Introduction to focus issue: dynamics in systems biology

The methods of nonlinear systems form an extensive toolbox for the study of biology, and systems biology provides a rich source of motivation for the development of new mathematical techniques and the furthering of understanding of dynamical systems. This Focus Issue collects together a large variety of work which highlights the complementary nature of these two fields, showing what each has to offer the other. While a wide range of subjects is covered, the papers often have common themes such as "rhythms and oscillations," "networks and graph theory," and "switches and decision making." There is a particular emphasis on the links between experimental data and modeling and mathematical analysis.     

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.     

Perceived sound quality of sound-reproducing systems.

Perceived sound quality of loudspeakers, headphones, and hearing aids was investigated by multivariate techniques from experimental psychology with the purpose (a) to find out and interpret the meaning of relevant dimensions in perceived sound quality, (b) find out the positions of the investigated systems in these dimensions, (c) explore the relations between the perceptual dimensions and the physical characteristics of the systems, and (d) explore the relations between the perceptual dimensions and overall evaluations of the systems. The resulting dimensions were interpreted as "clearness/distinctness," "sharpness/hardness softness," "brightness-darkness," "fullness-thinness," "feeling of space," "nearness," "disturbing sounds," and "loudness." Their relations to physical variables were explored by studying the positions of the investigated systems in the respective dimensions. Their relations to overall evaluations were studied, and the implications of the investigations for continued research are discussed.     

Driven alloys in the athermal limit

Static molecular simulations of binary alloys under extrinsic forcing show that complex ordered or segregated structures may evolve even in the absence of thermally activated diffusion. This result is in opposition to the standard theoretical framework for so-called "driven alloys," which assumes that extrinsic driving is an ideally randomizing process, and therefore predicts only random atomic configurations in the athermal limit. We propose a qualitative modification to the theory that introduces a new control parameter and use additional Monte Carlo simulations to demonstrate the physical plausibility of this modification. New research directions in nonequilibrium dynamic systems are also suggested by this analysis.     

Efficiency of autonomous soft nanomachines at maximum power

We consider nanosized artificial or biological machines working in steady state enforced by imposing nonequilibrium concentrations of solutes or by applying external forces, torques, or electric fields. For unicyclic and strongly coupled multicyclic machines, efficiency at maximum power is not bounded by the linear response value 1/2. For strong driving, it can even approach the thermodynamic limit 1. Quite generally, such machines fall into three different classes characterized, respectively, as "strong and efficient," "strong and inefficient," and "balanced." For weakly coupled multicyclic machines, efficiency at maximum power has lost any universality even in the linear response regime.     

A gentle introduction to the thermodynamics of biochemical stoichiometric networks in steady state

Reaction networks in thermodynamic equilibrium under isothermal and isobaric conditions minimize the Gibbs free energy, but chemical reactions in living organisms operate typically far from equilibrium. Currently, there is no general optimization principle for nonequilibrium systems which can be used in the analysis of biochemical networks. Motivated by the avalailabity of whole genome reconstructions of metabolic reactions, the thermodynamics of biochemical stoichiometric networks has made significant progress in the last decade. These include the consistent formulation of conservation conditions resembling Kirchhoff’s law for electrical networks. In addition, Beard and Qian suggested that the flow force relationship Δμ = RT log(J⁺/J⁻) between the forward and backward fluxes J⁺ and J⁻ and the chemical potential difference of a chemical reaction can be extended from mass action kinetics to more general reactions schemes. In this tutorial review we summarize the recent literature on reaction network thermodynamics and discuss its implications to the analysis of large biochemical systems. In addition, we discuss some recent work on flow-force relationships and global variational principles characterizing nonequilibrium steady states of reaction networks.     

Uncovering the underlying physical mechanisms of biological systems via quantification of landscape and flux

In this review, we explore the physical mechanisms of biological processes such as protein folding and recognition,ligand binding, and systems biology, including cell cycle, stem cell, cancer, evolution, ecology, and neural networks. Our approach is based on the landscape and flux theory for nonequilibrium dynamical systems. This theory provides a unifying principle and foundation for investigating the underlying mechanisms and physical quantification of biological systems.     

Sampling rare switching events in biochemical networks

Bistable biochemical switches are widely found in gene regulatory networks and signal transduction pathways. Their switching dynamics are difficult to study, however, because switching events are rare, and the systems are out of equilibrium. We present a simulation method for predicting the rate and mechanism of the flipping of these switches. We apply it to a genetic switch and find that it is highly efficient. The path ensembles for the forward and reverse processes do not coincide. The method is widely applicable to rare events and nonequilibrium processes.     

Nonequilibrium steady states in contact: approximate thermodynamic structure and zeroth law for driven lattice gases

We explore driven lattice gases for the existence of an intensive thermodynamic variable which could determine "equilibration" between two nonequilibrium steady-state systems kept in weak contact. In simulations, we find that these systems satisfy surprisingly simple thermodynamic laws, such as the zeroth law and the fluctuation-response relation between the particle-number fluctuation and the corresponding susceptibility remarkably well. However, at higher densities, small but observable deviations from these laws occur due to nontrivial contact dynamics and the presence of long-range spatial correlations.     

The interplay of diffusion and heterogeneity in nucleation of the networked Ising model

Nucleation underlies the dynamics of most first-order phase transitions in natural and man-made systems. However, most of the systems of interest are out of equilibrium. Little is known on the effect of nonequilibrium factors on the dynamics of nucleation. Here, we use the forward flux sampling method to investigate the effect of nonequilibrium diffusion on nucleation in small-world Ising networks, wherein spins can be exchanged between nearest-neighboring nodes. We introduce a parameter α to quantify the difference of nucleation rate with and without diffusion. We find that α shows a nonmonotonic dependence on the rewiring probability p of small-world networks. In particular, for different diffusion probability D, a crossover happens at p ≃ 0.17, below which the nucleation rate decreases as D increases, suggesting that the diffusion is against nucleation; while above which the nucleation rate increases with D, indicating that the diffusion is in favor of nucleation. By identifying the distinct features of nucleating clusters along the pathways for different randomness of networks, we reveal the underlying mechanism of such a nontrivial dependence.     

Statistical physics and physiology: monofractal and multifractal approaches.

Even under healthy, basal conditions, physiologic systems show erratic fluctuations resembling those found in dynamical systems driven away from a single equilibrium state. Do such "nonequilibrium" fluctuations simply reflect the fact that physiologic systems are being constantly perturbed by external and intrinsic noise? Or, do these fluctuations actually, contain useful, "hidden" information about the underlying nonequilibrium control mechanisms? We report some recent attempts to understand the dynamics of complex physiologic fluctuations by adapting and extending concepts and methods developed very recently in statistical physics. Specifically, we focus on interbeat interval variability as an important quantity to help elucidate possibly non-homeostatic physiologic variability because (i) the heart rate is under direct neuroautonomic control, (ii) interbeat interval variability is readily measured by noninvasive means, and (iii) analysis of these heart rate dynamics may provide important practical diagnostic and prognostic information not obtainable with current approaches. The analytic tools we discuss may be used on a wider range of physiologic signals. We first review recent progress using two analysis methods--detrended fluctuation analysis and wavelets--sufficient for quantifying monofractual structures. We then describe recent work that quantifies multifractal features of interbeat interval series, and the discovery that the multifractal structure of healthy subjects is different than that of diseased subjects.     

Quantum Thermodynamic Uncertainties in Nonequilibrium Systems from Robertson-Schrödinger Relations

Thermodynamic uncertainty principles make up one of the few rare anchors in the largely uncharted waters of nonequilibrium systems, the fluctuation theorems being the more familiar. In this work we aim to trace the uncertainties of thermodynamic quantities in nonequilibrium systems to their quantum origins, namely, to the quantum uncertainty principles. Our results enable us to make this categorical statement: For Gaussian systems, thermodynamic functions are functionals of the Robertson-Schrödinger uncertainty function, which is always non-negative for quantum systems, but not necessarily so for classical systems. Here, quantum refers to noncommutativity of the canonical operator pairs. From the nonequilibrium free energy, we succeeded in deriving several inequalities between certain thermodynamic quantities. They assume the same forms as those in conventional thermodynamics, but these are nonequilibrium in nature and they hold for all times and at strong coupling. In addition we show that a fluctuation-dissipation inequality exists at all times in the nonequilibrium dynamics of the system. For nonequilibrium systems which relax to an equilibrium state at late times, this fluctuation-dissipation inequality leads to the Robertson-Schrödinger uncertainty principle with the help of the Cauchy-Schwarz inequality. This work provides the microscopic quantum basis to certain important thermodynamic properties of macroscopic nonequilibrium systems.     

Time Evolution in Macroscopic Systems. II. The Entropy

The concept of entropy in nonequilibrium macroscopic systems is investigated in the light of an extended equation of motion for the density matrix obtained in a previous study. It is found that a time-dependent information entropy can be defined unambiguously, but it is the time derivative or entropy production that governs ongoing processes in these systems. The differences in physical interpretation and thermodynamic role of entropy in equilibrium and nonequilibrium systems is emphasized and the observable aspects of entropy production are noted. A basis for nonequilibrium thermodynamics is also outlined.     

Optical properties of nonequilibrium low-dimensional systems

The optical properties of low-dimensional carrier systems ("quantum wire" type) driven away from equilibrium are studied. The frequency and wave-vector-dependent dielectric function of a quasi-one-dimensional electron system under the action of an exciting external pumping source is derived. The optical responses of the system are obtained in terms of its nonequilibrium thermodynamic state, the latter characterized resorting to a nonequilibrium statistical ensemble formalism.     

Entropy production fluctuation theorem and the nonequilibrium work relation for free energy differences

There are only a very few known relations in statistical dynamics that are valid for systems driven arbitrarily far-from-equilibrium. One of these is the fluctuation theorem, which places conditions on the entropy production probability distribution of nonequilibrium systems. Another recently discovered far from equilibrium expression relates nonequilibrium measurements of the work done on a system to equilibrium free energy differences. In this paper, we derive a generalized version of the fluctuation theorem for stochastic, microscopically reversible dynamics. Invoking this generalized theorem provides a succinct proof of the nonequilibrium work relation.     

Dissipative Structures,Organisms and Evolution

Self-organization in nonequilibrium systems has been known for over 50 years. Under nonequilibrium conditions, the state of a system can become unstable and a transition to an organized structure can occur. Such structures include oscillating chemical reactions and spatiotemporal patterns in chemical and other systems. Because entropy and free-energy dissipating irreversible processes generate and maintain these structures, these have been called dissipative structures. Our recent research revealed that some of these structures exhibit organism-like behavior, reinforcing the earlier expectation that the study of dissipative structures will provide insights into the nature of organisms and their origin. In this article, we summarize our study of organism-like behavior in electrically and chemically driven systems. The highly complex behavior of these systems shows the time evolution to states of higher entropy production. Using these systems as an example, we present some concepts that give us an understanding of biological organisms and their evolution.     

Nonequilibrium Thermodynamics of the First and Second Kind: Averages and Fluctuations

We elaborate and compare two approaches to nonequilibrium thermodynamics, the two-generator bracket formulation of time-evolution equations for averages and the macroscopic fluctuation theory, for a purely dissipative isothermal driven diffusive system under steady state conditions. The fluctuation dissipation relations of both approaches play an important role for a detailed comparison. The nonequilibrium Helmholtz free energies introduced in these two approaches differ as a result of boundary conditions. A Fokker-Planck equation derived by projection operator techniques properly reproduces long range fluctuations in nonequilibrium steady states and offers the most promising possibility to describe the physically relevant fluctuations around macroscopic averages for time-dependent nonequilibrium systems.     

Asymmetric fluctuation–relaxation paths in FPU models

A recent theory by Bertini, De Sole, Gabrielli, Jona-Lasinio and Landim predicts a temporal asymmetry in the fluctuation–relaxation paths of certain observables of nonequilibrium systems in local thermodynamic equilibrium. We find temporal asymmetries in the fluctuation–relaxation paths of a form of local heat flow, in the nonequilibrium FPU-$\beta$ model of Lepri, Livi and Politi.