Initial rates of ATP hydrolysis by wild-type GroEL were measured as a function of ATP concentration from 0 to 0.8 mM. Two allosteric transitions are observed: one at relatively low ATP concentrations (< or = 100 microM) and the second at higher concentrations of ATP with respective midpoints of about 16 and 160 microM. Two allosteric transitions were previously observed also in the case of the Arg-196-->Ala GroEL mutant [Yifrach, O., & Horovitz, A. (1994) J. Mol. Biol. 243, 397-401]. On the basis of these observations a mathematical model for nested cooperativity in ATP hydrolysis by GroEL is developed in which there are two levels of allostery: one within each ring and the second between rings. In the first level, each hepatameric ring is in equilibrium between the T and R states, in accordance with the Monod-Wyman-Changeux (MWC) model of cooperativity [Monod et al. (1965) J. Mol. Biol. 12, 88-118]. A second level of allostery is between the rings of the GroEL particle which undergoes sequential Koshland-Némethy-Filmer (KNF)-type transitions from the TT state via the TR state to the RR state [Koshland et al. (1966) Biochemistry 5, 365-385]. Using our model, we estimate the values of the Hill coefficient for the negative cooperativity between rings in wild-type GroEL and the Arg-196-->Ala mutant to be 0.003 (+/- 0.001) and 0.07 (+/- 0.02), respectively. The inter-ring coupling free energies in wild-type GroEL and the Arg-196-->Ala mutant are -7.5 (+/- 0.4) and -3.9 (+/- 0.3) kcal mol-1, respectively.