Multiple types of spatio-temporal oscillations induced by differential diffusion in the Landolt reaction

The acid autoactivated iodate-sulfite redox reaction (Landolt reaction) exhibits bistability but no oscillatory dynamics when operated in a continuous stirred tank reactor (CSTR). However, it has been previously found experimentally that this reaction can exhibit both spatial bistability and oscillations when carried out in a one side diffusely fed spatial reactor. The precise origin of the oscillatory instability remained mainly elusive. We unambiguously show, in numerical simulations based of a kinetic model recently proposed by Csek?et al., J. Phys. Chem., 2008, 112, 5954), that the observed oscillations are due to the faster diffusion of the proton relative to the other feed species (long range activation instability). Furthermore, our calculations account for the previous experimental observation of two different oscillatory modes. The first one is associated to localized front oscillations, as already reported in another reaction. The other one is a periodic switch between the two states of the spatial bistability and affects the system as a whole. This oscillatory mode was undocumented in the previous studies of long range activation instabilities. More complex dynamical behaviors that mix these two types of oscillations are also reported.     

Oscillatory dynamics protect enzymes and possibly cells against toxic substances

We have used the oscillating peroxidase-oxidase (PO) reaction as a model system to study how oscillatory dynamics may affect the influence of toxic reaction intermediates on enzyme stability. In the peroxidase-oxidase reaction reactive intermediates, such as hydrogen peroxide, superoxide, and hydroxyl radical are formed. Such intermediates inactivate many cellular macromolecules such as proteins and nucleic acids. These reaction intermediates also react with peroxidase itself to form an inactive enzyme. The fact that the PO reaction shows bistability between an oscillatory and a steady state gives us a unique possibility to compare such inactivation when the system is in one of these two states. We show that inactivation of peroxidase is slower when the system is in an oscillatory state, and using numerical simulations we provide evidence that oscillatory dynamics lower the average concentration of the reactive intermediates.     

Electrochemical oscillations and bistability during anodic dissolution of vanadium electrode in acidic media��part II. The model

Following our experimental results on the oscillatory and bistable vanadium electrooxidation in phosphoric acid medium (Gorzkowski et al., 2011), we constructed a theoretical model of the V/H₃PO₄ passivation/dissolution system which allowed us to reproduce the basic dynamic instabilities observed. In this model, the formation of the negative resistance (N-NDR) region was ascribed to the potential-dependent formation/destruction of the inhibitive V₂O₅ layer on the electrode surface. The dynamic system was defined in terms of two variables: electrode potential and surface concentration of VO ₂ ⁺ species. Both relaxation oscillations and bistability for conditions similar to the experimental ones were obtained in numerical calculations and a theoretical bifurcation diagram was constructed. The analysis of the system??s dynamics in terms of nullclines allowed us to detect directly unattainable unstable steady states associated with both the oscillatory and bistable regimes.     

封闭型简单反应系统中复杂动力学行为的模拟

研究了一个只涉及双分子反应步骤(A+B→C+D, B+C→2B)和单分反应步骤(B→p, E→A, G→B, A→D, C→P)的封闭反应系统的动力学行为, 尤其是各种状态之间的跃迁行为. 在用准定态分析方法得到的准定态分岔图的基础上, 进行了大量的数值模拟, 模拟结果和难定态分岔图预言的结果基本一致. 这些结果能定性解释近年来在封闭反应体系中发现的各种动力学现象, 如化学振荡, 多重稳定性以及定态激发现象等。     

Electrochemical oscillations and bistability during anodic dissolution of vanadium electrode in acidic media—part II. The model

Following our experimental results on the oscillatory and bistable vanadium electrooxidation in phosphoric acid medium (Gorzkowski et al., 2011), we constructed a theoretical model of the V/H₃PO₄ passivation/dissolution system which allowed us to reproduce the basic dynamic instabilities observed. In this model, the formation of the negative resistance (N-NDR) region was ascribed to the potential-dependent formation/destruction of the inhibitive V₂O₅ layer on the electrode surface. The dynamic system was defined in terms of two variables: electrode potential and surface concentration of VO ⁺₂ species. Both relaxation oscillations and bistability for conditions similar to the experimental ones were obtained in numerical calculations and a theoretical bifurcation diagram was constructed. The analysis of the system’s dynamics in terms of nullclines allowed us to detect directly unattainable unstable steady states associated with both the oscillatory and bistable regimes.

     

Effect of enzyme concentration on the dynamic behavior of a membrane-bound enzyme system

The dynamic behavior of the reaction-diffusion system, composed of glucose oxidase (EC 1.1.3.4) immobilized at a uniform concentration in a membrane, used as a glucose electrode is represented by a diffusion equation with a nonlinear reaction-term in one-dimensional space. The mathematical model is analyzed by computer simulation, that is, numerical integration of the equation under various initial and boundary conditions, to examine the effect of enzyme concentration on the response characteristics (responsiveness and linearity in response) of the electrode. The analysis of the responses of the system to stepwise changes in the boundary value (glucose concentration in simple solution) infers that the enzyme concentration governs the patterns of the spatial distributions of the substrates (glucose and dissolved oxygen) in steady states and transient responses. It is also revealed that the response characteristics of the electrode are optimized with concentration of immobilized enzyme and that the system establishes the steady states at the same spatial distributions of the substrates, regardless of the boundary value. The diffusion of the substrates and the oxygen concentration also have significant effects on the response characteristics of the electrode.     

Modeling chlorite-iodide reaction dynamics using a chlorine dioxide-iodide reaction mechanism

The mechanism of Lengyel, Li, Kustin, and Epstein (J. Am. Chem. Soc. 1996, 118, 3708) for the oscillatory chlorine dioxide-iodide reaction accurately models the reaction in closed and open systems. We investigated whether this mechanism minus the single reaction involving chlorine dioxide models the chlorite-iodide reaction equally well. It agrees qualitatively with clock reaction results. As for open system dynamics, the mechanism predicts the existence of two steady states and bistability in very nearly the same regions where these features are found experimentally in the pH range 2-4. A discrepancy in the range of bistability emerges as pH decreases, and it cannot be remedied by taking into account chlorous acid decomposition. That we were unable to locate an oscillatory region is of greater significance. Because the chlorite-iodide reaction is sensitive to mixing effects, we incorporated a two-parameter model of imperfect mixing but still found no oscillations at physically reasonable parameter values. These discrepancies strongly suggest that to obtain predictive utility for the chlorite-iodide reaction, revision of the chlorine dioxide-iodide mechanism is required.     

硫氰酸盐电化学氧化的动力学复杂性

本文利用多种电化学方法研究了多晶电极上硫氰酸盐电化学氧化的动力学和机理,观察到电流和电位振荡。循环伏安测量表明氧化动力学分为二步过程。除了振荡现象,系统也展示双稳态,利用时间电位扰动,氧化反应可在高电位和低电位稳态之间转换,而且强烈吸附的惰性离子也可诱导从振荡转化为稳态。     

Spatial bifurcations of fixed points and limit cycles during the electrochemical oxidation of H2 on Pt ring-electrodes

Pattern formation during the oscillatory oxidation of H2 on Pt ring-electrodes in the presence of electrosorbing ions was studied under potentiostatic control for three different positions of the reference electrode (RE). The position of the RE crucially affects the degree of the global feedback which is imposed by the potentiostatic operation mode, and the three configurations selected corresponded to zero, maximum and intermediate global coupling. In the absence of global coupling, 'communication' among different positions occurs exclusively through migration coupling (the electrochemical counterpart to diffusion in reaction-diffusion systems). In this case, spatially inhomogeneous oscillations that were attributed to a spatial bifurcation of the homogeneous limit cycle were observed throughout. This implies that the system is Benjamin-Feir unstable. For the strongest global coupling adjustable, travelling pulses were found that emerged in a wave bifurcation with n = 1 from the homogeneous steady state. The pulses exhibited modulations in velocity and width that most likely resulted from the interaction between inhomogeneities of the catalytic surface and the nonlinear reaction dynamics. In the case of an intermediate global coupling strength, a diversity of spatio-temporal motions was observed. The dynamics ranged from pulses over target patterns and so-called asymmetric target patterns to mixed states where two or three of these states alternate. For some parameters these mixed states were in addition separated by bursts of the system to a nearly homogeneous unreactive state.     

Superlattice patterns and spatial instability induced by delay feedback

Various oscillatory superlattice patterns in a reaction-diffusion system are observed by means of delay feedback (DF) in the parametric domain where the system without DF displays uniform bulk oscillation. By varying DF parameters within an appropriate range, the system undergoes transitions to oscillatory hexagons, stripes and squares, and square superlattices with different wavenumbers are also obtained. Linear stability analysis reveals that the patterns do not result from the Turing instability and a possible mechanism of pattern formation is suggested and proved analytically: DF induces instability of a homogeneous limit cycle with respect to spatial perturbations even if the Turing instability does not occur, so that oscillatory patterns possessing the corresponding spatial modes are produced. The different behavior of the dominant characteristic multiplier seems to be connected to the pattern selection. Here it is clearly demonstrated that DF can play a destabilizing role in spatially extended system instead of stabilizing the periodic orbits or turbulent states, which most earlier works have usually focused on.     

从多重定态经复杂奇点向极限环型化学振荡的跃迁

在文Ⅰ~[1]的相平面分析的基础上, 利用线性稳定性分析和数值模拟的方法进一步分析了Schlögl模型在CSTR中的行为, 发现在适当条件下可以发生从定态向振荡态的不连续的突然跃迁现象。认为这种突然跃迁现象是通过鞍点-结点型复杂奇点的形成和消失而实现的。分析了这种鞍点-结点型跃迁与亚临界Hopf分支现象的差别, 并讨论了这种跃迁现象和实验上发现的某些突变型化学振荡现象之间的联系。     

Folding dynamics of Trp-cage in the presence of chemical interference and macromolecular crowding. I

Proteins fold and function in the crowded environment of the cell's interior. In the recent years it has been well established that the so-called "macromolecular crowding" effect enhances the folding stability of proteins by destabilizing their unfolded states for selected proteins. On the other hand, chemical and thermal denaturation is often used in experiments as a tool to destabilize a protein by populating the unfolded states when probing its folding landscape and thermodynamic properties. However, little is known about the complicated effects of these synergistic perturbations acting on the kinetic properties of proteins, particularly when large structural fluctuations, such as protein folding, have been involved. In this study, we have first investigated the folding mechanism of Trp-cage dependent on urea concentration by coarse-grained molecular simulations where the impact of urea is implemented into an energy function of the side chain and/or backbone interactions derived from the all-atomistic molecular dynamics simulations with urea through a Boltzmann inversion method. In urea solution, the folding rates of a model miniprotein Trp-cage decrease and the folded state slightly swells due to a lack of contact formation between side chains at the terminal regions. In addition, the equilibrium m-values of Trp-cage from the computer simulations are in agreement with experimental measurements. We have further investigated the combined effects of urea denaturation and macromolecular crowding on Trp-cage's folding mechanism where crowding agents are modeled as hard-spheres. The enhancement of folding rates of Trp-cage is most pronounced by macromolecular crowding effect when the extended conformations of Trp-cast dominate at high urea concentration. Our study makes quantitatively testable predictions on protein folding dynamics in a complex environment involving both chemical denaturation and macromolecular crowding effects.     

Atoms of multistationarity in chemical reaction networks

Chemical reaction systems are dynamical systems that arise in chemical engineering and systems biology. In this work, we consider the question of whether the minimal (in a precise sense) multistationary chemical reaction networks, which we propose to call ‘atoms of multistationarity,’ characterize the entire set of multistationary networks. Our main result states that the answer to this question is ‘yes’ in the context of fully open continuous-flow stirred-tank reactors (CFSTRs), which are networks in which all chemical species take part in the inflow and outflow. In order to prove this result, we show that if a subnetwork admits multiple steady states, then these steady states can be lifted to a larger network, provided that the two networks share the same stoichiometric subspace. We also prove an analogous result when a smaller network is obtained from a larger network by ‘removing species.’ Our results provide the mathematical foundation for a technique used by Siegal- Gaskins et al. of establishing bistability by way of ‘network ancestry.’ Additionally, our work provides sufficient conditions for establishing multistationarity by way of atoms and moreover reduces the problem of classifying multistationary CFSTRs to that of cataloging atoms of multistationarity. As an application, we enumerate and classify all 386 bimolecular and reversible two-reaction networks. Of these, exactly 35 admit multiple positive steady states. Moreover, each admits a unique minimal multistationary subnetwork, and these subnetworks form a poset (with respect to the relation of ‘removing species’) which has 11 minimal elements (the atoms of multistationarity).     

Turing patterns created by cross-diffusion for a Holling II and Leslie-Gower type three species food chain model

In this paper, we develop a theoretical framework for a research into spatial patterns in a three-species Holling II and Leslie-Gower type food chain model with cross-diffusion, the results of which show that the cross-diffusion induces the spatial patterns. When biological pattern formation has been concerned with the method of reaction-diffusion theory, in most of the previous works, as a precondition, the assumption of the existence of nonhomogeneous steady state is presented essentially. We give a rigorous proof to the assumption that the model has at least a nonhomogeneous stationary solution by the Leray-Schauder degree theory. Moreover, the numerical simulations for spatial pattern is also carried out, we propose a method to estimate the wavenumber of the spatial patterns.     

Conditions for Mixed Mode Oscillations and Deterministic Chaos in Nonlinear Chemical Systems

An approach has been proposed for finding the conditions for the existence of mixed-mode oscillations and deterministic chaos in a kinetic scheme after reduction to a simple system of equations. Analysis of the position and stability of the steady states of this system suggested simple conditions for the existence of mixed-mode oscillations and deterministic chaos. The boundaries for monostability, bistability, and oscillations were also found. The results obtained were completely confirmed by numerical modelling.     

Turing patterns in the chlorine dioxide-iodine-malonic acid reaction with square spatial periodic forcing

We use the photosensitive chlorine dioxide-iodine-malonic acid reaction-diffusion system to study wavenumber locking of Turing patterns to two-dimensional "square" spatial forcing, implemented as orthogonal sets of bright bands projected onto the reaction medium. Various resonant structures emerge in a broad range of forcing wavelengths and amplitudes, including square lattices and superlattices, one-dimensional stripe patterns and oblique rectangular patterns. Numerical simulations using a model that incorporates additive two-dimensional spatially periodic forcing reproduce well the experimental observations.     

Spatiotemporal dynamics of the Landolt reaction in an open spatial reactor with conical geometry

In a previous study, the iodate-sulfite proton autoactivated reaction (Landolt reaction) was shown to exhibit spatial bistability and spatiotemporal oscillations when operated in an open spatial reactor with fixed "thickness", i.e., feed boundary to core distance. Here, we show that the spatial reactors with conical geometry enable one to rapidly probe the sensitivity of the above phenomena over a large range of the "thickness" parameter. This often-neglected parameter in chemical pattern studies plays an important role on the selection and stability of states. We reveal that the quenching capacity of slow diffusing polyacrylate ions on the spatiotemporal oscillations depends on this "thickness". The presented results should be useful for further research on reaction diffusion patterns and chemomechanical structures.     

Spatial bistability and waves in a reaction with acid autocatalysis

The phenomenon of spatial bistability has recently been proposed for a comprehensive understanding of a number of chemical patterns observed in open spatial reactors consisting of thin films of gel diffusively fed from one side. We study experimentally and numerically this phenomenon in the tetrathionate-chlorite reaction characterized by an acid superautocatalysis. We focus on the similarities and differences with previous studies on the chlorine dioxide-iodide reaction. In addition, we show that this reaction, which is only bistable in a continuous stirred tank reactor, can exhibit oscillatory and traveling waves when diffusion comes into play. Our computations suggest that the nonstationary behaviour originates from differential diffusive transport.     

Anomalous kinetics in diffusion limited reactions linked to non-Gaussian concentration probability distribution function

We investigate anomalous reaction kinetics related to segregation in the one-dimensional reaction-diffusion system A + B → C. It is well known that spatial fluctuations in the species concentrations cause a breakdown of the mean-field behavior at low concentration values. The scaling of the average concentration with time changes from the mean-field t(-1) to the anomalous t(-1/4) behavior. Using a stochastic modeling approach, the reaction-diffusion system can be fully characterized by the multi-point probability distribution function (PDF) of the species concentrations. Its evolution is governed by a Fokker-Planck equation with moving boundaries, which are determined by the positivity of the species concentrations. The concentration PDF is in general non-Gaussian. As long as the concentration fluctuations are small compared to the mean, the PDF can be approximated by a Gaussian distribution. This behavior breaks down in the fluctuation dominated regime, for which anomalous reaction kinetics are observed. We show that the transition from mean field to anomalous reaction kinetics is intimately linked to the evolution of the concentration PDF from a Gaussian to non-Gaussian shape. This establishes a direct relationship between anomalous reaction kinetics, incomplete mixing and the non-Gaussian nature of the concentration PDF.     

Study of dynamics of glucose-glucose oxidase-ferricyanide reaction

This work is focused on dynamics of the glucose-glucose oxidase-ferricyanide enzymatic reaction with or without sodium hydroxide in a continuous-flow stirred tank reactor (CSTR) and in a batch reactor. This reaction exhibits pH-variations having autocatalytic character and is reported to provide nonlinear dynamic behavior (bistability, excitability). The dynamical behavior of the reaction was examined within a wide range of inlet parameters. The main inlet parameters were the ratio of concentrations of sodium hydroxide and ferricyanide and the flow rate. In a batch reactor we observed an autocatalytic drop of pH from slightly basic to medium acidic values. In a CSTR our aim was to find bistability in the presence of sodium hydroxide. However, only a basic steady state was found. In order to reach an acidic steady state, we investigated the system in the absence of sodium hydroxide. Under these conditions the transition from the basic to the acidic steady state was observed when inlet glucose concentration was increased.