The cost of sensitive response and accurate adaptation in networks with an incoherent type-1 feed-forward loop

Cascade-enhanced transport efficiency of biochemical systems

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

Improved estimation for energy dissipation in biochemical oscillations

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

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

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

KLF4, a Key Regulator of a Transitive Triplet, Acts on the TGF-β Signaling Pathway and Contributes to High-Altitude Adaptation of Tibetan Pigs

Tibetan pigs are native mammalian species on the Tibetan Plateau that have evolved distinct physiological traits that allow them to tolerate high-altitude hypoxic environments. However, the genetic mechanism underlying this adaptation remains elusive. Here, based on multitissue transcriptional data from high-altitude Tibetan pigs and low-altitude Rongchang pigs, we performed a weighted correlation network analysis (WGCNA) and identified key modules related to these tissues. Complex network analysis and bioinformatics analysis were integrated to identify key genes and three-node network motifs. We found that among the six tissues (muscle, liver, heart, spleen, kidneys, and lungs), lung tissue may be the key organs for Tibetan pigs to adapt to hypoxic environment. In the lung tissue of Tibetan pigs, we identified KLF4, BCL6B, EGR1, EPAS1, SMAD6, SMAD7, KDR, ATOH8, and CCN1 genes as potential regulators of hypoxia adaption. We found that KLF4 and EGR1 genes might simultaneously regulate the BCL6B gene, forming a KLF4-EGR1-BCL6B complex. This complex, dominated by KLF4, may enhance the hypoxia tolerance of Tibetan pigs by mediating the TGF-β signaling pathway. The complex may also affect the PI3K-Akt signaling pathway, which plays an important role in angiogenesis caused by hypoxia. Therefore, we postulate that the KLF4-EGR1-BCL6B complex may be beneficial for Tibetan pigs to survive better in the hypoxia environments. Although further molecular experiments and independent large-scale studies are needed to verify our findings, these findings may provide new details of the regulatory architecture of hypoxia-adaptive genes and are valuable for understanding the genetic mechanism of hypoxic adaptation in mammals.

Proofreading through spatial gradients

Key enzymatic processes use the nonequilibrium error correction mechanism called kinetic proofreading to enhance their specificity. The applicability of traditional proofreading schemes, however, is limited because they typically require dedicated structural features in the enzyme, such as a nucleotide hydrolysis site or multiple intermediate conformations. Here, we explore an alternative conceptual mechanism that achieves error correction by having substrate binding and subsequent product formation occur at distinct physical locations. The time taken by the enzyme–substrate complex to diffuse from one location to another is leveraged to discard wrong substrates. This mechanism does not have the typical structural requirements, making it easier to overlook in experiments. We discuss how the length scales of molecular gradients dictate proofreading performance, and quantify the limitations imposed by realistic diffusion and reaction rates. Our work broadens the applicability of kinetic proofreading and sets the stage for studying spatial gradients as a possible route to specificity.

Optimizing energetic cost of uncertainty in a driven system with and without feedback

Many biological functions require dynamics to be necessarily driven out of equilibrium. In contrast, in various contexts, a nonequilibrium dynamics at fast timescales can be described by an effective equilibrium dynamics at a slower timescale. In this work, we study two different aspects: (i) the energy-efficiency tradeoff for a specific nonequilibrium linear dynamics of two variables with feedback and (ii) the cost of effective parameters in a coarse-grained theory as given by the "hidden" dissipation and entropy production rate in the effective equilibrium limit of the dynamics. To meaningfully discuss the tradeoff between energy consumption and the efficiency of the desired function, a one-to-one mapping between function(s) and energy input is required. The function considered in this work is the variance of one of the variables. We get a one-to-one mapping by considering the minimum variance obtained for a fixed entropy production rate and vice versa. We find that this minimum achievable variance is a monotonically decreasing function of the given entropy production rate. When there is a timescale separation, in the effective equilibrium limit, the cost of the effective potential and temperature is the associated "hidden" entropy production rate.

Stochastic Dynamics of Gene Switching and Energy Dissipation for Gene Expression

Stochastic dynamics of gene switching and energy dissipation for gene expression are largely unknown, mainly due to the complexity of non-equilibrium mechanisms. Here, based on an important double-deck loop model, the stochastic mechanisms of gene switching and energy dissipation are studied. First, the probability distributions of steady states are calculated theoretically. It is found that the signal can strengthen the choice of gene switching between the "off" and "on" states. Our analysis of energy consumption illustrates that, compared with the synthesis and degradation of proteins, the process of gene switching costs little energy. Our theoretical analysis reveals some interesting insights into the determination of cell state and energy dissipation for gene expression.

Thermodynamic Cost, Speed, Fluctuations, and Error Reduction of Biological Copy Machines

Due to large fluctuations in cellular environments, transfer of information in biological processes without regulation is error-prone. The mechanistic details of error-reducing mechanisms in biological copying processes have been a subject of active research; however, how error reduction of a process is balanced with its thermodynamic cost and dynamical properties remain largely unexplored. Here, we study the error reducing strategies in light of the recently discovered thermodynamic uncertainty relation (TUR) that sets a physical bound to the cost-precision trade-off for dissipative processes. We found that the two representative copying processes, DNA replication by the exonuclease-deficient T7 DNA polymerase and mRNA translation by the E. coli ribosome, reduce the error rates to biologically acceptable levels while also optimizing the processes close to the physical limit dictated by TUR.

Trade-offs between effectiveness and cost in bifunctional enzyme circuit with concentration robustness

A fundamental trade-off in biological systems is whether they consume resources to perform biological functions or save resources. Bacteria need to reliably and rapidly respond to input signals by using limited cellular resources. However, excessive resource consumption will become a burden for bacteria growth. To investigate the relationship between functional effectiveness and resource cost, we study the ubiquitous bifunctional enzyme circuit, which is robust to fluctuations in protein concentration and responds quickly to signal changes. We show that trade-off relationships exist between functional effectiveness and protein cost. Expressing more proteins of the circuit increases concentration robustness and response speed but affects bacterial growth. In particular, our study reveals a general relationship between free-energy dissipation rate, response speed, and concentration robustness. The dissipation of free energy plays an important role in the concentration robustness and response speed. High robustness can only be achieved with a large amount of free-energy consumption and protein cost. In addition, the noise of the output increases with increasing protein cost, while the noise of the response time decreases with increasing protein cost. We also calculate the trade-off relationships in the EnvZ-OmpR system and the nitrogen assimilation system, which both have the bifunctional enzyme. Similar results indicate that these relationships are mainly derived from the specific feature of the bifunctional enzyme circuits and are not relevant to the details of the models. According to the trade-off relationships, bacteria take a compromise solution that reliably performs biological functions at a reasonable cost.

Adaptation of Living Systems

Adaptation refers to the biological phenomenon where living systems change their internal states in response to changes in their environments in order to maintain certain key functions critical for their survival and fitness. Adaptation is one of the most ubiquitous and arguably one of the most fundamental properties of living systems. It occurs throughout all biological scales, from adaptation of populations of species over evolutionary time to adaptation of a single cell to different environmental stresses during its life span. In this article, we review some of the recent progress made in understanding molecular mechanisms of cellular level adaptation. We take the minimalist (or the physicist) approach and study the simplest systems that exhibit generic adaptive behaviors. We focus on understanding the basic biochemical interaction networks in living matter that are responsible for adaptation dynamics. By combining theoretical modeling with quantitative experimentation, we demonstrate universal features in adaptation as well as important differences in different cellular systems, including chemotaxis in bacterium cells (Escherichia coli) and eukaryotic cells (Dictyostelium). Future work in extending the modeling framework to study adaptation in more complex systems such as sensory neurons are discussed.

Mutual inhibition of lateral inhibition: a network motif for an elementary computation in the brain

A series of classical studies in non-human primates has revealed the neuronal activity patterns underlying decision-making. However, the circuit mechanisms for such patterns remain largely unknown. Recent detailed circuit analyses in simpler neural systems have started to reveal the connectivity patterns underlying analogous processes. Here we review a few of these systems that share a particular connectivity pattern, namely mutual inhibition of lateral inhibition. Close examination of these systems suggests that this recurring connectivity pattern ('network motif') is a building block to enforce particular dynamics, which can be used not only for simple behavioral choice but also for more complex choices and other brain functions. Thus, a network motif provides an elementary computation that is not specific to a particular brain function and serves as an elementary building block in the brain.

Understanding Robust Adaptation Dynamics of Gene Regulatory Network

Robust adaptation is a critical attribute for gene regulatory network (GRN), understanding the relationship between adaptation and the GRN topology, and corresponding parameters is a challenging issue. The work in this paper includes: first, seven constraint multiobjective optimization algorithms are used to find sufficient solutions to get more reliable statistic rules. Meanwhile, the algorithms are compared to facilitate the future algorithm selection; second, a fuzzy c-mean algorithm is used to analyze solutions and to classify the solutions into different groups; third, the histogram analysis for all satisfactory solutions shows the preferred parameter range, i.e., parameter motif. The contributions of this paper includes: 1) Two new adaptation indices i.e., peak time and settle down time, are proposed for the first time to give more accurate description of the robust adaptation. Our conclusion is that some solutions even with satisfactory sensitivity and precision are not practically of robust adaptation because of too long time needed. 2) The relationship between topology, parameter set, and robust adaptation of GRN is discovered in the sense of both preferred topology and parameter motif. Our conclusion is that the robust adaptation depends more on the GRN topology than the model parameter set in two feasible topologies.

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.

The free energy cost of accurate biochemical oscillations

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 three different types of biochemical oscillations representing three basic oscillation motifs shared by all known oscillatory systems. In all the systems studied, we find that the phase diffusion constant depends on the free energy dissipation per period following the same inverse relation parameterized by system specific constants. This relationship and its range of validity are 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 general relationship and testable predictions are also presented.

Energy dissipation drives the gradient signal amplification through an incoherent type-1 feed-forward loop

We present here the analytical relation between the gain of eukaryotic gradient sensing network and the associated thermodynamic cost. By analyzing a general incoherent type-1 feed-forward loop, we derive the gain function (G) through the reaction network and explicitly show that G depends on the nonequilibrium factor (0≤γ≤1 with γ=0 and 1 representing irreversible and equilibrium reaction systems, respectively), the Michaelis constant (K_{M}), and the turnover ratio (r_{cat}) of the participating enzymes. We further find the maximum possible gain is intrinsically determined by K_{M}/G_{max}=(1/K_{M}+2)/4. Our model also indicates that the dissipated energy (measured by -lnγ), from the intracellular energy-bearing bioparticles (e.g., ATP), is used to generate a force field F_{γ}∝(1-sqrt[γ]) that reshapes and disables the effective potential around the zero gain region, which leads to the ultrasensitive response to external chemical gradients.

An Optimal Free Energy Dissipation Strategy of the MinCDE Oscillator in Regulating Symmetric Bacterial Cell Division

Sustained molecular oscillations are ubiquitous in biology. The obtained oscillatory patterns provide vital functions as timekeepers, pacemakers and spacemarkers. Models based on control theory have been introduced to explain how specific oscillatory behaviors stem from protein interaction feedbacks, whereas the energy dissipation through the oscillating processes and its role in the regulatory function remain unexplored. Here we developed a general framework to assess an oscillator’s regulation performance at different dissipation levels. Using the Escherichia coli MinCDE oscillator as a model system, we showed that a sufficient amount of energy dissipation is needed to switch on the oscillation, which is tightly coupled to the system’s regulatory performance. Once the dissipation level is beyond this threshold, unlike stationary regulators’ monotonic performance-to-cost relation, excess dissipation at certain steps in the oscillating process damages the oscillator’s regulatory performance. We further discovered that the chemical free energy from ATP hydrolysis has to be strategically assigned to the MinE-aided MinD release and the MinD immobilization steps for optimal performance, and a higher energy budget improves the robustness of the oscillator. These results unfold a novel mode by which living systems trade energy for regulatory function.

This paper presents a unique dissipation mode of converting biochemical free energy in ATP to regulatory function through the MinCDE bio-oscillator that marks the mid-cell position for symmetric bacterial cell division. Through assessing the oscillator’s performance-to-cost relation, we demonstrate that some dissipation threshold needs to be satisfied to switch on the oscillation, but the oscillator’s performance can be damaged by excess free energy dissipation, which is distinct from the known monotonic tradeoff relation of stationary regulators. An optimal dissipation strategy has been unveiled: the ATP free energy must be precisely allocated to specific reaction steps for accurate mid-cell recognition, which also coincides with the dynamic requirements for robust oscillation to occur. These discoveries identify an optimizable operation scheme of free energy consumption in biological systems and provide deep insights into the evolution of dynamic regulatory networks.

Adaptive Responses Limited by Intrinsic Noise

Sensory systems have mechanisms to respond to the external environment and adapt to them. Such adaptive responses are effective for a wide dynamic range of sensing and perception of temporal change in stimulus. However, noise generated by the adaptation system itself as well as extrinsic noise in sensory inputs may impose a limit on the ability of adaptation systems. The relation between response and noise is well understood for equilibrium systems in the form of fluctuation response relation. However, the relation for nonequilibrium systems, including adaptive systems, are poorly understood. Here, we systematically explore such a relation between response and fluctuation in adaptation systems. We study the two network motifs, incoherent feedforward loops (iFFL) and negative feedback loops (nFBL), that can achieve perfect adaptation. We find that the response magnitude in adaption systems is limited by its intrinsic noise, implying that higher response would have higher noise component as well. Comparing the relation of response and noise in iFFL and nFBL, we show that whereas iFFL exhibits adaptation over a wider parameter range, nFBL offers higher response to noise ratio than iFFL. We also identify the condition that yields the upper limit of response for both network motifs. These results may explain the reason of why nFBL seems to be more abundant in nature for the implementation of adaption systems.

Thermodynamic costs of information processing in sensory adaptation

Biological sensory systems react to changes in their surroundings. They are characterized by fast response and slow adaptation to varying environmental cues. Insofar as sensory adaptive systems map environmental changes to changes of their internal degrees of freedom, they can be regarded as computational devices manipulating information. Landauer established that information is ultimately physical, and its manipulation subject to the entropic and energetic bounds of thermodynamics. Thus the fundamental costs of biological sensory adaptation can be elucidated by tracking how the information the system has about its environment is altered. These bounds are particularly relevant for small organisms, which unlike everyday computers, operate at very low energies. In this paper, we establish a general framework for the thermodynamics of information processing in sensing. With it, we quantify how during sensory adaptation information about the past is erased, while information about the present is gathered. This process produces entropy larger than the amount of old information erased and has an energetic cost bounded by the amount of new information written to memory. We apply these principles to the E. coli's chemotaxis pathway during binary ligand concentration changes. In this regime, we quantify the amount of information stored by each methyl group and show that receptors consume energy in the range of the information-theoretic minimum. Our work provides a basis for further inquiries into more complex phenomena, such as gradient sensing and frequency response.

Modelling and analysis of a gene-regulatory feed-forward loop with basal expression of the second regulator

Efficient adaptation strategies to changing environmental conditions are essential for bacteria to survive and grow. Fundamental restructuring of their metabolism is usually mediated by corresponding gene regulation. Here, often several different environmental stimuli have to be integrated into a reasonable, energy-efficient response. Fast fluctuations and overshooting have to be filtered out. The gene regulatory network for the anaerobic adaptation of the pathogenic bacterium Pseudomonas aeruginosa is organized as a feed-forward loop (FFL), which is a three-gene network motif composed of two transcription factors (Anr for oxygen, NarxL for nitrate) and one target (Nar for nitrate reductase). The upstream transcription factor (Anr) induces the downstream transcription factor (NarXL). Both regulators act together positively by inducing the target (Nar) via a direct and indirect regulation path (coherent type-1 FFL). Since full promoter activity is only achieved when both transcription factors are present the target operon is expressed with a delay. Thus, in response to environmental stimuli (oxygen, nitrate), signals are mediated and processed in a way that short pulses are filtered out. In this study we analyze a special kind of FFL called FFLk by means of a family of ordinary differential equation models. The secondary FFL regulator (NarXL) is expressed constitutively but further induced in the presence of the upstream stimuli. This FFL modification has substantial influence on the response time and cost-benefit ratio mediated by environmental fluctuations. In order to find conditions where this regulatory network motif might be beneficial, we analyzed various models and environments. We describe the observed evolutional advantage of FFLk and its role in environmental adaptation and pathogenicity.