Theoretical Models in Force Spectroscopy - Dynamic Force Spectroscopy and Biomolecular Recognition - page 97

Jones 6-12 potential under the influence of a parabolic pulling potential. The figure
shows that the addition of a spring potential to a bond with a single metastable min-
imum can result in a monostable or bistable system depending on two important
control parameters: the location of the spring minimum x cant vt and the stiffness

of the spring k cant . The resulting surface of stability points contains a fold when the
spring constant is less than the largest gradient of the bond potential k cant

=

U max .
The fold designates the emergence of a barrier (unstable maximum) separating two
metastable minima. When translating the probe minimum x cant from left to right (i.e.,
away from the bond minimum), the system passes through the region of bistability
and can discontinuously jump from one metastable minimum to another. Such is the
case when a bond ruptures. The same is true when moving right to left (toward the
bond), but the jump to the bound state will typically occur at a smaller displace-
ment from the bond minimum than the jumping off location. Hence, a hysteresis is
observed when comparing the approach and retract jump positions during a nonequi-
librium measurement. On the other hand, when the spring constant is greater than this
critical value k cant

<

U max , the fold vanishes, and only a single monostable minimum
exists for all translations of the probe minimum. In most force spectroscopy experi-
ments, the probe stiffness is much weaker than the bond, leading to the discontinuous
snapping-on and snapping-off of the probe to and from the surface. Hence, we will
consider the use of soft probes to be a given assumption throughout this chapter.

>

3.1.3 D IFFUSION AND E SCAPE OVER A B ARRIER

The current theory of molecular bond rupture is a culmination of over 100 years of
discoveries that have provided experimental and theoretical insight into the physics
of molecules in condensed phases. It is, therefore, useful to begin by discussing the
principles underlying the microscopic motion of molecules. A particle in solution
undergoes rapid collisions with the surrounding solvent. Each impulse from a neigh-
boring molecule imparts some kinetic energy to the particle. But shortly thereafter
(

12 s), the particle experiences another kick of random strength and direction.
These collisions have two primary effects. The first is to induce irregular motion of
the particle. The second is to slow down the particle through friction. Since these
random and frictional forces arise from the same source, they are closely related
through Einstein's fluctuation-dissipation theorem . The motion observed is random
and known as diffusive (or Brownian) motion and is fundamental to particles that
are both free in solution, or under the influence of externally applied force fields.
Here, we will derive the basic laws describing the motion of particles, beginning
with Einstein's derivation of diffusion.
Assume that during an interval of time τ, a particle in solution makes a movement
Δ, which is independent of the movements of any other particle. As each movement
takes place, we also assume that it is independent of the previous movement. In
other words, the particle has no memory of the last step it took. Each step distance
is assumed to occur with some probability P

10 −

∼

. Note also that the step distance Δ
is the magnitude away from the current position, we are not concerned with which
direction it moves away. Consider a distribution of the concentration of many such
particles ρ

( Δ )

( x , t ) at time t . The above assumptions constitute a Markov process that is