Final answer:
Spontaneous growth of a cytoskeletal filament occurs when the concentration of free subunits is greater than the ratio of the on rate to the off rate of monomer addition, represented as C > kon/koff.
Step-by-step explanation:
For the growth of a cytoskeletal filament to proceed spontaneously, the concentration of free subunits (C) must exceed the critical concentration (Cc), which is the concentration at which the addition rate of subunits equals the rate of loss. At equilibrium, the rate of monomer addition (kon) and loss (koff) would be equal, leading to koff/kon = koff/kon. However, the growth conditions for the actin polymerization in cells are non-equilibrium due to ATP hydrolysis altering the dynamics. In non-equilibrium conditions, the growth occurs at a rate dependent on the differences in binding and unbinding of ATP and ADP bound actin monomers at both ends of the filament.
The critical concentration for spontaneous filament growth is thus influenced by ATP hydrolysis and monomer binding dynamics, resulting in the condition where C must be greater than a value related to the off and on rates of monomer addition. Considering the principle that growth will be spontaneous when the free subunit concentration C exceeds the net loss rate at the filament ends, the correct condition for spontaneous filament growth is C > kon/koff.