Parameters of two-site allosteric model are not SI when constrained by total binding data. (A) Diagram of two-site allosteric model. Large circles represent ligation states of the system. Small circles represent binding sites I (left) and II (right). States are designated by symbols R_ij, where i (j) is equal to 1 or 0 depending on whether site I (site II) is bound or not bound by ligand, respectively. States with zero, one, or two bound ligands are color coded black, red, and blue, respectively. Closed and open circles represent bound and unbound sites, respectively. Model parameters are the microscopic association equilibrium constants K_I and K_II for sites I and II, respectively, and cooperativity factor f. (B, top) Equation relating mean number of bound ligands, v, to concentrations of the ligated states. (bottom) Equation relating v to free ligand concentration, x, and model parameters. In the top and bottom, terms arising from states with zero, one, and two bound ligands are color coded black, red, and blue, respectively. (C, top) Simulated binding curve computed from the top equation in B using parameter values {f, K_I, K_II} = {10.017, 1.0034 × 10⁶ M⁻¹, 1.99 × 10⁸ M⁻¹}. Parameters were chosen to produce a relatively “unstructured” binding curve. (bottom) Locus of all parameter triples {f, K_I, K_II} that yield total binding curve identical to the curve shown above. The curve was computed using Eqs. 12a and 12b. Parameter triples are determined by taking vertical lines, which determine the value of f, and their intersections with the bold and light green curves, which determine parameters K_I and K_II. Because of the symmetric appearance of K_I and K_II in the binding equation (B, bottom), the bold and light curves may correspond to either K_I and K_II or K_II and K_I, respectively. Dashed lines marked “A,” “B,” and “C” correspond to parameter triples {0.2, 10⁸ M⁻¹, 10⁸ M⁻¹}, {1.0086, 1.0462 × 10⁷ M⁻¹, 1.8954 × 10⁸ M⁻¹}, and {100.86, 99,193 M⁻¹, 2.0 × 10⁸ M⁻¹}, respectively. Arrows on abscissa delineate regions of negative (f < 1), zero (f = 1), and positive (f > 1) cooperativity. (D, top) Simulated total binding curve computed from the top equation in B using parameter values {f, K_I, K_II} = {0.10086, 1.999 × 10¹⁰ M⁻¹, 9.9193 × 10⁶ M⁻¹}. Parameters were chosen to produce a more “structured” binding curve than the one in the top of C. (bottom) Locus of all parameter triples that yield total binding curves identical to curve shown above. The curve was computed using Eqs. 12a and 12b. Dashed lines marked “D,” “E,” and “F” correspond to parameter triples {0.0002, 10¹⁰ M⁻¹, 10¹⁰ M⁻¹}, {1.0017, 2.0 × 10¹⁰ M⁻¹, 9.9837 × 10⁵ M⁻¹}, and {100.17, 2 × 10¹⁰ M⁻¹, 9,983.2 M⁻¹}, respectively.