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I am reading Mathematical Physiology, Sneyd. I am a undergraduate mathematician interested in pursuing the field, so forgive me if my terminology is off. In this example, the writer considers a protein with two binding sites. The subscript $i$ indicates the number of bound ligands. This expression, called the saturation function, appears on page 18:

$$Y = \frac{r_1 + 2r_2 + t_1 + 2t_2}{2(r_0+r_1 + r_2 + t_0 + t_1 + t_2)}$$

Where, $r_i, t_i$ denote the concentrations of chemical species $R_i, T_i$ respectively.
It looks like the weighted average of the ezymes that have occupied binding sites.

The fact that $t_0, r_0$ do not appear in the numerator suggest that it is the saturation of enzyme binding sites, the $2$ in front of the $r_2,t_2$ indicates that the enzyme with both binding sites occupied is "double concentrated". The two on the denominator indicates the number of available binding sites in the enzyme. Perhaps, if we were considering an enzyme with three binding sites:

$$Y = \frac{r_1 + 2r_2 + 3r_3 + t_1 + 2t_2 + 3t_3}{3(r_0+r_1 + r_2+ r_3 + t_0 + t_1 + t_2 + t_3)}$$

Would this description be correct?

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  • $\begingroup$ can you give a more precise reference to the book (Sneyd has apparently authored/co-authored several books ...) ? $\endgroup$
    – Ben Bolker
    Oct 3 '16 at 0:14
  • $\begingroup$ ISBN 978-0-387-75846-6 Mathematical Physiology Volume 1: Cellular Physiology $\endgroup$
    – user21915
    Oct 3 '16 at 18:46
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I don't have access to Sneyd's Mathematical Physiology, but in Wikipedia's description of the Monod-Wyman-Changeux model, they call Y the "fractional occupancy of the ligand binding site". Other references refer to it as the "fractional saturation".

In other words, it's simply the fraction of the binding sites which currently have ligands bound to them. The denominator, then, is the total number of binding sites, which is equivalent (for a unit volume) to the total concentration of the protein (i.e. the sum of every protein form) times the number of binding sites per protein.

The numerator is the number of ligand-bound binding sites. Which is equivalent (for a unit volume) to the concentration of each form, multiplied by the number of ligands bound to it. So the zero-ligand forms are multiplied by zero, the one-ligand forms are multiplied by one, the two-ligand forms are multiplied by two, etc. (This doesn't reflect "doubly concentrated", rather it reflects the stoichiometry: for each protein molecule in the 2-bound state, there are two ligand molecules bound to it.)

So the answer to your boxed question is yes: that would be the appropriate form for a cooperative protein with three binding sites.

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