  In various cellular processes, biofilaments like F-actin and F-tubulin are
able to exploit chemical energy associated to polymerization to perform
mechanical work against an external load. The force-velocity relationship
quantitatively summarizes the nature of this process. By a stochastic dynamical
model, we give, together with the evolution of a staggered bundle of $N_f$
rigid living filaments facing a loaded wall, the corresponding force--velocity
relationship. We compute systematically the simplified evolution of the model
in supercritical conditions $\rho_1=U_0/W_0>1$ at $\epsilon=d^2W_0/D=0$, where
$d$ is the monomer size, $D$ is the obstacle diffusion coefficient, $U_0$ and
$W_0$ are the polymerization and depolymerization rates. Moreover, we see that
the solution at $\epsilon=0$ is valid for a good range of small non-zero
$\epsilon$ values. We consider two classical protocols: the bundle is opposed
either to a constant load or to an optical trap set-up, characterized by a
harmonic restoring force. The constant force case leads, for each $F$ value, to
a stationary velocity $V^{stat}(F;N_f,\rho_1)$ after a relaxation with
characteristic time $\tau_{micro}(F)$. When the bundle (initially taken as an
assembly of filament seeds) is subjected to a harmonic restoring force (optical
trap load), the bundle elongates and the load increases up to stalling
(equilibrium) over a characteristic time $\tau^{OT}$. Extracted from this
single experiment, the force-velocity $V^{OT}(F;N_f,\rho_1)$ curve is found to
coincide with $V^{stat}(F;N_f,\rho_1)$, except at low loads. We show that this
result follows from the adiabatic separation between $\tau_{micro}$ and
$\tau^{OT}$, i.e. $\tau_{micro}\ll\tau^{OT}$.
