56 research outputs found
From Individual to Collective Behavior of Unicellular Organisms: Recent Results and Open Problems
The collective movements of unicellular organisms such as bacteria or amoeboid (crawling) cells are often modeled by partial differential equations (PDEs) that describe the time evolution of cell density. In particular, chemotaxis equations have been used to model the movement towards various kinds of extracellular cues. Well-developed analytical and numerical methods for analyzing the time-dependent and time-independent properties of solutions make this approach attractive. However, these models are often based on phenomenological descriptions of cell fluxes with no direct correspondence to individual cell processes such signal transduction and cell movement. This leads to the question of how to justify these macroscopic PDEs from microscopic descriptions of cells, and how to relate the macroscopic quantities in these PDEs to individual-level parameters. Here we summarize recent progress on this question in the context of bacterial and amoeboid chemotaxis, and formulate several open problems
Multiscale modeling in biology
The 1966 science-fction film Fantastic Voyage captured the public imagination with a clever idea: what fantastic things might we see and do if we could minaturize ourselves and travel through the bloodstream as corpuscles do? (This being Hollywood, the answer was that we'd save a fellow scientist from evildoers.
Travelling waves in hyperbolic chemotaxis equations
Mathematical models of bacterial populations are often written as systems of partial differential equations for the densities of bacteria and concentrations of extracellular (signal) chemicals. This approach has been employed since the seminal work of Keller and Segel in the 1970s [Keller and Segel, J. Theor. Biol., 1971]. The system has been shown to permit travelling wave solutions which correspond to travelling band formation in bacterial colonies, yet only under specific criteria, such as a singularity in the chemotactic sensitivity function as the signal approaches zero. Such a singularity generates infinite macroscopic velocities which are biologically unrealistic. In this paper, we formulate a model that takes into consideration relevant details of the intracellular processes while avoiding the singularity in the chemotactic sensitivity. We prove the global existence of solutions and then show the existence of travelling wave solutions both numerically and analytically
Radial and spiral stream formation in Proteus mirabilis
The enteric bacterium Proteus mirabilis, which is a pathogen that forms
biofilms in vivo, can swarm over hard surfaces and form concentric ring
patterns in colonies. Colony formation involves two distinct cell types:
swarmer cells that dominate near the surface and the leading edge, and swimmer
cells that prefer a less viscous medium, but the mechanisms underlying pattern
formation are not understood. New experimental investigations reported here
show that swimmer cells in the center of the colony stream inward toward the
inoculation site and in the process form many complex patterns, including
radial and spiral streams, in addition to concentric rings. These new
observations suggest that swimmers are motile and that indirect interactions
between them are essential in the pattern formation. To explain these
observations we develop a hybrid cell-based model that incorporates a
chemotactic response of swimmers to a chemical they produce. The model predicts
that formation of radial streams can be explained as the modulation of the
local attractant concentration by the cells, and that the chirality of the
spiral streams can be predicted by incorporating a swimming bias of the cells
near the surface of the substrate. The spatial patterns generated from the
model are in qualitative agreement with the experimental observations
Pattern formation of a pathway-based diffusion model: linear stability analysis and an asymptotic preserving method
We investigate the linear stability analysis of a pathway-based diffusion
model (PBDM), which characterizes the dynamics of the engineered Escherichia
coli populations [X. Xue and C. Xue and M. Tang, P LoS Computational Biology,
14 (2018), pp. e1006178]. This stability analysis considers small perturbations
of the density and chemical concentration around two non-trivial steady states,
and the linearized equations are transformed into a generalized eigenvalue
problem. By formal analysis, when the internal variable responds to the outside
signal fast enough, the PBDM converges to an anisotropic diffusion model, for
which the probability density distribution in the internal variable becomes a
delta function. We introduce an asymptotic preserving (AP) scheme for the PBDM
that converges to a stable limit scheme consistent with the anisotropic
diffusion model. Further numerical simulations demonstrate the theoretical
results of linear stability analysis, i.e., the pattern formation, and the
convergence of the AP scheme
A pathway-based mean-field model for E. coli chemotaxis: Mathematical derivation and Keller-Segel limit
A pathway-based mean-field theory (PBMFT) was recently proposed for E. coli
chemotaxis in [G. Si, T. Wu, Q. Quyang and Y. Tu, Phys. Rev. Lett., 109 (2012),
048101]. In this paper, we derived a new moment system of PBMFT by using the
moment closure technique in kinetic theory under the assumption that the
methylation level is locally concentrated. The new system is hyperbolic with
linear convection terms. Under certain assumptions, the new system can recover
the original model. Especially the assumption on the methylation difference
made there can be understood explicitly in this new moment system. We obtain
the Keller-Segel limit by taking into account the different physical time
scales of tumbling, adaptation and the experimental observations. We also
present numerical evidence to show the quantitative agreement of the moment
system with the individual based E. coli chemotaxis simulator.Comment: 21 pages, 3 figure
Hybrid modelling of individual movement and collective behaviour
Mathematical models of dispersal in biological systems are often written in terms of partial differential equations (PDEs) which describe the time evolution of population-level variables (concentrations, densities). A more detailed modelling approach is given by individual-based (agent-based) models which describe the behaviour of each organism. In recent years, an intermediate modelling methodology β hybrid modelling β has been applied to a number of biological systems. These hybrid models couple an individual-based description of cells/animals with a PDEmodel of their environment. In this chapter, we overview hybrid models in the literature with the focus on the mathematical challenges of this modelling approach. The detailed analysis is presented using the example of chemotaxis, where cells move according to extracellular chemicals that can be altered by the cells themselves. In this case, individual-based models of cells are coupled with PDEs for extracellular chemical signals. Travelling waves in these hybrid models are investigated. In particular, we show that in contrary to the PDEs, hybrid chemotaxis models only develop a transient travelling wave
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