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Spontaneous pattern formation in homogeneous systems is ubiquitous in nature. Although Turing demonstrated that spatial patterns can emerge in reaction-diffusion (RD) systems when the homogeneous state becomes linearly unstable, it remains unclear whether the Turing mechanism is the only route for pattern formation. Here, we develop an efficient algorithm to systematically map the solution landscape to find all steady-state solutions. By applying our method to generic RD models, we find that stable spatial patterns can emerge via saddle-node bifurcations before the onset of Turing instability. Furthermore, by using a generalized action in functional space based on large deviation theory, our method is extended to evaluate stability of spatial patterns against noise. Applying this general approach in a three-species RD model, we show that though formation of Turing patterns only requires two chemical species, the third species is critical for stabilizing patterns against strong intrinsic noise in small biochemical systems.

Although optimal control (OC) has been studied in stochastic thermodynamics for systems with continuous state variables, less is known in systems with discrete state variables, such as Chemical Reaction Networks (CRNs). Here, we develop a general theoretical framework to study OC of CRNs for changing the system from an initial distribution of states to a final distribution with minimum dissipation. We derive a ``Kirchhoff's law" for the probability current in the adiabatic limit, from which the optimal kinetic rates are determined analytically for any given probability trajectory. By using the optimal rates, we show that the total dissipation is determined by a $L_2$-distance measure in the probability space and derive an analytical expression for the metric tensor that depends on the probability distribution, network topology, and capacity of each link. Minimizing the total dissipation leads to the geodesic trajectory in the probability space and the corresponding OC protocol is determined by the Kirchhoff's law. To demonstrate our general approach, we use it to find a lower bound for the minimum dissipation that is tighter than existing bounds obtained with only global constraints. We also apply it to simple networks, e.g., fully connected 3-state CRNs with different local constraints and show that indirect pathway and non-functional transient state can play a crucial role in switching between different probability distributions efficiently. Future directions in studying OC in CRNs by using our general framework are discussed., Comment: 6 pages,3 figures, with supplement

The integration of neural representations in the two hemispheres is an important problem in neuroscience. Recent experiments revealed that odor responses in cortical neurons driven by separate stimulation of the two nostrils are highly correlated. This bilateral alignment points to structured inter-hemispheric connections, but detailed mechanism remains unclear. Here, we hypothesized that continuous exposure to environmental odors shapes these projections and modeled it as online learning with local Hebbian rule. We found that Hebbian learning with sparse connections achieves bilateral alignment, exhibiting a linear trade-off between speed and accuracy. We identified an inverse scaling relationship between the number of cortical neurons and the inter-hemispheric projection density required for desired alignment accuracy, i.e., more cortical neurons allow sparser inter-hemispheric projections. We next compared the alignment performance of local Hebbian rule and the global stochastic-gradient-descent (SGD) learning for artificial neural networks. We found that although SGD leads to the same alignment accuracy with modestly sparser connectivity, the same inverse scaling relation holds. We showed that their similar performance originates from the fact that the update vectors of the two learning rules align significantly throughout the learning process. This insight may inspire efficient sparse local learning algorithms for more complex problems.

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 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 counter-intuitive 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 at regimes far from the onset where period-lengthening reactions exist., Comment: 19 pages, 11 figures (main text + supplementary information)

The Escherichia coli chemoreceptors form an extensive array that achieves cooperative and adaptive sensing of extracellular signals. The receptors control the activity of histidine kinase CheA, which drives a nonequilibrium phosphorylation-dephosphorylation reaction cycle for response regulator CheY. Cooperativity and dissipation are both important aspects of chemotaxis signaling, yet their consequences have only been studied separately. Recent single-cell FRET measurements revealed that kinase activity of the array spontaneously switches between active and inactive states, with asymmetric switching times that signify time-reversal symmetry breaking in the underlying dynamics. Here, we present a nonequilibrium lattice model of the chemosensory array, which demonstrates that the observed asymmetric switching dynamics can only be explained by an interplay between the dissipative reactions within individual core units and the cooperative coupling between neighboring units. Microscopically, the switching time asymmetry originates from irreversible transition paths. The model shows that strong dissipation enables sensitive and rapid signaling response by relieving the speed-sensitivity trade-off, which can be tested by future single-cell experiments. Overall, our model provides a general framework for studying biological complexes composed of coupled subunits that are individually driven by dissipative cycles and the rich nonequilibrium physics within., Comment: 11 pages, 5 figures. SI included as an ancillary PDF file

Granular flows in small-outlet hoppers exhibit several characteristic but poorly understood behaviors: temporary clogs (pauses) that last for an extended period before flow spontaneously restarts, permanent clogs that last indefinitely, and non-Gaussian, non-monotonic flow-rate statistics. These aspects have been extensively studied independently, but a model of hopper flow that includes all three has not been formulated. Here, we introduce such a phenomenological model that provides a unifying dynamical explanation of all three behaviors: a coupling between the flow rate and a hidden configurational mode that controls the stability of clogs. In the theory, flow rate evolves according to Langevin dynamics with multiplicative noise and an absorbing state at zero flow, conditional on the hidden mode. The model fully reproduces the statistics of pause and clog events taken from a large ($>40,000$ flows) experimental dataset, including non-exponentially distributed clogging times and non-Gaussian flow rate distribution, and explains the stretched-exponential growth of the average clogging time with outlet size. Further, we highlight several microscopic configurational features of this hidden mode, including size and smoothness of the static arch structure formed during pauses and clogs. Our work provides a unifying framework for several poorly understood granular phenomena, and suggests numerous new paths toward further understanding of this complex system., Comment: 7 pages, 4 figures; SM included as an ancillary PDF file

The most effective domain adaptation (DA) involves the decomposition of data representation into a domain independent representation (DIRep), and a domain dependent representation (DDRep). A classifier is trained by using the DIRep of the labeled source images. Since the DIRep is domain invariant, the classifier can be "transferred" to make predictions for the target domain with no (or few) labels. However, information useful for classification in the target domain can "hide" in the DDRep in current DA algorithms such as Domain-Separation-Networks (DSN). DSN's weak constraint to enforce orthogonality of DIRep and DDRep, allows this hiding and can result in poor performance. To address this shortcoming, we developed a new algorithm wherein a stronger constraint is imposed to minimize the DDRep by using a KL divergent loss for the DDRep in order to create the maximal DIRep that enhances transfer learning performance. By using synthetic data sets, we show explicitly that depending on initialization DSN with its weaker constraint can lead to sub-optimal solutions with poorer DA performance whereas our algorithm with maximal DIRep is robust against such perturbations. We demonstrate the equal-or-better performance of our approach against state-of-the-art algorithms by using several standard benchmark image datasets including Office. We further highlight the compatibility of our algorithm with pretrained models, extending its applicability and versatility in real-world scenarios.

The Escherichia coli chemotaxis signaling pathway has served as a model system for studying 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 this asymmetric shift in binding and kinase response is inconsistent with equilibrium allosteric models regardless of parameter choices. To resolve this inconsistency, we present a nonequilibrium allosteric model that explicitly includes the dissipative reaction cycles driven by ATP hydrolysis. The model successfully explains all existing measurements for both aspartate and serine receptors. Our results suggest that while ligand binding controls the equilibrium balance between the ON and OFF states of the kinase, receptor methylation modulates the kinetic properties (e.g., the phosphorylation rate) of the ON state. Furthermore, sufficient energy dissipation is necessary 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 new perspective on cooperative sensing by large protein complexes and opens up new research directions for understanding their microscopic mechanisms through simultaneous measurements and modeling of ligand binding and downstream responses., Comment: 12 (main text) + 4 (supplemental information) pages, 6+4 figures

Generative adversarial networks (GANs) are a class of machine-learning models that use adversarial training to generate new samples with the same (potentially very complex) statistics as the training samples. One major form of training failure, known as mode collapse, involves the generator failing to reproduce the full diversity of modes in the target probability distribution. Here, we present an effective model of GAN training, which captures the learning dynamics by replacing the generator neural network with a collection of particles in the output space; particles are coupled by a universal kernel valid for certain wide neural networks and high-dimensional inputs. The generality of our simplified model allows us to study the conditions under which mode collapse occurs. Indeed, experiments which vary the effective kernel of the generator reveal a mode collapse transition, the shape of which can be related to the type of discriminator through the frequency principle. Further, we find that gradient regularizers of intermediate strengths can optimally yield convergence through critical damping of the generator dynamics. Our effective GAN model thus provides an interpretable physical framework for understanding and improving adversarial training., Comment: 19 pages, 21 figures

Accurate and robust spatial orders are ubiquitous in living systems. In 1952, Alan Turing proposed an elegant mechanism for pattern formation based on spontaneous breaking of the spatial translational symmetry in the underlying reaction-diffusion system. Much is understood about dynamics and structure of Turing patterns. However, little is known about the energetic cost of Turing pattern. Here, we study nonequilibrium thermodynamics of a small spatially extended biochemical reaction-diffusion system by using analytical and numerical methods. We find that the onset of Turing pattern requires a minimum energy dissipation to drive the nonequilibrium chemical reactions. Above onset, only a small fraction of the total energy expenditure is used to overcome diffusion for maintaining the spatial pattern. We show that the positioning error decreases as energy dissipation increases following the same tradeoff relationship between timing error and energy cost in biochemical oscillatory systems. In a finite system, we find that a specific Turing pattern exists only within a finite range of total molecule number, and energy dissipation broadens the range, which enhances the robustness of the Turing pattern against molecule number fluctuations in living cells. These results are verified in a realistic model of the Muk system underlying DNA segregation in E. coli, and testable predictions are made for the dependence of the accuracy and robustness of the spatial pattern on the ATP/ADP ratio. In general, the theoretical framework developed here can be applied to study nonequilibrium thermodynamics of spatially extended biochemical systems., Comment: main text plus SI

Generalization is one of the most important problems in deep learning (DL). In the overparameterized regime in neural networks, there exist many low-loss solutions that fit the training data equally well. The key question is which solution is more generalizable. 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 how SGD drives the learning system to flat solutions, we construct a simple model whose loss landscape has a continuous set of degenerate (or near degenerate) minima. By solving the Fokker-Planck equation of the underlying stochastic learning dynamics, 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. Furthermore, a stronger SGD noise shortens the convergence time to the flat solutions. However, we identify an upper bound for the SGD noise beyond which the system fails to converge. Our results not only elucidate the role of SGD for generalization they may also have important implications for hyperparameter selection for learning efficiently without divergence., Comment: Main text: 11 pages, 3 figures; supplementary materials: 19 pages, 5 figures

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 tradeoff relationship provides a new perspective for understanding the dynamics of natural flocks and designing optimal artificial flocking systems., Comment: 5 (main text)+ 3 (appendix) + 15 (supplemental material) pages, 3+1+7 figures

One of the fundamental problems in machine learning is generalization. In neural network models with a large number of weights (parameters), many solutions can be found to fit the training data equally well. The key question is which solution can describe testing data not in the training set. Here, we report the discovery of an exact duality (equivalence) between changes in activities in a given layer of neurons and changes in weights that connect to the next layer of neurons in a densely connected layer in any feed forward neural network. The activity-weight (A-W) duality allows us to map variations in inputs (data) to variations of the corresponding dual weights. By using this mapping, we show that the generalization loss can be decomposed into a sum of contributions from different eigen-directions of the Hessian matrix of the loss function at the solution in weight space. The contribution from a given eigen-direction is the product of two geometric factors (determinants): the sharpness of the loss landscape and the standard deviation of the dual weights, which is found to scale with the weight norm of the solution. Our results provide an unified framework, which we used to reveal how different regularization schemes (weight decay, stochastic gradient descent with different batch sizes and learning rates, dropout), training data size, and labeling noise affect generalization performance by controlling either one or both of these two geometric determinants for generalization. These insights can be used to guide development of algorithms for finding more generalizable solutions in overparametrized neural networks.

In this article, we develop a mathematical model for the rotary bacterial flagellar motor (BFM) based on the recently discovered structure of the stator complex (MotA$_5$MotB$_2$). The structure suggested that the stator also rotates. The BFM is modeled as two rotating nano-rings that interact with each other. Specifically, translocation of protons through the stator complex drives rotation of the MotA pentamer ring, which in turn drives rotation of the FliG ring in the rotor via interactions between the MotA ring of the stator and the FliG ring of the rotor. Preliminary results from the structure-informed model are consistent with the observed torque-speed relation. More importantly, the model predicts distinctive rotor and stator dynamics and their load dependence, which may be tested by future experiments. Possible approaches to verify and improve the model to further understanding of the molecular mechanism for torque generation in BFM are also discussed.

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 (SSRG) 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: a family of nontrivial fixed points where the correlation exhibits power-law decay and 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. We also discuss a possible connection between the RG flows of flux correlation with those of the Kosterlitz-Thouless transition., Comment: 18 pages, 6 figures

Distributed Deep Learning (DDL) is essential for large-scale Deep Learning (DL) training. Synchronous Stochastic Gradient Descent (SSGD) 1 is the de facto DDL optimization method. Using a sufficiently large batch size is critical to achieving DDL runtime speedup. In a large batch setting, the learning rate must be increased to compensate for the reduced number of parameter updates. However, a large learning rate may harm convergence in SSGD and training could easily diverge. Recently, Decentralized Parallel SGD (DPSGD) has been proposed to improve distributed training speed. In this paper, we find that DPSGD not only has a system-wise run-time benefit but also a significant convergence benefit over SSGD in the large batch setting. Based on a detailed analysis of the DPSGD learning dynamics, we find that DPSGD introduces additional landscape-dependent noise that automatically adjusts the effective learning rate to improve convergence. In addition, we theoretically show that this noise smoothes the loss landscape, hence allowing a larger learning rate. We conduct extensive studies over 18 state-of-the-art DL models/tasks and demonstrate that DPSGD often converges in cases where SSGD diverges for large learning rates in the large batch setting. Our findings are consistent across two different application domains: Computer Vision (CIFAR10 and ImageNet-1K) and Automatic Speech Recognition (SWB300 and SWB2000), and two different types of neural network models: Convolutional Neural Networks and Long Short-Term Memory Recurrent Neural Networks.

Protein concentration in a living cell fluctuates over time due to noise in growth and division processes. From extensive single-cell experiments by using E. coli strains with different promoter strength (over two orders of magnitude) and under different nutrient conditions, we found that the variance of protein concentration fluctuations follows a robust square power-law dependence on its mean, which belongs to a general phenomenon called Taylor's law. To understand the mechanistic origin of this observation, we use a minimal mechanistic model to describe the stochastic growth and division processes in a single cell with a feedback mechanism for regulating cell division. The model reproduces the observed Taylor's law. The predicted protein concentration distributions agree quantitatively with those measured in experiments for different nutrient conditions and a wide range of promoter strength. By using a mean-field approximation, we derived a single Langevin equation for protein concentration with multiplicative noise, which can be solved exactly to prove the square Taylor's law and to obtain an analytical solution for the protein concentration distribution function that agrees with experiments. Analysis of experiments by using our model showed that noise in production rates dominates over noise from cell division in contributing to protein concentration fluctuations. In general, a multiplicative noise in the underlying stochastic dynamics may be responsible for Taylor's law in other systems., Comment: added appendix

Despite tremendous success of deep neural network in machine learning, the underlying reason for its superior learning capability remains unclear. Here, we present a framework based on statistical physics to study dynamics of stochastic gradient descent (SGD) that drives learning in neural networks. By using the minibatch gradient ensemble, we construct order parameters to characterize dynamics of weight updates in SGD. Without mislabeled data, we find that the SGD learning dynamics transitions from a fast learning phase to a slow exploration phase, which is associated with large changes in order parameters that characterize the alignment of SGD gradients and their mean amplitude. In the case with randomly mislabeled samples, SGD learning dynamics falls into four distinct phases. The system first finds solutions for the correctly labeled samples in phase I, it then wanders around these solutions in phase II until it finds a direction to learn the mislabeled samples during phase III, after which it finds solutions that satisfy all training samples during phase IV. Correspondingly, the test error decreases during phase I and remains low during phase II; however, it increases during phase III and reaches a high plateau during phase IV. The transitions between different phases can be understood by changes of order parameters that characterize the alignment of mean gradients for the correctly and incorrectly labeled samples and their (relative) strength during learning. We find that individual sample losses for the two datasets are most separated during phase II, which leads to a cleaning process to eliminate mislabeled samples for improving generalization.

The energy dissipation rate in a nonequilibirum 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 flux correlation. Implications of this inverse dissipation scaling law for active flow systems such as microtubule-kinesin mixture are discussed., Comment: 6 (main text)+ 21 (supplemental information) pages, 4+9 figures, 1 supplemental table

A model of coupled molecular oscillators is proposed to study nonequilibrium thermodynamics of synchronization. We find that synchronization of nonequilibrium oscillators costs energy even when the oscillator-oscillator coupling is conservative. By solving the steady state of the many-body system analytically, we show that the system goes through a nonequilibrium phase transition driven by energy dissipation, and the critical energy dissipation depends on both the frequency and strength of the exchange reactions. Moreover, our study reveals the optimal design for achieving maximum synchronization with a fixed energy budget. We apply our general theory to the Kai system in Cyanobacteria circadian clock and predict a relationship between the KaiC ATPase activity and synchronization of the KaiC hexamers. The theoretical framework can be extended to study thermodynamics of collective behaviors in other extended nonequilibrium active systems., Comment: 3 figures, plus Supplementary Material

Despite the tremendous success of Stochastic Gradient Descent (SGD) algorithm in deep learning, little is known about how SGD finds generalizable solutions in the high-dimensional weight space. By analyzing the learning dynamics and loss function landscape, we discover a robust inverse relation between the weight variance and the landscape flatness (inverse of curvature) for all SGD-based learning algorithms. To explain the inverse variance-flatness relation, we develop a random landscape theory, which shows that the SGD noise strength (effective temperature) depends inversely on the landscape flatness. Our study indicates that SGD attains a self-tuned landscape-dependent annealing strategy to find generalizable solutions at the flat minima of the landscape. Finally, we demonstrate how these new theoretical insights lead to more efficient algorithms, e.g., for avoiding catastrophic forgetting.

Synthesis of biopolymers such as DNA, RNA, and proteins are biophysical processes aided by enzymes. 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 paper, 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., Comment: PDF file consist of the main text (pages 1 to 5) and the supplementary material (pages 6 to 12). Overall, 7 figures split between main text and SI

Despite remarkable successes achieved by modern neural networks in a wide range of applications, these networks perform best in domain-specific stationary environments where they are trained only once on large-scale controlled data repositories. When exposed to non-stationary learning environments, current neural networks tend to forget what they had previously learned, a phenomena known as catastrophic forgetting. Most previous approaches to this problem rely on memory replay buffers which store samples from previously learned tasks, and use them to regularize the learning on new ones. This approach suffers from the important disadvantage of not scaling well to real-life problems in which the memory requirements become enormous. We propose a memoryless method that combines standard supervised neural networks with self-organizing maps to solve the continual learning problem. The role of the self-organizing map is to adaptively cluster the inputs into appropriate task contexts - without explicit labels - and allocate network resources accordingly. Thus, it selectively routes the inputs in accord with previous experience, ensuring that past learning is maintained and does not interfere with current learning. Out method is intuitive, memoryless, and performs on par with current state-of-the-art approaches on standard benchmarks., Comment: Continual Learning Workshop - NeurIPS 2018

Lack of performance when it comes to continual learning over non-stationary distributions of data remains a major challenge in scaling neural network learning to more human realistic settings. In this work we propose a new conceptualization of the continual learning problem in terms of a temporally symmetric trade-off between transfer and interference that can be optimized by enforcing gradient alignment across examples. We then propose a new algorithm, Meta-Experience Replay (MER), that directly exploits this view by combining experience replay with optimization based meta-learning. This method learns parameters that make interference based on future gradients less likely and transfer based on future gradients more likely. We conduct experiments across continual lifelong supervised learning benchmarks and non-stationary reinforcement learning environments demonstrating that our approach consistently outperforms recently proposed baselines for continual learning. Our experiments show that the gap between the performance of MER and baseline algorithms grows both as the environment gets more non-stationary and as the fraction of the total experiences stored gets smaller., Comment: ICLR 2019

Complex biological functions are carried out by the interaction of genes and proteins. Uncovering the gene regulation network behind a function is one of the central themes in biology. Typically, it involves extensive experiments of genetics, biochemistry and molecular biology. In this paper, we show that much of the inference task can be accomplished by a deep neural network (DNN), a form of machine learning or artificial intelligence. Specifically, the DNN learns from the dynamics of the gene expression. The learnt DNN behaves like an accurate simulator of the system, on which one can perform in-silico experiments to reveal the underlying gene network. We demonstrate the method with two examples: biochemical adaptation and the gap-gene patterning in fruit fly embryogenesis. In the first example, the DNN can successfully find the two basic network motifs for adaptation - the negative feedback and the incoherent feed-forward. In the second and much more complex example, the DNN can accurately predict behaviors of essentially all the mutants. Furthermore, the regulation network it uncovers is strikingly similar to the one inferred from experiments. In doing so, we develop methods for deciphering the gene regulation network hidden in the DNN "black box". Our interpretable DNN approach should have broad applications in genotype-phenotype mapping.

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 non-zero average velocity $\langle{\bf v}\rangle$) 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., Comment: 5 pages, 1 figure. The short paper on flocking with quenched noise

The effect of quenched (frozen) orientational disorder on the collective motion of active particles is analyzed. We find that, as with annealed disorder (Langevin noise), active polar systems are far more robust against quenched disorder than their equilibrium counterparts. In particular, long ranged order (i.e., the existence of a non-zero average velocity $\langle {\bf v} \rangle$) persists in the presence of quenched disorder even in spatial dimensions $d=3$, while it is destroyed even by arbitrarily weak disorder in $d \le 4$ in equilibrium systems. Furthermore, in $d=2$, quasi-long-ranged order (i.e., spatial velocity correlations that decay as a power law with distance) occurs when quenched disorder is present, in contrast to the short-ranged order that is all that can survive in equilibrium. These predictions are borne out by simulations in both two and three dimensions., Comment: 22 pages, 6 figures. The long paper on flocking with quenched noise

The performance of a molecular motor, characterized by its power output and energy efficiency, is investigated in the motor design space spanned by the stepping rate function and the motor-track interaction potential. Analytic results and simulations show that a gating mechanism that restricts forward stepping in a narrow window in configuration space is needed for generating high power at physiologically relevant loads. By deriving general thermodynamics laws for nonequilibrium motors, we find that the maximum torque (force) at stall is less than its theoretical limit for any realistic motor-track interactions due to speed fluctuations. Our study reveals a tradeoff for the motor- track interaction: while a strong interaction generates a high power output for forward steps, it also leads to a higher probability of wasteful spontaneous back steps. Our analysis and simulations show that this tradeoff sets a fundamental limit to the maximum motor efficiency in the presence of spontaneous back steps, i.e., loose-coupling. Balancing this tradeoff leads to an optimal design of the motor-track interaction for achieving a maximum efficiency close to 1 for realistic motors that are not perfectly coupled with the energy source.Comparison with existing data and suggestions for future experiments are discussed.

Oscillation is an important cellular process that regulates timing of different vital life cycles. However, in the noisy cellular environment, oscillations can be highly inaccurate due to phase fluctuations. It remains poorly understood how biochemical circuits suppress phase fluctuations and what is the incurred thermodynamic cost. Here, we study four different types of biochemical oscillations representing three basic oscillation motifs shared by all known oscillatory systems. We find that the phase diffusion constant follows the same inverse dependence on the free energy dissipation per period for all systems studied. This relationship between the phase diffusion and energy dissipation is shown analytically in a model of noisy oscillation. Microscopically, we find that the oscillation is driven by multiple irreversible cycles that hydrolyze the fuel molecules such as ATP; the number of phase coherent periods is proportional to the free energy consumed per period. Experimental evidence in support of this universal relationship and testable predictions are also presented.

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 non-equilibrium 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.

By integrating 4 lines of thoughts: symmetry breaking originally advanced by Anderson, bifurcation from nonlinear dynamics, Landau's theory of phase transition, and the mechanism of emergent rare events studied by Kramers, we introduce a possible framework for understanding mesoscopic dynamics that links (i) fast lower level microscopic motions, (ii) movements within each basin at the mid-level, and (iii) higher-level rare transitions between neighboring basins, which have rates that decrease exponentially with the size of the system. In this mesoscopic framework, multiple attractors arise as emergent properties of the nonlinear systems. The interplay between the stochasticity and nonlinearity leads to successive jump-like transitions among different basins. We argue each transition is a dynamic symmetry breaking, with the potential of exhibiting Thom-Zeeman catastrophe as well as phase transition with the breakdown of ergodicity (e.g., cell differentiation). The slow-time dynamics of the nonlinear mesoscopic system is not deterministic, rather it is a discrete stochastic jump process. The existence of these discrete states and the Markov transitions among them are both emergent phenomena. This emergent stochastic jump dynamics then serves as the stochastic element for the nonlinear dynamics of a higher level aggregates on an even larger spatial and slower time scales (e.g., evolution). This description captures the hierarchical structure outlined by Anderson and illustrates two distinct types of limit of a mesoscopic dynamics: A long-time ensemble thermodynamics in terms of time $t$ tending infinity followed by the size of the system $N$ tending infinity, and a short-time trajectory steady state with $N$ tending infinity followed by $t$ tending infinity. With these limits, symmetry breaking and cusp catastrophe are two perspectives of the same mesoscopic system on different time scales., Comment: 30 pages, 3 figures, 1 table

Two distinct mechanisms for filtering noise in an input signal are identified in a class of adaptive sensory networks. We find that the high frequency noise is filtered by the output degradation process through time-averaging; while the low frequency noise is damped by adaptation through negative feedback. Both filtering processes themselves introduce intrinsic noises, which are found to be unfiltered and can thus amount to a significant internal noise floor even without signaling. These results are applied to E. coli chemotaxis. We show unambiguously that the molecular mechanism for the Berg-Purcell time-averaging scheme is the dephosphorylation of the response regulator CheY-P, not the receptor adaptation process as previously suggested. The high frequency noise due to the stochastic ligand binding-unbinding events and the random ligand molecule diffusion is averaged by the CheY-P dephosphorylation process to a negligible level in E.coli. We identify a previously unstudied noise source caused by the random motion of the cell in a ligand gradient. We show that this random walk induced signal noise has a divergent low frequency component, which is only rendered finite by the receptor adaptation process. For gradients within the E. coli sensing range, this dominant external noise can be comparable to the significant intrinsic noise in the system. The dependence of the response and its fluctuations on the key time scales of the system are studied systematically. We show that the chemotaxis pathway may have evolved to optimize gradient sensing, strong response, and noise control in different time scales, Comment: 15 pages, 4 figures

Rotation of a single bacterial flagellar motor is powered by multiple stators tethered to the cell wall. In a "power-stroke" model the observed independence of the speed at low load on the number of stators is explained by a torque-dependent stepping mechanism independent of the strength of the stator tethering spring. On the other hand, in models that depend solely on the stator spring to explain the observed behavior, exceedingly small stator spring constants are required. To study the dynamics of the motor driven by external forces (such as those exerted by an optical tweezer), back-stepping is introduced when stators are driven far out of equilibrium. Our model with back-stepping reproduces the observed absence of a barrier to backward rotation, as well the behaviors in the high-speed negative-torque regime. Recently measured temperature dependence of the motor speed near zero load (Yuan & Berg 2010 Biophys J) is explained quantitatively by the thermally activated stepping rates in our model. Finally, we suggest that the general mechanical properties of all molecular motors (linear and rotary), characterized by their force(torque)-speed curve, can be determined by their power-stroke potentials and the dependence of the stepping rates on the mechanical state of the motor (force or speed). The torque-speed curve for the clockwise rotating flagellar motor has been observed for the first time recently (Yuan et al. 2010 PNAS). Its quasi-linear behavior is quantitatively reproduced by our model. In particular, we show that concave and convex shapes of the torque-speed curve can be achieved by changing the interaction potential from linear to quadratic form. We also show that reversing the stepping rate dependence on force (torque) can lead to non-monotonicity in the speed-load dependency., Comment: Article: 32 pages, 6 figures. Supplementary Information: 16 pages, 6 figures

A microscopic, stochastic, minimal model for collective and cohesive motion of identical self-propelled particles is introduced. Even though the particles interact strictly locally in a very noisy manner, we show that cohesion can be maintained, even in the zero-density limit of an arbitrarily large flock in an infinite space. The phase diagram spanned by the two main parameters of our model, which encode the tendencies for particles to align and to stay together, contains non-moving "gas", "liquid"' and "solid" phases separated from their moving counterparts by the onset of collective motion. The "gas/liquid" and "liquid/solid" are shown to be first-order phase transitions in all cases. In the cohesive phases, we study also the diffusive properties of individuals and their relation to the macroscopic motion and to the shape of the flock.

The signaling apparatus mediating bacterial chemotaxis can adapt to a wide range of persistent external stimuli. In many cases, the bacterial activity returns to its pre-stimulus level exactly and this "perfect adaptability" is robust against variations in various chemotaxis protein concentrations. We model the bacterial chemotaxis signaling pathway, from ligand binding to CheY phosphorylation. By solving the steady-state equations of the model analytically, we derive a full set of conditions for the system to achieve perfect adaptation. The conditions related to the phosphorylation part of the pathway are discovered for the first time, while other conditions are generalization of the ones found in previous works. Sensitivity of the perfect adaptation is evaluated by perturbing these conditions. We find that, even in the absence of some of the perfect adaptation conditions, adaptation can be achieved with near perfect precision as a result of the separation of scales in both chemotaxis protein concentrations and reaction rates, or specific properties of the receptor distribution in different methylation states. Since near perfect adaptation can be found in much larger regions of the parameter space than that defined by the perfect adaptation conditions, their existence is essential to understand robustness in bacterial chemotaxis., Comment: 16 pages, 9 figures