**Effect of increasing Hill exponents**. We consider a simple cascade between the four species

*X*_{1},

*X*_{2},

*X*_{3},

*X*_{4 }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

*X*_{1 }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

*X*_{3 }and

*X*_{4 }comes closer and closer to the time point at which their activators

*X*_{2 }and

*X*_{3}, respectively, cross the threshold

*k*.

**(D) **shows the input-output curve. Plotted is the (constant) input concentration

*c *of node

*X*_{1 }against the steady-state concentration of node

*X*_{4}. 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.

## PubMed Commons