Signal processing cascade increases the Hill coefficient. The parameters for each reaction were chosen randomly from a beta distribution, denoted as a random variable z∼B(α,β), which yields values in the range [0,1]. The parameters m=100z and g=5+10z were chosen randomly and independently for each reaction from a beta distribution with α=β=3. The parameter k for each reaction was obtained randomly as 1+z, yielding a range of coefficients 1≤k≤2. (a) In three separate trials, different combinations of (α,β) were used for the beta distribution that generated the Hill coefficient, k: in the first, shown as the left distribution, (α,β)=(1,6); in the second, shown in the middle, (α,β)=(4,4); in the third, on the right, (α,β)=(6,2). The plot shows the peak heights normalized for each curve to be the same to aid visual comparison. (b) The input-output relation over the full cascade. The curves from left to right correspond to the distributions for k from left to right in the prior panel. The input scale is normalized so that the maximum input value for each curve coincides at 80% of the maximum output that could be obtained at infinite input. The observed output curves have more strongly reduced sensitivity at low input than at high input compared with the Hill equation, but nonetheless match reasonably well. The best fit Hill equation for the three curves has a Hill coefficient of, from left to right, k=1.7,2.2,2.8. The average Hill coefficient for each reaction in a cascade is, from left to right, k=1.14,1.5,1.75. Each curve shows a single particular realization of the randomly chosen reaction parameters from the underlying distributions.