Biomedical Engineering Reference
In-Depth Information
collides with a strongly positively charged surface (see Figure 10.9b), exactly
those conditions in which such high rates are possible.
2
What about k
2f
5 1.0 6
10
13
M
21
s
21
and k
2b
5 1.0 6 10
7
M
21
s
21
? These rates appear much too high,
until one realises that these processes, as proposed, are not true bimolecular
collisions—they are diffusion over a surface between two binding sites that are
close in space. A more meaningful interpretation is a diffusion, which is only
productive when the other site is empty. As we calculated the fractions of all
species for a concentration of
1 mM, the maximum success rate for the IS to an open IC site is given by p
2f
(max) 5 k
2f
[C]
empty
5 1.0 6 10
13
6 10
23
5 1.0 6 10
10
s
21
.
Similarly, p
2b
(max) 5 k
2b
[S]
empty
5 1.0 6 10
4
s
21
.
In this interpretation, in place of eqn (10.25), one has,
p
1f
~k
1f
½
S
eq
p
1b
~k
1b
p
2f
~k
2f
p
empty
C
p
2b
~k
2b
p
empty
ð
10
:
31
Þ
S
p
3f
~k
3f
½
C
eq
p
3b
~k
3b
with p
empt
C
being the time fraction (0-1) that site C is empty. In this notation,
we thus have k
2f
51.0 6 10
10
s
21
and k
2b
51.0 6 10
4
s
21
. These rates are quite
reasonable, and correspond to the typical 100 ps timescales of local motions of
loops and tails in proteins, molecular entities of sizes similar to IP6.
In summary, a re-investigation of the computational aspects of the modeling
of a three-site exchange process in which IP6 can bind to two sites on
hemoglobin, S and C, and can migrate between these sites, confirms the
conclusions derived three decades ago. These conclusions are that very fast
exchange with hardly any line broadening for very high-affinity (nM) binding
can be modelled with the model presented in eqns (10.22) and following.
10.4 Non-Canonical Line Broadening in Slow Exchange
after Equivalence is Reached
We are studying the binding of metal ligands, Cd
2+
in particular, to designed
three-helix bundles with one cysteine per peptide, which are juxtaposed to form
a intramolecular metal binding site (see Figure 10.10).
18
The bundle is called
(GrandL26AL30C)
3
, where Grand 5 AcG-(LKALEEK)
5
-GNH
2
. UV spec-
troscopy shows that (GrandL26AL30C)
3
binds Cd
2+
with high affinity (K
D
,33 nM) with a stoichiometry of one Cd
2+
per trimer.
However, the
113
Cd NMR results shown in Figure 10.10 indicate that the
binding is more complicated than suggested by UV. The figure shows that no
