Figure 1. **Evidence that calcium total binding parameters for CaM are not PI.** (A, top) Four-site, sequential Adair–Klotz model. *R*_{i} represents CaM with *i* Ca^{2+} ions bound; *K*_{1}–*K*_{4} are macroscopic association equilibrium constants. (A, middle left) Solid curve represents total fraction of binding sites occupied by ligand (*Y*), computed using Adair parameters reported in Linse et al. (1991). The function *Y* (with a maximum value of 1) is the normalized version of the function *v* (maximum value of 2) appearing in Eq. 2. Parameters are (in M^{−1}): {*K*_{1}, *K*_{2}, *K*_{3}, *K*_{4}} = {79,433, 3.98 × 10^{6}, 25,119, 398,107}. Circles represent total binding curve computed using parameter set {160,344, 1.85 × 10^{6}, 37,678, 287,086}. (A, middle right) Same as A (middle left), except triangles represent total binding curve computed using parameter set {50.3, 6.6 × 10^{9}, 15,093, 617,904}. (A, bottom) Parameter fit-error plots. Each curve represents the results of 300 curve fits in which *K*_{1}, *K*_{2}, *K*_{3}, or *K*_{4} was held at one of 300 values ranging from 10 to 10^{10} M^{−1}. At each of these values, the chosen *K* was held constant while the other association constants were varied freely to achieve the best least-squares fit to the synthetic data (solid lines in middle graphs) using nonlinear regression. Curves depict rms error in best fit for systematic variation of *K*_{1} (red), *K*_{2} (blue), *K*_{3} (black), or *K*_{4} (green). Dashed line shows 5% rms error threshold. (B) Effect of noise on parameter estimates. (top left) Each curve is a histogram representing the distribution of estimates of parameter *K*_{1} obtained from 1,000 separate nonlinear regression fits in which all four parameters were allowed to vary freely. For each fit, random, Gaussian-distributed noise with a standard deviation of 0.05 (corresponding to 5% rms noise) was added to the noiseless, synthetic data (solid curve in middle graphs of A). Each histogram corresponds to a different starting value for *K*_{1}. Starting values are indicated by inverted arrows. (B, top right and bottom left and right) Same as top left graph of B, except the histograms represent distributions of estimates for parameter *K*_{2}, *K*_{3}, and *K*_{4}, respectively, for different starting values of these parameters (indicated by inverted arrows). Dashed vertical lines represent “correct” values of parameters (taken from Linse et al. [1991]).