A generalized discrete model linking rippling pattern formation and individual cell reversal statistics in colonies of myxobacteria

Self-organization processes in multicellular aggregates of bacteria and amoebae offer fascinating insights into the evolution of cooperation and differentiation of cells. During myxobacterial development a variety of spatio-temporal patterns emerges such as counterpropagating waves of cell density that are known as rippling. Recently, several models have been introduced that qualitatively reproduce these patterns. All models include active motion and a collision-triggered reversal of individual bacteria. Here, we present a systematic study of a generalized discrete model that is based on similar assumptions as the continuous model by Igoshin et al (2001 Proc. Natl Acad. Sci. USA 98 14913). We find counterpropagating as well as unidirectional rippling waves in extended regions of the parameter space. If the interaction strength and the degree of cooperativity are large enough, rippling patterns appear even in the absence of a refractory period. We show for the first time that the experimentally observed double peak in the reversal statistics of bacteria in rippling colonies (Welch and Kaiser 2001 Proc. Natl Acad. Sci. USA 98 14907) can be reproduced in simulations of counterpropagating rippling waves which are dominant in experiments. In addition, the reversal statistics in the pre-rippling phase is correctly reproduced.     

Soliton behaviour in a bistable reaction diffusion model

We analyze a generic reaction-diffusion model that contains the important features of Turing systems and that has been extensively used in the past to model biological interesting patterns. This model presents various fixed points. Analysis of this model has been made in the past only in the case when there is only a single fixed point, and a phase diagram of all the possible instabilities shows that there is a place where a Turing-Hopf bifurcation occurs producing oscillating Turing patterns. In here we focus on the interesting situation of having several fixed points, particularly when one unstable point is in between two equally stable points. We show that the solutions of this bistable system are traveling front waves, or solitons. The predictions and results are tested by performing extensive numerical calculations in one and two dimensions. The dynamics of these solitons is governed by a well defined spatial scale, and collisions and interactions between solitons depend on this scale. In certain regions of parameter space the wave fronts can be stationary, forming a pattern resembling spatial chaos. The patterns in two dimensions are particularly interesting because they can present a coherent dynamics with pseudo spiral rotations that simulate the myocardial beat quite closely. We show that our simple model can produce complicated spatial patterns with many different properties, and could be used in applications in many different fields.      

Clocks and patterns in myxobacteria: a remembrance of Art Winfree

At the beginning of their aggregation phase waves of cell density sweep across the surface of myxobacteria colonies. These waves are unlike any other in biology. Waves can be linear, concentric or spiral and when they collide, instead of annihilating one another they appear to pass through each other unchanged. Moreover, the wavelength determines the spacing and pattern of fruiting bodies that will rise up presaging sporulation. The explanation for these waves was suggested by the work of Art Winfree on cellular clocks, and confirmed by a mathematical model that explains all of the observed wave behavior. The story of how this model evolved illustrates the roles of chance and scientific networking in the search for the explanation of a new phenomenon.     

A master equation for a spatial population model with pair interactions

We derive a closed master equation for an individual-based population model in continuous space and time. The model and master equation include Brownian motion, reproduction via binary fission, and an interaction-dependent death rate moderated by a competition kernel. Using simulations we compare this individual-based model with the simplest approximation, the spatial logistic equation. In the limit of strong diffusion the spatial logistic equation is a good approximation to the model. However, in the limit of weak diffusion the spatial logistic equation is inaccurate because of spontaneous clustering driven by reproduction. The weak-diffusion limit can be partially analyzed using an exact solution of the master equation applicable to a competition kernel with infinite range. This analysis shows that in the case of a top-hat kernel, reducing the diffusion can increase the total population. For a Gaussian kernel, reduced diffusion invariably reduces the total population. These theoretical results are confirmed by simulation.     

From Biochemistry to Morphogenesis in Myxobacteria

Many aspects of metazoan morphogenesis find parallels in the communal behavior of microorganisms. The cellular slime mold D. discoideum has long provided a metaphor for multicellular embryogenesis. However, the spatial patterns in D.d. colonies are generated by an intercellular communication system based on diffusible morphogens, whereas the interactions between embryonic cells are more often mediated by direct cell contact. For this reason, the myxobacteria have emerged as a contending system in which to study spatial pattern formation, for their colony strutures rival those of D.d. in complexity, yet communication between cells in a colony is carried out by direct cell contacts. Here I sketch some of the progress my laboratory has made in modeling the life cycle of these organisms.     

Biology and global distribution of myxobacteria in soils

This review presents an overview of the present status of the biology of the myxobacteria, including the molecular biology of the systems that control and regulate myxobacterial gliding movement and morphogenesis. The present status of myxobacterial taxonomy and phylogeny is described. The evolutionary biology of the myxobacteria is emphasized with respect to their social behavior and the molecular basis of their signal chains. Most important within the metabolic physiology are the biologically active secondary metabolites of myxobacteria and their molecular mechanisms of action. The global distribution of myxobacteria in soils is described on the basis of data given in the literature as well as of comprehensive analyses of 1398 soil samples from 64 countries of all continents. The results are analyzed with respect to the spectrum and number of species depending on ecological and habitat-specific factors. The myxobacterial floras of different climate zones are compared. Included are myxobacterial species adapted to extreme biotopes. The efficiency of different methods used presently for isolation of myxobacteria is compared.     

A new interpretation of the Keller-Segel model based on multiphase modelling

In this paper an alternative derivation and interpretation are presented of the classical Keller-Segel model of cell migration due to random motion and chemotaxis. A multiphase modelling approach is used to describe how a population of cells moves through a fluid containing a diffusible chemical to which the cells are attracted. The cells and fluid are viewed as distinct components of a two-phase mixture. The principles of mass and momentum balance are applied to each phase, and appropriate constitutive laws imposed to close the resulting equations. A key assumption here is that the stress in the cell phase is influenced by the concentration of the diffusible chemical. By restricting attention to one-dimensional cartesian geometry we show how the model reduces to a pair of nonlinear coupled partial differential equations for the cell density and the chemical concentration. These equations may be written in the form of the Patlak-Keller-Segel model, naturally including density-dependent nonlinearities in the cell motility coefficients. There is a direct relationship between the random motility and chemotaxis coefficients, both depending in an inter-related manner on the chemical concentration. We suggest that this may explain why many chemicals appear to stimulate both chemotactic and chemokinetic responses in cell populations. After specialising our model to describe slime mold we then show how the functional form of the chemical potential that drives cell locomotion influences the ability of the system to generate spatial patterns. The paper concludes with a summary of the key results and a discussion of avenues for future research.     

Independent roles of Drosophila Moesin in imaginal disc morphogenesis and hedgehog signalling

The three ERM proteins (Ezrin, Radixin and Moesin) form a conserved family required in many developmental processes involving regulation of the cytoskeleton. In general, the molecular function of ERM proteins is to link specific membrane proteins to the actin cytoskeleton. In Drosophila, loss of moesin (moe) activity causes incorrect localisation of maternal determinants during oogenesis, failures in rhabdomere differentiation in the eye and alterations of epithelial integrity in the wing imaginal disc. Some aspects of Drosophila Moe are related to the activity of the small GTPase RhoA, because the reduction of RhoA activity corrects many phenotypes of moe mutant embryos and imaginal discs. We have analysed the phenotype of moesin loss-of-function alleles in the wing disc and adult wing, and studied the effects of reduced Moesin activity on signalling mediated by the Notch, Decapentaplegic, Wingless and Hedgehog pathways. We found that reductions in Moesin levels in the wing disc cause the formation of wing-tissue vesicles and large thickenings of the vein L3, corresponding to breakdowns of epithelial continuity in the wing base and modifications of Hedgehog signalling in the wing blade, respectively. We did not observe any effect on signalling pathways other than Hedgehog, indicating that the moe defects in epithelial integrity have not generalised effects on cell signalling. The effects of moe mutants on Hedgehog signalling depend on the correct gene-dose of rhoA, suggesting that the requirements for Moesin in disc morphogenesis and Hh signalling in the wing disc are mediated by its regulation of RhoA activity. The mechanism linking Moesin activity with RhoA function and Hedgehog signalling remains to be elucidated.     

Multiscale analysis of pattern formation via intercellular signalling

Lateral inhibition, a juxtacrine signalling mechanism by which a cell adopting a particular fate inhibits neighbouring cells from doing likewise, has been shown to be a robust mechanism for the formation of fine-grained spatial patterns (in which adjacent cells in developing tissues diverge to achieve contrasting states of differentiation), provided that there is sufficiently strong feedback. The fine-grained nature of these patterns poses problems for analysis via traditional continuum methods since these require that significant variation takes place only over lengthscales much larger than an individual cell and such systems have therefore been investigated primarily using discrete methods. Here, however, we apply a multiscale method to derive systematically a continuum model from the discrete Delta-Notch signalling model of Collier et al. (J.R. Collier, N.A.M. Monk, P.K. Maini, J.H. Lewis, Pattern formation by lateral inhibition with feedback: a mathematical model of Delta-Notch intercellular signalling, J. Theor. Biol., 183, 1996, 429-446) under particular assumptions on the parameters, which we use to analyse the generation of fine-grained patterns. We show that, on the macroscale, the contact-dependent juxtacrine signalling interaction manifests itself as linear diffusion, motivating the use of reaction-diffusion-based models for such cell-signalling systems. We also analyse the travelling-wave behaviour of our system, obtaining good quantitative agreement with the discrete system.     

Simple and complex spatiotemporal structures in a glycolytic allosteric enzyme model

Pattern formation in glycolysis is studied with a classical reaction-diffusion allosteric enzyme model. It is found that, similar to recent experimental reports in the yeast extracts, a small magnitude local perturbation can induce transient target waves in a two dimensional oscillatory medium. An above threshold stimulation generates target waves which eventually evolve into spatiotemporal chaos upon collisions with the boundary or other wave activities. Detailed simulation studies show that the studied simple glycolytic reaction-diffusion model can support three types of spatiotemporal behaviors which are independent of the boundary conditions: (1) a spatially uniform stable steady state, (2) periodic global oscillations and (3) spatiotemporal chaos.     

From snapshot information to long-term population dynamics of Acacias by a simulation model

The African Acacia species A. raddiana is believed to be endangered in the Negev desert of Israel. The ecology of this species is not well understood. The main idea of our study is to learn more about the long-term population dynamics of these trees using snapshot information in the form of size frequency distributions. These distributions are highly condensed indices of population dynamics acting over many years. In this paper, we analyse field data on recruitment, growth, and mortality and use an existing simulation model of the population dynamics of A. raddiana (SAM) to produce contrasting scenarios of these live history processes that are based on the analysed field evidence. The main properties of simulated as well as observed tree size frequency distributions are characterised with Simpson's index of dominance and a new permutation index. Finally, by running the SAM model under the different scenarios, we study the effect of these different processes on simulated size frequency distributions (pattern) and we compare them to size distributions observed in the field, in order to identify the processes acting in the field. Our study confirms rare recruitment events as a major factor shaping tree size frequency distributions and shows that the paucity of recruitment has been a normal feature of A. raddiana in the Negev over many years. Irregular growth, e.g., due to episodic rainfall, showed a moderate influence on size distributions. Finally, the size frequency distributions observed in the Negev reveal the information that, in this harsh environment, mortality of adult A. raddiana is independent of tree size (age).     

Principles of biological pattern formation: swarming and aggregation viewed as selforganization phenomena

Migration automaton models are introduced which offer the possibility to directly analyse essential selforganization properties of biological pattern formation at the cellular level. We present examples of migration automata as models of collective motion and cellular aggregation—patterns that are typical for example in the life cycle of Myxobacteria. Linear stability analysis of the corresponding automaton Boltzmann equation allows to distinguish orientation-dependent (collective motion) and density-dependent (aggregation) instabilities. Presented at the National Symposium on Evolution of Life.     

Generation of bilateral symmetry in Anthozoa: a model

Polyps of Anthozoa usually display bilateral symmetry with respect to their mouth opening, to their pharynx, and in particular to the arrangement of their mesenteries. Mesenteries, which are endodermal folds running from the apical to the basal end of the body, subdivide the gastric cavity into pouches. They form in a bilateral symmetric sequence. In this article I propose that early in polyp development the endoderm subdivides successively into three different types of compartments. A mesentery forms at the border between compartments. Two of the compartments are homologous to those of Scyphozoa. They form by mutual activation of cell states that locally exclude each other. The third compartment leads to siphonoglyph formation and is an evolutionary innovation of the Anthozoa. The mechanism that controls the number and spatial arrangement of the third type of compartment changes the radial symmetry into a bilateral one and occasionally into a different one. The dynamics of its formation indicate an activator-inhibitor mechanism. Computer models are provided that reproduce decision steps in the generation of the mesenteries.     

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.     

On the importance of dimensionality of space in models of space-mediated population persistence

Spatially explicit models have become widely used in today's mathematical ecology to study persistence of populations. For the sake of simplicity, population dynamics is often analyzed with 1-D models. An important question is: how adequate is such 1-D simplification of 2-D (or 3-D) dynamics for predicting species persistence. Here we show that dimensionality of the environment can play a critical role in the persistence of predator-prey interactions. We consider 1-D and 2-D dynamics of a predator-prey model with the prey growth damped by the Allee effect. We show that adding a second space coordinate into the 1-D model results in a pronounced increase of size of the domain in the parametric space where predator-prey coexistence becomes possible. This result is due to the possibility of formation of a number of 2-D patterns, which is impossible in the 1-D model. The 1-D and the 2-D models exhibit different qualitative responses to variations of system parameters. We show that in ecosystems having a narrow width (e.g. mountain valleys, vegetation patterns along canals in dry areas, etc.), extinction of species is more probable compared to ecosystems having a pronounced second dimension. In particular, the width of a long narrow natural reserve should be large enough to guarantee nonextinction of species via interaction of 2-D population patches.     

Global cell sorting is mediated by local cell-cell interactions in the C. elegans embryo

The Caenorhabditis elegans embryo achieves pattern formation by sorting cells into coherent regions before morphogenesis is initiated. The sorting of cells is coupled to their fate. Cells move extensively relative to each other to reach their correct position in the body plan. Analyzing the mechanism of cell sorting in in vitro culture experiments using 4D microscopy, we show that all AB-derived cells sort only according to their local neighbors, and that all cells are able to communicate with each other. The directions of cell movement do not depend on a cellular polarity but only on local cell-cell interactions; in experimental situations, this allows even the reversal of the polarity of whole regions of the embryo. The work defines a new mechanism of pattern formation we call "cell focusing".     

Bicoid mRNA diffusion as a mechanism of morphogenesis in Drosophila early development

We show that mRNA diffusion is the main morphogenesis mechanism that consistently explains the establishment of Bicoid protein gradients in the embryo of Drosophila, contradicting the current view of protein diffusion. Moreover, we show that if diffusion for both bicoid mRNA and Bicoid protein were assumed, a steady distribution of Bicoid protein with a constant concentration along the embryo would result, contradicting observations.     

Instabilities in spatially extended predator-prey systems: spatio-temporal patterns in the neighborhood of Turing-Hopf bifurcations

We investigate the emergence of spatio-temporal patterns in ecological systems. In particular, we study a generalized predator-prey system on a spatial domain. On this domain diffusion is considered as the principal process of motion. We derive the conditions for Hopf and Turing instabilities without specifying the predator-prey functional responses and discuss their biological implications. Furthermore, we identify the codimension-2 Turing-Hopf bifurcation and the codimension-3 Turing-Takens-Bogdanov bifurcation. These bifurcations give rise to complex pattern formation processes in their neighborhood. Our theoretical findings are illustrated with a specific model. In simulations a large variety of different types of long-term behavior, including homogenous distributions, stationary spatial patterns and complex spatio-temporal patterns, are observed.     

Analysis and computer simulation of accretion patterns in bacterial cultures

Patterned growth of bacteria created by interactions between the cells and moving gradients of nutrients and chemical buffers is observed frequently in laboratory experiments on agar pour plates. This has been investigated by several microbiologists and mathematicians usually focusing on some hysteretic mechanism, such as dependence of cell uptake kinetics on pH. We show here that a simpler mechanism, one based on cell torpor, can explain patterned growth. In particular, we suppose that the cell population comprises two subpopulations —one actively growing and the other inactive. Cells can switch between the two populations depending on the quality of their environment (nutrient availability, pH, etc.) We formulate here a model of this system, derive and analyze numerical schemes for solving it, and present several computer simulations of the system that illustrate various patterns formed. These compare favorably with observed experiments.     

Travelling waves and pattern formation in a model for fungal development

 Under a variety of conditions, the hyphal density within the expanding outer edge of growing fungal mycelia can be spatially heterogeneous or nearly uniform. We conduct an analysis of a system of reaction-diffusion equations used to model the growth of fungal mycelia and the subsequent development of macroscopic patterns produced by differing hyphal and hence biomass densities. Both local and global results are obtained using analytical and numerical techniques. The emphasis is on qualitative results, including the effects of changes in parameter values on the structure of the solution set. Received 22 November 1995; received in revised form 17 May 1996