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.
Citations
More filters
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]....
[...]
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]....
[...]
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
More filters
Journal Article•
TL;DR: A diffusion-reaction model is developed, in which density dependent cell movements are incorporated by the level of nutrient concentration available for the cell, which predicts the growth velocity of a colony as a function of the nutrient concentration.
210 citations
TL;DR: This work presents an additional model which includes a lubrication fluid excreted by the bacteria and adds fields of chemotactic agents to the other models, and presents a critique of this whole enterprise with focus on the models’ potential for revealing new biological features.
Abstract: Various bacterial strains exhibit colonial branching patterns during growth on poor substrates. These patterns reflect bacterial cooperative self-organization and cybernetic processes of communication, regulation and control employed during colonial development. One method of modeling is the continuous, or coupled reaction–diffusion approach, in which continuous time evolution equations describe the bacterial density and the concentration of the relevant chemical fields. In the context of branching growth, this idea has been pursued by a number of groups. We present an additional model which includes a lubrication fluid excreted by the bacteria. We also add fields of chemotactic agents to the other models. We then present a critique of this whole enterprise with focus on the models’ potential for revealing new biological features.
206 citations
"Models of Self-Organizing Bacterial..." refers background or methods in this paper
...Moreover, many models to explain swarming assume that the surface of the agar is covered by a thin liquid film [6, 2]....
[...]
...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....
[...]
...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....
[...]
...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]....
[...]
...The references [6, 22] also contain several related models....
[...]
TL;DR: The Marangoni effect, which is fluid flow induced by gradients in surface tension, drives the instability, which occurs at the spreading edge of a thin wetting film.
Abstract: We report on a new hydrodynamic instability which occurs at the spreading edge of a thin wetting film. A drop of aqueous surfactant solution placed on a glass surface moistened with a thin layer of water spreads by propagating fingers, whose velocity and shape depend on the thickness of the ambient water layer and on the surfactant concentration. The two fluids are miscible and show negligible viscosity difference, ruling out a Saffman-Taylor instability. We propose that the Marangoni effect, which is fluid flow induced by gradients in surface tension, drives the instability.
178 citations
"Models of Self-Organizing Bacterial..." refers background in this paper
...Since surfactin is a surfactant, concentration gradients give rise to Marangoni forces, which have been shown to create branching patterns in the spreading of liquid droplets [25]....
[...]
TL;DR: It is shown that when the diffusion of the fluid is governed by a nonlinear diffusion coefficient, branching patterns evolve and fields of chemotactic agents and food chemotaxis are needed in order to explain the observations.
Abstract: Various bacterial strains (e.g., strains belonging to the genera Bacillus, Paenibacillus, Serratia, and Salmonella) exhibit colonial branching patterns during growth on poor semisolid substrates. These patterns reflect the bacterial cooperative self-organization. A central part of the cooperation is the collective formation of a lubricant on top of the agar which enables the bacteria to swim. Hence it provides the colony means to advance towards the food. One method of modeling the colonial development is via coupled reaction-diffusion equations which describe the time evolution of the bacterial density and the concentrations of the relevant chemical fields. This idea has been pursued by a number of groups. Here we present an additional model which specifically includes an evolution equation for the lubricant excreted by the bacteria. We show that when the diffusion of the fluid is governed by a nonlinear diffusion coefficient, branching patterns evolve. We study the effect of the rates of emission and decomposition of the lubricant fluid on the observed patterns. The results are compared with experimental observations. We also include fields of chemotactic agents and food chemotaxis and conclude that these features are needed in order to explain the observations.
141 citations
"Models of Self-Organizing Bacterial..." refers background in this paper
...The effect of surfactant has, for instance, been analysed in [15] and affects the diffusion term by allowing variation in cell motility of cells depending on the height of the surface liquid....
[...]
...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]....
[...]
TL;DR: Strain 168, which does not produce surfactin, displayed less swarming activity, both qualitatively (pattern formation) and in speed of colonization, while early stages of swarming in 3610 are accompanied by the formation of large numbers of dendrites whose rapid advance involves packs of cells at the tips, strain 168 advanced more slowly as a continuous front.
Abstract: The natural wild-type Bacillus subtilis strain 3610 swarms rapidly on the synthetic B medium in symmetrical concentric waves of branched dendritic patterns. In a comparison of the behavior of the laboratory strain 168 (trp) on different media with that of 3610, strain 168 (trp), which does not produce surfactin, displayed less swarming activity, both qualitatively (pattern formation) and in speed of colonization. On E and B media, 168 failed to swarm; however, with the latter, swarming was arrested at an early stage of development, with filamentous cells and rafts of cells (characteristic of dendrites of 3610) associated with bud-like structures surrounding the central inoculum. In contrast, strain 168 apparently swarmed efficiently on Luria-Bertani (LB) agar, colonizing the entire plate in 24 h. However, analysis of the intermediate stages of development of swarms on LB medium demonstrated that, in comparison with strain 3610, initiation of swarming of 168 (trp) was delayed and the greatly reduced rate of expansion of the swarm was uncoordinated, with some regions advancing faster than others. Moreover, while early stages of swarming in 3610 are accompanied by the formation of large numbers of dendrites whose rapid advance involves packs of cells at the tips, strain 168 advanced more slowly as a continuous front. When sfp+ was inserted into the chromosome of 168 (trp) to reestablish surfactin production, many features observed with 3610 on LB medium were now visible with 168. However, swarming of 168 (sfp+) still showed some reduced speed and a distinctive pattern compared to swarming of 3610. The results are discussed in terms of the possible role of surfactin in the swarming process and the different modes of swarming on LB medium.
126 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])....
[...]