Effect of increasing Hill exponents. We consider a simple cascade between the four species X1, X2, X3, X4 as shown in the inset in (A). Each activation is modeled using a Hill function with threshold k = 0.5 and Hill coefficient n. The life-times τ
are set to 1. As initial conditions we take = c > 0, = 0, = 0, = 0, for some constant input concentration c. The input node X1 remains constant and the other concentrations change accordingly to the ODE , i = 2, 3, 4. We simulate the model for different Hill coefficients n = 1, 4, 16 and input level c = 1; the results are shown in (A), (B) and (C). All three time courses show qualitatively the same cascade-like pattern. With growing n, however, the onset of activation of X3 and X4 comes closer and closer to the time point at which their activators X2 and X3, respectively, cross the threshold k. (D) shows the input-output curve. Plotted is the (constant) input concentration c of node X1 against the steady-state concentration of node X4. For n > 1, we observe the typical sigmoid stimulus-response behavior of signaling cascades, see e.g. . With increasing n the steepness of the input-output curve increases, leading to an almost discrete (Boolean) output in the case n = 16.