Biomedical Engineering Reference
In-Depth Information
Bimolecular (Association/Dissociation)
Unimolecular (Conformation change)
H
(
x,t
) =
U
(
x
) +
V
(
x,t
)
H
(
x,t
) =
U
(
x
) +
V
(
x,t
)
V
(
x,t
)
V
(
x,t
)
U
(
x
)
U
(
x
)
v
v
Intermolecular separation
x
Intramolecular separation
x
FIGURE 3.1
The two primary systems encountered in force spectroscopy of biomolecules
involve intermolecular bonding and intramolecular conformational changes. While the inter-
molecular potential
U
of bimolecular bonds are typically characterized by a single well
defining the bound state, the addition of a spring-like external potential
V
(
x
)
can lead to a
double-well, bistable system. In a unimolecular system containing two states, the intramolec-
ular potential is inherently bistable and thus the external field acts to tilt the balance between
the two conformations.
(
x
)
A is in excess, while molecule B is present at low levels, then the fraction of AB
pairs found relative to the total number of molecules available will be small. As
the concentration of B is increased, the proportion of molecules reacted to form
AB will increase as well. While more complicated theoretically, this property places
bimolecular interactions in an exciting arena of their own. Investigating the strength
of bimolecular systems by force and finding clear connections between the single-
molecule measurement and ensemble bimolecular equilibria is still a relatively new
area of study.
3.1.2 P
ROBING
I
NTERACTIONS BY
F
ORCE
When the molecule of interest is linked to a force transducer, we will assume that the
primary degree of displacement occurs along the direction of the applied force. Thus,
the analysis in this chapter, unless otherwise noted, will be along one-dimensional
space
x
. When an inter- or intramolecular potential
U
(
x
)
is perturbed by an external
potential
V
(
x
,
t
)
, the total energy of the system at every position is their sum,
H
(
x
,
t
)=
U
(
x
)+
V
(
x
,
t
)
(3.1)
where
H
is the Hamiltonian of the system. Typically, the force transducer is
spring-like, and therefore applies a parabolic potential
V
(
x
,
t
)
1
2
k
cant
2
to
the system, where
k
cant
is the force constant of the spring. We use the symbol
k
cant
to refer to the commonly used cantilever in atomic force microscopes, however, the
treatment from here on is general to any force transducer, which is approximately
(
x
,
t
)=
(
x
−
vt
)
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