Additional file 2 of Cell adhesion heterogeneity reinforces tumour cell dissemination: novel insights from a mathematical model

LGCA transition dynamics. LGCA transition dynamics D : = R ∘ A $\mathcal {D}:=\mathcal {R} \, \circ \, \mathcal {A}$ in a von Neumann neighbourhood N r $\mathcal {N}_{\boldsymbol {r}}$ around node r (gray) followed by translocation T i $\mathcal {T}_{i}$ . (a) During reorientation R $\mathcal {R}$ cells are stochastically redistributed within nodes according to a probability function P. The highest probability is assigned to a resulting node configuration η ′(r)with post-reorientation momentum J:=J(η ′,a ″)(r)parallel to the pre-reorientation local adhesivity gradient G:=G(η,a ′)(r)of the neighbourhood excluding r [Fig. 1b]. (b) In our model, the newly introduced adhesivity update operator A $\mathcal {A}$ couples the time scales of the LGCA and ODE models. The graph shows an example for the analytical solution y r (t)[Eq. (1)] of the underlying ODE [Eq. 1] according to which A $\mathcal {A}$ decreases the adhesive states a i (r,k)of all cells as a i (r,k)>a i (r,k+τ) ∀ k; in this example there is no adhesion heterogeneity. The update from time-step k = t to k+τ = t+τ shown here is labelled red in the graph. After reorientation, cells are moved by the translocation operator T i $\mathcal {T}_{i}$ (see Additional file 1 for details). Note that all nodes have only one rest channel. Notation as in Fig. 1. (PDF 28 kb)