On the mechanochemical theory of biological pattern formation with application to vasculogenesis

We first describe the Murray-Oster mechanical theory of pattern formation, the biological basis of which is experimentally well documented. The model quantifies the interaction of cells and the extracellular matrix via the cell-generated forces. The model framework is described in quantitative detail. Vascular endothelial cells, when cultured on gelled basement membrane matrix, rapidly aggregate into clusters while deforming the matrix into a network of cord-like structures tessellating the planar culture. We apply the mechanical theory of pattern formation to this culture system and show that neither strain-biased anisotropic cell traction nor cell migration are necessary for pattern formation: isotropic, strain-stimulated cell traction is sufficient to form the observed patterns. Predictions from the model were confirmed experimentally.     

Reaction–diffusion wave model for self-assembled network formation of poly(dA)·poly(dT) DNA on mica and HOPG surfaces

Deoxyribonucleic acid (DNA) is a vital molecule for life since it contains genetic information. However, DNA has recently been reported to have unique properties that make it suitable for bionanoelectronic applications, such as the possibility of electrical conductivity and self-organisation. Self-assembled DNA network structures have been observed on several substrates, but the detailed self-assembly mechanism has yet to be determined. The present study investigates self-assembled structures of DNA both theoretically and experimentally. We developed a reaction–diffusion model and used it to investigate pattern formations observed by atomic force microscopy. The computational results qualitatively replicate the network patterns of DNA molecules based on a quantitative agreement with the surface size and timescale. The model can account for the effect of the DNA concentration on pattern formation. Furthermore, peculiar geometric patterns are simulated for mica and highly oriented pyrolytic graphite surfaces.     

Formation of band-type electric patterns in Characean cells

It is observed by experiments that band patterns of alternating acid and alkaline zones are formed on the surface of Characean cells under illumination. In order to understand theoretically such pattern formation, we employ a reaction-diffusion equation model with activator-inhibitor interaction. We study the existence problem of band patterns and hysteresis phenomena such as patterns appearing and disappearing with changing light intensity by using singular perturbation methods.     

Diffusion-driven instabilities and emerging spatial patterns in patchy landscapes

Spatial variation in population densities across a landscape is a feature of many ecological systems, from self-organised patterns on mussel beds to spatially restricted insect outbreaks. It occurs as a result of environmental variation in abiotic factors and/or biotic factors structuring the spatial distribution of populations. However the ways in which abiotic and biotic factors interact to determine the existence and nature of spatial patterns in population density remain poorly understood. Here we present a new approach to studying this question by analysing a predator–prey patch-model in a heterogenous landscape. We use analytical and numerical methods originally developed for studying nearest-neighbour (juxtacrine) signalling in epithelia to explore whether and under which conditions patterns emerge. We find that abiotic and biotic factors interact to promote pattern formation. In fact, we find a rich and highly complex array of coexisting stable patterns, located within an enormous number of unstable patterns. Our simulation results indicate that many of the stable patterns have appreciable basins of attraction, making them significant in applications. We are able to identify mechanisms for these patterns based on the classical ideas of long-range inhibition and short-range activation, whereby landscape heterogeneity can modulate the spatial scales at which these processes operate to structure the populations.     

Complex pattern formation by a self-destabilization of established patterns: chemotactic orientation and phyllotaxis as examples

Stable patterns can be generated by molecular interactions involving local self-enhancement and long-range inhibition. In contrast, highly dynamic patterns result if the maxima, generated in this way, become destabilized by a second antagonistic reaction. The latter must act local and must be long-lasting. Maxima either disappear and reappear at displaced positions or they move over the field as travelling waves. The wave can have unusual properties in that they can penetrate each other without annihilation. The resulting pattern corresponds to those observed in diverse biological systems. In the chemotactic orientation of cells, the temporary signals allow the localized extensions of protrusions under control of minute external asymmetries imposed by the chemoattractant. In phyllotaxis, these signals lead to successive leaf initiation, whereby the longer-lasting extinguishing reaction can cause a displacement of the subsequent leaf initiation site by the typical 137.5 degrees, the golden angle. On seashells, this patterns leads either to oblique lines that can cross each other or to oblique rows of dots. For some of the models animated simulations are available at http://www.eb.tuebingen.mpg.de/abt.4/meinhardt/theory.html.     

Symmetries,non-Euclidean metrics,and patterns in a Swift–Hohenberg model of the visual cortex

The aim of this work is to investigate the effect of the shift-twist symmetry on pattern formation processes in the visual cortex. First, we describe a generic set of Riemannian metrics of the feature space of orientation preference that obeys properties of the shift-twist, translation, and reflection symmetries. Second, these metrics are embedded in a modified Swift-Hohenberg model. As a result we get a pattern formation process that resembles the pattern formation process in the visual cortex. We focus on the final stable patterns that are regular and periodic. In a third step we analyze the influences on pattern formation using weakly nonlinear theory and mode analysis. We compare the results of the present approach with earlier models.     

Interactions Between Pattern Formation and Domain Growth

In this paper we develop a theoretical framework for investigating pattern formation in biological systems for which the tissue on which the spatial pattern resides is growing at a rate which is itself regulated by the diffusible chemicals that establish the spatial pattern. We present numerical simulations for two cases of interest, namely exponential domain growth and chemically controlled growth. Our analysis reveals that for domains undergoing rapid exponential growth dilution effects associated with domain growth influence both the spatial patterns that emerge and the concentration of chemicals present in the domain. In the latter case, there is complex interplay between the effects of the chemicals on the domain size and the influence of the domain size on the formation of patterns. The nature of these interactions is revealed by a weakly nonlinear analysis of the full system. This yields a pair of nonlinear equations for the amplitude of the spatial pattern and the domain size. The domain is found to grow (or shrink) at a rate that depends quadratically on the pattern amplitude, the particular functional forms used to model the local tissue growth rate and the kinetics of the two diffusible species dictating the resulting behaviour.     

Maintenance of heterocyst patterning in a filamentous cyanobacterium

In the absence of sufficient combined nitrogen, some filamentous cyanobacteria differentiate nitrogen-fixing heterocysts at approximately every 10th cell position. As cells between heterocysts grow and divide, this initial pattern is maintained by the differentiation of a single cell approximately midway between existing heterocysts. This paper introduces a mathematical model for the maintenance of the periodic pattern of heterocysts differentiated by Anabaena sp. strain PCC 7120 based on the current experimental knowledge of the system. The model equations describe a non-diffusing activator (HetR) and two inhibitors (PatS and HetN) that undergo diffusion in a growing one-dimensional domain. The inhibitors in this model have distinct diffusion rates and temporal expression patterns. These unique aspects of the model reflect recent experimental findings regarding the molecular interactions that regulate patterning in Anabaena. Output from the model is in good agreement with both the temporal and spatial characteristics of the pattern maintenance process observed experimentally.     

Changing clothes easily: connexin41.8 regulates skin pattern variation

The skin patterns of animals are very important for their survival, yet the mechanisms involved in skin pattern formation remain unresolved. Turing's reaction-diffusion model presents a well-known mathematical explanation of how animal skin patterns are formed, and this model can predict various animal patterns that are observed in nature. In this study, we used transgenic zebrafish to generate various artificial skin patterns including a narrow stripe with a wide interstripe, a narrow stripe with a narrow interstripe, a labyrinth, and a 'leopard' pattern (or donut-like ring pattern). In this process, connexin41.8 (or its mutant form) was ectopically expressed using the mitfa promoter. Specifically, the leopard pattern was generated as predicted by Turing's model. Our results demonstrate that the pigment cells in animal skin have the potential and plasticity to establish various patterns and that the reaction-diffusion principle can predict skin patterns of animals.     

Emergence of complex patterns induced by dynamic systems implemented in dynamic structures (DS)2

We present the evolution of the simple system of Meinhardt implemented in both static or dynamic two-dimensional structures of almost-squared cells. In a static structure of 8 x 4=32 to 128 x 128=16384 cells, the pattern observed is periodic. An algorithm allows us to divide the cells following the greater size, and to define a dynamic structure. The implementation of the same Meinhardt system in this dynamic structure varying from 32 to 16 384 cells and a context of the same genotypic complexity for the model provides aperiodic patterns, with a higher phenotypic complexity than those observed in static structures, while the number of computations is comparable. We define that emergence occurs each time the ratio of phenotypic/genotypic complexities increases.     

Beyond Turing: The response of patterned ecosystems to environmental change

Spatially periodic patterns can be observed in a variety of ecosystems. Model studies revealed that patterned ecosystems may respond in a nonlinear way to environmental change, meaning that gradual changes result in rapid degradation. We analyze this response through stability analysis of patterned states of an arid ecosystem model. This analysis goes one step further than the frequently applied Turing analysis, which only considers stability of uniform states. We found that patterned arid ecosystems systematically respond in two ways to changes in rainfall: (1) by changing vegetation patch biomass or (2) by adapting pattern wavelength. Minor adaptations of pattern wavelength are constrained to conditions of slow change within a high rainfall regime, and high levels of stochastic variation in biomass (noise). Major changes in pattern wavelength occur under conditions of either low rainfall, rapid change or low levels of noise. Such conditions facilitate strong interactions between vegetation patches, which can trigger a sudden loss of half the patches or a transition to a degraded bare state. These results highlight that ecosystem responses may critically depend on rates, rather than magnitudes, of environmental change. Our study shows how models can increase our understanding of these dynamics, provided that analyses go beyond the conventional Turing analysis.     

Alpha-alpha linking motifs and interhelical orientations

While the geometry and sequence preferences of turns that link two beta-strands have been exhaustively explored, the corresponding preferences for sequences that link helical structures have been less well studied. Here we examine the interhelical geometry of two connected helices as a function of their link's length. The interhelical geometry of a helical pair appears to be significantly influenced by the number of linking residues. Furthermore, for relatively short link lengths, a very limited number of predominant conformations are observed, which can be categorized by their phi/psi angles. No more than two predominant linking backbone conformations are observed for a given link length, and some linking backbone conformations correlate strongly with distinctive interhelical geometric parameters. In this study, sequence and hydrogen-bonding patterns were defined for predominant interhelical link motifs. These results should assist in both protein structure prediction and de novo protein design.     

Effects of Intrinsic and Extrinsic Noise Can Accelerate Juxtacrine Pattern Formation

Epithelial pattern formation is an important phenomenon that, for example, has roles in embryogenesis, development and wound-healing. The ligand Epithelial Growth Factor (EGF) and its receptor EGF-R, constitute a system that forms lateral induction patterns by juxtacrine signalling—binding of membrane-bound ligands to receptors on neighbouring cells. Owen et al. developed a generic ordinary differential equation model of juxtacrine lateral induction that exhibits stable patterning under some conditions. The model predicts relatively slow pattern formation. We examine here the effects of both intrinsic and extrinsic cellular noise arising from the stochastic treatment of this model, and show that this noise could have an accelerating effect on the patterning process.     

Why the leopard got its spots: relating pattern development to ecology in felids

A complete explanation of the diversity of animal colour patterns requires an understanding of both the developmental mechanisms generating them and their adaptive value. However, only two previous studies, which involved computer-generated evolving prey, have attempted to make this link. This study examines variation in the camouflage patterns displayed on the flanks of many felids. After controlling for the effects of shared ancestry using a fully resolved molecular phylogeny, this study shows how phenotypes from plausible felid coat pattern generation mechanisms relate to ecology. We found that likelihood of patterning and pattern attributes, such as complexity and irregularity, were related to felids' habitats, arboreality and nocturnality. Our analysis also indicates that disruptive selection is a likely explanation for the prevalence of melanistic forms in Felidae. Furthermore, we show that there is little phylogenetic signal in the visual appearance of felid patterning, indicating that camouflage adapts to ecology over relatively short time scales. Our method could be applied to any taxon with colour patterns that can reasonably be matched to reaction-diffusion and similar models, where the kinetics of the reaction between two or more initially randomly dispersed morphogens determines the outcome of pattern development.     

Comparison of reaction-diffusion simulations with experiment in self-organised microtubule solutions

This article deals with the physical chemical processes underlying biological self-organization by which an initially homogenous solution of reacting chemicals spontaneously self-organizes so as to give rise to a preparation of macroscopic order and form. Theoreticians have predicted that self-organization can arise from a coupling of reactive processes with molecular diffusion. In addition, the presence or absence of an external field, such as gravity, at a critical moment early in the self-organizing process may determine the morphology that subsequently develops. We have found that the formation in vitro of microtubules, a major element of the cellular skeleton, show this type of behaviour. The microtubule preparations spontaneously self-organise by way of reaction and diffusion, and the morphology of the state that forms depends on the presence of gravity at a critical moment early in the process. We have developed a numerical reaction-diffusion scheme, based on the chemical dynamics of a population of microtubules, which simulates the experimental self-organisation. In this article we outline the main features of these simulations and discuss the manner by which a permanent dialogue with experiment has helped develop a microscopic understanding of the collective behaviour.     

干旱半干旱区斑块状植被格局形成模拟研究进展

斑块状植被格局是世界上干旱半干旱区常见的景观类型,它们的形成、组成结构和演替过程研究,对于揭示区域生态系统变化的关键过程具有重要意义。鉴于基于地面调查和遥感技术的方法难以全面刻画斑块状植被格局的形成过程及机制,借助于模型模拟成为解决这一问题的有效方法。自20世纪90年代初至今,斑块状植被格局形成的连续和离散模拟研究不断涌现,然而,连续模拟侧重于植被格局形成的一般机理,缺乏与现实格局的对比和验证,离散模拟单元选择与规则制定等仍需不断研究。在简要回顾斑块状格局形成的反馈机制基础上,重点综述了斑块状植被格局形成的连续和离散模拟的最新研究进展,并指出了现有研究的不足。干旱半干旱区小尺度上植物和水的反馈作用决定了大尺度的斑块状植被格局,充分揭示植被-土壤水分相互作用机理是模型模拟研究的关键,放牧强度和降水格局等外部环境对干旱半干旱区斑块状植被格局特征具有重要影响。在未来研究中,应加强模型模拟结果与实际观测的植被格局比较和验证,重视局域环境条件、生态系统功能在模型中的表达,构建综合连续和离散模型各自优点的混合模型,注重斑块状植被格局形成过程中的标准子模型及模型开发和集成平台的研发,同时强调面向格局模拟和构建空间显式的斑块状植被格局形成模型。     

Surface topology assisted alignment of Min protein waves

Self-organization of proteins into large-scale structures is of pivotal importance for the organization of cells. The Min protein system of the bacterium Escherichia coli is a prime example of how pattern formation occurs via reaction–diffusion. We have previously demonstrated how Min protein patterns are influenced by compartment geometry. Here we probe the influence of membrane surface topology, as an additional regulatory element. Using microstructured membrane-clad soft polymer substrates, Min protein patterns can be aligned. We demonstrate that Min pattern alignment starts early during pattern formation and show that macroscopic millimeter-sized areas of protein patterns of well-defined orientation can be generated.     

Pattern selection in a cooperative biochemical in vitro amplification system: the role of parasites

Previously, numerical simulations have shown that evolving systems can be stabilized against emerging parasites by pattern formation in spatially extended flow reactors. Hence, it can be argued that pattern formation is a prerequisite for any experimental investigation of the biochemical evolution of cooperative function. Here, we study a model of an experimental biochemical system for the cooperative in vitro amplification of DNA strands and show that emerging parasites can induce a complex pattern formation even when no pattern formation occurs without parasites. In an adiabatic approximation where the cooperative amplification reaction is assumed to adapt fast to slowly emerging parasites, the parasite concentration itself acts as a Steuer parameter for the selection of various complex patterns. Without such an adiabatic approximation only transient patterns emerge. As any species can grow for very low concentrations, the parasite is able to infect the entire reactor and the system is finally diluted out. In the experimental biochemical system, however, the species are individual molecules and the growth of spatially separated, non-infected regions becomes feasible. Hence a cutoff threshold for the minimal concentration is applied. In these simulations the otherwise lethal infection by parasites induces the formation of spatiotemporal spirals, and this spatial structure help the host and parasitoid species to survive together. These theoretical results describe an inherent property of cooperative reactions and have an important impact on experimental investigations on the molecular evolution and complex function in spatially extended reactors. Since the formation of the complex pattern is restricted either to a rather large cutoff value or a special choice of the kinetic parameters, we, however, conclude that the persistence of evolving cooperative amplification is not possible in a simple reaction-diffusion reactor. Experimental set-ups with patchy environments, e.g. biomolecular amplification in coupled microstructured flow chambers or in microemulsion, are eligible candidates for the observation of such a self-organized pattern selection.     

Avoiding toxic prey may promote harmful algal blooms

Blooms of freshwater cyanobacteria are a worldwide spread environmental issue. Despite toxin producing planktonic species are generally expected to be poor competitors for resources, dense blooms of toxic cyanobacteria, such as Microcystis, do often occur in nature. Experimental results suggest that the formation of such blooms is promoted by the predatory activity of zooplankton. In fact, such predator grazes on both the nontoxic and toxic species despite the toxicity of the latter actually inhibits its growth. We model this phenomenon through a Lotka–Volterra reaction–diffusion system. Our goal is to investigate the coupled role of toxicity and zooplankton's predation in the persistence of the toxic prey and to study the mechanisms behind the formation of spatially local toxic blooms. It is known that the classical Lotka-Volterra system consisting of one prey and one predator never exhibits pattern formation. In this paper, we show that the introduction of a toxic prey may destabilize the spatially homogeneous coexistence and trigger spatial pattern formation. We also show that local blooms more likely occur when predators avoid the toxic prey and when the strength of the toxicity is of an intermediate level.     

Early tetrapodomorph biogeography: Controlling for fossil record bias in macroevolutionary analyses

The fossil record provides direct empirical data for understanding macroevolutionary patterns and processes. Inherent biases in the fossil record are well known to confound analyses of this data. Sampling bias proxies have been used as covariates in regression models to test for such biases. Proxies, such as formation count, are associated with paleobiodiversity, but are insufficient for explaining species dispersal owing to a lack of geographic context. Here, we develop a sampling bias proxy that incorporates geographic information and test it with a case study on early tetrapodomorph biogeography. We use recently-developed Bayesian phylogeographic models and a new supertree of early tetrapodomorphs to estimate dispersal rates and ancestral habitat locations. We find strong evidence that geographic sampling bias explains supposed radiations in dispersal rate (potential adaptive radiations). Our study highlights the necessity of accounting for geographic sampling bias in macroevolutionary and phylogenetic analyses and provides an approach to test for its effect.