  A microscopic theory of the free energy barriers and folding routes for
minimally frustrated proteins is presented, greatly expanding on the
presentation of the variational approach outlined previously [J. J. Portman, S.
Takada, P. G. Wolynes, Phys. Rev. Lett. {\bf 81}, 5237 (1998)]. We choose the
$\lambda$-repressor protein as an illustrative example and focus on how the
polymer chain statistics influence free energy profiles and partially ordered
ensembles of structures. In particular, we investigate the role of chain
stiffness on the free energy profile and folding routes. We evaluate the
applicability of simpler approximations in which the conformations of the
protein molecule along the folding route are restricted to have residues that
are either entirely folded or unfolded in contiguous stretches. We find that
the folding routes obtained from only one contiguous folded region corresponds
to a chain with a much greater persistence length than appropriate for natural
protein chains, while the folding route obtained from two contiguous folded
regions is able to capture the relatively folded regions calculated within the
variational approach. The free energy profiles obtained from the contiguous
sequence approximations have larger barriers than the more microscopic
variational theory which is understood as a consequence of partial ordering.
