Turing patterns beyond hexagons and stripes

Unravelling diverse spatiotemporal orders in chlorine dioxide-iodine-malonic acid reaction-diffusion system through circularly polarized electric field and photo-illumination

Designing and predicting self-organized pattern formation in out-of-equilibrium chemical and biochemical reactions holds fundamental significance. External perturbations like light and electric fields exert a crucial influence on reaction-diffusion systems involving ionic species. While the separate impacts of light and electric fields have been extensively studied, comprehending their combined effects on spatiotemporal dynamics is paramount for designing versatile spatial orders. Here, we theoretically investigate the spatiotemporal dynamics of chlorine dioxide-iodine-malonic acid reaction-diffusion system under photo-illumination and circularly polarized electric field (CPEF). By applying CPEF at varying intensities and frequencies, we observe the predominant emergence of oscillating hexagonal spot-like patterns from homogeneous stable steady states. Furthermore, our study unveils a spectrum of intriguing spatiotemporal instabilities, encompassing stripe-like patterns, oscillating dumbbell-shaped patterns, spot-like instabilities with square-based symmetry, and irregular chaotic patterns. However, when we introduce periodic photo-illumination to the hexagonal spot-like instabilities induced by CPEF in homogeneous steady states, we observe periodic size fluctuations. Additionally, the stripe-like instabilities undergo alternating transitions between hexagonal spots and stripes. Notably, within the Turing region, the interplay between these two external influences leads to the emergence of distinct superlattice patterns characterized by hexagonal-and square-based symmetry. These patterns include parallel lines of spots, target-like formations, black-eye patterns, and other captivating structures. Remarkably, the simple perturbation of the system through the application of these two external fields offers a versatile tool for generating a wide range of pattern-forming instabilities, thereby opening up exciting possibilities for future experimental validation.

Appearance and suppression of Turing patterns under a periodically forced feed

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.

Diffusive instability in hyperbolic reaction-diffusion equation with different inertia

This work considers a two-dimensional hyperbolic reaction-diffusion system with different inertia and explores criteria for various instabilities, like a wave, Turing, and Hopf, both theoretically and numerically. It is proven that wave instability may occur in a two-species hyperbolic reaction-diffusion system with identical inertia if the diffusion coefficients of the species are nonidentical but cannot occur if diffusion coefficients are identical. Wave instability may also arise in a two-dimensional hyperbolic reaction-diffusion system if the diffusivities of the species are equal, which is never possible in a parabolic reaction-diffusion system, provided the inertias are different. Interestingly, Turing instability is independent of inertia, but the stability of the corresponding local system depends on the inertia. Theoretical results are demonstrated with an example where the local interaction is represented by the Schnakenberg system.

Pattern selection in reaction diffusion systems

Turing's theory of pattern formation has been used to describe the formation of self-organized periodic patterns in many biological, chemical, and physical systems. However, the use of such models is hindered by our inability to predict, in general, which pattern is obtained from a given set of model parameters. While much is known near the onset of the spatial instability, the mechanisms underlying pattern selection and dynamics away from onset are much less understood. Here, we provide physical insight into the dynamics of these systems. We find that peaks in a Turing pattern behave as point sinks, the dynamics of which is determined by the diffusive fluxes into them. As a result, peaks move toward a periodic steady-state configuration that minimizes the mass of the diffusive species. We also show that the preferred number of peaks at the final steady state is such that this mass is minimized. Our work presents mass minimization as a potential general principle for understanding pattern formation in reaction diffusion systems far from onset.

Spatiotemporal antiresonance in coupled reaction-diffusion systems

We present a theoretical study of the spatiotemporal antiresonance in a system of two diffusively coupled chemical reactions, one of which is driven by an external periodic forcing. Although antiresonance is well known in various physical systems, the phenomenon in coupled chemical reactions has largely been overlooked. Based on the linearized dynamics around the steady state of the two-component coupled reaction-diffusion systems we have derived the general analytical expressions for the amplitude-frequency response functions of the driven and undriven components of the system. Our theoretical analysis is well corroborated by detailed numerical simulations on coupled Gray-Scott reaction-diffusion systems exhibiting antiresonance dip in the amplitude-frequency response curve as a result of destructive interference between the coupling and the periodic external forcing imparting differential stability of the two subsystems. This leads to the emergence of spatiotemporal patterns in an undriven subsystem, while the driven one settles down to a homogeneously stable steady state.

Sub-100 nm 2D nanopatterning on a large scale by ultrafast laser energy regulation

Coupling ultrafast light irradiation to surface nanoreliefs leads to periodic patterns, achieving record processing scales down to tens of nanometers. Driven by near-field interactions, the promising potential of the spontaneous pattern formation relies on the scaling up of one-step manufacturing processes. Here, we report the self-assembly of unconventional arrays of nanocavities of 20 nm diameter with a periodicity down to 60 nm upon ultrafast laser irradiation of a nickel surface. In stark contrast to laser-induced surface ripples, which are stochastic and suffer from a lack of regularity, the 2D patterns present an unprecedented uniformity on extreme scales. The onset of nanocavity arrays ordered in a honeycomb lattice is achieved by overcoming the anisotropic polarization response of the surface by a delayed action of cross-polarized laser pulses. The origin of this self-arrangement is identified as a manifestation of Marangoni convection instability in a nanoscale melt layer, destabilized by the laser-induced rarefaction wave.

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.

Turing patterns modulation by chemical gradient in isothermal and non-isothermal conditions

Chemical gradients imposed through boundary conditions induce spatial symmetry breaking of Turing patterns in small systems.

Theoretical analysis of spatial nonhomogeneous patterns of entomopathogenic fungi growth on insect pest

This paper presents the study of the dynamics of intrahost (insect pests)-pathogen [entomopathogenic fungi (EPF)] interactions. The interaction between the resources from the insect pest and the mycelia of EPF is represented by the Holling and Powell type II functional responses. Because the EPF's growth is related to the instability of the steady state solution of our system, particular attention is given to the stability analysis of this steady state. Initially, the stability of the steady state is investigated without taking into account diffusion and by considering the behavior of the system around its equilibrium states. In addition, considering small perturbation of the stable singular point due to nonlinear diffusion, the conditions for Turing instability occurrence are deduced. It is observed that the absence of the regeneration feature of insect resources prevents the occurrence of such phenomena. The long time evolution of our system enables us to observe both spot and stripe patterns. Moreover, when the diffusion of mycelia is slightly modulated by a weak periodic perturbation, the Floquet theory and numerical simulations allow us to derive the conditions in which diffusion driven instabilities can occur. The relevance of the obtained results is further discussed in the perspective of biological insect pest control.

Self-organization principles of intracellular pattern formation

Dynamic patterning of specific proteins is essential for the spatio-temporal regulation of many important intracellular processes in prokaryotes, eukaryotes and multicellular organisms. The emergence of patterns generated by interactions of diffusing proteins is a paradigmatic example for self-organization. In this article, we review quantitative models for intracellular Min protein patterns in Escherichia coli, Cdc42 polarization in Saccharomyces cerevisiae and the bipolar PAR protein patterns found in Caenorhabditis elegans. By analysing the molecular processes driving these systems we derive a theoretical perspective on general principles underlying self-organized pattern formation. We argue that intracellular pattern formation is not captured by concepts such as ‘activators’, ‘inhibitors’ or ‘substrate depletion’. Instead, intracellular pattern formation is based on the redistribution of proteins by cytosolic diffusion, and the cycling of proteins between distinct conformational states. Therefore, mass-conserving reaction–diffusion equations provide the most appropriate framework to study intracellular pattern formation. We conclude that directed transport, e.g. cytosolic diffusion along an actively maintained cytosolic gradient, is the key process underlying pattern formation. Thus the basic principle of self-organization is the establishment and maintenance of directed transport by intracellular protein dynamics.

This article is part of the theme issue ‘Self-organization in cell biology’.

Tristability between stripes, up-hexagons, and down-hexagons and snaking bifurcation branches of spatial connections between up- and down-hexagons

Third-order amplitude equations on hexagonal lattices can be used for predicting the existence and stability of stripes, up-hexagons, and down-hexagons in pattern-forming systems. These amplitude equations predict the nonexistence of bistable ranges between up- and down-hexagons and tristable ranges between stripes, up-, and down-hexagons. In the present work we use fifth-order amplitude equations for finding such bistable and tristable ranges for a generalized Swift-Hohenberg equation and discuss stationary front connections between up- and down-hexagons.

Formation of spiral waves with substructure in a bursting media

Formation of spiral waves in a bursting media is investigated. Due to the multiple timescale oscillation of the local dynamics, an interesting substructure of traveling wave (STW) is observed in the spiral arm. As a result of the special moving media formed by neurons in the spiral arm, STWs propagate from the spiral tip to far field with an increasing wave length and move faster along the front of the spiral arm than along the back, leading to the formation of fragments in STWs. Moreover, we find that a sharp change of stimulus current can lead to backfiring of STWs, which may break the spiral wave front and further result in the formation of a multi-spiral pattern.

Spatio-temporal dynamics induced by competing instabilities in two asymmetrically coupled nonlinear evolution equations

Pattern formation often occurs in spatially extended physical, biological, and chemical systems due to an instability of the homogeneous steady state. The type of the instability usually prescribes the resulting spatio-temporal patterns and their characteristic length scales. However, patterns resulting from the simultaneous occurrence of instabilities cannot be expected to be simple superposition of the patterns associated with the considered instabilities. To address this issue, we design two simple models composed by two asymmetrically coupled equations of non-conserved (Swift-Hohenberg equations) or conserved (Cahn-Hilliard equations) order parameters with different characteristic wave lengths. The patterns arising in these systems range from coexisting static patterns of different wavelengths to traveling waves. A linear stability analysis allows to derive a two parameter phase diagram for the studied models, in particular, revealing for the Swift-Hohenberg equations, a co-dimension two bifurcation point of Turing and wave instability and a region of coexistence of stationary and traveling patterns. The nonlinear dynamics of the coupled evolution equations is investigated by performing accurate numerical simulations. These reveal more complex patterns, ranging from traveling waves with embedded Turing patterns domains to spatio-temporal chaos, and a wide hysteretic region, where waves or Turing patterns coexist. For the coupled Cahn-Hilliard equations the presence of a weak coupling is sufficient to arrest the coarsening process and to lead to the emergence of purely periodic patterns. The final states are characterized by domains with a characteristic length, which diverges logarithmically with the coupling amplitude.

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.

Simulations of pattern dynamics for reaction-diffusion systems via SIMULINK

BACKGROUND

Investigation of the nonlinear pattern dynamics of a reaction-diffusion system almost always requires numerical solution of the system's set of defining differential equations. Traditionally, this would be done by selecting an appropriate differential equation solver from a library of such solvers, then writing computer codes (in a programming language such as C or Matlab) to access the selected solver and display the integrated results as a function of space and time. This "code-based" approach is flexible and powerful, but requires a certain level of programming sophistication. A modern alternative is to use a graphical programming interface such as Simulink to construct a data-flow diagram by assembling and linking appropriate code blocks drawn from a library. The result is a visual representation of the inter-relationships between the state variables whose output can be made completely equivalent to the code-based solution.

RESULTS

As a tutorial introduction, we first demonstrate application of the Simulink data-flow technique to the classical van der Pol nonlinear oscillator, and compare Matlab and Simulink coding approaches to solving the van der Pol ordinary differential equations. We then show how to introduce space (in one and two dimensions) by solving numerically the partial differential equations for two different reaction-diffusion systems: the well-known Brusselator chemical reactor, and a continuum model for a two-dimensional sheet of human cortex whose neurons are linked by both chemical and electrical (diffusive) synapses. We compare the relative performances of the Matlab and Simulink implementations.

CONCLUSIONS

The pattern simulations by Simulink are in good agreement with theoretical predictions. Compared with traditional coding approaches, the Simulink block-diagram paradigm reduces the time and programming burden required to implement a solution for reaction-diffusion systems of equations. Construction of the block-diagram does not require high-level programming skills, and the graphical interface lends itself to easy modification and use by non-experts.

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

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

Instabilities and patterns in coupled reaction-diffusion layers

We study instabilities and pattern formation in reaction-diffusion layers that are diffusively coupled. For two-layer systems of identical two-component reactions, we analyze the stability of homogeneous steady states by exploiting the block symmetric structure of the linear problem. There are eight possible primary bifurcation scenarios, including a Turing-Turing bifurcation that involves two disparate length scales whose ratio may be tuned via the interlayer coupling. For systems of n-component layers and nonidentical layers, the linear problem's block form allows approximate decomposition into lower-dimensional linear problems if the coupling is sufficiently weak. As an example, we apply these results to a two-layer Brusselator system. The competing length scales engineered within the linear problem are readily apparent in numerical simulations of the full system. Selecting a sqrt[2]:1 length-scale ratio produces an unusual steady square pattern.

Rearrangement dynamics of fishbonelike Turing patterns generated by spatial periodic forcing

External forcing can greatly affect the evolution of Turing patterns in reaction-diffusion (RD) media. Here, we employ spatially periodic illumination in a photosensitive RD system to create Turing structures that are repetitive in one direction. We then study their relaxation in the absence of light. These unforced, fishbonelike configurations undergo self-reorganization and establish stationary arrangements, some of which fully exhibit, while others partially or completely lose, the symmetry of the initially imposed pattern.

Three-dimensional patterns in the Lengyel-Rabai-Epstein model of the chlorine dioxide-iodine-malonic acid reaction

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.

Design and control of patterns in reaction-diffusion systems

We discuss the design of reaction-diffusion systems that display a variety of spatiotemporal patterns. We also consider how these patterns may be controlled by external perturbation, typically using photochemistry or temperature. Systems treated include the Belousov-Zhabotinsky (BZ) reaction, the chlorite-iodide-malonic acid and chlorine dioxide-malonic acid-iodine reactions, and the BZ-AOT system, i.e., the BZ reaction in a water-in-oil reverse microemulsion stabilized by the surfactant sodium bis(2-ethylhexyl) sulfosuccinate (AOT).