ABSTRACT
Chapter 2 outlined several methods through which pulling forces can be exerted on individual molecules. This chapter examines, in more detail, what can be learned from such measurements. A parallel between molecules and mechanical springs has already been invoked in Chapter 4 to discuss the behavior of a diatomic molecule: When the length of the bond between its two atoms is changed by a small amount x relative to the equilibrium value, the potential energy is increased by
V (x) ≈ (1/2)γ0x2, (8.1) where the parameter γ0 is an intrinsic property of the molecule. This potential energy is identical to that of a linear spring with the spring constant γ0. If we could devise a way to stretch a single diatomic, it would thus generate a restoring force obeying Hooke’s law:
f = −V ′(x) = −γ0x . (8.2) Eq. 8.2 can be derived by noting that, when the spring extension x is increased by an infinitesimal amount dx , the exerted force − f (which has the direction opposite that of the restoring force) performs a work equal to dW = − f dx , which causes the molecule’s potential energy to increase from V (x) to V (x + dx) ≈ V (x) + V ′(x)dx . Conservation of energy then requires that
− f dx = V ′(x)dx, which readily gives Eq. 8.2. The same reasoning, of course, also applies when x is not small and the potential V (x) is no longer quadratic. Therefore if we know V (x) we could always compute the restoring force as its derivative and, conversely, if we knew the dependence of the restoring force on the molecule’s extension x we could integrate this dependence to estimate V (x). Pursuing this line of investigation into diatomic molecules is, however, neither realistic nor particularly compelling, as the potentials of diatomics are already known quite accurately from other types of measurements and quantum mechanical calculations. Single-molecule pulling experiments are more commonly employed to study interactions within (or between) larger, polyatomic
molecules such as DNA, RNA, and proteins. A common property of those is that they all are chain molecules (i.e., polymers). Their spring action can, for example, be probed by applying opposing forces at the ends of the molecular chain (Fig.8.1). When applied to such systems, the above relationship between the potential energy and the extension is no longer correct. What is missing is a different source of molecular elasticity that is due to thermal motion.
