Fig. 1: Signal termination in a CMC: the importance of the relative concentrations between substrate and enzymes.

a Scheme representing a CMC cycle. S is the unmodified substrate, P the product (modified substrate), E₁ and E₂ are the modifying enzymes (kinase and phosphatase), C₁ is the substrate–kinase complex, C₂ is the product-phosphatase complex. Three kinetic parameters are associated with each enzymatic reaction, a_i, d_i, and k_i: the association, dissociation, and catalytic rates, respectively (more details in the Methods section). b Phosphorylated protein (P, solid green line), substrate–kinase complex (C₁, dotted pink line), and product-phosphatase complex (C₂, dashed blue line) are plotted versus time. Parameter values: a₁ = 3.5; d₁ = 1; k₁ = 50; a₂ = 0.3; d₂ = 0.25; k₂ = 0.25; E_1T = 100; E_2T = 100; S_T = 35. a_i, i = 1,2, are in units of 1/(time × concentration), d_i and k_i are in units of 1/time, the total enzymes (E_1T and E_2T) are in units of concentration (see Methods section). P_max and P_ss, the maximum and the steady-state levels of P, are indicated over the plot. The characteristic value 0.63 × P_max is indicated with an asterisk over the time course. c Graphs of Q (=P_max/P_ss) versus E_1T/S_T and E_2T/S_T, with Q in color scale. E_1T/S_T, E_2T/S_T, and Q are in logarithmic scale, E_1T/S_T and E_2T/S_T vary between 0.1 and 10, resulting in Q values in the range between 1 and ~46. The solid blue line over the plots corresponds to Q = 1.6 and separates signal termination from monotonic behavior and non-monotonic behavior not satisfying the criteria. The solid black line over the plot corresponds to P_ss = 0.2. The cases over the line (P_ss > 0.2) are termed strong signal termination. Dashed black lines correspond to P_ss = 0.1 or 0.3, indicating different boundaries for the strong signal termination region. On the upper left triangle, Q < 1.6 (no signal termination satisfying the criteria). The kinetic parameter values are as in b. d Representative time courses in different regions in c. Parameter set 1: E_1T/S_T = 0.13, E_2T/S_T = 0.13, parameter set 2: E_1T/S_T = 1, E_2T/S_T = 0_.13, parameter set 3: E_1T/S_T = 1, E_2T/S_T = 0_.79, parameter set 4: E_1T/S_T =1, E_2T/S_T = 1_.26, parameter set 5: E_1T/S_T = 0.13, E_2T/S_T = 7.9, parameter set 6: E_1T/S_T = 7.9, E_2T/S_T = 7.9.