Turing Conditions Research Articles

Turing Pattern Formation in Reaction-Cross-Diffusion Systems with a Bilayer Geometry

Conditions for self-organisation via Turing’s mechanism in biological systems represented by reaction-diffusion or reaction-cross-diffusion models have been extensively studied. Nonetheless, the impact of tissue stratification in such systems is under-explored, despite its ubiquity in the context of a thin epithelium overlying connective tissue, for instance the epidermis and underlying dermal mesenchyme of embryonic skin. In particular, each layer can be subject to extensively different biochemical reactions and transport processes, with chemotaxis - a special case of cross-diffusion - often present in the mesenchyme, contrasting the solely molecular transport typically found in the epidermal layer. We study Turing patterning conditions for a class of reaction-cross-diffusion systems in bilayered regions, with a thin upper layer and coupled by a linear transport law. In particular, the role of differential transport through the interface is explored together with the presence of asymmetry between the homogeneous equilibria of the two layers. A linear stability analysis is carried out around a spatially homogeneous equilibrium state in the asymptotic limit of weak and strong coupling strengths, where quantitative approximations of the bifurcation curve can be computed. Our theoretical findings, for an arbitrary number of reacting species, reveal quantitative Turing conditions, highlighting when the coupling mechanism between the layered regions can either trigger patterning or stabilize a spatially homogeneous equilibrium regardless of the independent patterning state of each layer. We support our theoretical results through direct numerical simulations, and provide an open source code to explore such systems further.

Spatial heterogeneity localizes Turing patterns in reaction-cross-diffusion systems

Motivated by bacterial chemotaxis and multi-species ecological interactions in heterogeneous environments, we study a general one-dimensional reaction-cross-diffusion system in the presence of spatial heterogeneity in both transport and reaction terms. Under a suitable asymptotic assumption that the transport is slow over the domain, while gradients in the reaction heterogeneity are not too sharp, we study the stability of a heterogeneous steady state approximated by the system in the absence of transport. Using a WKB ansatz, we find that this steady state can undergo a Turing-type instability in subsets of the domain, leading to the formation of localized patterns. The boundaries of the pattern-forming regions are given asymptotically by 'local' Turing conditions corresponding to a spatially homogeneous analysis parameterized by the spatial variable. We developed a general open-source code which is freely available, and show numerical examples of this localized pattern formation in a Schnakenberg cross-diffusion system, a Keller-Segel chemotaxis model, and the Shigesada-Kawasaki-Teramoto model with heterogeneous parameters. We numerically show that the patterns may undergo secondary instabilities leading to spatiotemporal movement of spikes, though these remain approximately within the asymptotically predicted localized regions. This theory can elegantly differentiate between spatial structure due to background heterogeneity, from spatial patterns emergent from Turing-type instabilities.

Liouville-Green approximation for linearly coupled systems: Asymptotic analysis with applications to reaction-diffusion systems

<p style='text-indent:20px;'>Asymptotic analysis has become a common approach in investigations of reaction-diffusion equations and pattern formation, especially when considering generalizations of the original model, such as spatial heterogeneity, where finding an analytic solution even to the linearized equations is generally not possible. The Liouville-Green approximation (also known as WKBJ method), one of the more robust asymptotic approaches for investigating dissipative phenomena captured by linear equations, has recently been applied to the Turing model in a heterogeneous environment. It demonstrated the anticipated modifications to the results obtained in a homogeneous setting, such as localized patterns and local Turing conditions. In this context, we attempt a generalization of the scalar Liouville-Green approximation to multicomponent systems. Our broader mathematical approach results in general approximation theorems for systems of ODEs without turning points. We discuss the cases of exponential and oscillatory behaviour first before treating the general case. Subsequently, we demonstrate the spectral properties utilized in the approximation theorems for a typical Turing system, hence showing that Liouville-Green approximation is plausible for an arbitrary number of coupled species outside of turning points and generally valid for fast growing modes as long as the diffusivities are distinct. Note that our line of approach is via showing that the solution is close (using suitable weight functions for measuring the error) to a linear combination of Airy-like functions.</p>

Turing conditions for pattern forming systems on evolving manifolds.

The study of pattern-forming instabilities in reaction-diffusion systems on growing or otherwise time-dependent domains arises in a variety of settings, including applications in developmental biology, spatial ecology, and experimental chemistry. Analyzing such instabilities is complicated, as there is a strong dependence of any spatially homogeneous base states on time, and the resulting structure of the linearized perturbations used to determine the onset of instability is inherently non-autonomous. We obtain general conditions for the onset and structure of diffusion driven instabilities in reaction-diffusion systems on domains which evolve in time, in terms of the time-evolution of the Laplace-Beltrami spectrum for the domain and functions which specify the domain evolution. Our results give sufficient conditions for diffusive instabilities phrased in terms of differential inequalities which are both versatile and straightforward to implement, despite the generality of the studied problem. These conditions generalize a large number of results known in the literature, such as the algebraic inequalities commonly used as a sufficient criterion for the Turing instability on static domains, and approximate asymptotic results valid for specific types of growth, or specific domains. We demonstrate our general Turing conditions on a variety of domains with different evolution laws, and in particular show how insight can be gained even when the domain changes rapidly in time, or when the homogeneous state is oscillatory, such as in the case of Turing-Hopf instabilities. Extensions to higher-order spatial systems are also included as a way of demonstrating the generality of the approach.

Turing Patterning in Stratified Domains

Reaction–diffusion processes across layered media arise in several scientific domains such as pattern-forming E. coli on agar substrates, epidermal–mesenchymal coupling in development, and symmetry-breaking in cell polarization. We develop a modeling framework for bilayer reaction–diffusion systems and relate it to a range of existing models. We derive conditions for diffusion-driven instability of a spatially homogeneous equilibrium analogous to the classical conditions for a Turing instability in the simplest nontrivial setting where one domain has a standard reaction–diffusion system, and the other permits only diffusion. Due to the transverse coupling between these two regions, standard techniques for computing eigenfunctions of the Laplacian cannot be applied, and so we propose an alternative method to compute the dispersion relation directly. We compare instability conditions with full numerical simulations to demonstrate impacts of the geometry and coupling parameters on patterning, and explore various experimentally relevant asymptotic regimes. In the regime where the first domain is suitably thin, we recover a simple modulation of the standard Turing conditions, and find that often the broad impact of the diffusion-only domain is to reduce the ability of the system to form patterns. We also demonstrate complex impacts of this coupling on pattern formation. For instance, we exhibit non-monotonicity of pattern-forming instabilities with respect to geometric and coupling parameters, and highlight an instability from a nontrivial interaction between kinetics in one domain and diffusion in the other. These results are valuable for informing design choices in applications such as synthetic engineering of Turing patterns, but also for understanding the role of stratified media in modulating pattern-forming processes in developmental biology and beyond.

From one pattern into another: analysis of Turing patterns in heterogeneous domains via WKBJ.

Pattern formation from homogeneity is well studied, but less is known concerning symmetry-breaking instabilities in heterogeneous media. It is non-trivial to separate observed spatial patterning due to inherent spatial heterogeneity from emergent patterning due to nonlinear instability. We employ WKBJ asymptotics to investigate Turing instabilities for a spatially heterogeneous reaction-diffusion system, and derive conditions for instability which are local versions of the classical Turing conditions. We find that the structure of unstable modes differs substantially from the typical trigonometric functions seen in the spatially homogeneous setting. Modes of different growth rates are localized to different spatial regions. This localization helps explain common amplitude modulations observed in simulations of Turing systems in heterogeneous settings. We numerically demonstrate this theory, giving an illustrative example of the emergent instabilities and the striking complexity arising from spatially heterogeneous reaction-diffusion systems. Our results give insight both into systems driven by exogenous heterogeneity, as well as successive pattern forming processes, noting that most scenarios in biology do not involve symmetry breaking from homogeneity, but instead consist of sequential evolutions of heterogeneous states. The instability mechanism reported here precisely captures such evolution, and extends Turing's original thesis to a far wider and more realistic class of systems.

Effect of kernels on spatio-temporal patterns of a non-local prey-predator model

Non-local interaction describes the effects of mobility of a population species in their spatial locations. Non-local interaction is incorporated into a prey-predator model by introducing an integral term with appropriate kernel function. The kernel function determines the nature and range of the accessibility of the resources. We consider a nonlocal spatio-temporal prey-predator model with additive weak Allee effect in prey growth and density-dependent predator mortality. The dynamics of the model is examined in presence of a parabolic and a triangular kernel functions. Comparisons are made between the resulting dynamics of the nonlocal model with these kernels for the same range of nonlocal interaction. A linear stability analysis is performed for both the kernels to derive Turing and spatial-Hopf bifurcation conditions. In general, Turing bifurcation curves are different for the two kernels and the same is true for spatial-Hopf bifurcation curves. This results in different dynamics for the two kernels for some parameter values. However, for a fixed range of nonlocal interaction, the dynamics of the model with these two kernels are almost similar for most of the parameter values explored. Increase in the range of nonlocal interaction up to a certain value stabilizes the system dynamics, and then, it destabilizes. Extensive numerical simulations are performed to illustrate the system dynamics.

A Bistable Switch Mechanism for Stem Cell Domain Nucleation in the Shoot Apical Meristem.

In plants, the stem cells residing in shoot apical meristems (SAM) give rise to above-ground tissues (Aichinger et al., 2012). Hence, the maintenance of stem cell niches is of central importance to a plant's continued growth and development (Fletcher and Meyerowitz, 2000; Gordon et al., 2009). For the flowering plant Arabidopsis thaliana, the genetic determinants of stem cell growth, division, and localization have been identified, and negative feedback between a homeodomain transcription factor, WUSCHEL (WUS), and a receptor kinase, CLAVATA (CLV), is known to play a crucial role in controlling the reservoir of stem cells in the central domain of a SAM. The morphology of plant stems and floral organs is controlled in large part by the size and stability of SAMs, which is controlled, in turn, by spatiotemporal patterns of WUS and CLV expression in meristems. For example, loss of restrictive signals in clv mutants of Arabidopsis leads to enlargement of shoot and floral meristems, resulting in extra floral organs and club-shaped siliques (Jonsson et al., 2005). The size, localization and stability of stem cell domains should be determined, in principle, by the interactions of WUS and CLV proteins, especially by their propensities to diffuse through the domain and by the rates of the molecular reactions that control their activities. Within this paradigm, reaction-diffusion (RD) models of WUS-CLV interactions have been popular mathematical models of SAM dynamics (Jonsson et al., 2005; Hohm et al., 2010; Fujita et al., 2011). In RD models, the spontaneous generation of inhomogeneous distributions of WUS and CLV in SAM domains is usually attributed to mechanisms based on a “Turing” instability (Turing, 1952; Segel and Jackson, 1972). The generic RD equations for spatiotemporal changes in the concentrations, u(x,t) and v(x,t), of two interacting proteins are ∂u∂t=f(u,v)+Du∂2u∂x2, ∂v∂t=g(u,v)+Dv∂2v∂x2, where f(u,v) and g(u,v) are nonlinear functions describing their local chemical interactions. A unique, uniform, steady-state solution, u(x,t) = u0 = constant and v(x,t) = v0, of these equations can become unstable with respect to small, non-uniform perturbations, u(x,t) = u0 + eλt·δu·cos(qx) and v(x,t) = v0 + eλt·δv·cos(qx), δu > Du, generating standing waves of wavelength l ≈ 2π/qcrit in the simulations of the RD system (Gierer and Meinhardt, 1972; Murray, 2003). At present, the diffusive lengths of CLV and WUS in SAMs have not been determined, and there is no evidence to suggest that the Turing condition (diffusivity of CLV >> diffusivity of WUS) is satisfied in the central zone of a SAM.

Pattern formation for reactive species undergoing anisotropic diffusion

Turing instabilities for a two species reaction-diffusion systems is studied under anisotropic diffusion. More specifically, the diffusion constants which characterize the ability of the species to relocate in space are direction sensitive. Under this working hypothesis, the conditions for the onset of the instability are mathematically derived and numerically validated. Patterns which closely resemble those obtained in the classical context of isotropic diffusion, develop when the usual Turing condition is violated, along one of the two accessible directions of migration. Remarkably, the instability can also set in when the activator diffuses faster than the inhibitor, along the direction for which the usual Turing conditions are not matched.

Investigating the Turing conditions for diffusion-driven instability in the presence of a binding immobile substrate.

Turing's diffusion-driven instability for the standard two species reaction-diffusion system is only achievable under well-known and rather restrictive conditions on both the diffusion rates and the kinetic parameters, which necessitates the pairing of a self-activator with a self-inhibitor. In this study we generalize the standard two-species model by considering the case where the reactants can bind to an immobile substrate, for instance extra-cellular matrix, and investigate the influence of this dynamics on Turing's diffusion-driven instability. Such systems have been previously studied on the grounds that binding of the self-activator to a substrate may effectively reduce its diffusion rate and thus induce a Turing instability for species with equal diffusion coefficients, as originally demonstrated by Lengyel and Epstein (1992) under the assumption that the bound state dynamics occurs on a fast timescale. We, however, analyse the full system without any separation of timescales and demonstrate that the full system also allows a relaxation of the standard constraints on the reaction kinetics for the Turing instability, increasing the type of interactions that could give rise to spatial patterning. In particular, we show that two self-activators can undertake a diffusively driven instability in the presence of a binding immobile substrate, highlighting that the interactions required of a putative biological Turing instability need not be associated with a self-activator-self-inhibitor morphogen pair.

Pattern formation by two-layer Turing system with complementarysynthesis

Many multicellular organisms have a layered structure. The interaction between these layers plays an essential role in many developmental processes, and key molecules involved in these processes are often expressed in a layer-specific manner. On the other hand, pattern formation of organisms has been frequently discussed in connection with the Turing system. However, the Turing system has so far been studied mainly in single-layered space. In this paper, we thus investigate a two-layer Turing system with complementary synthesis, in which two interacting molecules are exclusively synthesized in different layers. From a linear stability analysis, we determine the Turing condition of the complementary system, and show that this condition requires stronger regulatory interactions of the molecules than that of the system with usual ubiquitous synthesis. We then confirm that this complementary system affects pattern types in fixed and expanding two-dimensional spaces in a similar way to the system with ubiquitous synthesis. In addition, the two-layer system includes two types of diffusion, lateral and transversal, and these have distinct effects on pattern formation with lateral diffusion mainly determining the periodicity of patterns generated and transversal diffusion affecting pattern type. These results suggest that the transversal diffusion functions as a time delay in the two-layer system. Finally, we apply this complementary system to explain pattern formation of the shoot apical meristem of plants. These findings provide an understanding of pattern formation caused by the interaction between cell layers in multicellular organisms.

Nonlinear effects on Turing patterns: Time oscillations and chaos

We show that a model reaction-diffusion system with two species in a monostable regime and over a large region of parameter space produces Turing patterns coexisting with a limit cycle which cannot be discerned from the linear analysis. As a consequence, the patterns oscillate in time. When varying a single parameter, a series of bifurcations leads to period doubling, quasiperiodic, and chaotic oscillations without modifying the underlying Turing pattern. A Ruelle-Takens-Newhouse route to chaos is identified. We also examine the Turing conditions for obtaining a diffusion-driven instability and show that the patterns obtained are not necessarily stationary for certain values of the diffusion coefficients. These results demonstrate the limitations of the linear analysis for reaction-diffusion systems.

Effects of cross diffusion on Turing bifurcations in two-species reaction-transport systems

We study the Turing bifurcation in general two-species reaction-transport systems, where particle dispersal is governed by diffusion, including cross diffusion. By performing a linear stability analysis, we find conditions for the Turing instability and compare the results with the standard Turing conditions. We apply our results to two model systems, the Lengyel-Epstein model and the Brusselator, and find strong effects of cross diffusion on the Turing bifurcation.

Synthesis of nanocrystalline cubic zirconia using femtosecond laser ablation

We report on the synthesis of nanocrys- talline zirconia in liquid using femtosecond laser ablation. Nanocrystalline cubic zirconia has been prepared by femtosecond laser ablation of zirconium in ammonia, while nanocrystalline tetragonal and monoclinic zirconia was synthesized in water. The physical and chemical mechanisms of the formation of nanocrystalline metastable zirconia are discussed. The intrinsic properties of femtosecond laser ablation in liquid and OH -1 may be responsible for the synthesis of cubic zirconia. It is suggested that the femtosecond laser pulse can create higher tempera- ture and pressure conditions at a localized area in the liquid than the nanosecond laser pulse and the cooling is also faster in the femtosecond laser ablation process, which determined the difference between the products synthesized with femtosecond and nanosecond-pulsed laser ablation.

Stochastic Turing patterns in the Brusselator model

A stochastic version of the Brusselator model is proposed and studied via the system size expansion. The mean-field equations are derived and shown to yield to organized Turing patterns within a specific parameters region. When determining the Turing condition for instability, we pay particular attention to the role of cross-diffusive terms, often neglected in the heuristic derivation of reaction-diffusion schemes. Stochastic fluctuations are shown to give rise to spatially ordered solutions, sharing the same quantitative characteristic of the mean-field based Turing scenario, in term of excited wavelengths. Interestingly, the region of parameter yielding to the stochastic self-organization is wider than that determined via the conventional Turing approach, suggesting that the condition for spatial order to appear can be less stringent than customarily believed.

Turing bifurcation in a reaction–diffusion system with density-dependent dispersal

Motivated by the recent finding [N. Kumar, G.M. Viswanathan, V.M. Kenkre, Physica A 388 (2009) 3687] that the dynamics of particles undergoing density-dependent nonlinear diffusion shows sub-diffusive behaviour, we study the Turing bifurcation in a two-variable system with this kind of dispersal. We perform a linear stability analysis of the uniform steady state to find the conditions for the Turing bifurcation and compare it with the standard Turing condition in a reaction–diffusion system, where dispersal is described by simple Fickian diffusion. While activator–inhibitor kinetics are a necessary condition for the Turing instability as in standard two-variable systems, the instability can occur even if the diffusion constant of the inhibitor is equal to or smaller than that of the activator. We apply these results to two model systems, the Brusselator and the Gierer–Meinhardt model.

Stability analysis of non-autonomous reaction-diffusion systems: the effects of growing domains

By using asymptotic theory, we generalise the Turing diffusively-driven instability conditions for reaction-diffusion systems with slow, isotropic domain growth. There are two fundamental biological differences between the Turing conditions on fixed and growing domains, namely: (i) we need not enforce cross nor pure kinetic conditions and (ii) the restriction to activator-inhibitor kinetics to induce pattern formation on a growing biological system is no longer a requirement. Our theoretical findings are confirmed and reinforced by numerical simulations for the special cases of isotropic linear, exponential and logistic growth profiles. In particular we illustrate an example of a reaction-diffusion system which cannot exhibit a diffusively-driven instability on a fixed domain but is unstable in the presence of slow growth.

Optimal primary culture conditions of human gastric mucosal fibroblasts

Objective To optimize primary culture conditions of human gastric mucosal fibre- blasts for further investigations of gastric cancer associated fibroblasts. Methods The fibroblasts were cul-tured primarily by explant method, and identified by immunohistochemistry, morphology and electron mi-croscopy. The effects of different culture conditions on emigration ratios of ceils were investigated to opti-mize the culture surface, medium and pH in primary culture of gastric mucosal fibroblasts. Results When the other conditions were the same,no significandy different emigration rate was found on the mouse tail-coated surface,fetal bovine serum-coated surface ,and nothing coated surface (P > 0.05 ). As to the role of different culture media including RPMI 1640 medium, DMEM and high-gluco DMEM,there was signifi-cantly higher emigration rate in the RPMI 1640 medium compared with other two media (P < 0.01, stand- ardized estimate = 0. 4688 ). Different pH values(pH = 7.0,7.4 and 7.8)of culture solution led to differ-ent emigration rate,and the best emigration rate was found in pH = 7.4 solution(P < 0.01, standardized estimate =0.5857 ). Furthermore, there was no interaetional effect on the cell emigration rate among cul- ture conditions. Conclusion The primary culture of human gastric mucosal fibroblasts can be greatly im-proved by optimizing culture conditions such as kinds of medium and pH values of culture solution. Key words: Gastric mucosa; Fibroblasts; Primary culture

Marine diatom growth on different forms of particulate silica: evidence of cell/particle interaction

The influence of particulate silica sources on growth and silicon uptake of 3 diatom spe- cies: Cylindrotheca fusiformis, Navicula sp., and Skeletonema costatum, was investigated. Each diatom strain was incubated in controlled conditions, with mineral (quartz sand and 2 pure quartz dusts with variable degree of hydrophilicity/hydrophobicity) and biogenic (diatomaceous earth and sponge spicules) silica substrates. Mineral sources were all crystalline, while the biogenic substrates were mostly amorphous. Each diatom species showed a different growth pattern with the various particulate silica substrates, none of which related to the rate of silicon dissolution in the growth medium: S. costatum grew better in presence of the quartz sand; while Navicula sp. and C. fusiformis showed higher growth values with the hydrophobic quartz particles. In contrast, low levels of growth of the 3 diatoms were found in the presence of the biogenic amorphous silica substrates. The high values of the silicon uptake of all diatom species in the presence of the crystalline substrates in cul- ture conditions seemed to confirm the preferred exploitation of the dissolved silicon from crystalline sources with respect to the amorphous mineral substrates. Thus, it could be hypothesized that the dis- solved silicon uptake by marine diatoms was not only mediated by solubilized oligomeric silica, but a direct interaction between diatom cell and particulate mineral substrates. Such chemical/physiologi- cal interaction can be highly specific either for the different diatom species or silica particles. A chem- ical model of a possible role of some organic compounds involved in the uptake of dissolved silicon from particulate silica sources by marine diatoms was suggested, and the potential ecological signif- icance of these findings is discussed.

Can soil temperature direct the composition of high arctic plant communities?

Low temperatures exert a primary constraint on the growth of high arctic vascular plants. However, investigations into the impact of temperature on high arctic plants rarely separate out the role of air and soil temperatures, and few data exist to indicate whether soil temperatures alone can signifi- cantly influence the growth of high arctic vascular plants in a manner that might direct community composition. We examined the response of high arctic plants of three functional types (grasses, sedges/rushes and non-graminoids) to manipulated soil temperature under common air tempera- ture conditions. Target plants, within intact soil cores, were placed in water baths at a range of temperatures between 4.9 and 15.3 o C for one growing season. Grasses responded most rapidly to increased soil temperature, with increased total live plant mass, above-ground live mass and total below-ground live mass, with non-graminoids having the lowest, and sedges/ rushes an intermediate degree of response. The ratio of above- ground live mass to total live mass increased in all growth forms. Grasses, in particular, responded to enhanced soil temperatures by increasing shoot size rather than shoot number. In all growth forms the mass of root tissue beneath the moss layer increased significantly and to a similar extent with increasing soil temperature. These results clearly indicate that different growth forms, although collected from the same plant community, respond differently to changes in soil temperature. As a consequence, factors influencing soil temperature in high arctic ecosystems, such as global climate change or herbivory (which leads to reduced moss depth and increased soil temperatures), may also direct changes in vascular plant community composition.