Biomedical Engineering Reference
In-Depth Information
1.4.1 T
HE
A
FFINITY
C
ONSTANT
As previously acknowledged [195], the concept of affinity still dominated most
thinking about complex biological reactions only two decades ago. Starting from
the standard equation
=
[
AB
]
B
←
(
A
+
AB
)
;
K
a
(1.2)
[
A
][
B
]
where
A
and
B
are a ligand and a receptor molecule, [
A
], [
B
], and [
AB
] are, respec-
tively, the molar concentrations of isolated molecules
A
and
B
and of the molecular
complex
AB
,and
K
a
is the affinity constant, we can, in principle, calculate the amount
of complex if we know the total amounts of molecules
A
and
B
. Further, determining
the affinity constant between soluble receptors and ligands may be easily achieved
with powerful and widely available methods such as those based on optical biosen-
sors [174] (some caution is however warranted [166]). Finally, the thermodynamic
relationship
Δ
G
0
K
a
=
exp
(
−
/
RT
)
(1.3)
allows us to relate the affinity constant to the free enthalpy of reaction under standard
conditions (see standard treatises or [20] for more details). However, there are two
problems with this formalism.
Firstly, while Equation 1.2 is useful under equilibrium conditions, life works
out of equilibrium. As an example, the affinity constant may conveniently account
for the amount of occupied receptors on the cell membrane in a stable environ-
ment, but it is certainly insufficient to account for the evolution of rapid signaling
cascades.
Secondly, while Equation 1.2 can be used to deal with two soluble reactants,
or a cell receptor interacting with soluble ligands, it cannot account for interac-
tions between surface-attached molecules. A major problem is related to the reac-
tion entropy. As emphasized by Page and Jencks, the standard free enthalpy Δ
G
0
is the sum of an “intrinsic term” that represents the intrinsic binding energy and a
connecting term that represents the loss of entropy generated by complex forma-
tion [132] [139]. The problem is that both terms are of comparable order of mag-
nitude and they may be quite different when interacting molecules are bound to
surfaces, which may dramatically restrict their motion and number of degrees of
freedom.
Reasoning with kinetic parameters instead of affinity constants may suffice to
deal with out-of-equilibrium processes. As was emphasized, dealing with surface-
attached molecules will result in the replacement of two
numbers
, the reaction on-
rate and off-rate, with two
functions
, namely
k
off
(
)
F
, that is, the dissociation rate
(
)
as a function of applied force, and
k
on
, that is, the association frequency of two
molecules maintained at a fixed distance
d
[148]. The suitability of these functions
d
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