists that can bind Tar without inducing downstream signaling. Comparison between the antagonist and attractant binding patterns suggests that the key interactions for chemotaxis signaling are mediated by the hydrogen bonds formed between a donor group in the attractant and the main-chain carbonyls (Y149 and/or Q152) on the α4 helix of Tar. This molecular insight for signaling is verified by converting an antagonist to an attractant when introducing an N-H group into the antagonist to restore the hydrogen bond. Similar signal triggering effect by an O-H group is also confirmed. Our study suggests that the Tar chemoeffector binding pocket may be separated into two functional regions: region I mainly contributes to binding and region II contributes to both binding and signaling. This scenario of binding and signaling suggests that Tar may be rationally designed to respond to a nonnative ligand by altering key residues in region I to strengthen binding with the novel ligand while maintaining the key interactions in region II for signaling. Following this strategy, we have successfully redesigned Tar to respond to l-arginine, a basic amino acid that does not have chemotactic effect for WT Tar, by two site-specific mutations (R69'E and R73'E).

Lateral density of receptor arrays in the membrane plane influences sensitivity of the E. coli chemotaxis response.

In chemotactic bacteria, transmembrane chemoreceptors, CheA and CheW form the core signalling complex of the chemotaxis sensory apparatus. These complexes are organized in extended arrays in the cytoplasmic membrane that allow bacteria to respond to changes in concentration of extracellular ligands via a cooperative, allosteric response that leads to substantial amplification of the signal induced by ligand binding. Here, we have combined cryo-electron tomographic studies of the 3D spatial architecture of chemoreceptor arrays in intact E. coli cells with computational modelling to develop a predictive model for the cooperativity and sensitivity of the chemotaxis response. The predictions were tested experimentally using fluorescence resonance energy transfer (FRET) microscopy. Our results demonstrate that changes in lateral packing densities of the partially ordered, spatially extended chemoreceptor arrays can modulate the bacterial chemotaxis response, and that information about the molecular organization of the arrays derived by cryo-electron tomography of intact cells can be translated into testable, predictive computational models of the chemotaxis response.

Free energy cost of reducing noise while maintaining a high sensitivity.

Living systems need to be highly responsive, and also to keep fluctuations low. These goals are incompatible in equilibrium systems due to the fluctuation dissipation theorem (FDT). Here, we show that biological sensory systems, driven far from equilibrium by free energy consumption, can reduce their intrinsic fluctuations while maintaining high responsiveness. By developing a continuum theory of the E. coli chemotaxis pathway, we demonstrate that adaptation can be understood as a nonequilibrium phase transition controlled by free energy dissipation, and it is characterized by a breaking of the FDT. We show that the maximum response at short time is enhanced by free energy dissipation. At the same time, the low frequency fluctuations and the adaptation error decrease with the free energy dissipation algebraically and exponentially, respectively.