Allosteric conformational spread: exact results using a simple transfer matrix method.

A transfer matrix method is described for the conformational spread (CS) model of allosteric cooperativity within a one-dimensional arrangement of four-state binding sites. Each such binding site can realize one of two possible conformational states. Each of these states can either bind ligand or not bind ligand. Thus, analytical expressions that are exact within the context of the CS model are derived for the grand partition function, for the mean fraction of binding sites occupied by ligand versus ligand concentration, and for the mean fraction of binding sites in a given allosteric state versus ligand concentration. The utility of our analytical results is demonstrated by least-mean-square fitting of prior experimental results obtained on the bacterial flagellar motor for the fraction of FliM/FliG/FliN complexes with CheY-P bound [V. Sourjik and H. C. Berg, Proc. Natl. Acad. Sci. U.S.A. 99, 12669 (2002)] and for the cw bias [P. Cluzel, Science 287, 1652 (2000)], which plausibly may be identified as the fraction of protomers realizing state 2. Finally, the relationships between our analytical results and the classical Monod-Wyman-Changeaux, Koshland-Nemethy-Filmer, and McGhee-Von Hippel treatments of allosteric cooperativity are elucidated, as is the connection to an earlier approximate analytical treatment of the CS model.