Cell movement in vivo is typically characterized by strong confinement and heterogeneous, three-dimensional environments. Such external constraints on cell motility are known to play important roles in many vital processes e.g. during development, differentiation, and the immune response, as well as in pathologies like cancer metastasis. Here we develop a physics-driven three-dimensional computational modeling framework that describes lamellipodium-based motion of cells in arbitrarily shaped and topographically structured surroundings. We use it to investigate the primary in vitro model scenarios currently studied experimentally: motion in vertical confinement, confinement in microchannels, as well as motion on fibers and on imposed modulations of surface topography. We find that confinement, substrate curvature and topography modulate the cell's speed, shape and actin organization and can induce changes in the direction of motion along axes defined by the constraints. Our model serves as a benchmark to systematically explore lamellipodium-based motility and its interaction with the environment.

In the framework of a phase field model of a single cell crawling on a substrate, we investigate how the properties of the cell membrane affect the shape and motility of the cell. Since the membrane influences the cell dynamics on multiple levels and provides a nontrivial feedback, we consider the following fundamental interactions: (i) the reduction of the actin polymerization rate by membrane tension; (ii) area conservation of the cell's two-dimensional cross-section vs. conservation of its circumference (i.e. membrane inextensibility); and (iii) the contribution from the membrane's bending energy to the shape and integrity of the cell. As in experiments, we investigate two pertinent observables -the cell's velocity and its aspect ratio. We find that the most important effect is the feedback of membrane tension on the actin polymerization. Bending rigidity has only minor effects, visible mostly in dynamic reshaping events, as exemplified by collisions of the cell with an obstacle.

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Cellular protrusion-and lamellipodium waves are widespread for both non-motile and moving cells and observed for many cell types. They are involved in the cell's exploration of the substrate, its internal organization, as well as for the establishment of self-polarization prior to the onset of motion. Here we apply the recently developed phase field approach to model shape waves and their competition on the level of a whole cell, including all main physical effects (acto-myosin, cell membrane, adhesion formation and substrate deformation via traction) but ignoring specific biochemistry and regulation. We derive an analytic description of the emergence of a single wave deformation, which is of Burgers/Fisher-Kolmogorov type. Finally, we develop an amplitude equation approach to study multiple competing rotational waves and show how they allow the cell to transition from a non-moving state towards a polarized, steady moving state.