Editing Cooperative binding (section)

=== The Klotz equation ===
Working on calcium binding proteins, Irving Klotz deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages, and tried to express the cooperative binding in terms of elementary processes governed by mass action law.<ref name=Klotz1946a>{{cite journal | vauthors = Klotz IM | title = The application of the law of mass action to binding by proteins; interactions with calcium | journal = Archives of Biochemistry | volume = 9 | pages = 109–17 | date = January 1946 | pmid = 21009581 }}</ref><ref name=Klotz2004>{{cite journal | vauthors = Klotz IM | title = Ligand-receptor complexes: origin and development of the concept | journal = The Journal of Biological Chemistry | volume = 279 | issue = 1 | pages = 1–12 | date = January 2004 | pmid = 14604979 | doi = 10.1074/jbc.X300006200 | doi-access = free }}</ref> In his framework, <math>K_1</math> is the association constant governing binding of the first ligand molecule, <math>K_2</math> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives:

:<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>

It is worth noting that the constants <math>K_1</math>, <math>K_2</math> and so forth do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant <math>K</math>, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> (that is <math>K_i=\frac{n-i+1}{i}K</math>) in the absence of cooperativity. We have positive cooperativity if <math>K_i</math> lies above these expected values for <math>i>1</math>.

The Klotz equation (which is sometimes also called the Adair-Klotz equation) is still often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.<ref name=Dagher2011>{{cite journal | vauthors = Dagher R, Peng S, Gioria S, Fève M, Zeniou M, Zimmermann M, Pigault C, Haiech J, Kilhoffer MC | title = A general strategy to characterize calmodulin-calcium complexes involved in CaM-target recognition: DAPK and EGFR calmodulin binding domains interact with different calmodulin-calcium complexes | journal = Biochimica et Biophysica Acta (BBA) - Molecular Cell Research | volume = 1813 | issue = 5 | pages = 1059–67 | date = May 2011 | pmid = 21115073 | doi = 10.1016/j.bbamcr.2010.11.004 | doi-access = free }}</ref>