Turing Patterns In Reaction-diffusion Systems Research Articles

Cross-diffusion-induced transitions between Turing patterns in reaction-diffusion systems

Cross-diffusion is one of the most important factors affecting the formation and transition of Turing patterns in reaction diffusion systems. In this paper, cross-diffusion is introduced into a reaction diffusion Brusselator model to investigate the effects of the directivity and density-dependence of cross-diffusion on Turing pattern transition. Turing space is obtained by the standard linear stability analysis, and the amplitude equations are derived based on weakly nonlinear method, by which Turing pattern selection can be determined theoretically. It is found that the degree of deviation from the primary Turing bifurcation point plays an important role in determining the process of pattern selection in the Turing region. As the deviation from onset is increased, the system exhibits a series of pattern transitions from homogenous state to honeycomb hexagonal pattern, to stripe pattern, and then to hexagonal spot pattern. In the case of one-way cross-diffusion, the direction of cross-diffusion determines the order of Turing pattern transition. The cross-diffusion from the inhibitor to the activator enhances the Turing mode and drives the system far away from the primary bifurcation point, resulting in the forward order of Turing pattern transition. On the contrary, the cross-diffusion from the activator to the inhibitor suppresses the Turing mode and forces the pattern transition in a reverse order. In the case of two-way cross-diffusion, the cross-diffusion effect from inhibitors to activators is stronger than that from activators to inhibitors with the same diffusion coefficient. Essentially, the cross-diffusion coefficient is dependent on not only the local concentration of species itself, but also the concentrations of other species due to their interaction. It is found that concentration dependent cross diffusion also affects the transformation direction of Turing pattern. When the diffusion coefficient <inline-formula><tex-math id="M6">\begin{document}$ {D_{uv}} $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="19-20230333_M6.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="19-20230333_M6.png"/></alternatives></inline-formula> is linearly dependent on the concentration of retarders, the positive transformation of the Turing pattern is induced with the increase of the concentration linear adjustment parameter <inline-formula><tex-math id="M7">\begin{document}$ \beta $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="19-20230333_M7.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="19-20230333_M7.png"/></alternatives></inline-formula>. On the contrary, when the diffusion coefficient <inline-formula><tex-math id="M8">\begin{document}$ {D_{vu}} $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="19-20230333_M8.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="19-20230333_M8.png"/></alternatives></inline-formula> is linearly dependent on the concentration of active particles, the reverse transformation of the Turing pattern is induced. The numerical simulation results are consistent with the theoretical analysis.

Robust controlled formation of Turing patterns in three-component systems.

Over the past few decades, formation of Turing patterns in reaction-diffusion systems has been shown to be the underlying process in several examples of biological morphogenesis, confirming Alan Turing's hypothesis, put forward in 1952. However, theoretical studies suggest that Turing patterns formation via classical "short-range activation and long-range inhibition" concept in general can happen within only narrow parameter ranges. This feature seemingly contradicts the accuracy and reproducibility of biological morphogenesis given the stochasticity of biochemical processes and the influence of environmental perturbations. Moreover, it represents a major hurdle to synthetic engineering of Turing patterns. In this work it is shown that this problem can be overcome in some systems under certain sets of interactions between their elements, one of which is immobile and therefore corresponding to a cell-autonomous factor. In such systems Turing patterns formation can be guaranteed by a simple universal control under any values of kinetic parameters and diffusion coefficients of mobile elements. This concept is illustrated by analysis and simulations of a specific three-component system, characterized in absence of diffusion by a presence of codimension two pitchfork-Hopf bifurcation.

Widening the criteria for emergence of Turing patterns.

The classical concept for emergence of Turing patterns in reaction-diffusion systems requires that a system should be composed of complementary subsystems, one of which is unstable and diffuses sufficiently slowly while the other one is stable and diffuses sufficiently rapidly. In this work, the phenomena of emergence of Turing patterns are studied and do not fit into this concept, yielding the following results. (1) The criteria are derived, under which a reaction-diffusion system with immobile species should spontaneously produce Turing patterns under any diffusion coefficients of its mobile species. It is shown for such systems that under certain sets of types of interactions between their species, Turing patterns should be produced under any parameter values, at least provided that the corresponding spatially non-distributed system is stable. (2) It is demonstrated that in a reaction-diffusion system, which contains more than two species and is stable in absence of diffusion, the presence of a sufficiently slowly diffusing unstable subsystem is already sufficient for diffusion instability (i.e., Turing or wave instability), while its complementary subsystem can also be unstable. (3) It is shown that the presence of an immobile unstable subsystem, which leads to destabilization of waves within an infinite range of wavenumbers, in a spatially discrete case can result in the generation of large-scale stationary or oscillatory patterns. (4) It is demonstrated that under the presence of subcritical Turing and supercritical wave bifurcations, the interaction of two diffusion instabilities can result in the spontaneous formation of Turing structures outside the region of Turing instability.

Structure preserving reduced order modeling for gradient systems

Minimization of energy in gradient systems leads to formation of oscillatory and Turing patterns in reaction-diffusion systems. These patterns should be accurately computed using fine space and time meshes over long time horizons to reach the spatially inhomogeneous steady state. In this paper, a reduced order model (ROM) is developed which preserves the gradient dissipative structure. The coupled system of reaction-diffusion equations are discretized in space by the symmetric interior penalty discontinuous Galerkin (SIPG) method. The resulting system of ordinary differential equations (ODEs) are integrated in time by the average vector field (AVF) method, which preserves the energy dissipation of the gradient systems. The ROMs are constructed by the proper orthogonal decomposition (POD) with Galerkin projection. The nonlinear reaction terms are computed efficiently by discrete empirical interpolation method (DEIM). Preservation of the discrete energy of the FOMs and ROMs with POD-DEIM ensures the long term stability of the steady state solutions. Numerical simulations are performed for the gradient dissipative systems with two specific equations; real Ginzburg–Landau equation and Swift–Hohenberg equation. Numerical results demonstrate that the POD-DEIM reduced order solutions preserve well the energy dissipation over time and at the steady state.

Square Turing patterns in reaction-diffusion systems with coupled layers.

Square Turing patterns are usually unstable in reaction-diffusion systems and are rarely observed in corresponding experiments and simulations. We report here an example of spontaneous formation of square Turing patterns with the Lengyel-Epstein model of two coupled layers. The squares are found to be a result of the resonance between two supercritical Turing modes with an appropriate ratio. Besides, the spatiotemporal resonance of Turing modes resembles to the mode-locking phenomenon. Analysis of the general amplitude equations for square patterns reveals that the fixed point corresponding to square Turing patterns is stationary when the parameters adopt appropriate values.

Turing Instability of a Chemical System with Reactants Binding to a Substrate

Relevance of Turing mechanism for biology has been questioned due to not allowing pattern formation when the diffusion constants of the two reacting chemicals are identical. The idea that binding of the activator to a substrate may effectively reduce its diffusion rate and thus destabilize a system that would otherwise be stable was formulated in an article by Lengyel and Epstein [2], where the authors reduce the original system of three linear partial differential equations to a two-dimensional reaction-diffusion system that they analyse. We question relevance of this analysis due to lack of connection between the original and the reduced model and suggest that analysing the reduced model actually does not yield any possibilities beyond the standard setting. Nevertheless, our analysis of the three dimensional system shows that one can indeed relax the standard conditions on diffusion constants that are necessary for Turing instability, in particular allow identical diffusion coefficients. Another question that we raise is whether one can relax the condition on the effective diffusion rates in the three- or four-dimensional model. This idea is supported by a previous result of Klika, Baker, Headon and Gaffney [1] that one can significantly reduce the necessary conditions for Turing instability by adding more reactants into the system. Furthermore, the approach of studying the full four-dimensional system allows relaxing the kinetic contraints, for example by permitting two activators to generate a pattern. [1] V. Klika, R. E. Baker, D. Headon, E. A. Gaffney, The Influence of Receptor-Mediated Interactions on Reaction-Diffusion Mechanisms of Cellular Self-organisation , Bull Math Biol (2012), 74 935--957. [2] I. Lengyel, I. R. Epstein, A Chemical Approach to Designing Turing Patterns in Reaction-Diffusion Systems , Proc. Natl. Acad. Sci. USA (1992) 89 3977--3979.

Oscillatory Turing patterns in reaction-diffusion systems with two coupled layers.

A model reaction-diffusion system with two coupled layers yields oscillatory Turing patterns when oscillation occurs in one layer and the other supports stationary Turing structures. Patterns include "twinkling eyes," where oscillating Turing spots are arranged as a hexagonal lattice, and localized spiral or concentric waves within spot-like or stripe-like Turing structures. A new approach to generating the short-wave instability is proposed.

Pattern formation from defect chaos — a theory of chevrons

For over 25 years it is known that the roll structure of electroconvection (EC) in the dielectric regime in planarly aligned nematic liquid crystals has, after a transition to defect chaos, the tendency to form chevron structures. We show, with the help of a coarse-grained model, that this effect can generally be expected for systems with spontaneously broken isotropy, that is lifted by a small external perturbation. The linearized model as well as a nonlinear extension are compared to simulations of a system of coupled amplitude equations which generate chevrons out of defect chaos. The mechanism of chevron formation is similar to the development of Turing patterns in reaction-diffusion systems.

Effect of randomness and anisotropy on Turing patterns in reaction-diffusion systems

We study the effect of randomness and anisotropy on Turing patterns in reaction-diffusion systems. For this purpose, the Gierer-Meinhardt model of pattern formation is considered. The cases we study are: (i)randomness in the underlying lattice structure, (ii)the case in which there is a probablity p that at a lattice site both reaction and diffusion occur, otherwise there is only diffusion and lastly, the effect of (iii) anisotropic and (iv) random diffusion coefficients on the formation of Turing patterns. The general conclusion is that the Turing mechanism of pattern formation is fairly robust in the presence of randomness and anisotropy.

Tracking unstable turing patterns through mixed-mode spatiotemporal chaos.

A method is presented for stabilizing and tracking unstable Turing patterns in reaction-diffusion systems. The Gray-Scott model is used to simulate a chemical system exhibiting spatiotemporal chaos arising from the interaction between Turing and Hopf bifurcations. The local behavior of the unstable pattern is first approximated with a single-input, single-output linear model constructed from a time series. A recursive control algorithm is then used to stabilize and track the unstable pattern by monitoring a single point in space and making small adjustments to a global parameter.

Turing structures. Progress toward a room temperature, closed system

There are many parallels between the study of temporal oscillation in homogeneous chemical systems and of Turing patterns in reaction-diffusion systems. We discuss recent progress toward constructing a system that displays Turing structures under conditions suitable for a lecture demonstration. Such a system would play a role similar to that of the Belousov-Zhabotinsky oscillating reaction. We examine the prototype chlorite-iodide-malonic acid and related reactions, mathematical models, methods for designing new reactions that exhibit Turing structures and an approach to generating Turing structures in a closed system.

Application of Kronecker products to the analysis of systems with uniform linear coupling

In this paper, we show how the Kronecker product can be useful in simplifying the notation encountered in the analysis of linearly coupled systems where the cells are identical and the coupling is uniform. We give two applications where this is useful. First we use it in the analysis of Turing patterns in reaction-diffusion systems to obtain conditions for the coupling to destabilize the uniform equilibrium point. In the second application we use the Kronecker product to obtain simple sufficient conditions for an array of linearly coupled dynamical systems to synchronize. We discuss briefly extensions to additive nonlinear coupling.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

A chemical approach to designing Turing patterns in reaction-diffusion systems.

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.