Single-molecule mechanics: new insights from the escape-over-a-barrier problem.

T he early description of the living cell as a reaction vessel of molecules undergoing random collisions has evolved to a much more interesting picture of a small chemical factory of complex and dynamic molecular structures that carry out specialized functions in a highlycoordinated fashion. Traditional solution biochemistry has had extraordinary success in elucidating molecular structures of the cellular components. The task of capturing the dynamics and the distribution of states of these structures, however, seemed nearly impossible because the activities of the molecules in the solution are asynchronous as the result of their interactions with the surroundings. This situation has changed with the development of single-molecule biophysical methods (1, 2) that permit measurements of the response of macromolecules or individual molecular bonds to external load with the spectacular resolution of subnanometer distances and picoNewton forces, revealing details that are typically lost to ensemble averaging when studied by solution methods. Unprecedented information about the mechanisms that drive biological processes becomes encoded in the resulting wealth of unaveraged data. Recall, however, that these systems are of microscopic dimensions, which puts thermal fluctuations on equal footing with the external deterministic forces. This circumstance, along with the far from equilibrium conditions, makes a quantitative description of the molecular bond properties that can account for observed data a challenging task. The article by Freund in this issue of PNAS (3) offers a theoretical framework that does just that. A novel and concise derivation takes the reader from a very fundamental starting point to a set of analytical results that provide intriguing insights into the response of a molecular bond to an imposed mechanical load. To probe mechanical properties of macromolecules experimentally, the molecules are attached, one at a time, to an external probe, such as a m-sized bead, a cantilever, or a pipette. By pulling on the probe, and hence on the molecule, the response of the molecule (DNA, RNA, or protein) or a ligand– receptor pair to a controlled force or twist is examined. Given the same external load, the responses will differ from one measurement to another because of the stochastic nature of the system, but upon observing many of the responses, the entire distribution of the dynamic properties, such as dissociation times and rupture forces, can be collected. Such a systematic investigation of the dependence of molecular properties on external loading is at the heart of dynamic force spectroscopy method (4–6), brilliantly designed to probe the inner world of molecular interactions. Because the thermal noise from the environment is an integral part of the bond separation process, the response of the bond to an external force can only be described in probabilistic terms. Assume that the instantaneous configuration of the bond can be fully identified by a single variable x, which can, for instance, be the bond length. The bond can only adopt configurations confined to a 1D free-energy landscape U(x, t), which is the sum of the bond natural free energy of a general shape sketched in Fig. 1 and the mechanical work of (in general, time-dependent) external force acting in the direction of x (Fig. 1). Now consider an ensemble of nominallyidentical bonds that are simultaneously subjected to the same external force, responding independently of each other. It is the probability distribution of the bond configurations that is diffusing on the free energy landscape U(x, t) governed by the generalized diffusion (Smoluchowski) equation. The fraction of the bonds in the ensemble that remain intact at time t is the ‘‘survival probability’’ of the bond. To describe force-induced molecular rupture, a common starting point is the first-order rate equation governing the evolution of the survival probability (the first-order kinetics is a signature of the ‘‘rare’’ character of the rupture events). The time dependence of the rupture rate in this equation can be calculated by treating bond rupture under force as a diffusive barrier crossing, a generalized version of a classical problem studied by Kramers (7). Applied to a class of simple microscopic models of the bond free-energy landscape, Kramers’ theory has been used to derive analytical solutions for the rupture rate under constant external force and for the rupture-force distributions under constant