Fig 1.
Mathematical models can be used to bridge from intracellular signaling (left), to single cell shape and motility (center), to cell-cell interactions (right).
At the lowest scales, the goal is deciphering the interplay between stimuli to the cell (chemical, topographic, mechanical, etc.) and intracellular signaling networks that regulate F-actin (branched polymer) and the cytoskeleton (not drawn to scale). These interactions lead to protrusion or retraction, cell polarization, and shape changes that enable directed motility and chemotaxis. At a higher level, an aim is to link cell behavior and cell-cell interactions to the outcomes of cell collisions (e.g., CIL) and to the cohesion of tissues versus EMT, where cells break off. Interconnections exist between all layers, only 2 of which (white arrows) are shown here. CIL, Contact Inhibition of Locomotion; EMT, Epithelial Mesenchymal Transition.
Fig 2.
A mapping of computational models according to the number of cells (horizontal axis) and the level of detail for each cell (vertical axis).
Citations of papers in the diagram (starting from the upper left to lower right: [5–30]).
Fig 3.
Modeling goals can be classified into broad categories that span levels of hierarchy.
Some models attempt to span knowledge of single cell behavior plus interactions to predict emergent multicellular behavior (bottom up, left), whereas others start with observations of tissue dynamics and seek to infer underlying rules, feedbacks, and cell-cell (c-c) interactions that lead to those observations (top down, right).
Table 1.
Experimental summary: CIL, contact inhibition of locomotion; GEF, guanine nucleotide exchange factor; NCCs, neural crest cells; RDEs, reaction diffusion equations.
Fig 4.
A summary of common modeling (top, red box) and experimental (bottom, green box) methods used to study cell behaviors from single cells to cell groups and up to tissues (left to right in increasing number of cells and increasing cell-cell adhesion).
Table 2.
Summary of the modeling papers.
Papers classified by 3 levels of organization: S, SCB, and CCB or tissue behavior. SPP models represent the cell shape statistically; common cell shape choices are spherical, ellipsoidal, or cylindrical cells. Models are categorized into: (1) CPM; (2) phase-field models; (3) vertex models; (4) particle models; (5) continuum models. CPM and phase-field models resolve cell shape well. c-c, cell-cell; CCB, collective cell behavior; CPM, Cellular Potts Models; expts., experiments; FEM, finite element methods; ODE, ordinary differential equations; RDE, reaction diffusion equation; S, signaling; SCB, single cell behavior; SPP, self-propelled particles.
Fig 5.
Cell sorting (left) and wound-healing (right) captured by several distinct computational methods.
(A1) Cells represented by deformable ellipsoids in 3D. Simulations by Hildur Knutsdottir based on code originally created by Eirikur Palsson (A2) Vertex-based simulations using CHASTE open source platform, run by Dhananjay Bhaskar. (A3) CompuCell3D cell-sorting simulations run by Dhananjay Bhaskar. In (A1–A3), there are 2 cell types with differing adhesion strengths to self and other cell type. (B1) and (B2) show deformable ellipsoids and CPM scratch wound models with 2 cell types. Cells with weaker adhesion (red) tend to segregate to the edge of the monolayer, acting as leader cells, and causing fingering of the front. (Compare with [168]). Papers that compare distinct computational platforms include [143].