Cellular oscillators in animal segmentation

Kinetic modeling of developmental dynamics requires detailed knowledge about genetic and metabolic networks that underlie developmental processes. However, such knowledge is not available for a vast majority of developmental processes. Here, we present an coarse-grained, phenomenological model of periodic pattern formation in multicellular organisms based on cellular oscillators (CO) that can be applied to systems for which little or no molecular data is available. An oscillatory process within cells serves as a developmental clock whose period is tightly regulated by cell-autonomous and non-autonomous mechanisms. A spatial pattern is generated as a result of an initial temporal ordering of the cell oscillators freezing into spatial order as the clocks slow down and stop at different times or phases in their cycles. When applied to vertebrate somitogenesis, the CO model can reproduce the dynamics of periodic gene expression patterns observed in the presomitic mesoderm. Different somite lengths can be generated by altering the period of the oscillation. There is evidence that a CO-type mechanism might also underlie segment formation in certain invertebrates, such as annelids and short germ insects. This suggests that the dynamical principles of sequential segmentation might be equivalent throughout the animal kingdom although most of the genes involved in segment determination differ between distant phyla.     

Evaluation of the Oscillatory Interference Model of Grid Cell Firing through Analysis and Measured Period Variance of Some Biological Oscillators

Models of the hexagonally arrayed spatial activity pattern of grid cell firing in the literature generally fall into two main categories: continuous attractor models or oscillatory interference models. Burak and Fiete (2009, PLoS Comput Biol) recently examined noise in two continuous attractor models, but did not consider oscillatory interference models in detail. Here we analyze an oscillatory interference model to examine the effects of noise on its stability and spatial firing properties. We show analytically that the square of the drift in encoded position due to noise is proportional to time and inversely proportional to the number of oscillators. We also show there is a relatively fixed breakdown point, independent of many parameters of the model, past which noise overwhelms the spatial signal. Based on this result, we show that a pair of oscillators are expected to maintain a stable grid for approximately t = 5µ ³ /(4πσ) ² seconds where µ is the mean period of an oscillator in seconds and σ² its variance in seconds². We apply this criterion to recordings of individual persistent spiking neurons in postsubiculum (dorsal presubiculum) and layers III and V of entorhinal cortex, to subthreshold membrane potential oscillation recordings in layer II stellate cells of medial entorhinal cortex and to values from the literature regarding medial septum theta bursting cells. All oscillators examined have expected stability times far below those seen in experimental recordings of grid cells, suggesting the examined biological oscillators are unfit as a substrate for current implementations of oscillatory interference models. However, oscillatory interference models can tolerate small amounts of noise, suggesting the utility of circuit level effects which might reduce oscillator variability. Further implications for grid cell models are discussed.     

How Does Cellular Contact Affect Differentiation Mediated Pattern Formation?

In this paper, we present a two-population continuous integro-differential model of cell differentiation, using a non-local term to describe the influence of the local environment on differentiation. We investigate three different versions of the model, with differentiation being cell autonomous, regulated via a community effect, or weakly dependent on the local cellular environment. We consider the spatial patterns that such different modes of differentiation produce, and investigate the formation of both stripes and spots by the model. We show that pattern formation only occurs when differentiation is regulated by a strong community effect. In this case, permanent spatial patterns only occur under a precise relationship between the parameters characterising cell dynamics, although transient patterns can persist for biologically relevant timescales when this condition is relaxed. In all cases, the long-lived patterns consist only of stripes, not spots.     

Temporal and spatial oscillations in bacteria

Oscillations pervade biological systems at all scales. In bacteria, oscillations control fundamental processes, including gene expression, cell cycle progression, cell division, DNA segregation and cell polarity. Oscillations are generated by biochemical oscillators that incorporate the periodic variation in a parameter over time to generate an oscillatory output. Temporal oscillators incorporate the periodic accumulation or activity of a protein to drive temporal cycles such as the cell and circadian cycles. Spatial oscillators incorporate the periodic variation in the localization of a protein to define subcellular positions such as the site of cell division and the localization of DNA. In this Review, we focus on the mechanisms of oscillators and discuss the design principles of temporal and spatial oscillatory systems.     

Radial and spiral stream formation in Proteus mirabilis colonies

The enteric bacterium Proteus mirabilis, which is a pathogen that forms biofilms in vivo, can swarm over hard surfaces and form a variety of spatial patterns in colonies. Colony formation involves two distinct cell types: swarmer cells that dominate near the surface and the leading edge, and swimmer cells that prefer a less viscous medium, but the mechanisms underlying pattern formation are not understood. New experimental investigations reported here show that swimmer cells in the center of the colony stream inward toward the inoculation site and in the process form many complex patterns, including radial and spiral streams, in addition to previously-reported concentric rings. These new observations suggest that swimmers are motile and that indirect interactions between them are essential in the pattern formation. To explain these observations we develop a hybrid model comprising cell-based and continuum components that incorporates a chemotactic response of swimmers to a chemical they produce. The model predicts that formation of radial streams can be explained as the modulation of the local attractant concentration by the cells, and that the chirality of the spiral streams results from a swimming bias of the cells near the surface of the substrate. The spatial patterns generated from the model are in qualitative agreement with the experimental observations.     

Pattern formation of leaf veins by the positive feedback regulation between auxin flow and auxin efflux carrier

The generation of vascular pattern formation in plants is an interesting process of pattern formation in organisms. It is well known that the plant hormone auxin is involved in plant vascular differentiation and that the PIN1 protein, an auxin efflux carrier, localizes to one side of the cell membrane. Several hypotheses have been proposed to explain the formation of leaf venation. One is the canalization hypothesis that is based on the assumption that a positive feedback regulation exists between the flow of a signal molecule and the capacity of its flow. Here, we attempted to integrate the canalization hypothesis and experimental data. We investigated models of the positive feedback regulation between the auxin flow and PIN1 localization. Model 1, with conserved PIN1 amount in each cell, can generate a branching pattern similar to that of plant leaf venation. We introduced the diffusible enhancer "e" into the model as unknown factor. The obtained patterns show a quasi-periodic distribution of auxin flow paths, when the model dynamics includes domain growth. In order to understand the early initiation process that generates an inhomogeneity from an almost homogeneous distribution, we introduced model 2, a simplified version of model 1. Model 2 can generate inhomogeneity with a parameter dependency similar to that of model 1. To analyse parameter condition required for pattern development, approximated equations are obtained from model 2. The isocline analysis of the equations without spatial structure shows that the inhomogeneous distribution occurs from an almost homogeneous distribution. This parameter condition for generating inhomogeneity is consistent with the results of models 1 and 2.     

Oscillatory neural network model of attention focus formation and control

A new mechanism to control attention focus formation and switching in the model of selective attention is suggested and studied. The model is based on an oscillatory neural network (ONN) with the star-like architecture and phase shifts in connections between oscillators. Attention is modelled as a dynamical mode of partial synchronisation between a particular subgroup of oscillators and the central oscillator (CO). A new theoretical method to study full and partial synchronisation in the system is presented. Equations for the frequency of synchronisation are derived which allow the programming of the dynamical behaviour of the system depending on the parameters. In particular, we show that phase shifts in connections between oscillators provide an efficient mechanism of attention control.     

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.     

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.     

When are cellular oscillators sufficient for sequential segmentation?

Sequential segmentation during embryogenesis involves the generation of a repeated pattern along the embryo, which is concurrently undergoing axial elongation by cell division. Most mathematical models of sequential segmentation involve inherent cellular oscillators, acting as a segmentation clock. The cellular oscillation is assumed to be governed by the cell's physiological age or by its interaction with an external morphogen gradient. Here, we address the issue of when cellular oscillators alone are sufficient for predicting segmentation, and when a morphogen gradient is required. The key to resolving this issue lies in how cells determine positional information in the model - this is directly related to the distribution of cell divisions responsible for axial elongation. Mathematical models demonstrate that if axial elongation occurs through cell divisions restricted to the posterior end of the unsegmented region, a cell can obtain its positional information from its physiological age, and therefore cellular oscillators will suffice. Alternatively, if axial elongation occurs through cell divisions distributed throughout the unsegmented region, then positional information can be obtained through another mechanism, such as a morphogen gradient. Two alternative ways to establish a morphogen gradient in tissue with distributed cell divisions are presented - one with diffusion and the other without diffusion. Our model produces segment polarity and a distribution of segment size from the anterior-to-posterior ends, as observed in some systems. Furthermore, the model predicts segment deletions when there is an interruption in cell division, just as seen in heat shock experiments, as well as the growth and final shrinkage of the presomitic mesoderm during somitogenesis.     

Spatial phase pattern in one-dimensional arrays of limit cycle oscillators with discrete coupling

We present a model of limit cycle oscillators for collective oscillations in intracellular calcium concentration in cell communities. A phase-dependent discrete coupling between nearest neighbors is introduced into the model on the basis of the experimental observation that intercellular transmission of calcium or calcium mobilizing messenger is effected by gap junction and gap junctional permeability is affected by intracellular calcium concentration. The spatial phase pattern of several clusters in which oscillations are in phase is found with the phase-dependent discrete coupling.     

Density of founder cells affects spatial pattern formation and cooperation in Bacillus subtilis biofilms

In nature, most bacteria live in surface-attached sedentary communities known as biofilms. Biofilms are often studied with respect to bacterial interactions. Many cells inhabiting biofilms are assumed to express ‘cooperative traits'', like the secretion of extracellular polysaccharides (EPS). These traits can enhance biofilm-related properties, such as stress resilience or colony expansion, while being costly to the cells that express them. In well-mixed populations cooperation is difficult to achieve, because non-cooperative individuals can reap the benefits of cooperation without having to pay the costs. The physical process of biofilm growth can, however, result in the spatial segregation of cooperative from non-cooperative individuals. This segregation can prevent non-cooperative cells from exploiting cooperative neighbors. Here we examine the interaction between spatial pattern formation and cooperation in Bacillus subtilis biofilms. We show, experimentally and by mathematical modeling, that the density of cells at the onset of biofilm growth affects pattern formation during biofilm growth. At low initial cell densities, co-cultured strains strongly segregate in space, whereas spatial segregation does not occur at high initial cell densities. As a consequence, EPS-producing cells have a competitive advantage over non-cooperative mutants when biofilms are initiated at a low density of founder cells, whereas EPS-deficient cells have an advantage at high cell densities. These results underline the importance of spatial pattern formation for competition among bacterial strains and the evolution of microbial cooperation.     

Forest gap dynamics and the Ising model

The vegetation height in forest ecosystems is spatially heterogeneous. Canopy gaps (sites with low vegetation) are formed by treefalls, and they recover to canopy sites (with high vegetation) either by growth of small trees or by branch extension of surrounding trees. The dynamics of canopy gaps have been studied using a spatial Markov chain with nearest neighbor interaction. (1) If the canopy recovery rate is constant and if the gap formation rate for a site increases exponentially with the number of neighboring gap sites, the equilibrium distribution is the same as the one generated by the Ising model in statistical mechanics. Here, we extend the equivalence to the situation in which both the gap formation and canopy recovery depend on the neighborhood, as shown in recent forest data. (2) We develop a statistical test of whether a given spatial pattern is a random sample from the Ising model. The test is based on the conditional probability of configurations on a partial lattice. We apply the method to vegetation height data from the Ogawa forest reserve, Japan, measured on a 5x5 m grid in 1976, 1981, 1986, and 1991. The spatial pattern of the original forest data deviates significantly from the Ising model. We examine whether a larger sampling distance or the removal of the effects of the topography can reduce this deviation.     

Turing Pattern Formation with Two Kinds of Cells and a Diffusive Chemical

We study a three-variable Turing system with two kinds of cells and a diffusive chemical, considering the formation and maintenance of fish skin patterns with multiple pigment cells. The two types of cells are produced from undifferentiated cells. They inhibit the supply rate of the other cell type, forming local clusters of the same cell type. In addition, the cells of one type can be maintained only in the presence of a diffusive chemical produced by the other cell type, resulting in the coexistence of two cell types in heterogeneous spatial patterns. We assume linear interaction among cells and the chemical, and cell supply rates constrained to be positive or zero. We derive the condition for diffusion-driven instability. In one-dimensional model, we examine how the wavelength of the periodic pattern depends on parameters. In the two-dimensional model, we study the condition for spot, stripe or reversed spot pattern to emerge (pattern selection). We discuss heuristic criteria for the pattern selection.     

Multi-Stability and Pattern-Selection in Oscillatory Networks with Fast Inhibition and Electrical Synapses

A model or hybrid network consisting of oscillatory cells interconnected by inhibitory and electrical synapses may express different stable activity patterns without any change of network topology or parameters, and switching between the patterns can be induced by specific transient signals. However, little is known of properties of such signals. In the present study, we employ numerical simulations of neural networks of different size composed of relaxation oscillators, to investigate switching between in-phase (IP) and anti-phase (AP) activity patterns. We show that the time windows of susceptibility to switching between the patterns are similar in 2-, 4- and 6-cell fully-connected networks. Moreover, in a network (N = 4, 6) expressing a given AP pattern, a stimulus with a given profile consisting of depolarizing and hyperpolarizing signals sent to different subpopulations of cells can evoke switching to another AP pattern. Interestingly, the resulting pattern encodes the profile of the switching stimulus. These results can be extended to different network architectures. Indeed, relaxation oscillators are not only models of cellular pacemakers, bursting or spiking, but are also analogous to firing-rate models of neural activity. We show that rules of switching similar to those found for relaxation oscillators apply to oscillating circuits of excitatory cells interconnected by electrical synapses and cross-inhibition. Our results suggest that incoming information, arriving in a proper time window, may be stored in an oscillatory network in the form of a specific spatio-temporal activity pattern which is expressed until new pertinent information arrives.     

Simple models for excitable and oscillatory neural networks

 Chains of coupled oscillators of simple “rotator” type have been used to model the central pattern generator (CPG) for locomotion in lamprey, among numerous applications in biology and elsewhere. In this paper, motivated by experiments on lamprey CPG with brainstem attached, we investigate a simple oscillator model with internal structure which captures both excitable and bursting dynamics. This model, and that for the coupling functions, is inspired by the Hodgkin–Huxley equations and two-variable simplifications thereof. We analyse pairs of coupled oscillators with both excitatory and inhibitory coupling. We also study traveling wave patterns arising from chains of oscillators, including simulations of “body shapes” generated by a double chain of oscillators providing input to a kinematic musculature model of lamprey.. Received: 25 November 1996 / Revised version: 9 December 1997     

Indirect effects of cattle grazing on shrub spatial pattern in a mediterranean scrub community

Identifying the mechanisms and interactions that influence the spatial structure of vegetation is important for both scientific and practical purposes. Grazing is one of the most fundamental interactions in ecology but so far its effect on vegetation spatial pattern received little attention. In this study we propose a conceptual model that can be used to predict the effect of grazing on shrub spatial pattern in water-limited ecosystems where shrubs grow within a matrix of annual vegetation. According to the model, grazing may increase or decrease clumping in shrub distribution, depending on (1) the relative palatability of shrubs vs. annual plants to the herbivores, and (2) the manner (negative or positive) by which adult shrubs and annual plants affect the establishment of shrub seedlings. We tested our model in a Mediterranean scrub ecosystem by analyzing the development of shrub spatial pattern over a period of 40 years in plots characterized by contrasting intensities of cattle grazing. As predicted by the model, all plots showed a clumped pattern of shrub distribution in the absence of cattle grazing while intense cattle grazing reduced the clumpiness of the vegetation and generated a more random pattern of shrub distribution. Interestingly, plots representing the two grazing regimes did not differ significantly in their shrub cover, suggesting that shrub spatial pattern may be more sensitive to grazing than overall shrub cover.     

Spatial pattern switching enables cyclic evolution in spatial epidemics

Infectious diseases often spread as spatial epidemic outbreak waves. A number of model studies have shown that such spatial pattern formation can have important consequences for the evolution of pathogens. Here, we show that such spatial patterns can cause cyclic evolutionary dynamics in selection for the length of the infectious period. The necessary reversal in the direction of selection is enabled by a qualitative change in the spatial pattern from epidemic waves to irregular local outbreaks. The spatial patterns are an emergent property of the epidemic system, and they are robust against changes in specific model assumptions. Our results indicate that emergent spatial patterns can act as a rich source for complexity in pathogen evolution.     

Reaction-diffusion pattern in shoot apical meristem of plants

A fundamental question in developmental biology is how spatial patterns are self-organized from homogeneous structures. In 1952, Turing proposed the reaction-diffusion model in order to explain this issue. Experimental evidence of reaction-diffusion patterns in living organisms was first provided by the pigmentation pattern on the skin of fishes in 1995. However, whether or not this mechanism plays an essential role in developmental events of living organisms remains elusive. Here we show that a reaction-diffusion model can successfully explain the shoot apical meristem (SAM) development of plants. SAM of plants resides in the top of each shoot and consists of a central zone (CZ) and a surrounding peripheral zone (PZ). SAM contains stem cells and continuously produces new organs throughout the lifespan. Molecular genetic studies using Arabidopsis thaliana revealed that the formation and maintenance of the SAM are essentially regulated by the feedback interaction between WUSHCEL (WUS) and CLAVATA (CLV). We developed a mathematical model of the SAM based on a reaction-diffusion dynamics of the WUS-CLV interaction, incorporating cell division and the spatial restriction of the dynamics. Our model explains the various SAM patterns observed in plants, for example, homeostatic control of SAM size in the wild type, enlarged or fasciated SAM in clv mutants, and initiation of ectopic secondary meristems from an initial flattened SAM in wus mutant. In addition, the model is supported by comparing its prediction with the expression pattern of WUS in the wus mutant. Furthermore, the model can account for many experimental results including reorganization processes caused by the CZ ablation and by incision through the meristem center. We thus conclude that the reaction-diffusion dynamics is probably indispensable for the SAM development of plants.