OSCILLATIONS, BISTABILITY, AND ECHO WAVES IN MODELS OF THE BELOUSOV-ZHABOTINSKII REACTION*

From biological data to oscillator models using SINDy

Periodic changes in the concentration or activity of different molecules regulate vital cellular processes such as cell division and circadian rhythms. Developing mathematical models is essential to better understand the mechanisms underlying these oscillations. Recent data-driven methods like SINDy have fundamentally changed model identification, yet their application to experimental biological data remains limited. This study investigates SINDy's constraints by directly applying it to biological oscillatory data. We identify insufficient resolution, noise, dimensionality, and limited prior knowledge as primary limitations. Using various generic oscillator models of different complexity and/or dimensionality, we systematically analyze these factors. We then propose a comprehensive guide for inferring models from biological data, addressing these challenges step by step. Our approach is validated using glycolytic oscillation data from yeast.

Perspectives on Principles of Cellular Behavior from the Biophysics of Protists

Cells are the fundamental unit of biological organization. Although it may be easy to think of them as little more than the simple building blocks of complex organisms such as animals, single cells are capable of behaviors of remarkable apparent sophistication. This is abundantly clear when considering the diversity of form and function among the microbial eukaryotes, the protists. How might we navigate this diversity in the search for general principles of cellular behavior? Here, we review cases in which the intensive study of protists from the perspective of cellular biophysics has driven insight into broad biological questions of morphogenesis, navigation and motility, and decision making. We argue that applying such approaches to questions of evolutionary cell biology presents rich, emerging opportunities. Integrating and expanding biophysical studies across protist diversity, exploiting the unique characteristics of each organism, will enrich our understanding of general underlying principles.

Symmetry-breaking rhythms in coupled, identical fast-slow oscillators

Symmetry-breaking in coupled, identical, fast-slow systems produces a rich, dramatic variety of dynamical behavior-such as amplitudes and frequencies differing by an order of magnitude or more and qualitatively different rhythms between oscillators, corresponding to different functional states. We present a novel method for analyzing these systems. It identifies the key geometric structures responsible for this new symmetry-breaking, and it shows that many different types of symmetry-breaking rhythms arise robustly. We find symmetry-breaking rhythms in which one oscillator exhibits small-amplitude oscillations, while the other exhibits phase-shifted small-amplitude oscillations, large-amplitude oscillations, mixed-mode oscillations, or even undergoes an explosion of limit cycle canards. Two prototypical fast-slow systems illustrate the method: the van der Pol equation that describes electrical circuits and the Lengyel-Epstein model of chemical oscillators.

Artificial temperature-compensated biological clock using temperature-sensitive Belousov-Zhabotinsky gels

The circadian rhythm is a fundamental physiological function for a wide range of organisms. The molecular machinery for generating rhythms has been elucidated over the last few decades. Nevertheless, the mechanism for temperature compensation of the oscillation period, which is a prominent property of the circadian rhythm, is still controversial. In this study, we propose a new mechanism through a chemically synthetic approach (i.e., we realized temperature compensation by the Belousov-Zhabotinsky (BZ) gels). The BZ gels are prepared by embedding a metal catalyst of the BZ reaction into the gel polymer. We made the body of BZ gels using a temperature-sensitive polymer gel, which enabled temperature compensation of the oscillation by using temperature dependence of volume. Moreover, we constructed a simple mathematical model for the BZ oscillation in temperature-sensitive gels. The model can reproduce temperature compensation of BZ gels, even though all reactions are temperature sensitive according to the Arrhenius rule. Our finding hints that a soft body coupling may be underlying temperature-compensated biological functions, including circadian rhythms.

From the Belousov-Zhabotinsky reaction to biochemical clocks, traveling waves and cell cycle regulation

In the last 20 years, a growing army of systems biologists has employed quantitative experimental methods and theoretical tools of data analysis and mathematical modeling to unravel the molecular details of biological control systems with novel studies of biochemical clocks, cellular decision-making, and signaling networks in time and space. Few people know that one of the roots of this new paradigm in cell biology can be traced to a serendipitous discovery by an obscure Russian biochemist, Boris Belousov, who was studying the oxidation of citric acid. The story is told here from an historical perspective, tracing its meandering path through glycolytic oscillations, cAMP signaling, and frog egg development. The connections among these diverse themes are drawn out by simple mathematical models (nonlinear differential equations) that share common structures and properties.

Resonant Behavior in a Periodically Forced Nonisothermal Oregonator

Nonisothermal chemical oscillators are poorly studied systems because chemical oscillations are conventionally studied under isothermal conditions. Coupling chemical reactions with heat generation and removal in a nonisothermal oscillatory system can lead to a highly nontrivial nonlinear dynamic behavior. For the current study, we considered the three-variable Oregonator model with the temperature incorporated as a variable (not a parameter), thus adding an energy balance to the set of equations. The effect of temperature on reaction rates is included through the temperature-dependent reaction rate coefficients (Arrhenius law). To model a continuous operation in a laboratory environment, the system was subjected to external forcing through the coolant temperature and infrared irradiation. By conducting numerical simulations and parametric studies, we found that the system is capable of a resonant behavior exhibiting induced oscillations. Our findings indicate that an external source of heat (e.g., via an infrared light emitting diode) can be used to induce a Hopf bifurcation under resonant conditions in an experimental Belousov-Zhabotinsky reactor.

Fast propagation regions cause self-sustained reentry in excitable media

Self-sustained waves of electrophysiological activity can cause arrhythmia in the heart. These reentrant excitations have been associated with spiral waves circulating around either an anatomically defined weakly conducting region or a functionally determined core. Recently, an ablation procedure has been clinically introduced that stops atrial fibrillation of the heart by destroying the electrical activity at the spiral core. This is puzzling because the tissue at the anatomically defined spiral core would already be weakly conducting, and a further decrease should not improve the situation. In the case of a functionally determined core, an ablation procedure should even further stabilize the rotating wave. The efficacy of the procedure thus needs explanation. Here, we show theoretically that fundamentally in any excitable medium a region with a propagation velocity faster than its surrounding can act as a nucleation center for reentry and can anchor an induced spiral wave. Our findings demonstrate a mechanistic underpinning for the recently developed ablation procedure. Our theoretical results are based on a very general and widely used two-component model of an excitable medium. Moreover, the important control parameters used to realize conditions for the discovered phenomena are applicable to quite different multicomponent models.

Complex mixed-mode oscillatory patterns in a periodically forced excitable Belousov-Zhabotinsky reaction model

The Oregonator is the simplest chemically plausible model for the Belousov-Zhabotinsky reaction. We investigate the response of the Oregonator to sinusoidal inputs with amplitudes and frequencies within plausible ranges. We focus on a regime where the unforced Oregonator is excitable (with no sustained oscillations). We use numerical simulations and dynamical systems tools to both characterize the response patterns and explain the underlying dynamic mechanisms.

Oregonator generalization as a minimal model of quorum sensing in Belousov–Zhabotinsky reaction with catalyst confinement in large populations of particles

The Oregonator demonstrates that quorum sensing in populations of Belousov–Zhabotinsky oscillators arises from modification of the stoichiometry by catalyst confinement.

Stochastic resonance with a mesoscopic reaction-diffusion system

In a mesoscopic reaction-diffusion system with an Oregonator reaction model, we show that intrinsic noise can drive a resonant stable pattern in the presence of the initial subthreshold perturbations. Both spatially periodic and aperiodic stochastic resonances are demonstrated by employing the Gillespies stochastic simulation algorithm. The mechanisms for these phenomena are discussed.

Dynamic mechanisms of generation of oscillatory cluster patterns in a globally coupled chemical system

We use simulations and dynamical systems tools to investigate the mechanisms of generation of phase-locked and localized oscillatory cluster patterns in a globally coupled Oregonator model where the activator receives global feedback from the inhibitor, mimicking experimental results observed in the photosensitive Belousov-Zhabotinsky reaction. A homogeneous two-cluster system (two clusters with equal cluster size) displays antiphase patterns. Heterogenous two-cluster systems (two clusters with different sizes) display both phase-locked and localized patterns depending on the parameter values. In a localized pattern the oscillation amplitude of the largest cluster is roughly an order of magnitude smaller than the oscillation amplitude of the smaller cluster, reflecting the effect of self-inhibition exerted by the global feedback term. The transition from phase-locked to localized cluster patterns occurs as the intensity of global feedback increases. Three qualitatively different basic mechanisms, described previously for a globally coupled FitzHugh-Nagumo model, are involved in the generation of the observed patterns. The swing-and-release mechanism is related to the canard phenomenon (canard explosion of limit cycles) in relaxation oscillators. The hold-and-release and hold-and-escape mechanisms are related to the release and escape mechanisms in synaptically connected neural models. The methods we use can be extended to the investigation of oscillatory chemical reactions with other types of non-local coupling.

Disproportionation of bromous acid HOBrO by direct O-transfer and via anhydrides O(BrO)2 and BrO-BrO2. An ab initio study of the mechanism of a key step of the Belousov-Zhabotinsky oscillating reaction

The results are reported of an ab initio study of the thermochemistry and of the kinetics of the HOBrO disproportionation reaction 2HOBrO (2) ⇄ HOBr (1) + HBrO(3) (3), reaction ( R4' ), in gas phase (MP2(full)/6-311G*) and aqueous solution (SMD(MP2(full)/6-311G*)). The reaction energy of bromous acid disproportionation is discussed in the context of the coupled reaction system R2-R4 of the FKN mechanism of the Belousov-Zhabotinsky reaction and considering the acidities of HBr and HOBrO(2). The structures were determined of ten dimeric aggregates 4 of bromous acid, (HOBrO)(2), of eight mixed aggregates 5 formed between the products of disproportionation, (HOBr)(HOBrO(2)), and of four transition states structures 6 for disproportionation by direct O-transfer. It was found that the condensation of two HOBrO molecules provides facile access to bromous acid anhydride 7, O(BrO)(2). A discussion of the potential energy surface of Br(2)O(3) shows that O(BrO)(2) is prone to isomerization to the mixed anhydride 8, BrO-BrO(2), and to dissociation to 9, BrO, and 10, BrO(2), and their radical pair 11. Hence, three possible paths from O(BrO)(2) to the products of disproportionation, HOBr and HOBrO(2), are discussed: (1) hydrolysis of O(BrO)(2) along a path that differs from its formation, (2) isomerization of O(BrO)(2) to BrO-BrO(2) followed by hydrolysis, and (3) O(BrO)(2) dissociation to BrO and BrO(2) and their reactions with water. The results of the potential energy surface analysis show that the rate-limiting step in the disproportionation of HOBrO consists of the formation of the hydrate 12a of bromous acid anhydride 7 via transition state structure 14a. The computed activation free enthalpy ΔG(act)(SMD) = 13.6 kcal/mol for the process 2·2a → [14a](‡) → 12a corresponds to the reaction rate constant k(4) = 667.5 M(-1) s(-1) and is in very good agreement with experimental measurements. The potential energy surface analysis further shows that anhydride 7 is kinetically and thermodynamically unstable with regard to hydrolysis to HOBr and HOBrO(2) via transition state structure 14b. The transition state structure 14b is much more stable than 14a, and, hence, the formation of the "symmetrical anhydride" from bromous acid becomes an irreversible reaction for all practical purposes because 7 will instead be hydrolyzed as a "mixed anhydride" to afford HOBr and HOBrO(2). The mixed anhydride 8, BrO-BrO(2), does not play a significant role in bromous acid disproportionation.

Excitation functions of coupling

The responses of nonlinear dynamics of two classes to coupling are investigated. It is shown both analytically and numerically that coupling has an excitation ability in a network of the linearly coupled systems. That is, when an uncoupled system is degenerated to a stable steady state from a limit cycle but in the "marginal" state due to the system parameter, an appropriate coupling strength can excite the limit cycle such that the coupled systems exhibit synchronous oscillation; when the uncoupled system is in a stable limit cycle but close to a chaotic attractor, a certain coupling strength can induce the chaotic attractor such that the coupled systems reach chaotic synchronization. Such excitation functions of coupling are different from its traditional role where coupling mainly synchronizes the coupled systems with the original dynamics of the uncoupled system.

Out-of-phase oscillatory Turing patterns in a bistable reaction-diffusion system

A new type of out-of-phase oscillatory Turing pattern is found in simulations of a simple two-variable model of a bistable reaction-diffusion system consisting of an autocatalytic activator reacting with a substrate that is replenished by a flow. This class of models can describe pH oscillators or enzymatic reactions. No Hopf instability is necessary for this type of oscillatory Turing pattern.

Complex pattern formation by a self-destabilization of established patterns: chemotactic orientation and phyllotaxis as examples

Stable patterns can be generated by molecular interactions involving local self-enhancement and long-range inhibition. In contrast, highly dynamic patterns result if the maxima, generated in this way, become destabilized by a second antagonistic reaction. The latter must act local and must be long-lasting. Maxima either disappear and reappear at displaced positions or they move over the field as travelling waves. The wave can have unusual properties in that they can penetrate each other without annihilation. The resulting pattern corresponds to those observed in diverse biological systems. In the chemotactic orientation of cells, the temporary signals allow the localized extensions of protrusions under control of minute external asymmetries imposed by the chemoattractant. In phyllotaxis, these signals lead to successive leaf initiation, whereby the longer-lasting extinguishing reaction can cause a displacement of the subsequent leaf initiation site by the typical 137.5 degrees, the golden angle. On seashells, this patterns leads either to oblique lines that can cross each other or to oblique rows of dots. For some of the models animated simulations are available at http://www.eb.tuebingen.mpg.de/abt.4/meinhardt/theory.html.

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

An account of the experimental discovery of complex dynamical behavior in the continuous-flow, stirred tank reactor (CSTR) Belousov-Zhabotinsky (BZ) reaction, as well as numerical simulations based on the BZ chemistry are given. The most recent four- and three-variable models that are deduced from the well-accepted, updated chemical mechanism of the BZ reaction and which exhibit robust chaotic states are summarized. Chaos has been observed in experiments and simulations embedded in the regions of complexities at both low and high flow rates. The deterministic nature of the observed aperiodicities at low flow rates is unequivocally established. However, controversy still remains in the interpretation of certain aperiodicities observed at high flow rates.

Computer Simulations of the Belousov-Zhabotinsky Reaction

Morphological features of the two-dimensional Belousov-Zhabotinsky reaction were modeled with an algorithm involving only two simple parameters, one describing the productivity of the reaction on a local scale length and the other characterizing the delay or quiescent time after the localized reaction. Self-organizing wavelike structures, including single-and multiarmed spirals, were most easily generated.