Chapter 8 – Theoretical Biophysics  357

Often however, molecular stretches using force spectroscopy are relatively rapid, resulting

in hysteresis on a force-​extension trace due to the events such as molecular domain unfolding

(e.g., involving the breaking of chemical or other weaker molecular bonds), which drives the

system away from thermal equilibrium. In the case of domains unfolding, they may not have

enough time to refold again before the next molecular stretch cycle and so the transition is,

in effect, irreversible, resulting in a stiffer stretch half-​cycle compared to the relaxation half-​

cycle, (Figure 8.5c). In this instance, the Kramers theory can be applied. The Kramers theory

is adapted to model systems in nonequilibrium states diffusing out of a potential energy well

(interested readers should see the classic Kramers, 1940). The theory can be extended to

making and breaking adhesive bonds (characterized originally in Bell’s classic paper to model

the adhesive forces between cells; see Bell, 1978). This theory can also be adapted into a two-​

state transition model of folded/​unfolded protein domains in a single-​molecule mechanical

stretch-​release experiment (Evans and Ritchie, 1997). This indicates that the mean expected

unfolding force Fd is given by

(8.79)

F

k T

x

rx

k

k T

d

B

w

e

w

d

=

( )



^



^

log

B

0

where kd(0) is the spontaneous rate of domain unfolding at zero restoring force on the mol­

ecule, assuming a uniform rate of molecular stretch r, while xw is the width of potential energy

for the unfolding transition. Therefore, by plotting experimentally determined Fd values,

measured from either optical or magnetic tweezers or, more commonly, AFM, against the

logarithm of different values of r, the value for xw can be estimated.

A similar analysis may also be performed for refolding processes, and for typical domain

motifs that undergo such unfolding/​refolding transitions, a width value of ~0.2–​0.3 nm is

typical for the unfolding transition, while the width for the refolding transitions is an order of

magnitude or more greater. This is consistent with molecular simulations suggesting that the

unfolding is due to the unzipping of a few key hydrogen bonds that are of similar length scales

to the estimated widths of potential energy (see Chapter 2), whereas the refolding process is

far more complex requiring cooperative effects over a much longer length scales of the whole

molecule. An axiom of structural biology known as Levinthal’s paradox states that where

a denatured/​unfolded protein to refolded by exploring all possible refolding possibilities

through the available stable 3D conformations one by one to determine which is the most

stable folded conformation, this would take longer than the best estimate for the current age

of the universe. Thus, refolding mechanisms typically employ short cuts that, if rendered on

a plot of the free energy state of the molecule during the folding transition as a function of

two orthogonal coordinates that define the shape of the molecule in some way (e.g., mean

end-​to-​end extension of the molecule projected on the x and y axes), the free energy surface

resembles a funnel, such that refolding occurs by in effect spiraling down the funnel aperture.

An important relation of statistical mechanics that links the mechanical work done W

on a thermodynamic system of a nonequilibrium process to the free energy difference ΔG

between the states for the equivalent equilibrium process is the Jarzynski equality that states

(8.80)

exp

exp

−



^



^=

−



^



^

∆G

k T

W

k T

B

B

In other words, it relates the classical free energy difference between two states of a system

to the ensemble average of finite-​time measurements of the work performed in switching

from one state to the other. Unlike many theorems in statistical mechanics, this was derived

relatively recently (interested students should read Jarzynski, 1997). This is relevant to force

spectroscopy of single biopolymers, since the mean work in unfolding a molecular domain

is given by the work done in moving a small distance xw against a force Fd, that is, Fdxw, and