Calcium-Binding Protein Protocols Calcium-Binding Protein Protocols

30 Haiech and Kilhoffer
reporter groups with spectroscopic properties that are sensitive to the occupancy
of one specific site by the ligand.
Assume that to each complex depicted in Fig. 1B, we associate a signal,
namely s 0 for the protein without ligand, s 1 for the protein with ligand in site 1,
s 2 for the protein with ligand in site 2, and s 3 for the protein with two ligands. It
is straightforward to derive the equation describing the variation of the signal
as a function of the free-ligand concentration (L).
s 0 + s 1 * k 1 * (L) + s 2 * k 2 * (L) + s 3 * c * k 1 * k 2 * (L) 2
S = ———————————————————————— (11)
1 + k 1 * (L) + k 2 * (L) + c * k 1 * k 2 * (L) 2
Notice that s = s 0 when (L) = 0 and s = s 3 for (L) at saturating concentration.
This last point means that from an experimental point of view, we may increase
the concentration of (L) in such a way that c * k 1 * k 2 * (L) 2 >> 1 + (k 1 + k 2) * (L).
As the concentration of (L) may be limited to a given range, the unknown s 3
cannot be always determined independently.
If in a given experiment, the signal S we measure corresponds to the number
of ligand bound per protein, we have
s 0 = 0; s 1 = s 2 = 1
and s 3 = 2 (the signal corresponds to the number of bound ligands for each
complex).
Equation 11 can be rewritten:
k 1 * (L) + k 2 * (L) + 2 * c * k 1 * k 2 * (L) 2
v = —————————————————— (12)
1 + k 1 * (L) + k 2 * (L) + c * k 1 * k 2 * (L) 2
This equation is equivalent to Eq. 2 combined with Eq. 1.
Assume now that we are able to introduce, at a specific location in the protein,
a reporter group sensitive to the occupancy of the site 1 (respectively, a
reporter group sensitive to the occupancy of site 2). Therefore, for the signal
arising from the first reporter group, we have:
k 1 * (L) + c * k 1 * k 2 * (L) 2
S* = —————————————————— (13)
1 + k 1 * (L) + k 2 * (L) + c * k 1 * k 2 * (L) 2
as s 0 = 0, s 1 = 1, s 2 = 0, and s 3 = 1 (in relative units).
For the second reporter group, sensitive to the occupancy of site 2, we have:
k 2 * (L) + c * k 1 * k 2 * (L) 2
S** = —————————————————— (14)
1 + k 1 * (L) + k 2 * (L) + c * k 1 * k 2 * (L) 2
as s 0 = 0, s 1 = 0, s 2 = 1, and s 3 = 1 (in relative units).
Figure 2 presents a graphical representation of Eqs. 12–14 in the case of a
protein with two independent and equivalent sites (see Fig. 2A) and in the case