Reaction-diffusion Medium Research Articles

Self-Organized Target Wave Chimeras in Reaction-Diffusion Media

Self-Organized Target Wave Chimeras in Reaction-Diffusion Media

Drastic effects of an inert Pt wire on the redox behavior of the Belousov-Zhabotinsky reaction.

This research investigated responses of the Belousov-Zhabotinsky (BZ) reaction to the presence of a chemically inert Pt wire in solution. Experiments showed that connecting the Pt wire to a neutral ground caused a spontaneous drastic shift in the redox potential and might even induce complex behavior. Characterizations using an unstirred ferriin solution demonstrated the formation of a red colored propagating front at the grounded Pt wire, suggesting the reduction of ferriin to ferroin. Measurements with different combinations of electrodes in both stirred and reaction-diffusion media further confirmed the reduction of BZ metal catalysts at the Pt wire and the accompanying oxidation reaction at the reference electrode. The observed drastic change in redox potential and oscillation waveform can be understood based on the passive reduction reaction at the indicator electrode that is connected to the reference electrode through a potential meter. The obtained influence can be further manipulated by adding a resistor between the Pt wire and the neutral ground, making this convenient perturbation method attractive for the study of redox chemical reaction dynamics.

Two Domains of Meandering Spiral Waves in a Modified Barkley Model

The stability of rigidly rotating spiral waves is a very important topic in the study of nonlinear reaction-diffusion media. Computer experiments carried out with a slightly modified Barkley model showed that, in addition to one region of instability observed earlier in the original Barkley model, there is another one exhibiting completely different properties. The wave instability in the second region is not related to the Hopf bifurcation. Moreover, hysteresis effects are observed at the boundary of the region. This means that in the vicinity of this region of instability, direct integration of the model equations leads either to a rigidly rotating or meandering spiral, depending on the initial conditions.

Designing Stationary Reaction-Diffusion Patterns in pH Self-Activated Systems.

Since Alan Turing's 1952 pioneering work, reaction-diffusion (RD) processes are regarded as prototype mechanisms for pattern formation in living systems. Though suspected in many aspects of morphogenetic development, pure RD patterns have not yet been demonstrated in living organisms. The first observations of an autonomous development of stationary chemical patterns were made in the early 1990s. In this Account, we discuss the recent developments for producing stationary pH RD patterns in open spatial reactors. The theoretical analysis of the early experiments anticipated the possibility of finding Turing patterns in a wide range of oscillatory reactions if one could control the kinetic and diffusional rate of some key species. However, no experimentally effective method to produce stationary Turing patterns was attained before 2009, and the number of systems stagnated at two until then. The two precursor reaction systems benefited from unplanned favorable chemical properties of the RD media. Theoretical studies point out that appropriate diffusion rate differences are necessary to produce stationary patterns since a competition between an effective short distance self-activation and a long distance inhibitory process is required. This differential diffusion would naturally lead to differential exchange rates between the RD system and its feed environment, an aspect somewhat overlooked in theoretical and in primal experimental approaches. Our pattern design method takes this aspect into account. A slower diffusion of a self-activated species (here, protons), produced in the RD part of the spatial reactor, generates the accumulation of this species compared to the other species. This accumulation has to be at least partly compensated by an independent scavenging reaction. The above requirement naturally brought us to focus on two-substrate pH oscillatory reactions. Stationary RD patterns are now well documented in six pH driven reaction systems. Furthermore, the coupling with a pH dependent metal ion complexing agent led to stationary patterns in calcium ion concentration. Our effective semiempirical design method does not require a detailed knowledge of the reaction kinetics; thus it is applicable to a broad spectrum of reactions and even to synthetic biological systems. It is based on simple dynamic arguments and on general topological characteristics of a nonequilibrium phase diagram. We first illustrate our method with numerical simulations, based on a realistic but idealized general model of the two-substrate pH-oscillator reaction family, and provide a refined view of the topology of the resulting phase diagrams. Then, we exemplify its effectiveness by observations made in distinct pH self-activated systems. Analogies and differences between experiments and the model calculations are pointed out. Besides standard hexagonal arrays of spots and parallel stripes, hitherto undocumented dynamic phenomena, such as randomly blinking areas and complex dynamic and stationary filamentous structures, were observed. The main challenge, to find low-mobility complexing agents that would selectively and reversibly bind a species controlling the self-activatory kinetic path of the reaction, was readily overcome in multiple ways by anions of weak acids: not only by polymeric substances but in some cases by a pH color indicator or even smaller molecules, depending on their proton binding affinity.

Photoinduced Oscillations and Pulse Waves in the Hydrogen Peroxide-Sulfite-Ferrocyanide Reaction.

The hydrogen peroxide-sulfite-ferrocyanide reaction shows excellent photosensitivity under pulse light illumination, which can be used to efficiently control the spatiotemporal dynamics of the nonlinear chemical system. Here, we numerically simulated the photoinduced pH oscillations by integrating two models that describe the oscillatory and photosensitive behaviors, respectively. A dynamic transition from the low-pH steady state to oscillations can be obtained using light illumination. In accordance with the simulation results, oscillatory dynamics was experimentally obtained under light illumination through excitation of a low-pH steady state. In the reaction-diffusion medium, corresponding multiple pulse waves were observed under suitable acid concentration and illumination conditions. Hence, light illumination can be efficiently employed to tune pattern formation in pH dynamic systems. Especially, the observation indicates that the local oscillations and pulse waves were promoted by diffusion in the gel reactor when comparing with dynamics in a homogeneous system.

Kinetic and Diffusion-Driven Instabilities in the Bromate–Sulfite–Ferrocyanide System

The spatiotemporal dynamics of the bromate-sulfite-ferrocyanide (BSF) reaction-diffusion system in a open one-side-fed reactor (OSFR) is investigated by numerical simulations. The results of the simulations are compared with experiments performed in an annular shape OSFR. Both kinetic and diffusion-driven instabilities are identified in the model. There are two hydrogen ion consuming pathways in the mechanism: the partial oxidation of sulfite to dithionate and the oxidation of ferrocyanide by bromate ions. Their dynamical effects are similar, as they support the same negative feedback loop via sulfite ion. However, the time scale of the oxidation of ferrocyanide by bromate ions can be conveniently controlled by the input feed concentrations, thus it provides a more flexible way to find spatiotemporal oscillations. Long-range activation due to the relative fast diffusion of hydrogen ions compared to the other reactants can also result in oscillations in this mechanism. We show that the spatial extent of the reaction-diffusion medium along the direction of the diffusive feed (the thickness) acts as a general control parameter of the dynamics. Oscillations, either originated in kinetic or in diffusive instabilities, can only develop in a narrow range of the thickness. This property explains the experimentally often observed spatial localization of the oscillations. A reciprocal relationship is found between two main control parameters of the dynamics, which are the thickness and the hydrogen ion input feed concentration.

Chemical computing with reaction-diffusion processes.

Chemical reactions are responsible for information processing in living organisms. It is believed that the basic features of biological computing activity are reflected by a reaction-diffusion medium. We illustrate the ideas of chemical information processing considering the Belousov-Zhabotinsky (BZ) reaction and its photosensitive variant. The computational universality of information processing is demonstrated. For different methods of information coding constructions of the simplest signal processing devices are described. The function performed by a particular device is determined by the geometrical structure of oscillatory (or of excitable) and non-excitable regions of the medium. In a living organism, the brain is created as a self-grown structure of interacting nonlinear elements and reaches its functionality as the result of learning. We discuss whether such a strategy can be adopted for generation of chemical information processing devices. Recent studies have shown that lipid-covered droplets containing solution of reagents of BZ reaction can be transported by a flowing oil. Therefore, structures of droplets can be spontaneously formed at specific non-equilibrium conditions, for example forced by flows in a microfluidic reactor. We describe how to introduce information to a droplet structure, track the information flow inside it and optimize medium evolution to achieve the maximum reliability. Applications of droplet structures for classification tasks are discussed.

Generalized Minimal Principle for Rotor Filaments

To a reaction-diffusion medium with an inhomogeneous anisotropic diffusion tensor D, we add a fourth spatial dimension such that the determinant of the diffusion tensor is constant in four dimensions. We propose a generalized minimal principle for rotor filaments, stating that the scroll wave filament strives to minimize its surface area in the higher-dimensional space. As a consequence, stationary scroll wave filaments in the original 3D medium are geodesic curves with respect to the metric tensor G=det(D)D(-1). The theory is confirmed by numerical simulations for positive and negative filament tension and a model with a non-stationary spiral core. We conclude that filaments in cardiac tissue with positive tension preferentially reside or anchor in regions where cardiac cells are less interconnected, such as portions of the cardiac wall with a large number of cleavage planes.

Nanomotor dynamics in a chemically oscillating medium.

Synthetic nanomotors powered by chemical reactions have potential uses as cargo transport vehicles in both in vivo and in vitro applications. In many situations, motors will have to operate in out-of-equilibrium complex chemically reacting media, which supply fuel to the motors and remove the products they produce. Using molecular simulation and mean-field theory, this paper describes some of the new features that arise when a chemically powered nanomotor, operating through a diffusiophoretic mechanism, moves in an environment that supports an oscillatory chemical reaction network. It is shown how oscillations in the concentrations in chemical species in the environment give rise to oscillatory motor dynamics. More importantly, since the catalytic reactions on the motor that are responsible for its propulsion couple to the bulk phase reaction network, the motor can change its local environment. This process can give rise to distinctive spatiotemporal structures in reaction-diffusion media that occur as a result of active motor motion. Such locally induced nonequilibrium structure will play an important role in applications that involve motor dynamics in complex chemical media.

Rotation of a scroll wave reveals the metric tensor for the associated geodesic filament.

This study considers a reaction-diffusion medium that supports a rotating scroll wave with stationary filament and diffusivity matrix D. The filament is a geodesic curve in three-space. The metric tensor g associated with that geodesic is obtained in terms of D, resulting in g=(detD)D-1. Here detD (the determinant of D) is not necessarily constant in space. The detD factor, although strongly indicated previously, has not yet been derived deductively, as it is here. We take the opportunity to outline, starting from the reaction-diffusion equations, a sequence of mathematical steps leading to both geodesic and metric.

Environmental Sciences Informatics Based on Reaction-Diffusion Mechanisms of Nano-bio Material Systems Using Chemical Sensing and Computing Paradigms-A Novel Suggestion

Chemical reaction-diffusion media represents information processing means capable to efficiently solve problems of high complexity. It was found during the last decades that chemical reaction-diffusion media could be effectively used by the media by means of physical stimuli and fabrication of multi-level reaction-diffusion systems. An important challenge of the current nano-bio scenario is identifying and treating the environmental contaminants and preventing pollution, which are obligatory steps in environmental sensing and informatics. Although, we focused on the theoretical aspects, experimental implementation is also possible. Environmental informatics is a promising novel topic of interest to a wider audience.

Density wave propagation of a wave train in a closed excitable medium

A wave train in an excitable reaction-diffusion medium shows a variety of spatiotemporal patterns as a result of interactions between the individual waves. In this paper, we report a novel spatiotemporal pattern in a wave train in a closed excitable medium. We carried out experiments using a photosensitive Belousov-Zhabotinsky reaction with Ru(bpy)(3)(2+) as a catalyst and a numerical calculation using the FitzHugh-Nagumo equation. A wave train was locally distributed as an initial condition and the number of waves was systematically varied. In both the experiment and numerical calculation, density wave propagation was formed in a wave train during relaxation with a large number of waves. Our results suggest that density wave propagation originates from inhibitory interaction between the waves.

Entrainment of marginally stable excitation waves by spatially extended sub-threshold periodic forcing

We introduce a novel approach of stabilizing the dynamics of excitation waves by spatially extended sub-threshold periodic forcing. Entrainment of unstable primary waves has been studied numerically for different amplitudes and frequencies of additional sub-threshold stimuli. We determined entrainment regimes under which excitation blocks were transformed into consistent 1:1 responses. These responses were spatially homogeneous and synchronized in the entire excitable medium. Compared to primary pulses, pulses entrained by secondary stimulations were stable at considerably shorter periods which decreased at higher amplitudes and greater number of secondary stimuli. Our results suggest a practical methodology for stabilization of excitation in reaction-diffusion media such as nerve tissue with regions of reduced excitability.

Drifting solitary waves in a reaction-diffusion medium with differential advection

A distinct propagation of solitary waves in the presence of autocatalysis, diffusion, and symmetry-breaking (differential) advection, is being studied. These pulses emerge at lower reaction rates of the autocatalytic activator, i.e., when the advective flow overcomes the fast excitation and induces a fluid type "drifting" behavior, making the phenomenon unique to reaction-diffusion-advection class systems. Using the spatial dynamics analysis of a canonical model, we present the properties and the organization of such drifting pulses. The insights underly a general understanding of localized transport in simple reaction-diffusion-advection models and thus provide a background to potential chemical and biological applications.

Intelligent image sensor based on probing the evolution of redox potentials distributed in reaction–diffusion medium

Belousov–Zhabotinsky (BZ) reaction–diffusion medium, considered as a continuously distributed information processing system operating in a highly parallel mode, can be used to process images. A sol–gel method is used to prepare the light-sensitive BZ reaction–diffusion medium, which forms on a circuit board containing 8 × 8 channel matrix platinum electrodes. This medium ensures both reaction running and potential detecting. BZ reaction is excited by projecting an image (input-image) on the light-sensitive BZ medium, followed by the chemical waves’ (output-image) occurrence in the BZ medium. The imaging process in BZ medium is an evolutional one, including image inversion and boundary-enhancement function. Redox potentials distributed in the BZ medium reflect the evolution of the output-image and are detected synchronizing with the image evolution process. The output-image is reconstructed in the form of the distribution of redox potentials. The light-sensitive BZ medium and the potential detecting equipment can be developed as an intelligent image sensor. This system enabled us to open a new way to construct image sensors.

ON STOCHASTIC EVOLUTION EQUATIONS WITH NON-LIPSCHITZ COEFFICIENTS

In this paper, we study the existence and uniqueness of solutions for several classes of stochastic evolution equations with non-Lipschitz coefficients, that contains backward stochastic evolution equations, stochastic Volterra type evolution equations and stochastic functional evolution equations. In particular, the results can be used to treat a large class of quasi-linear stochastic equations, which includes the reaction diffusion and porous medium equations.

The electric charge and magnetisation distribution of the nucleon: evidence of a subatomic Turing wave pattern

Subquantum kinetics, a physics methodology that applies general systems theoretic concepts to the field of microphysics has gained the status of being a viable unified field theory. Earlier publications of this theory had proposed that a subatomic particle should consist of an electrostatic field that has the form of a radial Turing wave pattern whose form is maintained through the ongoing activity of a non-linear reaction–diffusion medium that fills all space. This subatomic Turing wave prediction now finds confirmation in recent nucleon scattering form factor data which show that the nucleon core has a Gaussian charge density distribution with a peripheral periodicity whose wavelength approximates the particle's Compton wavelength and which declines in amplitude with increasing radial distance. The subquantum kinetics explanation for the origin of charge correctly anticipates the observation that the proton's charge density wave pattern is positively biased while the neutron's is not. The phenomenon of beta decay is interpreted as the onset of a secondary bifurcation leading from the uncharged neutron solution to the charged proton solution. The Turing wave dissipative structure prediction is able to account in a unitary fashion for nuclear binding, particle diffraction, and electron orbital quantisation. The wave packet model is shown to be fundamentally flawed implying that quantum mechanics, although mathematically workable, does not realistically represent the microphysical world. This new conception points to the possible existence of orbital energy states below the Balmer ground state whose transitions may be tapped as a new source of energy.

Effect of noise on chemical waves in three-dimensional reaction-diffusion systems with gradient

The effect of noise on chemical waves in a quasi-three-dimensional reaction-diffusion medium with a gradient in the third dimension is studied using the FitzHugh-Nagumo model [R. FitzHugh, Biophysics J. 1, 445 (1961)]. Numerical simulations reveal that noise of appropriate intensity can postpone the onset of turbulence and stabilize the three-dimensional (3D) waves which would otherwise undergo the gradient-induced collapse. It is also found that the 3D waves can be interrupted by incident irregularities when the noise is not too strong; it can be induced into complete turbulence when the noise is strong enough. A mathematical analysis is given based on the dependence of the oscillation frequency on the control parameter. It agrees qualitatively with our numerical findings.

EFFECT OF NOISE AND STRUCTURAL INHOMOGENEITIES IN REACTION–DIFFUSION MEDIA

We investigate the pinning and depinning condition of a kink in both homogeneous and inhomogeneous reaction–diffusion media. We show that structural inhomogeneities hinder information propagation whereas noise enhances propagation.

Propagation and refraction of chemical waves generated by local periodic forcing in a reaction-diffusion model

We study wave propagation, interaction, and transmission across the boundary between two chemical media in a model of an oscillatory reaction-diffusion medium subjected to local periodic forcing. The forced waves can be either outwardly (OP) or inwardly propagating (IP), depending on the dispersion of the medium. Competition among forced waves, spontaneous spiral waves, and bulk oscillations is studied for both cases. We demonstrate development of a negatively refracted wave train when forced waves traverse the boundary between the OP medium and the IP medium.