  Using theoretical arguments and extensive Monte Carlo (MC) simulations of a
coarse-grained three-dimensional off-lattice model of a \beta-hairpin, we
demonstrate that the equilibrium critical force, $F_c$, needed to unfold the
biopolymer increases non-linearly with increasing volume fraction occupied by
the spherical macromolecular crowding agent. Both scaling arguments and MC
simulations show that the critical force increases as $F_c \approx
\varphi_c^{\alpha}$. The exponent $\alpha$ is linked to the Flory exponent
relating the size of the unfolded state of the biopolymer and the number of
amino acids. The predicted power law dependence is confirmed in simulations of
the dependence of the isothermal extensibility and the fraction of native
contacts on $\varphi_c$. We also show using MC simulations that $F_c$ is
linearly dependent on the average osmotic pressure ($\mathrm{P}$) exerted by
the crowding agents on the \beta-hairpin. The highly significant linear
correlation coefficient of 0.99657 between $F_c$ and $\mathrm{P}$ makes it
straightforward to predict the dependence of the critical force on the density
of crowders. Our predictions are amenable to experimental verification using
Laser Optical Tweezers.
