title: Extracting cellular automaton rules from physical Langevin equation models for single and collective cell migration.
abstract: Cellular automata (CA) are discrete time, space, and state models which are extensively used for modeling biological phenomena. CA are "on-lattice" models with low computational demands. In particular, lattice-gas cellular automata (LGCA) have been introduced as models of single and collective cell migration. The interaction rule dictates the behavior of a cellular automaton model and is critical to the model's biological relevance. The LGCA model's interaction rule has been typically chosen phenomenologically. In this paper, we introduce a method to obtain lattice-gas cellular automaton interaction rules from physically-motivated "off-lattice" Langevin equation models for migrating cells. In particular, we consider Langevin equations related to single cell movement (movement of cells independent of each other) and collective cell migration (movement influenced by cell-cell interactions). As examples of collective cell migration, two different alignment mechanisms are studied: polar and nematic alignment. Both kinds of alignment have been observed in biological systems such as swarms of amoebae and myxobacteria. Polar alignment causes cells to align their velocities parallel to each other, whereas nematic alignment drives cells to align either parallel or antiparallel to each other. Under appropriate assumptions, we have derived the LGCA transition probability rule from the steady-state distribution of the off-lattice Fokker-Planck equation. Comparing alignment order parameters between the original Langevin model and the derived LGCA for both mechanisms, we found different areas of agreement in the parameter space. Finally, we discuss potential reasons for model disagreement and propose extensions to the CA rule derivation methodology.
authors.affiliation: Center for Information Services and High Performance Computing, Technische Universität Dresden, Nöthnitzer Straße 46, 01062, Dresden, Germany. nava@mail.zih.tu-dresden.de.
authors.forename: J M
authors.identifier: http://orcid.org/0000-0001-9451-8676
authors.initials: JM
authors.lastname: Nava-Sedeño
authors.affiliation: Center for Information Services and High Performance Computing, Technische Universität Dresden, Nöthnitzer Straße 46, 01062, Dresden, Germany.|Department of Systems Immunology and Braunschweig Integrated Centre of Systems Biology, Helmholtz Center for Infection Research, Inhoffenstraße 7, 38124, Braunschweig, Germany.
authors.forename: H
authors.initials: H
authors.lastname: Hatzikirou
authors.affiliation: Laboratoire J. A. Dieudonné, UMR 7351 CNRS, Parc Valrose, Université Côte d'Azur, 06108, Nice Cedex 02, France.
authors.forename: F
authors.initials: F
authors.lastname: Peruani
authors.affiliation: Center for Information Services and High Performance Computing, Technische Universität Dresden, Nöthnitzer Straße 46, 01062, Dresden, Germany.
authors.forename: A
authors.initials: A
authors.lastname: Deutsch
mesh_terms: D002450:Cell Communication / Q000502:physiology; D002465:Cell Movement / Q000502:physiology*; D003198:Computer Simulation; D005583:Fourier Analysis; D007700:Kinetics; D055641:Mathematical Concepts; D008954:Models, Biological; D009068:Movement / Q000502:physiology; D013269:Stochastic Processes
keywords: Collective cell migration; Ferromagnetic interaction; Fokker-Planck equation; Langevin equations; Lattice-gas cellular automata; Liquid-crystal interaction; Nematic alignment; Polar alignment; Single cell migration