Restoration of Coherent Population Movement from Noise-Induced Chaos in the Chemotaxis of E. Coli: A Fractal Interpretation

Introduction

Many biological process may be analyzed usefully through fractal measures of the time-domain or spatial profiles of one or more key variables. The systems for which fractal analysis is applicable vary widely, as illustrated by applications to microbial flocs (Li & Leung, 2005), treatment regimens for cancer (Bizzari, Giuliani, Cucunia, Anselmi, Soto, & Sonnenschein, 2011), the dispersal of pathogenic organisms in the atmosphere (Kenkel & Irwin, 2004), and structure function analysis of complex biological molecules (Mirny, 2011).

Although the applications cited above differ in their scope, their spatial and time domains and in the microorganisms involved, two features are common to most of them. One is the regulated movement of a population of cells (Escherichia coli in this work) in the increasing direction of a chemical stimulant, and the other is the presence of fluctuations n the observed variables due to the impingement of noise from different sources. While these features have been tacitly acknowledged to exist in these examples, other studies have concentrated specifically on the mechanistic and/or quantitative analysis of chemically driven bacterial movement per se. This phenomenon is called chemotaxis, meaning the taxis of a swarm of cells in response to a chemical stimulus. Chemotaxis has been explicitly recognized to have important implications in diverse processes such as wound healing (Agyingi, Maggelakis, & Rose, 2010), the operations of microfluidic reactors (Ahmed, Shimizu, & Stocker, 2010), the diagnosis of Alzheimer’s disease (Magdalena, 2002), and the degradation of harmful chemicals in the soil or in water⁸. In these and many other situations the cells move toward increasing concentration of a chemical gradient. This observation has two significant features: (a) chemotaxis generally involves cellular movement toward, and not away from, a chemical stimulant and (b) the movement is “uphill” of a gradient, which is against the normal direction of Fickian motion. Thus, most useful examples of chemotaxis have chemoattractants, and gradient sensing is a critical feature that determines population movement.

Fluctuations (or “noise”) are a ubiquitous feature of many real biological processes, including bacterial chemotaxis. The noise may emanate either from within the cells or from outside. Intra-cellular noise has received greater attention in research studies, both for its mechanisms; Rao, Wolf, & Arkin, 2002; Shibata & Ueda, 2008; Patnaik, 2012a) and for certain quantitative effects (Paulsson, 2004; Swain, Elowitz, & Siggia, 2002; Elmonet & Cluzel, 2008), than noise from two other main sources. One of these is at the sites of binding of the ligands of the chemoattractant molecules to compatible sensory receptors attached to the cells. The other source is in the extra cellular environment, whose myriad disturbances are manifested in fluctuations in the concentration of the chemoattractant. Despite having fewer quantitative analyses than intra-cellular noise, results from models of ligand-receptor binding noise (Andrews, Yi, & Iglesias, 2006; Patnaik, 2010; Sartori & Yu, 2011) and environment noise (Patnaik, 2007b; Xu & Tao, 2006) provide persuasive arguments that their permeation into chemically motile cells and subsequent interaction with intra-cellular noise can significantly distort the chemotactic behavoior of a bacterial population, even driving the ensemble to stochastic chaos.