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Cell motility relies on the continuous reorganization of a dynamic actin-myosin-adhesion network at the leading edge of the cell, in order to create protrusive force at the leading edge membrane and traction force between the cell and its external environment. In motile cells, a broad and flat network of F-actin filaments polymerizes near the leading edge membrane, and undergoes coherent retrograde motion away from the leading edge toward the cell center, driven by a combination of leading-edge polymerization force and myosin contractile pulling forces. Dynamic assemblies of proteins called adhesions couple the F-actin cytoskeleton to the external environment, causing retrograde flow of actin to slow. The role of the adhesions is often likened to a molecular ‘clutch’, which attenuates retrograde flow and allows leading edge polymerization to translate into leading edge protrusion. We analyze experimentally measured spatial distributions of actin flow, traction force, myosin density, and adhesion density in control and myosin-inhibited epithelial cells in order to develop a mechanical model for the coordination of actin, myosin and adhesion dynamics. A model in which adhesion clutch engagement is strengthened by myosin force but weakened by actin flow can explain the characteristic measured molecular distributions.