Models of Self-Organizing Bacterial Communities and Comparisons with Experimental Observations
TL;DR: A critical anal ysis of the validity of the model based on recent observations of the swarming bacteria which show that nutrients are not limitating but distinct subpopulations growing at different rates are lik ely present is presented.
Abstract: Bacillus subtilis swarms rapidly over the surface of a synthetic medium creating remarkable hyperbranched dendritic communities. Models to reproduce such effects have been proposed under the form of parabolic Partial Differential Equations representing the dynamics of the active cells (both motile and multiplying), the passive cells (non-motile and non-growing) and nutrient concentration. We test the numerical behavior of such models and compare them to relevant experimental data together with a critical analysis of the validity of the models based on recent observations of the swarming bacteria which show that nutrients are not limitating but distinct subpopulations growing at different rates are likely present.
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TL;DR: A review and critical analysis of the mathematical literature concerning the modeling of vehicular traffic and crowd phenomena and a critical analysis focused on research perspectives that consider the development of a unified modeling strategy are presented.
Abstract: This paper presents a review and critical analysis of the mathematical literature concerning the modeling of vehicular traffic and crowd phenomena. The survey of models deals with the representation scales and the mathematical frameworks that are used for the modeling approach. The paper also considers the challenging objective of modeling complex systems consisting of large systems of individuals interacting in a nonlinear manner, where one of the modeling difficulties is the fact that these systems are difficult to model at a global level when based only on the description of the dynamics of individual elements. The review is concluded with a critical analysis focused on research perspectives that consider the development of a unified modeling strategy.
434 citations
Cites background from "Models of Self-Organizing Bacterial..."
...Finally, let us mention that although far from the specific contents of this paper, various interesting papers investigate crowding and swarming phenomena at the low scale (molecular and cellular) in biology, such as [155], [207], [212]....
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TL;DR: An analytical and computational analysis is performed to study pattern formation during the spreading of an initially circular bacterial colony on a Petri dish, finding the spreading colony is found to be always linearly unstable to perturbations of the interface, whereas branching instability arises in finite-element numerical simulations.
Abstract: Self-organization in developing living organisms relies on the capability of cells to duplicate and perform a collective motion inside the surrounding environment. Chemical and mechanical interactions coordinate such a cooperative behaviour, driving the dynamical evolution of the macroscopic system. In this work, we perform an analytical and computational analysis to study pattern formation during the spreading of an initially circular bacterial colony on a Petri dish. The continuous mathematical model addresses the growth and the chemotactic migration of the living monolayer, together with the diffusion and consumption of nutrients in the agar. The governing equations contain four dimensionless parameters, accounting for the interplay among the chemotactic response, the bacteria–substrate interaction and the experimental geometry. The spreading colony is found to be always linearly unstable to perturbations of the interface, whereas branching instability arises in finite-element numerical simulations. The typical length scales of such fingers, which align in the radial direction and later undergo further branching, are controlled by the size parameters of the problem, whereas the emergence of branching is favoured if the diffusion is dominant on the chemotaxis. The model is able to predict the experimental morphologies, confirming that compact (resp. branched) patterns arise for fast (resp. slow) expanding colonies. Such results, while providing new insights into pattern selection in bacterial colonies, may finally have important applications for designing controlled patterns.
66 citations
TL;DR: In this paper, the authors studied the branching instability in the hyperbolic Keller-Segel system with logistic sensitivity, where repulsive and attractive forces, acting on a conservative system, create stable traveling patterns or branching instabilities.
Abstract: How can repulsive and attractive forces, acting on a conservative system, create stable traveling patterns or branching instabilities? We have proposed to study this question in the framework of the hyperbolic Keller-Segel system with logistic sensitivity. This is a model system motivated by experiments on cell communities auto-organization, a field which is also called socio-biology. We continue earlier modeling work, where we have shown numerically that branching patterns arise for this system and we have analyzed this instability by formal asymptotics for small diffusivity of the chemo-repellent. Here we are interested in the more general situation, where the diffusivities of both the chemo-attractant and the chemo-repellent are positive. To do so, we develop an appropriate functional analysis framework. We apply our method to two cases. Firstly we analyze steady states. Secondly we analyze traveling waves when neglecting the degradation coefficient of the chemo-repellent; the unique wave speed appears through a singularity cancelation which is the main theoretical difficulty. This shows that in different situations the cell density takes the shape of a plateau. The existence of steady states and traveling plateaus are a symptom of how rich the system is and why branching instabilities can occur. Numerical tests show that large plateaus may split into smaller ones, which remain stable.
36 citations
TL;DR: In this paper, a simple model where a given population evolves feeded by a diffusing nutriment, but is subject to a density constraint is proposed, where particles (e.g., cells) of the population spontaneously stay passive at rest, and only move in order to satisfy the constraint by choosing the minimal correction velocity so as to prevent overloading.
Abstract: In order to observe growth phenomena in biology where dendritic shapes appear, we propose a simple model where a given population evolves feeded by a diffusing nutriment, but is subject to a density constraint. The particles (e.g., cells) of the population spontaneously stay passive at rest, and only move in order to satisfy the constraint $\rho\leq 1$, by choosing the minimal correction velocity so as to prevent overcongestion. We treat this constraint by means of projections in the space of densities endowed with the Wasserstein distance $W_2$, defined through optimal transport. This allows to provide an existence result and suggests some numerical computations, in the same spirit of what the authors did for crowd motion (but with extra difficulties, essentially due to the fact that the total mass may increase). The numerical simulations show, according to the values of the parameter and in particular of the diffusion coefficient of the nutriment, the formation of dendritic patterns in the space occupied by cells.
33 citations
Cites background from "Models of Self-Organizing Bacterial..."
...More complicated models include specific features, such as the presence of some lubricating fluid produced by the cells (see [11]), the effects of chemotaxis described for example in [11, 15], or different states of mobility for bacteria, as in [14]....
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TL;DR: It is shown that variations in the wettability and surfactant production are sufficient to reproduce four different types of colony growth, which have been described in the literature, namely, arrested and continuous spreading of circular colonies, slightly modulated front lines and the formation of pronounced fingers.
Abstract: The spreading of bacterial colonies at solid-air interfaces is determined by the physico-chemical properties of the involved interfaces. The production of surfactant molecules by bacteria is a widespread strategy that allows the colony to efficiently expand over the substrate. On the one hand, surfactant molecules lower the surface tension of the colony, effectively increasing the wettability of the substrate, which facilitates spreading. On the other hand, gradients in the surface concentration of surfactant molecules result in Marangoni flows that drive spreading. These flows may cause an instability of the circular colony shape and the subsequent formation of fingers. In this work, we study the effect of bacterial surfactant production and substrate wettability on colony growth and shape within the framework of a hydrodynamic thin film model. We show that variations in the wettability and surfactant production are sufficient to reproduce four different types of colony growth, which have been described in the literature, namely, arrested and continuous spreading of circular colonies, slightly modulated front lines and the formation of pronounced fingers.
31 citations
References
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TL;DR: The chemotactic response of unicellular microscopic organisms is viewed as analogous to Brownian motion, and a macroscopic flux is derived which is proportional to the chemical gradient.
1,660 citations
"Models of Self-Organizing Bacterial..." refers methods in this paper
...One class of model concerns auto-chemotaxis (attraction of cells by a chemical substance emitted by the cells themselves) and gives rise to a Fokker-Planck equation that is commonly called the Keller-Segel system after the seminal work [12]....
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Book•
05 Sep 2008
TL;DR: In this paper, the renewal equation is used to describe the structure of a population from a point of view of population balance equations, which is an asymptotic view of the population dynamics.
Abstract: From differential equations to structured population dynamics.- Adaptive dynamics an asymptotic point of view.- Population balance equations: the renewal equation.- Population balance equations: size structure.- Cell motion and chemotaxis.- General mathematical tools.
932 citations
"Models of Self-Organizing Bacterial..." refers background in this paper
...This model is mathematically very challenging and has motivated numerous studies (see [3, 23] and the references therein); in particular this class of model typically leads to cell aggregation in one or several discrete spots (blow-up of the system as a Dirac mass solution)....
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TL;DR: This work presents the complete three-dimensional branching pattern and lineage of the mouse bronchial tree, reconstructed from an analysis of hundreds of developmental intermediates, and proposes that each mode of branching is controlled by a genetically encoded subroutine, a series of local patterning and morphogenesis operations, which are themselvescontrolled by a more global master routine.
Abstract: Mammalian lungs are branched networks containing thousands to millions of airways arrayed in intricate patterns that are crucial for respiration. How such trees are generated during development, and how the developmental patterning information is encoded, have long fascinated biologists and mathematicians. However, models have been limited by a lack of information on the normal sequence and pattern of branching events. Here we present the complete three-dimensional branching pattern and lineage of the mouse bronchial tree, reconstructed from an analysis of hundreds of developmental intermediates. The branching process is remarkably stereotyped and elegant: the tree is generated by three geometrically simple local modes of branching used in three different orders throughout the lung. We propose that each mode of branching is controlled by a genetically encoded subroutine, a series of local patterning and morphogenesis operations, which are themselves controlled by a more global master routine. We show that this hierarchical and modular programme is genetically tractable, and it is ideally suited to encoding and evolving the complex networks of the lung and other branched organs.
720 citations
"Models of Self-Organizing Bacterial..." refers background in this paper
...Interestingly, this motif, termed domain branching, is observed in mouse lung tissue [18]....
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Journal Article•
TL;DR: The Keller-Segel system describes the collective motion of cells which are attracted by a chemical substance and are able to emit it in its simplest form it is a conservative drift-diffusion equation coupled to an elliptic equation for the chemo-attractant concentration as mentioned in this paper.
Abstract: The Keller-Segel system describes the collective motion of cells which are attracted by a chemical substance and are able to emit it In its simplest form it is a conservative drift-diffusion equation for the cell density coupled to an elliptic equation for the chemo-attractant concentration It is known that, in two space dimensions, for small initial mass, there is global existence of solutions and for large initial mass blow-up occurs In this paper we complete this picture and give a detailed proof of the existence of weak solutions below the critical mass, above which any solution blows-up in finite time in the whole euclidean space Using hypercontractivity methods, we establish regularity results which allow us to prove an inequality relating the free energy and its time derivative For a solution with sub-critical mass, this allows us to give for large times an ``intermediate asymptotics'' description of the vanishing In self-similar coordinates, we actually prove a convergence result to a limiting self-similar solution which is not a simple reflect of the diffusion
560 citations
"Models of Self-Organizing Bacterial..." refers background in this paper
...This model is mathematically very challenging and has motivated numerous studies (see [3, 23] and the references therein); in particular this class of model typically leads to cell aggregation in one or several discrete spots (blow-up of the system as a Dirac mass solution)....
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TL;DR: In this paper, the authors consider the simple case of uniform temperatures and concentrations in the isothermal CSTR and the simplest of reaction schemes: (i) quadratic autocatalysis (A + B →2 B ); and (ii) cubic autoccatalysis ( A + 2 B →3 B ).
526 citations
Additional excerpts
...Gray-Scott system [7] is a simple and classical example which writes as ∂...
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