High order numerical methods for myxobacteria pattern formation

Rippling patterns of myxobacteria appear in starving colonies before they aggregate to form fruiting bodies. These periodic traveling cell density waves arise from the coordination of individual cell reversals, resulting from an internal clock regulating them, and from contact signaling during bacterial collisions. Our main interest in this research is the numerical approximation with high order accuracy in space of the solutions of mathematical models proposed for myxobacteria rippling. We revisit the studies of Igoshin et al [Proc. Natl. Acad. Sci, USA 98, 14913 (2001) and Phys. Rev. E 70, 041911 (2004)] which describe the rippling phenomena of myxobacteria as a system of hyperbolic conservation laws (when the di¤usion is zero). Since the solution of systems of conservation laws develops jump discontinuities in time and space, it is important to use accurate numerical simulators in order to explain and predict the natural biological process, which is our approach. Previously, patterns for this model were obtained only by numerical methods of low order of accuracy and it was not possible to find their wavenumber analytically.En las colonias hambrientas de myxobacteria aparecen patrones ondulatorios antes de que las bacterias se agreguen para formar cuerpos fructíferos. Estas ondas periódicas de densidad celular itinerante surgen como resultado de la coordinación de las reversiones celulares, a causa de un reloj interno, y por el contacto de señalización durante las colisiones bacterianas. Nuestro principal interés en esta tesis es la aproximación numérica con alta precisión para las soluciones de los modelos matemáticos propuestos para la ondulación de las myxobacterias. Revisamos los estudios de Igoshin y coautores [Proc. Natl. Acad. Sci, EE.UU. 98, 14913 (2001) y Phys. Rev. E 70, 041911 (2004)], que describen los patrones ondulatorios de myxobacterias como un sistema de leyes de conservación hiperbólica (cuando la difusión es cero). Teniendo en cuenta que las propiedades de la solución de sistemas de leyes de conservación desarrollan discontinuidades de salto y fuertes gradientes en el tiempo y el espacio, consideramos importante utilizar simuladores numéricos precisos con el .n de explicar y predecir el proceso biológico natural, que es nuestro enfoque. Anteriormente, las pautas de este modelo se obtuvieron solamente por métodos numéricos de orden de precisión inferior y no fue posible encontrar su número de onda de forma analítica.Programa Oficial de Posgrado en Ingeniería MatemáticaPresidente: Ana María Carpio Rodríguez.- Secretario: Manuel Carretero Cerrajero.- Vocal: Gloria Platero Coell

Understanding morphogenesis in myxobacteria from a theoretical and experimental perspective

Several species of bacteria exhibit multicellular behaviour, with individuals cells cooperatively working together within a colony. Often this has communal benefit since multiple cells acting in unison can accomplish far more than an individual cell can and the rewards can be shared by many cells. Myxobacteria are one of the most complex of the multicellular bacteria, exhibiting a number of different spatial phenotypes. Colonies engage in multiple emergent behaviours in response to starvation culminating in the formation of massive, multicellular fruiting bodies. In this thesis, experimental work and theoretical modelling are used to investigate emergent behaviour in myxobacteria. Computational models were created using FABCell, an open source software modelling tool developed as part of the research to facilitate modelling large biological systems. The research described here provides novel insights into emergent behaviour and suggests potential mechanisms for allowing myxobacterial cells to go from a vegetative state into a fruiting body. A differential equation model of the Frz signalling pathway, a key component in the regulation of cell motility, is developed. This is combined with a three-dimensional model describing the physical characteristics of cells using Monte Carlo methods, which allows thousands of cells to be simulated. The unified model explains how cells can ripple, stream, aggregate and form fruiting bodies. Importantly, the model copes with the transition between stages showing it is possible for the important myxobacteria control systems to adapt and display multiple behaviours

Understanding morphogenesis in myxobacteria from a theoretical and experimental perspective

Several species of bacteria exhibit multicellular behaviour, with individuals cells cooperatively working together within a colony. Often this has communal benefit since multiple cells acting in unison can accomplish far more than an individual cell can and the rewards can be shared by many cells. Myxobacteria are one of the most complex of the multicellular bacteria, exhibiting a number of different spatial phenotypes. Colonies engage in multiple emergent behaviours in response to starvation culminating in the formation of massive, multicellular fruiting bodies. In this thesis, experimental work and theoretical modelling are used to investigate emergent behaviour in myxobacteria. Computational models were created using FABCell, an open source software modelling tool developed as part of the research to facilitate modelling large biological systems. The research described here provides novel insights into emergent behaviour and suggests potential mechanisms for allowing myxobacterial cells to go from a vegetative state into a fruiting body. A differential equation model of the Frz signalling pathway, a key component in the regulation of cell motility, is developed. This is combined with a three-dimensional model describing the physical characteristics of cells using Monte Carlo methods, which allows thousands of cells to be simulated. The unified model explains how cells can ripple, stream, aggregate and form fruiting bodies. Importantly, the model copes with the transition between stages showing it is possible for the important myxobacteria control systems to adapt and display multiple behaviours.EThOS - Electronic Theses Online ServiceEngineering and Physical Sciences Research Council (Great Britain) (EPSRC)GBUnited Kingdo

A hydrodynamic limit for chemotaxis in a given heterogeneous environment

In this paper, the first equation within a class of well-known chemotaxis systems is derived as a hydrodynamic limit from a stochastic interacting many particle system on the lattice. The cells are assumed to interact with attractive chemical molecules on a finite number of lattice sites, but they only directly interact among themselves on the same lattice site. The chemical environment is assumed to be stationary with a slowly varying mean, which results in a non-trivial macroscopic chemotaxis equation for the cells. Methodologically, the limiting procedure and its proofs are based on results by Koukkous (Stoch. Process. Appl. 84, 297–312, cite.​Kou99) and Kipnis and Landim (Scaling limits of interacting particle systems, cite.​KL99). Numerical simulations extend and illustrate the theoretical findings

Spikes and diffusion waves in one-dimensional model of chemotaxis

We consider the one-dimensional initial value problem for the viscous transport equation with nonlocal velocity $u_t = u_{xx} - \left(u (K^\prime \ast u)\right)_{x}$ with a given kernel $K'\in L^1(\R)$. We show the existence of global-in-time nonnegative solutions and we study their large time asymptotics. Depending on $K'$, we obtain either linear diffusion waves ({\it i.e.}~the fundamental solution of the heat equation) or nonlinear diffusion waves (the fundamental solution of the viscous Burgers equation) in asymptotic expansions of solutions as $t\to\infty$. Moreover, for certain aggregation kernels, we show a concentration of solution on an initial time interval, which resemble a phenomenon of the spike creation, typical in chemotaxis models

Swimming and Swarming of Self-Propelled Particles

A number of micro-organisms and cells, such as sperm and some spieces of roundworms (nematodes), employ a sinusoidal beating motion of their rod-like body to swim though a fluid medium. For the motion of these microscopic swimmers, the viscosity is dominating and the inertia is negligible. They cooperate with each other through hydrodynamic interactions and exhibit complex swarm behaviors, such as aggregation near surfaces and clustering at high density. These interesting and surprising phenomena indicate that, in addition to the individual motion of wandering and struggling alone, there are more efficient cooperative ways for the swimmers to overcome long distance and obstacles to reach their ultimate goal. This applies especially for sperm as one of the most important cells for the reproduction of high animals. The goal of this work is to explain the importance of hydrodynamic interaction and volume exclusion for the cooperation and swarm behavior of micro-swimmers which employ sinusoidal beating, like sperm and nematodes. We classify the swimmers as rod-like self-propelled particles (rSPP) in a viscous environment, and compare the swarm behaviors of straight self-propelled rods and sinusoidal beating swimmers by simulations. The hydrodynamic interaction between the swimmers is simulated by multi-particle collision dynamics (MPC), a particle-based meso-scopic simulation method for fluid dynamics. We also perform the simulations with anisotropic frictions (AF), an approximation of hydrodynamics, which neglects hydrodynamic interactions between swimmers. The contributions of hydrodynamic interaction and volume exclusion are distinguished by comparing results in a MPC fluid and with AF. Volume exclusion of the elongated particles is the key factor to induce the alignment and clustering behavior of self-propelled rods in viscous environment in two dimensions. Two kinds of clusters are found: motile clusters with all of their components polarized, which are found for low rod density and strong environmental noise; giant, immobile clusters of blocked rods, which are found for high rod density and weak environmental noise. A stable distribution function of cluster size is reached when the system is balanced between the formation rate and break-up rate. Three types of the distribution functions, corresponding to three states of the system, are found. For systems of motile clusters, the distribution function always has a power-law-decay part. The average cluster size shows a power-law relation with the variance of environmental noise. Giant density fluctuations, which are a characteristic fingerprint of aggregating systems of self-propelled particles, are also found in our rod simulations. The main difference between self-propelled rods and flagella systems is that the sinusoidal beating flagella have synchronization and attraction through hydrodynamic interaction. The hydrodynamic synchronization and attraction make the flagella in the same cluster tightly packed and locked in phase. The clusters extend strongly in the direction of motion, and the probability to find small clusters is decreased. Hydrodynamic interaction between clusters acts as the environmental background noise. The swarm behavior of sinusoidal undulating flagella is basically the same as the self-propelled rods. The distribution function of cluster size has a power-law decay. In nature, sperm and nematodes can have a wide distribution of beat frequencies, which can be considered as noise due to internal property. The average cluster size has a power-law dependence on the variance of distribution of beating frequencies. A sperm is a sinusoidal beating flagellum with a head attached in front. Although the heads generate strong viscous resistance, the hydrodynamic interaction - synchronization and attraction - between beating tails is still dominating. The swarming behavior of a multi-sperm system is the same as a multi-flagellum system. However, the heads make the cluster configuration much looser, thus the stability of large clusters decreases. Thus we conclude that, in two dimensions, the fundamental elements for the swarming behavior of active rod-like particles like sperm and nematodes are the anisotropic shape and the self-propelled motion. The volume exclusion is a strong mechanism to induce the alignment. The hydrodynamic interaction due to the sinusoidal beating motion regulates the shape of the clusters and the distribution function of cluster size. Our results are in good agreements with experimental observations of the swarming of sperm and nematodes in a thin layer of fluid medium near surfaces. Interesting experimental phenomena, such as the elongated cluster of rodent sperm and the vortices of sea-urchin sperm, are reproduced in the simulations

Learning differential equation models from stochastic agent-based model simulations

Agent-based models provide a flexible framework that is frequently used for modelling many biological systems, including cell migration, molecular dynamics, ecology, and epidemiology. Analysis of the model dynamics can be challenging due to their inherent stochasticity and heavy computational requirements. Common approaches to the analysis of agent-based models include extensive Monte Carlo simulation of the model or the derivation of coarse-grained differential equation models to predict the expected or averaged output from the agent-based model. Both of these approaches have limitations, however, as extensive computation of complex agent-based models may be infeasible, and coarse-grained differential equation models can fail to accurately describe model dynamics in certain parameter regimes. We propose that methods from the equation learning field provide a promising, novel, and unifying approach for agent-based model analysis. Equation learning is a recent field of research from data science that aims to infer differential equation models directly from data. We use this tutorial to review how methods from equation learning can be used to learn differential equation models from agent-based model simulations. We demonstrate that this framework is easy to use, requires few model simulations, and accurately predicts model dynamics in parameter regions where coarse-grained differential equation models fail to do so. We highlight these advantages through several case studies involving two agent-based models that are broadly applicable to biological phenomena: a birth-death-migration model commonly used to explore cell biology experiments and a susceptible-infected-recovered model of infectious disease spread