Stall, spiculate or runaway - the fate of fibers growing towards fluctuating membranes

We solve the dynamic equations of motion for a growing semi-flexible polymer, or fiber, approaching a fluctuating membrane at an angle. At late times we find three different regimes: fiber {\em stalling}, when fiber growth stops due to membrane resistence, {\em run-away}, in which the polymer bends away from the membrane, and another regime in which the membrane response is nonlinear and tubular membrane {\em spicules} are formed. We discuss which regions of the resulting `phase diagram' are explored by (i) single and bundled actin fibers in living cells, (ii) sickle hemoglobin fibers in red blood cells, and (iii) microtubules growing within artificial vesicles. We complement our analysis with full 3-dimensional stochastic simulations.