Work probability distribution in single molecule experi - ments

– We derive and solve a differential equation satisfied by the probability distribution of the work done on a single biomolecule in a mechanical unzipping experiment. The unzipping is described as a thermally activated escape process in an energy landscape. The Jarzynski equality is recovered as an identity, independent of the pulling protocol. This approach allows one to evaluate easily, by numerical integration, the work distribution, once a few parameters of the energy landscape are known. Introduction. – The introduction of micromanipulation techniques has dramatically improved our knowledge of physical and chemical properties of biological molecules. Such techniques have been used to probe the structure of proteins [1–5] and nucleic acids [6]. A typical experiment consists of pulling the free end of a biomolecule with a controlled force, while its end-to-end distance is measured at the same time. It has been suggested that the study of the kinetics of bond breakage under different loading rates can provide many informations on the molecule internal structure, and in particular allows one to measure the strength of the molecular bonds, and to associate to them a position along the molecular structure [7]. The loading-rate dependent kinetics experiments on biomolecules have been interpreted in terms of thermally activated escape from bond states over a succession of energy barriers, along a one-dimensional energy landscape [8–11]. Usually, because of technical limitations, the molecule pulling process is characterized by time scales much faster than the typical molecular relaxation time. This prevents the possibility to perform the experiment in quasiequilibrium conditions and thus to obtain direct measurements of the thermodynamic state variables. This difficulty can be overcome by exploiting the remarkable equality derived by Jarzynski [12, 13] (Jarzynski equality, JE), and extended by Crooks [14], which allows one to obtain the free energy difference ∆F between two equilibrium states by evaluating the average of exp(−βW), where W is the work performed on the system during the thermodynamic transformation: