Fig 1.
The key question is to identify the interaction rules underlying a particular collective phenomenon in a population of cells. BIO-LGCA interaction rules can be chosen ad hoc, extracted from experimental single cell migration data, or derived from biophysical equations for single cell migration.
Fig 2.
Lattice and neighbourhood in the BIO-LGCA.
Example of square lattice (left). The node state is represented by the occupation of velocity channels (right); in the example, there are four velocity channels c1, c2, c3, c4, corresponding to the lattice directions, and one “rest channel” c5. Filled dots denote the presence of a cell in the respective velocity channel; right: von Neumann neighbourhood (green) of the red node. Black dots represent interaction partners of the red dot. Gray dots lie outside of the interaction neighbourhood and do not interact with the red dot.
Fig 3.
Operator-based dynamics of the BIO-LGCA.
Propagation , reorientation
, phenotypic switch
, and birth/death operators
(top); conservation laws maintained by the different operators (middle); sketches of the operator dynamics (bottom), see text for explanations.
Fig 4.
Rule generation in BIO-LGCA models.
A: Starting from the Langevin equations of a self-propelled particle model, the interaction rule can be obtained from the steady-state distribution of the associated Fokker-Planck equation. B: Alternatively, experimental observables can be used as input for the maximum caliber theory to derive the probabilities of cell tracks and the corresponding interaction rule.
Fig 5.
Basic interactions with neighbourhood impact.
Node configuration (red) before and after application of stochastic interaction rule: cell-cell attraction (top), polar cell alignment (bottom). Gray dots represent cells outside the interaction neighbourhood.
Fig 6.
Pattern formation in the LGCA aggregation model.
Left: critical wave length obtained from the mean-field analysis. The critical wave length diverges for , and
. Right: emergence of a periodic pattern from a homogeneous initial state. The horizontal distance of the orange dashed lines is equal to the critical wave length predicted by mean-field analysis. Parameters:
.
Fig 7.
Migration modes in the BIO-LGCA model.
A: Phase diagram for low (left) and high (right) values of βrest. B-E: Snapshots of the different phases of cell density (left) and local flux (right): B: diffusive motion, if all sensitivities β are low: cells interact weakly with one another and with the ECM; C: collective motion for high βalign; D: pattern formation for high βagg; E: jammed state for high βrest.