Topic #3: Cooperative Binding Calculations
Model reaction:
PA2 <=> P + 2 A
Dissociation constant: Kd = [P][A]nH/[PA2] (and nH < 2, for this example)
Fractional occupancy: Y = [A]nH/(KdnH + [A]nH)
[Note: P and A are used on this page (instead of the more general, M and L) to indicate a particular binding reaction under study. Any symbols can be (and are) used to represent the macromolecule and ligand, respectively.]
- The Hill equation is derived from the expressions for Kd and Y. In logarithmic form:
log(Y/1-Y) = nHlogA - nHlogKd
, where
- nH = the Hill coefficient (maximum slope of the Hill plot);
- A = [A]free;
- Kd = the dissociation constant.
We have a 0.4 mg/ml solution of a protein (Mr = 40,000 Da). The stoichiometry of A binding is 2.0. When bound, the UV absorption of the ligand changes. Thus, we can determine [A]bound directly. We want to determine Kd and the Hill coefficient, nH. This is done in three steps.
(Each calculated [A]bound value has a small "experimental error" added to it.
Tabbing out of the volume entry slot or clicking anywhere on the page will also calculate [L]free.)
3. Record the values of [A]bound you obtain. Then, calculate [A]free, Y, and (Y/1-Y). Finally, graph the values on a Hill plot to determine Kd and nH. You should get enough data so as to have 3 or 4 values of Y, both above and below the Kd value.
(Hint: Graph the values you obtain as they are calculated; then as the shape of the curve becomes apparent, choose values for [A]total that fall into the appropriate range.)
Answers to this problem.
A sample Answer Sheet for a similar problem shows the format of the results and the graph required.
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