From Individual to Collective Behavior in Bacterial Chemotaxis

Abstract

Bacterial chemotaxis is widely studied from both the microscopic (cell) and macroscopic (population) points of view, and here we connect these different levels of description by deriving the classical macroscopic description for chemotaxis from a microscopic model of the behavior of individual cells. The analysis is based on the velocity jump process for describing the motion of individuals such as bacteria, wherein each individual carries an internal state that evolves according to a system of ordinary differential equations forced by a time-and/or space-dependent external signal. In the problem treated here the turning rate of individuals is a functional of the internal state, which in turn depends on the external signal. Using moment closure techniques in one space dimension, we derive and analyze a macroscopic system of hyperbolic differential equations describing this velocity jump process. Using a hyperbolic scaling of space and time we obtain a single second-order hyperbolic equation for the populations density, and using a parabolic scaling we obtain the classical chemotaxis equation, wherein the chemotactic sensitivity is now a known function of parameters of the internal dynamics. Numerical simulations show that the solutions of the macroscopic equations agree very well with the results of Monte Carlo simulations of individual movement.

Lyapunov–Schmidt and Centre Manifold Reduction Methods for Nonlocal PDEs Modelling Animal Aggregations

“…Moreover, the coupling of PDEs for cell movement with ODEs for the dynamics of cell receptors has been considered not only for cancer cells but also for other types of cells, such as Dictyostelium cells (Höfer et al, 1995a,b). However, the past 10-15 years have seen the development of structured cell population models (described by mesoscale models) where the macroscopic cell dynamics is dependent on a microscopic variable that characterises the cells at molecular level (Erban & Othmer, 2004Xue et al, 2011;Lorenz & Surulescu, 2014;Engwer et al, 2015;Perthame et al, 2016;Engwer et al, 2017;Hunt & Surulescu, 2017). In Appendix B we present a modified mesoscale version of model (2.13)-(2.16) where we assume that the two cancer cell populations are structured by an internal variable that describes the integrin level.…”

Reaction-Diffusion Processes

“…We devoted the most space to the derivation of formula (9) which is (to our knowledge) a new mathematical result. Derivation of formulas (7) and (13) is simple and we included the mathematical arguments for completeness.…”

“…Comparing this formula with (9) or (13), we see that the Robin boundary condition is different for Position Jump Process I and Position Jump Process II even if we scale the adsorption probability with the same factor √ ∆t. The velocity jump processes can also be further compared.…”

“…Using standard methods [15,9], one can show that the density of molecules n(x, t) satisfies the damped wave (telegrapher's) equation 1 2λ…”

“…Equation (51) is solved in the interval [0, 3] together with Robin boundary condition (3), with K given by (9), and with no-flux boundary condition at x = 3. Using P 2 = 5, formula (9) implies that K .…”