Turing's Diffusive Threshold in Random Reaction-Diffusion Systems

Turing Instabilities are Not Enough to Ensure Pattern Formation

Symmetry-breaking instabilities play an important role in understanding the mechanisms underlying the diversity of patterns observed in nature, such as in Turing's reaction-diffusion theory, which connects cellular signalling and transport with the development of growth and form. Extensive literature focuses on the linear stability analysis of homogeneous equilibria in these systems, culminating in a set of conditions for transport-driven instabilities that are commonly presumed to initiate self-organisation. We demonstrate that a selection of simple, canonical transport models with only mild multistable non-linearities can satisfy the Turing instability conditions while also robustly exhibiting only transient patterns. Hence, a Turing-like instability is insufficient for the existence of a patterned state. While it is known that linear theory can fail to predict the formation of patterns, we demonstrate that such failures can appear robustly in systems with multiple stable homogeneous equilibria. Given that biological systems such as gene regulatory networks and spatially distributed ecosystems often exhibit a high degree of multistability and nonlinearity, this raises important questions of how to analyse prospective mechanisms for self-organisation.

Forceful patterning: theoretical principles of mechanochemical pattern formation

Biological pattern formation is essential for generating and maintaining spatial structures from the scale of a single cell to tissues and even collections of organisms. Besides biochemical interactions, there is an important role for mechanical and geometrical features in the generation of patterns. We review the theoretical principles underlying different types of mechanochemical pattern formation across spatial scales and levels of biological organization.

Stochastic Turing patterns of trichomes in Arabidopsis leaves

Biological patterns that emerge during the morphogenesis of multicellular organisms can display high precision at large scales, while at cellular scales, cells exhibit large fluctuations stemming from cell-cell differences in molecular copy numbers also called demographic noise. We study the conflicting interplay between high precision and demographic noise in trichome patterns on the epidermis of wild-type Arabidopsis thaliana leaves, as a two-dimensional model system. We carry out a statistical characterization of these patterns and show that their power spectra display fat tails-a signature compatible with noise-driven stochastic Turing patterns-which are absent in power spectra of patterns driven by deterministic instabilities. We then present a theoretical model that includes demographic noise stemming from birth-death processes of genetic regulators which we study analytically and by stochastic simulations. The model captures the observed experimental features of trichome patterns.

Dynamical Pattern Formation without Self-Attraction in Quorum-Sensing Active Matter: The Interplay between Nonreciprocity and Motility

We study a minimal model involving two species of particles interacting via quorum-sensing rules. Combining simulations of the microscopic model and linear stability analysis of the associated coarse-grained field theory, we identify a mechanism for dynamical pattern formation that does not rely on the standard route of intraspecies effective attractive interactions. Instead, our results reveal a highly dynamical phase of chasing bands induced only by the combined effects of self-propulsion and nonreciprocity in the interspecies couplings. Turning on self-attraction, we find that the system may phase separate into a macroscopic domain of such chaotic chasing bands coexisting with a dilute gas. We show that the chaotic dynamics of bands at the interfaces of this phase-separated phase results in anomalously slow coarsening.

Effects of square spatial periodic forcing on oscillatory hexagon patterns in coupled reaction-diffusion systems

One of the central issues in pattern formation is understanding the response of pattern-forming systems to an external stimulus. While significant progress has been made in systems with only one instability, much less is known about the response of complex patterns arising from the interaction of two or more instabilities. In this paper, we consider the effects of square spatial periodic forcing on oscillatory hexagon patterns in a two-layer coupled reaction diffusion system which undergoes both Turing and Hopf instabilities. Two different types of additive forcings, namely direct and indirect forcing, have been applied. It is shown that the coupled system exhibits different responses towards the spatial forcing under different forcing types. In the indirect case, the oscillatory hexagon pattern transitions into other oscillatory Turing patterns or resonant Turing patterns, depending on the forcing wavenumber and strength. In the direct forcing case, only non-resonant Turing patterns can be obtained. Our results may provide new insight into the modification and control of spatio-temporal patterns in multilayered systems, especially in biological and ecological systems.

Nonideal Reaction-Diffusion Systems: Multiple Routes to Instability

We develop a general classification of the nature of the instabilities yielding spatial organization in open nonideal reaction-diffusion systems, based on linear stability analysis. This encompasses dynamics where chemical species diffuse, interact with each other, and undergo chemical reactions driven out of equilibrium by external chemostats. We find analytically that these instabilities can be of two types: instabilities caused by intermolecular energetic interactions (E type), and instabilities caused by multimolecular out-of-equilibrium chemical reactions (R type). Furthermore, we identify a class of chemical reaction networks, containing unimolecular networks but also extending beyond them, that can only undergo E-type instabilities. We illustrate our analytical findings with numerical simulations on two reaction-diffusion models, each displaying one of the two types of instability and generating stable patterns.

Physical interactions in non-ideal fluids promote Turing patterns

Turing's mechanism is often invoked to explain periodic patterns in nature, although direct experimental support is scarce. Turing patterns form in reaction-diffusion systems when the activating species diffuse much slower than the inhibiting species, and the involved reactions are highly nonlinear. Such reactions can originate from cooperativity, whose physical interactions should also affect diffusion. We here take direct interactions into account and show that they strongly affect Turing patterns. We find that weak repulsion between the activator and inhibitor can substantially lower the required differential diffusivity and reaction nonlinearity. By contrast, strong interactions can induce phase separation, but the resulting length scale is still typically governed by the fundamental reaction-diffusion length scale. Taken together, our theory connects traditional Turing patterns with chemically active phase separation, thus describing a wider range of systems. Moreover, we demonstrate that even weak interactions affect patterns substantially, so they should be incorporated when modelling realistic systems.

The emergence of spatial patterns for compartmental reaction kinetics coupled by two bulk diffusing species with comparable diffusivities

Originating from the pioneering study of Alan Turing, the bifurcation analysis predicting spatial pattern formation from a spatially uniform state for diffusing morphogens or chemical species that interact through nonlinear reactions is a central problem in many chemical and biological systems. From a mathematical viewpoint, one key challenge with this theory for two component systems is that stable spatial patterns can typically only occur from a spatially uniform state when a slowly diffusing 'activator' species reacts with a much faster diffusing 'inhibitor' species. However, from a modelling perspective, this large diffusivity ratio requirement for pattern formation is often unrealistic in biological settings since different molecules tend to diffuse with similar rates in extracellular spaces. As a result, one key long-standing question is how to robustly obtain pattern formation in the biologically realistic case where the time scales for diffusion of the interacting species are comparable. For a coupled one-dimensional bulk-compartment theoretical model, we investigate the emergence of spatial patterns for the scenario where two bulk diffusing species with comparable diffusivities are coupled to nonlinear reactions that occur only in localized 'compartments', such as on the boundaries of a one-dimensional domain. The exchange between the bulk medium and the spatially localized compartments is modelled by a Robin boundary condition with certain binding rates. As regulated by these binding rates, we show for various specific nonlinearities that our one-dimensional coupled PDE-ODE model admits symmetry-breaking bifurcations, leading to linearly stable asymmetric steady-state patterns, even when the bulk diffusing species have equal diffusivities. Depending on the form of the nonlinear kinetics, oscillatory instabilities can also be triggered. Moreover, the analysis is extended to treat a periodic chain of compartments. This article is part of the theme issue 'New trends in pattern formation and nonlinear dynamics of extended systems'.

Complex Network Evolution Model Based on Turing Pattern Dynamics

Complex network models are helpful to explain the evolution rules of network structures, and also are the foundations of understanding and controlling complex networks. The existing studies (e.g., scale-free model, small-world model) are insufficient to uncover the internal mechanisms of the emergence and evolution of communities in networks. To overcome the above limitation, in consideration of the fact that a network can be regarded as a pattern composed of communities, we introduce Turing pattern dynamic as theory support to construct the network evolution model. Specifically, we develop a Reaction-Diffusion model according to Q-Learning technology (RDQL), in which each node regarded as an intelligent agent makes a behavior choice to update its relationships, based on the utility and behavioral strategy at every time step. Extensive experiments indicate that our model not only reveals how communities form and evolve, but also can generate networks with the properties of scale-free, small-world and assortativity. The effectiveness of the RDQL model has also been verified by its application in real networks. Furthermore, the depth analysis of the RDQL model provides a conclusion that the proportion of exploration and exploitation behaviors of nodes is the only factor affecting the formation of communities. The proposed RDQL model has potential to be the basic theoretical tool for studying network stability and dynamics.

conditions for Turing and wave instabilities in reaction-diffusion systems

Necessary and sufficient conditions are provided for a diffusion-driven instability of a stable equilibrium of a reaction-diffusion system with n components and diagonal diffusion matrix. These can be either Turing or wave instabilities. Known necessary and sufficient conditions are reproduced for there to exist diffusion rates that cause a Turing bifurcation of a stable homogeneous state in the absence of diffusion. The method of proof here though, which is based on study of dispersion relations in the contrasting limits in which the wavenumber tends to zero and to [Formula: see text], gives a constructive method for choosing diffusion constants. The results are illustrated on a 3-component FitzHugh-Nagumo-like model proposed to study excitable wavetrains, and for two different coupled Brusselator systems with 4-components.

Design centering enables robustness screening of pattern formation models

MOTIVATION

Access to unprecedented amounts of quantitative biological data allows us to build and test biochemically accurate reaction-diffusion models of intracellular processes. However, any increase in model complexity increases the number of unknown parameters and, thus, the computational cost of model analysis. To efficiently characterize the behavior and robustness of models with many unknown parameters remains, therefore, a key challenge in systems biology.

RESULTS

We propose a novel computational framework for efficient high-dimensional parameter space characterization of reaction-diffusion models in systems biology. The method leverages the Lp-Adaptation algorithm, an adaptive-proposal statistical method for approximate design centering and robustness estimation. Our approach is based on an oracle function, which predicts for any given point in parameter space whether the model fulfills given specifications. We propose specific oracles to efficiently predict four characteristics of Turing-type reaction-diffusion models: bistability, instability, capability of spontaneous pattern formation and capability of pattern maintenance. We benchmark the method and demonstrate that it enables global exploration of a model's ability to undergo pattern-forming instabilities and to quantify robustness for model selection in polynomial time with dimensionality. We present an application of the framework to pattern formation on the endosomal membrane by the small GTPase Rab5 and its effectors, and we propose molecular mechanisms underlying this system.

AVAILABILITY AND IMPLEMENTATION

Our code is implemented in MATLAB and is available as open source under https://git.mpi-cbg.de/mosaic/software/black-box-optimization/rd-parameter-space-screening.

SUPPLEMENTARY INFORMATION

Supplementary data are available at Bioinformatics online.

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.

Modern perspectives on near-equilibrium analysis of Turing systems

In the nearly seven decades since the publication of Alan Turing's work on morphogenesis, enormous progress has been made in understanding both the mathematical and biological aspects of his proposed reaction-diffusion theory. Some of these developments were nascent in Turing's paper, and others have been due to new insights from modern mathematical techniques, advances in numerical simulations and extensive biological experiments. Despite such progress, there are still important gaps between theory and experiment, with many examples of biological patterning where the underlying mechanisms are still unclear. Here, we review modern developments in the mathematical theory pioneered by Turing, showing how his approach has been generalized to a range of settings beyond the classical two-species reaction-diffusion framework, including evolving and complex manifolds, systems heterogeneous in space and time, and more general reaction-transport equations. While substantial progress has been made in understanding these more complicated models, there are many remaining challenges that we highlight throughout. We focus on the mathematical theory, and in particular linear stability analysis of 'trivial' base states. We emphasize important open questions in developing this theory further, and discuss obstacles in using these techniques to understand biological reality. This article is part of the theme issue 'Recent progress and open frontiers in Turing's theory of morphogenesis'.