Theoretical Models in Force Spectroscopy - Dynamic Force Spectroscopy and Biomolecular Recognition - page 122

N = 5

Δ G N

F

M = 5

FIGURE 3.9 Equilibrium free energy of a multivalent cluster under force. N ligand-receptor
interactions are considered to occur on average at equilibrium. M receptors are assumed to
be within reach of any of the N receptors. This allows for “mixing” of the ligands on the
surface receptors and contributes an entropic term in the free energy favoring the bound state.
The illustration depicts the case of a cluster bearing five ligands, while three are bound to the
surface receptors on average. The absolute free energy of the bound state alone depends only
on the number of combinations of N ligands on M receptors, which in this example is 10.
However, the free energy change between the N -bonded cluster and the free cluster depends
on the number of permutations of N ligands with M receptors that is 60. This large degeneracy
enhances the free energy cost to forcibly driving the system to the completely unbound state
near equilibrium.

The free energy difference between the N -bonded cluster and the free cluster is
given by

k B T ln Z N , off

Δ G N ( F )= −

Z N , on ,

(3.92)

which when bound and unbound states are balanced by the applied force

Δ G N ( F N )= 0, and we find the equilibrium force

, eq

N 2 2 k poly Δ G 1

N 2 k poly k B T ln

M !

F N = + (3.93)

, eq

( M − N ) !

where

k B T ln κ 0
k poly

= −

Δ G 1 Δ U 0 (3.94)

is the free energy of detaching a single ligand-receptor pair by transferring the
applied force with the polymer linker of force constant k poly . In Equation 3.93, the
second term under the root accounts for the entropy imparted by total number of
configurations, or permutations

M !

! of associating N ligands with N receptors
out of the M available. This can be a considerably large number when M

( M − N )

> N .Even

=

when M N , an entropic enhancement is found due to the different configurations

that can occur between N ligands on N receptors (i.e., N !

= ( − )( − ) ···

1).
All such configurations are identical in energy, which is important near equilibrium
when spontaneous unbinding and rebinding events occur faster than the timescale of

N N 1 N 2