Diffusion driven instability in an inhomogeneous domain

Diffusion driven instability in reaction-diffusion systems has been proposed as a mechanism for pattern formation in numerous embryological and ecological contexts. However, the possible effects of environmental inhomogeneities has received relatively little attention. We consider a general two species reaction-diffusion model in one space dimension, with one diffusion coefficient a step function of the spatial coordinate. We derive the dispersion relation and the solution of the linearized system. We apply our results to Turing-type models for both embryogenesis and predator-prey interactions. In the former case we derive conditions for pattern to be isolated in one part of the domain, and in the latter we introduce the concept of “environmental instability”. Our results suggest that environmental inhomogeneity could be an important regulator of biological pattern formation.     

Quasi-Local Competition in Stage-Structured Metapopulations: A New Mechanism of Pattern Formation

A central question of ecology is what determines the presence and abundance of species at different locations. In cases of ecological pattern formation, population sizes are largely determined by spatially distributed interactions and may have very little to do with the habitat template. We find pattern formation in a single-species metapopulation model with quasi-local competition, but only if the populations have (at least) two age or stage classes. Quasi-local competition is modeled using an explicit resource competition model with fast resource dynamics, and assuming that adults, but not juveniles, spend a fraction of their foraging time in habitat patches adjacent to their home patch. Pattern formation occurs if one stage class depletes the common resource but the shortage of resource affects mostly the other stage. When the two stages are spatially separated due to quasi-local competition, this results in competitive exclusion between the populations. We find deep similarity between spatial pattern formation and population cycles due to competitive exclusion between cohorts of biennial species, and discuss the differences between the present mechanism and established ways of pattern formation such as diffusive instability and distributed competition with local Allee-effects.     

A new assembly pathway for the cytokinetic Z ring from a dynamic helical structure in vegetatively growing cells of Bacillus subtilis

The earliest event in bacterial cell division is the formation of a Z ring, composed of the tubulin-like FtsZ protein, at the division site at midcell. This ring then recruits several other division proteins and together they drive the formation of a division septum between two replicated chromosomes. Here we show that, in addition to forming a cytokinetic ring, FtsZ localizes in a helical-like pattern in vegetatively growing cells of Bacillus subtilis. FtsZ moves rapidly within this helix-like structure. Examination of FtsZ localization in individual live cells undergoing a single cell cycle suggests a new assembly mechanism for Z ring formation that involves a cell cycle-mediated multistep remodelling of FtsZ polymers. Our observations suggest that initially FtsZ localizes in a helical pattern, with movement of FtsZ within this structure occurring along the entire length of the cell. Next, movement of FtsZ in a helical-like pattern is restricted to a central region of the cell. Finally the FtsZ ring forms precisely at midcell. We further show that another division protein, FtsA, shown to interact with FtsZ prior to Z ring formation in B. subtilis, also localizes to similar helical patterns in vegetatively growing cells.     

Pigmentation pattern formation on snakes

We consider a cell-chemotaxis model mechanism for generating some of the common, simple and complex, patterns found on the skin of snakes. By investigating the pattern generation potential of the model we show that many of the more complex patterns might result from growth of the integument during the pattern formation process. We suggest that many of the diverse elaborate patterns on snakes, and other species, can be generated by a single mechanism if the time scale of the pattern process is commensurate with the time scale associated with significant embryonic growth.     

Instability in a generalized Keller-Segel model

We present a generalized Keller-Segel model where an arbitrary number of chemical compounds react, some of which are produced by a species, and one of which is a chemoattractant for the species. To investigate the stability of homogeneous stationary states of this generalized model, we consider the eigenvalues of a linearized system. We are able to reduce this infinite dimensional eigenproblem to a parametrized finite dimensional eigenproblem. By matrix theoretic tools, we then provide easily verifiable sufficient conditions for destabilizing the homogeneous stationary states. In particular, one of the sufficient conditions is that the chemotactic feedback is sufficiently strong. Although this mechanism was already known to exist in the original Keller-Segel model, here we show that it is more generally applicable by significantly enlarging the class of models exhibiting this instability phenomenon which may lead to pattern formation.     

Instability in a generalized Keller–Segel model

We present a generalized Keller–Segel model where an arbitrary number of chemical compounds react, some of which are produced by a species, and one of which is a chemoattractant for the species. To investigate the stability of homogeneous stationary states of this generalized model, we consider the eigenvalues of a linearized system. We are able to reduce this infinite dimensional eigenproblem to a parametrized finite dimensional eigenproblem. By matrix theoretic tools, we then provide easily verifiable sufficient conditions for destabilizing the homogeneous stationary states. In particular, one of the sufficient conditions is that the chemotactic feedback is sufficiently strong. Although this mechanism was already known to exist in the original Keller–Segel model, here we show that it is more generally applicable by significantly enlarging the class of models exhibiting this instability phenomenon which may lead to pattern formation.     

Reaction-diffusion-advection model for pattern formation of rhythmic contraction in a giant amoeboid cell of the physarum plasmodium

The plasmodium of Physarum polycephalum is a large amoeboid organism showing rhythmic contraction everywhere within an organism, and moves by forming spatio-temporal patterns of the rhythmic contraction. We propose a reaction-diffusion-advection model for the pattern formation. This model is constructed under physiological suggestions that the chemical oscillator acts as a clock regulating the rhythmic contraction and interacts spatially not only by diffusion but also by advection of protoplasm. Behavior of the model is studied by numerical calculation, especially the effects of the advection term on a simple reaction-diffusion system. The advection effect reproduces experimentally observed phenomena of fluctuating propagation of the contraction wave. Concept of the reaction-diffusion-advection system is promising for modeling the mechanism of amoeboid behaviour in the Physarum plasmodium. Copyright 1999 Academic Press.     

Reaction and diffusion on growing domains: Scenarios for robust pattern formation

We investigate the sequence of patterns generated by a reaction—diffusion system on a growing domain. We derive a general evolution equation to incorporate domain growth in reaction—diffusion models and consider the case of slow and isotropic domain growth in one spatial dimension. We use a self-similarity argument to predict a frequency-doubling sequence of patterns for exponential domain growth and we find numerically that frequency-doubling is realized for a finite range of exponential growth rate. We consider pattern formation under different forms for the growth and show that in one dimension domain growth may be a mechanism for increased robustness of pattern formation.     

Phase Transitions in a Logistic Metapopulation Model with Nonlocal Interactions

The presence of one or more species at some spatial locations but not others is a central matter in ecology. This phenomenon is related to ecological pattern formation. Nonlocal interactions can be considered as one of the mechanisms causing such a phenomenon. We propose a single-species, continuous time metapopulation model taking nonlocal interactions into account. Discrete probability kernels are used to model these interactions in a patchy environment. A linear stability analysis of the model shows that solutions to this equation exhibit pattern formation if the dispersal rate of the species is sufficiently small and the discrete interaction kernel satisfies certain conditions. We numerically observe that traveling and stationary wave-type patterns arise near critical dispersal rate. We use weakly nonlinear analysis to better understand the behavior of formed patterns. We show that observed patterns arise through both supercritical and subcritical bifurcations from spatially homogeneous steady state. Moreover, we observe that as the dispersal rate decreases, amplitude of the patterns increases. For discontinuous transitions to instability, we also show that there exists a threshold for the amplitude of the initial condition, above which pattern formation is observed.     

Automata with hierarchical control and evolutionary learning

We propose an automata-theoretical framework for structured hierarchical control, in terms of rules and meta-rules, for sequences of moves on a graph. This leads to a notion of a "universal" hierarchically structured automaton mu which can move on a given graph in such a way as to emulate any automaton which moves on that graph in response to inputs. This emulation is achieved via a mapping of the inputs in the given automaton to those of mu, and we think of such a mapping as an encoding of the given automaton. We see in several examples that efficient encodings of graph-search algorithms correspond to their natural hierarchical structure (in terms of rules and meta-rules), and this leads one to a precise notion of the "depth" of an automaton which moves on a given graph. By way of application, we discuss a proposed structure of a series of stochastic neural networks which can learn, by example, to encode a given sequence of moves on a graph, so that the encoding obtained is structurally the "natural" one for the given sequence of moves. Thus, such a learning system would perform both structural pattern recognition (in terms of "patterns" of moves), and encoding based on a desired outcome.     

Chemically Based Mathematical Model for Development of Cerebral Cortical Folding Patterns

The mechanism for cortical folding pattern formation is not fully understood. Current models represent scenarios that describe pattern formation through local interactions, and one recent model is the intermediate progenitor model. The intermediate progenitor (IP) model describes a local chemically driven scenario, where an increase in intermediate progenitor cells in the subventricular zone correlates to gyral formation. Here we present a mathematical model that uses features of the IP model and further captures global characteristics of cortical pattern formation. A prolate spheroidal surface is used to approximate the ventricular zone. Prolate spheroidal harmonics are applied to a Turing reaction-diffusion system, providing a chemically based framework for cortical folding. Our model reveals a direct correlation between pattern formation and the size and shape of the lateral ventricle. Additionally, placement and directionality of sulci and the relationship between domain scaling and cortical pattern elaboration are explained. The significance of this model is that it elucidates the consistency of cortical patterns among individuals within a species and addresses inter-species variability based on global characteristics and provides a critical piece to the puzzle of cortical pattern formation.     

Interspecific influence on mobility and Turing instability

In this paper we formulate a multi-patch multi-species model in which the percapita emigration rate of one species depends on the density of some other species. We then focus on Turing instability to examine if and when this cross-emigration response has crucial effects. We find that the type of interaction matters greatly. In the case of competition a cross-emigration response promotes pattern formation by exercising a destabilizing influence; in particular, it may lead to diffusive instability provided that the response is sufficiently strong, which contrasts sharply with the well-known fact that the standard competition system does not exhibit Turing instability. In the case of prey-predator or activator-inhibitor interaction it acts against pattern formation by exerting a stabilizing effect; in particular, the diffusive instability, even though it may happen in a standard system, never occurs when the response is sufficiently strong. We conclude that the cross-emigration response is an important factor that should not be ignored when pattern formation is the issue.     

A role for a novel centrosome cycle in asymmetric cell division

Tissue stem cells play a key role in tissue maintenance. Drosophila melanogaster central brain neuroblasts are excellent models for stem cell asymmetric division. Earlier work showed that their mitotic spindle orientation is established before spindle formation. We investigated the mechanism by which this occurs, revealing a novel centrosome cycle. In interphase, the two centrioles separate, but only one is active, retaining pericentriolar material and forming a "dominant centrosome." This centrosome acts as a microtubule organizing center (MTOC) and remains stationary, forming one pole of the future spindle. The second centriole is inactive and moves to the opposite side of the cell before being activated as a centrosome/MTOC. This is accompanied by asymmetric localization of Polo kinase, a key centrosome regulator. Disruption of centrosomes disrupts the high fidelity of asymmetric division. We propose a two-step mechanism to ensure faithful spindle positioning: the novel centrosome cycle produces a single interphase MTOC, coarsely aligning the spindle, and spindle-cortex interactions refine this alignment.     

Pattern Selection by Dynamical Biochemical Signals

The development of multicellular organisms involves cells to decide their fate upon the action of biochemical signals. This decision is often spatiotemporally coordinated such that a spatial pattern arises. The dynamics that drive pattern formation usually involve genetic nonlinear interactions and positive feedback loops. These complex dynamics may enable multiple stable patterns for the same conditions. Under these circumstances, pattern formation in a developing tissue involves a selection process: why is a certain pattern formed and not another stable one? Herein we computationally address this issue in the context of the Notch signaling pathway. We characterize a dynamical mechanism for developmental selection of a specific pattern through spatiotemporal changes of the control parameters of the dynamics, in contrast to commonly studied situations in which initial conditions and noise determine which pattern is selected among multiple stable ones. This mechanism can be understood as a path along the parameter space driven by a sequence of biochemical signals. We characterize the selection process for three different scenarios of this dynamical mechanism that can take place during development: the signal either 1) acts in all the cells at the same time, 2) acts only within a cluster of cells, or 3) propagates along the tissue. We found that key elements for pattern selection are the destabilization of the initial pattern, the subsequent exploration of other patterns determined by the spatiotemporal symmetry of the parameter changes, and the speeds of the path compared to the timescales of the pattern formation process itself. Each scenario enables the selection of different types of patterns and creates these elements in distinct ways, resulting in different features. Our approach extends the concept of selection involved in cellular decision-making, usually applied to cell-autonomous decisions, to systems that collectively make decisions through cell-to-cell interactions.     

A novel mechanism for the formation of actin-filament bundles by a nonprocessive formin

Actin-filament bundles (or cables) have a structural role during cell division and morphogenesis, but also serve as important "tracks" for the transport of materials during cytokinesis and polarized cell growth. However, the dynamic formation of these longitudinal actin-filament higher-order structures is not understood. Recently, several lines of evidence suggest that formins provide one avenue for the initiation of actin cables in vivo. A popular model for the mechanism of polymerization of actin filaments by formin involves the processive movement of formin attached at the barbed end of an elongating filament. In the present study, we use an in vitro system to reconstitute the dynamic formation of actin-filament bundles generated by Arabidopsis FORMIN1 (AFH1). To be able to visualize individual events in such a complex system, we used real-time evanescent-wave microscopy. Surprisingly, we find that AFH1 is a nonprocessive formin that moves from the barbed end to the side of an actin filament after the nucleation event. We show why this new mechanism of nucleation by a member of the formin family is important for bundle formation. Finally, we analyze the different parameters controlling the dynamic formation of such longitudinal actin-filament bundles.     

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.     

Polyploidy induces centromere association

Many species exhibit polyploidy. The presence of more than one diploid set of similar chromosomes in polyploids can affect the assortment of homologous chromosomes, resulting in unbalanced gametes. Therefore, a mechanism is required to ensure the correct assortment and segregation of chromosomes for gamete formation. Ploidy has been shown to affect gene expression. We present in this study an example of a major effect on a phenotype induced by ploidy within the Triticeae. We demonstrate that centromeres associate early during anther development in polyploid species. In contrast, centromeres in diploid species only associate at the onset of meiotic prophase. We propose that this mechanism provides a potential route by which chromosomes can start to be sorted before meiosis in polyploids. This explains previous reports indicating that meiotic prophase is shorter in polyploids than in their diploid progenitors. Even artificial polyploids exhibit this phenotype, suggesting that the mechanism must be present in diploids, but only expressed in the presence of more than one diploid set of chromosomes.     

欧亚大陆东部毛茛科植物多样性格局及主导因子

毛茛科是真双子叶植物的基部类群之一, 包含多种药用植物, 具有较高的保护价值, 但关于毛茛科物种多样性和谱系多样性大尺度格局及其影响因子的研究还比较匮乏, 特别是以较高分辨率分布数据为基础的物种多样性格局研究尚未见报道。本文旨在: (1)建立欧亚大陆东部毛茛科植物分布数据库, 估算不同生活型物种多样性和谱系多样性格局, 并探究格局的形成机制。(2)分析毛茛科物种多样性和谱系多样性的相关关系, 确定多样性热点地区, 为毛茛科保护规划提供依据。根据中国、哈萨克斯坦、吉尔吉斯斯坦、塔吉克斯坦、土库曼斯坦、乌兹别克斯坦、蒙古和俄罗斯等国家的区域和地方植物志, 建立了“欧亚大陆东部地区毛茛科物种分布数据库”。该数据库包含了欧亚大陆东部地区1,688种毛茛科物种的分布数据, 空间分辨率为100 km × 100 km。在此基础上, 估算了毛茛科全部及不同生活型的物种多样性和谱系多样性格局, 并利用广义线性模型和等级方差分离方法分析了毛茛科物种和谱系多样性格局与环境因子的关系。最后比较了物种多样性和谱系多样性的相关关系, 确定了毛茛科的古热点地区。结果显示: (1)欧亚大陆东部毛茛科植物物种和谱系多样性均呈明显的纬度格局, 且在山区具有较高的多样性。(2)毛茛科植物物种和谱系多样性受现代气候、地形异质性和末次冰期以来的气候变化的共同影响, 但不同影响因子的相对贡献率在物种和谱系多样性及不同生活型之间差异显著。(3)中高纬度地区的谱系多样性高于给定物种数的预期, 是毛茛科的古热点地区, 在毛茛科保护规划中应受到重视。     

A model for the prestalk/prespore patterning in the slug of the slime mold Dictyostelium discoideum

Abstract. We propose that the prestalk/prespore pattern in Dictyostelium is generated in two steps: In a first process, an intermingled, non-position dependent prestalk/prespore pattern is generated by a cell-restricted autocatalysis and the antagonistic action of a long-ranging substrate which becomes depleted during this autocatalysis. By computer simulations we show that the assumed interaction accounts for several experimentally observed features of the prestalk/ prespore pattern: The size-independent ratio of both cell types, the pattern regulation after removal of one cell type, the development towards one or the other pathway before the slug obtains its final shape or even before aggregation is completed. Our hypothetical substrate may be identical with an experimentally found differentiation-inducing factor (DIF). Alternative molecular realizations of the basic mechanism are discussed. A second process leads to the aggregation of the prestalk cells in a particular region of the aggregate, the future tip region. Interactions which en-able tip formation and the coupling between the prestalk/prespore and the tip-forming system are discussed. Our model shows that the formation of a single large patch of differentiated cells and its size regulation requires conflicting parameters. By a separation into a mechanism which determines the position and a second one which determines the size of a structure, each mechanism can be optimized individually without requiring compromises for the other. Such a separation also seems to occur in other developmental systems.     

Universal ecological patterns in college basketball communities

The rank abundance of common and rare species within ecological communities is remarkably consistent from the tropics to the tundra. This invariant patterning provides one of ecology's most enduring and unified tenets: most species rare and a few very common. Increasingly, attention is focused upon elucidating biological mechanisms that explain these species abundance distributions (SADs), but these evaluations remain controversial. We show that college basketball wins generate SADs just like those observed in ecological communities. Whereas college basketball wins are structured by competitive interactions, the result produces a SAD pattern indistinguishable from random wins. We also show that species abundance data for tropical trees exhibits a significant-digit pattern consistent with data derived from complex structuring forces. These results cast doubt upon the ability of SAD analysis to resolve ecological mechanism, and their patterning may reflect statistical artifact as much as biological processes.