# The fractional diffusion limit of a kinetic model with biochemical pathway

@article{Perthame2017TheFD, title={The fractional diffusion limit of a kinetic model with biochemical pathway}, author={B. Perthame and W. Sun and M. Tang}, journal={Zeitschrift f{\"u}r angewandte Mathematik und Physik}, year={2017}, volume={69}, pages={1-15} }

Kinetic-transport equations that take into account the intracellular pathways are now considered as the correct description of bacterial chemotaxis by run and tumble. Recent mathematical studies have shown their interest and their relations to more standard models. Macroscopic equations of Keller–Segel type have been derived using parabolic scaling. Due to the randomness of receptor methylation or intracellular chemical reactions, noise occurs in the signaling pathways and affects the tumbling… Expand

#### 10 Citations

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#### References

SHOWING 1-10 OF 38 REFERENCES

Derivation of the bacterial run-and-tumble kinetic equation from a model with biochemical pathway

- Physics, Medicine
- Journal of mathematical biology
- 2016

This program shows how the first class of equations can be derived from the second class with molecular content after appropriate rescaling, and how Randomness of receptor methylation events can be included and used to compute the tumbling frequency in presence of such a noise. Expand

Macroscopic Limits of Pathway-Based Kinetic Models for E. coli Chemotaxis in Large Gradient Environments

- Physics, Mathematics
- Multiscale Model. Simul.
- 2017

This work derives macroscopic models for E.coli chemotaxis that match quantitatively with the agent-based model (SPECS) for all ranges of the spacial gradient, in particular when the chemical gradient is large such that the standard Keller-Segel model is no longer valid. Expand

Macroscopic equations for bacterial chemotaxis: integration of detailed biochemistry of cell signaling

- Biology, Medicine
- Journal of mathematical biology
- 2015

It is analytically show that the macroscopic bacterial density can be approximated by the Patlak–Keller–Segel equation in response to signals that change slowly in space and time. Expand

A Pathway-Based Mean-Field Model for E. coli Chemotaxis: Mathematical Derivation and Its Hyperbolic and Parabolic Limits

- Mathematics, Computer Science
- Multiscale Model. Simul.
- 2014

A new kinetic system of PBMFT is derived under the assumption that the methylation level is locally concentrated, whose turning operator takes into account the dynamical intracellular pathway and hence is more physically relevant. Expand

From Individual to Collective Behavior in Bacterial Chemotaxis

- Mathematics, Computer Science
- SIAM J. Appl. Math.
- 2004

This work derives and analyzes a macroscopic system of hyperbolic differential equations describing the motion of individuals such as bacteria from a microscopic model of the behavior of individual cells using moment closure techniques in one space dimension. Expand

A pathway-based mean-field model for E. coli chemotaxis: Mathematical derivation and Keller-Segel limit

- Mathematics, Biology
- 2013

A new moment system of PBMFT is derived by using the moment closure technique in kinetic theory under the assumption that the methylation level is locally concentrated and is hyperbolic with linear convection terms. Expand

ABOUT THE KINETIC DESCRIPTION OF FRACTIONAL DIFFUSION EQUATIONS MODELING CHEMOTAXIS

- Mathematics
- 2016

In this paper, we are interested in the microscopic description of fractional diffusion chemotactic models. We will use the kinetic framework of collisional equations having a heavy-tailed… Expand

Kinetic models for chemotaxis: Hydrodynamic limits and spatio-temporal mechanisms

- Physics, Medicine
- Journal of mathematical biology
- 2005

We study kinetic models for chemotaxis, incorporating the ability of cells to assess temporal changes of the chemoattractant concentration as well as its spatial variations. For prescribed smooth… Expand

Quantitative Modeling of Escherichia coli Chemotactic Motion in Environments Varying in Space and Time

- Physics, Medicine
- PLoS Comput. Biol.
- 2010

The model can be used to study E. coli chemotaxis behavior in arbitrary spatiotemporally varying environments and agrees quantitatively with the classical capillary assay experiments where the attractant concentration changes both in space and time. Expand

Pathway-based mean-field model for Escherichia coli chemotaxis.

- Physics, Medicine
- Physical review letters
- 2012

A mean-field theory for Escherichia coli chemotaxis is developed based on the coupled spatiotemporal dynamics of the cell population and the mean receptor methylation level field that reveals a simple scaling dependence of the chemot axis velocity on the adaptation rate in exponential gradients. Expand