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Turing instability in a boundary-fed system
Abstract
The formation of localized structures in the chlorine dioxide-idodine-malonic acid (CDIMA) reaction-diffusion system is investigated numerically using a realistic model of this system. We analyze the one-dimensional patterns formed along the gradients imposed by boundary feeds, and study their linear stability to symmetry-breaking perturbations (Turing instability) in the plane transverse to these gradients. We establish that an often-invoked simple local linear analysis that neglects longitudinal diffusion is inappropriate for predicting the linear stability of these patterns. Using a fully nonuniform analysis, we investigate the structure of the patterns formed along the gradients and their stability to transverse Turing pattern formation as a function of the values of two control parameters: the malonic acid feed concentration and the size of the reactor in the dimension along the gradients. The results from this investigation are compared with existing experiments.
... 2,[9][10][11][12][13][14] This can typically occur in experiments when non uniform profiles of concentrations of the main reactants result from the feeding of the system from the sides. 8,9,[14][15][16][17][18] These concentration gradients can lead to localized waves (such as excyclons for instance 19,20 ), Turing patterns spatially confined in a part of the reactor 8,14-18 or coexistence in different parts of the reactor of waves and Turing patterns. 18 In parallel, the Brusselator has also been largely used to decipher new spatio-temporal dynamics arising from the interplay between Turing and Hopf modes in the vicinity of a codimension-two bifurcation point in conditions where the concentration of reactants A and B is uniform in space. ...
... The qualitative information extracted from the morphology of the density profiles can be compared to the spatiotemporal dynamics obtained from the nonlinear simulations of the full equation system (13)(14)(15)(16)(17)(18)(19)(20) for the general cases considered above. An overview of these chemo-hydrodynamic patterns is presented in Fig. 4. Following the discussion in Sec. ...
The interplay of reaction and diffusion processes can trigger localized spatiotemporal patterns when two solutions containing separate reactants A and B of an oscillating reaction are put in contact. Using the Brusselator, a classical model for chemical oscillations, we show numerically that localized waves and Turing patterns as well as reaction-diffusion (RD) patterns due to an interaction between these two kinds of modes can develop in time around the reactive contact zone depending on the initial concentration of reactants and diffusion coefficients of the intermediate species locally produced. We further explore the possible hydrodynamic destabilization of an initially buoyantly stable stratification of such an A + B → oscillator system, when the chemical reaction provides a buoyant periodic forcing via localized density changes. Guided by the properties of the underlying RD dynamics, we predict new chemo-hydrodynamic instabilities on the basis of the dynamic density profiles which are here varying with the concentration of one of the intermediate species of the oscillator. Nonlinear simulations of the related reaction-diffusion-convection equations show how the active coupling between the localized oscillatory kinetics and buoyancy-driven convection can induce pulsatile convective fingering and pulsatile plumes as well as rising or sinking Turing spots, depending on the initial concentration of the reactants and their contribution to the density.
... Initially, the gel contains no reactant and dissipative spatiotemporal dynamics develop in the central part of the gel once the reactants meet by diffusion. Either localized waves (such as excyclons for instance [12,13]) or Turing patterns spatially confined in a part of the reactor [14][15][16][17][18] are then obtained experimentally depending on which * mabudroni@uniss.it; http://physchem.uniss.it/cnl.dyn/budroni. ...
... html † adewit@ulb.ac.be instability is controlling the system. From a theoretical point of view, several works have already characterized the properties of RD patterns localized by ramps of concentrations of the main reactants maintained by the feeding of the system from the sides [10,11,[16][17][18][19][20][21][22][23][24]. These concentration gradients can lead to coexistence in different parts of the reactor of waves and Turing patterns or to composite stationary spatial structures [17,[23][24][25][26][27][28]. ...
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.
... Our scheme formally resembles the construction of an augmented system in pseudo-arclength continuation [2,3,4,5]. Its formulation was motivated by the structure of the linear stability analysis of a particular chemical reaction-diffusion system, known as the chlorine-dioxide-iodine-malonic-acid system [6,7,8,9] (henceforth referred to as CDIMA), which exhibits the Turing instability. ...
... It is clear from the formulation given by Eq. 27, that the steady state of the mobile subsystem is independent of the immobile reactions. However, as illustrated in previous works [6,7,8,9], the immobile subsystem affects the stability of the mobile system. The stability of the steady state to a mode with transverse wavenumber k (k ⊥ẑ) leads to the following linear operator: ...
Consider an N x N matrix A for which zero is a defective eigenvalue. In this case, the algebraic multiplicity of the zero eigenvalue is greater than the geometric multiplicity. We show how an inflated (N+1) x (N+1) matrix L can be constructed as a rank one perturbation to A, such that L is singular but no longer defective, and the nullvectors of L can be easily related to the nullvectors of A. The motivation for this construction comes from linear stability analysis of an experimental reaction-diffusion system which exhibits the Turing instability. The utility of this scheme is accurate numerical computation of nullvector(s) corresponding to a defective zero eigenvalue. We show that numerical computations on L yield more accurate eigenvectors than direct computation on A.
... Therefore, reduced two-dimensional simulations show reasonable qualitative agreement with the experimental observations 43 . This does not mean that a three-dimensional effect cannot occur under some conditions, as numerical simulations made in parameter gradients show the possibility of forming different patterns at different locations along the gradients 44,45 . We have not observed clear signs of threedimensional effects in our experiments. ...
Turing instability is a general and straightforward mechanism of pattern formation in reaction–diffusion systems, and its relevance has been demonstrated in different biological phenomena. Still, there are many open questions, especially on the robustness of the Turing mechanism. Robust patterns must survive some variation in the environmental conditions. Experiments on pattern formation using chemical systems have shown many reaction–diffusion patterns and serve as relatively simple test tools to study general aspects of these phenomena. Here, we present a study of sinusoidal variation of the input feed concentrations on chemical Turing patterns. Our experimental, numerical and theoretical analysis demonstrates that patterns may appear even at significant amplitude variation of the input feed concentrations. Furthermore, using time-dependent feeding opens a way to control pattern formation. The patterns settled at constant feed may disappear, or new patterns may appear from a homogeneous steady state due to the periodic forcing. The generation of stationary patterns is often studied under constant experimental conditions, but in biological systems parameters such as chemical flow are not stationary. Here, the authors use experiments and numerical analyses to elucidate the mechanisms controlling Turing patterns under periodic variations in chemical feed concentration.
... In the classical review of pattern selection (Borckmans et al. 1995), for instance, the authors explicitly state that they will not be concerned with the role of boundary conditions. While many authors describe the impact of boundary conditions on such non-equilibrium phenomena (Cross and Hohenberg 1993;Murray 2004), relatively few consider more exotic conditions beyond the standard Neumann, Dirichlet, or periodic settings (Setayeshgar and Cross 1998;Dillon et al. 1994;Klika et al. 2018). Physically, periodic boundary conditions can be justified for a flat approximation of a curved manifold, which can be appropriate for patterns appearing across the entire surface of an organism. ...
Realistic examples of reaction–diffusion phenomena governing spatial and spatiotemporal pattern formation are rarely isolated systems, either chemically or thermodynamically. However, even formulations of ‘open’ reaction–diffusion systems often neglect the role of domain boundaries. Most idealizations of closed reaction–diffusion systems employ no-flux boundary conditions, and often patterns will form up to, or along, these boundaries. Motivated by boundaries of patterning fields related to the emergence of spatial form in embryonic development, we propose a set of mixed boundary conditions for a two-species reaction–diffusion system which forms inhomogeneous solutions away from the boundary of the domain for a variety of different reaction kinetics, with a prescribed uniform state near the boundary. We show that these boundary conditions can be derived from a larger heterogeneous field, indicating that these conditions can arise naturally if cell signalling or other properties of the medium vary in space. We explain the basic mechanisms behind this pattern localization and demonstrate that it can capture a large range of localized patterning in one, two, and three dimensions and that this framework can be applied to systems involving more than two species. Furthermore, the boundary conditions proposed lead to more symmetrical patterns on the interior of the domain and plausibly capture more realistic boundaries in developmental systems. Finally, we show that these isolated patterns are more robust to fluctuations in initial conditions and that they allow intriguing possibilities of pattern selection via geometry, distinct from known selection mechanisms.
... In the classical review of pattern selection [4], for instance, the authors explicitly state that they will not be concerned with the role of boundary conditions. While many authors describe the impact of boundary conditions on such non-equilibrium phenomena [10,50], relatively few consider more exotic conditions beyond the standard Neumann, Dirichlet, or periodic settings [65,15,35]. Physically, periodic boundary conditions can be justified for a flat approximation of a curved manifold, which can be appropriate for patterns appearing across the entire surface of an organism. ...
Realistic examples of reaction-diffusion phenomena governing spatial and spatiotemporal pattern formation are rarely isolated systems, either chemically or thermodynamically. However, even formulations of `open' reaction-diffusion systems often neglect the role of domain boundaries. Most idealizations of closed reaction-diffusion systems employ no-flux boundary conditions, and often patterns will form up to or along these boundaries. Motivated by boundaries of patterning fields related to emergence of spatial form in embryonic development, we propose a set of mixed boundary conditions for a two-species reaction-diffusion system which forms inhomogeneous solutions away from the boundary of the domain for a variety of different reaction kinetics, with a prescribed uniform state near the boundary. We show that these boundary conditions can be derived from a larger heterogeneous field, indicating that these conditions can arise naturally if cell signalling or other properties of the medium vary in space. We explain the basic mechanisms behind this pattern localization, and demonstrate that it can capture a large range of localized patterning in one, two, and three dimensions, and that this framework can be applied to systems involving more than two species. Furthermore, the boundary conditions proposed lead to more symmetrical patterns on the interior of the domain, and plausibly capture more realistic boundaries in developmental systems. Finally, we show that these isolated patterns are more robust to fluctuations in initial conditions, and that they allow intriguing possibilities of pattern selection via geometry, distinct from known selection mechanisms.
... This phenomenon in which a nonlinear system is asymptotically stable in the absence of diffusion but unstable in the presence of diffusion, known as Turing or diffusion-driven instability, happened to be very general and is found across many disciplines. This concept has been playing significant roles in theoretical ecology, embryology and other branches of science [6,7,14,21,22,25,27,28,36,38,40,44,55]. In the history of population ecology, the stability behavior of a system of interacting populations by taking into account the effect of diffusion in the ratio-dependent predator-prey models has received much attention by both ecologists and mathematicians [1,32,39,53]. ...
In this paper, spatial patterns of a diffusive predator-prey model with sigmoid (Holling type III) ratio-dependent functional response which concerns the influence of logistic population growth in prey and intra-species competition among predators are investigated. The (local and global) asymptotic stability behavior of the corresponding nonspatial model around the unique positive interior equilibrium point in homogeneous
steady state is obtained. In addition, we derive the conditions for Turing instability and the consequent parametric Turing space in spatial domain. The results of spatial pattern analysis through numerical simulations are depicted and analyzed. Furthermore, we perform a series of numerical simulations and find that the proposed model dynamics exhibits complex pattern replication. The feasible results obtained in this paper indicate that the effect of diffusion in Turing instability plays an important role to understand better the pattern formation in ecosystem.
... Triangles would result from the superposition of clear hexagons of the first layer with dark hexagons of the second layer, while hexa-bands would correspond to the superposition of stripes and dark hexagons. Indeed, numerical simulations Setayeshgar & Cross, 1998 show that, in the presence of parameter gradients, patterns with different symmetries can develop at different locations along the gradients. It is however noteworthy that, in experiments, the invoked dark hexagon patterns, were never observed as asymptotic monolayer patterns Ouyang & Swinney, 1991] which may question their origin in "multilayer structures". ...
Diffusive instabilities provide the engine for an ever increasing number of dissipative structures. In this class autocatalytic chemical systems are prone to generate temporal and spatial self-organization phenomena. The development of open spatial reactors and the subsequent discovery in 1989 of the stationary reaction–diffusion patterns predicted by Turing [1952] have triggered a large amount of research. This review aims at a comparison between theoretical predictions and experimental results obtained with various type of reactors in use. The differences arising from the use of reactions exhibiting either bistability of homogeneous steady states or a single one in a CSTR are emphasized.
... Heterogeneity is an obvious cause of localized oscillations, since a spatially varying medium can lead to spatially varying patterns, or patterns appearing only in localized regions of space; for example, see [8], [9] and [10]. Heterogeneity is not a necessary condition for such patterns; localized oscillations can also occur in homogeneous systems, where they have been referred to as stable coexisting patterns [11,12,13], or where pinned interfaces or grain boundaries between rolls and steady states have been studied [14,15]. ...
A new asymptotic multiple scale expansion is used to derive envelope equations for localized spatially periodic patterns in the context of the generalized Swift-Hohenberg equation. An analysis of this envelope equation results in parametric conditions for localized patterns. Furthermore, it yields corrections for wave number selection which are an order of magnitude larger for asymmetric nonlinearities than for the symmetric case. The analytical results are compared with numerical computations which demonstrate that the condition for localized patterns coincides with vanishing Hamiltonian and Lagrangian for periodic solutions. One striking feature of the choice of scaling parameters is that the derived condition for localized patterns agrees with the numerical results for a significant range of parameters which are an O(1) distance from the bifurcation, thus providing a novel approach for studying these localized patterns.
The modulation of Turing patterns through Dirichlet boundary conditions has been studied through the isothermal and non-isothermal versions of Brusselator-like model in a small-size domain reactor. We considered the Minkowski functional and the rate of entropy production to characterize the morphological aspects of the patterns and to indicate transitions of spatial states. We find that boundary conditions can induce the spatial symmetry breaking of Turing patterns when it is defined around the equilibrium points of the homogeneous dynamical system. As a result, two different Turing patterns can emerge in a reactor under an imposed gradient of chemicals that contains the equivalent concentration of the equilibrium points, at some point in the boundary.
Many exciting frontiers of science and engineering require understanding the spatiotemporal properties of sustained nonequilibrium systems such as fluids, plasmas, reacting and diffusing chemicals, crystals solidifying from a melt, heart muscle, and networks of excitable neurons in brains. This introductory textbook for graduate students in biology, chemistry, engineering, mathematics, and physics provides a systematic account of the basic science common to these diverse areas. This book provides a careful pedagogical motivation of key concepts, discusses why diverse nonequilibrium systems often show similar patterns and dynamics, and gives a balanced discussion of the role of experiments, simulation, and analytics. It contains numerous worked examples and over 150 exercises. This book will also interest scientists who want to learn about the experiments, simulations, and theory that explain how complex patterns form in sustained nonequilibrium systems.
We investigate activator–inhibitor systems in two spatial dimensions with a non-local coupling, for which the interaction strength decreases with the lattice distance as a power-law. By varying a single parameter we can pass from a local (Laplacian) to a global (all-to-all) coupling type. We derived, from a linear stability analysis of the Fourier spatial modes, a set of conditions for the occurrence of a Turing instability, by which a spatially homogeneous pattern can become unstable. In nonlinear systems the growth of these modes is limited and pattern formation is possible. We have studied some qualitative features of the patterns formed in non-local coupled activator–inhibitor systems described by the Meinhardt–Gierer equations.
Self-organized patterns are sensitive to microscopic external perturbations that modify the diffusion process. We find that Turing instability formed in a compartmented medium, a Belousov–Zhabotinski–aerosol-OT micelle reaction, responds sensitively to a change in the diffusion process. In order to modify the diffusion mechanism, we apply a centrifugal force that generates a perturbation with an anisotropic character. We find experimentally and numerically that the perturbation is able to modify the pattern and even force its disappearance. For different values of the perturbation significant changes can be seen in both the pattern wavelength and its morphology. Furthermore, for strong perturbations, the orientation of the patterns couples with the symmetry of the perturbation.
We consider a generic reaction-diffusion system in which one of the parameters is subjected to dichotomous noise by controlling the flow of one of the reacting species in a continuous-flow-stirred-tank reactor (CSTR) -membrane reactor. The linear stability analysis in an extended phase space is carried out by invoking Furutzu-Novikov procedure for exponentially correlated multiplicative noise to derive the instability condition in the plane of the noise parameters (correlation time and strength of the noise). We demonstrate that depending on the correlation time an optimal strength of noise governs the self-organization. Our theoretical analysis is corroborated by numerical simulations on pattern formation in a chlorine-dioxide-iodine-malonic acid reaction-diffusion system.
This article is about resettled Afghan Hazaras in Australia, many of whom are currently undergoing a complex process of transition (from transience into a more stable position) for the first time in their lives. Despite their permanent residency status, we show how resettlement can
be a challenging transitional experience. For these new migrants, we argue that developing a sense of belonging during the transition period is a critical rite of passage in the context of their political and cultural identity. A study of forced migrants such as these, moving out of one transient
experience into another transitional period (albeit one that holds greater promise and permanence) poses a unique intellectual challenge. New understandings about the ongoing, unpredictable consequences of ‘transience’ for refugee communities is crucial as we discover what might
be necessary, as social support structures, to facilitate the process of transition into a distinctly new environment. The article is based on a doctoral ethnographic study of 30 resettled Afghan Hazara living in the region of Dandenong in Melbourne, Australia. Here, we include four of these
participants’ reflections of transition during different phases of their resettlement. These reflections were particularly revealing of the ways in which some migrants deal with change and acquire a sense of belonging to the community. Taking a historical view, and drawing on Bourdieu’s
notion of symbolic social capital to highlight themes in individual experiences of belonging, we show how some new migrants adjust and learn to ‘embody’ their place in the new country. Symbolic social capital illuminates how people access and use resources such as social networks
as tools of empowerment, reflecting how Hazara post-arrival experiences are tied to complex power relations in their everyday social interactions and in their life trajectories as people in transition. We learned that such tools can facilitate the formation of Hazara migrant identities and
are closely tied to their civic community participation, English language development, and orientation in, as well as comprehension of local cultural knowledge and place. This kind of theorization allows refugee, post-refugee and recent migrant narratives to be viewed not merely as static
expressions of loss, trauma or damage, but rather as individual experiences of survival, adaptation and upward mobility.
We study the competition between the Turing instability to steady patterns and the Hopf instability to oscillations in diffusively coupled open reactors. Our approach is based on exact, analytical criteria for the occurrence of these instabilities in arrays of coupled reactors. We consider a general two-variable kinetic model that represents an activator−inhibitor scheme with a complexing agent or substrate for the activator. We apply our results to the Lengyel−Epstein model of the chlorine dioxide−iodine−malonic acid reaction. Using symbolic computation software, we derive exact conditions for the Turing and Hopf bifurcations in small, linear arrays of coupled reactors with an inhomogeneous concentration profile of the substrate. Our main result is the determination of the critical substrate concentration profile, above which the Turing instability occurs before the Hopf instability. This provides the condition for stationary Turing patterns to be experimentally observable in arrays of coupled reactors.
We review on the effects of the feed mode on pattern selection observed in chemical systems operated in open spatial reactors. In two-side-fed reactors, strong parameter ramps naturally confine patterns in a stratum. The reactor thickness acts both as a genuine bifurcation parameter and on the pattern dimensionality. Depending on that thickness, standard 2D hexagon and stripe Turing patterns or more complex 3D planforms are observed. In thin one-side-fed reactors, patterning process must escape the imposed fixed boundary conditions either by devices introducing mixed boundary conditions or by an intrinsic phenomenon dubbed spatial bistability. We show that in most cases, for a comprehensive understanding of experimental observations, the full 3D aspects have to be taken into account.
Some basic principles of theoretical nonlinear chemistry are introduced to explain an origin of temporal oscillations and
spatial bistability, that have both been used in other chapters in conjunction with gels sensitive to such chemical behavior.
Because the use of inert gels played such importance in their experimental discovery, elements of the theory of Turing stationary
spatial periodic patterns are also discussed.
The phenomenon of spatial bistability has recently been proposed to understand a number of paradoxical results obtained in experiments on nonequilibrium chemical patterns performed in open reactors made of a thin film of gel fed from one side. On the basis of a realistic kinetic model, we predict that the chlorine-dioxide–iodide reaction, taken as a prototypic example of a large class of reactions, should exhibit spatial bistability. The theoretical and numerical results are supported by experiments performed in specially designed reactors. This spatial bistability introduces an additional geometric dimension in the system which is generally overlooked. We elaborate on the role that this additional complexity can play in the observation of patterns associated to fronts in such reactors.
We investigate analytically and numerically the conditions for the Turing
instability to occur in a one-dimensional chain of nonlinear oscillators
coupled non-locally in such a way that the coupling strength decreases with the
spatial distance as a power-law. A range parameter makes possible to cover the
two limiting cases of local (nearest-neighbor) and a global (all-to-all)
couplings. We consider an example from a non-linear auto-catalytic
reaction-diffusion model.
We present computational solutions to the Lengyel-Rabai-Epstein model in three space dimensions. The results show that three-dimensional patterns exist and that they differ significantly from the two-dimensional patterns. Patterns occur at three locations in the reactor corresponding to peaks in the one-dimensional concentration of the starch tri-iodide concentration. Each pattern possesses its own intrinsic wavelength and is neither striped nor hexagonal, the two types that have been shown to exist in two dimensions. Computations suggest a bifurcation exists as a function of the reactor thickness. Solutions are computed using a high-order adaptive finite element method coupled with a multistep integrator in time. Linear systems generated in the multistep solver are solved using the iterative method GMRES with a Jacobi preconditioner. Matrix storage is reduced by incomplete assembly via thresholding. Preconditioner factorization and matrix-vector multiplication efficiency are enhanced by the use of OPENMP.
What is believed to be the first experimental evidence for Turing patterns was observed in the CIMA reaction by De Kepper and colleagues. Ouyang and Swinney performed further experiments in a "thin" layer of gel. Patterns observed at onset were basically two-dimensional. However, beyond onset a structure that does not typically occur in two-dimensional domains was observed-the black-eye pattern. In this paper we use the full three-dimensionality of the patterned layer to find a setting where black-eye patterns naturally occur. We propose that black-eye patterns have the symmetry of a body-centered-cubic lattice.
The development of a Turing instability to a spatially modulated state in a photoexcited electron-hole system is proposed as a novel signature of exciton Bose statistics. We show that such an instability, driven by kinetics of exciton formation, can result from stimulated processes that build up near quantum degeneracy. The stability of an electron-hole interface which describes recently observed exciton rings is analyzed. Interface instability occurs below a critical temperature, with a periodic 1D pattern developing via a continuous (type II) transition, in a qualitative agreement with observations.
We investigate three-dimensional Turing patterns in two-component reaction diffusion systems. The FitzHugh-Nagumo equation, the Brusselator, and the Gray-Scott model are solved numerically in three dimensions. Several interconnected structures of domains as well as lamellar, hexagonal, and spherical domains are obtained as stable motionless equilibrium patterns. The relative stability of these structures is studied analytically based on the reduction approximation. The relation with the microphase-separated structures in block copolymers is also discussed.
The standing concentration patterns recently discovered in open gel-filled reactors, with the chlorite-iodide-malonic acid (CIMA) oscillating reaction in the presence of starch, were ascribed to a Turing-type reaction-diffusion symmetry breaking instability. Here we extend the investigations to other regions of parameters, with a particular emphasis to the role played by the chemical nature of the gel matrix and by the starch concentration on the onset of stationary patterns. Stationary Turing patterns are shown to develop in gel-free systems. Transitions between stationary Turing structures and wave patterns are presented. The first evidence of an anti-symmetric “homogeneous” wave source is presented.
Pattern selection, localized structure formation, and front propagation are analyzed within the framework of a model for the chlorine dioxide-iodine-malonic acid reaction that represents a key to understanding recently obtained Turing structures. This model is distinguished from previously studied, simple reaction-diffusion models by producing a strongly subcritical transition to stripes. The wave number for the modes of maximum linear gain is calculated and compared with the dominant wave number for the finally selected, stationary structures grown from the homogeneous steady state or developed behind a traveling front. The speed of propagation for a front between the homogeneous steady state and a one-dimensional (1D) Turing structure is obtained. This velocity shows a characteristic change in behavior at the crossover between the subcritical and supercritical regimes for the Turing bifurcation. In the subcritical regime there is an interval where the front velocity vanishes as a result of a pinning of the front to the underlying structure. In 2D, two different nucleation mechanisms for hexagonal structures are illustrated on the Lengyel-Epstein and the Brusselator model. Finally, the observation of 1D and 2D spirals with Turing-induced cores is reported.
We report the experimental observation of a sustained standing nonequilibrium chemical pattern in a single-phase open reactor. Considering the properties of the pattern (symmetry breaking, intrinsic wavelength), it is interpreted as the first unambiguous experimental evidence of a Turing structure.
A systematic approach is suggested to design chemical systems capable of displaying stationary, symmetry-breaking reaction diffusion patterns (Turing structures). The technique utilizes the fact that reversible complexation of an activator species to form an unreactive, immobile complex reduces the effective diffusion constant of the activator, thereby facilitating the development of Turing patterns. The chlorine dioxide/iodine/malonic acid reaction is examined as an example, and it is suggested that a similar phenomenon may occur in some biological pattern formation processes.
Recent experiments on the chlorite-iodide-malonic acid-starch reaction in a gel reactor give the first evidence of the existence of the symmetry breaking, reaction-diffusion structures predicted by Turing in 1952. A five-variable model that describes the temporal behavior of the system is reduced to a two-variable model, and its spatial behavior is analyzed. Structures have been found with wavelengths that are in good agreement with those observed experimentally. The gel plays a key role by binding key iodine species, thereby creating the necessary difference in the effective diffusion coefficients of the activator and inhibitor species, iodide and chlorite ions, respectively.
It is suggested that a system of chemical substances, called morphogens, reacting together and diffusing through a tissue, is adequate to account for the main phenomena of morphogenesis. Such a system, although it may originally be quite homogeneous, may later develop a pattern or structure due to an instability of the homogeneous equilibrium, which is triggered off by random disturbances. Such reaction-diffusion systems are considered in some detail in the case of an isolated ring of cells, a mathematically convenient, though biologically unusual system. The investigation is chiefly concerned with the onset of instability. It is found that there are six essentially different forms which this may take. In the most interesting form stationary waves appear on the ring. It is suggested that this might account, for instance, for the tentacle patterns on Hydra and for whorled leaves. A system of reactions and diffusion on a sphere is also considered. Such a system appears to account for gastrulation. Another reaction system in two dimensions gives rise to patterns reminiscent of dappling. It is also suggested that stationary waves in two dimensions could account for the phenomena of phyllotaxis. The purpose of this paper is to discuss a possible mechanism by which the genes of a zygote may determine the anatomical structure of the resulting organism. The theory does not make any new hypotheses; it merely suggests that certain well-known physical laws are sufficient to account for many of the facts. The full understanding of the paper requires a good knowledge of mathematics, some biology, and some elementary chemistry. Since readers cannot be expected to be experts in all of these subjects, a number of elementary facts are explained, which can be found in text-books, but whose omission would make the paper difficult reading.
The formation of localized structures in the chlorine dioxide-idodine-malonic acid (CDIMA) reaction-diffusion system is investigated numerically using a realistic model of this system. We analyze the one-dimensional patterns formed along the gradients imposed by boundary feeds, and study their linear stability to symmetry- breaking perturbations (the Turing instability) in the plane transverse to these gradients. We establish that an often-invoked simple local linear analysis which neglects longitudinal diffusion is inappropriate for predicting the linear stability of these patterns. Using a fully nonuniform analysis, we investigate the structure of the patterns formed along the gradients and their stability to transverse Turing pattern formation as a function of the values of two control parameters: the malonic acid feed concentration and the size of the reactor in the dimension along the gradients. The results from this investigation are compared with existing experimental results. We also verify that the two-variable reduction of the chemical model employed in the linear stability analysis is justified. Finally, we present numerical solution of the CDIMA system in two dimensions which is in qualitative agreement with experiments. This result also confirms our linear stability analysis, while demonstrating the feasibility of numerical exploration of realistic chemical models.
The influence on a reaction-diffusion system of slow spatial variations possessing a minimum is analyzed. We show that local nonuniform steady states or phase gradient waves may be generated. Their onset is studied using a two-scaling approach.
CHEMICAL travelling waves have been studied experimentally for more than two decades1-5, but the stationary patterns predicted by Turing6 in 1952 were observed only recently7-9, as patterns localized along a band in a gel reactor containing a concentration gradient in reagents. The observations are consistent with a mathematical model for their geometry of reactor10 (see also ref. 11). Here we report the observation of extended (quasi-two-dimensional) Turing patterns and of a Turing bifurcation-a transition, as a control parameter is varied, from a spatially uniform state to a patterned state. These patterns form spontaneously in a thin disc-shaped gel in contact with a reservoir of reagents of the chlorite-iodide-malonic acid reaction12. Figure 1 shows examples of the hexagonal, striped and mixed patterns that can occur. Turing patterns have similarities to hydrodynamic patterns (see, for example, ref. 13), but are of particular interest because they possess an intrinsic wavelength and have a possible relationship to biological patterns14-17.
We describe experimental observations of symmetry breaking stationary patterns. These patterns are interpreted as the first unambiguous evidence of Turing-type structures in a single-phase isothermal chemical reaction system. Experiments are conducted with the versatile chlorite-iodide-malonic acid reaction in open spatial reactors filled with hydrogel. A phase diagram gathering the domain of existence of symmetry breaking and no-symmetry breaking standing patterns is discussed.
At pH 0.5-5.0, a closed system containing an aqueous mixture of chlorine dioxide, iodine, and a species such as malonic acid (MA) or ethyl acetoacetate, which reacts with iodine to produce iodide, shows periodic changes in the light absorbance of I3-. This behavior can be modeled by a simple scheme consisting of three component reactions: (1) the reaction between MA and iodine, which serves as a continuous source of I-; (2) the reaction between ClO2• and I-, which acts as a source of ClO2-; and (3) the self-inhibited reaction of chlorite and iodide that kinetically regulates the system. The fast component reaction between chlorine dioxide and iodide ion was studied by stopped-flow spectrophotometry. The rate law is -[ClO2•]/df = 6 × 103 (M-2 s-1)[ClO2•][I-]. A two-variable model obtained from the empirical rate laws of the three component reactions gives a good description of the dynamics of the system. The oscillatory behavior results not from autocatalysis but from the self-inhibitory character of the chlorite-iodide reaction.
The time development of the chlorine dioxide-iodine-malonic acid-starch reaction, which shows transient Turing patterns, traveling waves, and oscillations in an inhomogeneous closed system, is simulated with a six-variable model in one and two spatial dimensions. The results of numerical simulations are compared with experiments, and the agreement is found to be excellent.
A comprehensive review of spatiotemporal pattern formation in systems driven away from equilibrium is presented, with emphasis on comparisons between theory and quantitative experiments. Examples include patterns in hydrodynamic systems such as thermal convection in pure fluids and binary mixtures, Taylor-Couette flow, parametric-wave instabilities, as well as patterns in solidification fronts, nonlinear optics, oscillatory chemical reactions and excitable biological media. The theoretical starting point is usually a set of deterministic equations of motion, typically in the form of nonlinear partial differential equations. These are sometimes supplemented by stochastic terms representing thermal or instrumental noise, but for macroscopic systems and carefully designed experiments the stochastic forces are often negligible. An aim of theory is to describe solutions of the deterministic equations that are likely to be reached starting from typical initial conditions and to persist at long times. A unified description is developed, based on the linear instabilities of a homogeneous state, which leads naturally to a classification of patterns in terms of the characteristic wave vector q0 and frequency 0 of the instability.
We elaborate on the transition from quasi-two-dimensional to three-dimensional Turing patterns in a chemical reaction-diffusion system confined in gradients of chemicals between two feed boundaries. This transition is observed in open spatial reactors specially designed to make possible the unfolding of a pattern sequence in one direction of the plane of observation. In this direction, the confinement of the structure is progressively relaxed. Complementary observations from two reactor geometries allow the dimensionality of the structure to be elucidated: quasi-two-dimensional and three-dimensional patterns, respectively, correspond to patterns developing in monolayers and in bilayers. Beyond the now classical hexagonal and stripe patterns, various new stable planforms are shown to result from the coupling of these two classical pattern modes which develop in two adjacent layers, with well-defined phase relations between the two pattern modes.
We present some theoretical concepts that have been used in the study of chemical disspative structures together with a brief description of recent experimental work on Turing patterns and their interaction with travelling waves.
The existence of localized structures is discussed within the framework of the pattern selection problem for a model for the chlorine dioxide-iodine-malonic acid reaction that represents a key to the understanding of the recently obtained Turing structures.
It is shown that for systems developing stationary periodic patterns there exists at most one stable wavelength state if a supercritical region is connected to a subcritical one by the imposition of a slow spatial variation of the external parameters. In nonpotential systems the selected wavelength depends on the particular combination of parameters that vary but not on the (slow) rate of spatial variation. Suitable parameter variations force the system into a dynamic state.
We perform simulations of Turing patterns confined to a monolayer by a gradient of parameters in a three-dimensional system. The results provide a more comprehensive basis for the interpretation of the actual experimental results than the usual, but disputable, interpretation in terms of ideal two-dimensional systems. Systematic comparison of the bifurcation behavior in genuine two-dimensional systems and in such monolayers is achieved with a theoretical model. We show that in the monolayers, hexagonal phases are restabilized as a result of the longitudinal instability.
In experiments on quasi-two-dimensional Turing structures, patterns form perpendicular to a concentration gradient imposed by the boundary conditions. Using linear stability analysis, with the ClO2-I2-MA (malonic acid) reaction as an example, we obtain conditions on the position along the gradient direction and possible three dimensionality of the structures. Experiments on the effects of MA and starch concentrations on the position of the structures support the theory. Simulations taking into account the starch indicator yield Turing patterns even with equal diffusion coefficients for the activator and inhibitor species.
A general N+Q component reaction-diffusion system is analyzed with regard to pattern forming instabilities (Turing bifurcations). The system consists of N mobile species and Q immobile species. The Q immobile species form in response to reactions between the N mobile species and an immobile substrate and allow the Turing instability to occur. These results are valid both for bifurcations from a spatially uniform state and for systems with an externally imposed gradient as in the experimental systems in which Turing patterns have been observed. It is shown that the critical wave number and the location of the instability in parameter space are independent of the substrate concentration. It is also found that the system necessarily undergoes a Hopf bifurcation as the total substrate concentration is decreased. Further, in the case that all the mobile species diffuse at identical rates we show that if the full system is at a point of Turing bifurcation then the N component mobile subsystem is at transition from an unstable focus to an unstable node, and the critical wave number is simply related to the degenerate positive eigenvalue of the mobile subsystem. A sequence of bifurcations that occur in the eigenspectra as the total substrate concentration is decreased to zero is also discussed.
Experiments have been conducted on Turing-type chemical spatial patterns and their variants in a quasi-two-dimensional open spatial reactor with a chlorite-iodide-malonic acid reaction. A variety of stationary spatial structures-hexagons, stripes, and mixed states-were observed, and transitions to these states were studied. For conditions beyond those corresponding to the emergence of patterns, a transition was observed from stationary spatial patterns to chemical turbulence, which is marked by a continuous motion of the pattern within a domain and of the grain boundaries between domains. The transition to chemical turbulence was analyzed by measuring the correlation length, the average pattern speed, and the total length of the domain boundaries. The emergence of chemical turbulence is accompanied by a large increase in the defects in the pattern, which suggests that this is an example of defect-mediated turbulence.
Transient, symmetry-breaking, spatial patterns were obtained in a closed, gradient-free, aqueous medium containing chlorine dioxide, iodine, malonic acid, and starch at 4 degrees to 5 degrees C. The conditions under which these Turing-type structures appear can be accurately predicted from a simple mathematical model of the system. The patterns, which consist of spots, stripes, or both spots and stripes, require about 25 minutes to form and remain stationary for 10 to 30 minutes.
- G Dewel
- D Walgraef
- P Borckmans
G. Dewel, D. Walgraef, and P. Borckmans, J. Chim. Phys. (France) 84, 1335 (1987).
- E Dulos
- P Davies
- B Rudovics
- P De Kepper
E. Dulos, P. Davies, B. Rudovics, and P. De Kepper, Physica D, 98, 53 (1996).
- O Jensen
- V O Pannbacker
- E Mosekilde
- G Dewel
- P Borckmans
O. Jensen, V. O. Pannbacker, E. Mosekilde, G. Dewel, and P. Borckmans, Phys. Rev. E
50, 736 (1993).
- M C Cross
- P C Hohenberg
M. C. Cross and P. C. Hohenberg, Rev. Mod. Phys. 65, 854 (1993).
- I Lengyel
- S Kadar
- I R Epstein
I. Lengyel, S. Kadar, and I. R. Epstein, Phys. Rev. Lett. 69, 2729 (1992).
- W H Press
- S A Teukolsky
- W T Vetterling
- B P Flannery
W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes
in C (Cambridge Univ. Press, 1992).
Chemical Waves and Patterns
- I Lengyel
- I R Epstein
I. Lengyel and I. R. Epstein, Chemical Waves and Patterns, edited by R. Kapral and
K. Showalter (Klewer, 1995), p. 297.
- L Kramer
- E Benjacob
- H Brand
- M C Cross
L. Kramer, E. Benjacob, H. Brand, and M. C. Cross, Phys. Rev. Lett. 49, 1891 (1982).
- J D Murray
J. D. Murray, Mathematical Biology, (Springer-Verlag, Berlin, 1989), Chp. 15.
- I Lengyel
- G Rabai
- I R Epstein
I. Lengyel, G. Rabai, and I. R. Epstein, J. Am. Chem. Soc. 112, 4606 (1990).
- J F Auchmuty
- G Nicolis
J. F. Auchmuty, and G. Nicolis, Bulletin of Mathematical Biology 37, 323 (1975).
- A M Turing
Turing, A.M., Phil. Trans. R. Soc. London, Ser. B 327, 37 (1952).
- B T Smith
- J M Boyle
- J J Dongarra
- B S Garbow
- Y Ikebe
- V C Klema
- C B Moler
B. T. Smith, J. M. Boyle, J. J. Dongarra, B. S. Garbow, Y. Ikebe, V. C. Klema, and
C. B. Moler, Matrix Eigensystem Routines – EISPACK Guide (Springer-Verlag, Berlin,
1976).
- S Kadar
- I Lengyel
- I R Epstein
S. Kadar, I. Lengyel, and I. R. Epstein, J. Phys. Chem. 99, 4504 (1995).