Cross-diffusion and pattern formation in reaction–diffusion systems

Social media enhances languages differentiation: a mathematical description

Understanding and predicting the evolution of competing languages is a topic of high interest in a world with more than 6000 languages competing in a highly connected environment. We consider a reasonable mathematical model describing a situation of competition between two languages and analyse the effect of the speakers' connectivity (i.e. social networks). Surprisingly, instead of homogenizing the system, a high degree of connectivity helps to introduce differentiation for the appropriate parameters.

Comb-like Turing patterns embedded in Hopf oscillations: Spatially localized states outside the 2:1 frequency locked region

A generic mechanism for the emergence of spatially localized states embedded in an oscillatory background is demonstrated by using a 2:1 frequency locking oscillatory system. The localization is of Turing type and appears in two space dimensions as a comb-like state in either π phase shifted Hopf oscillations or inside a spiral core. Specifically, the localized states appear in absence of the well known flip-flop dynamics (associated with collapsed homoclinic snaking) that is known to arise in the vicinity of Hopf-Turing bifurcation in one space dimension. Derivation and analysis of three Hopf-Turing amplitude equations in two space dimensions reveal a local dynamics pinning mechanism for Hopf fronts, which in turn allows the emergence of perpendicular (to the Hopf front) Turing states. The results are shown to agree well with the comb-like core size that forms inside spiral waves. In the context of 2:1 resonance, these localized states form outside the 2:1 resonance region and thus extend the frequency locking domain for spatially extended media, such as periodically driven Belousov-Zhabotinsky chemical reactions. Implications to chlorite-iodide-malonic-acid and shaken granular media are also addressed.

Chemical communication and dynamics of droplet emulsions in networks of Belousov-Zhabotinsky micro-oscillators produced by microfluidics

Chemical communication leading to synchronization and collective behaviour of dynamic elements, such as cell colonies, is a widespread phenomenon with biological, physical and chemical importance. Such synchronization between elements proceeds via chemical communication by emmision, interdiffusion and reception of specific messenger molecules. On a lab scale, these phenomena can be modeled by encapsulating an oscillating chemical reaction, which serves as a signal (information) sender/receiver element, inside microcompartments such as droplet emulsions, liposomes and polymersomes. Droplets can thus be regarded as single units, able to generate chemical messengers that diffuse in the environment and hence can interact with other compartments. The Belousov-Zhabotinsky (BZ) reaction is a well-known chemical oscillator largely used as a model for complex nonlinear phenomena, including chemical, physical and biological examples. When the BZ-reaction is encapsulated inside microcompartments, its chemical intermediates can serve as messengers by diffusing among different microcompartments, to trigger specific reactions leading to a collective behavior between the elements. The geometry and constitution of the diffusion pathways play an important role in governing the collective behaviour of the system. In this context, microfluidics is not only a versatile tool for mastering the encapsulation process of the BZ-reaction in monodisperse microcompartments, but also for creating geometries and networks with well defined boundaries. The individual compartments can be engineered with selected properties using different surfactants in the case of simple emulsions, or with specific membrane properties in the case of liposomes. Furthermore, it enables the arrangement of these microcompartments in various geometric configurations, where the diffusive coupling pathways between individual compartments are both spatially and chemically well-defined. In this tutorial paper, we review a number of articles reporting various approaches to generate networks of compartmentalized Belousov-Zhabotinsky (BZ) chemical oscillators using microfluidics. In contrast to biological cellular networks, the dynamical characteristics of the BZ-reaction is well-known and, when confined in microcompartments arranged in different configurations with a pure interdiffusive coupling, these communicative microreactors can serve to mimic various types of bio-physical networks, aiding to comprehend the concept of chemical communication.

Oscillatory pulses and wave trains in a bistable reaction-diffusion system with cross diffusion

We study waves with exponentially decaying oscillatory tails in a reaction-diffusion system with linear cross diffusion. To be specific, we consider a piecewise linear approximation of the FitzHugh-Nagumo model, also known as the Bonhoeffer-van der Pol model. We focus on two types of traveling waves, namely solitary pulses that correspond to a homoclinic solution, and sequences of pulses or wave trains, i.e., a periodic solution. The effect of cross diffusion on wave profiles and speed of propagation is analyzed. We find the intriguing result that both pulses and wave trains occur in the bistable cross-diffusive FitzHugh-Nagumo system, whereas only fronts exist in the standard bistable system without cross diffusion.

Large amplitude excitations traveling along the interface in bistable catalytic methanol oxidation on Rh(110)

A PEEM image and an x–t-plot showing traveling interface modulations in a bistable surface reaction.

Quasisolitons in self-diffusive excitable systems, or Why asymmetric diffusivity obeys the Second Law

Solitons, defined as nonlinear waves which can reflect from boundaries or transmit through each other, are found in conservative, fully integrable systems. Similar phenomena, dubbed quasi-solitons, have been observed also in dissipative, "excitable" systems, either at finely tuned parameters (near a bifurcation) or in systems with cross-diffusion. Here we demonstrate that quasi-solitons can be robustly observed in excitable systems with excitable kinetics and with self-diffusion only. This includes quasi-solitons of fixed shape (like KdV solitons) or envelope quasi-solitons (like NLS solitons). This can happen in systems with more than two components, and can be explained by effective cross-diffusion, which emerges via adiabatic elimination of a fast but diffusing component. We describe here a reduction procedure can be used for the search of complicated wave regimes in multi-component, stiff systems by studying simplified, soft systems.

Multifold Increases in Turing Pattern Wavelength in the Chlorine Dioxide-Iodine-Malonic Acid Reaction-Diffusion System

Turing patterns in the chlorine dioxide-iodine-malonic acid reaction were modified through additions of sodium halide salt solutions. The range of wavelengths obtained is several times larger than in the previously reported literature. Pattern wavelength was observed to significantly increase with sodium bromide or sodium chloride. A transition to a uniform state was found at high halide concentrations. The observed experimental results are qualitatively well reproduced in numerical simulations with the Lengyel-Epstein model with an additional chemically realistic kinetic term to account for the added halide and an adjustment of the activator diffusion rate to allow for interhalogen formation.

Localized stationary and traveling reaction-diffusion patterns in a two-layer A+B→ oscillator system

When two solutions containing separate reactants A and B of an oscillating reaction are put in contact in a gel, localized spatiotemporal patterns can develop around the contact zone thanks to the interplay of reaction and diffusion processes. Using the Brusselator model, we explore analytically the deployment in space and time of the bifurcation diagram of such an A+B→ oscillator system. We provide a parametric classification of possible instabilities as a function of the ratio of the initial reactant concentrations and of the reaction intermediate species diffusion coefficients. Related one-dimensional reaction-diffusion dynamics are studied numerically. We find that the system can spatially localize waves and Turing patterns as well as induce more complex dynamics such as zigzag spatiotemporal waves when Hopf and Turing modes interact.

Diffusive instabilities in hyperbolic reaction-diffusion equations

We investigate two-variable reaction-diffusion systems of the hyperbolic type. A linear stability analysis is performed, and the conditions for diffusion-driven instabilities are derived. Two basic types of eigenvalues, real and complex, are described. Dispersion curves for both types of eigenvalues are plotted and their behavior is analyzed. The real case is related to the Turing instability, and the complex one corresponds to the wave instability. We emphasize the interesting feature that the wave instability in the hyperbolic equations occurs in two-variable systems, whereas in the parabolic case one needs three reaction-diffusion equations.

Convective flow reversal in self-powered enzyme micropumps

Surface-bound enzymes can act as pumps that drive large-scale fluid flows in the presence of their substrates or promoters. Thus, enzymatic catalysis can be harnessed for “on demand” pumping in nano- and microfluidic devices powered by an intrinsic energy source. The mechanisms controlling the pumping have not, however, been completely elucidated. Herein, we combine theory and experiments to demonstrate a previously unreported spatiotemporal variation in pumping behavior in urease-based pumps and uncover the mechanisms behind these dynamics. We developed a theoretical model for the transduction of chemical energy into mechanical fluid flow in these systems, capturing buoyancy effects due to the solution containing nonuniform concentrations of substrate and product. We find that the qualitative features of the flow depend on the ratios of diffusivities δ=D(P)/D(S) and expansion coefficients β=β(P)/β(S) of the reaction substrate (S) and product (P). If δ>1 and δ>β (or if δ<1 and δ<β ), an unexpected phenomenon arises: the flow direction reverses with time and distance from the pump. Our experimental results are in qualitative agreement with the model and show that both the speed and direction of fluid pumping (i) depend on the enzyme activity and coverage, (ii) vary with the distance from the pump, and (iii) evolve with time. These findings permit the rational design of enzymatic pumps that accurately control the direction and speed of fluid flow without external power sources, enabling effective, self-powered fluidic devices.

Cross-diffusion-driven hydrodynamic instabilities in a double-layer system: General classification and nonlinear simulations

Cross diffusion, whereby a flux of a given species entrains the diffusive transport of another species, can trigger buoyancy-driven hydrodynamic instabilities at the interface of initially stable stratifications. Starting from a simple three-component case, we introduce a theoretical framework to classify cross-diffusion-induced hydrodynamic phenomena in two-layer stratifications under the action of the gravitational field. A cross-diffusion-convection (CDC) model is derived by coupling the fickian diffusion formalism to Stokes equations. In order to isolate the effect of cross-diffusion in the convective destabilization of a double-layer system, we impose a starting concentration jump of one species in the bottom layer while the other one is homogeneously distributed over the spatial domain. This initial configuration avoids the concurrence of classic Rayleigh-Taylor or differential-diffusion convective instabilities, and it also allows us to activate selectively the cross-diffusion feedback by which the heterogeneously distributed species influences the diffusive transport of the other species. We identify two types of hydrodynamic modes [the negative cross-diffusion-driven convection (NCC) and the positive cross-diffusion-driven convection (PCC)], corresponding to the sign of this operational cross-diffusion term. By studying the space-time density profiles along the gravitational axis we obtain analytical conditions for the onset of convection in terms of two important parameters only: the operational cross-diffusivity and the buoyancy ratio, giving the relative contribution of the two species to the global density. The general classification of the NCC and PCC scenarios in such parameter space is supported by numerical simulations of the fully nonlinear CDC problem. The resulting convective patterns compare favorably with recent experimental results found in microemulsion systems.

Diffusion-driven instabilities by immobilizing the autocatalyst in ionic systems

Spatiotemporal coupling of an autocatalytic chemical reaction between ions with diffusion yields various types of reaction-diffusion patterns. The driving force is short range activation and long range inhibition which can be achieved by selective binding of the autocatalyst even for ions with equal mobility. For Turing and lateral instability, we show that identical charge on the autocatalyst and its counterpart has a stabilizing effect on the base state, while opposite charge on them favors the formation of spatial patterns with reversible binding.

Interfacial hydrodynamic instabilities driven by cross-diffusion in reverse microemulsions

When two microemulsions are put in contact in the gravity field along a horizontal contact line, cross-diffusion can trigger the transport of one species in the presence of a gradient in concentration of another species. We show here theoretically that such cross-diffusion effects can induce buoyancy-driven convective instabilities at the interface between two solutions of different compositions even when initially the less dense solution lies on top of the denser one. Two different sources of convective modes are identified depending whether positive or negative cross-diffusion is involved. We evidence the two predicted cross-diffusion driven instabilities experimentally using a two-layer stratification of Aerosol-OT (AOT) water-in-oil microemulsions solutions with different water or AOT composition.

Standing wave-like patterns in the Gray-Scott model

Standing wave-like patterns are obtained in the Gray-Scott model when the dynamics that correspond to defect-mediated turbulence for equal diffusivities interact with a Turing instability. The Turing instability can be caused by either differential or cross-diffusion. We compare results with the Oregonator model, for which standing wave-like patterns are also observed under similar conditions.

Cross-diffusion-induced convective patterns in microemulsion systems

Cross-diffusion phenomena are experimentally and theoretically shown to be able to induce convective fingering around an initially stable stratification of two microemulsions with different compositions.

Classification of wave regimes in excitable systems with linear cross diffusion

We consider principal properties of various wave regimes in two selected excitable systems with linear cross diffusion in one spatial dimension observed at different parameter values. This includes fixed-shape propagating waves, envelope waves, multienvelope waves, and intermediate regimes appearing as waves propagating at a fixed shape most of the time but undergoing restructuring from time to time. Depending on parameters, most of these regimes can be with and without the "quasisoliton" property of reflection of boundaries and penetration through each other. We also present some examples of the behavior of envelope quasisolitons in two spatial dimensions.

Amplification of salt-induced polymer diffusiophoresis by increasing salting-out strength

The role of salting-out strength on (1) polymer diffusiophoresis from high to low salt concentration, and (2) salt osmotic diffusion from high to low polymer concentration was investigated. These two cross-diffusion phenomena were experimentally characterized by Rayleigh interferometry at 25 °C. Specifically, we report ternary diffusion coefficients for polyethylene glycol (molecular weight, 20 kg·mol(-1)) in aqueous solutions of several salts (NaCl, KCl, NH4Cl, CaCl2, and Na2SO4) as a function of salt concentration at low polymer concentration (0.5% w/w). We also measured polymer diffusion coefficients by dynamic light scattering in order to discuss the interpretation of these transport coefficients in the presence of cross-diffusion effects. Our cross-diffusion results, primarily those on salt osmotic diffusion, were utilized to extract N(w), the number of water molecules in thermodynamic excess around a macromolecule. This preferential-hydration parameter characterizes the salting-out strength of the employed salt. For chloride salts, changing cation has a small effect on N(w). However, replacing NaCl with Na2SO4 (i.e., changing anion) leads to a 3-fold increase in N(w), in agreement with cation and anion Hofmeister series. Theoretical arguments show that polymer diffusiophoresis is directly proportional to the difference N(w) - n(w), where n(w) is the number of water molecules transported by the migrating macromolecule. Interestingly, the experimental ratio, n(w)/N(w), was found to be approximately the same for all investigated salts. Thus, the magnitude of polymer diffusiophoresis is also proportional to salting-out strength as described by N(w). A basic hydrodynamic model was examined in order to gain physical insight on the role of n(w) in particle diffusiophoresis and explain the observed invariance of n(w)/N(w). Finally, we consider a steady-state diffusion problem to show that concentration gradients of strong salting-out agents such as Na2SO4 can produce large amplifications and depletions of macromolecule concentration. These effects may be exploited in self-assembly and adsorption processes.

Exhibiting cross-diffusion-induced patterns for reaction-diffusion systems on evolving domains and surfaces

The aim of this manuscript is to present for the first time the application of the finite element method for solving reaction-diffusion systems with cross-diffusion on continuously evolving domains and surfaces. Furthermore we present pattern formation generated by the reaction-diffusion system with cross-diffusion on evolving domains and surfaces. A two-component reaction-diffusion system with linear cross-diffusion in both u and v is presented. The finite element method is based on the approximation of the domain or surface by a triangulated domain or surface consisting of a union of triangles. For surfaces, the vertices of the triangulation lie on the continuous surface. A finite element space of functions is then defined by taking the continuous functions which are linear affine on each simplex of the triangulated domain or surface. To demonstrate the role of cross-diffusion to the theory of pattern formation, we compute patterns with model kinetic parameter values that belong only to the cross-diffusion parameter space; these do not belong to the standard parameter space for classical reaction-diffusion systems. Numerical results exhibited show the robustness, flexibility, versatility, and generality of our methodology; the methodology can deal with complicated evolution laws of the domain and surface, and these include uniform isotropic and anisotropic growth profiles as well as those profiles driven by chemical concentrations residing in the domain or on the surface.

Effect of particle size on salt-induced diffusiophoresis compared to Brownian mobility

For ternary polymer-salt-water systems at low polymer concentration (0.5%, w/w), we have experimentally investigated the effect of polymer size on polymer diffusiophoresis (i.e., polymer migration induced by a salt concentration gradient) and salt osmotic diffusion (i.e., salt migration induced by a polymer concentration gradient). Specifically, Rayleigh interferometry was employed to measure ternary diffusion coefficients for aqueous solutions of poly(ethylene glycol) (PEG) and KCl at 25 °C. Our investigation focused on four polymer molecular masses (from 10 to 100 kg mol(-1)) and two salt concentrations (0.25 and 0.50 M). To describe and examine our experimental results, we introduced a normalized diffusiophoresis coefficient as the ratio of polymer diffusiophoresis to polymer Brownian mobility. This coefficient was found to increase with polymer molecular mass, thereby demonstrating that the relative importance of polymer diffusiophoresis compared to its intrinsic Brownian mobility increases with particle size. The observed behavior was linked to preferential hydration (water thermodynamic excess) and hydration (bound water) of the macromolecule. The ratio of salt osmotic diffusion to binary salt-water diffusion approximately describes the nonuniform spatial distribution of salt along a static polymer concentration gradient at equilibrium. The significance of polymer diffusiophoresis, especially at high PEG molecular mass, was examined by considering a steady-state diffusion problem showing that salt concentration gradients can produce large enhancements and depletions of polymer concentration. This work is valuable for understanding and modeling the effect of salt concentration gradients on diffusion-based transport of polymers with applications to interfacial processes.

Cross-diffusion-driven instability for reaction-diffusion systems: analysis and simulations

By introducing linear cross-diffusion for a two-component reaction-diffusion system with activator-depleted reaction kinetics (Gierer and Meinhardt, Kybernetik 12:30-39, 1972; Prigogine and Lefever, J Chem Phys 48:1695-1700, 1968; Schnakenberg, J Theor Biol 81:389-400, 1979), we derive cross-diffusion-driven instability conditions and show that they are a generalisation of the classical diffusion-driven instability conditions in the absence of cross-diffusion. Our most revealing result is that, in contrast to the classical reaction-diffusion systems without cross-diffusion, it is no longer necessary to enforce that one of the species diffuse much faster than the other. Furthermore, it is no longer necessary to have an activator-inhibitor mechanism as premises for pattern formation, activator-activator, inhibitor-inhibitor reaction kinetics as well as short-range inhibition and long-range activation all have the potential of giving rise to cross-diffusion-driven instability. To support our theoretical findings, we compute cross-diffusion induced parameter spaces and demonstrate similarities and differences to those obtained using standard reaction-diffusion theory. Finite element numerical simulations on planary square domains are presented to back-up theoretical predictions. For the numerical simulations presented, we choose parameter values from and outside the classical Turing diffusively-driven instability space; outside, these are chosen to belong to cross-diffusively-driven instability parameter spaces. Our numerical experiments validate our theoretical predictions that parameter spaces induced by cross-diffusion in both the [Formula: see text] and [Formula: see text] components of the reaction-diffusion system are substantially larger and different from those without cross-diffusion. Furthermore, the parameter spaces without cross-diffusion are sub-spaces of the cross-diffusion induced parameter spaces. Our results allow experimentalists to have a wider range of parameter spaces from which to select reaction kinetic parameter values that will give rise to spatial patterning in the presence of cross-diffusion.

Cross-diffusion in the two-variable Oregonator model

We explore the effect of cross-diffusion on pattern formation in the two-variable Oregonator model of the Belousov-Zhabotinsky reaction. For high negative cross-diffusion of the activator (the activator being attracted towards regions of increased inhibitor concentration) we find, depending on the values of the parameters, Turing patterns, standing waves, oscillatory Turing patterns, and quasi-standing waves. For the inhibitor, we find that positive cross-diffusion (the inhibitor being repelled by increasing concentrations of the activator) can induce Turing patterns, jumping waves and spatially modulated bulk oscillations. We qualitatively explain the formation of these patterns. With one model we can explain Turing patterns, standing waves and jumping waves, which previously was done with three different models.

Spectromicroscopy of pulses transporting alkali metal in a surface reaction

The NO + H2 reaction on a potassium promoted Rh(110) surface is shown to sustain the formation of spatio-temporal periodic patterns leading to mass transport phenomena. The excitation of pulses and the mass transport mechanism are studied in the 10(-7) and 10(-6) mbar pressure range, with the potassium coverage varying between θK = 0.05 and θK = 0.12 ML. Using spectroscopic photoemission and spectroscopic low energy electron microscopy (SPELEEM) as well as related microprobe diffraction techniques, we show that the excitation mechanism comprises a cyclic structural transformation: K + O-coadsorbate → (2 × 1)-N → c(2 × 4)-2O,N → K + O coadsorbate. Laterally resolved spectroscopy demonstrates that potassium is accumulated in front of the nitrogen pulses, suggesting that adsorbed nitrogen acts as a diffusion barrier for potassium.

Spinodal decomposition and the emergence of dissipative transient periodic spatio-temporal patterns in acentrosomal microtubule multitudes of different morphology

We have studied a spontaneous self-organization dynamics in a closed, dissipative (in terms of guansine 5'-triphosphate energy dissipation), reaction-diffusion system of acentrosomal microtubules (those nucleated and organized in the absence of a microtubule-organizing centre) multitude constituted of straight and curved acentrosomal microtubules, in highly crowded conditions, in vitro. Our data give experimental evidence that cross-diffusion in conjunction with excluded volume is the underlying mechanism on basis of which acentrosomal microtubule multitudes of different morphologies (straight and curved) undergo a spatial-temporal demix. Demix is constituted of a bifurcation process, manifested as a slow isothermal spinodal decomposition, and a dissipative process of transient periodic spatio-temporal pattern formation. While spinodal decomposition is an energy independent process, transient periodic spatio-temporal pattern formation is accompanied by energy dissipative process. Accordingly, we have determined that the critical threshold for slow, isothermal spinodal decomposition is 1.0 ± 0.05 mg/ml of microtubule protein concentration. We also found that periodic spacing of transient periodic spatio-temporal patterns was, in the overall, increasing versus time. For illustration, we found that a periodic spacing of the same pattern was 0.375 ± 0.036 mm, at 36 °C, at 155th min, while it was 0.540 ± 0.041 mm at 31 °C, and at 275th min after microtubule assembly started. The lifetime of transient periodic spatio-temporal patterns spans from half an hour to two hours approximately. The emergence of conditions of macroscopic symmetry breaking (that occur due to cross-diffusion in conjunction with excluded volume) may have more general but critical importance in morphological pattern development in complex, dissipative, but open cellular systems.

Particle dynamics simulations of Turing patterns

The direct simulation Monte Carlo method is used to reproduce Turing patterns at the microscopic level in reaction-diffusion systems. In order to satisfy the basic condition for the development of such a spatial structure, we propose a model involving a solvent, which allows for disparate diffusivities of individual reactive species. One-dimensional structures are simulated in systems of various lengths. Simulation results agree with the macroscopic predictions obtained by integration of the reaction-diffusion equations. Additional effects due to internal fluctuations are observed, such as temporal transitions between structures of different wavelengths in a confined system. For a structure developing behind a propagating wave front, the fluctuations suppress the induction period and accelerate the formation of the Turing pattern. These results support the ability of reaction-diffusion models to robustly reproduce axial segmentation including the formation of early vertebrae or somites in noisy biological environments.

Experimental assessment of the sensitiveness of an electrochemical oscillator towards chemical perturbations

In this study we address the problem of the response of a (electro)chemical oscillator towards chemical perturbations of different magnitudes. The chemical perturbation was achieved by addition of distinct amounts of trifluoromethanesulfonate (TFMSA), a rather stable and non-specifically adsorbing anion, and the system under investigation was the methanol electro-oxidation reaction under both stationary and oscillatory regimes. Increasing the anion concentration resulted in a decrease in the reaction rates of methanol oxidation and a general decrease in the parameter window where oscillations occurred. Furthermore, the addition of TFMSA was found to decrease the induction period and the total duration of oscillations. The mechanism underlying these observations was derived mathematically and revealed that inhibition in the methanol oxidation through blockage of active sites was found to further accelerate the intrinsic non-stationarity of the unperturbed system. Altogether, the presented results are among the few concerning the experimental assessment of the sensitiveness of an oscillator towards chemical perturbations. The universal nature of the complex chemical oscillator investigated here might be used for reference when studying the dynamics of other less accessible perturbed networks of (bio)chemical reactions.

Segmented waves in a reaction-diffusion-convection system

The interaction of traveling waves, with both Marangoni and buoyancy driven flows, can generate an extraordinary rich array of patterns ranging from stationary structures to chaotic waves. However, the inherent complexity of reaction-diffusion-convection (RDC) systems makes the explanation of the patterning mechanisms very difficult, both numerically and experimentally. In this paper, we describe the appearance of segmented waves in a shallow layer of an excitable Belousov-Zhabotinsky solution. The segmentation process was found to be dependent both on the depth of the solution and on the excitability of the reaction. We caught the essential features of the system through a RDC model, where the chemical waves were coupled both with surface and bulk fluid motions and we found that by varying the excitability of the reaction, and in turn the wavelength of the chemical fronts, it is possible to create a sort of hydrodynamic resonance structures (corridors), which are responsible for the segmentation process.

Envelope quasisolitons in dissipative systems with cross-diffusion

We consider two-component nonlinear dissipative spatially extended systems of reaction-cross-diffusion type. Previously, such systems were shown to support "quasisoliton" pulses, which have a fixed stable structure but can reflect from boundaries and penetrate each other. Herein we demonstrate a different type of quasisolitons, with a phenomenology resembling that of the envelope solitons in the nonlinear Schrödinger equation: spatiotemporal oscillations with a smooth envelope, with the velocity of the oscillations different from the velocity of the envelope.

Amplitude equations for reaction-diffusion systems with cross diffusion

Using Taylor series expansion, multiscaling, and further expansion in powers of a small parameter, we develop general amplitude equations for two-variable reaction-diffusion systems with cross-diffusion terms in the cases of Hopf and Turing instabilities. We apply this analysis to the Oregonator and Brusselator models and find that inhibitor cross diffusion induced by the activator and activator cross diffusion induced by the inhibitor have opposite effects in the two models as a result of the different structure of their community matrices. Our analysis facilitates finding regions of supercritical and subcritical bifurcations, as well as wave and antiwave domains and domains of turbulent waves in the case of Hopf instability.

Effects of cross diffusion on Turing bifurcations in two-species reaction-transport systems

We study the Turing bifurcation in general two-species reaction-transport systems, where particle dispersal is governed by diffusion, including cross diffusion. By performing a linear stability analysis, we find conditions for the Turing instability and compare the results with the standard Turing conditions. We apply our results to two model systems, the Lengyel-Epstein model and the Brusselator, and find strong effects of cross diffusion on the Turing bifurcation.

Diffusive coupling can discriminate between similar reaction mechanisms in an allosteric enzyme system

BACKGROUND

A central question for the understanding of biological reaction networks is how a particular dynamic behavior, such as bistability or oscillations, is realized at the molecular level. So far this question has been mainly addressed in well-mixed reaction systems which are conveniently described by ordinary differential equations. However, much less is known about how molecular details of a reaction mechanism can affect the dynamics in diffusively coupled systems because the resulting partial differential equations are much more difficult to analyze.

RESULTS

Motivated by recent experiments we compare two closely related mechanisms for the product activation of allosteric enzymes with respect to their ability to induce different types of reaction-diffusion waves and stationary Turing patterns. The analysis is facilitated by mapping each model to an associated complex Ginzburg-Landau equation. We show that a sequential activation mechanism, as implemented in the model of Monod, Wyman and Changeux (MWC), can generate inward rotating spiral waves which were recently observed as glycolytic activity waves in yeast extracts. In contrast, in the limiting case of a simple Hill activation, the formation of inward propagating waves is suppressed by a Turing instability. The occurrence of this unusual wave dynamics is not related to the magnitude of the enzyme cooperativity (as it is true for the occurrence of oscillations), but to the sensitivity with respect to changes of the activator concentration. Also, the MWC mechanism generates wave patterns that are more stable against long wave length perturbations.

CONCLUSIONS

This analysis demonstrates that amplitude equations, which describe the spatio-temporal dynamics near an instability, represent a valuable tool to investigate the molecular effects of reaction mechanisms on pattern formation in spatially extended systems. Using this approach we have shown that the occurrence of inward rotating spiral waves in glycolysis can be explained in terms of an MWC, but not with a Hill mechanism for the activation of the allosteric enzyme phosphofructokinase. Our results also highlight the importance of enzyme oligomerization for a possible experimental generation of Turing patterns in biological systems.

Spontaneous traveling waves in oscillatory systems with cross diffusion

We identify a type of pattern formation in spatially distributed active systems. We simulate one-dimensional two-component systems with predator-prey local interaction and pursuit-evasion taxis between the components. In a sufficiently large domain, spatially uniform oscillations in such systems are unstable with respect to small perturbations. This instability, through a transient regime appearing as spontaneous focal sources, leads to establishment of periodic traveling waves. The traveling wave regime is established even if boundary conditions do not favor such solutions. The stable wavelength is within a range bounded both from above and from below, and this range does not coincide with instability bands of the spatially uniform oscillations.