Sigmoidicity in Allosteric Models

We show results complementary to papers by Bardsley and Waight [2, 3], Gibson and Levin [13], Goldbeter [14], and Karlin [19] on sigmoidicity as an essential feature of allosteric models, possibly leading to a criterion of choice between these. In particular, we give the explicit form for the second derivative of the saturation function in the MWC case, and also the calculation of the binding polynomial in the circular KNF case. First, we give analytical conditions of sigmoidicity for each characteristic function in the MWC and KNF allosteric models. In the MWC model, the regions of sigmoidicity are different for the state and saturation functions, and when the catalytic activities of the two conformational states are different, the area of sigmoidicity is significantly larger for the steady-state rate function than for the saturation function. Furthermore, we rigorously prove the existence of mixed kinetic cooperativity in certain conditions. In the KNF model, the state and saturation functions are the same and their sigmoidicity depends only on the degree of coupling between subunits and on the relative stability of the asymmetrically induced subunit interactions. Finally, we suggest a theoretical criterion for discrimination between allosteric models.