Theoretical Models in Force Spectroscopy - Dynamic Force Spectroscopy and Biomolecular Recognition - page 119

Numerical solution to Equation 3.86 is shown in Figure 3.8. Notice that the enhance-
ment in rupture force is modest in this nonequilibrium regime. That is, the rupture
force of N bonds is less than N times the rupture force of one! The reason is because
of the increased probability of detachment when multiple bonds are available. Con-
sider the limit of vanishing force, such that the lifetime of one ligand-receptor com-
plex is τ 1 and we assume the same Markov process for unbinding of each ligand
takes place. The total mean lifetime of two ligands is

τ 2 = τ 1 / 2 + τ 1 = 1 . 5τ 1

which is not even twice the lifetime of one bond! The total lifetime of the three gives
τ 3 1τ 1 . Under the irreversible scheme

we are considering here, the total lifetime of N bonds at zero force is the harmonic

≈ 1 . 8τ 1 and that of four gives only τ 4 ≈ 2 .

500

N = 3

400

300

N = 2

200

N = 1

100

0

-30 -28 -26 -24 -22 -20 -18

ln( r f )

FIGURE 3.8 (See color insert.) Rupture of multiple parallel bonds for 1, 2, and 3 multivalent
clusters bewteen the cancer marker Mucin-1 and its antigen fragment. Data points reproduced
with permission from Sulchek, T. et al. 2005. Proc. Nat. Acad. Sci. USA , 102 (46), 16638-
16643. Solid lines are numerical solutions of Equation 3.86. Only the single-bond, N 1,

curve is fit to the data to determine the kinetic unbinding rate and transition state, while the
N

=

3 curves are predictions for bivalent and trivalent bonding. The dashed lines
illustrate that, for a given loading rate, including two or three bonds to the cluster does not
add a significant increase in total rupture force. In fact, at the chosen loading rate, the rupture
force for N

= 2and N =

1 case. This is due to the increased
probability of rupture when multiple bonds are present. The enhancement over a single bond
improves for increased loading rate, but even at extremely fast loading, the rupture force for
N bonds is limited to less than N times the single-bond rupture force. As discussed in the text,
binding enhancement due to multivalency is most prominent in the equilibrium regime where
rebinding and entropic effects stabilize the bound state.

= 3 bonds is just twice that of the single N =