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Cells often migrate on curved surfaces inside the body, such as curvedtissues, blood vessels or highly curved protrusions of other cells. Recent\textit{in-vitro} experiments provide clear evidence that motile cells areaffected by the curvature of the substrate on which they migrate, preferringcertain curvatures to others, termed ``curvotaxis". The origin and underlyingmechanism that gives rise to this curvature sensitivity are not wellunderstood. Here, we employ a ``minimal cell" model which is composed of avesicle that contains curved membrane protein complexes, that exert protrusiveforces on the membrane (representing the pressure due to actin polymerization).This minimal-cell model gives rise to spontaneous emergence of a motilephenotype, driven by a lamellipodia-like leading edge. By systematicallyscreening the behaviour of this model on different types of curved substrates(sinusoidal, cylinder and tube), we show that minimal ingredients and energyterms capture the experimental data. The model recovers the observed migrationon the sinusoidal substrate, where cells move along the grooves (minima), whileavoiding motion along the ridges. In addition, the model predicts the tendencyof cells to migrate circumferentially on convex substrates and axially onconcave ones. Both of these predictions are verified experimentally, on severalcell types. Altogether, our results identify the minimization ofmembrane-substrate adhesion energy and binding energy between the membraneprotein complexes as key players of curvotaxis in cell migration.

A minimal physical model for curvotaxis driven by curved protein complexes at the cell's leading edge