  In the current AFM experiments the distribution of unfolding times, P(t), is
measured by applying a constant stretching force f_s from which the apparent
unfolding rate is obtained. To describe the complexity of the underlying energy
landscape requires additional probes that can incorporate the dynamics of
tension propagation and relaxation of the polypeptide chain upon force quench.
We introduce a theory of force correlation spectroscopy (FCS) to map the
parameters of the energy landscape of proteins. In the FCS the joint
distribution, P(T,t) of folding and unfolding times is constructed by repeated
application of cycles of stretching at constant fs, separated by release
periods T during which the force is quenched to f_q<f_s. During the release
period, the protein can collapse to a manifold of compact states or refold. We
show that P(T,t) can be used to resolve the kinetics of unfolding as well as
formation of native contacts and to extract the parameters of the energy
landscape using chain extension as the reaction coordinate and P(T,t). We
illustrate the utility of the proposed formalism by analyzing simulations of
unfolding-refolding trajectories of a coarse-grained protein S1 with beta-sheet
architecture for several values of f_s, T and f_q=0. The simulations of
stretch-relax trajectories are used to map many of the parameters that
characterize the energy landscape of S1.
