The results of the signal-response relationship classification for the 18 responsive topologies.

For each topology, we have sampled 1000 parameter sets (rate constants and total protein concentrations) from a biologically permissible range (Supporting Text S6), derived the signal-response curve for each parameter set and classified this curve as hyperbolic or sigmoidal (see Methods). The topologies are indicated with a binary identification code that indicates the presence (1) or absence (0) of reverse phosphotransfer reactions along the relay, and the presence (1) or absence (0) of hydrolysis reactions at layers 2 and 4 (see Supporting Text S1 for all possible topologies). The classification results are given as the fraction of parameter sets resulting in a sigmoidal signal-response curve (the rest being classified as hyperbolic) and assuming a monofunctional HK. Classifications are based on the second derivative of the signal-response curve at zero and from sampling all parameters (with equal total protein concentrations at different layers) (see Methods). For additional results using alternative classification and sampling schemes (different total protein concentrations at different layers) or assuming bifunctional HK, see Supporting Text S3.