Figure 6

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 _i 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. [28]. With increasing n the steepness of the input-output curve increases, leading to an almost discrete (Boolean) output in the case n = 16.