Fig 1.
The concept of integral control.
In single-E controllers the variable E is proportional to the integrated error ϵ, ∫ϵdt, which is used to correct for perturbations in A. In dual-E (antithetic) controllers the difference between variables E1 and E2 is proportional to the integrated error (see S1 Text). In both cases integral control will move A precisely to its set-point Aset when A is perturbed by step-wise perturbations [3].
Fig 2.
Single-E and dual-E (antithetic) representations of integral control using a motif 5 negative feedback structure.
Left panel: Single-E controller where error integration occurs by zero-order kinetics (low k5) removing E [4, 6]. Right panel: Dual-E controller [7, 8, 10] with controller pairs E1 and E2. Error integration occurs by the (here second-order) reaction between E1 and E2. In the single-E controller the concentration of E is proportional to the integrated error . In the antithetic (dual-E) controller, the difference E1−E2 is proportional to the integrated error
. The colorings of the reaction schemes relate to the different parts in the general control loop shown in Fig 1.
Fig 3.
Dual-E (antithetic) integral control in combination with the eight negative feedback structures m1-m8.
In the calculations the removal of E1 and E2 is catalyzed by enzyme Ez using different mechanisms. The signaling between A and the manipulated variables E1/E2 occurs either by an “inner loop” between A and E1 (motifs m2, m3, m5, and m8), or by an “outer-loop” signaling between A and E2 (motifs m1, m4, m6, and m7).
Fig 4.
Overview (Cleland notation [19]) of the enzymatic mechanisms removing E1 and E2 (Eq 4).
(a) Ternary complex mechanism with random binding of E1 and E2 to the enzyme. (b) Ternary complex mechanism with compulsory binding order. Here E1 binds first to free enzyme Ez then E2 binds to the E1⋅Ez complex. Alternatively, E2 can bind first to Ez and then E1 to form the ternary complex. (c) Ping-pong mechanism. E1 (or E2) bind first to Ez leading to the alternate enzyme form Ez*, which then can bind E2 (or E1). (d) Single-substrate Michaelis-Menten mechanism used in single-E controllers.
Fig 5.
Motif 2 antithetic controller: Removal of E1 and E2 by enzyme Ez using a ternary-complex mechanism with random binding order.
Fig 6.
Motif 2 single-E controller: Removal of E by enzyme Ez using a Michaelis-Menten mechanism.
Fig 7.
Behavior of the catalyzed dual-E controller (Fig 5) and the single-E controller (Fig 6) towards step-wise perturbations in k2.
Total enzyme concentration Eztot=1×10−6. Upper left panel: Behavior of controlled variable A of the dual-E controller. Phase 1: k2=10.0; phase 2: 1, k2 = 1×102; 2, k2 = 1×103; 3, k2 = 2×104, note the offset in A from Aset. Upper right panel: Behavior of E1 and Ez⋅E2 as a function of k2-perturbations 1–3. Note that for perturbation 3 the enzyme is saturated with E2. Rate constants: k1=0.0, k3=1×105, k4=1.0, k5=10.0, k6=20.0, k7=1×109, k8=0.1, k9=1×108, k10=1×103, k11=1×108, k12=1×103, k13=1×108, k14=1×103, k15=1×108, k16=1×103. Initial concentrations: A0=2.0, E1,0=454.4, E2,0=0.204, Ez0=4.4×10−10, (E1⋅Ez)0=9.796×10−7, (E1⋅Ez⋅E2)0=2.0×10−8, (Ez⋅E2)0=1.98×10−13. Lower left panel: Behavior of controlled variable A for the single-E controller. Same step-wise k2 perturbations 1–3 as for the dual-E controller. Lower right panel: Behavior of E as a function of k2-perturbations. Rate constant values are the same as for the dual-E controller, except that k5=50.0, and k7=1×108. Initial concentrations: A0=1.995, E0=455.5, Ez0=2.19×10−9, (Ez⋅E)0=9.976×10−7.
Fig 8.
Same system as in Fig 7 with perturbation 3 applied, i.e., during phase 1 (0–10 time units) k2=10.0, while during phase 2 k2 = 2×104. All other rate constants are as in Fig 7, except that in panel (a) the total amount of enzyme Ez has been increased by one order of magnitude to Ez0=10−5, while in panel (b) Ez0=10−6, but k5 and k6 have been decreased in phase 2 by one order of magnitude to 1.0 and 2.0, respectively. Initial concentrations: (a) A0=2.0, E1,0=454.4, E2,0=0.0204, Ez0=4.4×10−10, (E1⋅Ez)0=9.98×10−6, (E1⋅Ez⋅E2)0=2.0×10−8, (Ez⋅E2)0=1.98×10−14; (b) as in Fig 7.
Fig 9.
Switch between dual-E and single-E control in the motif 2 antithetic controller with a random-order ternary-complex mechanism removing E1 and E2 (Fig 5).
(a) Ass (steady state in A) as a function of k6. Red and blue lines indicate the respective set-point values for single-E and dual-E control. Gray solid points show the numerically calculated steady state levels. The outlined red and blue circles (indicated by the vertical arrows) show the k6 values (10.0 and 0.4) used in panels c and d when changes in k2 are applied. (b) Steady state values of v (Eq 13) obtained by the King-Altman method (inner red dots, S1 Text) and numerically calculated velocities (gray dots). (c) and (d) Single-E and dual-E control when k6 values are respectively 10.0 and 0.4, and k2 changes step-wise from 10.0 to 500. Other rate constants: k3=1×105, k4=1.0, k5=0.4, k7=1×106, k8=0.1, k9=1×108, k10=1×103, k11=1×108, k12=1×103, k13=1×108, k14=1×103, k15=1×108, k16=1×103. Initial concentrations, panel c: A0=2.5, E1,0=363.5, E2,0=4.5×104, Ez0=3.04×10−13, (E1⋅Ez)0=4.3×10−13, (E1⋅Ez⋅E2)0=1.0×10−6, (Ez⋅E2)0=2.7×10−11. Initial concentrations, panel d: A0=1.0, E1,0=905.3, E2,0=6.7×10−3, Ez0=4.4×10−12, (E1⋅Ez)0=6.0×10−7, (E1⋅Ez⋅E2)0=4.0×10−7, (Ez⋅E2)0=4.5×10−15. (e) Outlined in red: the active part of the network during single-E control. E2 is continuously increasing (wind-up). (f) In dual-E control the entire network participates in the control of A (outlined in blue).
Fig 10.
Behaviors of single-E control and dual-E control for the schemes in Fig 9e and 9f when going from zero-order to nonzero-order conditions.
In panels (a) and (b), k9=k11=k13=k15=1 × 108 (zero-order condition); in panels (c) and (d), k9=k11=k13=k15=1 × 106 (weak nonzero-order); in panels (e) and (f), k9=k11=k13=k15=1 × 104 (strong nonzero-order). Panels b, d, and f to the right show the time-dependent kinetics of A for a step-wise perturbation in k2 from 10 (phase 1) to 500 (phase 2) applied at t = 500. The k6 values in these calculations were 0.4. Other rate constants as in Fig 9. Initial concentrations for panels (a), (c), and (e), dual-E controller: A0=2.0, E1,0=4.5 × 102, E2,0=2.0 × 10−1, Ez0=4.4 × 10−10, (E1⋅Ez)0=9.7 × 10−7, (E1⋅Ez⋅E2)0=2.0 × 10−8, (Ez⋅E2)0=2.0 × 10−13; single-E controller: A0=2.5, E0=3.6 × 102, Ez0=2.8 × 10−11, (E⋅Ez)0=1.0 × 10−8; steady state concentrations were obtained after 2000 time units. Initial concentrations panels (b) and (d): dual-E controller: A0=1.0, E1,0=9.1 × 102, E2,0=6.7 × 10−1, Ez0=4.4 × 10−10, (E1⋅Ez)0=6.0 × 10−7, (E1⋅Ez⋅E2)0=4.0 × 10−7, (Ez⋅E2)0=7.6 × 10−13; single-E controller: A0=2.5, E0=3.6 × 102, Ez0=2.7 × 10−9, (E⋅Ez)0=1.0 × 10−8. Initial concentrations panels panel (f): dual-E controller: A0=1.0, E1,0=9.1 × 102, E2,0=6.7 × 10−1, Ez0=4.4 × 10−10, (E1⋅Ez)0=6.0 × 10−7, (E1⋅Ez⋅E2)0=4.0 × 10−7, (Ez⋅E2)0=7.6 × 10−13; single-E controller: A0=2.04, E0=4.5 × 102, Ez0=1.8 × 10−7, (E⋅Ez)0=8.1 × 10−7.
Fig 11.
Motif 2 dual-E controller when E1 and E2 are removed enzymatically by compulsory-order ternary-complex mechanisms.
Panel a: E1 binds first to free enzyme Ez. Panel b: E2 binding first to Ez.
Fig 12.
Dual- and single-E control mode of the m2 feedback loop when E1 and E2 are removed by a compulsory-order ternary complex mechanism and when E1 binds first to Ez (Fig 11a).
Panel a, outlined in blue, shows the concentration of A for the mechanism of Fig 11a with a step-wise change of k2 from 10.0 (phase 1) to 500.0 (phase 2). For comparison, outlined in orange, the results of Fig 10b for the random-order ternary complex mechanism working in dual-E mode are shown. Rate constant k6=0.4 for both phases. Other rate constants and initial concentrations are the same as for Fig 10b. Panel b shows the concentration of A for the compulsory-order ternary complex mechanism from panel a, but k6 is changed in phase 2 from 0.4 to 10.0. The controller switches in phase 2 from dual-E mode to single-E mode with the associated change of Aset from 1.0 (Eq 14) to 2.5 (Eq 15). Initial concentrations and rate constants as in panel a.
Fig 13.
Switch between single-E and dual-E control for the m2 controller when E1 and E2 are removed by a compulsory-order ternary-complex mechanism with E2 binding first to Ez (Fig 11b).
Panel a: steady state values of A (Ass) as a function of k6. Gray dots show numerical results. The line outlined in red describes the set-point of A (k7Eztot/k5) at single-E control. The blue line shows the set-point of A (k6/k5) when the system is in dual-E control mode. Panel b: corresponding numerical (gray dots) and steady state values (red small dots, calculated by King-Altman method, S1 Text) of the degradation rate v of the ternary-complex (Eq 13). Rate constants: k1=0.0, k2=100.0, k3=1 × 105, k4=1.0, k5=4.0, k6 varies between 40.0 and 0.05, k7=1 × 107, k8=0.1, k13=k15=1 × 108, k14=k16=1 × 103. Initial concentrations: A0=1.0, E1,0=9.1 × 102, E2,0=6.7 × 10−2, Ez0=6.0 × 10−7, (E1⋅Ez⋅E2)0=4.0 × 10−7, (Ez⋅E2)0=4.4 × 10−11. Eztot=1.0 × 10−6. Steady state values were obtained after 10000 time units.
Fig 14.
Critical slowing down in the transition from single-E to dual-E control in the negative feedback loop of Fig 11b.
The set-point of A during single-E control is 2.5, but 1.0 during dual-E control. Panel a: Time profiles of A and E2 for k6=4.0 (solid lines) and k6=8.0 (dotted lines). T, the transition time, is the time difference from t = 0 until E2 has reached steady state. Panel b: T as a function of k6. When k6→10.0 the steady state of the dual-E control mode vanishes and T→∞. Rate constants (for each data point): k1=0.0, k2=10.0, k3=1 × 105, k4=1.0, k5=4.0, k6 takes the values 1.0, 2.0, …, 9.0, 9.5 and 9.75, k7=1 × 107, k8=0.1, k13=k15=1 × 108, k14=k16=1 × 103. Initial concentrations: A0=2.5, E1,0=3.635 × 102, E2,0=4.38 × 104, Ez0=2.28 × 10−13, (E1⋅Ez⋅E2)0=1.0 × 10−6, (Ez⋅E2)0=2.75 × 10−11.
Fig 15.
Enzymatic ping-pong mechanisms removing E1 and E2 in m2 dual-E controller.
(a) E1 binds first to Ez. (b) E2 binds first to Ez.
Fig 16.
Influence of total enzyme concentration Eztot on the switch between dual-E and single-E control in the m2 controller with ping-pong mechanism of Fig 15a.
(a) Eztot=1 × 10−6, (b) Eztot=1 × 10−5, (c) Eztot=2 × 10−5, (d) Eztot=1 × 10−4. Rate constants: k1=0.0, k2=500.0, k3=1 × 105, k4=1.0, k5=0.4, k6 varies between 40.0 and 0.05, k7=1 × 106, k8=0.1, k9=k11=k13=1 × 108, k10=k12=k14=1 × 103. Initial concentrations: A0=2.0, E1,0=4.5 × 102, E2,0=2.0 × 10−1, Ez0=Eztot, (E1⋅Ez⋅)0=0.0, (Ez*)0=0.0, (Ez*E2)0=0.0. Steady state values were obtained after 4000 time units.
Fig 17.
Change of the switch point between dual-E and single-E control with decreasing values of k9, k11, and k13.
(a) k9=k11=k13=1 × 108; (b) k9=k11=k13=1 × 107; (c) k9=k11=k13=1 × 106; (d) k9=k11=k13=1 × 105. Other rate constants: k1=0.0, k2=500.0, k3=1 × 105, k4=1.0, k5=0.4, k6 takes values between 0.1 and 40.0 (indicated by the gray dots), k7=1 × 106, and k8=0.1. Initial concentrations: A0=2.0, E1,0=4.5 × 102, E2,0=2.0 × 10−1, Ez0=Eztot=2.0 × 10−5, (EzE2)0=0.0, (Ez*)0=0.0, (Ez*E1)0=0.0. Steady state values were obtained after 4000 time units.
Fig 18.
Influence of step-wise k2 for catalyzed m2 controller under nonzero-order conditions.
The mechanism considered in that of Fig 15b. Small colored dots indicate Ass levels for different k2 values when k9=k11=k13=1 × 107 and Eztot=2.0 × 10−5. For comparison, large blue dots show the Ass values under zero-order conditions when k9=k11=k13=1 × 109 and k2=1.0. Other rate constant values and initial concentrations are as in Fig 17.
Fig 19.
Motif 4 dual-E/antithetic controller using an enzymatic random-order ternary-complex mechanism for the removal of E1 and E2.
Fig 20.
Response of the m4 random-order ternary-complex controller (Fig 19) with respect to step-wise changes in k2.
(a) Phase 1: k2=10. At time t = 50 phase 2 starts with the following changes in k2: (1) k2=20, (2) k2=50, (3) k2=100. (b) Phase 1: k2=10. At time t = 50 phase 2 starts with the following changes in k2: (4) k2=500, (5) k2=1 × 103, (6) k2=1 × 104, (7) k2=2 × 104. Other rate constants: k1=0.0, k3=1 × 105, k4=1.0, k5=31.0, k6=1.0, k7=1 × 108, k8=0.1, k9=k11=k13=k15=1 × 108, k10=k12=k14=k16=1 × 103, k17=0.1. Initial concentrations: A0=3.0, E1,0=1.0 × 10−2, E2,0=3.0 × 102, Ez0=3.3 × 10−11, (E1⋅Ez)0=1.4 × 10−15, (E1⋅Ez⋅E2)0=1.0 × 10−8, (EzE2)0=9.9 × 10−7. Total enzyme concentration Eztot=1.0 × 10−6.
Fig 21.
Operational range of m4 controller with upper defendable limit of k2 (=k1+ k3−k4Aset/Aset).
(a) Blue area indicates the range as a function of Aset in which the controller can defend Aset. Black solid curve:
as a function of Aset when k1=0.0, k3=1 × 105, and k4=1.0. Red area:
where Ass is lower than Aset. (b) Computation showing the partial loss of homeostasis when k2 becomes larger than
. Phase 1 (0–50 time units): k2=10; phase 2 (50–250 time units): k2=2 × 104; phase 3 (250–450 time units): k2=5 × 104. Other rate constants and initial conditions as in Fig 20. For further descriptions, see S1 Text.
Fig 22.
Operational range of the controller from Fig 19 as a function of k5 and the ratios (k10/k9)=(k12/k11)=(k14/k13)=(k16/k15).
Aset in the left panels is the theoretical set-point described by Eq 56. Ass (gray dots) are the numerically calculated steady state values of A. Middle panels show the concentrations of E1 and E2 indicated by blue and orange dots, respectively. Panels to the right show the flux j5 (small blue dots) which generates E1 by A-repression (Eq 56). vnum (yellow dots) is the numerically calculated degradation velocity of the ternary-complex. Dark red dots show k6. Turquoise areas indicate the controllers operational range when Eq 56 is satisfied. (a) k9=k11=k13=k15=1 × 108. (b) k9=k11=k13=k15=1 × 105. (c) k9=k11=k13=k15=1 × 104. Remaining rate constants and initial concentrations are as in Fig 20.
Fig 23.
Influence of k2 on the operational range of the m4 controller Fig 19.
See Fig 22 for explanation of symbols. (a) k2 = 50.0, (b) k2 = 500.0, (c) k2 = 5000.0. Other rate constants and initial concentrations are as in Fig 20.
Fig 24.
Reaction schemes when E1 and E2 in a m4-type of control structure (Fig 3) are removed by enzyme Ez with two compulsory-order ternary-complex mechanism.
In (a) E1 binds first to Ez, while in (b) E2 binds first.
Fig 25.
Comparison between the three m4-controllers when E1 and E2 are removed by enzymatic ternary-complex mechanisms (Figs 19 and 24) upon step-wise changes at time t = 50 from k2=10 to (a) k2=500, (b) k2=1 × 103, (c) k2=2 × 104, (d) k2=5 × 104.
Color coding: Thick blue line, compulsory-order mechanism with E2 binding first to Ez; overlaid red line, compulsory-order mechanism with E1 binding first to Ez; top overlaid yellow line, random-order mechanism. Rate constants and initial concentrations as for the random-order ternary-complex mechanism (Fig 20).
Fig 26.
Concentration profiles of E1, E2, and enzyme species with respect to the controllers’ breakdown shown in Fig 25d. Column a: Random-order mechanism (Fig 19).
Column b: Compulsory-order mechanism (Fig 24a). Column c: Compulsory-order mechanism (Fig 24b). Rate constants and initial concentrations as in Fig 25.
Fig 27.
Reaction schemes when E1 and E2 in a m4-type of control structure (Fig 3) are removed by enzyme Ez following two ping-pong mechanisms.
In (a) E1 binds first to the free enzyme Ez, while in (b) E2 binds first.
Fig 28.
Enzyme species profiles of the m4 ternary-complex (Figs 19 and 24) and ping-pong mechanisms (Fig 27) when k2=10 in phase 1, and k2=2 × 104 in phase 2.
(a) random-order ternary-complex mechanism, (b) compulsory-order ternary-complex mechanism with E1 binding first to Ez, (c) compulsory-order ternary-complex mechanism with E2 binding first to Ez, (d) ping-pong mechanism with E1 binding first to Ez, (e) ping-pong mechanism with E2 binding first to Ez. Rate constants (if applicable) are as in Fig 20. Initial concentrations: (a) A0=3.0, E1,0=1.0 × 10−2, E2,0=3.0 × 102, Ez0=3.3 × 10−11, (E1⋅Ez)0=1.4 × 10−15, (E1⋅Ez⋅E2)0=1.0 × 10−8, (EzE2)0=9.9 × 10−7. (b) A0=3.0, E1,0=1.0 × 10−2, E2,0=3.0 × 102, Ez0=9.9 × 10−7, (E1⋅Ez)0=3.3 × 10−11, (E1⋅Ez⋅E2)0=1.0 × 10−8. (c) A0=3.0, E1,0=1.0 × 10−2, E2,0=3.0 × 102, Ez0=3.3 × 10−11, (EzE2)0=9.9 × 10−7, (E1⋅Ez⋅E2)0=1.0 × 10−8. (d) A0=3.0, E1,0=1.0 × 10−2, E2,0=3.0 × 102, Ez0=9.8 × 10−7, (E1⋅Ez)0=1.0 × 10−8, =3.3 × 10−11, (Ez*E2)0=1.0 × 10−8. (e) A0=3.0, E1,0=1.0 × 10−2, E2,0=3.0 × 102, Ez0=3.3 × 10−11, (Ez*E1)0=1.0 × 10−8,
× 10−7, (EzE2)0=1.0 × 10−8.
Fig 29.
Example of m5 feedback loop where E1 and E2 are removed by a random-order ternary-complex mechanism which works under dual-E control.
(a) Reaction scheme. (b) Step-wise change of k1 from 500.0 to 1000.0 at time t = 50. (c) In dual-E mode the set-point is Aset=k6/k5 (= 1.0) which is defended. The panel shows the response of A with respect to the step-wise change of k1 in panel (a). (d) Change of E1 and E2 in response to the step-wise change of k1 in panel (a). Rate constants: k1=500.0 (phase 1), k1=1000.0 (phase 2), k2=1.0, k3=0.0, k4=1.0, k5=40.0, k6=40.0 k7=1 × 108, k9=k11=k13=k15=1 × 109, k10=k12=k14=k16=1 × 103. Initial concentrations: A0=1.0, E1,0=499.0, E2,0=6.67 × 10−2, Ez0=8.02 × 10−11, (E1⋅Ez)0=5.99 × 10−7, (E1⋅Ez⋅E2)0=4.0 × 10−7, (EzE2)0=1.15 × 10−14.
Fig 30.
Reaction scheme of the catalyzed single-E m5-type of controller.
Fig 31.
Example of the m5 feedback loop with E1 and E2 being removed by a random-order ternary-complex mechanism working in single-E control mode.
(a) Scheme outlined in red shows part of the network participating in the control of A. (b) Step-wise change of k1 from 500.0 to 1000.0 at time t = 50.0. (c) Homeostatic response of A, i.e. the controller defends its set-point (=2.5) defined by Eq 99. (d) Change of E1 and wind-up of E2. Rate constants as in Fig 29, except that k6=200. Initial concentrations: A0=2.5, E1,0=199.2, E2,0=2.0 × 103, Ez0=4.54 × 10−11, (E1⋅Ez)0=4.51 × 10−12, (E1⋅Ez⋅E2)0=9.995 × 10−7, (EzE2)0=4.56 × 10−10.
Fig 32.
Switching between single-E control (Fig 31) and dual-E control (Fig 29) as a function of k6 for different values of k9, k11, k13, and k15.
Panel (a): high value (1 × 109) of k9, k11, k13, and k15. The dual-E controller shows its maximum operational range. In this case the switch occurs when k6 > k7Eztot. Panels (b)-(d): for the lower values of k9, k11, k13, and k15 (indicated inside the figure) the ternary-complex concentration (E1⋅Ez⋅E2) is lower than Eztot and the switch occurs at lower k6 values, which leads to a decreased operational range of the dual-E controller. Due to the lower (E1⋅Ez⋅E2) concentration the single-E control mode (which occurs analogous to the red-outlined part in Fig 9e) shows an offset below k7Eztot/k5. Note however, that Ass will depend on the perturbation k1 and move towards Aset with increasing k1, thereby reducing the single-E controller’s offset. Other rate constants: k1=500.0, k2=1.0, k3=0.0, k4=1.0, k5=40.0, k7=1 × 108, k10=k12=k14=k16=1 × 103. Initial concentrations: A0=2.0, E1,0=5.49 × 10−2, E2,0=5.21 × 103, Ez0=7.4 × 10−14, (E1⋅Ez)0=9.09 × 10−8, (E1⋅Ez⋅E2)0=9.09 × 10−7, (EzE2)0=1.66 × 10−10.
Fig 33.
Switching between dual-E and single-E control in the random-order ternary-complex m5 controller at different Eztot concentrations.
Rate constants are as in Fig 32a with Eztot values as indicated in the four panels. With increasing Eztot values (from panel (a) to panel (d)) the operational range of the dual-E controller increases together with increasing set-point values of the single-E controller (Eq 99). Initial concentrations, panel (a): A0=2.0, E1,0=5.49 × 10−2, E2,0=5.21 × 103, Ez0=7.4 × 10−14, (E1⋅Ez)0=9.09 × 10−8, (E1⋅Ez⋅E2)0=9.09 × 10−7, (EzE2)0=1.66 × 10−10. Initial concentrations panel (b): A0=2.5, E1,0=5.49, E2,0=5.21 × 101, Ez0=1.18 × 10−12, (E1⋅Ez)0=9.96 × 10−6, (E1⋅Ez⋅E2)0=4.5 × 10−8, (EzE2)0=1.19 × 10−17. Initial concentrations panel (c): A0=2.5, E1,0=5.49, E2,0=5.21 × 101, Ez0=1.21 × 10−12, (E1⋅Ez)0=1.995 × 10−5, (E1⋅Ez⋅E2)0=4.5 × 10−8, (EzE2)0=1.21 × 10−17. Initial concentrations panel (d): A0=97.63, E1,0=4.12, E2,0=9.87 × 103, Ez0=3.95 × 10−10, (E1⋅Ez)0=1.66 × 10−13, (E1⋅Ez⋅E2)0=3.905 × 10−5, (EzE2)0=9.47 × 10−7.
Fig 34.
Metastable single-E controller and critical slowing down in the autonomous transition from single-E to dual-E control mode.
(a) Step-wise change of k1 at t = 100.0 from 500.0 (phase 1) to 1000.0 (phase 2). (b) Metastable single-E control mode. The single-E controller defends its set-point (=2.5), but transition to dual-E control mode (indicated by arrow) occurs at approximately 1600 time units when E2 reaches its steady state. (c) The metastable single-E control mode is operative as long as E2 is above its steady state value. The transition from single-E to dual-E control mode occurs when E2 has reached its steady state (indicated by arrow). Rate constants: k1=500.0 (phase 1), k1=1000.0 (phase 2), k2=1.0, k3=0.0, k4=1.0, k5=40.0, k6=40.0 k7=1 × 108, k9=k11=k13=k15=1 × 109, k10=k12=k14=k16=1 × 103. Initial concentrations: A0=2.5, E1,0=199.1, E2,0=9.52 × 104, Ez0=1.08 × 10−12, (E1⋅Ez)0=2.37 × 10−15, (E1⋅Ez⋅E2)0=9.995 × 10−7, (EzE2)0=5.01 × 10−10. (d) Transition time T as a function of k6. k1=500.0; all other rate constants and initial concentrations as for (a)-(c).
Fig 35.
Scheme of the m5 controller when E1 and E2 are removed by a compulsory ternary-complex mechanism with E1 binding first to the free enzyme Ez.
Fig 36.
Switch from dual-E control to single-E in the compulsory ternary-complex mechanism of motif 5 when E1 binds first to free Ez (Fig 35).
(a) Perturbation k1 as a function of time. (b) Change of the controlled variable A’s concentration as a function of time. Phase 1: dual-E control; phases 2 and 3: single-E control. (c) Concentration of E1 and E2 as a function of time. (d) Concentration of the enzymatic species Ez, E1⋅Ez, and E1⋅Ez⋅E2 as a function of time. Rate constants: k1=1000.0 (phases 1 and 2), k1=2000.0 (phase 3), k2=1.0, k3=0.0, k4=1.0, k5=40.0, k6=40.0 (phase 1), k6=200.0 (phases 2 and 3) k7=1 × 108, k9=k11=1 × 109, k10=k12=1 × 103. Initial concentrations: A0=1.0, E1,0=993.4, E2,0=6.67 × 10−2, Ez0=4.02 × 10−11, (E1⋅Ez)0=5.999 × 10−7, and (E1⋅Ez⋅E2)0=4.00 × 10−7.
Fig 37.
Switch between dual-E and single-E control in the m5 compulsory-order ternary-complex mechanisms (E1 binding first to Ez) as a function of k6 and total enzyme concentration Eztot.
(a) Eztot=1 × 10−6. (b) Eztot=1 × 10−5; (c) Eztot=2 × 10−5; (d) Eztot=4 × 10−5. Set-points for dual-E and single-E control are indicated in blue and red, respectively. Numerical values are shown as gray filled dots. Rate constants: k1=1000.0, k2=1.0, k3=0.0, k4=1.0, k5=40.0, k6 variable, k7=1 × 108, k9=k11=1 × 109, k10=k12=1 × 103. Initial concentrations: A0=1.0, E1,0=993.4, E2,0=6.67 × 10−2 Panel (a): Ez0=1 × 10−6, (E1⋅Ez)0=0, and (E1⋅Ez⋅E2)0=0. Panel (b): Ez0=1 × 10−5, (E1⋅Ez)0=0, and (E1⋅Ez⋅E2)0=0. Panel (c): Ez0=2 × 10−5, (E1⋅Ez)0=0, and (E1⋅Ez⋅E2)0=0. Panel (d): Ez0=4 × 10−5, (E1⋅Ez)0=0, and (E1⋅Ez⋅E2)0=0. Ass values were taken after a simulation time of 20000 time units.
Fig 38.
Switch between dual-E and single-E control in the m5 compulsory-order ternary-complex mechanisms (E1 binding first to Ez) as a function of k6, k9, and k11.
The total enzyme concentration is 1 × 10−6 and constant. (a) k9=k11=1 × 109. Aset of the single-E controller is 2.5=k7⋅Eztot/k5. (b) k9=k11=1 × 108. Also in this case Aset of the single-E controller is still close to 2.5. (c) k9=k11=1 × 106. v () is no longer zero-order but is described by Eq 112, and Ass of the single-E controller is described by Eq 113 with E1=4.82 × 102 and
=2.07. (d) k9=k11=1 × 105. At single-E control conditions we have E1=4.82 × 102 and
=1.16 (Eq 113). Other rate constant values and initial concentrations as for Fig 37a.
Fig 39.
Under single-E control an increased perturbation k1 moves Ass in the m5 compulsory-order ternary-complex mechanisms (E1 binding first to Ez) towards Aset=k7⋅Eztot/k5.
(a) Phase 1: the system is that from Fig 38d with k6=200 and k1=1000.0. In phases 2 and 3 k1 is stepwise increased to respectively 2000.0 and 3000.0. In phases 2 and 3 A is moved to Aset=k7⋅Eztot/k5=2.5. Rate constant values as in Fig 38d. (b) Corresponding changes in E1 and E2. Note the wind-up of E2 and that only E1 is the controller species. Initial concentrations: A0=1.153, E1,0=866.2, E2,0=7.728 × 106, Ez0=4.027 × 10−11, (E1⋅Ez)0=5.999 × 10−7, and (E1⋅Ez⋅E2)0=4.000 × 10−7.
Fig 40.
Scheme of the m5 controller when E1 and E2 are removed by a compulsory ternary-complex mechanism with E2 binding first to Ez.
Fig 41.
Switch from dual-E to single-E control by increase of k6 in the compulsory ternary-complex mechanism of motif 5 when E2 binds first to free Ez (Fig 40).
An increase of k1 in phase 3 shows that the set-point of A under single-E control is defended. (a) Perturbation k1 as a function of time. (b) Change of the controlled variable A’s concentration as a function of time. Phase 1: dual-E control; phases 2 and 3: single-E control. (c) Concentration of E1 and E2 as a function of time. (d) Concentration of the enzymatic species Ez, Ez⋅E2, and E1⋅Ez⋅E2 as a function of time. Rate constants: k1=1000.0 (phases 1 and 2), k1=2000.0 (phase 3), k2=1.0, k3=0.0, k4=1.0, k5=40.0, k6=40.0 (phase 1), k6=200.0 (phases 2 and 3) k7=1 × 108, k13=k15=1 × 109, k14=k16=1 × 103. Initial concentrations: A0=1.0, E1,0=9.99 × 102, E2,0=6.67 × 10−2, Ez0=5.999 × 10−7, (Ez⋅E2)0=4.004 × 10−11, and (E1⋅Ez⋅E2)0=4.00 × 10−7.
Fig 42.
Switch between dual-E and single-E control in the m5 compulsory-order ternary-complex mechanisms (E2 binding first to Ez, Fig 40) as a function of k6, k13, and k15.
The total enzyme concentration is 1 × 10−6 and constant. (a) k13=k15=1 × 109. Aset of the single-E controller is 2.5 (=k7⋅Eztot/k5, analogous to Eq 99. (b) k13=k15=1 × 108. Also in this case Aset of the single-E controller is still close to 2.5. (c) k9=k11=1 × 106. v () is no longer zero-order with respect to E1 and E2, but is described by Eq 123, and Ass of the single-E controller is described by Eq 124 with E1,ss=4.82 × 102 and
=2.07. (d) k9=k11=1 × 105. At single-E control conditions we have E1,ss=8.63 × 102 and
=1.16 (Eq 124). Other rate parameters as for Fig 37a.
Fig 43.
Under single-E control an increased perturbation k1 moves Ass in the m5 compulsory-order ternary-complex mechanisms (E1 binding first to Ez) towards Aset=k7⋅Eztot/k5.
(a) Phase 1: the system is that from Fig 42d with k6=200 and k1=1000.0. In phases 2 and 3 k1 is stepwise increased to respectively 1 × 104 and 5 × 105. In phase 3 A is moved close to Aset=k7⋅Eztot/k5=2.5. Rate constant values as in Fig 42d. (b) Corresponding changes in E1 and E2. Note the wind-up of E2 and that only E1 is the active controller species. Initial concentrations: A0=1.156, E1,0=866.4, E2,0=1.543 × 105, Ez0=2.995 × 10−9, Ez⋅(E2)0=5.347 × 10−7, and (E1⋅Ez⋅E2)0=4.622 × 10−7.
Fig 44.
Scheme of the m5 controller when E1 and E2 are removed by a ping-pong mechanism with E1 binding first to the free enzyme Ez.
Fig 45.
Ass (=Aset) as a function of k6 when (a) k17=100.0 and (b) k17=0.0. Gray solid points show the numerically calculated values of Ass, while red and blue curves show the values of k7(Eztot)/k5 and k6/k5, respectively. Other rate constant values: k1=800.0, k2=1.0, k3=0.0, k4=1.0, k5=40.0, k7=1 × 108, k9 = k11 = k13=1 × 108, and k10 = k12 = k14=1 × 103. Initial concentrations: A0=1.0, E1,0=9.9 × 101, E2,0=5.04 × 10−1, Ez0=6.03 × 10−9, (E1⋅Ez)0=4.97 × 10−7, (Ez*⋅E2)0=1.0 × 10−7, and =3.97 × 10−7. Simulation time: 5000 time units, step-length: 0.01 time units.
Fig 46.
Demonstration of homeostatic behavior in A (Fig 45) when k6=10.0 and k17=100.0 (panels a and b) or k17=0.0 (panel c and d). Step-wise perturbations are applied with values k1=100 (phase 1), k1=400 (phase 2), and k1=800 (phase 3). Other rate constants as in Fig 45. Initial concentrations, (a) and (b): A0=0.1189, E1,0=8.40 × 102, E2,0=5.25 × 10−2, Ez0=5.66 × 10−11, (E1⋅Ez)0=4.75 × 10−8, (Ez*⋅E2)0=4.75 × 10−8, and =9.05 × 10−7. Initial concentrations, (c) and (d): A0=0.25, E1,0=3.99 × 102, E2,0=1.25 × 10−1, Ez0=2.51 × 10−10, (E1⋅Ez)0=1.00 × 10−7, (Ez*⋅E2)0=1.00 × 10−7, and
=9.76 × 10−7.
Fig 47.
Demonstration of the homeostatic behavior of Ass in Fig 45 when k6=100.0 and k17=100.0 (panels a and b) or k17=0.0 (panel c and d). Step-wise perturbations are applied with values k1=100 (phase 1), k1=400 (phase 2), and k1=800 (phase 3). Other rate constants as in Fig 45. The linear increase of E2 is seen as a concave line due to the logarithmic scale of the E2-axis. Initial concentrations, (a) and (b): A0=0.7309, E1,0=1.36 × 102, E2,0=7.08 × 10−1, Ez0=2.15 × 10−9, (E1⋅Ez)0=2.95 × 10−7, (Ez*⋅E2)0=2.93 × 10−7, and =4.13 × 10−7. Initial concentrations, (c) and (d): A0=1.24, E1,0=7.96 × 101, E2,0=3.50 × 10−2, Ez0=6.23 × 10−9, (E1⋅Ez)0=4.96 × 10−7, (Ez*⋅E2)0=4.96 × 10−7, and
=1.41 × 10−9.
Fig 48.
Scheme of the m5 controller when E1 and E2 are removed by a ping-pong mechanism with E2 binding first to the free enzyme Ez.
Fig 49.
Demonstration of the homeostatic behavior of Ass (scheme Fig 48) when k6=100.0 and k17=100.0 (panels a and b) or k17=0.0 (panel c and d). The behavior is analogous to that shown in Fig 47. Step-wise perturbations are applied with values k1=100 (phase 1), k1=400 (phase 2), and k1=800 (phase 3). Other rate constants as in Fig 45. The linear increase of E2 is seen as a concave line due to the logarithmic scale of the E2-axis. vmax=50 (Eq 151). Initial concentrations, (a) and (b): A0=0.7309, E1,0=1.36 × 102, E2,0=7.08 × 10−1, Ez0=2.15 × 10−9, (E2⋅Ez)0=2.924 × 10−7, (Ez*⋅E1)0=2.924 × 10−7, and =4.13 × 10−7. Initial concentrations, (c) and (d): A0=1.242, E1,0=7.95 × 101, E2,0=2.52 × 104, Ez0=1.97 × 10−11, (E2⋅Ez)0=4.97 × 10−7, (Ez*⋅E1)0=4.97 × 10−7, and
=6.25 × 10−9.
Fig 50.
Ass (=Aset) as a function of k6 when (a) k17=100.0 and (b) k17 = 0.0. Gray solid points show the numerically calculated values of Ass, while red and blue curves show the values of k7(Eztot)/k5 and k6/k5, respectively. Other rate constant values: k1=800.0, k2=1.0, k3=0.0, k4=1.0, k5=40.0, k7=1 × 108, k9=k11=k13=1 × 108, and k10=k12=k14=1 × 103. Initial concentrations: A0=1.0, E1,0=9.9 × 101, E2,0=5.04 × 10−1, Ez0=6.03 × 10−9, (E2⋅Ez)0=4.97 × 10−7, (Ez*⋅E1)0=1.0 × 10−7, and =3.97 × 10−7. Simulation time: 3000 time units, step-length: 0.01 time units.
Fig 51.
Reaction scheme of the m7-type of controller when E1 and E2 are removed by enzyme Ez with a random-order ternary-complex mechanism.
Fig 52.
Homeostatic behavior towards step-wise perturbations of k1 in the scheme of Fig 51. (a) stepwise changes of k1, (b) homeostatic control of A, (c) Variation of controller variables E1 and E − 2, (d) changes in the enzymatic species Ez, E1⋅Ez, Ez⋅E2, and E1⋅Ez⋅E2. Rate constants: k1=100.0 in phase 1, 400.0 in phase 2, and 800.0 in phase 3. k2=k3=0.0, k4=1.01 × 101, k5=31.0, k6=1.0, k7=1 × 108, k8 not used, k9=k11=k13=k15=1 × 108, k10=k12=k14=k16=1 × 103, k17=0.1. Initial concentrations: A0=3.000, E1,0=1.01 × 10−2, E2,0=3.333, Ez0=2.994 × 10−9, (E1⋅Ez)0=9.102 × 10−12, (E1⋅Ez⋅E2)0=1.0 × 10−8, (EzE2)0=9.871 × 10−7, Eztot=1.0 × 10−6.
Fig 53.
Loss of A-homeostasis in the m7 controller with a random-order ternary-complex mechanism (Fig 51) when k6 > k5.
(a) Ass as a function of k6. (b) vnum (gray dots) and vsteady state (King-Altman) (red line and small red dots) as a function of k6. (c) Decrease of A as a function of time when k6=40.0. (d) Steady state in E1 and wind-up in E2 when k6=40.0. (e) Time profiles of the different enzyme species. Rate constants: k1=100.0, k2=1.0, k3=0.0, k4=1 × 101, k5=31.0, k7=1 × 108, k8=0.1, k9=k11=k13=k15=1 × 108, k10=k12=k14=k16=1 × 103, k17=0.1. Initial concentrations: A0=2.5, E1,0=5.5, E2,0=52.1, Ez0=1 × 10−6, (E1⋅Ez)0=0.0, (E1⋅Ez⋅E2)0=0.0, (EzE2)0=0.0, Eztot=1.0 × 10−6. Steady state values are determined after a simulation time of 48000 time units.
Fig 54.
Decrease in the operational range of the enzymatic controller of Fig 51 (in dual-E control mode) as a function to decreased values of the forward enzymatic rate constants k9, k11, k13, and k15.
The k6 range for which homeostasis is observed is outlined as turquoise areas. (a) k9=k11=k13=k15=1 × 107. (b) k9=k11=k13=k15=1 × 106. (c) k9=k11=k13=k15=1 × 104. The reverse rate constants k10, k12, k13, k15 are in (a)-(c) kept constant at 1 × 103. (d) k9=k11=k13=k15=1 × 103, while k10=k12=k14=k16=0. Despite the irreversibility of the system the k9, k11, k13, and k15 values are too low to enable homeostasis. Other rate constants (a)-(d): k1=100, k2=k3=0, k6=5.0, k7=1 × 108, k17=0.1. Eztot=1 × 10−6. Initial concentrations (a)-(d): A0=3.000, E1,0=1.01 × 10−2, E2,0=3.333, Ez0=2.994 × 10−9, (E1⋅Ez)0=9.102 × 10−12, (E1⋅Ez⋅E2)0=1.0 × 10−8, (EzE2)0=9.871 × 10−7, Eztot=1.0 × 10−6.
Fig 55.
Influence of total enzyme concentration Eztot on the performance of the system in Fig 54c.
(a) Eztot=1.0 × 10−6; (b) Eztot=1.0 × 10−5; (c) Eztot=1.0 × 10−4; (d) Eztot=1.0 × 10−3. Rate constant values as for Fig 54c. Initial concentrations: (a) A0=3.000, E1,0=1.01 × 10−2, E2,0=3.333, Ez0=1 × 10−6, (E1⋅Ez)0=0.0, (E1⋅Ez⋅E2)0=0.0, (EzE2)0=0.0. (b) A0=3.000, E1,0=1.01 × 10−2, E2,0=3.333, Ez0=1 × 10−5, (E1⋅Ez)0=0.0, (E1⋅Ez⋅E2)0=0.0, (EzE2)0=0.0. (c) A0=3.000, E1,0=1.01 × 10−2, E2,0=3.333, Ez0=1 × 10−4, (E1⋅Ez)0=0.0, (E1⋅Ez⋅E2)0=0.0, (EzE2)0=0.0. (d) A0=3.000, E1,0=1.01 × 10−2, E2,0=3.333, Ez0=1 × 10−3, (E1⋅Ez)0=0.0, (E1⋅Ez⋅E2)0=0.0, (EzE2)0=0.0.
Fig 56.
The two compulsory-order ternary-complex mechanisms with feedback motif m7.
In (a) E1 binds first to the free enzyme Ez, while in (b) E2 binds first.
Fig 57.
Influence of k9, k11, and Eztot on the operational range of the controller from Fig 56a.
(a) Optimum controller behavior for large k9 and k11 values (both 1 × 107) at Eztot=1 × 10−6. (b) Reducing k9 and k11 to 1 × 104 leads to a complete loss of the controller’s homeostatic behavior. Although the steady state values of A (gray circles) are independent and constant for k5 > k6, they depend on the perturbation k1, which will be illustrated below for scheme Fig 56b. (c) Increasing the total enzyme concentration to 1 × 10−4 partially improves the controller’s performance. (d) Increasing the total Ez concentration to 1 × 10−3 restores the homeostatic behavior as the increased k9 and k11 values in (a) at low Eztot. Other rate constants (a)-(d): k1=100, k2=k3=0, k4=10.0, k6=5.0, k7=1 × 108, k9=k11=k13=k15=1 × 108, k10=k12=k14=k16=1 × 103, k17=0.1. Initial concentrations (a)-(d): A0=0.08, E1,0=5.27 × 10−2, E2,0=125.0; (a)-(b) Ez0=1.0 × 10−6, (E1⋅Ez)0=(E1⋅Ez⋅E2)0=0.0; (c) Ez0=1.0 × 10−4, (E1⋅Ez)0=(E1⋅Ez⋅E2)0=0.0; (d) Ez0=1.0 × 10−3, (E1⋅Ez)0=(E1⋅Ez⋅E2)0=0.0. The steady state values of A were determined after 6000 time units.
Fig 58.
Influence of k1 on the operational range of the m7 ternary-complex controllers.
The results using the scheme of Fig 56b are shown. (a) k1=1 × 102. (b) k1=1 × 103. (c) k1=1 × 104. (d) k1=1 × 105. Other rate constants (a)-(d): k2=k3=0, k4=10.0, k6=5.0, k7=1 × 108, k13=k15=1 × 104, k14=k16= k17=1 × 103. Initial concentrations (a)-(d): A0=1.88, E1,0=5.39 × 102, E2,0=5.315, Ez0=1.0 × 10−6, (E1⋅Ez)0=(E1⋅Ez⋅E2)0=0.0. Due to a slow response (large response time) of the controller at higher k1, steady state values of A were determined after 1 × 106 time units.
Fig 59.
Reaction schemes of the two m7-type of controllers when E1 and E2 (Fig 3) are removed by enzyme Ez using ping-pong mechanisms with (a) E1 binding first, or (b) when E2 binds first.
Fig 60.
Influence of k6, k9, k11, k13 and Eztot on the homeostatic behavior of the m7 ping-pong controller when E1 binds first to free enzyme Ez.
Numerical A values calculated after 104 time units are compared with corresponding analytical expressions of Aset as a function of k5. (a) k6=5.0, k9=k11=k13=1 × 107, and Eztot=1 × 10−6. (b) k6=5.0, k9=k11=k13=1 × 107, and Eztot=1 × 10−3. (c) k6=5.0, k9=k11=k13=1 × 108, and Eztot=1 × 10−6. (d) k6=1.0, k9=k11=k13=1 × 108, and Eztot=1 × 10−6. Other rate constants (a)-(d): k1=100.0, k2=k3=0, k4=10.0, k7=1 × 108, k12=k14=1 × 103. Initial concentrations (a), (c), and (d): A0=3.0, E1,0=1.0 × 10−2, E2,0=3.0 × 102, Ez0=1.0 × 10−6, (E1⋅Ez)0==(Ez*⋅E2)0=0.0. Initial concentrations (b): A0=3.0, E1,0=1.0 × 10−2, E2,0=3.0 × 102, Ez0=1.0 × 10−3, (E1⋅Ez)0=
=(Ez*⋅E2)0=0.0.
Fig 61.
Influence of k1 on the homeostatic behavior of the m7 ping-pong controller when E1 binds first to free enzyme Ez.
Since the response of the controller at higher k1 values becomes significantly slower the numerical A values are calculated after 106 time units and compared with positive Aset (blue lines) as a function of k5. (a) k1=100.0, k6=5.0. The controller looses homeostatic control in the k5 range from 460–1000. (b) Increasing k1 from 100.0 to 10000.0 moves Ass to Aset for the higher k5 values, but not for the lower k5 values. (c) A decrease of k6 from 5.0 to 1.0 while k1 is kept at 100.0 gives a general improvement of the homeostatic performance of the ping-pong controller, except for the higher end k5 range between 900–1000, where Ass becomes constant (indicated by the red circle). (d) Low k6 (1.0) and higher k1 (1000.0) shows Ass values that match Aset. Other rate constants (a)-(d): k2=k3=0, k4=10.0, k7=1 × 108, k9=k11=k13=1 × 107, k12=k14=1 × 103, k17=0.1. Initial concentrations (a)-(d): A0=3.0, E1,0=1.0 × 10−2, E2,0=3.0 × 102, Ez0=1.0 × 10−6, (E1⋅Ez)0==(Ez*⋅E2)0=0.0.
Fig 62.
The m7 ping-pong mechanisms (Fig 59) show identical homeostatic responses for step-wise changes in k1.
Phase 1: k1=100.0, phase 2: k1=400.0, phase 3: k1=800.0. (a) A as a function of the step-wise changes in k1 for both controllers. (b) Concentration profiles of E1 and E2 for both controllers. (c) Concentration profiles of the enzyme species for the mechanism in Fig 59a. (d) Concentration profiles of the enzyme species for the mechanism in Fig 59b. Other rate constants: k2=k3=0, k4=10.0, k5=31.0, k6=1.0, k7=1 × 108, k9=k11=k13=1 × 108, k12=k14=1 × 103, k17=0.1. Initial concentrations for the controller of Fig 59a: A0=3.0, E1,0=1.0 × 10−2, E2,0=3.33, Ez0=9.77 × 10−7, (E1⋅Ez)0=1.0 × 10−8, =3 × 10−9, (Ez*⋅E2)0=1.0 × 10−8. Initial concentrations for the controller of Fig 59b: A0=3.0, E1,0=1.0 × 10−2, E2,0=3.33, Ez0=3 × 10−9, (Ez⋅E2)0=1.0 × 10−8,
=9.77 × 10−7, (Ez*⋅E1)0=1.0 × 10−8.
Fig 63.
Concentration profiles of reaction species of the two m7 ping-pong mechanisms (Fig 59) as a function of k5.
(a) Left ordinate: Numerical steady state values of A (gray dots) in comparison with the theoretical set-point Aset (Eq 160, blue line). Ordinate to the right: k6 (red dots), steady state values of j5=k5k17/(k17 + A) (blue dots), and numerically calculated reaction rate vnum=dP/dt (orange dots). (b) Steady state values of E1 (orange dots) and E2 (blue dots). (c) Steady state profiles of Ez*E2 (Fig 59a) or Ez*E1 (Fig 59b). (d) Steady state profiles of E1Ez (Fig 59a) or EzE2 (Fig 59b). (e) Steady state profile of Ez when E1 binds first to it (Fig 59a) or profile for Ez* when E2 binds first to Ez (Fig 59b). (f) Steady state profile of Ez when E2 binds first to Ez (Fig 59b) or profile for Ez* when E1 binds first to Ez (Fig 59a). Rate constants: k1=100.0, other rate constants as in Fig 62. Initial concentrations (a)-(f): A0=3.0, E1,0=1.0 × 10−2, E2,0=3.0 × 102, Ez0=1.0 × 10−6, (E1⋅Ez)0==(Ez*⋅E2)0=0.0.
Fig 64.
Central transcriptional-translational negative feedback loop of the Neurospora circadian clock.
In the presence of FRQ the transcription factor White Collar Complex (WCC) is phosphorylated, which leads to its inhibition by FRQ and thereby suppressing FRQ synthesis. FRQ on its side is phosphorylated, which moves the inhibitory FRQ form out of the loop and leads to its eventually to its degradation. The dual-E controller suggests that frq-mRNA is under homeostatic control with respect to variable frq-mRNA degradation.
Fig 65.
M2 dual-E control loop of Brassinosteroid homeostasis.
Brassinosteroid genes are transcribed where TF indicate a set of transcription factors. When BRs bind to their receptors unphosphorylated BZ1 is produced which binds to the transcription factor and thereby inhibits Brassinosteroid transcription. The GSK3-like kinase BIN2 phosphorylates BZR1 and removes it from the negative feedback loop. Phosphorylated BZR1 is finally degraded by the proteasome.
Fig 66.
Suggested mechanism for the inflow control regulation of iron in mammalian cells.
IRP2 activtes and stabilizes reactions promoting the inflow of iron into the cell. Iron activates the enzyme FBXL5 (FBXL5i is an inactive form) which enables the binding of IRP2 and UB leading to ubiquitinated IRP2. In this way IRP2 is moved out of the negative feedback loop.