Branching instability in expanding bacterial colonies
06 Mar 2015-Journal of the Royal Society Interface (The Royal Society)-Vol. 12, Iss: 104, pp 20141290-20141290
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.
Citations
More filters
Journal Article•
TL;DR: In this paper, the authors report simultaneous measurements of the positions, velocities, and orientations of up to a thousand wild-type Bacillus subtilis bacteria in a colony.
Abstract: Flocking birds, fish schools, and insect swarms are familiar examples of collective motion that plays a role in a range of problems, such as spreading of diseases. Models have provided a qualitative understanding of the collective motion, but progress has been hindered by the lack of detailed experimental data. Here we report simultaneous measurements of the positions, velocities, and orientations as a function of time for up to a thousand wild-type Bacillus subtilis bacteria in a colony. The bacteria spontaneously form closely packed dynamic clusters within which they move cooperatively. The number of bacteria in a cluster exhibits a power-law distribution truncated by an exponential tail. The probability of finding clusters with large numbers of bacteria grows markedly as the bacterial density increases. The number of bacteria per unit area exhibits fluctuations far larger than those for populations in thermal equilibrium. Such “giant number fluctuations” have been found in models and in experiments on inert systems but not observed previously in a biological system. Our results demonstrate that bacteria are an excellent system to study the general phenomenon of collective motion.
394 citations
TL;DR: The physical basis elaborated through this study provides a useful framework for understanding the swarming behavior of numerous species of bacteria, supporting the conclusion that swarming motility is restricted by the surface tension at the swarm front and swarm growth is limited by the rate of water supply from within the agar gel.
65 citations
Cites background or methods from "Branching instability in expanding ..."
...(34,35) draw upon a strikingly similar physical phenomenon, classically known as viscous fingering (59,60), to suggest its relevance to the dendritic spread of bacterial swarm fronts....
[...]
...aeruginosa strains under varied physicochemical conditions (25,26,28–33), the exact mechanism of dendritic growth remains an open question despite several models proposed (25,28,29,34,35)....
[...]
...Examples include a reaction-diffusion model based on the idea that branched protrusions may be driven by opposing biochemical agents of competing drives (57), a model based on Marangoni flow (29,58), and a new analysis of branching instability assuming either volumetric or chemotactic expansion (34,35)....
[...]
...Although several models invoking reaction-diffusion (34,35,57), chemotaxis (33), or quorum sensing (31), are able to yield dendritic or fingering growth at the swarm front, results from the simple manipulation of physical factors presented in this report suggest that biologically specific molecular mechanisms, such as chemotaxis and quorum sensing, are not required for swarming....
[...]
...Indeed, theoretical models based on the reaction-diffusion mechanism can yield dendritic growth both via a linear stability analysis and through computer simulations (34,35,57)....
[...]
TL;DR: Computation of microbial ecosystems in time and space (COMETS) as mentioned in this paper extends dynamic flux balance analysis to generate simulations of multiple microbial species in molecularly complex and spatially structured environments.
Abstract: Genome-scale stoichiometric modeling of metabolism has become a standard systems biology tool for modeling cellular physiology and growth. Extensions of this approach are emerging as a valuable avenue for predicting, understanding and designing microbial communities. Computation of microbial ecosystems in time and space (COMETS) extends dynamic flux balance analysis to generate simulations of multiple microbial species in molecularly complex and spatially structured environments. Here we describe how to best use and apply the most recent version of COMETS, which incorporates a more accurate biophysical model of microbial biomass expansion upon growth, evolutionary dynamics and extracellular enzyme activity modules. In addition to a command-line option, COMETS includes user-friendly Python and MATLAB interfaces compatible with the well-established COBRA models and methods, as well as comprehensive documentation and tutorials. This protocol provides a detailed guideline for installing, testing and applying COMETS to different scenarios, generating simulations that take from a few minutes to several days to run, with broad applicability to microbial communities across biomes and scales.
46 citations
TL;DR: It is proved that the new setting allows an explicit solution to the problem of Operatorial Ordinary Kriging, and the relation of the novel predictor with the key concept of conditional expectation of a Gaussian measure is established.
40 citations
TL;DR: In this article, the spatial field of soil particle-size distributions within a heterogeneous aquifer system is characterized by spatially varying soil textural properties associated with diverse geomaterials.
Abstract: This work addresses the problem of characterizing the spatial field of soil particle-size distributions within a heterogeneous aquifer system. The medium is conceptualized as a composite system, characterized by spatially varying soil textural properties associated with diverse geomaterials. The heterogeneity of the system is modeled through an original hierarchical model for particle-size distributions that are here interpreted as points in the Bayes space of functional compositions. This theoretical framework allows performing spatial prediction of functional compositions through a functional compositional Class-Kriging predictor. To tackle the problem of lack of information arising when the spatial arrangement of soil types is unobserved, a novel clustering method is proposed, allowing to infer a grouping structure from sampled particle-size distributions. The proposed methodology enables one to project the complete information content embedded in the set of heterogeneous particle-size distributions to unsampled locations in the system. These developments are tested on a field application relying on a set of particle-size data observed within an alluvial aquifer in the Neckar river valley, in Germany.
32 citations
References
More filters
[...]
TL;DR: The functions, properties and constituents of the EPS matrix that make biofilms the most successful forms of life on earth are described.
Abstract: The microorganisms in biofilms live in a self-produced matrix of hydrated extracellular polymeric substances (EPS) that form their immediate environment. EPS are mainly polysaccharides, proteins, nucleic acids and lipids; they provide the mechanical stability of biofilms, mediate their adhesion to surfaces and form a cohesive, three-dimensional polymer network that interconnects and transiently immobilizes biofilm cells. In addition, the biofilm matrix acts as an external digestive system by keeping extracellular enzymes close to the cells, enabling them to metabolize dissolved, colloidal and solid biopolymers. Here we describe the functions, properties and constituents of the EPS matrix that make biofilms the most successful forms of life on earth.
7,041 citations
TL;DR: A comprehensive review of spatiotemporal pattern formation in systems driven away from equilibrium is presented in this article, with emphasis on comparisons between theory and quantitative experiments, and a classification of patterns in terms of the characteristic wave vector q 0 and frequency ω 0 of the instability.
Abstract: A comprehensive review of spatiotemporal pattern formation in systems driven away from equilibrium is presented, with emphasis on comparisons between theory and quantitative experiments. Examples include patterns in hydrodynamic systems such as thermal convection in pure fluids and binary mixtures, Taylor-Couette flow, parametric-wave instabilities, as well as patterns in solidification fronts, nonlinear optics, oscillatory chemical reactions and excitable biological media. The theoretical starting point is usually a set of deterministic equations of motion, typically in the form of nonlinear partial differential equations. These are sometimes supplemented by stochastic terms representing thermal or instrumental noise, but for macroscopic systems and carefully designed experiments the stochastic forces are often negligible. An aim of theory is to describe solutions of the deterministic equations that are likely to be reached starting from typical initial conditions and to persist at long times. A unified description is developed, based on the linear instabilities of a homogeneous state, which leads naturally to a classification of patterns in terms of the characteristic wave vector q0 and frequency ω0 of the instability. Type Is systems (ω0=0, q0≠0) are stationary in time and periodic in space; type IIIo systems (ω0≠0, q0=0) are periodic in time and uniform in space; and type Io systems (ω0≠0, q0≠0) are periodic in both space and time. Near a continuous (or supercritical) instability, the dynamics may be accurately described via "amplitude equations," whose form is universal for each type of instability. The specifics of each system enter only through the nonuniversal coefficients. Far from the instability threshold a different universal description known as the "phase equation" may be derived, but it is restricted to slow distortions of an ideal pattern. For many systems appropriate starting equations are either not known or too complicated to analyze conveniently. It is thus useful to introduce phenomenological order-parameter models, which lead to the correct amplitude equations near threshold, and which may be solved analytically or numerically in the nonlinear regime away from the instability. The above theoretical methods are useful in analyzing "real pattern effects" such as the influence of external boundaries, or the formation and dynamics of defects in ideal structures. An important element in nonequilibrium systems is the appearance of deterministic chaos. A greal deal is known about systems with a small number of degrees of freedom displaying "temporal chaos," where the structure of the phase space can be analyzed in detail. For spatially extended systems with many degrees of freedom, on the other hand, one is dealing with spatiotemporal chaos and appropriate methods of analysis need to be developed. In addition to the general features of nonequilibrium pattern formation discussed above, detailed reviews of theoretical and experimental work on many specific systems are presented. These include Rayleigh-Benard convection in a pure fluid, convection in binary-fluid mixtures, electrohydrodynamic convection in nematic liquid crystals, Taylor-Couette flow between rotating cylinders, parametric surface waves, patterns in certain open flow systems, oscillatory chemical reactions, static and dynamic patterns in biological media, crystallization fronts, and patterns in nonlinear optics. A concluding section summarizes what has and has not been accomplished, and attempts to assess the prospects for the future.
6,145 citations
TL;DR: In this paper, it was shown that a flow is possible in which equally spaced fingers advance steadily at very slow speeds, such that behind the tips of the advancing fingers the widths of the two columns of fluid are equal.
Abstract: When a viscous fluid filling the voids in a porous medium is driven forwards by the pressure of another driving fluid, the interface between them is liable to be unstable if the driving fluid is the less viscous of the two. This condition occurs in oil fields. To describe the normal modes of small disturbances from a plane interface and their rate of growth, it is necessary to know, or to assume one knows, the conditions which must be satisfied at the interface. The simplest assumption, that the fluids remain completely separated along a definite interface, leads to formulae which are analogous to known expressions developed by scientists working in the oil industry, and also analogous to expressions representing the instability of accelerated interfaces between fluids of different densities. In the latter case the instability develops into round-ended fingers of less dense fluid penetrating into the more dense one. Experiments in which a viscous fluid confined between closely spaced parallel sheets of glass, a Hele-Shaw cell, is driven out by a less viscous one reveal a similar state. The motion in a Hele-Shaw cell is mathematically analogous to two-dimensional flow in a porous medium.
Analysis which assumes continuity of pressure through the interface shows that a flow is possible in which equally spaced fingers advance steadily. The ratio λ = (width of finger)/(spacing of fingers) appears as the parameter in a singly infinite set of such motions, all of which appear equally possible. Experiments in which various fluids were forced into a narrow Hele-Shaw cell showed that single fingers can be produced, and that unless the flow is very slow λ = (width of finger)/(width of channel) is close to , so that behind the tips of the advancing fingers the widths of the two columns of fluid are equal. When λ = 1/2 the calculated form of the fingers is very close to that which is registered photographically in the Hele-Shaw cell, but at very slow speeds where the measured value of λ increased from 1/2 to the limit 1.0 as the speed decreased to zero, there were considerable differences. Assuming that these might be due to surface tension, experiments were made in which a fluid of small viscosity, air or water, displaced a much more viscous oil. It is to be expected in that case that λ would be a function of μU/T only, where μ is the viscosity, U the speed of advance and T the interfacial tension. This was verified using air as the less viscous fluid penetrating two oils of viscosities 0.30 and 4.5 poises.
2,634 citations
TL;DR: Comparing different types of collective migration at the molecular and cellular level reveals a common mechanistic theme between developmental and cancer research.
Abstract: The collective migration of cells as a cohesive group is a hallmark of the tissue remodelling events that underlie embryonic morphogenesis, wound repair and cancer invasion. In such migration, cells move as sheets, strands, clusters or ducts rather than individually, and use similar actin- and myosin-mediated protrusions and guidance by extrinsic chemotactic and mechanical cues as used by single migratory cells. However, cadherin-based junctions between cells additionally maintain 'supracellular' properties, such as collective polarization, force generation, decision making and, eventually, complex tissue organization. Comparing different types of collective migration at the molecular and cellular level reveals a common mechanistic theme between developmental and cancer research.
2,397 citations
TL;DR: In this article, the authors examined several common modes of crystal growth and identified a few new theoretical ideas and a larger number of outstanding problems, including sidebranching and tip-splitting instabilities.
Abstract: Several common modes of crystal growth provide particularly simple and elegant examples of spontaneous pattern formation in nature. Phenomena of interest here are those in which an advancing nonfaceted solidification front suffers an instability and subsequently reorganizes itself into a more complex mode of behavior. The purpose of this essay is to examine several such situations and, in doing this, to identify a few new theoretical ideas and a larger number of outstanding problems. The systems studied are those in which solidification is controlled entirely by a single diffusion process, either the flow of latent heat away from a moving interface or the analogous redistribution of chemical constituents. Convective effects are ignored, as are most effects of crystalline anisotropy. The linear theory of the Mullins-Sekerka instability is reviewed for simple planar and spherical cases and also for a special model of directional solidification. These techniques are then extended to the case of a freely growing dendrite, and it is shown how this analysis leads to an understanding of sidebranching and tip-splitting instabilities. A marginal-stability hypothesis is introduced; and it is argued that this intrinsically nonlinear theory, if valid, permits aone to use results of linear-stability analysis to predict dendritic growth rates. The review concludes with a discussion of nonlinear effects in directional solidication. The nonplanar, cellular interfaces which emerge in this situation have much in common with convection patterns in hydrodynamics. The cellular stability problem is discussed briefly, and some preliminary attempts to do calculations in the strongly nonlinear regime are summarized.
1,969 citations