Theoretical Models in Force Spectroscopy - Dynamic Force Spectroscopy and Biomolecular Recognition - page 117

Given the total number of domains is fixed N N off , we can use Equation 3.75

to write down the fraction of folded and unfolded states as

= N on +

N on ( F ) 1

= (3.78)

N 1 + exp [ Δ G ( F ) / k B T ]

( )

N off F 1

= (3.79)

N 1 + exp [ − Δ G ( F ) / k B T ]

whereΔ G . Convention is important here. We take the unper-

turbed free energy difference between the folded and unfolded states to be a negative
numberΔ G 0

( F )= Δ G 0 + F ( x β + x α )

0 to indicate the folded state is favored at zero force. Oesterhelt et al.
(1999) applied the relations in Equations 3.78 and 3.79 in a rather elegant way to
model the force-extension behavior of PEG in aqueous solution. PEG monomers
have been found to be stabilized in a helical, trans-trans-gauche conformation in
water. In organic solvent, this stabilization is lost and PEG monomers take a planar,
trans-trans-trans conformation. Oesterhelt et al. found that the water-mediated heli-
cal form of PEG could be driven into the planar form by stretching. Therefore, using
the extended freely-jointed-chain model (eFJC) to describe the polymer's extension
z with force, the contour lenth L C was taken to depend on force as well:

<

coth FL k
k B T

L C

k B T
FL k

N k F
κ s

z ( F )= − ( F )+ (3.80)

where L k is the Kuhn length, N k is the number of segments of length L k ,andκ s is
the segment elasticity. Thus, if a folded or helical segment takes on a length l f and
an unfolded or planar segment takes a length l u , then the contribution to the total
contour length from each of the two forms are given by l f N on

/ N k and l u N off / N k or

N k

l f l u

L C ( F )=

e Δ G ( F ) / k B T +

(3.81)

e −

( ) /

1 + 1 +

Δ G F k B T

3.5.3 P ARALLEL I NDEPENDENT B ONDS :K INETIC R EGIME

A multivalent cluster is typically viewed as a construct of identical, noninteracting
ligands joined together by flexible polymer linkers. Each ligand binds specifically
to one of many receptors on a surface. Here, we assume the cluster to be linked to
the end of a force transducer of stiffness k cant . We will consider the case in which
the linker end-to-end extension is small (small force approximation) such that the
Gaussian chain model for a random flight polymer is adequate (Kratky & Porod,
1949; Flory, 1969). Thus, the polymer linker has a linear force-extension relation
with force constant k poly . Given the fact that typical polymers are much softer than
mechanical force transducers ( k poly

k cant ), we assume the applied force on the
bonds to be dominated by the force-extension relation of the polymer.
The rate of transition from N to N

1 bonds is enhanced by the increased number
of bonds available to unbinding, but the load on each bond is divided over the whole

−