Kellermodellen

Kellermodellen, usually referred to in English as the Keller–Segel model, is a mathematical framework used to describe chemotaxis, the directed movement of cells in response to chemical signals. It was introduced by Evelyn Keller and Leo Segel in the 1970s to explain how cells such as Dictyostelium discoideum aggregate in response to chemoattractants. In its classical form the model describes the evolution of a cell density n(x,t) and a chemical concentration c(x,t) on a spatial domain Ω with boundary conditions.

The standard form consists of a coupled system of partial differential equations. A common parabolic form is: ∂t n = ∇·(D ∇n - χ n ∇c) and ∂t c = D_c ∆c + α n − β c, where D is the cell diffusion coefficient, χ the chemotactic sensitivity, D_c the chemical diffusion, α the production rate, and β the decay rate. A parabolic-elliptic reduction replaces the second equation by 0 = ∆c − κ c + γ n. The model captures chemotactic drift toward higher concentrations of the chemical, balanced by random diffusion.

Key features include the possibility of cell aggregation and pattern formation, and, in two dimensions, a threshold phenomenon where a critical mass can lead to finite-time blow-up (in the mathematical sense) representing concentration of cells. Global existence holds for small mass, while larger mass can produce sharp peaks.

Variants of Kellermodellen incorporate logistic growth, volume-filling effects, multiple chemicals, or more complex kinetics. The model has been applied to bacterial colonies, amoebae, and embryonic development studies.

See also: Keller–Segel equations, chemotaxis, mathematical biology. References: Keller and Segel (1970) JTB; reviews in mathematical biology.