Turing Systems Research Articles

Effects of intrinsic stochasticity on delayed reaction-diffusion patterning systems

Cellular gene expression is a complex process involving many steps, including the transcription of DNA and translation of mRNA; hence the synthesis of proteins requires a considerable amount of time, from ten minutes to several hours. Since diffusion-driven instability has been observed to be sensitive to perturbations in kinetic delays, the application of Turing patterning mechanisms to the problem of producing spatially heterogeneous differential gene expression has been questioned. In deterministic systems a small delay in the reactions can cause a large increase in the time it takes a system to pattern. Recently, it has been observed that in undelayed systems intrinsic stochasticity can cause pattern initiation to occur earlier than in the analogous deterministic simulations. Here we are interested in adding both stochasticity and delays to Turing systems in order to assess whether stochasticity can reduce the patterning time scale in delayed Turing systems. As analytical insights to this problem are difficult to attain and often limited in their use, we focus on stochastically simulating delayed systems. We consider four different Turing systems and two different forms of delay. Our results are mixed and lead to the conclusion that, although the sensitivity to delays in the Turing mechanism is not completely removed by the addition of intrinsic noise, the effects of the delays are clearly ameliorated in certain specific cases.

The finite volume spectral element method to solve Turing models in the biological pattern formation

It is well known that reaction–diffusion systems describing Turing models can display very rich pattern formation behavior. Turing systems have been proposed for pattern formation in various biological systems, e.g. patterns in fish, butterflies, lady bugs and etc. A Turing model expresses temporal behavior of the concentrations of two reacting and diffusing chemicals which is represented by coupled reaction–diffusion equations. Since the base of these reaction–diffusion equations arises from the conservation laws, we develop a hybrid finite volume spectral element method for the numerical solution of them and apply the proposed method to Turing system generated by the Schnakenberg model. Also, as numerical simulations, we study the variety of spatio-temporal patterns for various values of diffusion rates in the problem.

Stochastic reaction and diffusion on growing domains: Understanding the breakdown of robust pattern formation

Many biological patterns, from population densities to animal coat markings, can be thought of as heterogeneous spatiotemporal distributions of mobile agents. Many mathematical models have been proposed to account for the emergence of this complexity, but, in general, they have consisted of deterministic systems of differential equations, which do not take into account the stochastic nature of population interactions. One particular, pertinent criticism of these deterministic systems is that the exhibited patterns can often be highly sensitive to changes in initial conditions, domain geometry, parameter values, etc. Due to this sensitivity, we seek to understand the effects of stochasticity and growth on paradigm biological patterning models. In this paper, we extend spatial Fourier analysis and growing domain mapping techniques to encompass stochastic Turing systems. Through this we find that the stochastic systems are able to realize much richer dynamics than their deterministic counterparts, in that patterns are able to exist outside the standard Turing parameter range. Further, it is seen that the inherent stochasticity in the reactions appears to be more important than the noise generated by growth, when considering which wave modes are excited. Finally, although growth is able to generate robust pattern sequences in the deterministic case, we see that stochastic effects destroy this mechanism for conferring robustness. However, through Fourier analysis we are able to suggest a reason behind this lack of robustness and identify possible mechanisms by which to reclaim it.

Friction-Induced Pattern Formation and Turing Systems

We investigate the possibility of Turing-type pattern formation during friction. Turing or reaction-diffusion systems describe variations of spatial concentrations of chemical components with time due to local chemical reactions coupled with diffusion. Turing systems can lead to a variety of complex spatial patterns evolving with time. During friction, the patterns can form at the sliding interface due to the mass transfer (diffusion), heat transfer, various tribochemical reactions, and wear. We present simulation data showing the possibility of such pattern formation. On the other hand, existing experimental data suggest that in situ tribofilms can form at the frictional interface due to a variety of friction-induced chemical reactions (oxidation, the selective transfer of Cu ions, etc.). These tribofilms as well as other frictional "secondary structures" can form various patterns (islands or honeycomb domains). This mechanism of pattern formation can be attributed to the Turing systems.

The Dynamics of Turing Patterns for Morphogen-Regulated Growing Domains with Cellular Response Delays

Since its conception in 1952, the Turing paradigm for pattern formation has been the subject of numerous theoretical investigations. Experimentally, this mechanism was first demonstrated in chemical reactions over 20years ago and, more recently, several examples of biological self-organisation have also been implicated as Turing systems. One criticism of the Turing model is its lack of robustness, not only with respect to fluctuations in the initial conditions, but also with respect to the inclusion of delays in critical feedback processes such as gene expression. In this work we investigate the possibilities for Turing patterns on growing domains where the morphogens additionally regulate domain growth, incorporating delays in the feedback between signalling and domain growth, as well as gene expression. We present results for the proto-typical Schnakenberg and Gierer-Meinhardt systems: exploring the dynamics of these systems suggests a reconsideration of the basic Turing mechanism for pattern formation on morphogen-regulated growing domains as well as highlighting when feedback delays on domain growth are important for pattern formation.

An efficient and robust numerical algorithm for estimating parameters in Turing systems

We present a new algorithm for estimating parameters in reaction–diffusion systems that display pattern formation via the mechanism of diffusion-driven instability. A Modified Discrete Optimal Control Algorithm (MDOCA) is illustrated with the Schnakenberg and Gierer–Meinhardt reaction–diffusion systems using PDE constrained optimization techniques. The MDOCA algorithm is a modification of a standard variable step gradient algorithm that yields a huge saving in computational cost. The results of numerical experiments demonstrate that the algorithm accurately estimated key parameters associated with stationary target functions generated from the models themselves. Furthermore, the robustness of the algorithm was verified by performing experiments with target functions perturbed with various levels of additive noise. The MDOCA algorithm could have important applications in the mathematical modeling of realistic Turing systems when experimental data are available.

Superlattice Patterns in Coupled Turing Systems

In this paper, superlattice patterns have been investigated by using a two linearly coupled Brusselator model. It is found that superlattice patterns can only be induced in the sub-system with the short wavelength. Three different coupling methods have been used in order to investigate the mode interaction between the two Turing modes. It is proved in the simulations that interaction between activators in the two sub-systems leads to spontaneous formation of black eye pattern and/or white eye patterns while interaction between inhibitors leads to spontaneous formation of super-hexagonal pattern. It is also demonstrated that the same symmetries of the two modes and suitable wavelength ratio of the two modes should also be satisfied to form superlattice patterns.

Turing Systems, Entropy, and Kinetic Models for Self-Healing Surfaces

The paper addresses the methods of description of friction-induced self-healing at the interface between two solid bodies. A macroscopic description of self-healing is based on a Turing system for the transfer of matter that leads to self-organization at the interface in the case of an unstable state. A microscopic description deals with a kinetic model of the process and entropy production during self-organization. The paper provides a brief overview of the Turing system approach and statistical kinetic models. The relation between these methods and the description of the self-healing surfaces is discussed, as well as results of their application. The analytical considerations are illustrated by numerical simulations.

A Theoretical Model for ROP Localisation by Auxin in Arabidopsis Root Hair Cells

BackgroundLocal activation of Rho GTPases is important for many functions including cell polarity, morphology, movement, and growth. Although a number of molecules affecting Rho-of-Plants small GTPase (ROP) signalling are known, it remains unclear how ROP activity becomes spatially organised. Arabidopsis root hair cells produce patches of ROP at consistent and predictable subcellular locations, where root hair growth subsequently occurs.Methodology/Principal FindingsWe present a mathematical model to show how interaction of the plant hormone auxin with ROPs could spontaneously lead to localised patches of active ROP via a Turing or Turing-like mechanism. Our results suggest that correct positioning of the ROP patch depends on the cell length, low diffusion of active ROP, a gradient in auxin concentration, and ROP levels. Our theory provides a unique explanation linking the molecular biology to the root hair phenotypes of multiple mutants and transgenic lines, including OX-ROP, CA-rop, aux1, axr3, tip1, eto1, etr1, and the triple mutant aux1 ein2 gnom eb.Conclusions/SignificanceWe show how interactions between Rho GTPases (in this case ROPs) and regulatory molecules (in this case auxin) could produce characteristic subcellular patterning that subsequently affects cell shape. This has important implications for research on the morphogenesis of plants and other eukaryotes. Our results also illustrate how gradient-regulated Turing systems provide a particularly robust and flexible mechanism for pattern formation.

Modeling the skin pattern of fishes

Complicated patterns showing various spatial scales have been obtained in the past by coupling Turing systems in such a way that the scales of the independent systems resonate. This produces superimposed patterns with different length scales. Here we propose a model consisting of two identical reaction-diffusion systems coupled together in such a way that one of them produces a simple Turing pattern of spots or stripes, and the other traveling wave fronts that eventually become stationary. The basic idea is to assume that one of the systems becomes fixed after some time and serves as a source of morphogens for the other system. This mechanism produces patterns very similar to the pigmentation patterns observed in different species of stingrays and other fishes. The biological mechanisms that support the realization of this model are discussed.

THE IDENTIFICATION OF COMPLEX SPATIOTEMPORAL PATTERNS USING COUPLED MAP LATTICE MODELS

Many complex and interesting spatiotemporal patterns have been observed in a wide range of scientific areas. In this paper, two kinds of spatiotemporal patterns including spot replication and Turing systems are investigated and new identification methods are proposed to obtain Coupled Map Lattice (CML) models for this class of systems. Initially, a new correlation analysis method is introduced to determine an appropriate temporal and spatial data sampling procedure for the identification of spatiotemporal systems. A new combined Orthogonal Forward Regression and Bayesian Learning algorithm with Laplace priors is introduced to identify sparse and robust CML models for complex spatiotemporal patterns. The final identified CML models are validated using correlation-based model validation tests for spatiotemporal systems. Numerical results illustrate the identification procedure and demonstrate the validity of the identified models.

Complex turing patterns in non-linearly coupled systems

Recently pattern formation in layered structures, showing complicated superimposed patterns, has been modeled by coupling two Turing systems linearly, i.e., passively, such that the characteristic length scales of the independent systems are well separated. Here we propose a model of two non-linearly coupled Turing systems to study pattern formation in layered membrane-like structures, where the coupling plays an active role and changes the kinetics of the uncoupled systems. Extensive numerical simulations show that non-linear coupling generates a number of new regular patterns different from the ones observed earlier with linearly coupled systems. Some of them turn out to be superimposed patterns with different length scales, but many are not. Also, contrary to the linear coupling case, the strength of the non-linear coupling is found to play an important role in the formation and selection of patterns.

Soliton behaviour in a bistable reaction diffusion model

We analyze a generic reaction-diffusion model that contains the important features of Turing systems and that has been extensively used in the past to model biological interesting patterns. This model presents various fixed points. Analysis of this model has been made in the past only in the case when there is only a single fixed point, and a phase diagram of all the possible instabilities shows that there is a place where a Turing-Hopf bifurcation occurs producing oscillating Turing patterns. In here we focus on the interesting situation of having several fixed points, particularly when one unstable point is in between two equally stable points. We show that the solutions of this bistable system are traveling front waves, or solitons. The predictions and results are tested by performing extensive numerical calculations in one and two dimensions. The dynamics of these solitons is governed by a well defined spatial scale, and collisions and interactions between solitons depend on this scale. In certain regions of parameter space the wave fronts can be stationary, forming a pattern resembling spatial chaos. The patterns in two dimensions are particularly interesting because they can present a coherent dynamics with pseudo spiral rotations that simulate the myocardial beat quite closely. We show that our simple model can produce complicated spatial patterns with many different properties, and could be used in applications in many different fields.

Most stable patterns among three-dimensional Turing patterns

It is known that Turing systems in two dimensions produce spotted, striped, and labyrinthine patterns. In three dimensions, a greater variety of patterns is possible. By numerical simulation of the FitzHugh-Nagumo type of reaction-diffusion system, we have obtained not only lamellar, hexagonal and spherical structures (BCC and FCC) but also gyroid, Fddd, and perforated lamellar structures. The domains of these three structures constitute interconnected regular networks, a characteristic occurring in three dimensions. Moreover, we derive the Lyapunov functional by reducing the system, and we evaluate this functional by introducing the asymptotic solutions of each structure by the mode-expansion method and direct simulation of the time evolution equation.

Group classification of systems of non-linear reaction–diffusion equations with general diffusion matrix. II. Generalized Turing systems

Group classification of systems of two coupled non-linear reaction–diffusion equation with a diagonal diffusion matrix is carried out. Symmetries of diffusion systems with singular diffusion matrix and additional first order derivative terms are described.

Labyrinthine versus straight-striped patterns generated by two-dimensional Turing systems

Striped patterns are often observed on fish skin. Such patterns have been accounted for by reaction-diffusion (RD) Turing-type models, in which two substances can spontaneously form a spatially heterogeneous pattern in a homogeneous field. Among the striped patterns generated by Turing-type models, some are “straight-striped patterns,” with many stripes running in parallel, while others are “labyrinthine patterns,” in which the stripes often change direction, merge with each other, and frequently branch out. RD models differ in terms of their tendency to generate either labyrinthine or straight-striped patterns. Here, we studied the conditions under which either a labyrinthine or straight-striped pattern would emerge. First, we defined an index for stripe clearness, S h . Straight-striped patterns (large S h ) are formed if only a narrow range of spatial periods corresponds to an unstable mode. Labyrinthine patterns (small S h ) are formed when a wide range of spatial periods is unstable. More specifically, labyrinthine patterns are formed when the maximum spatial period of unstable modes is more than twice that of the minimum spatial period of unstable modes; otherwise, straight-striped patterns are formed. We then examined RD models with nonlinear reaction terms, including both activator–inhibitor and substrate-depletion models, and we demonstrated that the same conclusions hold with respect to the conditions required for labyrinthine versus straight-striped patterns.

Morphological transitions and bistability in Turing systems

It is well known that in two dimensions Turing systems produce spots, stripes and labyrinthine patterns, and in three dimensions lamellar and spherical structures, or their combinations, are observed. In this paper we study transitions between these states in both two and three dimensions. First, we derive the regions of stability for different patterns using nonlinear bifurcation analysis. Then, we apply large scale computer simulations to analyze the pattern selection in a bistable system by studying the effect of parameter selection on morphological clustering and the appearance of topological defects. The method elaborated in this paper presents a probabilistic approach for studying pattern selection in a bistable reaction-diffusion system.

Travelling Turing patterns with anomalous diffusion

We investigate the formation of Turing patterns (arising from a diffusion driven instability)when diffusion is anomalous. We analyse a model that contains the importantfeatures of Turing systems and that has been extensively used in the past tomodel interesting biological patterns. We concentrate on the case of asymmetricanomalous diffusion, and we cast a version of the model, suitable for numericalcalculations, using a discrete version of the fractional derivatives defining theanomalous diffusion operator. The results are interesting in the sense that patternsare no longer stationary but acquire a velocity that depends on the exponentof the fractional derivatives. Extensive numerical calculations in one and twodimensions exhibit many of the features predicted by the analysis of the equations.

Stripes, spots, or reversed spots in two-dimensional Turing systems

Two-dimensional Turing models can generate stationary striped patterns or spotted patterns, and are used to explain the body pattern formation of animals. We studied the effects of the choice of reaction terms on pattern selection, i.e., which pattern is likely to be formed. We examined in detail a model with linear reaction terms and additional constraint terms that confine two variables within a finite range. In the one-dimensional model, a periodic stationary pattern can be formed only when the activator level is constrained both from below and from above. In the two-dimensional model, the relative distance of the equilibrium level of the activator between the upper and lower limitations determines the pattern selection. Striped patterns are produced when the equilibrium is equally distant from the upper and the lower limitations, but spotted patterns are produced when the equilibrium is clearly closer to one than to the other of two limitations. We then examined models with nonlinear reaction terms, including both activator-inhibitor and activator-depletion substrate type models; we attempted to explain the pattern selection of these nonlinear models based on the results of linear models with constraints. The distribution of the activator level is skewed positively and negatively for spotted patterns and reversed spotted patterns, respectively. In contrast, the skew of the distribution of the activator level was close to zero in the case of striped patterns. This observation provides a heuristic argument of how the location of the equilibrium between the constraints leads to pattern selection.

The Effect of Noise on Turing Patterns

The effect of noise on pattern formation in Turing systems is studied. It is shown how robustness of 2D patterns and 3D structures against noise depends on the characteristics of the morphology. The effect of noise is of particular interest since Turing systems are often used for explaining biological patterns or structures, which have to be stable against noise.