MACROSCALE MATHEMATICAL MODELS FOR E.COLI MOTION

Last modified: 2014-03-28

#### Abstract

Chemotaxis of E. coli bacteria in a chemoattractant field is characterized by run-and-tumble motion. The bacteria are too small to sense spatial gradients along the body axis, instead they use memory to sense changes in concentration over time. Our aim is to develop a mathematical model in which chemotaxis is achieved through sensing temporal gradients. We start with a microscopic description of the motion of a single bacterium. The decision of a twiddle is done comparing the concentration registered at present with an average of the concentration sensed in the past. From this starting point we specifically describe the movement of E. coli via different models: 1) a piecewise deterministic Markov process with exponentially distributed delays; 2) a piecewise deterministic process whose dynamic is determined by the hitting time of a stochastic functional differential equation - at the time of jumps the bacteria change its direction of movement according to a probability distribution derived by experimental data; and 3) a Langevin-type stochastic functional differential equation with time-inhomogeneous coefficients. The simulation of the process shows that the probability distribution of the population follows the time-varying landscape of the chemical attractant. Thereby the model captures not only chemotaxis along spatial gradients, but also other features of E. coli motion like chemokinesis and adaptation.