Testing optimization with model constraints. The model shown in Fig. 2 A was optimized to fit the data in Fig. 3 (time course and steady-state curves), subject to the six sets of constraints shown in Fig. 2 B. (A) The convergence of the overall cost function (left) and penalty component (right). (B) Parameter convergence in each of the six test runs. Only the model parameters K are shown, but note that the optimizer searches in the free parameter space defined by $X ¯$⁠. To reduce clutter, some model parameters are not displayed, as they are defined by constraints (e.g., $k 1,2 0 =a×k 2,3 0$⁠). For better visualization, the exponential factors $k ij 1$ are plotted on the right axis (dotted lines), whereas all the other quantities are on the left axis: preexponential factors $k ij 0$ (log scale, solid lines), channel count N_C (log scale, dashed black line), and allosteric factor a₁ (dashed magenta line). The dashed gray horizontal lines and arrows indicate the boundaries of inequality linear constraints for $k 2,1 1$ and $k 4,3 1$ (runs II through IV) and the boundaries of the range constraint for N_C (run III). Note how $k 4,3 1$ is estimated as a positive value in run I, but it remains less than zero under the inequality constraint in runs II through VI. In each panel, the symbols aligned with the last iteration mark the true parameter values.