https://orcid.org/0000-0002-7566-2506
Figures
Abstract
Homeostasis plays a central role in our understanding how cells and organisms are able to oppose environmental disturbances and thereby maintain an internal stability. During the last two decades there has been an increased interest in using control engineering methods, especially integral control, in the analysis and design of homeostatic networks. Several reaction kinetic mechanisms have been discovered which lead to integral control. In two of them integral control is achieved, either by the removal of a single control species E by zero-order kinetics (“single-E controllers”), or by the removal of two control species by second-order kinetics (“antithetic or dual-E control”). In this paper we show results when the control species E1 and E2 in antithetic control are removed enzymatically by ping-pong or ternary-complex mechanisms. Our findings show that enzyme-catalyzed dual-E controllers can work in two control modes. In one mode, one of the two control species is active, but requires zero-order kinetics in its removal. In the other mode, both controller species are active and both are removed enzymatically. Conditions for the two control modes are put forward and biochemical examples with the structure of enzyme-catalyzed dual-E controllers are discussed.
Citation: Waheed Q, Zhou H, Ruoff P (2022) Kinetics and mechanisms of catalyzed dual-E (antithetic) controllers. PLoS ONE 17(8): e0262371. https://doi.org/10.1371/journal.pone.0262371
Editor: Rafael Vazquez-Duhalt, Universidad Nacional Autonoma de Mexico Centro de Nanociencias y Nanotecnologia, MEXICO
Received: December 16, 2021; Accepted: August 2, 2022; Published: August 18, 2022
Copyright: © 2022 Waheed et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Data Availability: All relevant data are within the paper and its Supporting information files.
Funding: The authors received no specific funding for this work.
Competing interests: The authors have declared that no competing interests exist.
Introduction
During the last twenty years there has been an increasing interest in the design of molecular models that can exhibit integral control and show robust homeostasis/perfect adaptation. [1–11]. Integral control, which is part of many industrial regulation processes works in the following way (Fig 1): the controlled variable A, outlined in blue, is compared with the controller’s set-point Aset (shown in red). The difference (or error) between Aset and the actual value of A, ϵ = Aset−A, is calculated and integrated in time. The time integral of ϵ, described as the variable E, is then used to correct for perturbations acting on A. It can be shown that for step-wise perturbations an integral feedback will move A precisely to Aset [3].
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].
Mustafa Khammash’s group recently suggested an interesting alternative approach, termed antithetic control, where instead of one controller molecule E there are two (E1 and E2) [7, 8, 10, 11]. In the single-E control case the condition of integral control is given by
(1)
where K is a constant.
In the antithetic/dual-E case integral control is achieved by
(2)
with K′ being a constant. Fig 2 shows, as an example, how integral control in single- and dual-E controllers can be achieved in a negative feedback structure termed motif 5. Motif 5, an outflow controller, is one of eight basic negative feedback structures, which divide equally into two sets of inflow and outflow controllers [6]. Briefly, in inflow controllers the compensatory flux opposes an uncontrolled removal of the controlled variable (here A), while in outflow controllers an uncontrolled inflow of the controlled variables is compensated.
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.
As indicated in Fig 2, left panel, and by Eq 1 the steady state condition of E () for a single-E controller determines its set-point. Since the antithetic controller is based on a reaction between E1 and E2 with speed v and rate constant k17 (Fig 2, right panel), i.e.,
(3)
the set-point for this controller is determined by the difference of the steady state conditions between E1 and E2 (Eq 2).
Aim of this work
As practically all processes within a cell are catalyzed by enzymes, we asked the question what influence enzymes may have on dual-E controllers, specifically when the reaction between controller species E1 and E2 is catalyzed. We here show the behaviors of a set of catalyzed antithetic/dual-E controllers. The enzymatic mechanisms for the removal of E1 and E2 include ping-pong, as well as random-order and compulsory order ternary-complex mechanisms [12, 13]. The role of total enzyme concentration is investigated and how the negative feedback structure of the motifs influence controller performance. Fig 3 shows the incorporation of dual-E integral control into the eight negative feedback motifs [6] with enzyme Ez catalyzing the reaction between E1 and E2.
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).
We will show that the performance of the catalyzed dual-E controllers, like response time, depends to a certain degree on the feedback structure/motif and on the enzymatic processing mechanism of E1 and E2. In comparison with single-E control [4, 6] the enzymatic dual-E controllers have the advantage that robust homeostasis is not bound to the requirement of zero-order kinetics, but can also work in its presence.
Materials and methods
Computations were performed by using the Fortran subroutine LSODE [14]. Plots were generated with gnuplot (www.gnuplot.info) and edited with Adobe Illustrator (adobe.com). To make notations simpler, concentrations of compounds are generally denoted by compound names without square brackets. Time derivatives are indicated by the ‘dot’ notation. Concentrations and rate parameter values are given in arbitrary units (au). Set-point values are arbitrarily chosen. For certain feedback structures we observe a switch between dual-E and single-E control when a set-point determining parameter is changed. In these cases the homeostatic properties of the two control modes were studied at different set-points.
Perturbations were applied as single steps without considering (more realistic) time-dependent perturbations [15–17]. The reason for applying steps is that when integral control is operational dual-E (and single-E) controllers will show robust perfect adaptation upon step perturbations, but will principally differ in their speed of resetting.
Enzymatic mechanisms considered
There are two major mechanisms [12, 13] when E1 and E2 are processed by an enzyme Ez, i.e.,
(4)
In one of them, a ternary complex E1⋅Ez⋅E2 between enzyme and substrates E1 and E2 is formed, either via a random binding order (Fig 4a) or by a compulsory binding order (Fig 4b). The other mechanism, termed “ping-pong”, contains two compulsory order binding events. During the first step one of the substrates E1 or E2 binds to the enzyme Ez, releases a possible first product and creates an alternative enzymatic form Ez*, which is able to bind the second substrate. In the final step the enzymatic species Ez is regenerated and a possible second product is released. A new enzymatic cycle can start again (Fig 4c).
(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.
In the case of single-E controllers E is removed by enzyme Ez
(5)
by using (single-substrate) Michaelis-Menten kinetics (Fig 4d). Although single-E controllers have already been analyzed to a large extent before [4, 6, 18], we will encounter their catalyzed versions also here, because some of the dual-E controllers can switch between single-E and dual-E control mode.
We noted that a necessary condition for robust homeostasis to occur is that the involved negative feedback loops need to be described as irreversible processes. Therefore, the enzymatic reactions in Eqs 4 and 5 need to be irreversible. Already in 1925 Lotka [20] investigated whether certain biological phenomena, such as oscillations and homeostasis, could be based on Le Chatelier’s principle, since at that time biologists attempted to apply the principle to biological systems [21]. Lotka concluded in the negative. Today we regard life as an overall irreversible process, a “dissipative structure” being far from chemical equilibrium [22, 23] and which allows for self-maintenance [24].
For each of the three mechanisms in Fig 4a–4c steady state expressions for v of reaction 4 have been found numerically with LSODE and by using the King-Altman method [25] (S1 Text). The King-Altman method has the advantage that v can be expressed as an analytical function of the concentrations of E1 and E2 and the other rate parameters. Our calculations showed that the steady state expressions of v were always in excellent agreement with the corresponding numerical results.
Feedback motifs considered
From the eight feedback structures of Fig 3 we have analyzed four of them: two of the four “inner-loop” motifs m2 and m5 and the two “outer-loop” motifs m4 and m7. The remaining four motifs have similar feedback symmetries and we do not expect significant differences to those considered here.
Results
For each of the motifs m2, m4, m5, and m7 we describe how the controllers perform under step-wise perturbations when the above mentioned ternary-complex and ping-pong mechanisms are applied to remove E1 and E2.
Controllers based on motif 2
This motif’s performance has been found to be remarkably good, especially with respect to perturbations which increase their strength with time [15, 16]. Motif 2 is an inflow type of controller which opposes outflow perturbations in the controlled variable.
Motif 2 dual-E controller removing E1 and E2 by an enzymatic random-order ternary-complez mechanism.
Fig 5 shows the reaction scheme when E1 and E2 are removed enzymatically by using a ternary-complex mechanism with random binding order and E1 as the derepressing agent.
The rate equations are:
(6)
(7)
(8)
(9)
(10)
(11)
(12)
An analytical expression for the reaction velocity
(13)
can be obtained by the steady state approximation (S1 Text), which has been found (see below) to be in excellent agreement with the numerical results.
We observed that the enzymatic controller in Fig 5 can show two set-points of A. One is given by
(14)
when both E1 and E2 participate in the regulation of A (dual-E control).
The other set-point is given by
(15)
In this case only E1 participates in the control of A. (single-E control). The switching between the two control modes is described in more detail below.
Motif 2 single-E controller with Michaelis-Menten removal of E.
Due to the above indicated switch between catalyzed dual-E and single-E control mode we here show the catalyzed single-E m2 controller (Fig 6), which will also be compared with the catalyzed m2 dual-E controller.
The rate equations for the scheme in Fig 6 are:
(16)
(17)
(18)
(19)
In this case the set-point of the controller is described by Eq 15.
The catalyzed m2-controllers: Failure at larger perturbation strengths and enzyme limitation.
Fig 7 shows a comparison of the single-E controller of Fig 5 and the dual-E controller of Fig 6 for step-wise perturbations in k2. While rate constants have been more or less arbitrarily set, for comparison reasons the set-points of the controllers are both put at 2.0. Due to the two different set-point expressions for the dual-E and the single-E controllers (Eqs 14 and 15) k5 and k7 values differ slightly as indicated in the legend of Fig 7. Perturbations are applied as follows: During phase 1 (0–10 time units) A is at the controllers’ set-points (2.0) with a k2-value of 10.0. During phase 2 k2 is increased step-wise using the three perturbations: 1, k2 = 1×102; 2, k2 = 1×103; 3, k2 = 2×104. By comparing the left panels of Fig 7 it is seen that one of the advantages of the dual-E controller is that it can maintain irs set-point even under enzymatic non-zero conditions, which means that Ez is not saturated by its substrates E1 and E2. The single-E controller, however, has problems to defend its set-point as with increasing k2 values the E⋅Ez complex shows increased dissociation (Fig 7, lower right panel) leading to an increasingly poorer performance and thereby increased offsets in A from Aset.
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.
However, with perturbation 3 also the enzyme-catalyzed dual-E controller starts to break down. The reason for the breakdown is related to the total amount of enzyme, Eztot, and the values of k5 and k6. In the present settings k5 and k6 are relatively high, which leads in the dual-E controller to a saturation of Ez by E2, i.e. the concentration of EzE2 approaches that of the total enzyme concentration Eztot. Under these conditions, however, the relationship
(20)
is still obeyed leading to the A steady state
(21)
Thus, E1 still exerts control over A in the dual-E controller, but now in form of a single-E (i.e. E1) control mode. In case Ez works under zero-order conditions ((E1EzE2)≈Eztot), Eq 21 becomes Eq 15. In this mode E2 shows wind-up: E2 increases linearly in time with slope
increasing with increasing k2 values. It is the increase of E2 which leads to the saturation (poisoning) of Ez by E2 (see Fig 7, upper right panel).
The above described limitation of the the dual-E controller can still be circumvented by either decreasing k5 and k6, or by increasing Eztot (see next section). It should however be pointed out that there is another way of a (dual-E) controller breakdown which cannot be opposed by either increasing Eztot) or by decreasing k5 and k6. This type of breakdown occurs when E1 is driven by k2 to such a low concentration that the compensatory flux jcomp approaches its maximum value k3, i.e.
(22)
By setting in Eq 6 the term k3k8/(k8+ E1) to k3 and A to Aset we can calculate the upper limit of k2,
,
(23)
Whenever
the controller breaks down irrespective of the values of k5, k6, and Eztot. Note, that in curve 3 of the upper right panel of Fig 7 we have that
(24)
which is the reason why the dual-E’s homeostatic behavior can be restored as described in the next section.
A more detailed description of this type of breakdown is given in the section Dual-E controllers based on motif 4.
Avoiding enzyme limitations.
Enzyme overload can be avoided by two means, either by increasing the total amount of enzyme, or by decreasing the reaction rates k5 and k6 by which E, E1, and E2 are formed.
Fig 8 illustrates the behavior of the controlled variable A for the antithetic controller (, outlined in orange) in Fig 7 when perturbation 3 is applied. In panel a the total amount of enzyme has been increased from 10−6 to 10−5. In panel b the enzyme concentration is kept at 10−6, but k5 and k6 are in phase 2 decreased by one order of magnitude to respectively 1.0 and 2.0. In comparison, the behavior of the controlled variable A for the zero-order controller (Azo, outlined in black) is also shown. For the higher total enzyme concentrations both controllers behave identical, while for the decreased values of k5 and k6 the antithetic controller is less aggressive, but eventually moves A to the controller’s set-point.
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.
Switching between dual-E and single-E control mode at zero-order conditions.
We found that a change in the control mode of the dual-E controller (Fig 5) occurs in dependence to the relative values of k5 and k6. When k6 is lower than the rate k7Eztot the controller works in an antithetic/dual-E mode. We assume here that the dual-E controller works under zero-order conditions with large values of k9 and k11 relative to k10 and k12 (Fig 5) leading that v is at its maximum velocity, i.e.
(25)
In dual-E mode both E1 and E2 participate in the regulation of A and Aset is given by Eq 14. However, when k6 is larger than k7Eztot the system switches to a single-E control mode where only E1 takes part in the regulation of A. Aset is now described by Eq 15. Fig 9 illustrates the behavior. Panel a shows the steady state values of A (Ass, gray solid circles) as a function of k6 when k5=0.4, k7=1×106 and Eztot=1×10−6. For k6 values lower than k7Eztot (=1.0) the system shows dual-E control with a set-point of k6/k5, while when k6 is larger than k7Eztot=1 single-E control is observed with Aset being k7Eztot/k5 (=2.5). In such a setting the system behaves precisely as a single-E controller (Fig 6) where E is replaced by E1 and Ez is replaced by Ez⋅E2. Under single-E mode conditions E2 does not participate in the control of A and its concentration rises continuously (showing wind-up).
(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 9b shows numerical and steady state values of v (Eq 13); they are in excellent agreement.
As illustrations, Fig 9c and 9d show that the set-points of single- and dual-E control are indeed defended. The two panels show the homeostatic responses when k6=10 (vertical downward red arrow in Fig 9a) and when k6=0.4 (vertical upright blue arrow in Fig 9a).
Fig 9e shows the part of the network (outlined in red) when single-E control is active. At the steady state in A, the rate k5Aset becomes equal to the degradation rate v=k7(E1⋅Ez⋅E2)= k7 Eztot. Typical for the dual-E control (Fig 9f) is that k5Aset and k6 are equal to v=k7(E1⋅Ez⋅E2).
Switching between dual-E and single-E control mode at nonzero-order conditions.
In this section we compare the dual-E and single-E control modes when v = k7(E1⋅Ez⋅E2) is not zero-order with respect to (E1⋅Ez⋅E2).
For single-E control (Fig 9e) nonzero-order conditions imply that
(26)
For the dual-E control (Fig 9f) Ass is given by Eq 14 independent whether the removal of the ternary complex is zero-order or not. However, dual-E mode will switch to single-E mode when
(27)
In this case E2 will show wind-up (i.e., continuously increase unless there is a removal of E2) and Ass is determined by the relationship:
(28)
where Vmax=k7(Eztot). D is the sum of all King-Altman numerator terms described in S1 Text.
Fig 10 illustrates the behavior going from zero-order to nonzero-order conditions. To impose nonzero-order conditions we have for the sake of simplicity, changed the values of k9, k11, k13, and k15 from 1 × 108 (practical zero-order, panels a and b) to 1 × 106 (panels c and d) and 1 × 104 (panels e and f), while other rate constants are kept unchanged. In all panels single-E control responses are outlined in red, while dual-E control is outlined in blue. Fig 10 clearly shows that when the system moves into a nonzero-order kinetics regime (by lowering k9, k11, k13, and k15) the performance by single-E control gets successively worse. However, although dual-E control can maintain/defend its set-point (Eq 14) the range of the dual-E working mode shrinks with increasing nonzero-order kinetics (i.e., with decreasing values of k9, k11, k13, and k15).
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.
Motif 2 dual-E controller removing E1 and E2 by enzymes using compulsory-order ternary complex mechanisms.
In the compulsory-order ternary-complex mechanisms E1 and E2 bind in an ordered manner to enzyme Ez, either E1 first (Fig 11a), or E2 first (Fig 11b).
Panel a: E1 binds first to free enzyme Ez. Panel b: E2 binding first to Ez.
Both mechanisms in Fig 11 can show single-E (E1) or dual-E control dependent on the value of k6.
We found that the mechanism when E1 binds first (Fig 11a) behaves analogous to the random-order ternary complex mechanism of Fig 5. Fig 12a shows the identical responses of the compulsory-order (E1 binds first) and the random-order ternary complex mechanisms when both controllers work in dual-E mode and both are subject to the same step-wise changes in k2 from 10.0 to 500.0. The switch of the compulsory-order (E1 binds first) controller (Fig 11a) from dual-E to single-E mode is shown in Fig 12b when in phase 2, besides the step-wise increase of k2, k6 is increased from 0.4 to 10.0.
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 shows the single-E and dual-E control mode when E1 and E2 are removed by a compulsory-order ternary complex mechanism, but E2 binds first to Ez (Fig 11b). Panel a shows Ass as a function of k6 while panel b shows the numerical and the King-Altman steady state values of the degradation rate v of the ternary complex (S1 Text). In single-E mode the controller of Fig 11b behaves precisely as the single-E controller of Fig 6.
Also for this compulsory-order ternary-complex mechanism (Fig 11b) single-E control is observed when k6 is getting larger than k7(E1⋅Ez⋅E2) or, as in Fig 13, k6 is larger than k7(Eztot) in the case v is zero-order with respect to (E1⋅Ez⋅E2). When k6 is smaller than k7(E1⋅Ez⋅E2) (or k7(Eztot)) the controller of Fig 11b will work in dual-E mode.
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.
Critical slowing down at spontaneous single-E to dual-E mode transitions.
We have seen above that when E1 and E2 are removed by an enzymatic ternary-complex mechanism then, dependent on k6, the m2-controller can work either in a single-E or in a dual-E mode, where each of the control modes can have separate set-points. However, even when the condition for dual-E control mode is fulfilled, i.e. when
(29)
the system can still stay in single-E mode whenever E2 is kept at a high value. In this situation the single-E control mode is “metastable”, i.e., A will be kept at the set-point of the single-E control mode until E2 has reached its steady state. Then A changes abruptly to the set-point of the dual-E controller. This “metastability” of the single-E control mode, with the condition of Eq 29 fulfilled, is illustrated in Fig 14a with two values of k6. For this purpose we have chosen the controller described by Fig 11b, but the other mechanism (Fig 11a) also shows this phenomenon. Outlined in red are the traces of A, while blue lines indicate the concentrations of E2. Continuous lines have a k6 of 4.0 while the dotted lines relate to a k6 value of 8.0. Calculations start with a high initial values of E2 (see legend of Fig 14). While E2 gradually decreases A remains at the set-point of the single-E control mode until it ubruptly changes to the set-point of the dual-E control mode. Also note that even when the single-E controller is metastable, it can still defend its set-point (see the m5 motif below for an explicit example).
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.
The transition time T (Fig 14a) denotes the time span A is kept at the set-point of the single-E controller until its transition to dual-E control. With increasing k6 the system shows the behavior of critical slowing down [26], i.e. T increases and approaches infinity when
(30)
and the set-point for the dual-E control mode vanishes (Fig 14b).
Ping-pong mechanism: Influence of total enzyme concentration on single-E and dual-E control mode.
In this section we turn, for completeness, to the ping-pong type of mechanisms (Fig 15). However, we should mention that no significant differences between the behaviors of ternary-complex mechanisms and ping-pong mechanisms have been observed. Although we could have used one of the ternary-complex mechanisms to illustrate how total enzyme concentration influences m2-controller dynamics and the transitions between single-E and dual-E control modes, we use here the ping-pong mechanism of Fig 15a. While in ternary-complex mechanisms E1 and E2 need both to bind to enzyme Ez to undergo catalysis, in ping-pong mechanisms one of the substrates (E1 or E2) binds first and creates an alternative enzyme form Ez* after forming a first product (for the sake of simplicity we have omitted it). Then Ez* can bind the second substrate which leads to the final product, and regenerates Ez (Fig 4a and 4b). The two mechanisms in Fig 15 differ in the binding order of E1 and E2. When E1 binds first to Ez (Fig 15a) the rate equations become:
(31)
(32)
(33)
(34)
(35)
(36)
(37)
(a) E1 binds first to Ez. (b) E2 binds first to Ez.
Fig 16 shows the effect of total enzyme concentration (Eztot) when in Fig 15a k9, k11, and k13 values are such high that the removal rate of E1 and E2, given by
(38)
becomes zero order with respect to E1 and E2, i.e., v ≈ Vmax = k7Eztot.
(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.
Panels a-d of Fig 16 show the steady state of A (Ass, gray dots) as a function of k6 when Eztot increases from 1 × 10−6 (panel a) up to 1 × 10−4. One sees clearly the increase of the operational range for the dual-E control mode to higher k6 values, while the set-point corresponding to the single-E control mode increases with increasing Eztot concentration.
Ping-pong mechanism: Influence of nonzero-order conditions on single-E and dual-E control mode.
In this section we show how nonzero-order conditions of v = k7(Ez*E1) with respect to E1 and E2 influence the ping-pong mechanism. For this purpose we show the results for the mechanism of Fig 15b when E2 binds first to Ez.
The rate equations are:
(39)
(40)
(41)
(42)
(43)
(44)
(45)
Fig 17 shows the switching behavior from dual-E control, gray dots on blue lines) to single-E control (horizontal gray dots) with changing k6 as a function of the rate constants k9, k11, and k13. The red lines indicate the steady state of A when single-E control mode works under zero-order conditions, i.e. at high values of k9, k11, and k13. In this case we have that
(46)
In the calculations of Fig 17 the total enzyme concentration Eztot is 2 × 10−5. With decreasing values of k9, k11, and k13 (from panel a to d), the system moves towards nonzero-order kinetics (with respect to E1 and E2) and the steady state value of (Ez*E1) decreases. The switch-point in Ass (
) from dual-E control to single-E control occurs now at lower Ass values, described by the equation
(47)
showing that nonzero-order conditions diminish the operational range of dual-E control.
(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.
Also increased values of the perturbation k2 reduces the operational range and moves to lower values (Fig 18).
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.
Summary of the catalyzed m2 controllers.
The catalyzed m2 controller works for all the four basic enzymatic mechanisms shown in Fig 4. Zero-order conditions for v (=dP/dt) with respect to E1 and E2 provide optimum controller performance, which, however, becomes limited at low enzyme concentrations and high perturbation (k2) values. Catalyzed antithetic controllers (i.e. controllers working in dual-E mode) become more aggressive by increased turnover numbers (k7 values). Switch to single-E control mode is observed when the rate forming E2 by k6 exceeds the degradation rate of the controller species E1 and E2. For nonzero-order conditions Ass in single-E control mode decreases with increasing k2 values. While this is also true for the dual-E control mode, in dual-E mode Ass is still determined by the ratio k6/k5 and thereby, unlike a single-E controller, shows robust control even for nonzero-order conditions.
Controllers based on motif 4
Motif 4 is based on double inhibition. In the antithetic/dual-E setting (Fig 3), A is inhibiting the synthesis of E1, while E2 is now activating the compensatory flux by derepression.
Motif 4 dual-E controller removing E1 and E2 by a random-order ternary-complex mechanism.
Fig 19 shows the dual-E m4-controller removing E1 and E2 by a random-order ternary-complex mechanism. It is, like the corresponding m2-controller, also an inflow type of controller, where the compensatory flux, is based on derepression, now by E2, i.e
(48)
The rate equations for the m4-controller are:
(49)
(50)
(51)
(52)
(53)
(54)
(55)
When the controller works in dual-E mode, its set-point is calculated from the following relationship
(56)
In comparison with the corresponding m2 dual-E controller (Fig 7) also for the m4 feedback arrangement the response time decreases with increased levels of step-wise perturbations in k2. Fig 20 shows the controller’s homeostatic behavior upon step-wise perturbations in k2 (curves 1–7) applied at time t = 50 from k2=10 (phase 1) up to k2=2 × 104 (curve 7, phase 2).
(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.
For a given set-point Aset the steady state condition of Eq 49 determines the range of k2 perturbations the controller can defend. By setting in Eq 49 E2=0 and A = Aset the upper limit of k2, , can be determined, i.e.,
(57)
For the m4 controller will defend the set-point described by Eq 56, i.e., Ass = Aset. This is indicated by the the blue area in Fig 21a. The red area in Fig 21 shows the k2 values when
for a given set-point Aset. In this case Ass < Aset and an offset in A concentration from Aset will be observed. Fig 21b illustrates this. During phase 1 (time between 0 and 50) k2=10.0 and the value of A is at its set-point Aset = 3.0. At time t = 50.0 (indicated by the blue downward arrow 1) k2 is increased to 2 × 104. The controller is able to defend the perturbation and is still within the blue area as indicated in Fig 21a by point 1. At time t = 250.0 phase 3 starts with a k2 of 5 × 104 (red downward arrow 2). Now
and the controller shows an offset in the controlled variable, i.e. the steady state value of A is below Aset. With increasing k2 values the offset will increase.
(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.
Another influence on the operational range of the m4 controller is the reaction-order by which the enzyme Ez removes E1 and E2. The reaction order is closely related to the ratios of k10/k9, k12/k11, k14/k13, and k16/k15. The ratios can be interpreted as KM values (in a rapid-equilibrium approach). For example, in the single-E m2 controller (Fig 6), an offset from Aset=k7(Eztot)/k5 (Eq 15) is observed when k10/k9 is relatively large, i.e. not small enough for the degradation of E to become zero-order (see Ref [4, 6] for more details). For the m4 controller (Fig 19) increasing values of the ratios k10/k9, k12/k11, k14/k13, and k16/k15 will lead to a reduction of the controller’s operational range. Fig 22 illustrates this. For the sake of simplicity, all odd-numbered rate constants k9,…k15 and all equal-numbered rate constants k10,…k16 have among themselves the same values, respectively. The turquoise areas in Fig 22 show the fully functional range of the controller as a function of k5, i.e. when the condition of Eq 56 is fulfilled and the controller works in dual-E mode. With increasing values of (k10/k9)=(k12/k11)=(k14/k13)=(k16/k15) the operational range of the controller is clearly reduced.
Interestingly, also in these calculations critical slowing down is observed, similar as in Fig 14b, when the border between dual-E control (turquoise area) and constant Ass values is approached with increasing k5 values.
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.
Importantly, unlike the corresponding m2-controller which goes into a regime of defended single-E control under zero-order conditions (Fig 9), the constant Ass regime of the m4 controller is not defended, but Ass decreases with increasing k2 (perturbation) values. This is shown in Fig 23, where the (k10/k9)=(k12/k11)=(k14/k13)=(k16/k15) ratios are kept constant at 1 × 10−4, while k2 is changed from 50 (panel a) to 500 (panel b). Finally, in panel c k2=5000. With increasing k2 values a reduction in Ass and the controller’s operational range is observed.
Motif 4 dual-E controller removing E1 and E2 by compulsory-order ternary-complex mechanisms.
Fig 24 shows the two mechanisms when the removal of E1 and E2 goes through a compulsory-order ternary-complex. In panel a E1 binds first to the free enzyme Ez, while in panel b E2 binds first.
In (a) E1 binds first to Ez, while in (b) E2 binds first.
In case E1 binds first to Ez (Fig 24a), the rate equations are:
(58)
(59)
(60)
(61)
(62)
(63)
When E2 is binding first to free Ez (Fig 24b), the rate equations are:
(64)
(65)
(66)
(67)
(68)
(69)
For both reaction schemes the set-point for the dual-E controller
(70)
is given by the same balance conditions as for the m4 random-order ternary-complex mechanism, i.e., we have a balance between the two inflow rates j5 = k5k17/(k17 + A)=k6, and the outflow rate k7(E1⋅Ez⋅E2) (see Eq 56).
As already seen for the m2-controller (Fig 12) when working in dual-E mode, the random-order and compulsory-order ternary-complex mechanisms show for the m4-feedback schemes the same kinetic behavior on step-wise changes in k2. Fig 25 illustrates this for the three m4-controllers removing E1 and E2 by enzymatic ternary-complex mechanisms (Figs 19 and 24). Even the breakdown at large k2 values show identical kinetics in A (Fig 25d).
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 shows the concentration profiles of E1, E2 and the different enzyme species for the three m4 controller arrangements in case of their breakdown described in Fig 25d.
Column b: Compulsory-order mechanism (Fig 24a). Column c: Compulsory-order mechanism (Fig 24b). Rate constants and initial concentrations as in Fig 25.
Although the concentration profiles of A, E1, E2, and the ternary-complex (E1⋅Ez⋅E2) are identical for the three controller configurations the other enzyme species replace each other in their functions. For example, when E1 binds first in the compulsory-order mechanisms of Fig 24a the complex (E1⋅Ez) is low during phase 1 but becomes close to the total enzyme concentration Eztot during the breakdown in phase 2 (Fig 26b). In the compulsory-order mechanism when E2 binds first (Fig 24b) the role of (E1⋅Ez) is now taken over by the free enzyme Ez (Fig 26c). In the random-order mechanism the role of the enzyme species is slightly more complex: during phase 1 Ez and (Ez⋅E2) have the same concentration profiles as in the compulsory-order mechanism where E2 binds first to Ez (Fig 24b). However, in phase 2 the Ez profile of the random-order mechanism is that of the other compulsory-order mechanism (Fig 26b)!
Motif 4 dual-E enzymatic controller in which E1 and E2 are removed by ping-pong mechanisms.
Fig 27 shows the two possibilities when enzyme Ez removes E1 and E2 by a ping-pong mechanism. In panel a E1 binds to the free enzyme and creates the alternative enzymatic form Ez*, which then bind the derepressing controller species E2. In panel b this is reversed. Here E2 binds first and forms Ez*, which can bind E1. As for the m2 controller case we have, for the sake of simplicity, omitted the release of a product prior to the formation of Ez*.
In (a) E1 binds first to the free enzyme Ez, while in (b) E2 binds first.
For the scheme of Fig 27a the rate equations are:
(71)
(72)
(73)
(74)
(75)
(76)
(77)
In case E2 binds first to Ez (Fig 27b), the rate equations are:
(78)
(79)
(80)
(81)
(82)
(83)
(84)
We have compared the two m4 ping-pong mechanisms (Fig 27) with the three m4 ternary-complex mechanisms (Figs 19 and 24) and found that their homeostatic behavior in A as well as the concentration profiles in E1 and E2 have identical dynamics with those shown in Fig 25 and the upper row in Fig 26, respectively (data not shown). However, despite their identical dynamical behaviors in the controlled variable A as well as in the controller variables E1 and E2 the different enzyme species show, like in the lower row of Fig 26, a mechanism-dependent restructuring of the enzyme species’ concentration profiles. This indicates that in the different mechanisms different enzyme species take over the tasks to decrease E2 (causing an increase in the compensatory flux when k2 is increased) and to increase E1, thereby leading to homeostasis in A. Specifically, for the intact m4 dual-E ternary-complex controllers (i.e. no breakdown occurs) the condition of Eq 56 defines the profiles of the enzyme species, while for the m4 ping-pong controllers the conditions
(85)
or
(86)
determine the enzyme species concentration profiles when E1 or E2 bind first to Ez, respectively (see Fig 27). In both cases the set-point is
(87)
Fig 28 illustrates the concentration profiles of the enzyme species when all five mechanisms show the same homeostatic behavior in A as in Fig 25c with identical changes in E1 and E2.
(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.
As an example, in the ping-pong mechanisms the role of the ternary-complex E1⋅Ez⋅E2 (Fig 28a–28c, outlined in green) is replaced by Ez*E2 (Fig 28d, E1 binding first to Ez) or by Ez*E1 (Fig 28e, E2 binding first to Ez) as implied by Eqs 56, 85 and 86. Likewise, the steady state concentrations of the other enzyme species can be derived from the above rate equations (see the King-Altman expressions in the S1 Text), but are not further elaborated here.
Controllers based on motif 5
As indicated in Fig 2, motif m5 is an outflow controller [6] and opposes inflow perturbations on the controlled variable A. Like the m2 controller the dual-E (antithetic) version of m5 has an “inner-loop” signaling (Fig 3).
Motif 5 dual-E controller removing E1 and E2 by a random-order ternary-complex mechanism.
Fig 29a shows the reaction scheme when in a m5 controller configuration E1 and E2 are removed by an enzymatic random-order ternary-complex mechanism.
(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.
The rate equations are:
(88)
(89)
(90)
(91)
(92)
(93)
(94)
The corresponding single-E controller is shown in Fig 30
with the corresponding rate equations:
(95)
(96)
(97)
(98)
The single-E feedback loop in Fig 30 shows robust homeostatic control when enzyme Ez works under zero-order conditions, i.e., KM=(k10 + k7)/k9 is low and k9 ≫ k10 + k7. In this case the set-point for A is given by the condition
(99)
For the dual-E m5 controller (Fig 29a) robust homeostasis in A is obtained by the condition
(100)
Fig 29b–29d show the controller’s behavior upon a step-wise change in k1 when working in dual-E mode, i.e., when the homeostatic set-point for A is given by Eq 100.
A switch from dual-E to single-E control mode occurs when k6 becomes larger than k7Eztot. For large k9/k10, k11/k12, k13/k14, and k15/k16 ratios Aset of the single-E controller is given by the condition
(101)
Fig 31 gives an example, where k6 has been increased to 200.0, while the other rate constant values are as in Fig 29. Fig 31a shows the operative part of the single-E controller outlined in red. The grayed part does not participate in the control of A, but shows a steady increase of E2 (wind-up). The controller is subject to the same step-wise increase as in k1 (panel b) as in Fig 29, but has now changed its set-point to 2.5 as described by Eq 99 (panel c). Fig 31d shows the wind-up behavior of E2 along with the change of E1, which activates the compensatory flux removing A and compensating for the increasing inflow of A by k1.
(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.
At low k9/k10, k11/k12, k13/k14, and, k15/k16 ratios the operational range of the dual-E controller decreases and the single-E controller’s steady state in A drops below Aset. Under these conditions the dual-E controller will defend its set-point Aset = k6/k5 exactly, while the single-E controller shows an offset, i.e. Ass < Aset = k7⋅Eztot/k5. Fig 32 illustrates this behavior when the total enzyme concentration is kept constant at 1 × 10−6.
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.
When Eztot increases the operational range of the dual-E controller increases as a function of k6. This is shown in Fig 33 when k9=k11=k13=k15=1 × 109 and Eztot varies from 1 × 10−6 to 4 × 10−5. In agreement with Eq 99 we observe that with changing Eztot the set-point of the single-E controller changes accordingly.
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.
Transition from single-E to dual-E control and critical slowing down.
In the previous section we saw that dual-E control occurs in the m5 random-order ternary-complex mechanism when the condition k6 < k7Eztot is met. In this case, both E1 and E2 are engaged in the control of A. On the other hand, single-E control is observed when k6 > k7Eztot. Here, only E1 acts as the controller variable while E2 shows wind-up. i.e., increases continuously. However, even when the condition for dual-E control is fulfilled, i.e., k6 < k7Eztot, single-E control can be temporarily present, as observed for the m2 controller (Fig 14), when the initial concentration of E2 is above its steady state value for a given perturbation value of k1. In this case, the single-E controller is metastable: E2 will decrease and approach its steady state, but during this period the set-point of the single-E controller will be defended when working under zero-order conditions. Fig 34a–34c illustrates the behavior. In this example E2 concentration starts out high at 9.5 × 104, while its steady state value is 6.7 × 10−2. At time t = 100 k1 is changed from 500.0 to 1000.0 (Fig 34a) and the set-point of A for the single-E controller (=2.5) is defended as long as E2 > E2,ss (Fig 34b). Note that during single-E control the E1 value is responsible for keeping A at its set-point, but the E1 level changes once E2 is at its steady state and the controller has reached dual-E control mode (Fig 34b and 34c).
(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).
We have further tested how the transition time T depends on k6 for this controller at constant k1 (for a definition of T see Fig 14a). As for the m2 controller (Fig 14b) T increases with increasing k6 values. For each value of k6 it takes T time units until A settles at the set-point of the dual-E controller (=k6/k5). T → ∞ as k6 approaches 100.0, the value at which the set-point of the dual-E controller approaches the set-point of the single-E controller.
Motif 5 dual-E controller removing E1 and E2 by a compulsory-order ternary-complex mechanism with E1 binding first to Ez.
The scheme of this mechanism is shown in Fig 35.
The rate equations are:
(102)
(103)
(104)
(105)
(106)
(107)
The reaction velocity producing P and recycling Ez is:
(108)
The conditions for the set-point of the dual-E controller for this mechanism are the same as for the random-order ternary-complex case (Eq 100), i.e.
As shown for the random-order ternary-complex case (Eq 100), also for this compulsory ternary-complex mechanism dual-E control requires that
(110)
In case k6 > k7⋅Eztot the feedback switches to single-E control, with E2 showing wind-up. The set-point switching and E2 wind-up is illustrated in Fig 36. There, during phase 1, A is under dual-E control with set-point of 1.0, where k5=k6=40.0 at perturbation k1=1000.0 (Fig 36a). At the beginning of phase 2 k6 is increased to 200.0, which leads to a change in the set-point of A to 2.5 (=k7⋅Eztot/k5, analogous to Eq 99, Fig 36b) and to wind-up of E2 (Fig 36c). In phase 3 k1 is increased to 2000.0 showing that the single-E controller defends its set-point. During single-E control the E1Ez enzyme species rapidly depletes (Fig 36d) and the concentration of the ternary-complex E1⋅Ez⋅E2 becomes practically equal to the total enzyme concentration Eztot.
(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 shows the switching between dual-E and single-E control as a function of k6 at four different total enzyme concentrations. Clearly, as previously observed for the other mechanisms, an increase in the enzyme concentration leads to an extended range upon which the dual-E controller is able to act. In addition, the k6 switch-point for the transition to dual-E control occurs at higher values as total enzyme concentration increases. The set-point for the single-E controller increases accordingly.
(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.
The velocity how fast P is produced by this mechanism can be expressed analytically using the King-Altman method [25]. The King-Altman treatment leads to
(111) Eq 111 shows that when k9 and k11 are much larger than k7, k10, and k12 the velocity becomes zero-order with respect to E1 and E2 such that v = k7Eztot.
However, when k9 and k11 become equal or lower than k7, k10, and k12 the zero-order condition with respect to E1 and E2 does no longer hold. In such a case, and when the mechanism shows single-E control, the wind-up of E2 makes the E2 terms in Eq 111 disappear, such that at steady state, we have
(112)
The condition k5⋅Ass = v defines the steady-state of A at single-E (single-E) control, i.e.
(113)
The influence of decreased k9 and k11 values in this mechanism is shown in Fig 38. At single-E control, Eq 113 shows excellent agreement with the numerical steady-state values of A.
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.
Note, however, that Ass will depend on the perturbation k1. With increasing k1 values (at single-E control) E1 will increase such that
(114)
and
(Eq 113) will move towards the Aset value at zero-order conditions.
Fig 39 shows that increasing k1 perturbations move Ass towards the set-point of the single-E controller, as with perturbation-induced increases of the controller variable E1 the factor E1/((k7/k9)+E1) in Eq 114 is getting close to 1.
(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.
Motif 5 dual-E controller removing E1 and E2 by a compulsory-order ternary-complex mechanism with E2 binding first to Ez.
The scheme of this mechanism is shown in Fig 40.
The rate equations are:
(115)
(116)
(117)
(118)
(119)
(120)
The set-points for dual-E and single-E control are as described in the previous section for the m5-compulsory-order ternary-complex mechanism when E1 binds first (Eq 41), i.e., =k6/k5 and
=k7Eztot/k5.
The velocity v how fast P is produced by this mechanism can be expressed analytically using the King-Altman steady-state method
(121)
Comparison with numerical results show that Eq 121 gives an excellent description of v as a function of E1 and E2. Eq 121 also shows that when k13 and k15 are much larger than k7, k14, and k16 v becomes zero-order with respect to E1 and E2 such that v = k7Eztot.
This mechanism’s behavior is in many respects identical to the other compulsory-order ternary-complex mechanism of Fig 35. Fig 41 shows as an example a calculation when the change from dual-E to single-E control occurs in an analogous way to that of the other m5 compulsory-order ternary-complex controller shown in Fig 36. For the same rate constant values and the same perturbation profile (Fig 41a) the behaviors of A, E1, and E2 are precisely the same when Fig 41b and 41c are compared with Fig 36b and 36c. Interestingly, Ez in this controller (Fig 41d) has now taken the role of E1⋅Ez in the other controller (Fig 36d), while the function/concentration profile of Ez in Fig 36d is identical to that of Ez⋅E2 in Fig 41d.
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 shows another example of identical behaviors between the two (compulsory-order ternary-complex) m5 controllers when the switching between dual-E and single-E control is investigated as a function of k6 and when zero-order conditions of v with respect to E1 and E2 are relaxed. Under single-E control wind-up of E2 is observed (Fig 41c) such that the expression of v (Eq 121) in this case is reduced to
(122)
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.
For a given perturbation k1 the steady states in A and E1 satisfy, under single-E control, the condition
(123)
which results in the steady state for A:
(124)
The switch between single-E and dual-E control occurs at
(125)
where
is the smallest k6 value which is equal to v from Eq 123. In the case
the controller is in dual-E control mode with v=k6=
.
As already addressed above in Fig 39, a typical property of m5 single-E control is that with increasing perturbation strength the controller species (E1) increases and Ass moves towards . This is also observed for this controller, although higher k1 values are needed here to reach
. Fig 43 shows the behavior when the Fig 42d parametrization is used with k6=200.
(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.
Motif 5 dual-E controller removing E1 and E2 by a ping-pong mechanism with E1 binding first to Ez.
The scheme of this mechanism is shown in Fig 44.
We have included a first-order degradation term of E2 with rate constant k17. The reason for this is the observation that for this controller only E1 acts as a control species while E2 remains to be constant. To see the influence of E2 on the set-point the first-order degradation of E2 is included.
The rate equations are:
(126)
(127)
(128)
(129)
(130)
(131)
(132)
The numerically calculated velocity vnum by which P is formed is calculated as
(133) vnum is in excellent agreement when
is calculated by using the steady-state approach with the help of the King-Altman method (see S1 Text).
In this case, vss is
(134)
with
(135)
From the rate equation of E2 (Eq 128) we see that the concentrations of E2 are related to the concentrations of Ez* and Ez*⋅E2. Since Ez* and Ez*⋅E2 show constant steady-state values the concentration of E2 is constant in time, but its value is dependent on the values of the other rate constants.
The relationship
(136)
determines the set-point for A, Aset, as
(137)
whenever
is constant and independent of the perturbation k1. Independence of
from k1 occurs when the terms αi/(E1) in the first line of Eq 135 become zero, either by sufficient large E1 values or/and by the αi’s being close to zero (large k9 and k11 values in comparison to k7 and k10). We found robust homeostasis in A for a large range of rate constant values. The rate constant values used here have been chosen such that comparisons with the other controllers can be made and for getting controller response times which are not too large.
A striking observation in comparison with the m5-based controllers based on ternary-complex mechanisms is that E2 has apparently no control function and that the ping-pong mechanism appears to be entirely controlled by E1, even if Ass = Aset = k6/k5, when k17=0, described by the set-point condition
(138)
Fig 45 shows steady state values Ass as a function of k6 when k17=100.0 and 0.0, at a total enzyme concentration of 1.0 × 10−6. Each Ass values represents an actual set-point of A, which is defended against step-wise perturbations by k1.
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 shows the homeostatic behavior of Ass in Fig 45 for k6=10.0 when k17=100.0 (panels a and b), or when k17=0.0 (panels c and d). Note that only E1 acts as the controller variable while E2 is constant independent of the values of the perturbation k1.
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.
For k6=100.0 the controller’s behavior is shown in Fig 47. Also in this case robust homeostasis in A is observed due to the high values of E1 and the constancy in E2. Due to the large values in both E1 and E2 the set-point of the controller is (k17=0):
(139)
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.
Motif 5 dual-E controller removing E1 and E2 by a ping-pong mechanism with E2 binding first to Ez.
Fig 48 shows the scheme of this mechanism.
The rate equations are:
(140)
(141)
(142)
(143)
(144)
(145)
(146)
The numerically calculated velocity is calculated as
(147)
which has been found to be in excellent agreement with the steady-state expression
(148)
with D being
(149)
Eq 148 is derived along the same lines as for Eq 134, i.e. by using the King-Altman method (S1 Text).
The set-point Aset for the controller is determined by its steady-state, Ass, due to the condition
(150)
where vss is independent of k1. The maximum velocity is reached for zero-order conditions with respect to E1 and E2, and is given by
(151)
analogous to the condition by Eq 139.
The relationship between Aset and (Ez*⋅E1)ss is also independent of k6. Like for the m5 ping-pong based controller when E1 binds first to Ez, E2 has also here no control function and remains constant as long as the inflow of k6 can be compensated by . However, when k6 > vmax, and for example k17 = 0, E2 will increase linearly. Fig 49 shows the controller’s behavior for three different k1 perturbation values analogous as in Fig 47 when k6=100.0. For this high value of k6 (>vmax and k17 = 0) E2 shows a (linear) increase.
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 shows how Ass (which defines the set-point Aset) depends on k6. Only when k17=0 and k6 < vmax the set-point is defined by k6/k5. Despite the formal agreement with the set-point of a dual-E control when comparison is made to the above ternary-complex motif 5 mechanisms, the ping-pong mechanism shows robust single-E control conducted by E1.
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.
Controllers based on motif 7
Motif 7 dual-E controller removing E1 and E2 by a random-order ternary-complex mechanism.
Fig 51 shows the reaction scheme when in a m7 controller configuration E1 and E2 are removed by an enzymatic random-order ternary-complex mechanism.
The rate equations are:
(152)
(153)
(154)
(155)
(156)
(157)
(158)
Under dual-E conditions the steady state concentration of A is determined by setting inflow rates k6 and j5 = k5k17/(k17 + Ass) equal to the outflow rate v = k7(E1⋅Ez⋅E2), i.e.,
(159)
Solving for Ass, which is equal to Aset, gives
(160)
Fig 52 illustrates the controller’s homeostatic behavior, following Eq 160, for step-wise perturbations in k1. The controller operates by increasing the controller variable E2, which activates the compensatory flux k4⋅(E2)(A). Although E1 undergoes an excursion during the perturbation, its steady-state value remains unchanged at the different k1 values.
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.
Operational range and irreversibility of the controller.
Dual-E control is enabled as long as the condition by Eq 159 is obeyed, i.e., k6 values need to be lower than k5, together with k5 < k7⋅Eztot. For these conditions the rates v, k6, and j5 = k5k17/(k17 + Ass) are equal. However, when k6 → k5, then Ass → 0, and j5 → k5. In the limit, when k5 = k6, the feedback is broken and A does not exert inhibition on j5.
In the case when k6 > k5, E2 will continuously increase, because v = k7(E1⋅Ez⋅E2) balances with k5, but not with k6. Due to the continuous increase of E2 the compensatory flux k4⋅(E2)(A) will also increase and drive A to zero.
The loss of homeostasis in A when k6 < k5 is described in Fig 53 where panels (a) and (b) show the numerically calculated A and v values (gray dots) as a function of k6 after a simulation time of 48000 time units. In these calculations k5=31.0 and k17=0.1. The blue line in panel (a) shows the calculated Aset values by Eq 160. When k6 ≥ 31.0 Aset becomes (formally) negative. In this case A is found to decrease as a function of time due to the continuous increase of E2 as a result of the negative feedback loss. The changes of A and E2 concentrations are shown in panels (c) and (d) when k6=40.0 (indicated by the red dot and red arrow). Panel (e) shows the concentrations of the different enzymatic species. Indicated in panel (b) is the loss of the negative feedback loop when k6 ≥ 31.0 leading to A → 0 and a constant vnum=k5=31.0.
(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.
As mentioned before a necessary condition to obtain robust control is the presence of a sufficient irreversible flux within the controller. This is indicated in Fig 54a–54c by using different values of the forward enzymatic rate constants k9, k11, k13, and k15, while the corresponding reverse rate constants k10, k12, k14, and k16 are kept constant (1 × 103). In panel d the enzymatic process is entirely irreversible (k10, k12, k14, and k16 are all zero), but due to the low value of the forward enzymatic rate constants k10, k12, k14, and k16 (all 1 × 103), the controller does not show homeostasis at all, despite being completely irreversible.
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.
The reason behind this failure to show homeostasis at high k5 values is the incapability of the enzymatic system to absorb the high j5 inflows. As a result the enzymatic system becomes saturated and E1 increases continuously.
Effect of enzyme concentration.
The above incapability of a saturated enzymatic system to maintain homeostatic behavior at large j5 values can be counteracted by increasing the total enzyme concentration. This is shown in Fig 55, where total enzyme concentration changes from 1 × 10−6 to 1 × 10−3. Clearly, the total amount of the enzyme plays an important role in the performance of catalyzed homeostatic controllers.
(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.
Motif 7 controller using compulsory-order ternary-complex mechanisms.
As for the other ternary-complex controller motifs there are two compulsory-order mechanisms, one in which E1 binds first to free Ez (Fig 56a), while in the other one (Fig 56b) E2 binds first to Ez.
In (a) E1 binds first to the free enzyme Ez, while in (b) E2 binds first.
The two compulsory-order mechanisms behave quite similar compared with the random-order mechanism. In the case when E1 is binding first to the free enzyme Ez (Fig 56a), the rate equations are:
(161)
(162)
(163)
(164)
(165)
(166)
The set-point for the controller in dual-E mode is derived in an analogous way as for the random-order mechanism, i.e. the condition for the operative controller is given by Eqs 159 and 160 for the set-point. Also here we have explored how the controller’s performance changes in response to rate constant k5 and find identical behaviors in response to the enzyme system’s behavior to “absorb” the flux j5=k5k17/(k17 + A). High values of k9 and k11, as seen in Fig 57a, promote the functionality of the controller, while low k9 and k11 values (Fig 57b) lead to a breakdown. As in the random-order case (Fig 55), an increase of the total enzyme concentration leads to an improvement of the controller’s homeostatic behavior. In Fig 57c and 57d the total enzyme concentrations are increased from 1 × 10−6 to 1 × 10−4 and 1 × 10−3. This allows the controller to maintain the homeostatic Aset for larger k5 values.
(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.
In the case E2 binds first to the free enzyme Ez (Fig 56b), the rate equations become:
(167)
(168)
(169)
(170)
(171)
(172)
As for the other m7 ternary-complex mechanisms the set-point of A is determined by the balance between reaction rates k6, j5=k5k17/(k17 + A), and v = k7(E1⋅Ez⋅E2) (see Eqs 159 and 160). With respect to varying values of k1 and k5 the controller’s steady state values in A behave precisely as shown in Fig 57 for the other ternary-complex mechanisms, i.e., the loss of homeostasis by low forward rate constants k13 and k15 can be counteracted by an increase in the total enzyme concentration.
As we already saw from the previous controllers (see for example the m5 controller, Fig 38) an increase in the perturbation strength (here k1) will drive the steady-state of the regulated variable A towards its theoretical set-point Aset. This is also observed for the m7-type of controllers for dual-E control. As seen in Fig 58, increasing k1 values extend the homeostatic region of the controller. This same pattern of A as a function of k5 for different k1 values is also observed for the other ternary-complex mechanisms (Figs 51 and 56a).
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.
Motif 7 dual-E controller removing E1 and E2 by ping-pong mechanisms.
Fig 59 shows the reaction scheme when in a m7 controller configuration E1 and E2 are removed by the two enzymatic ping-pong mechanisms when E1 binds first to Ez (Fig 59a) or when E2 binds first to Ez (Fig 59b).
In the case E1 binds first to Ez (Fig 59a) the rate equations are:
(173)
(174)
(175)
(176)
(177)
(178)
(179)
Minor differences between the m7 ping-pong and ternary-complex mechanisms.
The dynamic behaviors of the ping pong-mechanisms are very similar to the (m7) ternary-complex mechanisms. Also here Aset is determined by the balancing of the three fluxes j5=k5k17/(k17 + A), the inflow described by k6, and the rate v=k7(Ez*⋅E2) making P. Accordingly, Aset is described by Eq 160. Also, an increase of total enzyme concentration and an increase of the forward enzymatic rate constants k9, k11,… will improve the homeostatic performance of the ping-pong controllers.
However, since the ping-pong mechanisms have a slightly longer enzymatic reaction chain in comparison with the ternary-complex mechanisms, in the ping-pong case larger forward enzymatic rate constants values are needed together with lower k6 values to match the fluxes j5, k6, and v to achieve moving Ass to its set-point. The influences of the forward enzymatic rate constants and the total Ez concentration are illustrated in Fig 60 where numerically calculated A values are compared with the theoretical set-point Aset. In comparison with the ternary-complex mechanism results from Figs 57a and 60a show the behavior of the ping-pong mechanism whenEztot=1 × 10−6, and k9=k11=k13=1 × 107. Unlike in the ternary-complex mechanism, in the ping-pong case deviations between the numerically calculated A values and Aset are observed at the higher (k5 > 460) and lower k5 < 10 ends of the k5 scale. When in Fig 60a k5 gets higher than 460 the enzymatic system cannot absorb the inflow flux j5=k5k17/(k17+ A). As a result, E1 shows a linear increase in time, with a slope which is dependent on the value of k5, but where A becomes constant and independent of k5. At the lower end of the k5 scale (k5 < 10) the value of k6 is too high to get absorbed by Ez*⋅E2. This has the result that E2 shows a liner increase in time and an increasing compensatory flux k4⋅A⋅E2 with A decreasing continuously without reaching a steady state. The loss of homeostasis at high k5 values can be overcome by either increasing the total amount of enzyme (Fig 60b) or by increasing the values of the forward rate constants k9, k11, and/or k13 (Fig 60c).
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.
An increase of the perturbation k1 leads also in the ping-pong controllers to an improvement in the homeostatic accuracy as for example observed in the m5 controllers, but not at low k5 values. This is indicated in Fig 61. In panel a we have rate constant values for k1, k6, Eztot, and the forward enzymatic rate constants (k9, k11, and k13) as in Fig 60a and as in the ternary-complex mechanisms of Fig 57a. An increase of k1 from 1 × 102 to 1 × 104 in the ping-pong mechanism (Fig 61b) does improve the homeostatic response of the controller at high k5 values, but not at low k5, where the high inflow rate by k6 cannot be absorbed. In fact, a decrease of k6 from 5.0 to 1.0 leads to homeostasis for all k5 values with Aset > 0 (Fig 61c).
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.
An analysis of the two m7 ping-pong mechanisms in Fig 59 shows that their responses in A, E1, and E2 are, for different k1 perturbation strengths, identical (see Fig 62a and 62b). Both use E2 as the variable which controls the compensatory flux jcomp=k4⋅A⋅E2. The roles of the enzymatic species Ez, Ez*, E1⋅Ez and Ez*E2 in the mechanism of Fig 59a are in the mechanism of Fig 59b replaced by the respective species Ez*, Ez, Ez⋅E2, and Ez*E1; see Fig 62c and 62d. While the concentrations in A, E1, E2, E1⋅Ez, Ez*E2, Ez⋅E2, and Ez*E1 are identical as a function of k5, the concentrations of Ez and Ez* are different, but interchange in dependence whether E1 or E2 binds first to free Ez. The numerical results are shown in Fig 63 for the individual reaction species.
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.
(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.
Discussion
There are presently three kinetic approaches how error integration (Fig 1) can be achieved leading to perfect adaptation or homeostasis. One approach is the use of applying zero-order kinetics in the removal of the controller variable E [1, 2, 4–6, 18, 27]. A second approach [8, 28, 29] is based on a first-order autocatalytic production of E combined with its first-order removal. Finally, a third approach is based on antithetic control (described here also as dual-E control) [7, 8, 10, 11], where one of the controller variables (for example E1) participates in a negative feedback and reacts with a second controller variable E2, for example as described by Eq 3.
The advantage of the antithetic approach is that the removal of E1 and E2 does not necessarily need to be precisely a second-order process as formulated by Eq 3, but can in principle be of any type of kinetics. As practically all biochemical processes are catalyzed by enzymes, we have focussed here on mechanisms which remove E1 and E2 by classical two-substrate enzyme kinetics [12, 13]. In addition, taking a previously suggested basic set of negative feedback loops (controller motifs m1-m8) as a starting point, we have extended in Fig 3 this set for dual-E/antithetic control. Using enzyme kinetics with E1 and E2 as substrates allows for a large variety of processes as candidates for robust homeostasis.
Before we discuss a few examples where robust regulation appears to be associated with enzymatic dual-E controllers we would like to comment on cooperativity. Cooperativity by multisite binding or other mechanisms [30, 31], and conveniently described by a Hill function, is observed in many enzymatic systems [12, 13]. Although the influence of possible cooperative behaviors was not considered in this study, cooperativity may have significant effects on the controllers’ resetting kinetics and set-points. For example, calculations by Drobac et al. [17] on multisite derepression controllers showed that a difference in cooperativity (Hill coefficients) in the inhibition mechanism had a significant effect on the speed how fast a set-point is approached, while in this case the set-point value itself was not affected. In general, one may expect that cooperativity in feedback signaling or in the enzymatic removal of the control species E1 and E2 will possibly lead to changed resetting kinetics. To what extent set-points of dual-E (or single-E controllers) are influenced by cooperativity seems to depend on how the controlled variable A’s signaling will affect the manipulated variables E1/E2. These aspects, which are briefly mentioned here will need further and more systematic investigations.
Protein phosphorylation
Regulation by phosphorylating enzymes is observed in practically all aspects of life [32]. The enzymes, protein kinases, use as substrate a target protein and MgATP. A general feature of protein kinases is that they follow compulsory-order or random-order ternary-complex mechanisms [33]. In the following we give two examples that describe m2 control where ATP and the target protein are processed by a kinase using a ternary-complex mechanism.
Circadian rhythms.
Circadian rhythms play an important part in the adaptation of organisms to their environment, in particular to the day/night and seasonal changes on earth. The molecular bases of circadian rhythms are transcriptional-translational negative feedback loops which oscillate with a period of circa 24 hours [34]. Using the model organism Neurospora crassa phosphorylation was found to serve two functions: firstly, to close the negative feedback loop by phosphorylating the transcription factor WCC (White Collar Complex). The WCC phosphorylation leads to its inhibition by its gene product, the protein FREQUENCY (FRQ) [35, 36], Secondly, FRQ, which is central to the Neurospora circadian pacemaker [37] is phosphorylated by CK1 with the result that phosphorylated FRQ is no longer able to inhibit WCC. Fig 64 indicates the central negative feedback loop in the Neurospora circadian clock describing the phosphorylation of FRQ by CK1 as a random-order ternary-complex mechanism, thereby moving FRQ out of the negative feedback loop. FRQ is phosphorylated at multiple sites [38] and hyper-phosphorylated FRQ is finally degraded. It should be noted that analogous feedback loops with post-translational phosphorylation have also been observed for the Drosophila circadian clock [39, 40].
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.
In comparison with our m2 calculations above, Fig 64 predicts that frq-mRNA appears to be under homeostatic regulation with respect to its degradation. This prediction is indeed in agreement with experimental findings by Liu et al. [41]. Their results indicate that the level of frq-mRNA, although changing on a circadian time scale, is on average not altered at different temperatures. Since, furthermore, the circadian period is compensated towards variations in temperature (temperature-compensation) [42–44], it will be interesting to investigate how FRQ phosphorylation by CK1, leading to putative frq-mRNA homeostasis, also contributes to temperature compensation in the Neurospora circadian clock as indicated by recent experiments [45].
Brassinosteroid homeostasis.
Brassinosteroids (BRs) are plant hormones which have influence on plant growth and development, and adapt plants to environmental stresses. Plants lacking BRs show dwarf growth and abnormal organs [46]. BRs bound to their receptor BZR1 produce BZ1, which inhibits the transcription of the BR genes by binding to promoter regions of different genes in the BR synthesis pathway [47, 48]. The GSK3-like kinase BIN2 phosphorylates BZR1, which then leads to its proteasomal degradation [49]. Fig 65 shows the removal of BZR1 out of the negative feedback loop by BIN2 phosphorylation using a random-order ternary-complex mechanism [33].
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.
Ubiquitination and proteasomal degradation
In metal-ion homeostasis many of the controller molecules are subject to proteasomal degradation in a metal-ion dependent fashion (for a summary, see the Supporting Material of Ref [6]). In proteasomal degradation, ubiquitin, a small protein, is moved through a cascade of three ligases (E1-E3; not to be confused with the controllers E1 and E2 above) and then added on to the target protein [50]. Repeated ubiquitin ligation of the target protein leads then finally to its degradation by the proteasome.
A relatively well understood example is mammalian iron homeostasis. At low iron levels IRP2 together with IRP1 promote the inflow of iron by stabilizing mRNAs which code for proteins that are necessary for iron supply. Results by Vashisht et al. [51] indicate that IRP2 is degraded in an iron-dependent manner where the F-box protein FBXL5 catalyzes IRP2 ligation with ubiquitin. While in this case three substrates are involved (iron, IRP2 and ubiquitin), dual-E control as described above cannot directly applied. However, the indication by Vashisht et al. [51] that iron stabilizes/activates FBXL5 leads to the following m1 dual-E mechanism (Fig 66) where iron activates FBXL5 from a pool of inactive enzyme. This allows the binding of IRP2 forming a SCF complex [51, 52]. For simplicity, the other components of the SCF complex are not shown and the pool of inactive enzyme (FBXL5i) is considered to be constant.
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.
Under these assumptions iron is homeostatically controlled with set-point Feset, which is determined by the condition
(180)
resulting in
(181)
Thus, the level of iron under iron-deficient conditions is given by the ratio between the rate of IRP2 generation and the rate of FBXL5 activation.
Iron and zinc homeostasis in yeast follow analogous strategies (see Supporting Material in Ref [6]).
Conclusion
We showed that antithetic/dual-E control can be incorporated into eight basic negative feedback motifs m1-m8. For four of them we have explicitly shown that robust antithetic control is possible when the removal of the two controller molecules E1 and E2 is catalyzed by an enzyme. Antithetic control has the advantage that it does not require specific kinetics, like zero-order kinetics is required for robust single-E control. Enzymatic dual-E control allows for the possibility that many enzymatic processes which take part in feedback regulations (like phosphorylation) may be better understood in terms of their contributions to obtain robust control. Although dual-E controllers based on ternary-complex or ping-pong mechanisms have similar (and often identical) dynamics with respect to the controlled variable, the kinetics of the participating enzymatic species are generally different for the different mechanisms. Low enzyme concentrations may limit robust homeostatic performance of catalyzed dual-E (and single-E) controllers. Transition between dual-E and single-E control may occur, but robust homeostasis for the resulting single-E controller is generally bound to zero-order kinetics. Single-E control within a dual-E network may show metastability, i.e. single-E control will switch spontaneously to dual-E control and “critical slowing down” may be observed.
Irreversibility of catalyzed (or uncatalyzed) controllers is one of the necessary conditions to obtain robust homeostasis. The work by Prigogine and coworkers [22] showed that organisms, as dissipative structures, exist as steady states far from chemical equilibrium [53]. In view of Cannon’s definition [21, 54] homeostasis preserves these steady states and thereby contributes to the stability of organisms and cells.
Supporting information
S1 Text. Steady state (King-Altman) expressions for enzyme-catalyzed ternary-complex and ping-pong reactions using E1 and E2 as substrates.
https://doi.org/10.1371/journal.pone.0262371.s001
(PDF)
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