  We present a method to investigate the kinetics of protein folding on a long
time-scale and the dynamics underlying the formation of secondary and tertiary
structures during the entire reaction. The approach is based on the formal
analogy between thermal and quantum diffusion: by writing the solution of the
Fokker-Planck equation for the time-evolution of a protein in a viscous
heat-bath in terms of a path integral, we derive a Hamilton-Jacobi variational
principle from which we are able to compute the most probable pathway of
folding. The method is applied to the folding of the Villin Headpiece
Subdomain, in the framework of a Go-model. We have found that, in this model,
the transition occurs through an initial collapsing phase driven by the
starting coil configuration and a later rearrangement phase, in which secondary
structures are formed and all computed paths display strong similarities. This
method is completely general, does not require the prior knowledge of any
reaction coordinate and represents an efficient tool to perfom ab-initio
simulations of the entire folding process with available computers.
