# Chemotactic Patterns without Chemotaxis

Entry: Chia Wei Hsu, AP 225, Fall 2010

M.P. Brenner, Chemotactic Patterns without Chemotaxis, Proc. Natl. Acad. Sci, **107**, 11653–11654 (2010).

## Contents

## Summary

This commentary addresses the study by Cates et al (ref 1) where an effective model is used to describe the pattern formation in chemotaxis. In particular, the effective model considers not the mechanisms of the chemotaxis but the averaged effect of it, thus greatly reducing the complexity of the modeling. Brenner uses this study as an example to show the promise of using effective models to understand biological systems.

## Pattern Formation in Chemotaxis

Chemotaxis is the phenomenon where bacteria and other organisms direct their motion based on the environment. For example, bacteria can swim up the food concentration gradient to get to a region with more food, or swim down the poison concentration gradient to avoid poisoning.

The chemotaxis phenomenon results in pattern formation. Is has been found that (ref 2) when chemotactic bacteria swim through a small tube of rich medium, the bacteria form a dense band that moves at constant velocity. Subsequent theoretical works concluded that in order to qualitatively reproduce this behavior, detailed knowledge of the nutrient consumption and the production/depletion of chemo-attractant is required. It is disturbing that such a simple collective behavior requires such a detailed level of understanding. This motivates the research for a simpler description.

## Effective Model

The model of Cates et al (ref 1) boldly ignores the direct interaction between bacteria and attractant fields. Rather, they assume a density-dependent swim speed <math>v=v(\rho)</math>, where <math>\rho</math> is the density of bacteria. This swim speed is assumed to decrease with increasing <math>\rho</math>. Thus, there is a net drift in the direction of increasing bacteria density. Since bacteria density tends to be higher in regions with more attractant, this models creates the same effect: that bacteria drift toward high attractant concentration. However, there is a fundamental difference in the modeling: this effective model neglects chemotaxis completely and instead asserts that the fundamental quantity is the density-dependent velocity <math>v=v(\rho)</math>.

Cates et al demonstrate that this model quantitatively reproduces the results of a set of experiments on bacteria pattern formation. Same agreement can be found in more detailed modeling, but this effective model reduces the complexity of the model to only two dimensionless parameters.

## Connection to Soft Matter

Brenner's comment and this study by Cates et al point out the potential strength of effective modeling. With effective modeling, we can understand the most important underpinning driving forces in phenomenon we observe, rather than being overwhelmed by the complexity of the modeling and only being able to examine the numbers we get out of the simulations or numerical solutions. Such approach can help us to understand many biological and soft matter systems. Of course there is a trade-off. Such effective models have to be tested more rigorously in order to ascertain that it not only captures the correct result, but also have the right assumption.

## References

[1] Cates ME, Marenduzzo D, Pagonabarraga I, and Tailleur J, "Arrested phase separation in reproducing bacteria: A generic route to pattern formation," Proc Natl Acad Sci USA **107**, 11715–11720 (2010).

[2] Adler J, "Chemotaxis in bacteria," Science **153**, 708–716 (1966).