Editing Cooperative binding (section)

=== The KNF model===
Based on results showing that the structure of cooperative proteins changed upon binding to their ligand, [[Daniel E. Koshland, Jr.|Daniel Koshland]] and colleagues<ref name=Koshland1966>{{cite journal | vauthors = Koshland DE, Némethy G, Filmer D | title = Comparison of experimental binding data and theoretical models in proteins containing subunits | journal = Biochemistry | volume = 5 | issue = 1 | pages = 365–85 | date = January 1966 | pmid = 5938952 | doi = 10.1021/bi00865a047 }}</ref> refined the biochemical explanation of the mechanism described by Pauling.<ref name=Pauling1936/> The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit".<ref name=Koshland1958>{{cite journal | vauthors = Koshland DE | title = Application of a Theory of Enzyme Specificity to Protein Synthesis | journal = Proceedings of the National Academy of Sciences of the United States of America | volume = 44 | issue = 2 | pages = 98–104 | date = February 1958 | pmid = 16590179 | pmc = 335371 | doi = 10.1073/pnas.44.2.98 | bibcode = 1958PNAS...44...98K | doi-access = free }}</ref> Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved. For a tetrahedric structure (they also considered linear and square structures), they proposed the following formula:

:<math>
\bar{Y} = \frac{K_{AB}^3(K_XK_t[X])+3K_{AB}^4K_{BB}(K_XK_t[X])^2+3K_{AB}^3K_{BB}^3(K_XK_t[X])^3+K_{BB}^6(K_XK_t[X])^4}{1+4K_{AB}^3(K_XK_t[X])+6K_{AB}^4K_{BB}(K_XK_t[X])^2+4K_{AB}^3K_{BB}^3(K_XK_t[X])^3+K_{BB}^6(K_XK_t[X])^4} 
</math>

Where <math>K_X</math> is the constant of association for X, <math>K_t</math> is the ratio of B and A states in the absence of ligand ("transition"), <math>K_{AB}</math> and <math>K_{BB}</math> are the relative stabilities of pairs of neighbouring subunits relative to a pair where both subunits are in the A state (Note that the KNF paper actually presents <math>N_s</math>, the number of occupied sites, which is here 4 times <math>\bar{Y}</math>).