Experimentally measured distribution of actin network velocity and adhesion strength in migrating keratocyte. (A) Two-dimensional actin velocity map obtained as described in Schaub et al. (17). Bar, 10 μm. (B) Distribution profile of posterior-anterior component of actin velocity along posterior-anterior direction. Zero position corresponds to the back of the cell; velocity values for each distance from the back of the cell were averaged within the vertical rectangular region indicated on velocity map in panel A. (C) Distribution profile of lateral component of actin velocity along lateral direction. Zero position corresponds to the left side of the cell; velocity values for each distance from the left side were averaged within the horizontal rectangular region indicated on velocity map in panel A. Posterior-anterior (D) and lateral (E) distributions of adhesion strength parameter in, respectively, vertical and horizontal rectangular regions shown in panel A. Experimental determination of adhesion parameter will be described elsewhere. Briefly, actin network velocity and stress exerted on the substrate were measured simultaneously, and adhesion parameter was computed at every position of the cell as a ratio of the substrate stress to the actin network speed.

Simplified one-dimensional model of the actin-myosin flow. If the two-dimensional effects are neglected in the rectangular domain with uniform adhesion and myosin linearly biased to the rear (darker shading corresponds to higher myosin), then the anterior-posterior and lateral distributions of the flow velocities can be found analytically. (Left/right bottom line plots) Flow rates from the left to the right side and from the rear to the front, respectively. The flow rate is in units of the maximum rate at the side; the unit of distance is the half-width of the lamellipodial domain.

Two-dimensional model of the actin network flow. (A) Motile keratocyte cell, view from above. The cargo of the cell body is depicted by the gray ellipse at the rear. The lamellipod glides with the steady shape and velocity V, while the branched actin network (crisscross segments) grows at the leading edge and retracts at the rear due to the contractile action of myosin (thick double-arrows). (Inset) Mechanical model of the actin-myosin lamellipodial network. Actin filaments (segments, density ρ) glide with velocity u, undergo relative viscoelastic sliding (spring and dashpot in series, viscoelastic stress τ), and are contracted by working myosins (density m₁). Adhesion results in effective viscous drag (dashpot, density ξ). Myosin molecules cycle between the working and free (density m₀) states. (B) Hypothesized maintenance of the lamellipodial shape: The actin network grows at the front and sides (solid arrows), while the centripetal actin flow is fast at the rear and sides and slow at the front (dashed arrows). At the sides, the outward growth and inward flow cancel each other. At the rear, there is no actin growth, and the centripetal actin flow is equal to the cell speed (dotted arrow). The front advances with the cell speed equal to the difference between the local rates of actin growth and centripetal flow. To maintain the steady shape, the net local normal extension rate at the lamellipodial boundary has to obey the equation (V_n − V^c · n) = V × cosθ, where V_n is the local actin growth rate, V^c is the local actin centripetal flow velocity, n is the local normal unit vector, and θ is the angle between n and the direction of movement. (C) Characteristic map of the actin centripetal flow, u(r), in the lab coordinate system. (D) In the framework of the steadily moving cell, there is the kinematic flow of F-actin to the rear with constant rate V equal to the cell speed. (E) In the framework of the cell, the actin-myosin network drift is the geometric sum, (u(r) − V), of the flows shown in panels C and D. If V > |u(r)|, then the resulting drift sweeps the myosin to the rear.

Actin flow and traction forces. (A) Spatial distribution of the adhesion strength used in the calculations (dark shading corresponds to stronger adhesion). (B) Open arrows show the computed actin flow map. Shading shows the myosin density (light corresponds to high density). (C) Computed distribution of the traction forces. (D) Actin growth rate distribution at the lamellipodial boundary required to maintain the steady shape.