Many-body theory of chemotactic cell-cell interactions

We consider an individual-based stochastic model of cell movement mediated by chemical signaling fields. This model is formulated using Langevin dynamics, which allows an analytic study using methods from statistical and many-body physics. In particular we construct a diagrammatic framework within which to study cell-cell interactions. In the mean-field limit, where statistical correlations between cells are neglected, we recover the deterministic Keller-Segel equations. Within exact perturbation theory in the chemotactic coupling ϵ, statistical correlations are non-negligible at large times and lead to a renormalization of the cell diffusion coefficient D_R—an effect that is absent at mean-field level. An alternative closure scheme, based on the necklace approximation, probes the strong coupling behavior of the system and predicts that D_R is renormalized to zero at a critical value of ϵ, indicating self-localization of the cell. Stochastic simulations of the model give very satisfactory agreement with the perturbative result. At higher values of the coupling simulations indicate that D_R∼ϵ⁻², a result at odds with the necklace approximation. We briefly discuss an extension of our model, which incorporates the effects of short-range interactions such as cell-cell adhesion.