Table 2. Values of rates, feedback and receptor control parameters obtained after optimising the different channel architectures with $n_T$ tiers and $n_B$ branches.

The optimised values of the chemical rates quoted below are scaled by the unbinding rate r_u, ${\kappa }_u$ taken to be 1. The symbols $r_{-}$ and $r_{+}$ denote positive and negative feedbacks, respectively, on the rates following the equals sign; $\{\}$ implies absence of feedback.

Parameter (Symbol)	Value obtained in the optimised channel ($n_T,n_B$)
	(1,1)	(1,2)	(2,2)	(2,2)	(2,2)
	$r_{+}=r_d^{(1)}$	$r_{-}={\kappa }_C$	$r_{-}={\kappa }_I$	$r_{-}={\kappa }_C$	$r_{-}=\{\}$
Chemical rates
Signalling branch
Binding rate ($r_b,{\text{nM}}^{-1}$)	0.0898	0.0949	0.0932	0.0893	0.0787
Degradation rate in tier 1 ($r_d^{(1)}$)	0.0013	0.0081	0.0086	0.0098	0.0038
Degradation rate in tier 2 ($r_d^{(2)}$)	-	-	0.0066	0.0087	0.0016
Internalisation rate ($r_I$)	-	-	0.0531	0.0784	0.0363
Recycling rate ($r_R$)	-	-	0.0681	0.0359	0.0758
Non-signalling branch
Binding rate (${\kappa }_b,{\text{nM}}^{-1}$)	-	0.0590	0.0954	0.0835	0.0288
Degradation rate in tier 1 (${\kappa }_d^{(1)}$)	-	0.0086	0.001	0.0043	0.0068
Degradation rate in tier 2 (${\kappa }_d^{(2)}$)	-	-	0.0037	0.0031	0.0033
Internalisation rate (${\kappa }_I$)	-	-	0.0741	0.0846	0.0559
Recycling rate (${\kappa }_R$)	-	-	0.0123	0.0134	0.0998
Conjugation rate (${\kappa }_C,{\text{nM}}^{-1}$)	-	0.9926	0.9823	0.9722	0.6019
Splitting rate (${\kappa }_S$)	-	0.1285	0.1545	0.1350	0.7512
Feedback control
Amplification ($\alpha$)	3.2085	-	-	-	-
Feedback Sensitivity ($\gamma$)	0.2491	0.1831	0.5535	0.8259	-
Feedback strength ($n$)	2.6825	2.3683	2.0953	2.1880	-
Tier-wise weights
weight of tier 1 (w1)	1	1	0.0018	0.1232	0.9259
weight of tier 2 (w2)	-	-	0.9982	0.8768	0.0741
Receptor control
Signalling receptors
Hill coefficient ($a$)	4.9231	1.9974	3.8363	3.5251	3.3835
Minimum concentration ($A_0,\text{nM}$)	51.8130	51.0960	69.6940	51.9770	51.2
Maximum concentration ($A_0+A_1,\text{nM}$)	298.283	290.356	304.114	134	301
Position of half-maximum (A2)	0.4752	0.7818	0.9405	0.8344	0.4091
Non-signalling receptors
Hill coefficient ($b$)	-	4.8951	1.0802	1.7472	3.1821
Minimum concentration ($B_0,\text{nM}$)	-	192.32	248.69	192.4	94.1850
Maximum concentration ($B_0+B_1,\text{nM}$)	-	442	489.77	441.67	305
Position of half-maximum (B2)	-	0.7428	0.5177	0.3196	0.0902