Mapping the energy landscape of biomolecules using single molecule force correlation spectroscopy: theory and...

We present, to our knowledge, a new theory that takes internal dynamics of proteins into account to describe forced-unfolding and force-quench refolding in single molecule experiments. In the current experimental setup (using either atomic force microscopy or laser optical tweezers) the distribution of unfolding times, P(t), is measured by applying a constant stretching force f(S) from which the apparent f(S)-dependent 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 to map the parameters of the energy landscape of proteins. In force correlation spectroscopy, the joint distribution P(T, t) of folding and unfolding times is constructed by repeated application of cycles of stretching at constant f(S) 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) at various f(S) and f(Q) values can be used to resolve the kinetics of unfolding as well as formation of native contacts. We also present methods to extract the parameters of the energy landscape using chain extension as the reaction coordinate and P(T, t). The theory and a wormlike chain model for the unfolded states allows us to obtain the persistence length l(p) and the f(Q)-dependent relaxation time, giving us an estimate of collapse timescale at the single molecular level, in the coil states of the polypeptide chain. Thus, a more complete description of landscape of protein native interactions can be mapped out if unfolding time data are collected at several values of f(S) and f(Q). 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.