A scallop theorem for cells moving in 3D.

The famous scallop theorem proposed by Purcell in 1977 states that self-propelled objects swimming at low Reynolds number must follow a cycle of shape changes that breaks temporal symmetry. This should hold true for crawling cells as well. However a clear mechanism for this symmetry breaking is still elusive. Here we show that cells embedded in 3D matrix form at both sides of the nucleus force dipoles driven by myosin that locally and periodically pinch the matrix. Using a combination of 3D live cell imaging, traction force microscopy and a minimal model with multipolar expansion, we show that the existence of a phase shift between the two dipoles involves mainly the microtubular network and is required for directed cell motion. We confirm this mechanism by triggering local dipolar contractions with a laser, which leads to directed motion. Our study reveals that the cell controls its motility by synchronising dipolar forces distributed at front and back. This result opens new strategies to externally control cell motion.