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
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TL;DR: In this paper, the authors show that finite fluctuations in particle number can be essential for such an instability to occur, and suggest that fluctuations can provide a new and general pattern-forming mechanism in non-equilibrium growth.
Abstract: The formation of complex patterns in many non-equilibrium systems, ranging from solidifying alloys to multiphase flow1, nonlinear chemical reactions2 and the growth of bacterial colonies3,4, involves the propagation of an interface that is unstable to diffusive motion. Most existing theoretical treatments of diffusive instabilities are based on mean-field approaches, such as the use of reaction–diffusion equations, that neglect the role of fluctuations. Here we show that finite fluctuations in particle number can be essential for such an instability to occur. We study, both analytically and with computer simulations, the planar interface separating different species in the simple two-component reaction A+ B → 2A (which can also serve as a simple model of bacterial growth in the presence of a nutrient). The interface displays markedly different dynamics within the reaction–diffusion treatment from that when fluctuations are taken into account. Our findings suggest that fluctuations can provide a new and general pattern-forming mechanism in non-equilibrium growth.
111 citations
"Models of Self-Organizing Bacterial..." refers methods in this paper
...Variations around this model can also be interpreted in terms of bacterial motion as proposed in Kessler and Levine [13], and Golding et al [6]; they replace the growth term unv by h(u)v where h(·) is a truncation function for small values of u and h ≈ 1 for large values....
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...Variations around this model can also be interpreted in terms of bacterial motion as proposed in Kessler and Levine [13], and Golding et al [6]; they replace the growth term uv by h(u)v where h(·) is a truncation function for small values of u and h ≈ 1 for large values....
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...Rather than a limitation on growth for small values of u as in the Kessler and Levine model, Mimura et al [19] proposed a limitation on the transition rate to the passive state for large values of u or v....
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TL;DR: The modelling shows that Proteus colony geometries arise as a consequence of macroscopic rules governing collective motility, not as an independent biological function, and Kinetic models similar to this one may be applicable to periodic phenomena displayed by other biological systems with differentiated components of defined lifetimes.
Abstract: Proteus mirabilis colonies display striking symmetry and periodicity. Based on experimental observations of cellular differentiation and group motility, a kinetic model has been developed to describe the swarmer cell differentiation-dedifferentiation cycle and the spatial evolution of swimmer and swarmer cells during Proteus mirabilis swarm colony development. A key element of the model is the age dependence of swarmer cell behaviour, in particular specifying a minimal age for motility and maximum age for septation and dedifferentiation to swimmer cells. Density thresholds for collective motility by mature swarmer cells serve to synchronize the movements of distinct swarmer cell groups and thus help provide temporal coherence to colony expansion cycles. Numerical computations show that the model fits experimental data by generating a complete swarming plus consolidation cycle period that is robust to changes in parameters which affect other aspects of swarmer cell migration and colony development. The kinetic equations underlying this model provide a different mathematical basis for a temporal oscillator from reaction-diffusion partial differential equations. The modelling shows that Proteus colony geometries arise as a consequence of macroscopic rules governing collective motility. Thus, in this case, pattern formation results from the operation of an adaptive bacterial system for spreading on solid substrates, not as an independent biological function. Kinetic models similar to this one may be applicable to periodic phenomena displayed by other biological systems with differentiated components of defined lifetimes.
102 citations
"Models of Self-Organizing Bacterial..." refers background in this paper
...Similarly, swarming in Proteus mirabilis is not controlled by nutrient limitation [4, 24]....
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...Many additional factors have been incorporated into models, such as the observed higher motility of cells at the tip of the dendrites (region of higher population density and higher nutrient concentration) in [11], a surfactant secreted by the cells that may change the liquid surface and thus the migration speed of cells [15, 6], or differentiation from swimmers to swarmers for Proteus mirabilis as modelled in [4, 5]....
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TL;DR: In this paper, the existence, stability, and pulse-splitting behavior of spike patterns in the one-dimensional Gray-Scott model on a finite domain is analyzed in the semi-strong spike-interaction regime.
89 citations
"Models of Self-Organizing Bacterial..." refers background in this paper
...These concentration points are traveling pulses that undergo secondary instabilities which explain their branching, see [21, 14]....
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TL;DR: A detailed microscopic in situ analysis of swarms 1 and 2 revealed varied cell morphologies and a remarkable series of events, with cells assembling into different 'structures', as the architecture of the swarm developed.
Abstract: After optimizing the conditions, including nutrients and temperature, swarming of Bacillus subtilis 3610 was obtained on a synthetic, fully defined medium. The swarms formed highly branched (dendritic) patterns, generated by successive waves of moving cells. A detailed microscopic in situ analysis of swarms 1 and 2 revealed varied cell morphologies and a remarkable series of events, with cells assembling into different 'structures', as the architecture of the swarm developed. Long filamentous cells begin to form before the onset of the first swarming (11 h) and are again observed at later stages in the interior of individual mature dendrites. Swarm 2, detected at 18-22 h, is accompanied by the rapid movement of a wave of dispersed (non-filamentous) cells. Subsequently at the forward edge of this swarm, individual cells begin to cluster together, gradually forming de novo the shape of a dendrite tip with progressive lengthening of this new structure 'backwards' towards the swarm centre. In both swarms 1 and 2, after the initial clustering of cells, there is the progressive appearance of a spreading monolayer of rafts (4-5 non-filamented cells, neatly aligned). The alternative possible roles of the rafts in the development of the swarm are discussed.
87 citations
"Models of Self-Organizing Bacterial..." refers background in this paper
...subtilis over a fully defined medium (B-medium) in a Petri dish (a swarm plate), in which the bacteria migrate from a central inoculum as hyper-branching dendrites, forming radiating patterns covering several square centimeters in a few hours ([9, 10])....
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TL;DR: In this article, a new formulation of the system of partial differential equations is obtained by the introduction of a new variable (this new variable is similar to the quasi-Fermi level in the framework of semiconductor modelling).
Abstract: We start from a mathematical model which describes the collective motion of bacteria taking into account the underlying biochemistry. This model was first introduced by Keller-Segel [13]. A new formulation of the system of partial differential equations is obtained by the introduction of a new variable (this new variable is similar to the quasi-Fermi level in the framework of semiconductor modelling). This new system of P.D.E. is approximated via a mixed finite element technique. The solution algorithm is then described and finally we give some preliminary numerical results. Especially our method is well adapted to compute the concentration of bacteria.
79 citations