  Propulsion by growing actin networks is a universal mechanism used in many
different biological systems. Although the core molecular machinery for actin
network growth is well preserved in most cases, the geometry of the propelled
obstacle can vary considerably. In recent years, filament orientation
distribution has emerged as an important observable characterizing the
structure and dynamical state of the growing network. Here we derive several
continuum equations for the orientation distribution of filaments growing
behind stiff obstacles of various shapes and validate the predicted steady
state orientation patterns by stochastic computer simulations based on discrete
filaments. We use an ordinary differential equation approach to demonstrate
that for flat obstacles of finite size, two fundamentally different orientation
patterns peaked at either +35/-35 or +70/0/-70 degrees exhibit mutually
exclusive stability, in agreement with earlier results for flat obstacles of
very large lateral extension. We calculate and validate phase diagrams as a
function of model parameters and show how this approach can be extended to
obstacles with piecewise straight contours. For curved obstacles, we arrive at
a partial differential equation in the continuum limit, which again is in good
agreement with the computer simulations. In all cases, we can identify the same
two fundamentally different orientation patterns, but only within an
appropriate reference frame, which is adjusted to the local orientation of the
obstacle contour. Our results suggest that two fundamentally different network
architectures compete with each other in growing actin networks, irrespective
of obstacle geometry, and clarify how simulated and electron tomography data
have to be analyzed for non-flat obstacle geometries.
