Turing instabilities with nearly equal diffusion coefficients

The role of a reaction-diffusion system in the initiation of skin organ primordia. I. The first wave of initiation

Reaction-Diffusion (RD) systems provide a computational framework that governs many pattern formation processes in nature. Current RD system design practices boil down to trial-and-error parameter search. We propose a differentiable optimization method for learning the RD system parameters to perform example-based texture synthesis on a 2D plane. We do this by representing the RD system as a variant of Neural Cellular Automata and using task-specific differentiable loss functions. RD systems generated by our method exhibit robust, non-trivial 'life-like' behavior.

The stability of a one-spike solution to a general class of reaction-diffusion (RD) system with both regular and anomalous diffusion is analyzed. The method of matched asymptotic expansions is used to construct a one-spike equilibrium solution and to derive a nonlocal eigenvalue problem (NLEP) that determines the stability of this solution on an O (1) time-scale. For a particular sub-class of the reaction kinetics, it is shown that the discrete spectrum of this NLEP is determined in terms of the roots of certain simple transcendental equations that involve two key parameters related to the choice of the nonlinear kinetics. From a rigorous analysis of these transcendental equations by using a winding number approach and explicit calculations, sufficient conditions are given to predict the occurrence of Hopf bifurcations of the one-spike solution. Our analysis determines explicitly the number of possible Hopf bifurcation points as well as providing analytical formulae for them. The analysis is implemented for the shadow limit of the RD system defined on a finite domain and for a one-spike solution of the RD system on the infinite line. The theory is illustrated for two specific RD systems. Finally, in parameter ranges for which the Hopf bifurcation is unique, it is shown that the effect of sub-diffusion is to delay the onset of the Hopf bifurcation.

The interplay between reaction kinetics and diffusion leads to a wide range of spatiotemporal behaviors in reaction–diffusion (RD) systems. This article presents a theoretical and computational study of a RD system inspired by experimental observations of the Rho-GEF-Myosin signaling network controlling cell contraction dynamics. The temporal reaction system dynamics range from periodicity to bistability. By employing a dual hybrid dynamical systems approach of numerical bifurcation analysis and Floquet theory, we characterize the spatiotemporal dynamics when stable limit-cycle oscillators or homogeneous time-periodic solutions undergo the process of Floquet–Turing-diffusion-driven-instability (FTDDI). FTDDI refers to the emergence of spatially nonuniform patterns when diffusion destabilizes an otherwise temporally stable limit cycle of the underlying reaction kinetics. For the temporal reaction system, numerical bifurcation allows us to identify regions in a two-parameter space defining the stability of the uniform steady states and regions where the system exhibits limit cycles, which are characterized by employing the Floquet theory. In the presence of spatial variations, Floquet theory classifies regions where diffusion destabilizes the limit cycle or maintains their homogeneous stability and the emerging spatiotemporal dynamics of the full system, far from equilibrium. In the bistable regime, diffusion differentiates regions into those that exhibit classical Turing diffusion-driven instability, leading to pattern formation; and those that exhibit FTDDI, leading to space-time periodic patterns, spatially inhomogeneous patterns, oscillatory pulses, and wave propagation, and those that remain unaffected by diffusion. These findings provide theoretical insights into understanding complex spatiotemporal dynamics relevant to biological, chemical, and ecological spatiotemporal systems.

The stability and slow dynamics of spot patterns in the 2D Brusselator model: The effect of open systems and heterogeneities

We introduce a DNA-based reaction-diffusion (RD) system in which reaction and diffusion terms can be precisely and independently controlled. The effective diffusion coefficient of an individual reaction component, as we demonstrate on a traveling wave, can be reduced up to 2.7-fold using a self-assembled hydrodynamic drag. The intrinsic programmability of this RD system allows us to engineer, for the first time, orthogonal autocatalysts that counterpropagate with minimal interaction. Our results are in excellent quantitative agreement with predictions of the Fisher-Kolmogorov-Petrovskii-Piscunov model. These advances open the way for the rational engineering of pattern formation in pure chemical RD systems.

Time delays, modelling the process of intracellular gene expression, have been shown to have important impacts on the dynamics of pattern formation in reaction–diffusion systems. In particular, past work has shown that such time delays can shrink the Turing space, thereby inhibiting patterns from forming across large ranges of parameters. Such delays can also increase the time taken for pattern formation even when Turing instabilities occur. Here, we consider reaction–diffusion models incorporating fixed or distributed time delays, modelling the underlying stochastic nature of gene expression dynamics, and analyse these through a systematic linear instability analysis and numerical simulations for several sets of different reaction kinetics. We find that even complicated distribution kernels (skewed Gaussian probability density functions) have little impact on the reaction–diffusion dynamics compared to fixed delays with the same mean delay. We show that the location of the delay terms in the model can lead to changes in the size of the Turing space (increasing or decreasing) as the mean time delay, tau , is increased. We show that the time to pattern formation from a perturbation of the homogeneous steady state scales linearly with tau , and conjecture that this is a general impact of time delay on reaction–diffusion dynamics, independent of the form of the kinetics or location of the delayed terms. Finally, we show that while initial and boundary conditions can influence these dynamics, particularly the time-to-pattern, the effects of delay appear robust under variations of initial and boundary data. Overall, our results help clarify the role of gene expression time delays in reaction–diffusion patterning, and suggest clear directions for further work in studying more realistic models of pattern formation.

Robust point control for a class of fractional-order reaction–diffusion systems via non-collocated point measurement

By discretising space into compartments and letting system dynamics be governed by the reaction–diffusion master equation, it is possible to derive and simulate a stochastic model of reaction and diffusion on an arbitrary domain. However, there are many implementation choices involved in this process, such as the choice of discretisation and method of derivation of the diffusive jump rates, and it is not clear a priori how these affect model predictions. To shed light on this issue, in this work we explore how a variety of discretisations and methods for derivation of the diffusive jump rates affect the outputs of stochastic simulations of reaction–diffusion models, in particular using Turing’s model of pattern formation as a key example. We consider both static and uniformly growing domains and demonstrate that, while only minor differences are observed for simple reaction–diffusion systems, there can be vast differences in model predictions for systems that include complicated reaction kinetics, such as Turing’s diffusion-driven instability model of pattern formation. Our work highlights that care must be taken in using the reaction–diffusion master equation framework to make predictions as to the dynamics of stochastic reaction–diffusion systems.

For differential equations with a compact symmetry group Г some results on global bifurcation of time periodic orbits are presented. In particular it is investigated how the spatial and temporal action of the group Г on a time periodic orbit may vary along global bifurcation branches, e.g. at period doubling bifurcations. The results are obtained geometrically by generic but equivariant approximation, rather than by topological techniques. As an application periodic solutions in reaction diffusion systems with symmetry Dn or O(3) are discussed.

Cooperative behaviors arising from bacterial cell-to-cell communication can\nbe modeled by reaction-diffusion equations having only a single diffusible\ncomponent. This paper presents the following three contributions for the\nsystematic analysis of Turing instability in such reaction-diffusion systems.\n(i) We first introduce a unified framework to formulate the reaction-diffusion\nsystem as an interconnected multi-agent dynamical system. (ii) Then, we\nmathematically classify biologically plausible and implausible Turing\ninstabilities and characterize them by the root locus of each agent's dynamics,\nor the local reaction dynamics. (iii) Using this characterization, we derive\nanalytic conditions for biologically plausible Turing instability, which\nprovide useful guidance for the design and the analysis of biological networks.\nThese results are demonstrated on an extended Gray-Scott model with a single\ndiffuser.\n

We generalize our recent discrete cellular automata (CA) model of excitable media [Y. B. Chernyak, A. B. Feldman, and R. J. Cohen, Phys. Rev. E 55, 3215 (1997)] to incorporate the effects of inhibitory processes on the propagation of the excitation wave front. In the common two variable reaction-diffusion (RD) models of excitable media, the inhibitory process is described by the $v$ ``controller'' variable responsible for the restoration of the equilibrium state following excitation. In myocardial tissue, the inhibitory effects are mainly due to the inactivation of the fast sodium current. We represent inhibition using a physical model in which the ``source'' contribution of excited elements to the excitation of their neighbors decreases with time as a simple function with a single adjustable parameter (a rate constant). We sought specific solutions of the CA state transition equations and obtained (both analytically and numerically) the dependence of the wave-front speed $c$ on the four model parameters and the wave-front curvature $\ensuremath{\kappa}$. By requiring that the major characteristics of $c(\ensuremath{\kappa})$ in our CA model coincide with those obtained from solutions of a specific RD model, we find a unique set of CA parameter values for a given excitable medium. The basic structure of our CA solutions is remarkably similar to that found in typical RD systems (similar behavior is observed when the analogous model parameters are varied). Most notably, the ``turn-on'' of the inhibitory process is accompanied by the appearance of a solution branch of slow speed, unstable waves. Additionally, when $\ensuremath{\kappa}$ is small, we obtain a family of ``eikonal'' relations $c(\ensuremath{\kappa})$ that are suitable for the kinematic analysis of traveling waves in the CA medium. We compared the solutions of the CA equations to CA simulations for the case of plane waves and circular (target) waves and found excellent agreement. We then studied a spiral wave using the CA model adjusted to a specific RD system and found good correspondence between the shapes of the RD and CA spiral arms in the region away from the tip where kinematic theory applies. Our analysis suggests that only four physical parameters control the behavior of wave fronts in excitable media.

The Brusselator model is widely used to illustrate and study basic features of models of chemical reactions involving trimolecular steps. We provide the necessary mathematical theory related to Reaction-Diffusion systems in general and to the Brusselator model in particular. Specifically, the issues of local and global existence of solutions, their uniqueness and regularity are discussed in detail. The theoretical and numerical investigation presented futher in the thesis provides an insight into the asymptotic behavior of the solutions of this model, characterizing the parameter region for each of the three qualitatively different cases: homogeneous steady state, Turing pattern and bulk oscillations. Particular attention is given to the supercritical Hopf bifurcation parameter domain where no substantial theory is available. This study was largely motivated by the observations of Young, Zhabotinsky and Epstein that Turing patterns eventually (for sufficiently small ratio of the diffusion coefficients) dominate the Hopf bifurcation induced bulk oscillations. In this work we confirm this observation and further establish more precisely the shape of the boundary separating the Turing pattern domain and the bulk oscillations domain in the parameter space. The obtained results are used in revealing an essential mechanism generating oscillating patterns in the coupled Brusselator model. It can be considered as a model of the reaction sequences in two thin layers of gel that meet at an interface. Each layer contains the same reactants with the same kinetics but with different diffusion coefficients. The occurrence of oscillating patterns is due to the fact that for the same values of the parameters of the model but with different diffusion coefficients the one system can be in the Turing pattern domain while the other is in the bulk oscillations domain. Hence, roughly speaking, one layer provides a pattern while the other layer drives the oscillations.

It is demonstrated that slowly travelling pulses arising in a reaction-diffusion (RD) system with the FitzHugh-Nagumo type nonlinearity do not necessarily annihilate but reflect off of each other before they collide. This phenomenon is in contrast with the well-known annihilation of travelling pulses on nerve axon and expanding rings in the Belousov-Zhabotinsky chemical reaction. By using singular perturbation methods, we derive a fourth order system of ODEs from the RD system, and study non-annihilation phenomenon of very slowly travelling pulses.

Views Icon Views Article contents Figures & tables Video Audio Supplementary Data Peer Review Share Icon Share Twitter Facebook Reddit LinkedIn Tools Icon Tools Reprints and Permissions Cite Icon Cite Search Site Citation Jonathan D. Evans, Andrea Fernández; Sharp-interface models for concrete carbonation. AIP Conf. Proc. 26 September 2012; 1479 (1): 835–838. https://doi.org/10.1063/1.4756268 Download citation file: Ris (Zotero) Reference Manager EasyBib Bookends Mendeley Papers EndNote RefWorks BibTex toolbar search Search Dropdown Menu toolbar search search input Search input auto suggest filter your search All ContentAIP Publishing PortfolioAIP Conference Proceedings Search Advanced Search |Citation Search