Physical origins of entropy production, free energy dissipation, and their mathematical representations

Neural mechanisms balancing accuracy and flexibility in working memory and decision tasks

The living system follows the principles of physics, yet distinctive features, such as adaptability, differentiate it from conventional systems. The cognitive functions of decision-making (DM) and working memory (WM) are crucial for animal adaptation, but the underlying mechanisms are still unclear. To explore the mechanism underlying DM and WM functions, here we applied a general non-equilibrium landscape and flux approach to a biophysically based model that can perform decision-making and working memory functions. Our findings reveal that DM accuracy improved with stronger resting states in the circuit architecture with selective inhibition. However, the robustness of working memory against distractors was weakened. To address this, an additional non-selective input during the delay period of decision-making tasks was proposed as a mechanism to gate distractors with minimal increase in thermodynamic cost. This temporal gating mechanism, combined with the selective-inhibition circuit architecture, supports a dynamical modulation that emphasizes the robustness or flexibility to incoming stimuli in working memory tasks according to the cognitive task demands. Our approach offers a quantitative framework to uncover mechanisms underlying cognitive functions grounded in non-equilibrium physics.

Speed-Energy-Efficiency Trade-off in Hsp70 Chaperone System

Proteins must fold into their native structure to carry out cellular functions. However, they can sometimes misfold into non-native structures, leading to reduced efficiency or malfunction. Chaperones help prevent misfolding by guiding proteins to their active state using energy from ATP hydrolysis. Experiments have revealed numerous kinetic and structural aspects of how various chaperones facilitate the folding of proteins into their native structure. However, what remains missing is a fundamental theoretical understanding of their operational mechanisms, especially the limits and constraints imposed on their efficiency by energy flow and dissipation. To address this, we built a kinetic model of the Hsp70 chaperone system by incorporating all key structural and kinetic details. Then, using the chemical kinetic equations, we investigate how energy expenditure shapes the efficiency of Hsp70 chaperones in the proper folding of misfolded proteins. We show that ATP consumption by chaperones significantly enhances the folding of proteins into their native states. Our investigations reveal that a chaperone achieves optimal efficiency when its binding to misfolded proteins is much faster than the misfolding kinetics of that protein. We also demonstrate the presence of an upper bound on a chaperone's efficiency of protein folding and its overall rescue rate. This upper bound increases with energy dissipation until it reaches a saturation point. Furthermore, we show a speed-energy-efficiency trade-off in chaperone action, demonstrating that it is impossible to simultaneously optimize the efficiency of chaperone-assisted protein folding and the energy efficiency of the process.

New insights into the prediction for the potential of soil organic carbon accumulation: From the perspective of non-equilibrium statistical mechanics

The accumulation of soil organic carbon (SOC) is significant for soil health and ecosystem services. Numerous studies have assessed the dynamic changes of SOC by considering the microbial system as an equilibrium system. However, they failed to reveal the complexity of the SOC accumulation/loss process, as the microbial system is a non-equilibrium system affected by stochastic fluctuations from the external environment. This study is the first to explore the complex non-equilibrium relationship between microbial carbon use efficiency (CUE) and SOC by using potential landscape and flux in non-equilibrium statistical mechanics. Nitrogen (N) was identified as the most critical environmental factor influencing CUE on a global scale, with the transition between the carbon loss state and the carbon sequestration state observed along N gradients. Random perturbations of other environmental factors could also trigger transition. Non-equilibrium thermodynamic quantities indicated that carbon sequestration had the potential to be achieved when N = 0.5 g/kg, where active soil management measures should be taken. Furthermore, the non-equilibrium relationship between CUE and SOC was clarified through potential energy analysis, where the average deviation between predictions and actual observations of SOC is about 1.9792 g/kg. This study provides an effective framework for predicting SOC accumulation.

Exact time-dependent thermodynamic relations for a Brownian particle moving in a ratchet potential coupled with quadratically decreasing temperature

The thermodynamic relations for a Brownian particle moving in a discrete ratchet potential coupled with quadratically decreasing temperature are explored as a function of time. We show that this thermal arrangement leads to a higher velocity (lower efficiency) compared to a Brownian particle operating between hot and cold baths, and a heat bath where the temperature linearly decreases along with the reaction coordinate. The results obtained in this study indicate that if the goal is to design a fast-moving motor, the quadratic thermal arrangement is more advantageous than the other two thermal arrangements. In contrast, the entropy, entropy production rate, and entropy extraction rate are significantly larger in the case of a quadratically decreasing temperature compared to the linearly decreasing temperature case and piecewise constant temperature case. Furthermore, the thermodynamic features of a system consisting of several Brownian ratchets arranged in a complex network are explored. The theoretical findings exhibit that as the network size increases, the entropy, entropy production, and entropy extraction of the system also increase, showing that these thermodynamic quantities exhibit extensive property. As a result, as the number of lattice sizes increases, thermodynamic relations such as entropy, entropy production, and entropy extraction also step up, confirming that these complex networks cannot be reduced to a corresponding one-dimensional lattice. However, in the long time limit, thermodynamic relations such as velocity, entropy production rate, and entropy extraction rate become independent of the network size. These results are also confirmed via a continuum Fokker-Planck model for the overdamped case.

A topological mechanism for robust and efficient global oscillations in biological networks

Long and stable timescales are often observed in complex biochemical networks, such as in emergent oscillations. How these robust dynamics persist remains unclear, given the many stochastic reactions and shorter time scales demonstrated by underlying components. We propose a topological model that produces long oscillations around the network boundary, reducing the system dynamics to a lower-dimensional current in a robust manner. Using this to model KaiC, which regulates the circadian rhythm in cyanobacteria, we compare the coherence of oscillations to that in other KaiC models. Our topological model localizes currents on the system edge, with an efficient regime of simultaneously increased precision and decreased cost. Further, we introduce a new predictor of coherence from the analysis of spectral gaps, and show that our model saturates a global thermodynamic bound. Our work presents a new mechanism and parsimonious description for robust emergent oscillations in complex biological networks.

Probabilistic and maximum entropy modeling of chemical reaction systems: Characteristics and comparisons to mass action kinetic models

We demonstrate and characterize a first-principles approach to modeling the mass action dynamics of metabolism. Starting from a basic definition of entropy expressed as a multinomial probability density using Boltzmann probabilities with standard chemical potentials, we derive and compare the free energy dissipation and the entropy production rates. We express the relation between entropy production and the chemical master equation for modeling metabolism, which unifies chemical kinetics and chemical thermodynamics. Because prediction uncertainty with respect to parameter variability is frequently a concern with mass action models utilizing rate constants, we compare and contrast the maximum entropy model, which has its own set of rate parameters, to a population of standard mass action models in which the rate constants are randomly chosen. We show that a maximum entropy model is characterized by a high probability of free energy dissipation rate and likewise entropy production rate, relative to other models. We then characterize the variability of the maximum entropy model predictions with respect to uncertainties in parameters (standard free energies of formation) and with respect to ionic strengths typically found in a cell.

Early warning indicators of war and peace through the landscapes and flux quantifications

War and peace, spanning history, deeply affect society, economy, and individuals. Grasping their dynamics is vital to lessen harm and foster global peace. Yet, quantifying them remains hard. Our goal is to create a simple qualitative model using landscape and flux theory, exploring war and peace mechanisms. In this symmetric network, they appear as separate attraction basins, dynamically shifting. Analyzing landscape shape gives insights into global stability. Near critical points, indicators like cross correlations, autocorrelation times, and flickering frequency surge, as warnings. We also calculate the irreversible path between war and peace due to rotational flux. Global sensitivity analysis identifies history's role in system stability. In summary, our research unveils a way to understand war and peace complexities, enhancing knowledge of key elements that lead to conflict, aiding resolution.

Fluctuation theorems and thermodynamic inequalities for nonequilibrium processes stopped at stochastic times

We investigate the thermodynamics of general nonequilibrium processes stopped at stochastic times. We propose a systematic strategy for constructing fluctuation-theorem-like martingales for each thermodynamic functional, yielding a family of stopping-time fluctuation theorems. We derive second-law-like thermodynamic inequalities for the mean thermodynamic functional at stochastic stopping times, the bounds of which are even stronger than the thermodynamic inequalities resulting from the traditional fluctuation theorems when the stopping time is reduced to a deterministic one. Numerical verification is carried out for three well-known thermodynamic functionals, namely, entropy production, free energy dissipation, and dissipative work. These universal equalities and inequalities are valid for arbitrary stopping strategies, and thus provide a comprehensive framework with insights into the fundamental principles governing nonequilibrium systems.

Global dynamics, thermodynamics and non-equilibrium origin of bifurcations for single neuron dynamics

The understanding of neural excitability and oscillations in single neuron dynamics remains incomplete in terms of global stabilities and the underlying mechanisms for phase formation and associated phase transitions. In this study, we investigate the mechanism of single neuron excitability and spontaneous oscillations by analyzing the potential landscape and curl flux. The topological features of the landscape play a crucial role in assessing the stability of resting states and the robustness/coherence of oscillations. We analyze the excitation characteristics in Class I and Class II neurons and establish their relation to biological function. Our findings reveal that the average curl flux and associated entropy production exhibit significant changes near bifurcation or phase transition points. Moreover, the curl flux and entropy production offer insights into the dynamical and thermodynamical origins of nonequilibrium phase transitions and exhibit distinct behaviors in Class I and Class II neurons. Additionally, we quantify time irreversibility through the difference in cross-correlation functions in both forward and backward time, providing potential indicators for the emergence of nonequilibrium phase transitions in single neurons.

Quantifying nonequilibrium dynamics and thermodynamics of cell fate decision making in yeast under pheromone induction

Cellular responses to pheromone in yeast can range from gene expression to morphological and physiological changes. While signaling pathways are well studied, the cell fate decision-making during cellular polar growth is still unclear. Quantifying these cellular behaviors and revealing the underlying physical mechanism remain a significant challenge. Here, we employed a hidden Markov chain model to quantify the dynamics of cellular morphological systems based on our experimentally observed time series. The resulting statistics generated a stability landscape for state attractors. By quantifying rotational fluxes as the non-equilibrium driving force that tends to disrupt the current attractor state, the dynamical origin of non-equilibrium phase transition from four cell morphological fates to a single dominant fate was identified. We revealed that higher chemical voltage differences induced by a high dose of pheromone resulted in higher chemical currents, which will trigger a greater net input and, thus, more degrees of the detailed balance breaking. By quantifying the thermodynamic cost of maintaining morphological state stability, we demonstrated that the flux-related entropy production rate provides a thermodynamic origin for the phase transition in non-equilibrium morphologies. Furthermore, we confirmed that the time irreversibility in time series provides a practical way to predict the non-equilibrium phase transition.

Prevalence of multistability and nonstationarity in driven chemical networks

External flows of energy, entropy, and matter can cause sudden transitions in the stability of biological and industrial systems, fundamentally altering their dynamical function. How might we control and design these transitions in chemical reaction networks? Here, we analyze transitions giving rise to complex behavior in random reaction networks subject to external driving forces. In the absence of driving, we characterize the uniqueness of the steady state and identify the percolation of a giant connected component in these networks as the number of reactions increases. When subject to chemical driving (influx and outflux of chemical species), the steady state can undergo bifurcations, leading to multistability or oscillatory dynamics. By quantifying the prevalence of these bifurcations, we show how chemical driving and network sparsity tend to promote the emergence of these complex dynamics and increased rates of entropy production. We show that catalysis also plays an important role in the emergence of complexity, strongly correlating with the prevalence of bifurcations. Our results suggest that coupling a minimal number of chemical signatures with external driving can lead to features present in biochemical processes and abiogenesis.

Analysis of entropy production in finitely slow processes between nonequilibrium steady states

We investigate entropy production in finitely slow transitions between nonequilibrium steady states in Markov jump processes using the improved adiabatic approximation method, which has a close relationship with the slow driving perturbation. This method provides systematic improvement of the adiabatic approximation on infinitely slow transitions from which we obtain nonadiabatic corrections. We analyze two types of excess entropy production and confirm that the leading adiabatic contribution reproduces known results, and then obtain nonadiabatic corrections written in terms of thermodynamic metrics defined in protocol parameter spaces. We also numerically study the resulting excess entropy production in a two-state system.

Emergent second law for non-equilibrium steady states

The Gibbs distribution universally characterizes states of thermal equilibrium. In order to extend the Gibbs distribution to non-equilibrium steady states, one must relate the self-information \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{{{{{{\mathcal{I}}}}}}}}(x)=-\!\log ({P}_{{{{{{{{\rm{ss}}}}}}}}}(x))$$\end{document}I(x)=−log(Pss(x)) of microstate x to measurable physical quantities. This is a central problem in non-equilibrium statistical physics. By considering open systems described by stochastic dynamics which become deterministic in the macroscopic limit, we show that changes \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\Delta }}{{{{{{{\mathcal{I}}}}}}}}={{{{{{{\mathcal{I}}}}}}}}({x}_{t})-{{{{{{{\mathcal{I}}}}}}}}({x}_{0})$$\end{document}ΔI=I(xt)−I(x0) in steady state self-information along deterministic trajectories can be bounded by the macroscopic entropy production Σ. This bound takes the form of an emergent second law \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\Sigma }}+{k}_{b}{{\Delta }}{{{{{{{\mathcal{I}}}}}}}}\,\ge \,0$$\end{document}Σ+kbΔI≥0, which contains the usual second law Σ ≥ 0 as a corollary, and is saturated in the linear regime close to equilibrium. We thus obtain a tighter version of the second law of thermodynamics that provides a link between the deterministic relaxation of a system and the non-equilibrium fluctuations at steady state. In addition to its fundamental value, our result leads to novel methods for computing non-equilibrium distributions, providing a deterministic alternative to Gillespie simulations or spectral methods.

Contrary to states of thermal equilibrium, there is no universal characterization of non-equilibrium steady states displaying constant flows of energy and/or matter. Here, the authors make progress in this direction by deriving an emergent and stricter version of the second law of thermodynamics.

Observation of broken detailed balance in polymorphic transformation of bacterial flagellar filament

Living systems operate far from thermodynamic equilibrium, which usually manifests as broken detailed balance at the molecular scale. At larger scales with collective function of many molecules, the presence of non-equilibrium thermodynamics may not be evident. In bacterial motility, the switching dynamics of the flagellar rotary motor was recently discovered to be operating in non-equilibrium. However, the resulting motility pattern at the mesoscale, the run-and-tumble behavior, was normally considered to be a Poisson process that can be described by a two-state equilibrium model. Here, we studied the details of the run-and-tumble behavior by following the polymorphic transformation of the flagellar filaments, observing broken detailed balance that reveals its non-equilibrium nature. Evaluation of entropy production provided a direct measure of the lack of detailed balance, and a quantification of the rate of energy dissipation for bacterial run-and-tumble regulation.

The rate of thermodynamic cost against adiabatic and nonadiabatic fluctuations of a single gene circuit in Drosophila embryos

We study the stochastic dynamics of the externally regulating　gene circuit as an example of eve-skipped gene stripe in the development of Drosophila. Three gene regulation regimes are　considered: adiabatic phase when the switching rate of the gene from the OFF to the ON state is faster than the rate of mRNA degradation; nonadiabatic phase when the switching rate from the OFF to the ON state is slower than that of the mRNA degradation; the bursting phase when the gene switching is fast and transcription is very fast, while the ON state probability is very low. We found that the rate of　thermodynamic cost quantified by the entropy production rate can suppress the fluctuations of the gene circuit. Higher(lower) rate of thermodynamic cost leads to reduced (increased) fluctuations on the number of gene products in the adiabatic (nonadiabatic) regime. We also found that higher thermodynamic cost is often required to sustain the emergence of more gene states and therefore more heterogeneity coming from genetic mutations or epigenetics. We also study the stability of the gene state using the mean first passage time from one state to another. We found the monotonic decrease in time, i.e. on the stability of the state, in the transition from the nonadiabatic to the adiabatic regimes. Therefore, as the higher rate of thermodynamic cost suppresses the fluctuations, higher stability requires higher thermodynamics cost to maintain.

Exact time-dependent analytical solutions for entropy production rate in a system operating in a heat bath in which temperature varies linearly in space

The nonequilibrium thermodynamics feature of a Brownian motor is investigated by obtaining exact time-dependent solutions. This in turn enables us to investigate not only the long time property (steady state) but also the short time the behavior of the system. The general expressions for the free energy, entropy production e[over ̇]_{p}(t) as well as entropy extraction h[over ̇]_{d}(t) rates are derived for a system that is genuinely driven out of equilibrium by time-independent force as well as by spatially varying thermal background. We show that for a system that operates between hot and cold reservoirs, most of the thermodynamics quantities approach a nonequilibrium steady state in the long time limit. The change in free energy becomes minimal at a steady state. However, for a system that operates in a heat bath where its temperature varies linearly in space, the entropy production and extraction rates approach a nonequilibrium steady state while the change in free energy varies linearly in space. This reveals that unlike systems at equilibrium, when systems are driven out of equilibrium, their free energy may not be minimized. The thermodynamic properties of a system that operates between the hot and cold baths are further compared and contrasted with a system that operates in a heat bath where its temperature varies linearly in space along with the reaction coordinate. We show that the entropy, entropy production, and extraction rates are considerably larger for the linearly varying temperature case than a system that operates between the hot and cold baths revealing such systems are inherently irreversible. For both cases, in the presence of load or when a distinct temperature difference is retained, the entropy S(t) monotonously increases with time and saturates to a constant value as t further steps up. The entropy production rate e[over ̇]_{p} decreases in time and at steady state, e[over ̇]_{p}=h[over ̇]_{d}>0, which agrees with the results shown in M. Asfaw's [Phys. Rev. E 89, 012143 (2014)1539-375510.1103/PhysRevE.89.012143; Phys. Rev. E 92, 032126 (2015)10.1103/PhysRevE.92.032126]. Moreover, the velocity, as well as the efficiency of the system that operates between the hot and cold baths, are also collated and contrasted with a system that operates in a heat bath where its temperature varies linearly in space along with the reaction coordinate. A system that operates between the hot and cold baths has significantly lower velocity but a higher efficiency in comparison with a linearly varying temperature case.

Metastable dynamics of neural circuits and networks

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

The onset of dissipative chaos driven by nonequilibrium conditions

Dissipative chaos appears widely in various nonequilibrium systems; however, it is not clear how dissipative chaos originates from nonequilibrium. We discuss a framework based on the potential-flux approach to study chaos from the perspective of nonequilibrium dynamics. In this framework, chaotic systems possess a wide basin on the potential landscape, in which the rotational flux dominates the system dynamics, and chaos occurs with the appearance of this basin. In contrast, the probability flux is particularly associated with the detailed balance-breaking in nonequilibrium systems. This implies that the appearance of dissipative chaos is driven by nonequilibrium conditions.

Non-Equilibrium Entropy and Irreversibility in Generalized Stochastic Loewner Evolution from an Information-Theoretic Perspective

In this study, we theoretically investigated a generalized stochastic Loewner evolution (SLE) driven by reversible Langevin dynamics in the context of non-equilibrium statistical mechanics. Using the ability of Loewner evolution, which enables encoding of non-equilibrium systems into equilibrium systems, we formulated the encoding mechanism of the SLE by Gibbs entropy-based information-theoretic approaches to discuss its advantages as a means to better describe non-equilibrium systems. After deriving entropy production and flux for the 2D trajectories of the generalized SLE curves, we reformulated the system’s entropic properties in terms of the Kullback–Leibler (KL) divergence. We demonstrate that this operation leads to alternative expressions of the Jarzynski equality and the second law of thermodynamics, which are consistent with the previously suggested theory of information thermodynamics. The irreversibility of the 2D trajectories is similarly discussed by decomposing the entropy into additive and non-additive parts. We numerically verified the non-equilibrium property of our model by simulating the long-time behavior of the entropic measure suggested by our formulation, referred to as the relative Loewner entropy.

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.

Unifying deterministic and stochastic ecological dynamics via a landscape-flux approach

The frequency distributions can characterize the population-potential landscape related to the stability of ecological states. We illustrate the practical utility of this approach by analyzing a forest-savanna model. Savanna and forest states coexist under certain conditions, consistent with past theoretical work and empirical observations. However, a grassland state, unseen in the corresponding deterministic model, emerges as an alternative quasi-stable state under fluctuations, providing a theoretical basis for the appearance of widespread grasslands in some empirical analyses. The ecological dynamics are determined by both the population-potential landscape gradient and the steady-state probability flux. The flux quantifies the net input/output to the ecological system and therefore the degree of nonequilibriumness. Landscape and flux together determine the transitions between stable states characterized by dominant paths and switching rates. The intrinsic potential landscape admits a Lyapunov function, which provides a quantitative measure of global stability. We find that the average flux, entropy production rate, and free energy have significant changes near bifurcations under both finite and zero fluctuation. These may provide both dynamical and thermodynamic origins of the bifurcations. We identified the variances in observed frequency time traces, fluctuations, and time irreversibility as kinematic measures for bifurcations. This framework opens the way to characterize ecological systems globally, to uncover how they change among states, and to quantify the emergence of quasi-stable states under stochastic fluctuations.

Thermodynamic Optimality of Glycolytic Oscillations

Temporal order in living matters reflects the self-organizing nature of dynamical processes driven out of thermodynamic equilibrium. Because of functional reasons, the period of a biochemical oscillation must be tuned to a specific value with precision; however, according to the thermodynamic uncertainty relation (TUR), the precision of the oscillatory period is constrained by the thermodynamic cost of generating it. After reviewing the basics of chemical oscillations using the Brusselator as a model system, we study the glycolytic oscillation generated by octameric phosphofructokinase (PFK), which is known to display a period of several minutes. By exploring the phase space of glycolytic oscillations, we find that the glycolytic oscillation under the cellular condition is realized in a cost-effective manner. Specifically, over the biologically relevant range of parameter values of glycolysis and octameric PFK, the entropy production from the glycolytic oscillation is minimal when the oscillation period is (5-10) min. Furthermore, the glycolytic oscillation is found at work near the phase boundary of limit cycles, suggesting that a moderate increase of glucose injection rate leads to the loss of oscillatory dynamics, which is reminiscent of the loss of pulsatile insulin release resulting from elevated blood glucose level.

Effect of viscous friction on entropy, entropy production, and entropy extraction rates in underdamped and overdamped media

Considering viscous friction that varies spatially and temporally, the general expressions for entropy production, free energy, and entropy extraction rates are derived to a Brownian particle that walks in overdamped and underdamped media. Via the well known stochastic approaches to underdamped and overdamped media, the thermodynamic expressions are first derived at a trajectory level then generalized to an ensemble level. To study the nonequilibrium thermodynamic features of a Brownian particle that hops in a medium where its viscosity varies on time, a Brownian particle that walks on a periodic isothermal medium (in the presence or absence of load) is considered. The exact analytical results depict that in the absence of load f=0, the entropy production rate e[over ̇]_{p} approaches the entropy extraction rate h[over ̇]_{d}=0. This is reasonable since any system which is in contact with a uniform temperature should obey the detail balance condition in a long time limit. In the presence of load and when the viscous friction decreases either spatially or temporally, the entropy S(t) monotonously increases with time and saturates to a constant value as t further steps up. The entropy production rate e[over ̇]_{p} decreases in time and at steady state (in the presence of load) e[over ̇]_{p}=h[over ̇]_{d}>0. On the contrary, when the viscous friction increases either spatially or temporally, the rate of entropy production as well as the rate of entropy extraction monotonously steps up showing that such systems are inherently irreversible. Furthermore, considering a spatially varying viscosity, the nonequilibrium thermodynamic features of a Brownian particle that hops in a ratchet potential with load is explored. In this case, the direction of the particle velocity is dictated by the magnitude of the external load of f. Far from the stall load, e[over ̇]_{p}=h[over ̇]_{d}>0 and at stall force e[over ̇]_{p}=h[over ̇]_{d}=0 revealing the system is reversible at this particular choice of parameter. In the absence of load, e[over ̇]_{p}=h[over ̇]_{d}>0 as long as a distinct temperature difference is retained between the hot and cold baths. Moreover, considering a multiplicative noise, we explore the thermodynamic features of the model system.

Exploring dissipative sources of non-Markovian biochemical reaction systems

Many biological processes including important intracellular processes are governed by biochemical reaction networks. Usually, these reaction systems operate far from thermodynamic equilibrium, implying free-energy dissipation. On the other hand, single reaction events happen often in a memory manner, leading to non-Markovian kinetics. A question then arises: how do we calculate free-energy dissipation (defined as the entropy production rate) in this physically real case? We derive an analytical formula for calculating the energy consumption of a general reaction system with molecular memory characterized by nonexponential waiting-time distributions. It shows that this dissipation is composed of two parts: one from broken detailed balance of an equivalent Markovian system with the same topology and substrates, and the other from the direction-time dependence of waiting-time distributions. But, if the system is in a detailed balance and the waiting-time distribution is direction-time independent, there is no energy dissipation even in the non-Markovian case. These general results provide insights into the physical mechanisms underlying nonequilibrium processes. A continuous-time random-walk model and a generalized model of stochastic gene expression are chosen to clearly show dissipative sources and the relationship between energy dissipation and molecular memory.

Improved bounds on entropy production in living systems

Living systems maintain or increase local order by working against the second law of thermodynamics. Thermodynamic consistency is restored as they consume free energy, thereby increasing the net entropy of their environment. Recently introduced estimators for the entropy production rate have provided major insights into the efficiency of important cellular processes. In experiments, however, many degrees of freedom typically remain hidden to the observer, and, in these cases, existing methods are not optimal. Here, by reformulating the problem within an optimization framework, we are able to infer improved bounds on the rate of entropy production from partial measurements of biological systems. Our approach yields provably optimal estimates given certain measurable transition statistics. In contrast to prevailing methods, the improved estimator reveals nonzero entropy production rates even when nonequilibrium processes appear time symmetric and therefore may pretend to obey detailed balance. We demonstrate the broad applicability of this framework by providing improved bounds on the energy consumption rates in a diverse range of biological systems including bacterial flagella motors, growing microtubules, and calcium oscillations within human embryonic kidney cells.

Nonequilibrium Thermodynamics in Biochemical Systems and Its Application

Living systems are open systems, where the laws of nonequilibrium thermodynamics play the important role. Therefore, studying living systems from a nonequilibrium thermodynamic aspect is interesting and useful. In this review, we briefly introduce the history and current development of nonequilibrium thermodynamics, especially that in biochemical systems. We first introduce historically how people realized the importance to study biological systems in the thermodynamic point of view. We then introduce the development of stochastic thermodynamics, especially three landmarks: Jarzynski equality, Crooks' fluctuation theorem and thermodynamic uncertainty relation. We also summarize the current theoretical framework for stochastic thermodynamics in biochemical reaction networks, especially the thermodynamic concepts and instruments at nonequilibrium steady state. Finally, we show two applications and research paradigms for thermodynamic study in biological systems.

Application of Advanced Light Microscopy to the Study of HIV and Its Interactions with the Host

This review highlights the significant observations of human immunodeficiency virus (HIV) assembly, release and maturation made possible with advanced light microscopy techniques. The advances in technology which now enables these light microscopy measurements are discussed with special emphasis on live imaging approaches including Total Internal Reflection Fluorescence (TIRF), high-resolution light microscopy techniques including PALM and STORM and single molecule measurements, including Fluorescence Resonance Energy Transfer (FRET). The review concludes with a discussion on what new insights and understanding can be expected from these measurements.

Intrinsic and Extrinsic Thermodynamics for Stochastic Population Processes with Multi-Level Large-Deviation Structure

A set of core features is set forth as the essence of a thermodynamic description, which derive from large-deviation properties in systems with hierarchies of timescales, but which are not dependent upon conservation laws or microscopic reversibility in the substrate hosting the process. The most fundamental elements are the concept of a macrostate in relation to the large-deviation entropy, and the decomposition of contributions to irreversibility among interacting subsystems, which is the origin of the dependence on a concept of heat in both classical and stochastic thermodynamics. A natural decomposition that is known to exist, into a relative entropy and a housekeeping entropy rate, is taken here to define respectively the intensive thermodynamics of a system and an extensive thermodynamic vector embedding the system in its context. Both intensive and extensive components are functions of Hartley information of the momentary system stationary state, which is information about the joint effect of system processes on its contribution to irreversibility. Results are derived for stochastic chemical reaction networks, including a Legendre duality for the housekeeping entropy rate to thermodynamically characterize fully-irreversible processes on an equal footing with those at the opposite limit of detailed-balance. The work is meant to encourage development of inherent thermodynamic descriptions for rule-based systems and the living state, which are not conceived as reductive explanations to heat flows.

Enzyme activities predicted by metabolite concentrations and solvent capacity in the cell

Experimental measurements or computational model predictions of the post-translational regulation of enzymes needed in a metabolic pathway is a difficult problem. Consequently, regulation is mostly known only for well-studied reactions of central metabolism in various model organisms. In this study, we use two approaches to predict enzyme regulation policies and investigate the hypothesis that regulation is driven by the need to maintain the solvent capacity in the cell. The first predictive method uses a statistical thermodynamics and metabolic control theory framework while the second method is performed using a hybrid optimization-reinforcement learning approach. Efficient regulation schemes were learned from experimental data that either agree with theoretical calculations or result in a higher cell fitness using maximum useful work as a metric. As previously hypothesized, regulation is herein shown to control the concentrations of both immediate and downstream product concentrations at physiological levels. Model predictions provide the following two novel general principles: (1) the regulation itself causes the reactions to be much further from equilibrium instead of the common assumption that highly non-equilibrium reactions are the targets for regulation; and (2) the minimal regulation needed to maintain metabolite levels at physiological concentrations maximizes the free energy dissipation rate instead of preserving a specific energy charge. The resulting energy dissipation rate is an emergent property of regulation which may be represented by a high value of the adenylate energy charge. In addition, the predictions demonstrate that the amount of regulation needed can be minimized if it is applied at the beginning or branch point of a pathway, in agreement with common notions. The approach is demonstrated for three pathways in the central metabolism of E. coli (gluconeogenesis, glycolysis-tricarboxylic acid (TCA) and pentose phosphate-TCA) that each require different regulation schemes. It is shown quantitatively that hexokinase, glucose 6-phosphate dehydrogenase and glyceraldehyde phosphate dehydrogenase, all branch points of pathways, play the largest roles in regulating central metabolism.

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.

Validity of path thermodynamics in reactive systems

Path thermodynamic formulation of nonequilibrium reactive systems is considered. It is shown through simple practical examples that this approach can lead to results that contradict well established thermodynamic properties of such systems. Rigorous mathematical analysis confirming this fact is presented.

Entropy production estimation with optimal current

Entropy production characterizes the thermodynamic irreversibility and reflects the amount of heat dissipated into the environment and free energy lost in nonequilibrium systems. According to the thermodynamic uncertainty relation, we propose a deterministic method to estimate the entropy production from a single trajectory of system states. We explicitly and approximately compute an optimal current that yields the tightest lower bound using predetermined basis currents. Notably, the obtained tightest lower bound is intimately related to the multidimensional thermodynamic uncertainty relation. By proving the saturation of the thermodynamic uncertainty relation in the short-time limit, the exact estimate of the entropy production can be obtained for overdamped Langevin systems, irrespective of the underlying dynamics. For Markov jump processes, because the attainability of the thermodynamic uncertainty relation is not theoretically ensured, the proposed method provides the tightest lower bound for the entropy production. When entropy production is the optimal current, a more accurate estimate can be further obtained using the integral fluctuation theorem. We illustrate the proposed method using three systems: a four-state Markov chain, a periodically driven particle, and a multiple bead-spring model. The estimated results in all examples empirically verify the effectiveness and efficiency of the proposed method.

Curl Flux as a Dynamical Origin of the Bifurcations/Phase Transitions of Nonequilibrium Systems: Cell Fate Decision Making

The underlying interactions in physical and biological systems often lead to a variety of behaviors and emergent states or phases. Under certain conditions, these phases can be transformed from one to another. The phase transition behaviors can be described by the bifurcation or catastrophe where different stable/unstable states can branch out or meet together with the birth of the new and death of the old states. Despite significant efforts, how the bifurcation and catastrophe actually occur dynamically and the associated mechanisms for nonequilibrium systems are still not very clear. As an example, we study the underlying mechanism of cell differentiation through bifurcations. Cell differentiation is one of the key fate decision-making processes that a cell faces. It is crucial for the development of multicellular organisms. Under induction, gene regulation changes, or stochastic fluctuations, the cell fate decision-making processes can exhibit different types of bifurcations or phase transitions. In order to understand the underlying mechanism, it is crucial to find out where and how the bifurcation occurs. However, this is still largely unknown. In this study, we found that the average of the curl flux as a major component of the driving force for the dynamics in addition to the landscape gradient and the associated entropy production rate both reach maximum near the bifurcation. This indicates that the curl flux and entropy production rate may provide the dynamical and thermodynamic origins of the bifurcation/catastrophe or phase transitions for cell differentiation and this possibly applies to many other nonequilibrium active systems.

The role of fluctuations in determining cellular network thermodynamics

The steady state distributions of phenotypic responses within an isogenic population of cells result from both deterministic and stochastic characteristics of biochemical networks. A biochemical network can be characterized by a multidimensional potential landscape based on the distribution of responses and a diffusion matrix of the correlated dynamic fluctuations between N-numbers of intracellular network variables. In this work, we develop a thermodynamic description of biological networks at the level of microscopic interactions between network variables. The Boltzmann H-function defines the rate of free energy dissipation of a network system and provides a framework for determining the heat associated with the nonequilibrium steady state and its network components. The magnitudes of the landscape gradients and the dynamic correlated fluctuations of network variables are experimentally accessible. We describe the use of Fokker-Planck dynamics to calculate housekeeping heat from the experimental data by a method that we refer to as Thermo-FP. The method provides insight into the composition of the network and the relative thermodynamic contributions from network components. We surmise that these thermodynamic quantities allow determination of the relative importance of network components to overall network control. We conjecture that there is an upper limit to the rate of dissipative heat produced by a biological system that is associated with system size or modularity, and we show that the dissipative heat has a lower bound.

Unified formalism for entropy production and fluctuation relations

Stochastic entropy production, which quantifies the difference between the probabilities of trajectories of a stochastic dynamics and its time reversals, has a central role in nonequilibrium thermodynamics. In the theory of probability, the change in the statistical properties of observables due to reversals can be represented by a change in the probability measure. We consider operators on the space of probability measures that induce changes in the statistical properties of a process, and we formulate entropy production in terms of these change-of-probability-measure (CPM) operators. This mathematical underpinning of the origin of entropy production allows us to achieve an organization of various forms of fluctuation relations: All entropy production has a nonnegative mean value, admit the integral fluctuation theorem, and satisfy a rather general fluctuation relation. Other results such as the transient fluctuation theorem and detailed fluctuation theorems then are derived from the general fluctuation relation with more constraints on the operator of entropy production. We use a discrete-time, discrete-state-space Markov process to draw the contradistinction among three reversals of a process: time reversal, protocol reversal, and the dual process. The properties of their corresponding CPM operators are examined, and the domains of validity of various fluctuation relations for entropy production in physics and chemistry are revealed. We also show that our CPM operator formalism can help us rather easily extend other fluctuation relations for excess work and heat, discuss the martingale properties of entropy production, and derive the stochastic integral formulas for entropy production in constant-noise diffusion process with Girsanov theorem. Our formalism provides a general and concise way to study the properties of entropy-related quantities in stochastic thermodynamics and information theory.

Entropy production and entropy extraction rates for a Brownian particle that walks in underdamped medium

The expressions for entropy production, free energy, and entropy extraction rates are derived for a Brownian particle that walks in an underdamped medium. Our analysis indicates that as long as the system is driven out of equilibrium, it constantly produces entropy at the same time it extracts entropy out of the system. At steady state, the rate of entropy production e[over ̇]_{p} balances the rate of entropy extraction h[over ̇]_{d}. At equilibrium both entropy production and extraction rates become zero. The entropy production and entropy extraction rates are also sensitive to time. As time progresses, both entropy production and extraction rates increase in time and saturate to constant values. Moreover, employing microscopic stochastic approach, several thermodynamic relations for different model systems are explored analytically and via numerical simulations by considering a Brownian particle that moves in overdamped medium. Our analysis indicates that the results obtained for underdamped cases quantitatively agree with overdamped cases at steady state. The fluctuation theorem is also discussed.

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.

Thermodynamics of Markov processes with nonextensive entropy and free energy

Statistical thermodynamics of small systems shows dramatic differences from normal systems. Parallel to the recently presented steady-state thermodynamic formalism for master equation and Fokker-Planck equation, we show that a "thermodynamic" theory can also be developed based on Tsallis' generalized entropy S^{(q)}=∑_{i=1}^{N}(p_{i}-p_{i}^{q})/[q(q-1)] and Shiino's generalized free energy F^{(q)}=[∑_{i=1}^{N}p_{i}(p_{i}/π_{i})^{q-1}-1]/[q(q-1)], where π_{i} is the stationary distribution. dF^{(q)}/dt=-f_{d}^{(q)}≤0 and it is zero if and only if the system is in its stationary state. dS^{(q)}/dt-Q_{ex}^{(q)}=f_{d}^{(q)}, where Q_{ex}^{(q)} characterizes the heat exchange. For systems approaching equilibrium with detailed balance, f_{d}^{(q)} is the product of Onsager's thermodynamic flux and force. However, it is discovered that the Onsager's force is nonlocal. This is a consequence of the particular transformation invariance for zero energy of Tsallis' statistics.

Approach to equilibrium and nonequilibrium stationary distributions of interacting many-particle systems that are coupled to different heat baths

A Hamiltonian-based model of many harmonically interacting massive particles that are subject to linear friction and coupled to heat baths at different temperatures is used to study the dynamic approach to equilibrium and nonequilibrium stationary states. An equilibrium system is here defined as a system whose stationary distribution equals the Boltzmann distribution, the relation of this definition to the conditions of detailed balance and vanishing probability current is discussed both for underdamped as well as for overdamped systems. Based on the exactly calculated dynamic approach to the stationary distribution, the functional that governs this approach, which is called the free entropy S_{free}(t), is constructed. For the stationary distribution S_{free}(t) becomes maximal and its time derivative, the free entropy production S[over ̇]_{free}(t), is minimal and vanishes. Thus, S_{free}(t) characterizes equilibrium as well as nonequilibrium stationary distributions by their extremal and stability properties. For an equilibrium system, i.e., if all heat baths have the same temperature, the free entropy equals the negative free energy divided by temperature and thus corresponds to the Massieu function which was previously introduced in an alternative formulation of statistical mechanics. Using a systematic perturbative scheme for calculating velocity and position correlations in the overdamped massless limit, explicit results for few particles are presented: For two particles localization in position and momentum space is demonstrated in the nonequilibrium stationary state, indicative of a tendency to phase separate. For three elastically interacting particles heat flows from a particle coupled to a cold reservoir to a particle coupled to a warm reservoir if the third reservoir is sufficiently hot. This does not constitute a violation of the second law of thermodynamics, but rather demonstrates that a particle in such a nonequilibrium system is not characterized by an effective temperature which equals the temperature of the heat bath it is coupled to. Active particle models can be described in the same general framework, which thereby allows us to characterize their entropy production not only in the stationary state but also in the approach to the stationary nonequilibrium state. Finally, the connection to nonequilibrium thermodynamics formulations that include the reservoir entropy production is discussed.

Spontaneous mirror symmetry breaking: an entropy production survey of the racemate instability and the emergence of stable scalemic stationary states

Stability of non-equilibrium stationary states and spontaneous mirror symmetry breaking, provoked by the destabilization of the racemic thermodynamic branch, is studied for enantioselective autocatalysis in an open flow system, and for a continuous range n of autocatalytic orders.

Revealing the thermodynamics of individual catalytic steps based on temperature-dependent single-particle nanocatalysis

Due to the intrinsic heterogeneity of nanocatalysis, many underlying catalytic details on nanocatalysts are hidden in ensemble-averaged measurements.

On Differences between Deterministic and Stochastic Models of Chemical Reactions: Schlögl Solved with ZI-Closure

Deterministic and stochastic models of chemical reaction kinetics can give starkly different results when the deterministic model exhibits more than one stable solution. For example, in the stochastic Schlögl model, the bimodal stationary probability distribution collapses to a unimodal distribution when the system size increases, even for kinetic constant values that result in two distinct stable solutions in the deterministic Schlögl model. Using zero-information (ZI) closure scheme, an algorithm for solving chemical master equations, we compute stationary probability distributions for varying system sizes of the Schlögl model. With ZI-closure, system sizes can be studied that have been previously unattainable by stochastic simulation algorithms. We observe and quantify paradoxical discrepancies between stochastic and deterministic models and explain this behavior by postulating that the entropy of non-equilibrium steady states (NESS) is maximum.

Detailed Fluctuation Theorems: A Unifying Perspective

We present a general method to identify an arbitrary number of fluctuating quantities which satisfy a detailed fluctuation theorem for all times within the framework of time-inhomogeneous Markovian jump processes. In doing so, we provide a unified perspective on many fluctuation theorems derived in the literature. By complementing the stochastic dynamics with a thermodynamic structure (i.e., using stochastic thermodynamics), we also express these fluctuating quantities in terms of physical observables.

Decomposition of the entropy production rate and nonequilibrium thermodynamics of switching diffusion processes

A switching diffusion process (SDP) is a widely used stochastic model in physics and biology, especially for molecular motors that exhibit a discrete internal chemical kinetics as well as a continuous external mechanical motion. The nonequilibrium thermodynamics of switching diffusion processes has not been extensively studied yet. In the present paper, we propose the decomposition of the entropy production rate in one-dimensional SDPs, based on the flux decomposition. However, similar decompositions of the housekeeping heat dissipation rate and free energy dissipation rate cannot guarantee the non-negativity of each decomposed component. Hence, we modify this decomposition with the flow of exponential relative information under steady-state fluxes, resulting in another decomposition with all non-negative components. Furthermore, we also provide the nonequilibrium thermodynamics of one-dimensional SDPs under the perspectives of coarse -graining and exchange of information between the chemical kinetics and mechanical motion, resulting in several other decompositions of entropy production rate. Finally, we generalize all the results to high-dimensional SDPs with a more general mathematical treatment.

Landscape and flux for quantifying global stability and dynamics of game theory

Game theory has been widely applied to many research areas including economics, biology and social sciences. However, it is still challenging to quantify the global stability and global dynamics of the game theory. We developed a landscape and flux framework to quantify the global stability and global dynamics of the game theory. As an example, we investigated a model of three-strategy game: a special replicator mutator game termed as the repeated Prison Dilemma model. In this model, one stable state, two stable states and limit cycle can emerge under different parameters. The repeated Prisoner's Dilemma system has Hopf bifurcation from one stable state to limit cycle state, and then to another one stable state or two stable states, and vice versa. We quantified the global stability of the repeated Prisoner's Dilemma system and identified optimal kinetic paths between the basins of attractor. The optimal paths are irreversible due to the non-zero flux. We also quantified the interplay between Peace and War.

Generalized Onsager's reciprocal relations for the master and Fokker-Planck equations

The Onsager's reciprocal relation plays a fundamental role in the nonequilibrium thermodynamics. However, unfortunately, its classical version is valid only within a narrow region near equilibrium due to the linear regression hypothesis, which largely restricts its usage. In this paper, based on the conservation-dissipation formalism, a generalized version of Onsager's relations for the master equations and Fokker-Planck equations was derived. Nonlinear constitutive relations with nonsymmetric and positively stable operators, which become symmetric under the detailed balance condition, constitute key features of this new generalization. Similar conclusions also hold for many other classical models in physics and chemistry, which in turn make the current study as a benchmark for the application of generalized Onsager's relations in nonequilibrium thermodynamics.

Balancing Non-Equilibrium Driving with Nucleotide Selectivity at Kinetic Checkpoints in Polymerase Fidelity Control

High fidelity gene transcription and replication require kinetic discrimination of nucleotide substrate species by RNA and DNA polymerases under chemical non-equilibrium conditions. It is known that sufficiently large free energy driving force is needed for each polymerization or elongation cycle to maintain far-from-equilibrium to achieve low error rates. Considering that each cycle consists of multiple kinetic steps with different transition rates, one expects that the kinetic modulations by polymerases are not evenly conducted at each step. We show that accelerations at different kinetic steps impact quite differently to the overall elongation characteristics. In particular, for forward transitions that discriminate cognate and non-cognate nucleotide species to serve as kinetic selection checkpoints, the transition cannot be accelerated too quickly nor retained too slowly to obtain low error rates, as balancing is needed between the nucleotide selectivity and the non-equilibrium driving. Such a balance is not the same as the speed-accuracy tradeoff in which high accuracy is always obtained at sacrifice of speed. For illustration purposes, we used three-state and five-state models of nucleotide addition in the polymerase elongation and show how the non-equilibrium steady state characteristics change upon variations on stepwise forward or backward kinetics. Notably, by using the multi-step elongation schemes and parameters from T7 RNA polymerase transcription elongation, we demonstrate that individual transitions serving as selection checkpoints need to proceed at moderate rates in order to sustain the necessary non-equilibrium drives as well as to allow nucleotide selections for an optimal error control. We also illustrate why rate-limiting conformational transitions of the enzyme likely play a significant role in the error reduction.

Derivation of Markov processes that violate detailed balance

Time-reversal symmetry of the microscopic laws dictates that the equilibrium distribution of a stochastic process must obey the condition of detailed balance. However, cyclic Markov processes that do not admit equilibrium distributions with detailed balance are often used to model systems driven out of equilibrium by external agents. I show that for a Markov model without detailed balance, an extended Markov model can be constructed, which explicitly includes the degrees of freedom for the driving agent and satisfies the detailed balance condition. The original cyclic Markov model for the driven system is then recovered as an approximation at early times by summing over the degrees of freedom for the driving agent. I also show that the widely accepted expression for the entropy production in a cyclic Markov model is actually a time derivative of an entropy component in the extended model. Further, I present an analytic expression for the entropy component that is hidden in the cyclic Markov model.

Processes on the emergent landscapes of biochemical reaction networks and heterogeneous cell population dynamics: differentiation in living matters

The notion of an attractor has been widely employed in thinking about the nonlinear dynamics of organisms and biological phenomena as systems and as processes. The notion of a landscape with valleys and mountains encoding multiple attractors, however, has a rigorous foundation only for closed, thermodynamically non-driven, chemical systems, such as a protein. Recent advances in the theory of nonlinear stochastic dynamical systems and its applications to mesoscopic reaction networks, one reaction at a time, have provided a new basis for a landscape of open, driven biochemical reaction systems under sustained chemostat. The theory is equally applicable not only to intracellular dynamics of biochemical regulatory networks within an individual cell but also to tissue dynamics of heterogeneous interacting cell populations. The landscape for an individual cell, applicable to a population of isogenic non-interacting cells under the same environmental conditions, is defined on the counting space of intracellular chemical compositions x = (x₁,x₂, … ,x_N ) in a cell, where x_ℓ is the concentration of the ℓth biochemical species. Equivalently, for heterogeneous cell population dynamics x_ℓ is the number density of cells of the ℓth cell type. One of the insights derived from the landscape perspective is that the life history of an individual organism, which occurs on the hillsides of a landscape, is nearly deterministic and 'programmed', while population-wise an asynchronous non-equilibrium steady state resides mostly in the lowlands of the landscape. We argue that a dynamic 'blue-sky' bifurcation, as a representation of Waddington's landscape, is a more robust mechanism for a cell fate decision and subsequent differentiation than the widely pictured pitch-fork bifurcation. We revisit, in terms of the chemostatic driving forces upon active, living matter, the notions of near-equilibrium thermodynamic branches versus far-from-equilibrium states. The emergent landscape perspective permits a quantitative discussion of a wide range of biological phenomena as nonlinear, stochastic dynamics.