Leading edge protrusion is one of the critical events in the cell motility cycle and it is believed to be driven by the assembly of the actin network. The concept of dendritic nucleation of actin filaments provides a basis for understanding the organization and dynamics of the actin network at the molecular level. At a larger scale, the dynamic geometry of the cell edge has been described in terms of the graded radial extension model, but this level of description has not yet been linked to the molecular dynamics. Here, we measure the graded distribution of actin filament density along the leading edge of fish epidermal keratocytes. We develop a mathematical model relating dendritic nucleation to the long-range actin distribution and the shape of the leading edge. In this model, a steady-state graded actin distribution evolves as a result of branching, growth and capping of actin filaments in a finite area of the leading edge. We model the shape of the leading edge as a product of the extension of the actin network, which depends on actin filament density. The feedback between the actin density and edge shape in the model results in a cell shape and an actin distribution similar to those experimentally observed. Thus, we explain the stability of the keratocyte shape in terms of the self-organization of the branching actin network.