(a–b) Model schematic. Black lines, membrane; green circles, actin; purple flowers, Arp2/3 complex; blue crescents, capping protein. Rates: kon, polymerization; koff, depolymerization; kbranch, branching; kcap, capping. θbranch, branching angle. dbranch, branching window. Physical parameters: Fspring, forces between membrane segments; FBR, force of filaments on the membrane (Brownian ratchet); Fdrag, viscous drag. (b) Schematic demonstrating filament angle evolution. Filaments growing perpendicular to the leading edge (I) outcompete their progeny (branches), leading to a reduction in filament density; filaments growing at an angle (II and III) make successful progeny. Filaments spreading down a membrane positional gradient (II) are more evolutionarily successful than those spreading up (III). (c) Simulation snapshot: Black lines, membrane; colored lines, filament equilibrium position and shape; gray dots, barbed ends; black dots, capped ends; filament color, x-position of membrane segment filament is pushing (increasing across the x-axis from blue to red). (d-h) For a representative simulation, mean (solid line) and standard deviation (shading) of various membrane and actin filament properties as a function of simulation time. Note for linear filament density (d) lamellipodia are ~10 filament stacks tall along the z-axis, giving mean filament spacing of 10/density, or ~30 nm. (i–l) Autocorrelation analysis and fitting for a representative simulation (i–j) as well as best fit parameters averaged over 40 simulations (k–l). Data from this figure can be found in Figure 2—source data 1.