Deterministic chaos in the Belousov-Zhabotinsky reaction: Experiments and simulations

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.

An Overview of Emergent Order in Far-from-Equilibrium Driven Systems: From Kuramoto Oscillators to Rayleigh-Bénard Convection

Soft-matter systems when driven out of equilibrium often give rise to structures that usually lie in between the macroscopic scale of the material and microscopic scale of its constituents. In this paper we review three such systems, the two-dimensional square-lattice Ising model, the Kuramoto model and the Rayleigh–Bénard convection system which when driven out of equilibrium give rise to emergent spatio-temporal order through self-organization. A common feature of these systems is that the entities that self-organize are coupled to one another in some way, either through local interactions or through a continuous media. Therefore, the general nature of non-equilibrium fluctuations of the intrinsic variables in these systems are found to follow similar trends as order emerges. Through this paper, we attempt to find connections between these systems, and systems in general which give rise to emergent order when driven out of equilibrium. This study, thus acts as a foundation for modeling a complex system as a two-state system, where the states: order and disorder can coexist as the system is driven away from equilibrium.

Deciphering Dynamical Nonlinearities in Short Time Series Using Recurrent Neural Networks

Surrogate testing techniques have been used widely to investigate the presence of dynamical nonlinearities, an essential ingredient of deterministic chaotic processes. Traditional surrogate testing subscribes to statistical hypothesis testing and investigates potential differences in discriminant statistics between the given empirical sample and its surrogate counterparts. The choice and estimation of the discriminant statistics can be challenging across short time series. Also, conclusion based on a single empirical sample is an inherent limitation. The present study proposes a recurrent neural network classification framework that uses the raw time series obviating the need for discriminant statistic while accommodating multiple time series realizations for enhanced generalizability of the findings. The results are demonstrated on short time series with lengths (L = 32, 64, 128) from continuous and discrete dynamical systems in chaotic regimes, nonlinear transform of linearly correlated noise and experimental data. Accuracy of the classifier is shown to be markedly higher than ≫50% for the processes in chaotic regimes whereas those of nonlinearly correlated noise were around ~50% similar to that of random guess from a one-sample binomial test. These results are promising and elucidate the usefulness of the proposed framework in identifying potential dynamical nonlinearities from short experimental time series.

Coexisting Ordered States, Local Equilibrium-like Domains, and Broken Ergodicity in a Non-turbulent Rayleigh-Bénard Convection at Steady-state

A challenge in fundamental physics and especially in thermodynamics is to understand emergent order in far-from-equilibrium systems. While at equilibrium, temperature plays the role of a key thermodynamic variable whose uniformity in space and time defines the equilibrium state the system is in, this is not the case in a far-from-equilibrium driven system. When energy flows through a finite system at steady-state, temperature takes on a time-independent but spatially varying character. In this study, the convection patterns of a Rayleigh-Bénard fluid cell at steady-state is used as a prototype system where the temperature profile and fluctuations are measured spatio-temporally. The thermal data is obtained by performing high-resolution real-time infrared calorimetry on the convection system as it is first driven out-of-equilibrium when the power is applied, achieves steady-state, and then as it gradually relaxes back to room temperature equilibrium when the power is removed. Our study provides new experimental data on the non-trivial nature of thermal fluctuations when stable complex convective structures emerge. The thermal analysis of these convective cells at steady-state further yield local equilibrium-like statistics. In conclusion, these results correlate the spatial ordering of the convective cells with the evolution of the system’s temperature manifold.

Complexity: the organizing principle at the interface of biological (dis)order

The term complexity means several things to biologists.When qualifying morphological phenotype, on the one hand, it is used to signify the sheer complicatedness of living systems, especially as a result of the multicomponent aspect of biological form. On the other hand, it has been used to represent the intricate nature of the connections between constituents that make up form: a more process-based explanation. In the context of evolutionary arguments, complexity has been defined, in a quantifiable fashion, as the amount of information, an informatic template such as a sequence of nucleotides or amino acids stores about its environment. In this perspective, we begin with a brief review of the history of complexity theory. We then introduce a developmental and an evolutionary understanding of what it means for biological systems to be complex.We propose that the complexity of living systems can be understood through two interdependent structural properties: multiscalarity of interconstituent mechanisms and excitability of the biological materials. The answer to whether a system becomes more or less complex over time depends on the potential for its constituents to interact in novel ways and combinations to give rise to new structures and functions, as well as on the evolution of excitable properties that would facilitate the exploration of interconstituent organization in the context of their microenvironments and macroenvironments.

Physical constraints in cell fate specification. A case in point: Microgravity and phenotypes differentiation

Data obtained by studying mammalian cells in absence of gravity strongly support the notion that cell fate specification cannot be understood according to the current molecular model. A paradigmatic case in point is provided by studying cell populations growing in absence of gravity. When the physical constraint (gravity) is 'experimentally removed', cells spontaneously allocate into two morphologically different phenotypes. Such phenomenon is likely enacted by the intrinsic stochasticity, which, in turn, is successively 'canalized' by a specific gene regulatory network. Both phenotypes are thermodynamically and functionally 'compatibles' with the new, modified environment. However, when the two cell subsets are reseeded into the 1g gravity field the two phenotypes collapse into one. Gravity constraints the system in adopting only one phenotype, not by selecting a pre-existing configuration, but more precisely shaping it de-novo through the modification of the cytoskeleton three-dimensional structure. Overall, those findings highlight how macro-scale features are irreducible to lower-scale explanations. The identification of macroscale control parameters - as those depending on the field (gravity, electromagnetic fields) or emerging from the cooperativity among the field's components (tissue stiffness, cell-to-cell connectivity) - are mandatory for assessing boundary conditions for models at lower scales, thus providing a concrete instantiation of top-down effects.

Gravity Constraints Drive Biological Systems Toward Specific Organization Patterns: Commitment of cell specification is constrained by physical cues

Different cell lineages growing in microgravity undergo a spontaneous transition leading to the emergence of two distinct phenotypes. By returning these populations in a normal gravitational field, the two phenotypes collapse, recovering their original configuration. In this review, we hypothesize that, once the gravitational constraint is removed, the system freely explores its phenotypic space, while, when in a gravitational field, cells are "constrained" to adopt only one favored configuration. We suggest that the genome allows for a wide range of "possibilities" but it is unable per se to choose among them: the emergence of a specific phenotype is enabled by physical constraints that drive the system toward a preferred solution. These findings may help in understanding how cells and tissues behave in both development and cancer.

Technique for Determining Fluids Motion Characteristics in the Vicinity of Ferromagnetic Solids Under Magneto-Chemical Treatment

We report on new technique for determining and visualization of fluids motion in the vicinity of magnetized ferromagnetic surfaces under chemical dissolution and autocatalytic formation of spatiotemporal structures. The proposed method of obtaining data on the motion of fluids does not require the use of additional inclusions and allows to avoid the distortions caused by such inclusions or other changes in the nature of the reaction. The developed technique allows to obtain both the integral dependences and time frequencies distributions of the fluids over the volume of the medium under investigation.

ON THE CHAOS IN GENE NETWORKS

The methods for constructing "chaotic" nonlinear systems of differential equations modeling gene networks of arbitrary structure and dimensionality with various types of symmetry are considered. It has been shown that an increase in modality of the functions describing the control of gene expression efficiency allows for a decrease in the dimensionality of these systems with retention of their chaotic dynamics. Three-dimensional "chaotic" cyclic systems are considered. Symmetrical and asymmetrical attractors with "narrow" chaos having a Moebius-like structure have been detected in such systems. As has been demonstrated, a complete symmetry of the systems with respect to permutation of variables does not prevent the emergence of their chaotic dynamics.

Global consensus theorem and self-organized criticality: unifying principles for understanding self-organization, swarm intelligence and mechanisms of carcinogenesis

Complex biological systems manifest a large variety of emergent phenomena among which prominent roles belong to self-organization and swarm intelligence. Generally, each level in a biological hierarchy possesses its own systemic properties and requires its own way of observation, conceptualization, and modeling. In this work, an attempt is made to outline general guiding principles in exploration of a wide range of seemingly dissimilar phenomena observed in large communities of individuals devoid of any personal intelligence and interacting with each other through simple stimulus-response rules. Mathematically, these guiding principles are well captured by the Global Consensus Theorem (GCT) equally applicable to neural networks and to Lotka-Volterra population dynamics. Universality of the mechanistic principles outlined by GCT allows for a unified approach to such diverse systems as biological networks, communities of social insects, robotic communities, microbial communities, communities of somatic cells, social networks and many other systems. Another cluster of universal laws governing the self-organization in large communities of locally interacting individuals is built around the principle of self-organized criticality (SOC). The GCT and SOC, separately or in combination, provide a conceptual basis for understanding the phenomena of self-organization occurring in large communities without involvement of a supervisory authority, without system-wide informational infrastructure, and without mapping of general plan of action onto cognitive/behavioral faculties of its individual members. Cancer onset and proliferation serves as an important example of application of these conceptual approaches. In this paper, the point of view is put forward that apparently irreconcilable contradictions between two opposing theories of carcinogenesis, that is, the Somatic Mutation Theory and the Tissue Organization Field Theory, may be resolved using the systemic approaches provided by GST and SOC.

Chaotic motifs in gene regulatory networks

Chaos should occur often in gene regulatory networks (GRNs) which have been widely described by nonlinear coupled ordinary differential equations, if their dimensions are no less than 3. It is therefore puzzling that chaos has never been reported in GRNs in nature and is also extremely rare in models of GRNs. On the other hand, the topic of motifs has attracted great attention in studying biological networks, and network motifs are suggested to be elementary building blocks that carry out some key functions in the network. In this paper, chaotic motifs (subnetworks with chaos) in GRNs are systematically investigated. The conclusion is that: (i) chaos can only appear through competitions between different oscillatory modes with rivaling intensities. Conditions required for chaotic GRNs are found to be very strict, which make chaotic GRNs extremely rare. (ii) Chaotic motifs are explored as the simplest few-node structures capable of producing chaos, and serve as the intrinsic source of chaos of random few-node GRNs. Several optimal motifs causing chaos with atypically high probability are figured out. (iii) Moreover, we discovered that a number of special oscillators can never produce chaos. These structures bring some advantages on rhythmic functions and may help us understand the robustness of diverse biological rhythms. (iv) The methods of dominant phase-advanced driving (DPAD) and DPAD time fraction are proposed to quantitatively identify chaotic motifs and to explain the origin of chaotic behaviors in GRNs.

Stochastic oscillations in genetic regulatory networks: application to microarray experiments

We analyze the stochastic dynamics of genetic regulatory networks using a system of nonlinear differential equations. The system of S-functions is applied to capture the role of RNA polymerase in the transcription-translation mechanism. Using probabilistic properties of chemical rate equations, we derive a system of stochastic differential equations which are analytically tractable despite the high dimension of the regulatory network. Using stationary solutions of these equations, we explain the apparently paradoxical results of some recent time-course microarray experiments where mRNA transcription levels are found to only weakly correlate with the corresponding transcription rates. Combining analytical and simulation approaches, we determine the set of relationships between the size of the regulatory network, its structural complexity, chemical variability, and spectrum of oscillations. In particular, we show that temporal variability of chemical constituents may decrease while complexity of the network is increasing. This finding provides an insight into the nature of "functional determinism" of such an inherently stochastic system as genetic regulatory network.

Stochastic cooperativity in non-linear dynamics of genetic regulatory networks

Two major approaches are known in the field of stochastic dynamics of genetic regulatory networks (GRN). The first one, referred here to as the Markov Process Paradigm (MPP), places the focus of attention on the fact that many biochemical constituents vitally important for the network functionality are present only in small quantities within the cell, and therefore the regulatory process is essentially discrete and prone to relatively big fluctuations. The Master Equation of Markov Processes is an appropriate tool for the description of this kind of stochasticity. The second approach, the Non-linear Dynamics Paradigm (NDP), treats the regulatory process as essentially continuous. A natural tool for the description of such processes are deterministic differential equations. According to NDP, stochasticity in such systems occurs due to possible bistability and oscillatory motion within the limit cycles. The goal of this paper is to outline a third scenario of stochasticity in the regulatory process. This scenario is only conceivable in high-dimensional, highly non-linear systems, and thus represents an adequate framework for conceptually modeling the GRN. We refer to this framework as the Stochastic Cooperativity Paradigm (SCP). In this approach, the focus of attention is placed on the fact that in systems with the size and link density of GRN ( approximately 25000 and approximately 100, respectively), the confluence of all the factors which are necessary for gene expression is a comparatively rare event, and only massive redundancy makes such events sufficiently frequent. An immediate consequence of this rareness is 'burstiness' in mRNA and protein concentrations, a well known effect in intracellular dynamics. We demonstrate that a high-dimensional non-linear system, despite the absence of explicit mechanisms for suppressing inherent instability, may nevertheless reside in a state of stationary pseudo-random fluctuations which for all practical purposes may be regarded as a stochastic process. This type of stochastic behavior is an inherent property of such systems and requires neither an external random force as in the Langevin approach, nor the discreteness of the process as in MPP, nor highly specialized conditions of bistability as in NDP, nor bifurcations with transition to chaos as in low-dimensional chaotic maps.

On the joint time-frequency characteristics of chemical oscillations

A method of time-frequency transformations of nonstationary signals has been discussed and applied to the analysis of oscillatory Belousov-Zhabotinsky (BZ) reaction. Selected registers of BZ reactions, that represent regular and complex dynamic behavior, have been submitted to the detailed spectral analysis using the short time Fourier transform (STFT). It has been demonstrated that the STFT is an accurate and suitable method for decomposition of chemical oscillations in the joint time and frequency domain. The energy distribution via STFT, presented in this article, revealed some specific characteristics of transitions between different regimes in the closed system of BZ reaction.

Modelling complex transient oscillations for the BZ reaction in a batch reactor

The recent observations of transient complex oscillations in the Belousov-Zhabotinsky (BZ) reaction in a batch reactor are confirmed and an attempt is made to model this behavior using the Gyorgyi-Field mechanism (an extension of the classic Field-Koros-Noyes model). It is seen that the concentration of bromomalonic acid plays an important role, acting somewhat like a "slowly-varying parameter," causing the system to sweep through a region of oscillatory response even in the absence of reactant consumption. Complex behavior is not observed in the model if the usual "pre-equilibrium" assumption in made for the intermediate species BrO(2) but does arise if a full steady-state approximation is applied to this radical. These results may bring into question some assumptions made in the usual treatment of the cerium-catalyzed BZ system in closed reactors. (c) 1997 American Institute of Physics.