508 research outputs found
Derivation of the bacterial run-and-tumble kinetic equation from a model with biochemical pathway
Kinetic-transport equations are, by now, standard models to describe the
dynamics of populations of bacteria moving by run-and-tumble. Experimental
observations show that bacteria increase their run duration when encountering
an increasing gradient of chemotactic molecules. This led to a first class of
models which heuristically include tumbling frequencies depending on the
path-wise gradient of chemotactic signal.
More recently, the biochemical pathways regulating the flagellar motors were
uncovered. This knowledge gave rise to a second class of kinetic-transport
equations, that takes into account an intra-cellular molecular content and
which relates the tumbling frequency to this information. It turns out that the
tumbling frequency depends on the chemotactic signal, and not on its gradient.
For these two classes of models, macroscopic equations of Keller-Segel type,
have been derived using diffusion or hyperbolic rescaling. We complete this
program by showing how the first class of equations can be derived from the
second class with molecular content after appropriate rescaling. The main
difficulty is to explain why the path-wise gradient of chemotactic signal can
arise in this asymptotic process.
Randomness of receptor methylation events can be included, and our approach
can be used to compute the tumbling frequency in presence of such a noise
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
Overview of mathematical approaches used to model bacterial chemotaxis II: bacterial populations
We review the application of mathematical modeling to understanding the behavior of populations of chemotactic bacteria. The application of continuum mathematical models, in particular generalized Keller–Segel models, is discussed along with attempts to incorporate the microscale (individual) behavior on the macroscale, modeling the interaction between different species of bacteria, the interaction of bacteria with their environment, and methods used to obtain experimentally verified parameter values. We allude briefly to the role of modeling pattern formation in understanding collective behavior within bacterial populations. Various aspects of each model are discussed and areas for possible future research are postulated
Volcano effect in chemotactic aggregation and an extended Keller-Segel mode
Aggregation of chemotactic bacteria under a unimodal distribution of chemical
cues was investigated by Monte Carlo simulation and asymptotic analysis based
on a kinetic transport equation, which considers an internal adaptation
dynamics as well as a finite tumbling duration. It was found that there exist
two different regimes of the adaptation time, between which the effect of the
adaptation time on the aggregation behavior is reversed; that is, when the
adaptation time is as small as the running duration, the aggregation becomes
increasingly steeper as the adaptation time increases, while, when the
adaptation time is as large as the diffusion time of the population density,
the aggregation becomes more diffusive as the adaptation time increases.
Moreover, notably, the aggregation profile becomes bimodal (volcano) at the
large adaptation-time regime while it is always unimodal at the small
adaptation-time regime. The volcano effect occurs in such a way that the
population of tumbling cells considerably decreases in a diffusion layer which
is created near the peak of the external chemical cues due to the coupling of
diffusion and internal adaptation of the bacteria. Two different
continuum-limit models are derived by the asymptotic analysis according to the
scaling of the adaptation time; that is, at the small adaptation-time regime,
the Keller-Segel model while, at the large adaptation-time regime, an extension
of KS model, which involves both the internal variable and the tumbling
duration. Importantly, either of the models can accurately reproduce the MC
results at each adaptation-time regime, involving the volcano effect. Thus, we
conclude that the coupling of diffusion, adaptation, and finite tumbling
duration is crucial for the volcano effect
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
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