Fig 1.
Enzymatic futile cycle with retroactivity.
M and Mp are the inactive and active forms of the protein substrate. Kinase E and phosphatase P, respectively are the enzymes for the phosphorylation and dephosphorylation biochemical reactions. While S1 and S2 are the downstream targets, respectively of M and Mp, MS1 and MpS2 are the corresponding sequestered complexes.
Fig 2.
Schematic showing the steady-state dose response curve corresponding to the nominal profiles of (A) Hyperbolic (), (B) Signal transducing (
), (C) Threshold hyperbolic (
), (D) Ultrasensitive (
). The conditions employed for simulating these dose-response curves are in Table 1.
Table 1.
The nature of the state of the two biochemical reactions corresponding to four operating regimes.
Fig 3.
Retroactivity in Mp inducing operating regime transition from (A) U at (λ = 0) = 7,
(α = 0) = 70 to ST at (
(0),
(27)) = (7,1960) and (B) H at (
(λ = 0) = 389,
(α = 0) = 751) to ST at (
(0),
(15)) = (389,12016). Inset: Zoom in of the dose-response curves. For ease of comparison, the nominal profiles
and
from Fig 2D and 2B, respectively are included in (A). Similarly,
and
from Fig 2A and 2B, respectively are included in (B). Parameters (
(0),
(0)) used for simulating the nominal profiles are in Table 1.
Fig 4.
Boundaries of the four operating regimes hyperbolic (H, blue), signal transducing (ST, green), threshold-hyperbolic (TH, red), and ultrasensitive (U, yellow) in the planes of and
for λ = α = 0.
All boundaries of each of the regimes except the upper boundary of ST satisfy the relative distance criterion in Eq (8). For the case of upper boundary of ST the rhs of Eq (11) was set to 0.02. (,
) on the dotted lines extending the solid line boundaries were sourced directly from the database. While dose-response curves corresponding to parameter sets at green dots A and B were used as example of transition from U to TH regime, those curves at C and D of transition from H to ST.
Fig 5.
Effect of retroactivity strength on the operating regimes and the associated sensitivity for (i) U to TH, (ii) U to ST, and (iii) H to ST transitions. While panel (i) corresponds to effects due to load on M quantified by λ, panel (ii) and (iii) captures those due to load on Mp quantified by α. Dependence of dose-response curves on the load corresponding to (i), (ii) and (iii) are in (A), (D) and (G), respectively. Sensitivity of steady-state level for different retroactivity strengths for (i), (ii), and (iii) are in (B), (E) and (H), respectively. Sensitivity curves in (B) was estimated using Eq (9), Eq (10) was employed for those in (E) and (H). While rate-balance plot showing the effect of retroactivity strength on modulation of steady-state levels corresponding to (i) at et = 1000nM is in (C), that for (ii) and (iii) at et = 150nM are in (F) and (I), respectively. Colorbar in each of the panels display the retroactivity strengths. Dotted line in (A), (B) and (C) in panel (i) corresponds to the dose-response, sensitivity and Rd(λ) curves, respectively at the transition where λ = 960. Dotted line in panels (ii) and (iii) captured these curves at the corresponding transition where α = 9.44 and α = 5, respectively.
Fig 6.
Dose-response curves capturing the retroactivity driven transition of operating regimes from (A) ST to H and (B) TH to H. Colorbar displays the retroactivity strength corresponding to the dose-response curves. Dotted lines in (A) and (B), respectively correspond to the retroactivity strength λ = 400 and α = 578 at which the regime transition occurs.