Calcium-Binding Protein Protocols Calcium-Binding Protein Protocols

28 Haiech and Kilhoffer
where γ is the number of moles of ligand bound per mole of protein and (L) is
the free-ligand concentration. The denominator of Eq. 2 is the binding polynomial.
The degree of this polynomial corresponds to the number of sites.
Nonlinear regression on the experimental data using Eq. 2 allows to determine
the two macroscopic constants. Determination of the degree of the binding
polynomial is also possible. However, for obvious mathematical reasons, it
would be much better to determine the number of sites for a given ligand using
an independent technique (for instance, mass spectrometry). Determination of
the macroscopic constants would then be much more precise.
From the macroscopic constants and using Eq. 1, we would like to determine
the individual or microscopic constants and the coupling factor. Unfortunately,
we have two equations with three unknowns. To interpret our
“macroscopic data” in a “microscopic or molecular” scheme, we have to make
some simplifying hypothesis. In others words, we have either to fix the value of
at least one of the unknown parameters or to add a third equation.
Two hypotheses are classically found in the literature:
• One considers the sites to be independent; c is then equal to 1;
• The second uses a principle of symmetry and considers that the sites are identical;
their individual association constants are then equal (k 1 = k 2 = k).
Using the first hypothesis, Eq. 1 may be solved if
K 1 ≥ 4 * K 2
If this inequality does not hold, the system cannot be interpreted with c equal to
1. That implies that the two sites are not independent.
Using the second hypothesis, three equations with three unknowns are
obtained. The solution of this system is
(3)
k 1 = k 2 = k = K 1/2 (4)
Combining the two hypotheses, we assume that the two sites are independent
and equivalent. This strong assumption is called the Scatchard hypothesis. With
this assumption, Eq. 1 becomes
Equation 5 can be solved if and only if
With this hypothesis, Eq. 2 becomes
K 1 = 2 * k
K 1 * K 2 = k 2 (5)
K 1 = 4 * K 2 = 2 * k (6)
v = [2 * k * (L)] / [l + k * (L)] (7)