Additional File 1: Cell adhesion heterogeneity reinforces tumour cell dissemination: novel insights from a mathematical model - Model details

Correspondence: david [email protected] Department of Evolutionary Genetics, Max Planck Institute for Evolutionary Anthropology, Deutscher Platz 6, 04103 Leipzig, Germany Center for Information Services and High Performance Computing, Technische Universität Dresden, Nöthnitzer Str. 46, 01062 Dresden, Germany Full list of author information is available at the end of the article Formal description of the LGCA model LGCA model The LGCA model is defined on a discrete 2-dimensional square lattice L with periodic boundary conditions [2, 3]. The lattice-gas model used in our work is an extension of cellular automata with binary states that has first been used in statistical physics and fluid mechanics (see [1] for an overwiew). Each lattice node r ∈ L is connected to its four nearest neighbours, forming its von Neumann neighbourhood Nr, by unit vectors ci, i = 0, ..., 3, called velocity channels. The total number of channels per node is defined by κ, ands β := κ − 4 is an arbitrary number of channels with zero velocity, called rest channels, in which ci = 0, 4 ≤ i < κ. Each channel can be occupied by at most one cell at a time. In occupied channels, the occupation state ηi(r) = 1, i = 1, ..., κ, whereas for empty channels ηi(r) = 0. If ηi(r) = 1, the occupying cell’s adhesive state is described by the variable ai(r) ∈ R := [0,∞). Occupation states ηi(r) and adhesive states ai(r) of all channels in a node r give the node configuration (η,a)(r), formally defined as (η,a)(r) := ((η0, ..., ηκ−1), (a0, ..., aκ−1))(r) ∈ Ea := {0, 1} × R κ . Fig. 1 (a) illustrates the state space of the LGCA model. LGCA dynamics are characterised by a transition operatorD : Ea → Ea, (η,a)(r) 7→ (η′,a′′)(r), that updates a given node configuration (η,a)(r, k) := (η,a)(r) to a subsequent node configuration (η,a)(r, k+τ) := (η′,a′′)(r) at time k+τ ∈ K and is simultaneously applied to each node r ∈ L at discrete time k ∈ K := {j τ | j ∈ N}. The time-step length τ ∈ R, τ > 0 is constant. We define D as the composition of two operators: • The deterministic adhesivity change operator