  Cells moving on a two dimensional substrate generate motion by polymerizing
actin filament networks inside a flat membrane protrusion. New filaments are
generated by branching off existing ones, giving rise to branched network
structures. We investigate the force-extension relation of branched filaments,
grafted on an elastic structure at one end and pushing with the free ends
against the leading edge cell membrane. Single filaments are modeled as
worm-like chains, whose thermal bending fluctuations are restricted by the
leading edge cell membrane, resulting in an effective force. Branching can
increase the stiffness considerably; however the effect depends on branch point
position and filament orientation, being most pronounced for intermediate tilt
angles and intermediate branch point positions. We describe filament networks
without cross-linkers to focus on the effect of branching. We use randomly
positioned branch points, as generated in the process of treadmilling, and
orientation distributions as measured in lamellipodia. These networks reproduce
both the weak and strong force response of lamellipodia as measured in
force-velocity experiments. We compare properties of branched and unbranched
networks. The ratio of the network average of the force per branched filament
to the average force per unbranched filament depends on the orientation
distribution of the filaments. The ratio exhibits compression dependence and
may go up to about 4.5 in networks with a narrow orientation distribution. With
orientation distributions measured in lamellipodia, it is about two and
essentially independent from network compression, graft elasticity and filament
persistence length.
