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Bacteria such Escherichia coli and Bacillus subtilis common occur in ecological and biological environments, where they influence the nature and functional processes of their habitats through their movements and metabolic processes. The cells do not move randomly but are guided by chemical stimuli in the environments; bacteria detect and respond to these stimuli. The responses are manifested statistically as the net movement of a population of cells either toward or away from the stimulus. The term net movement is significant because, as the model considered here shows, at any given time some cells are moving toward the stimulus and some are traveling away; however, the overall macroscopic movement of the population, measured in terms of population density, is decidedly biased in one direction. Important natural situations where such guided movements (or chemotaxis) include wound healing (Agyingi et al. 2010), the operations of microfluidic systems sustaining biochemical reactions (Ahmed et al. 2010), and the degradation of undesirable chemicals in the soil or in water (Singh and Olson 2008). These examples indicate the variety of situations in which chemotaxis has a significant role and the fact that most applications involve the movement of cells toward a stimulus, i.e. a chemoattractant.
In a chemoattractant-containing environment, as stated above, at any given time some cells are moving away from the stimulus and some toward it. Even the cells that are swimming toward the attractant do no travel directly in the most favorable direction but somewhat tangentially. Thus, the cells that move in unfavorable directions as well as those moving favorably need to have their directions corrected frequently to maintain them on the right paths. These correctional movements involve reorientations of the cells, and they are called tumbles. The straight line motions between consecutive tumbles are called runs. Runs and tumbles therefore alternate, and their relative durations determine how efficiently a population moves toward a chemoattractant. Details of the mechanisms of runs and tumbles are well known in both E. coli and B. subtilis (Berg 2000; Blake et al. 2006) but they are not relevant to the scope of this study. What is important is to recognize the simultaneous presence of three kinds of cells: those traveling away from the stimulus, those traveling toward it, and cells that are tumbling.
Unlike the carefully controlled conditions of closed vessels used in laboratory-scale experiments, most real environments are perpetually under the influence of fluctuations (or noise). Moreover, the chemoattractant is not present either as a point source or homogenously distributed, as in a well-stirred vessel, but as a mobile distribution that fluctuates under the impact of external noise (Xu and Tao 2006; Patnaik 2007a). These fluctuating molecules bind to the chemoreceptors on the cell surface, where an additional source of noise is present in the binding process itself (Andrews et al. 2006). Both sources of noise together permeate the cells, where they encounter a third source in the genetic processes (Rao et al. 2002; Kaern et al. 2005; Patnaik 2012a). Since binding with the chemoattractant initiates a chain of events in the chemosensory network, the multiple sources of noise inevitably have an impact on the kinetic parameters characterizing the chemosensory reactions.