The Global Turning Probability Density Function for Motile Bacteria and its Applications

The angular turning probability density distribution for motile bacteria is usually measured in local coordinates and is therefore inconvenient for global analyses of the chemotactic bacterial migration. In this paper we present analytical derivations that convert the local angular turning probability density distribution into a global one. The explicit expression of a reduced global turning probability density function for motile bacteria was derived and its relevant properties were investigated. Depending on the angle variable being intergrated and the integration range, three types of cosine moments were separately defined and studied. Some statistical indices and parameters such as the directional persistence, persistence number, and one-dimensional reversal probability were found to be embedded in the various moments of the reduced global turning probability density function. Applications of the reduced global turning probability and its integrated moments to a three-dimensional cell balance equation in an axisymmetric system were also discussed.