Figure 1.
Candidate schemes for the sigmoidal switch.
Case a (upper panel within the shaded box): the full pathway of the zero-order ultrasensitive switch studied by Goldbeter and Koshland. The protein R is phosphorylated and dephosphorylated by enzyme S and A respectively. The phosphorylated form RP represents the response and the enzyme S the signal. Case a2 (the entire upper panel): R and S form a loosely bound complex, then a tightly bound one preceding the phosphorylation step. Case b (lower panel, the shaded area is the same as case a): RS can bind another S molecule, which accelerate the phosphorylation reaction k1.
Figure 2.
SR curves of the candidate schemes.
(a) Example SR curves obtained with the schemes shown in Figure 1. The G–K function is the result Goldbeter and Koshland obtained for the zero-order region. Parameters of other cases are obtained by fitting the steady state solutions to the G–K function (see Methods), and are listed in Table 1. (b) Concentrations of various protein forms as a function of the total signal concentration in case b, the only scheme showing the switch behavior.
Figure 3.
SR curves of case b in the presence of dynamic disorder in k1′.
A set of k1′ of 10 different conformations are computed from a gamma distribution p(k) = [1/(baΓ(a))ka−1exp(−k/b) (see Methods). For the results shown here, a = 4, b = 10/a, so that the mean rate constant is 10, the k1′ value used in the first section (Table 1). (a) The ensemble averaged response curves with the conformation k1′ interconversion rate rint = 10−5 (solid line) and rint = 10−1 (dashed line). (b) Responses by individual conformers with rint = 10−5 (dotted lines). The ensemble averaged response is also shown in comparison as the solid line. (c) Same as b with rint = 10−1.
Figure 4.
Variances of the SR curves change with the conformation interconversion rate, rint, with different sets of disordered parameters.
Upper: rint vs. relative variances of the critical signal level. The critical signal level is defined as [S]ti at half the plateau responses (i.e. [RP]i = [RP]i([S]t→∞)/2). Bottom: rint vs. relative variances of the plateau response (i.e. [RP]i([S]t→∞). Different sets of disordered parameters denoted in the legend: a) k2 is computed from the gamma distribution with a = 4, b2 = 1/a; b) k1′ is computed from the gamma distribution with a = 4, b1′ = 10/a; c) k1 is computed from the gamma distribution with a = 4, b1 = 0.008/a; d) all the enzymatic reaction rates, k1, k1′ and k2, are computed from the gamma distribution with (a,b1;a,b1′;a,b2); e) and k2, come from the gamma distribution with (a,b1;a,b2), but k1′ comes from the gamma distribution with (a/2,2b1′). The parameters of the gamma distributions in case a, b, c are chosen so that the mean rates equal the ones used in absence of the dynamic disorder. Note that the parameter a, which determines the width of the distribution, were chosen the same except in the last case for k1′.
Table 1.
Model parameters.