Turing's Theory Research Articles

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.

Nanomorphogenesis of template-induced crumpled polyamide nanofiltration membranes

Tailored thin-film composite (TFC) nanofiltration (NF) membranes can potentially achieve high permeance and effective salt separation during wastewater reclamation and water reuse. Recently, customized polyamide (PA) nanofilms with crumpled structures have received considerable attention owing to their outstanding separation performance; however, the underlying mechanism of the formation of crumpled features at the nanoscale remains poorly understood. Herein, we report the use of a particulate polydopamine (PDA) interlayer as a regulator for modulating chemical inhomogeneity (i.e., the diffusion and distribution of amine monomers) during the interfacial polymerization (IP) process, in which the characteristic spatial patterns of PA nanostructures are formed following a diffusion–reaction dynamics model in the Turing theory. Furthermore, the nanomorphologies and separation performance of PA membranes can be effectively controlled by tuning the physical inhomogeneity (i.e., the roughness of the organic–aqueous reaction interface) of the particulate interlayer. The interlayer-mediated TFC NF membrane exhibits crumpled nanosized stripes with three times higher permeance and four times higher Cl−/SO42− selectivity (up to 82.1) than those of pristine PA membranes. We have also identified an array of topographical parameters for both the PDA interlayer and PA nanostructures based on experimental observations and multiscale simulations (i.e., molecular dynamics and dissipative particle dynamics) to establish quantitative relationships between the interlayer morphology, PA morphology, and performance characteristics of NF membranes. Our work provides molecular insights into the nanomorphogenesis of crumpled PA nanofilms and interlayer effects on the IP reaction and final separation performance.

Turing patterns in systems with high-order interactions

Turing theory of pattern formation is among the most popular theoretical means to account for the variety of spatio-temporal structures observed in Nature and, for this reason, finds applications in many different fields. While Turing patterns have been thoroughly investigated on continuous support and on networks, only a few attempts have been made towards their characterization in systems with higher-order interactions. In this paper, we propose a way to include group interactions in reaction–diffusion systems, and we study their effects on the formation of Turing patterns. To achieve this goal, we rewrite the problem originally studied by Turing in a general form that accounts for a microscopic description of interactions of any order in the form of a hypergraph, and we prove that the interplay between the different orders of interaction may either enhance or repress the emergence of Turing patterns. Our results shed light on the mechanisms of pattern-formation in systems with many-body interactions and pave the way for further extensions of Turing original framework.

Unraveling the real magnitude of illegal wildlife poisoning to halt cryptic biodiversity loss

Illegal wildlife poisoning is a global threat for biodiversity, yet the magnitude of its impact on ecosystems is largely underestimated as most of poisoning episodes remain undetected. Here, we conducted a large-scale field experiment to better understand the real dimension of the illegal wildlife poisoning in terms of composition and number of species and abundance of impacted individuals, as well as the ecological factors driving it. We used camera traps to monitor simulated poison baits placed in 25 study areas in SW Europe and applied Good–Turing theory to estimate the richness of species of the entire assemblage (observed plus undetected). We recorded 3095 individuals from 39 vertebrate species that consumed 94 % of the baits (N = 590). Yet, using sample completeness to estimate the entire species assemblage yielded a total of 47 species exposed to illegal poisoning. The observed assemblage included different trophic and functional groups (from lizards and snakes to apex species among birds and mammals), as well as a 38 % of threatened and near threatened species (according to Spanish and Portuguese vertebrate red list and UICN list). The size (weight) of the bait outstands as a reliable predictor of the number of species (0–8 species/bait, mean = 2) and individuals (0–99 individuals/bait, mean = 5) susceptible to poisoning. The habitat where the bait was placed modulated the abundance of individuals affected (greater in open than in closed habitats). Type of bait and habitat drove the compositional variation of species. Our approach enables uncover entire species assemblages prone to illegal poisoning and their ecological drivers associated, advancing the understanding of the impact of wildlife poisoning in ecosystems.

Thermodynamic principles for system biology and the patterns of flower pigmentation.

The thermodynamic principles for system biology are reviewed and formulated, and then basic patterns of flower pigmentation are interpreted. Main thoughts: (1) any biological trait (color or function of a cell) is logically related to a thermodynamic system (or physiological system, signaling network of the cell), (2) the striped, speckled and circle are three basic patterns of flower pigmentation, the development of flowers is an irreversible process, (3) the patterns of flower pigmentation are formed in flower development, (4) the flower cells can change its color in a period of development and this process is controlled thermodynamically, (5) there is giant space of physiology within an organism and within its numerous thermal states can appear under different conditions. In this theory, the dominant inheritance means that a gene contributes great to the thermodynamic stability of a trait related system; different genes can be interacted or integrated thermodynamically according to their contribution to the stability of its related system. By combination of Turing theory and our views, complex patterns of pigmentation could be explained theoretically.

Effect of a Membrane on Diffusion-Driven Turing Instability

Biological, physical, medical, and numerical applications involving membrane problems on different scales are numerous. We propose an extension of the standard Turing theory to the case of two domains separated by a permeable membrane. To this aim, we study a reaction–diffusion system with zero-flux boundary conditions on the external boundary and Kedem-Katchalsky membrane conditions on the inner membrane. We use the same approach as in the classical Turing analysis but applied to membrane operators. The introduction of a diagonalization theory for compact and self-adjoint membrane operators is needed. Here, Turing instability is proven with the addition of new constraints, due to the presence of membrane permeability coefficients. We perform an explicit one-dimensional analysis of the eigenvalue problem, combined with numerical simulations, to validate the theoretical results. Finally, we observe the formation of discontinuous patterns in a system which combines diffusion and dissipative membrane conditions, varying both diffusion and membrane permeability coefficients.

Stray Dogs and Public Health: Population Estimation in Punjab, India.

The overpopulation of stray dogs is a serious public health and animal welfare concern in India. Neglected zoonotic diseases such as rabies and echinococcosis are transmitted at the stray–dog human interface, particularly in low to middle-income countries. The current study was designed to estimate the stray dog populations in Punjab to enhance the implementation of animal birth and disease (for example, rabies vaccination) control programs. This is the first systematic estimation of the stray dog population using a recommended method (mark–re-sight) in Punjab, India. The study was conducted from August 2016 to November 2017 in selected villages or wards in Punjab. For the rural areas, 22 sub-districts in each district were randomly selected, then one village from each of the 22 selected sub-districts was selected (by convenience sampling). For urban areas, 3 towns (less than 100,000 human population) and 2 large cities (more than or equal to 100,000 human population) were randomly selected, followed by convenience selection of two wards from each of the 5 selected towns/cities. To estimate the dog population size, we used a modified mark–re-sight procedure and analysed counts using two methods; the Lincoln–Petersen formula with Chapman’s correction, and an application of Good–Turing theory (SuperDuplicates method; estimated per km2 and per 1000 adult humans and were compared between localities (villages vs. towns), dog sex (male vs. female) and age group (young vs. adult) using linear mixed models with district as a random effect. The predicted mean (95% CI) count of the dogs per village or ward were extrapolated to estimate the number of stray dogs in Punjab based on (a) the number of villages and wards in the state; (b) the adult human population of the state and (c) the built-up area of the state. Median stray dog populations per village and per ward using the Lincoln–Petersen formula with Chapman’s correction were estimated to be 33 and 65 dogs, respectively. Higher estimates of 61 per village and 112 per ward are reported using the SuperDuplicates method. The number of males was significantly higher than the number of females and the number of adult dogs was about three times the number of young dogs. Based on different methods, estimates of the mean stray dog population in the state of Punjab ranged from 519,000 to 1,569,000. The current study revealed that there are a substantial number of stray dogs and a high number reside in rural (versus urban) areas in Punjab. The estimated stray dog numbers pose a potential public health hazard in Punjab. This impact requires assessment. The estimated stray dog numbers will help develop a dog population and rabies control program in which information about the logistics required as well as costs of implementing such programmes in Punjab can be incorporated.

Morphogenesis in Synthetic Chemical Cells.

We present an efficient model for describing morphogenesis and the emergence of spatiotemporal structures in synthetic chemical cells. This work is motivated by an experimental setup used for testing Turing's theory of morphogenesis. The model developed is based on the general theory of chemically active droplets, which combines the classical theory of phase separation with reaction-diffusion systems. Through the 2D calculations, we find the six spatiotemporal structures predicted by Turing in 1952 and experimentally observed, in a 1D array of droplets. Moreover, under Turing instability, with a determined chemical wavelength, the system undergoes morphogenesis. This theoretical approach provides a useful tool for understanding the physical differentiation through the direct calculation of the osmotic pressure in each cell as the chemical reaction occurs.

Pattern formation on a growing oblate spheroid. an application to adult sea urchin development

<p style='text-indent:20px;'>In this study, the formation of the adult sea urchin shape is rationalized within the Turing's theory paradigm. The emergence of protrusions from the expanding underlying surface is described through a reaction-diffusion model with Gray-Scott kinetics on a growing oblate spheroid. The case of slow exponential isotropic growth is considered. The model is first studied in terms of the spatially homogenous equilibria and of the bifurcations involved. Turing diffusion-driven instability is shown to occur and the impact of the slow exponential growth on the resulting Turing regions adequately discussed. Numerical investigations validate the theoretical results showing that the combination between an inhibitor and an activator can result in a distribution of spot concentrations that underlies the development of ambulacral tentacles in the sea urchin's adult stage. Our findings pave the way for a model-driven experimentation that could improve the current biological understanding of the gene control networks involved in patterning.</p>

Spatio-temporal analysis of coinfection using wavefronts of Escherichia coli O157: H7 in a dairy cattle farm

This study proposes a new methodology to analyze the impacts of host and pathogen dispersal on coinfection dynamics using the example of Escherichia coli O157:H7 in a dairy farm. A multi-strain Susceptible–Infected–Susceptible model is extended to a Reaction–Diffusion (RD) coinfection model of E. coli O157:H7 transmission, which includes intermittent shedding and pathogen growth in the environment. Analysis of the new RD coinfection model consists of existence and stability of constant equilibria, calculation of the basic reproduction number, and existence of traveling and stationary wave solutions. A stationary wave solution of the RD model defines a nonconstant endemic equilibrium. Whereas a traveling wavefront represents an epidemic wave of infection passing through the farm with a specific speed, direction, and amplitude. The numerical simulations of the RD model demonstrate stable traveling and stationary wavefronts of the RD coinfection model. The stationary wavefront connects two constant coexistence equilibria and the traveling wavefront represents the gradual establishment of an endemic coexistence equilibrium in the dairy farm. The significance of this study lies in utilizing the nonlinear wave theory to analyze dynamics of coinfection in a dairy farm both with respect to location and time. Hence, in addition to Turing theory, the nonlinear wave theory can be used to determine the likelihood of acquiring the infection based on the location of cattle at each specific time.

Introduction to 'Recent progress and open frontiers in Turing's theory of morphogenesis'.

Elucidating pattern forming processes is an important problem in the physical, chemical and biological sciences. Turing's contribution, after being initially neglected, eventually catalysed a huge amount of work from mathematicians, physicists, chemists and biologists aimed towards understanding how steady spatial patterns can emerge from homogeneous chemical mixtures due to the reaction and diffusion of different chemical species. While this theory has been developed mathematically and investigated experimentally for over half a century, many questions still remain unresolved. This theme issue places Turing's theory of pattern formation in a modern context, discussing the current frontiers in foundational aspects of pattern formation in reaction-diffusion and related systems. It highlights ongoing work in chemical, synthetic and developmental settings which is helping to elucidate how important Turing's mechanism is for real morphogenesis, while highlighting gaps that remain in matching theory to reality. The theme issue also surveys a variety of recent mathematical research pushing the boundaries of Turing's original theory to more realistic and complicated settings, as well as discussing open theoretical challenges in the analysis of such models. It aims to consolidate current research frontiers and highlight some of the most promising future directions.This article is part of the theme issue ‘Recent progress and open frontiers in Turing’s theory of morphogenesis’.

Systems for intricate patterning of the vertebrate anatomy.

Periodic patterns form intricate arrays in the vertebrate anatomy, notably the hair and feather follicles of the skin, but also internally the villi of the gut and the many branches of the lung, kidney, mammary and salivary glands. These tissues are composite structures, being composed of adjoined epithelium and mesenchyme, and the patterns that arise within them require interaction between these two tissue layers. In embryonic development, cells change both their distribution and state in a periodic manner, defining the size and relative positions of these specialized structures. Their placement is determined by simple spacing mechanisms, with substantial evidence pointing to a variety of local enhancement/lateral inhibition systems underlying the breaking of symmetry. The nature of the cellular processes involved, however, has been less clear. While much attention has focused on intercellular soluble signals, such as protein growth factors, experimental evidence has grown for contributions of cell movement or mechanical forces to symmetry breaking. In the mesenchyme, unlike the epithelium, cells may move freely and can self-organize into aggregates by chemotaxis, or through generation and response to mechanical strain on their surrounding matrix. Different modes of self-organization may coexist, either coordinated into a single system or with hierarchical relationships. Consideration of a broad range of distinct biological processes is required to advance understanding of biological pattern formation.This article is part of the theme issue 'Recent progress and open frontiers in Turing's theory of morphogenesis'.

Studies of Turing pattern formation in zebrafish skin.

Skin patterns are the first example of the existence of Turing patterns in living organisms. Extensive research on zebrafish, a model organism with stripes on its skin, has revealed the principles of pattern formation at the molecular and cellular levels. Surprisingly, although the networks of cell–cell interactions have been observed to satisfy the ‘short-range activation and long-range inhibition’ prerequisites for Turing pattern formation, numerous individual reactions were not envisioned based on the classical reaction–diffusion model. For example, in real skin, it is not an alteration in concentrations of chemicals, but autonomous migration and proliferation of pigment cells that establish patterns, and cell–cell interactions are mediated via direct contact through cell protrusions. Therefore, the classical reaction–diffusion mechanism cannot be used as it is for modelling skin pattern formation. Various studies are underway to adapt mathematical models to the experimental findings on research into skin patterns, and the purpose of this review is to organize and present them. These novel theoretical methods could be applied to autonomous pattern formation phenomena other than skin patterns.This article is part of the theme issue ‘Recent progress and open frontiers in Turing's theory of morphogenesis’.

Pattern forming systems coupling linear bulk diffusion to dynamically active membranes or cells.

Some analytical and numerical results are presented for pattern formation properties associated with novel types of reaction-diffusion (RD) systems that involve the coupling of bulk diffusion in the interior of a multi-dimensional spatial domain to nonlinear processes that occur either on the domain boundary or within localized compartments that are confined within the domain. The class of bulk-membrane system considered herein is derived from an asymptotic analysis in the limit of small thickness of a thin domain that surrounds the bulk medium. When the bulk domain is a two-dimensional disk, a weakly nonlinear analysis is used to characterize Turing and Hopf bifurcations that can arise from the linearization around a radially symmetric, but spatially non-uniform, steady-state of the bulk-membrane system. In a singularly perturbed limit, the existence and linear stability of localized membrane-bound spike patterns is analysed for a Gierer-Meinhardt activator-inhibitor model that includes bulk coupling. Finally, the emergence of collective intracellular oscillations is studied for a class of PDE-ODE bulk-cell model in a bounded two-dimensional domain that contains spatially localized, but dynamically active, circular cells that are coupled through a linear bulk diffusion field. Applications of such coupled bulk-membrane or bulk-cell systems to some biological systems are outlined, and some open problems in this area are discussed. This article is part of the theme issue 'Recent progress and open frontiers in Turing's theory of morphogenesis'.

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.

Influence of the Turing instability on the motion of domain boundaries.

Turing's theory of pattern formation has provided crucial insights into the behavior of various biological, geographical, and chemical systems over the last few decades. Existing studies have focused on moving-boundary Turing systems for which the motion of the boundary is prescribed by an external agent. In this paper, we present an extension of this theory to a class of systems in which the front motion is governed by the physical processes that occur within the domain. Biological systems exhibiting apically dominant growth and corrosion of metals and alloys highlight some of the noteworthy examples of such systems. In this study, we characterize the nature of interaction between the moving front and the Turing-instability for both an activator-inhibitor and an activator-substrate model. Behavioral regimes of periodic, as well as nonperiodic (nonconstant), growth rates are obtained. Furthermore, the trends in the first show striking similarities with the cyclic-boundary-kinetics observed in experimental systems. In general, a stationary, periodic structure is also left behind the moving front. If the periodicity of the boundary kinetics agrees with the allowed range of the stable-periodic solutions, the pattern formed tends to persist. Otherwise, it evolves to a nearby energy-minimum either by peak-splitting, peak-decay, or by settling down to a spatially homogeneous state.

Key Features of Turing Systems are Determined Purely by Network Topology

Turing's theory of pattern formation is a universal model for self-organization, applicable to many systems in physics, chemistry and biology. Essential properties of a Turing system, such as the conditions for the existence of patterns and the mechanisms of pattern selection are well understood in small networks. However, a general set of rules governing how network topology determines fundamental system properties and constraints has not be found. Here we provide a first general theory of Turing network topology, which proves why three key features of a Turing system are directly determined by the topology: the type of restrictions that apply to the diffusion rates, the robustness of the system, and the phase relations of the molecular species.

Stochastic Turing patterns in a synthetic bacterial population

The origin of biological morphology and form is one of the deepest problems in science, underlying our understanding of development and the functioning of living systems. In 1952, Alan Turing showed that chemical morphogenesis could arise from a linear instability of a spatially uniform state, giving rise to periodic pattern formation in reaction-diffusion systems but only those with a rapidly diffusing inhibitor and a slowly diffusing activator. These conditions are disappointingly hard to achieve in nature, and the role of Turing instabilities in biological pattern formation has been called into question. Recently, the theory was extended to include noisy activator-inhibitor birth and death processes. Surprisingly, this stochastic Turing theory predicts the existence of patterns over a wide range of parameters, in particular with no severe requirement on the ratio of activator-inhibitor diffusion coefficients. To explore whether this mechanism is viable in practice, we have genetically engineered a synthetic bacterial population in which the signaling molecules form a stochastic activator-inhibitor system. The synthetic pattern-forming gene circuit destabilizes an initially homogenous lawn of genetically engineered bacteria, producing disordered patterns with tunable features on a spatial scale much larger than that of a single cell. Spatial correlations of the experimental patterns agree quantitatively with the signature predicted by theory. These results show that Turing-type pattern-forming mechanisms, if driven by stochasticity, can potentially underlie a broad range of biological patterns. These findings provide the groundwork for a unified picture of biological morphogenesis, arising from a combination of stochastic gene expression and dynamical instabilities.

A hybrid discrete-continuum approach to model Turing pattern formation.

Since its introduction in 1952, with a further refinement in 1972 by Gierer and Meinhardt, Turing's (pre-)pattern theory (the chemical basis of morphogenesis) has been widely applied to a number of areas in developmental biology, where evolving cell and tissue structures are naturally observed. The related pattern formation models normally comprise a system of reaction-diffusion equations for interacting chemical species (morphogens), whose heterogeneous distribution in some spatial domain acts as a template for cells to form some kind of pattern or structure through, for example, differentiation or proliferation induced by the chemical pre-pattern. Here we develop a hybrid discrete-continuum modelling framework for the formation of cellular patterns via the Turing mechanism. In this framework, a stochastic individual-based model of cell movement and proliferation is combined with a reaction-diffusion system for the concentrations of some morphogens. As an illustrative example, we focus on a model in which the dynamics of the morphogens are governed by an activator-inhibitor system that gives rise to Turing pre-patterns. The cells then interact with the morphogens in their local area through either of two forms of chemically-dependent cell action: Chemotaxis and chemically-controlled proliferation. We begin by considering such a hybrid model posed on static spatial domains, and then turn to the case of growing domains. In both cases, we formally derive the corresponding deterministic continuum limit and show that that there is an excellent quantitative match between the spatial patterns produced by the stochastic individual-based model and its deterministic continuum counterpart, when sufficiently large numbers of cells are considered. This paper is intended to present a proof of concept for the ideas underlying the modelling framework, with the aim to then apply the related methods to the study of specific patterning and morphogenetic processes in the future.

Testing Turing's theory of morphogenesis in chemical cells.

Alan Turing, in "The Chemical Basis of Morphogenesis" [Turing AM (1952) Philos Trans R Soc Lond 237(641):37-72], described how, in circular arrays of identical biological cells, diffusion can interact with chemical reactions to generate up to six periodic spatiotemporal chemical structures. Turing proposed that one of these structures, a stationary pattern with a chemically determined wavelength, is responsible for differentiation. We quantitatively test Turing's ideas in a cellular chemical system consisting of an emulsion of aqueous droplets containing the Belousov-Zhabotinsky oscillatory chemical reactants, dispersed in oil, and demonstrate that reaction-diffusion processes lead to chemical differentiation, which drives physical morphogenesis in chemical cells. We observe five of the six structures predicted by Turing. In 2D hexagonal arrays, a seventh structure emerges, incompatible with Turing's original model, which we explain by modifying the theory to include heterogeneity.