Abstract
Cells and organisms have developed homeostatic mechanisms to maintain internal stabilities which protect them against a changing environment. How cellular growth and homeostasis interact is still not well understood, but of increasing interest to the synthetic and molecular biology community where molecular control circuits are sought and tried to maintain homeostasis that opposes the diluting effects of cell growth. In this paper we describe the performance of four negative feedback (inflow) controllers, which, for different observed growth laws (time-dependent increase in the cellular volume V) are able to compensate for various time-dependent removals of the controlled variable A. The four implementations of integral control are based on zero-order, first-order autocatalytic, second-order (antithetic), and derepressing inhibition kinetics. All controllers behave ideal in the sense that they for step-wise perturbations in V and A are able to drive the controlled variable precisely back to the controller’s theoretical set-point The applied increase in cellular volume includes linear, exponential and saturating growth and reflect experimentally observed growth laws of single cell organisms and other cell types. During the increase in V, additional linear or exponential time-dependent perturbations which remove A are applied, and controllers are tested with respect to their ability to compensate for both the increase in volume V and the applied perturbations removing A. Our results show that the way how integral control is kinetically implemented and the structure of the negative feedback loop are essential determinants of a controller’s performance. The results provide a ranking between the four tested controller types. Considering exponential volume increases together with an exponentially increasing removal rate of A controllers based on derepression kinetics perform best, but break down when the control-inhibitor’s concentration gets too low. The first-order autocatalytic controller is able to defend time-dependent exponential growth and removals in A, but generally with a certain offset below its theoretical set-point The controllers based on zero-order and second-order (antithetic) integral feedback can only manage linear increases in V and removals in A, in dependence of the controllers’ aggressiveness. Our results provide a theoretical basis what controller kinetics are needed in order to compensate for different growth laws.
Introduction
The term homeostasis was defined by Walter B. Cannon [1] to describe the coordinated ability of organisms and cells to maintain an internal stability by keeping concentrations of cellular components within certain tolerable limits [2]. Cannon’s emphasis on homeo indicates that he considered the internal physiological state not as a constant, as suggested earlier by Benard’s concept of a fixed “milieu intérieur” [2, 3], but conceives homeostasis as a dynamic adaptable system which allows variations within certain limits. Dependent on the controlled components, the homeostatic limits in which one or several controllers operate can vary considerably. For example, while the negative feedback regulation of cellular sodium shows an apparently changing (rheostatic) and less well-defined set-point [4, 5], the regulation of other metal ions including cytosolic calcium have more strict limits [6–8].
Growth, an essential aspect of all living beings is a highly regulated process. Although the protective functions of homeostasis need to be in place during growth, the interacting mechanisms between homeostasis and growth are not well understood. In principle, there are two aspects to consider. The first aspect, which is considered in this paper, is how homeostatic mechanisms can compensate for growth (and additional perturbations) without influencing growth. The other aspect, considered in a following paper, is how homeostatic mechanisms can influence growth. In this paper we consider growth as an increase of the cellular volume. We investigate by testing four negative feedback loop structures/motifs [7] with different kinetic implementations of integral control how these controllers are capable to oppose the dilution effects of growth along with additional perturbations (removals) in a controlled variable. Integral control is a concept from control engineering [9] enabling robust control for step-wise perturbations, and has been implicated to occur in a variety of homeostatic regulated systems [5, 10–12]
The growth kinetics that will be considered include linear (constant) as well as saturating and exponential growth laws. According to Bertalanffy [13, 14], the different observed growth kinetics of organisms are related to the organisms’ metabolism. For example, when respiration is proportional to the surface of the organism linear growth kinetics are obtained. On the other hand, if respiration is proportional to the organism’s weight/volume, exponential growth occurs. Growth kinetics of bacteria [15, 16] appear closely related to the bacterial form or shape. Rod-shaped bacteria show exponential growth rates, i.e. whereas spherical bacteria increase their cellular volume by a rate law related to the surface to volume ratio, i.e., where η and ξ are constants reflecting anabolism and catabolism, respectively [17].
During growth the controllers have also been tested with respect to outflow perturbations acting on a controlled variable A, i.e., by removing A in a time-dependent (linear or exponential) manner. We focus here primarily on outflow perturbations, because together with the diluting effects of the different growth laws these perturbations represent the most severe conditions for testing the controllers.
Materials, methods, and controllers tested
For the sake of simplicity we assume that compounds inside a growing cellular volume V undergo ideal mixing. Computations were performed by using the Fortran subroutine LSODE [18]. Plots were generated with gnuplot (www.gnuplot.info) and Adobe Illustrator (adobe.com). To make notations simpler, concentrations of compounds are 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). Rate parameters are presented as ki’s (i=1, 2, 3, …) irrespective of their kinetic nature, i.e. whether they represent turnover numbers, Michaelis constants, or inhibition constants. A set of MATLAB (mathworks.com) calculations with instructions are provided in the Supporting Information as a combined zip-file (S1 Matlab). In addition, the Supporting Information contains analytical expressions/estimates of steady state values.
We have previously described eight different two-component negative feedback loop arrangements (motifs) dependent on how the two components A (the controlled variable) and E (controller variable) activate or inhibit each other [7]. Half of the motifs describe inflow controllers, i.e., the compensatory flux adds A to the system from a certain source, while the other half are outflow controllers in which the compensatory flux removes A in order to maintain homeostasis in A. Since our interest here is to study controller performances that compensate for the dilution effects of growth together with (time-dependent) perturbations which remove A, we focus here entirely on inflow controllers. From the four inflow controller motifs we have chosen two, one based on only activation (motif 1, [7]), while the other is based on both activation and inhibition/derepression (motif 2). The reason for this choice comes from an earlier study [19] on time-dependent perturbations which indicated that a motif 1 controller with an autocatalytic (positive feedback) implementation of integral control [6, 9, 20] and a zero-order motif 2 controller with derepression kinetics showed good performances in comparison with a zero-order based motif 1 controller. In addition, we have included a motif 1 based controller with a second-order (antithetic, [21]) implementation of integral control.
Fig. 1 gives an overview of the tested controller types. All controllers behave ideal in the sense that they for step-wise removals in A, and in the absence of growth, are able to keep A precisely at their defined theoretical set-points The controllers were investigated with respect to their capabilities to compensate for time-dependent outflow perturbations in A and in the presence of different growth laws (increase in the reaction volume V) according to Bertalanffy’s classifications [13, 14, 17].
Reaction kinetics during volume changes
To get a correct description of the cellular concentration changes during cell growth we have to consider the concentration changes due to the increasing reaction volume V. If A denotes the concentration of nA moles of compound A in volume V, the overall change of concentration A is composed of two terms, one that describes the changes of A while V is kept constant, ȦV and of a second term, A(V̇/V), which describes the influence of the volume changes on the concentration of A, i.e.,
The structure of Eq. 3 will be used when formulating the rate equations of cellular compounds in the presence of a changing V. Before we turn to the actual controller examples we show how growth (V̇) affects the concentration of a given species A (which will be later our controlled variable) when A is unreactive, being produced internally within the cell, or being produced by a transporter-mediated process.
Unreactive A
In this example (Fig. 2) nA is kept constant, but the volume V increases with the rate V̇.
As V increases the concentration of A will decrease, i.e.,
Since we assume that nA is constant, we have that and the concentration of A decreases according to
Integrating Eq. 5 leads to: which can be rewritten as
Eq. 7 can also be derived by noting that A0=nA/V0 and A(t)=nA/V (t). Solving for nA from one of the equations and inserting it into the other leads to Eq. 7.
Cell internal generated A
Compensatory fluxes to counteract diminishing levels of a controlled compound A can be generated by a cell internal compound (assumed here to be homogeneously distributed inside V) or by the help of transporters from stores outside of the cell or from cell-internal (organelle) stores. We will investigate both ways to generate compensatory fluxes.
To achieve a constant level of A from a cell internal source, despite increasing V, we consider first a zero-order enzymatic reaction where enzyme E converts a species S (assumed to be present in sufficiently high amounts) to A, where V is assumed to increase by a constant rate (Fig. 3).
Let’s also assume that E is not subject to any synthesis, but that during the increase of V, E remains always saturated with S and produces A by zero-order kinetics with respect to A. The initial production rate of A at time t=0 is given as
Since E is considered to be saturated by S at all times we have that KM ≪ S(t) leading to where k2 is the turnover number of the enzymatic process generating A, and E0 is the enzyme concentration at time t = 0. As V increases, the concentrations of E and A are subject to dilution as described by the rate equations
For V̇ = k1=constant, E(t) and A(t) are described by the equations (S1 Text)
From Eq. 13 we see that A will approach a final concentration Afinal = k2·E0·α even when V continues to grow. The time needed of A to approach Afinal is determined by the term α/(t+α).
Fig. 4 shows that Afinal is independent of the initial values of A. However, the system is not stable against perturbations which remove A. In such a case A will go to zero (S1 Text).
Transporter generated A
Alternatively, A may be imported into the cell by a transporter T (Fig. 5).
Also here we consider that the transporter works under saturation (zero-order) conditions adding moles of A per time unit into the cellular volume V where T denotes the (surface/membrane) concentration of the transporter, is a dissociation constant between external A (Aext) and T, and k2 is the turnover number of the transporter-mediated uptake of A.
The change in the concentration of A inside an expanding cell is given by (see Eq. 3)
For constant V̇, k2, and T the steady state of A (Ȧ=0) is k2T/V̇ independent of the initial concentration of A. However, also in the transporter-based inflow of A, the steady state in A is not stable against perturbations removing A. Any reaction within the cell removing A while growth occurs will drive A to zero (S2 Text). To get a steady state that is stable against perturbations a negative feedback controller needs to be included.
Controllers with transporter-based compensatory fluxes and linear time-dependent perturbations
In this section the four controller motifs (Fig. 1) use transporter-based compensatory fluxes. The controllers’ performances are tested with respect to constant growth V̇ =k1 and a linear increase in the outflow perturbation rate parameter k3.
Motif 1 zero-order controller
Fig. 6 shows the motif 1 controller with zero-order implementation of integral control [7]. A is the controlled compound and E is the controller molecule which concentration (in the ideal controller case) is proportional to the integrated error between A and M i a precursor/store or E, which when consumed limits the operational life time of the controller.
The rate equations for this system are:
For simplicity T and Aext/( + Aext) are set to 1 leading to an inflow rate in A of k2E/V. When k̇3=V̇ =0, the set-point of the controller is (Ref. [7], S3 Text) independent of the inflow rate constant k2 and the perturbation strength parameter k3.
When V̇ =constant the zero-order controller maintains a steady state below (S3 Text): which becomes dependent of V̇, and the rate constants k2 and k3.
In testing the performance of this controller we consider three phases (see Fig. 7). During the first phase the volume and the perturbation k3 are kept constant. The controller is able to compensate for the perturbation rate k3·A and keeps A at its theoretical set-point In the second phase the volume increases linearly, while k3 remains constant. The zero-order controller is now no longer able to maintain homeostasis at but shows a V̇-dependent offset below as described by Eq. 20. When in phase 3 also k3 increases linearly the controller breaks down and A goes to zero.
Motif 1 antithetic controller
The antithetic controller [21] uses two controller molecules, E1 and E2 (Fig. 8). Compound E1 is activated by A but is removed by compound E2 by a second-order process. E2 is formed by a zero-order process which acts as a constant reference rate. In addition, E2 also acts as a signaling molecule, which closes the negative feedback loop by activating the transporter-based compensatory inflow of A.
Assuming, as in the previous two examples that T and Aext/( +Aext) are both 1, the rate equations are where k5 ≪ M and k9 ≪ O such that the generation of E1 and E2 are zero-order processes with respect to M and O.
In case V̇ =0 and k̇3=0 the set-point of the controller is given by setting Eq. 22 and Eq. 23 to zero. Eliminating the second-order term k6·E1·E2 leads to which is shown in phase 1 of Fig. 9. In phase 2 the volume increases linearly with V̇ =2.0 (Fig. 9, left panel) while k3 remains to be constant at k3=2.0. The controller is no longer be able to keep A at its theoretical set-point (Eq. 28). When V̇ and k3 are constant an analytical expression of Ass can be derived in good agreement with the numerical calculations (S4 Text): which is analogous to the Ass expression of the motif 1 zero-order controller (Eq. 20). Finally, in phase 3 k3 increases linearly with k̇3=1 together with V̇ =2.0. As indicated by Eq. 29 and shown by the numerical calculations (Fig. 9) the antithetic controller, like the zero-order controller, breaks down and A goes to zero (S4 Text).
Although not shown explicitly here, the following mass balances are obeyed: where ni,0 and ni are respectively the initial number of moles and the number of moles at time t of compound i.
As described above, when using a transporter mediated compensation in A the antithetic and the motif 1 zero-order controller have to increase their controller variables E2 or E in order to keep Ass constant, as indicated by the equation where E(2) represents E2 or E.
Motif 1 autocatalytic controller
Similar to controllers based on double integral action [22] an autocatalytic design [20] is able to keep the controlled species at its set-point even when perturbations become linearly time dependent and rapid [19]. However, in contrast to double integral action the autocatalytic controller is able to compensate for time-dependent perturbations of the form a·tn where n is larger than 1.
Fig. 10 shows the reaction scheme. The controller compound E is produced autocatalytically, i.e., its rate is proportionally to the concentration of E, while M, present in relative large amounts, produces E by an enzyme-catalyzed reaction which is zero-order with respect to M. E increases the activity of transporter T and leads to an increased import of external A into the cell. The negative feedback is closed by an A-induced recycling of E to M. Rate constant k3 represents a variable perturbation which removes A by a first-order process with respect to A. The rate equations are:
For simplicity, in Eq. 33, T and Aext/( +Aext) are set to 1. The last two terms in Eq. 34, 1 × 10−5 − 1 × 10−5 · E, represent small background reactions keeping E at a sufficiently high level such that the autocatalysis in E can start out from zero initial E concentrations (see also Ref. [19]).
To determine the controller’s set-point at constant V and k3 we set Eq. 34 to zero. Neglecting the 1 × 10−5 − 1 × 10−5 · E term and setting V̇ =0, we can solve for the steady state value of A, which defines the controller’s theoretical set-point
Since M/(k5+M)=1 (ideal zero-order conditions), we get from Eq. 36
In the case both V̇ and k̇3 are nonzero and constant, an (analytical) estimation of the steady state in A requires to consider Ä (”acceleration” in A). For constant V̇ and k̇3 values the set-point is calculated to be (S5 Text)
According to previous findings on the autocatalytic controller [19], any time-dependent function k3(t) = k3,0 + a·tn where a, n>0 will lead to the set-point conditions described by Eq. 38 (S5 Text).
The recycling scheme between E and M implies that E and M obey a mass balance of the form with nE (t)=E(t)·V (t), nM (t)=M (t)·V (t), and where nE,0 and nM,0 are the initial number of moles of respectively E and M. The rates how nE and nM change at a given time t are given as (S5 Text)
Fig. 11 shows the results. During the first phase no volume change occurs and k3 is a constant. The controller keeps A at =2.0 as described by Eq. 37. During the second phase both V and k3 increase linearly and the controller still keeps A at =2.0 according to Eq. 38. To keep A at its set-point during increasing V and/or k3 the concentration of E has to increase in order to maintain the steady state condition given by Eq. 33 when Ȧ =0 and V̇/V → 0, i.e.,
From the initial conditions (see legend of Fig. 11) we have that nE (t)+nM (t) = V0 · M0=2.5 × 107.
Motif 2 zero-order controller
The reaction scheme of this controller is shown in Fig. 12. The transporter-based compensatory flux is regulated by E through repression or derepression and E is removed by a zero-order reaction.
The rate equations are
Also here we keep, for the sake of simplicity, T · Aext/( +Aext)=1. In presence of growing V and k3 the motif 2 zero-order controller successfully defends its theoretical set-point given by (S6 Text)
However, since an increase of the compensatory flux is based on derepression by E (decreasing E), the controller will break down when E ≪ k4 and k4/(k4+E)≈1. Neglecting the A·V̇/V term, the point when the breakdown occurs can be estimated by setting Eq. 42 to zero
Fig. 13 shows that the motif 2 based controller is able to defend successfully against linear growth in both V and k3 and keeping A at Prolonged time intervals with increasing V and k3 will lead to controller breakdown when the condition of Eq. 47 is met.
Controllers with transporter-based compensatory fluxes and exponential time-dependent perturbations
Here we describe the performance of the four controller motifs (Fig. 1) with transporter-based compensatory fluxes when exposed to exponential growth, V̇ =k1·V, and an exponential increase in the outflow perturbation rate parameter k3 (Fig. 14).
There are three phases the controllers are exposed to. During the first phase the controllers are at their steady states and V and k3 are kept constant at respectively 25.0 and 2.0. During the second phase V increases exponentially according to V̇ =k1V (k1=0.1), while k3 is kept constant at 2.0. During phase 3, V continues to grow exponentially and k3 starts to increase according to where k3,p3 and tp3 are the values of respectively k3 and time t at the beginning of phase 3.
Fig. 15 shows that only the motif 2 based controller with de-repression kinetics (panel d) is able to counteract both exponential increases in V and k3. However, due to the de-repression kinetics and due to the transporter based kinetics (see Eq. 47) the controller breaks down when the product of the perturbations, k3V reaches k6/ The motif 1 autocatalytical controller (panel c) shows slight constant offsets below , as expected [19], both for the exponential increase of V and when both V and k3 increase exponentially. Since E increases with increasing perturbation strengths the controller is limited by the supply for E via M as indicated in Fig 11. Neither the motif 1 based zero-order controller (panel a) nor the antithetic controller based on motif 1 (panel b) are able to compensate for exponentially increasing perturbation strengths. They behave very similar, as already seen in Figs. 7 and 9 for linear time-dependent perturbations.
Controllers with a cell-internal compensatory fluxes and linear time-dependent perturbations
We consider here the four controllers, but the compensatory fluxes are now generated from cell-internal and homogeneously distributed sources.
Motif 1 zero-order controller
Fig. 16 shows the motif 1 zero-order controller using a cell-internal compensatory flux. The homogenously distributed compound N serves as a source for A, which is activated by E. Compound M serves as a source for E, while by the activation of A, M is recycled from E.
The rate equations are
The steady state of A when both V̇ and k̇3 are constant is given by the following expression (S3 Text)
When k̇3=0 and V̇ =constant Ass becomes =k4/k6 and the motif 1 zero-order controller is able to compensate for a constant growth rate (Fig. 17, phases 1 and 2). However, when k3 increases linearly, Ass is below and remains constant as long as sufficient M and N are present (Fig. 17, phase 3). Thus, in comparison with a transporter-mediated compensatory fluxes, the motif 1 zero-order controller with an internally generated compensatory flux shows an improved performance by being able to compensate for a constant growth rate in the absence of other outflow perturbations in A.
Motif 1 antithetic controller
When the antithetic integral controller is equipped with an internally generated compensatory flux (Fig. 18) its performance towards constant growth and linearly increasing outflow perturbations k3 is significantly improved in comparison with a controller having a transporter generated compensatory flux (Fig. 9). The rate equation for A is now changed to while the other rate equations (Eqs. 22-27) remain the same.
When V̇ is constant Ass becomes (S4 Text)
As indicated by Eq. 56 numerical results show (Fig. 19, phase 2) that the antithetic controller is now able to compensate for linear volume increases by moving A to =(k8/k4). However, an offset in Ass below is observed when, in addition, k3 increases linearly with time, i.e., when k3 is constant.
Although not explicitly shown here, during the volume increase, the mass (mole) balances described by Eqs. 30-31 are obeyed in addition to the mass balance connecting N, A, and Q where nN,0 is the number of moles of initial N at t = 0 with nA,0=nQ,0=0.
Motif 1 autocatalytic controller
Fig. 20 shows the autocatalytic controller but now with an internally generated compensatory flux. As for the motif 1 zero-order controller the compensatory flux originates from compound N, which is present in high concentration and forms A by a zero-order process with respect to N and activated by E.
The rate equation for the controlled variable A is while the rate equations for E and M remain the same as Eqs. 34 and 35. Species P is included with the rate equation to test that the mass (mole) balance between N, A, and P is preserved.
The controller’s steady state in A, Ass, and its theoretical set-point is also in case of a cell internal compensatory flux described by Eq. 38 (S5 Text). In contrast to the other controllers, even when V̇ and k̇3 are constant the autocatalytic controller is able to move A to =(k4/k6) (Fig. 21).
Motif 2 zero-order controller
The rate equations for the motif 2 based controller are
Fig. 23 shows the performance of the motif 2 controller with zero-order integral control. The controller is able to successfully defend against a linear increase in V (phase 2) as well as against linear increase in V and a simultaneous linear increase in k3 (phase 3). For both cases the controller will move A precisely to =k9/k8 without any offset (see S6 Text for details).
Controllers with cell-internal compensatory fluxes and exponential time-dependent perturbations
The controllers are exposed to the exponential perturbation profiles as shown in Fig. 14. The exponential growth of V is written as V̇ = κ · V, where κ (>0) is a constant.
Fig. 24a shows the performance of the motif 1 zero-order controller while Fig. 24b shows the responses of the motif 1 antithetic controller. During exponential growth and constant k3 the motif 1 zero-order and the antithetic controller show slight offsets from the theoretical set-point , while during phase 3 when both V and k3 increase exponentially, both controllers break down. Besides their different kinetic implementation of integral control both the motif 1 zero-order and the modtif 1 antithetic controller show similar responses due to analogous steady stae expressions in A (for details, see S3 Text and S4 Text). Fig. 24c shows the response of the autocatalytic controller, which is able to keep A at during exponential growth while k3 is kept constant. Only when V and k3 both increase exponentially then there is an offset from , which can be estimated analytically from the steady state condition for A where the theoretical set-point =k4/k6 and κ and ζ describe the doubling times ln 2/κ and ln 2/ζ of the exponential increases for V and k3, respectively (see S5 Text).
The motif 2 based controller shows in phase 2 a significant overcompensation from when exposed to exponential growth only. The overcompensated steady state in A at constant k3 and exponential growth can be expressed as where =k9/k8 and (κ/k8)Ess is the overcompensated offset (S6 Text).
The response kinetics of the motif 2 based controller is mostly determined by k4, which reflects the derepression property by E. For large k4 the derepression by E is slow and less effective.
Remarkable, when both k3 and V increase exponentially in phase 3 the controller is able to move A close to For this case Ass can be written as (S6 Text) where and
Note that during phase 3 E is not in a steady state, but decreases due to the controller’s derepression, while k̇3 increases exponentially. However, the derepression kinetics by E is faster than the exponential increase of k̇3 (Eq. 68), such that γ0 increases and and Ass approach (S6 Text). A disadvantage of compensating by derepression is that during rapidly increasing V and k3 E becomes eventually so low that the negative feedback loop cannot be maintained and the controller breaks down even when sufficient N and M are available.
Growth related to the surface to volume ratio
Here we investigate how the four controllers having cell internal compensatory fluxes perform with respect to a surface to volume ratio related growth law as found for spherical bacteria ([13, 14, 17], Eq. 2). We consider again three phases as in the previous sections, but with the difference that V now grows according to Eq. 2 with η=1 and ξ=0.2 (Fig. 25a). The values of η and ξ are arbitrarily chosen. The outflow perturbation, described by k3, is kept constant during phases 2 and 3, but increases during phase 3. The response behaviors of the controllers towards increasing volume (k3 is kept constant) is initially very similar to that when V increases linearly. However, the controllers gain control more and more control with decreasing V̇, provided that there is sufficient material in the cell to generate enough E’ (for the motif 1 controllers) or that there is still sufficient E left (for the motif 2 controller) to keep the negative feedback loop operating.
As an example Fig. 25 shows the behavior of the motif 1 antithetic and autocatalytic controllers and the motif 2 zero-order controller when k3 in phase 3 increases exponentially as described by Eq. 14 and compensatory fluxes are generated cell internally. The motif 1 zero-order controller’s behavior (not shown) is again very similar in comparison with the motif 1 antithetic controller.
Overview of results
Table 1 and Table 2 gives an overview of controller performances. Performances are described by the four categories perfect adaptation, partial adaptation, over-adaptation, and breakdown. Perfect adaptation means that the controller is able to keep A at A controller with partial adaptation can maintain a constant A value during an applied outflow perturbation, but below A controller showing over-adaptation keeps A above even when the perturbation should lead to a decrease in A. Controller breakdown means that the controller is unable to withstand the perturbation and A goes to zero.
Clearly, the motif 1 controllers when integral control is based on zero-order or a bimolecular (antithetic) mechanism cannot oppose an exponential volume increase or an additional exponential increase in k3. The motif 1 autocatalytic controller shows good performances with a constant offset in A below The motif 2 controller using zero-order based integral control shows best performance, is able to maintain A at even when V and k3 increase exponentially. However, the drawback of the motif 2 controller is that it is based on derepression by decreasing E and that the controller breaks down when E becomes too low.
Discussion
From Tables 1 and 2 it is seen that the motif 2 controller outperforms the other controllers. This has already been observed in a previous study [19], where time-dependent inflow and outflow perturbations for A at constant V were applied. However, a clear disadvantage of the motif 2 controller is its breakdown at low E value A somewhat surprising behavior of the motif 2 controller is its over-compensation whe growth increases exponentially at constant k3 (see phase 2 in Fig. 24). The over-compensation can be described analytically (Eq. 69). Its origin appears to be due to the rapid derepression kinetics. Previous results showed that the derepression kinetics are hyperbolic in nature, i.e., they can oppose growth processes with an exponentially increasing doubling time [19].
Performance improvement by increased controller aggressiveness
Although the motif 1 zero-order and the antithetic controllers break down when expose to exponential growth and perturbations (Figs. 15 and 24), their performance can be significantly improved at constant V̇ by increasing of what can be described as the controllers’ aggressiveness. By aggressiveness of a controller we mean loosely the controller’s response to a perturbation in terms of (mainly) quickness and precision. Increasing the aggressiveness of a controller will generally lead to a quicker controller response and an improved controller precision.
The aggressiveness of an integral controller can be varied by the controller’s gain. The gain is a factor in front of the error integral. For an ideal motif 1 zero-order integral controller (working at constant V and k3) Ė is proportional to the error e = ( − A) [7], i.e., where k6 is the controller gain and k4/k6 is the controller’s theoretical set-point, As indicated by Eq. 70 the concentration of E is proportional to the integrated error with respect to time. By increasing k6 and k4 such that remains unchanged the gain of the controller is increased and the controller becomes more aggressive.
For constant V̇ and k3 the steady state of A for the motif 1 zero-order controller is given by Eq. 20 where the offset in Ass below is due to the term 2V̇k3/k2. This term indicates that for increasing V̇ and/or increasing k3 values the controller will break down and A will go to zero as observed in Fig. 15. There are two ways the controller’s aggressiveness can be increased. One way, as indicated above, is by increasing k4 and k6 such that =k4/k6 is preserved with k6 becoming much larger than 2V̇ k3/k2. As a result the controller’s response kinetics become quicker and Ass moves closer to =k4/k6. The other way is to increase k2, which means to increase the activity of the transporter. This could be done by over-expressing the genes which code for the transporter.
Similar arguments apply also for the antithetic controller. Qian et al. [23] have shown that when the controller dynamics become faster than growth this leads to an improved controller performance.
Fig. 26 shows the results of increasing the aggressiveness of the motif 1 zero-order and antithetic controllers by increasing k2 from 1.0 to 1 × 103. The perturbation is divided into three phases. During the first phase the volume V is kept constant at 25.0 and the controllers are at their set-points. In phase 2 the volume increases with a constant rate (V̇ = 1.0). Finally, in phase 3 V continues to grow with V̇ = 1.0 but k2 is increased to 1 × 103. Both controllers show improved precisions, but show different kinetics in their way to reach
Similar is the situation when the compensatory flux is internally generated. Eq. 54 shows the steady state in A for the motif 1 zero-order controller. Also here increasing k2 values will move Ass towards the theoretical set-point =k4/k6.
Roles of kinetic implementations of integral control and negative feedback structures
The increased aggressiveness of the motif 1 zero-order and antithetic controllers allows them to defend their theoretical set-points as long as
However, for exponentially increasing V and V̇ this can be achieved only for a certain (often short) time period. The motif 1 zero-order controller will break down when Eq. 71 is no longer fulfilled. On the other hand, as shown above, the autocatalytic motif 1 controller is able to maintain a stable steady state in A, although with an offset from , when V and k3 increase exponentially. As Eq. 38 (for the transporter-based compensatory flux) indicates, any time-dependent perturbation of the type k3(t) = k3,0 + a·t (a > 0) will be successfully defended by the autocatalytic controller, because k̇3/k3 → 0 and thereby restoring the controller’s theoretical set-point. However, breakdown may occur if no sufficient supply of E (for example via M, Fig. 11) can be maintained.
Our results indicate that the type of kinetics realizing integral control (for example by autocatalysis) and the structure of the negative feedback loop (motifs 1-8, [7]) play essential roles in how a controller will perform. For example, the occurrence of autocatalytic steps/positive feedback loops in signaling and within homeostatic regulated feedback loops are becoming recognized [24–27]. As an illustration, in cortisol homeostasis ACTH signaling from the brain-pituitary system to the cortisol producing adrenals occurs by autocatalysis/positive feedback. In blood sugar homeostasis at low insulin concentrations insulin acts on its own secretion and leads to an autocatalytic production of insulin [27]. These examples indicate the importance of additional “helper kinetics” (such as autocatalysis/positive feedback) to obtain a homeostatic regulation with optimum response and precision properties.
In control engineering the Internal Model Principle [28–30] states that in order adapt to an environmental perturbation the controller needs to have the capability generate that type of perturbation internally. For synthetic biology, this indicates that knowledge about controller kinetics and controller structure (single or combined controller motifs) will be important for succeeding in the design and implementation of artificial regulatory units in order to oppose the dilution effect of growth or other time-dependent perturbations.
Acknowledgments
This research was financed in part by Program Area Funds from the University of Stavanger to PR and TD.