Theoretical Models in Force Spectroscopy - Dynamic Force Spectroscopy and Biomolecular Recognition - page 129

Unimolecular, bistable U ( x ) Bimolecular, bistable H ( x,t )

F f

W = W mol + W spring

F s

F s

W

W spring

W mol

0

0

t s

t f

t s t f

Time Time

FIGURE 3.11 Idealized schematic of the work done on (a) a unimolecular system such as a
protein or RNA molecule and (b) a bimolecular system such as a ligand-receptor pair. Symbols
correspond to those found in the text. In (a), the molecular potential U

is already bistable,
which leads to a total work done on the combined molecule/probe system W that depends
on the final pulling time t f . The two contributions to the total work, the work done on the
molecule W mol and the work done on stretching the spring W spring are separable, and illustrated
by different shading. In (b), only the total Hamiltonian H

( x )

of the combined intermolecular
and probe potentials is bistable, which means the probing spring is an integral component of
the two-state system. Once the bond ruptures, the particle resides around the minimum of the
pulling spring for the remainder of the pulling process, and no further work is done on the
spring. Thus, the work done on the system is the area under the force-distance trajectory, but
it is not dependent on the final observation time t f .

( x , t )

Thus, the dissipated energy W d will be independent of the final work done on the
spring:

W d = W − Δ G (3.116)

= W molecule + W spring ( t f ) − ( Δ G 0 + W spring ( t f )) (3.117)

= W molecule − Δ G 0 (3.118)

which is typically the quantity of interest when probing molecular systems.

Bimolecular system with single metastable U(x). Here, we consider a bimolec-
ular bond with a single minimum in U 0. The total Hamilto-

nian evolves into a bistable system when the minimum of the pulling potential
V

( x ) located at x =

1
2 k cant

2 is pulled sufficiently far past the barrier to rupture. The
minimum of the pulling potential defines the second state and is located at vt .There-
fore, assuming negligible displacement of the bound state minimum, the system will
switch from x

( x , t )= ( x − vt )

= 0to x = vt upon unbinding. Again, assuming unbinding occurs at