S6 Metabolic cost of the controller. 21 .... where k, η are the controller parameters and ν, θ define the desired set-point for the intracellular level of biofuel at ...
Perfect adaptation and optimal equilibrium productivity in a simple microbial biofuel metabolic pathway using dynamic integral control – Supplementary information Corentin Briat and Mustafa Khammash
Contents S1 Biofuel production model and controllers S1.1 Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S1.2 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S1.3 Antithetic integral controller . . . . . . . . . . . . . . . . . . . . . . . . . S1.3.1 Reaction network and model . . . . . . . . . . . . . . . . . . . . . S1.3.2 Potential implementation of the controller and related discussions S1.3.3 Properties of integral controllers . . . . . . . . . . . . . . . . . . . S1.3.4 Structural comparison with the standard integral controller . . . . S1.3.5 Comparison with proportional and Hill-type controllers . . . . . .
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2 2 2 3 3 4 4 5 5
S2 Preliminaries S2.1 Closed-loop system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S2.2 Equilibrium point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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S3 Local stability analysis of the S3.1 Preliminary results . . . . . S3.2 Main result . . . . . . . . . S3.3 Numerical results . . . . . .
equilibrium . . . . . . . . . . . . . . . . . . . . . . . .
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point 7 . . . . . . . . . . . . . . . . . . . . . . . . . . 7 . . . . . . . . . . . . . . . . . . . . . . . . . . 9 . . . . . . . . . . . . . . . . . . . . . . . . . . 12
S4 Analysis of the tracking and perfect adaptation properties S4.1 Perfect adaptation with respect to system parameters . . . . . . . . . . . . . . . . . . . . S4.2 Perfect adaptation with respect to (local) constant inputs . . . . . . . . . . . . . . . . . S4.3 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13 13 13 15
S5 Optimal biofuel production S5.1 Preliminaries . . . . . . . . . . . . . S5.2 Optimal set-point and its properties S5.3 The productivity-toxicity formula . S5.4 Fundamental optimality theorem .
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S6 Metabolic cost of the controller 21 S6.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 S6.2 Main result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 S7 Sensitivity analysis with respect to implementation errors in µ
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S8 Robustness analysis
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S8.1 Theoretical analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 S9 Approximate optimality and robustness S9.1 Approximate optimality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S9.2 Approximate robustness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S9.3 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
24 24 25 25
S10 Discussion about model modifications S10.1 Capturing the dynamics of the RNA polymerase . . . S10.2 Capturing the unbinding of the σ-factors . . . . . . . S10.3 Capturing the dilution of the σ-factors . . . . . . . . S10.4 Effect of the saturation of protein pump production . S10.5 Effect of the saturation of the measurement reaction
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References
S1
28
Biofuel production model and controllers
We describe here the network and its model as well as the controller that will be considered. Some additional discussions are also provided.
S1.1
Network
The biofuel production model proposed in [8] can be represented by the following set of reactions: N P +N
α n
n −−−→ αn np p+γp
−−−→ u
∅ −−−→
2N , P,
N P
α n2
n −−−→ N,
Bi + N
α n
b −−−→ N + Bi ,
βp
−−−→ ∅,
Bi + N P + Bi + N
δ nb
n i −−−→
δb pbi
−−−→
Bi P
V δb pbi n
−−−→
P + N + Be (S1.1) where the propensity functions (note the rates) of each reaction are given just above the arrows. Above, N corresponds to the (abstract) species associated with the cell-density, Bi to the species associated with the intracellular biofuel, P to the species associated with the pump proteins and Be to the species associated with the extracellular biofuel.
S1.2
P,
2N
Model
The (deterministic) model corresponding to the above reaction network is given by n˙
=
αn n(1 − n) − δn bi n −
b˙ i p˙ b˙ e
= = =
αb n − δb pbi u − βp p V δb pbi n
αn np p + γp (S1.2)
where n is the normalized cell density, bi is the intracellular level of biofuel, p is the concentration of efflux pump proteins, u is the control where n is the normalized cell density, bi is the concentration of intracellular biofuel, p is the concentration of efflux pump proteins, u is the rate of production of pump proteins, which will also be our control input, and be is the extracellular concentration of biofuel. The parameter αn is the specific growth rate of the cells, δn is the biofuel toxicity coefficient, αb is the biofuel production coefficient, δb is the biofuel export rate per pump, βp is the pump protein degradation rate, V is ratio of intra to extracellular volume and γp is the constant that sets the pump toxicity threshold. These parameters have been fitted to experimental data in previous works (see [8] and references therein) and their estimated values are given in Table 1.
2
Parameters αn δb
Table 1: Parameters of the model (S1.2) taken from [8] Values Parameters Values Parameters 0.66 [1/h] δn 0.91 [1/(Mh)] αb 0.5 [1/(Mh)] βp 0.66 [1/h] γp
Values 0.1 [1/h] 0.14 [M]
It is interesting to note the following limitations captured by the model. First of all, high concentrations of intracellular biofuel and protein pumps is detrimental to cell density because of toxicity. This is modeled by the two last terms in the dynamical model of the cell density. Secondly, high concentrations of pumps will act negatively of the concentration of biofuel and, therefore, limit the efficiency of the production of extracellular biofuel which is proportional to the product pbi . Finally, a high cell-density will result in a larger production of intracellular biofuel. All these interactions are summarized in Fig. S1. Due to the quite intricate pattern of interactions between the different components of the network, if one wants to control and optimize the production of biofuel while limiting, at the same time, the toxicity to the cell, a control scheme is clearly needed. Various control schemes have been proposed in [8], namely 1) by the constitutive production of pumps, 2) by a biofuel-responsive (i.e. negative Hill feedback) controller, 2) by a repressor cascade and 4) by a feedforward loop. Several properties for the controllers are studied in [8], notably, their efficiency in terms of the overall biofuel production, their overall toxicity and their sensitivity with respect to the parameters of the system. We propose here to use another type of controllers, that we refer to as antithetic integral controllers, which have been introduced in [4] in the stochastic setting.
Figure S1: Summary of the interactions in the open-loop network (S1.1).
S1.3 S1.3.1
Figure S2: Summary of the interactions in the closed-loop network (S1.1)-(S1.3).
Antithetic integral controller Reaction network and model
We propose the following reaction network controller consisting of the propensity function u = kz2 and the reactions η θ ν Bi −−−→ Z2 + Bi , ∅ −−−→ Z1 , Z1 + Z2 −−−→ ∅ (S1.3) where Z1 and Z2 are the controller species. The first reaction is the measurement reaction that catalytically creates one molecule of species Z2 from Bi with rate θ > 0. The second one is the reference reaction that implements part of the set-point value µ = ν/θ > 0 for the concentration of the molecules of the species Bi . The third reaction is the comparison reaction that correlates the species Z1 and Z2 u and closes the overall control loop. Finally, by combining the reaction ∅ −−−→ P in (S1.1) and the propensity function u = kz2 , we get the following actuation reaction k
Z2 −−−→ Z2 + P
(S1.4)
that acts on the system in order to control it. The topology of the interconnection is depicted in Fig. S2. We can see that there is a negative feedback loop involving the species Z1 , Z2 , Bi and P , as it is customary in control theory when regulation is the main goal; see e.g. [2, 7].
3
The model corresponding to the reaction network (S1.3) is given by z˙1 z˙2 u
= ν − ηz1 z2 , = θbi − ηz1 z2 , = kz2
(S1.5)
where k, η are the controller parameters and ν, θ define the desired set-point for the intracellular level of biofuel at equilibrium. All these parameters are degrees of freedom that can be tuned in order to ensure certain desired properties for the closed-loop network consisting of the interconnection of the open-loop network (S1.1) and the controller network (S1.3)-(S1.4). The associated model is hence given by (S1.2)-(S1.5). It is interesting to note that even though the above controller is of the class of antithetic integral controllers, the interconnection topology slightly differs from the one in [4] where the actuation reaction was k Z1 −−−→ Z1 + P . (S1.6) In other words, the species Z2 plays the role of both the sensing and the actuating species in the present paper whereas it was only the sensing species in [4], Z1 being the actuating one. The reason for this difference comes from the fact that the (local incremental gain) gain of the system is negative while it was positive in [4]. This can be understood the following way: increasing u will increase the concentration of pump proteins which will, at the same time, 1) decrease the cell-density by toxicity and hence decrease the production rate of the intracellular biofuel as it is proportional to the cell-density and 2) directly increase the extraction rate of the intracellular biofuel. The fact that the gain is negative will be theoretically proved in Proposition S3.3 in the next section. S1.3.2
Potential implementation of the controller and related discussions
The crucial element of the antithetic integral controller is the comparison reaction. In this regard, any proper implementation should rely upon molecules that can automatically implement this reaction (through their natural affinity). As already emphasized in [4], σ-factors and anti σ-factors, which are naturally present in bacteria, present such a complementarity property and spontaneously strongly bind with each other. By identifying Z1 with an anti-σ-factor and Z2 with the associated σ-factor, the comparison reaction will be automatically and implicitly implemented. In this regard, it seems reasonable to propose an implementation based on those factors; see Figure S3. In this setup, the reference reaction which constitutively produces anti σ-factor molecules can be easily implemented. Moreover, this reaction can be made easily tunable using chemical induction or optogenetics techniques. The measurement reaction can be approximately implemented through the use of a biosensor that will sense the quantity of intracellular biofuel and produce a measure of this quantity through the production of a transcription factor that will trigger the production of σ-factor molecules. Finally, the actuation reaction can be implemented by adjoining to the pump gene a promoter domain which is specific to the holoenzyme complex formed by a RNA polymerase molecule and a σ-factor molecule. Note that even if the actual implementation is more complex than the elementary reactions in (S1.3)(S1.4), this implementation is quite faithful. It will indeed be shown later in Section S10 that the unmodeled elements have very low impact on the conclusions of the paper.
S1.3.3
Properties of integral controllers
Integral controllers [1] have been widely used in many engineering disciplines such as electrical and mechanical engineering, and even biology [5, 6], as they can be used to ensure the tracking of the output of a given system to a specific set-point (or reference). This control structure has also been shown to be naturally present in living organisms and to be related to some perfect adaptation schemes; see e.g. [9, 14, 17, 18, 20]. A key feature of this control scheme is the so-called constant disturbance rejection property, which essentially means that if a constant signal is applied to the output or the input of the system, the integral controller will react in a way that drives back the output of the system to the desired set-point.
4
Figure S3: Practical implementation of the antithetic integral controller using σ-factors and anti σfactors (denoted by σ ¯ ). The reference reaction is implemented as the constitutive production of anti σ-factor molecules. This reaction can be tuned using chemical inducers or optogenetics. The production of σ-factor molecules is activated by the intracellular biofuel, possibly through the use of an intermediary biosensor. The comparison reaction is automatically realized through the complementarity property of the σ-factor and the anti σ-factor molecules. The free σ-factor molecules can then bind to the RNA polymerase to form a holoenzyme complex that specifically binds to the promoter region of the pump genes to initiate transcription.
S1.3.4
Structural comparison with the standard integral controller
The considered integral controller is a not a standard integral controller in the sense that it is not linear. Indeed, a general form of the standard integral controller is given by I˙ = θbi − ν, u = kI
(S1.7)
where I is the state of the integrator, θ can be seen as some sort of sensor gain, ν is part of the set-point µ = ν/θ and k is the gain of the controller. Note that these parameters are playing the exact same roles as in (S1.5). Note also that a slightly less general version of the controller where θ = 1 and ν = µ is usually considered in the literature. There are reasons for considering (S1.5) instead of (S1.7). The first one is that we need an integral controller that can be implemented in terms of chemical reactions, that is, whose representation is in terms of nonnegative variables and realistic propensities; i.e. mass-action or Hill kinetics. The second one is that we need a control input u(t) to be nonnegative at all times. These constraints are clearly not satisfied by the controller (S1.7) as 1) the term −ν clearly arises from a 0th-order reaction that is not mass-action nor Hill-type and 2) because both I and u in (S1.7) may take negative values. Yet, these controllers are very similar as the dynamical expression in (S1.7) can be retrieved from (S1.5) by considering the change of variables I = z2 − z1 . Note, however, that the expressions for the control input u remain quite different.
S1.3.5
Comparison with proportional and Hill-type controllers
The Hill-type controllers considered in [8] and proportional controllers are structurally not able to achieve perfect adaptation; see e.g. [2, 7]. However, it was demonstrated in [8] that it may still be possible to approach a target behavior through an appropriate tuning of the controller parameters. Unfortunately, this behavior will not be robust with respect to parametric and model uncertainties since suitable values for the controller parameters (i.e. those for which we can approach the desired behavior for the closedloop network) will depend on the model parameters, which are are uncertain by definition. The same problem will also arise in the presence of disturbances where the closed-loop system may fail to exhibit any good adaptation properties. Finally, the possible interplay between the different parameters of the controller network may result in additional difficulties in the tuning task. These problems do not arise when the integral controller (S1.7) is considered. Indeed, the integral action will automatically ensure that perfect adaptation holds in the presence of parametric uncertainties, 5
model uncertainties and constant disturbances, provided that the closed-loop system is asymptotically stable. The tuning part is also simplified thanks to the small number of free parameters. Finally, it seems important to mention that, when dilution terms are considered and, a fortiori, the controller species dilute, then the integral action will be lost – which will result in a steady-state tracking error. The influence of this error on the performance of the process can be easily analyzed using a sensitivity analysis; this will be detailed in Section S7.
S2
Preliminaries
In this section, we show the following results. First, we show that there exists a unique equilibrium point in the positive orthant for the closed-loop network provided that the set-point µ = ν/θ is achievable and we give a simple characterization of all the achievable set-points. We also show that given an achievable set-point µ = ν/θ, we can find parameters k, η > 0, such that the equilibrium point of the closed-loop network is locally asymptotically stable.
S2.1
Closed-loop system
The closed-loop system obtained from the interconnection of the system (S1.2)-(S1.5) is given by αn np n˙ = αn n(1 − n) − δn bi n − p + γp b˙ i = αb n − δb pbi (S2.1) p˙ = kz2 − βp p z˙1 = ν − ηz1 z2 z˙2 = θbi − ηz1 z2 . In the following, we will denote the full state of the above model by x := (n, bi , p, z1 , z2 ) for compactness. Note that we have dropped the extracellular biofuel be from the state as it does not contribute to the dynamics of the other states.
S2.2
Equilibrium point
We establish here the existence of a unique equilibrium point for the closed-loop network (S2.1). Proposition S2.1. The closed-loop system (S1.2)-(S1.5) has a unique equilibrium point in the positive orthant if and only if the growth-toxicity condition 0 0. (S3.5) αb −δb µ 0 (p∗ + γp )2 0 −βp 1 Therefore, H(0) < 0. This proves the result. The following result is a technical result that will be needed to prove the main stability result: Proposition S3.4. For any ν, θ > 0 such that ν/θ ∈ (0, ρ), there exists a k > 0 such that the matrix M (k, θ) is Hurwitz stable. Moreover, when k > 0 is chosen as such, then the matrix M (k, θ) − $e4 eT4 is Hurwitz stable for all $ ≥ 0. Proof. First note that when ν/θ ∈ (0, ρ), then Σ is Hurwitz stable and hence it has nonzero eigenvalues. Hence, M (0, θ) is marginally stable with a single eigenvalue at 0. The associated normalized left- and right-eigenvectors are given by 0 −θCΣ−1 u= and v = . (S3.6) 1 1 8
Hence, under an infinitesimal perturbation of the parameter k in the direction dk ∈ R, the zero-eigenvalue T 0 Bdk of M (0, θ) moves in the direction u v = −θCΣ−1 Bdk = θdk H(0); see e.g. [16]. Since H(0) < 0 0 0 then by choosing dk > 0, the zero-eigenvalue of M (0, θ) will be perturbed negatively, which will stabilize the matrix. Therefore, we can conclude that for any sufficiently small k > 0, the matrix M (k, θ) is Hurwitz stable. This proves the first statement of the result. To prove the second statement, assume that k > 0 is such that M (k, θ) is Hurwitz stable and note that P$ (s) := det(sI − M (k, θ) + $e4 eT4 ) = sD(s) − kθN (s) + $D(s) (S3.7) where the polynomials D(s) := det(sI − Σ) and N (s) := D(s)H(s). We know that the polynomial P0 (s) is Hurwitz stable since it is the characteristic polynomial of the matrix M (k, θ). By continuity of the eigenvalues, we also know that the polynomial P$ (s) remains Hurwitz stable for any sufficiently small $ > 0. We also know that since Σ is Hurwitz stable then so is the polynomial D(s). We now use a root locus argument and note that the relative degree of the transfer function D(s) sD(s) − kθN (s)
(S3.8)
is equal to 1. Hence, the only asymptotes for the single infinite zero is along the real axis and travels towards −∞. This implies that, as $ increases from 0 to ∞, 3 stable roots of P0 (s) converge to the stable roots of D(s) while the remaining one move towards −∞ along the real axis. This proves that the matrix M (k, θ) − $e4 eT4 is Hurwitz stable for all $ ≥ 0.
S3.2
Main result
We have the following result regarding the positive-density equilibrium point: Theorem S3.5. Assume that the growth-toxicity condition (S2.2) holds. If µ, k, θ > 0 are such that the matrix M (k, θ) is Hurwitz stable then for any η > 0, the equilibrium point given by ∗ δb p ν ν ∗ νk βp p∗ x∗ = , (S3.9) , ,p , θαb θ ηβp p∗ k where p∗ is the unique positive solution of the quadratic equation αn δb µ ∗ 2 αn γp δb − (p ) − µ + δn p∗ + γp (αn − δn µ) = 0 αb αb
(S3.10)
is locally exponentially stable. On the other hand, if µ, k, θ > 0 are such that M (k, θ) is not Hurwitz stable, then there will exist an ηc > 0 for which the system undergoes a Hopf bifurcation whenever η > ηc ; i.e. a (stable or unstable) limit cycle arises. Proof. Part 1. We prove first that, under the assumptions of Theorem S3.5, the matrix J (k, η) is Hurwitz stable for any sufficiently small η > 0. To this aim, note that the matrix −αn n∗ αb 0 J(k, 0) := 0 0
−δn n∗
−
−δb p∗ 0
αn γp n∗ (p∗ + γp )2 −δb µ −βp
0 0 0
0
0
0
θ
0
0
9
0
0 k νk − βp p∗ νk − βp p∗
(S3.11)
νk e4 eT4 . From Proposition S3.4, this βp p ∗ matrix is Hurwitz stable. Therefore, J(k, 0) is marginally stable with 0 as the only eigenvalue on the imaginary axis. The left- and right-eigenvectors associated with the zero-eigenvalue are given by αb ∗ αn n 0 ∗ αn n αb γp 0 − + δ µ b (p∗ + γp)2 βp and v = (S3.12) u = −ζ −1 0 ∗ αn p∗ 1 α b γp ∗ 1 −n + δ µ + (α δ + α δ p ) b b n n b ν (p∗ + γp )2 θ 0 n∗ (αb δn + αn δb p∗ ) θ has the eigenvalue 0 and the eigenvalues of the matrix M (k, θ) −
where ζ = n∗
αn p∗ ν
1 αb γp ∗ + δ µ + (α δ + α δ p ) b b n n b (p∗ + γp )2 θ
(S3.13)
is a normalizing constant such that uT v = 1. Hence, when η is perturbed in the direction dη , then the zero eigenvalue of J(k, 0) moves in the direction 0 0 ∗ βp p∗ n∗ dη αn p∗ αb γp ∗ T β p d p η + δb µ . (S3.14) u − 0 0 0 1 0 v=− ζk ν (p∗ + γp )2 β pk∗ d p η − k Hence, when dη > 0, the above perturbation direction for the zero eigenvalue is negative which implies that under the assumptions of the theorem, then there exists a sufficiently small ηm > 0 for which the matrix J(k, η) is Hurwitz stable for all η ∈ (0, ηm ). This concludes the first part of the proof. Part 2. We prove now that, under the assumptions of Theorem S3.5, the matrix J (k, η) is Hurwitz stable for any sufficiently large η > 0. We first note that eigenvalues bifurcation directions and the stability of the matrix J(k, η) are invariant with respect to any positive scaling of the matrix J(k, η). To this aim, let us consider the matrix J˜ (k) := J(k, −1 ) = J(k, η)/η where = η −1 . Now, we would like to perturb the matrix when transitions from zero to a small positive value. First note that the matrix 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 J˜0 (k) = (S3.15) βp p∗ 0 0 0 − 0 k βp p∗ 0 0 0 0 − k is marginally stable with a semi-simple eigenvalue at 0 of multiplicity 4. The associated left- and righteigenvectors are given by 0 1 0 0 1 0 0 0 0 0 1 0 0 0 1 0 0 0 T and v = 0 0 1 0 u = (S3.16) . 0 0 1 0 0 0 0 0 0 0 0 0 −1 1 0 0 0 1 Note that we have uT v = I and hence the eigenvectors are normalized. From [16], we know that the zero eigenvalues of J˜0 (k) bifurcate under the action of a small > 0 in the direction given by the eigenvalues
10
of the matrix −αn n∗ −δn n∗ −δb p∗ αb 0 0 uT 0 0 0 θ
−
αn γp n∗ (p∗ + γp )2 −δb µ −βp
0 0 0
0
0
0
0
0
∗ 0 −αn n k v= αb νk 0 − βp p ∗ 0 νk − βp p ∗
−δn n∗ −δb p∗ 0 θ
−
αn γp n∗ (p∗ + γp )2 −δb µ −βp 0
0 0 = M (k, η) k 0
(S3.17) which is Hurwitz stable by assumption. Hence, we can conclude that there exists a sufficiently small c > 0 for which the matrix J˜ (k) is Hurwitz stable for all 0 < < c . This is then equivalent to saying that for any ηM = −1 < η < ∞, the matrix J(k, η)/η, or equivalently, the matrix J(k, η) is Hurwitz c stable. Part 3. We prove now that, under the assumptions of Theorem S3.5, the matrix J (k, η) is Hurwitz stable for all intermediary values for η > 0. The characteristic polynomial of J(k, η) is given by αn γp n∗ −αn n∗ −δn n∗ − ∗ 0 0 (p + γp )2 0 0 −δb p∗ −δb µ αb 0 −βp 0 k 0 P (s, η) := det sI − ∗ ηβp p νk 0 0 − − 0 ∗ k βp p (S3.18) ηβp p∗ νk − − 0 θ 0 ∗ βp p k ηβp p∗ νk s + k βp p∗ = det(sI − Σ) det ηβp p∗ νk s+ − kθH(s) ∗ k βp p where Σ is the upper-left 3 × 3 submatrix of J(k, η) and H(s) := C(sI − Σ)−1 B. Then, we have that ηβp p∗ νk D(s) − kθN (s) + (sD(s) − kθN (s)) (S3.19) P (s, η) = s s + ∗ βp p k where D(s) = det(sI − Σ) and N (s) = H(s)D(s). Note that the only zero of P (s, 0) in the closed right-half plane is 0 and we know that, from the first part of the proof that there exists an ηm > 0 such that for all η ∈ (0, ηm ), the polynomial P (s, η) is Hurwitz stable. In a similar way, we know from the second part of the proof that there exists an ηM > 0 such that for all η ∈ (ηM , ∞), the polynomial P (s, η) is Hurwitz stable as well (note also, for completeness, that sD(s) − kθN (s) is the characteristic polynomial of M (k, θ)). To prove that the polynomial P (s, η) is Hurwitz stable for all η > 0, we use a root locus argument [7, 10, 11] and consider the transfer function βp p ∗ (sD(s) − kθN (s)) . νk k s s+ D(s) − kθN (s) βp p ∗
(S3.20)
Clearly, the above transfer function has relative degree one since the numerator has degree 4 whereas the denominator has degree 5. Hence, the only asymptote (in the root locus argument) is the real axis and the infinite zero of the above function is equal to −∞. Consequently, the polynomial P (s, η) and, equivalently, the matrix J(k, η) is Hurwitz stable for all η > 0 provided that k and θ are such that the matrix M (k, θ) is Hurwitz stable. This proves the stability result. Part 4. We prove now the existence of a Hopf bifurcation for some η = ηc when µ, k, θ > 0 are such that M (k, θ) is not Hurwitz stable. This can be achieved by showing that the matrix 11
J(k, η) transitions to instability through a pair of complex conjugate eigenvalues. This is equivalent to saying that the transition to instability never occurs through the zero eigenvalues and hence that the matrix M (k, θ) remains invertible for all η > 0. Noting again that the matrix J(k, η) is affine in η we can use the determinant formula to get that det(J(k, η)) = det(J(k, 0)) − η where uT = 0 0 0 1 0 , v = 0 0 0 Since J(k, 0) is singular, then we have that
βp p ∗ T u Adj(J(k, 0))v k
(S3.21)
1 and where Adj(J(k, 0)) is the adjugate of J(k, 0).
1
det(J(k, η)) = −η
βp p ∗ T u Adj(J(k, 0))v. k
(S3.22)
Clearly, the only way to make this expression equal to 0 would be to have η = 0 or uT Adj(J(k, 0))v = 0. Let ηc be the value for which J(k, ηc ) has its rightmost eigenvalues on the imaginary axis. Clearly, we need that uT Adj(J(k, 0))v = 0 for J(k, η) to be singular. However, this cannot be the case since we know that regardless the value of k, the matrix J(k, η) is Hurwitz stable for some sufficiently small η, then we must have that det(J(k, η)) < 0 for some sufficiently small η and, hence, this implies that uT Adj(J(k, 0)) > 0. As a consequence, the Jacobian matrix transitions to instability through a pair of complex conjugate eigenvalues indicating then the emergence of a stable or an unstable limit cycle for the closed-loop dynamics. The proof is completed.
S3.3
Numerical results
Using the numerical values in Table 1, we find that ρ = 0.7252. For µ = 0.1762, ν = µθ, we find that the matrix M (k, θ) is Hurwitz stable provided that k < kmax (µ, θ) = 0.2193θ. The evolution of the eigenvalues of J(k, η) for θ = 1 and k = kmax (µ, 1)/2 = 0.2193/2 = 0.1096 is depicted in Fig. S4 whereas the case k = 3kmax (µ, 1)/2 = 0.3289 is depicted in Fig. S5. 0.2 0.15
=(6(J(k; 2)))
0.1 0.05 0
-0.05 -0.1 -0.15 -0.2 -1
-0.9
-0.8
-0.7
-0.6
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
Figure S4: Evolution of the eigenvalues of J(k, η) with θ = 1, ν = 0.1762, k = kmax (µ, 1)/2 = 0.1096 when η moves from 0 (circles) to infinity (crosses). We can see that, in this case, the crosses are all located in the open left half-plane, emphasizing that for this value of k, the eigenvalues J(k, η) are stable for all η > 0. 12
0.25 0.2 0.15
=(6(J(k; 2)))
0.1 0.05 0
-0.05 -0.1 -0.15 -0.2 -0.25 -1
-0.9
-0.8
-0.7
-0.6
-0.5
-0.4
0 det(J(k, η)) (p + γp )2 −[J(k, η)−1 ]24 .
Therefore, we have that −1 ˜ CJ(k, η) = 0 0 0
αn n∗ ηβp p∗ det(J(k, η))
δb µ +
αb γp ∗ (p + γp )2
αn n∗ ηβp p∗ − det(J(k, η))
αb γp δb µ + ∗ (p + γp )2
(S4.7)
(S4.8)
and the conclusion follows.
S4.3
Numerical results
We illustrate here the theoretical results obtained above. The responses of the closed-loop to positive and negative constant disturbances on the control input u are depicted in Fig. S6 whereas the behavior of the disturbed control input itself is depicted in Fig. S7. Finally, the responses of the system when subject to sudden changes in the parameter values are depicted in Fig. S8.
15
Figure S6: Response of bi (t) to a constant input disturbance with initial condition x(0) = (0.01, 0, 0, 0, 0), θ = 0.3, ν = 0.0529 (i.e. µ = 0.1762), k = 0.12 and η = 0.1. The blue curve corresponds to an input disturbance of 0.05 whereas the red one to an input disturbance of −0.05. Relatively to the unperturbed control input kz2∗ , the value ±0.05 corresponds to a variation of approximately ±40%.
Figure S7: Response of u(t) to a constant input disturbance with initial condition x(0) = (0.01, 0, 0, 0, 0),, θ = 0.3, ν = 0.0529 (i.e. µ = 0.1762), k = 0.12 and η = 0.1. The blue curve corresponds to an input disturbance of 0.05 whereas the red one to an input disturbance of −0.05. Relatively to the unperturbed control input kz2∗ , the value ±0.05 corresponds to a variation of approximately ±40%.
16
A
B
C
D
Figure S8: Response of bi (t) to a variation in the parameter αb (A), δb (B), γp (C) and βp (D) at t = 200h with initial condition x(0) = (0.01, 0, 0, 0, 0), θ = 0.3, ν = 0.0529 (i.e. µ = 0.1762), k = 0.12 and η = 0.1. The red curve corresponds to a variation of −50% of the parameter while the blue one corresponds to a variation of +100% of the parameter. In all these scenarios, the process exhibits perfect adaptation, thereby demonstrating the importance of integral control.
S5
Optimal biofuel production
The performance measure considered in [8] is the concentration of extracellular biofuel at some finite time T > 0. Unfortunately, this measure may be highly biased as it is sensitive to initial conditions and the transient phase of the system. This is the reason why we propose to consider instead the rate of fuel production at equilibrium as performance measure. This quantity will be referred to as the equilibrium productivity. This equilibrium productivity can be related to the standard notions of productivity and titer (see e.g. [19]) widely used in metabolic engineering to evaluate the efficiency of metabolic networks. We show that the equilibrium productivity is a concave function in the variable µ, a property that directly implies that there exists a unique global maximum. We provide a closed-form expression for the value of µ that yields this maximum. We also unveil a conservation relation between the equilibrium toxicity and the equilibrium productivity that is intrinsic to the system and is independent of the controller.
S5.1
Preliminaries
Let us define the productivity and the toxicity as follows: Definition S5.1 (Instantaneous productivity and instantaneous toxicity). The instantaneous productivity P (t) and the instantaneous toxicity T (t) are defined as P (t) := V δb bi (t)p(t)n(t) and T (t) := Tb (t) + Tp (t)
17
(S5.1)
where Tb (t) := δn bi (t) is the instantaneous biofuel toxicity and Tp (t) :=
αn p(t) is the instantaneous p(t) + γp
pump toxicity. It is interesting to note that the above instantaneous toxicity concept coincides with the toxicity concept considered in [8]. We also introduce the equilibrium versions of the above notions: Definition S5.2 (Equilibrium productivity and equilibrium toxicity). The equilibrium productivity Pe∗ (µ) and the equilibrium toxicity Te∗ (µ) are defined as Pe∗ (µ) := V δb b∗i p∗ n∗ =
αn p∗ αn p∗ V 2 2 ∗ 2 δb µ (p ) and Te∗ (µ) := δn b∗i + ∗ = δn µ + ∗ . αb p + γp p + γp
(S5.2)
Note that when the positive-density equilibrium point is locally asymptotically stable and the system trajectories converge to it, then we get that Pi (t) → Pe∗ (µ) as t → ∞. Additionally, we also have the following relationship (assuming be (0) = 0) Z 1 T be (T ) T →∞ Pi (s)ds −−−→ Pe∗ (µ) (S5.3) = T T 0 which reconciles the concepts of titer be (T ) (here be (T ) ≈ T Pe∗ (µ) for any sufficiently large time T ), productivity be (T )/T , instantaneous productivity Pi (t) and equilibrium productivity Pe∗ (µ) all together. An immediate advantage of the equilibrium productivity is that it only depends on the set-point µ which makes it generic (no dependence on the transient behavior nor initial conditions of the model) and easy to analyze since it has a closed-form solution, as shown below: Proposition S5.3. The following expression holds Pe∗ (µ) =
V ˜∗ 2 P (µ) αb e
where
s
(S5.4)
2 αb µ2 γp δb − + 4αb γp µδb . ρ
(S5.5)
= δb b∗i p∗ q 2 −µ(αn γp δb + αb δn ) + µ2 (αn γp δb − αb δn ) + 4αb γp αn2 µδb = δb µ 2αn δb µ q 2 −µ(αn γp δb + αb δn ) + µ2 (αn γp δb − αb δn ) + 4αb γp αn2 µδb = s 2αn 2 µ αb 1 αb 2 = − γp δ b + + µ γp δb − + 4αb γp µδb . 2 ρ 2 ρ
(S5.6)
µ P˜e∗ (µ) := δb b∗i p∗ = − 2
αb γp δ b + ρ
1 + 2
Proof. We have that P˜e∗ (µ)
The proof is complete.
S5.2
Optimal set-point and its properties
We have the following result about the optimal set-point that maximizes the productivity of the process: Proposition S5.4. The value for µ that maximizes the equilibrium productivity Pe∗ (µ) is given by µopt :=
αb δb γp ρ3 1/2
(αb
1/2
+ (δb γp ρ)1/2 )2
18
(S5.7)
and the maximal value for the productivity is given by ∗ Pmax
V := αb
αb δb γp ρ 1/2
(αb
!2
+ (δb γp ρ)1/2 )2
(S5.8)
Proof. Using standard algebra, we get that the values of µ for which the derivative of P˜e∗ (µ)2 with respect to µ is zero are given by µ1 = 0 1/2 αb δb γp ρ3 µ2 = 1/2 (S5.9) (αb + (δb γp ρ)1/2 )2 3 1/2 − αb δb γp ρ . µ3 = 1/2 (αb − (δb γp ρ)1/2 )2 Clearly, the first value does not yield maximum since, in this case, we have Pe∗ (0) = 0, and the last one is negative. The only potential candidate for yielding a maximum is the second one. The evaluation of the Hessian at that point gives −4(δb αb γp )1/2 (S5.10) 1/2 ρ1/2 (αb + (δb γp ρ)1/2 )2 which is negative. Therefore, this is a maximum. To verify that the maximum is global, we simply have to note that Pe∗ (0) = Pe∗ (ρ) = 0. This completes the proof. We have the following interesting properties: Proposition S5.5. The following properties hold: ∗ 1. Pmax and µopt are related to each other by the relationship ∗ Pmax V δ b γp = . µ2opt ρ
(S5.11)
2. For any given ρ > 0, we have that µopt ≤ µmax opt :=
αb 4δb γp
(S5.12)
which is attained whenever αb − δb γp ρ = 0. 3. The following bound µopt ≤ µρ :=
ρ 4
(S5.13)
holds and is attained whenever αb − δb γp ρ = 0. Proof. The proofs are as follows: 1. The proof of the first statement follows from simple substitutions. 2. To prove p the second statement, let us first consider the bijective change of variables X := Y := δb γp ρ where ρ is assumed to be fixed. Then, we have that µopt =
XY ρ =: ρF (X, Y ). (X + Y )2
19
√
αb ,
(S5.14)
We then have the following expressions for the gradient and the Hessian of the function: Y (Y − X) Y (X − Y ) , ∇F (X, Y ) = X +Y X +Y 2Y (2Y − X) −(X − Y )2 + 2XY − (X + Y )4 (X + Y )4 ∇2 F (X, Y ) = 4 . −(X − Y )2 + 2XY 2X(2X − Y ) − (X + Y )4 (X + Y )
(S5.15)
We can then clearly see that the gradient is 0 along the line X = Y . Along X = Y , the Hessian becomes 1 −1 1 (S5.16) 8X 2 1 −1 and has one 0 eigenvalue and one negative eigenvalue equal to −1/(4X∗2 ). The eigenvector associ T ated with the 0-eigenvalue is 1 1 , which means that the value does not change along the line T X − Y = 0. The eigenvector associated with the negative eigenvalue is 1 −1 and is perpendicular to the other. Along this eigenvector the value of the function decreases. So, the function F (X, Y ) takes its maximum value on the line X − Y = 0 or, equivalently, when αb = δb γp ρ. Moreover, this maximum is given by 1/4, which leads to the inequality (S5.12). 3. To prove the inequality (S5.13), first note that √ p 4µopt αb , δb γp ρ = 4F ρ
(S5.17)
where the function F is defined in (S5.14). We just saw that this function is maximum on the line X = Y and, in such a case, we have that 4F (X, X) = 1, which proves the result.
S5.3
The productivity-toxicity formula
We provide an interesting intrinsic conservation relation of the system that is independent of the controller. Let us recall that Pe∗ (µ) is the equilibrium productivity whereas Te∗ (µ) is the equilibrium total toxicity. We then have the following result: Proposition S5.6. The conservation relation Te∗ (µ) + αn
Pe∗ (µ) V αb
1/2 =1
(S5.18)
holds for all 0 < µ < ρ. Proof. First note that, from the first equation of (S2.6), we have that 1 − n∗ −
p∗ δn µ − =0 αn γp + p∗
(S5.19)
Te∗ (µ) =0 αn
(S5.20)
Te∗ (µ) . αn
(S5.21)
and, therefore, that 1 − n∗ − or, equivalently,
n∗ = 1 −
Using now the fact that, from the second equation of (S2.6), we have that Pe∗ (µ) =
V (δb p∗ b∗i )2 αb 20
(S5.22)
then, we get that
we finally get that
α
b
V
Pe∗ (µ)
1/2
α 1/2 b ∗ αb n∗ = δb p∗ b∗i = Pe (µ) (S5.23) V T ∗ (µ) = αb 1 − e , which yields the result. The proof is complete. αn
This formula is very insightful as it clearly states that when the equilibrium productivity Pe∗ (µ) increases, then the equilibrium toxicity Te∗ (µ) necessarily decreases. Even more interestingly, it also says that when the equilibrium productivity Pe∗ (µ) is maximum, the associated equilibrium toxicity Te∗ (µ) is minimum. It is finally important to stress that this formula is only valid at equilibrium.
S5.4
Fundamental optimality theorem
We saw that the equilibrium productivity and the equilibrium toxicity obey a conservation relation. The following result summarizes and unifies all the previously mentioned stability and optimality results. Theorem S5.7. Assume that µ = ν/θ = µopt and that θ, k > 0 are such that the matrix Σ Bk Cθ 0
(S5.24)
is Hurwitz stable. Then, for all η > 0, the positive-density equilibrium point of the closed-loop network is locally exponentially stable and the controlled biofuel process exhibit tracking and perfect adaptation. Moreover, in this particular scenario, we have that 1. the equilibrium productivity Pe∗ (µopt ) is maximum; 2. the equilibrium normalized cell density n∗ (µopt ) is maximum; 3. the equilibrium total toxicity Te∗ (µopt ) is minimum;
Proof. For the second statement, we have used the expression n∗ =
1 ∗ P (µopt ) and the productivityαb e
toxicity formula (S5.18).
S6 S6.1
Metabolic cost of the controller Preliminaries
An interesting way to evaluate the metabolic cost induced by the introduction of the additional reactions is to evaluate the long-run time-averages of the propensity functions associated with the controller reactions. Clearly, four reactions are involved, namely, the reference reaction ν
∅ −−−→ Z1 , the measurement reaction
θ
Bi −−−→ Z2 + Bi , the comparison reaction η
Z1 + Z2 −−−→ ∅, and the actuation reaction
k
Z2 −−−→ Z2 + P to which we associate the positive elementary metabolic costs κr , κm , κc and κa , respectively. The total energy consumed until time t is given by Z t Z t Z t Z t E(t) = κr νds + κm θbi (s)ds + κc ηz1 (s)z2 (s)ds + κa kz2 (s)ds (S6.1) 0
0
0
21
0
where κr , κm , κc and κa (unit is [J/M] (joule per molar)) are the elementary metabolic costs associated with the reference reaction, the measurement reaction, the comparison reaction and the actuation reaction, respectively. The key idea behind the definition for the energy is that each time a reaction fires some energy is consumed under the form of ATP or the use of intermediary resources (this is modeled by the elementary metabolic costs). However, since we are in the deterministic setting, the reactions fire in a continuous fashion but the value of their propensity function is proportional to how often they fire. In this regard, the integral of the propensities denotes a quantity that is analogous to the number of firing times up to time t. However, since the energy grows without bound as time grows, it is not relevant to look at directly and rather we look at its rate of variation, called the instantaneous power, hence defined as dE(t) (S6.2) Wi (t) = dt and, more especially, its equilibrium value We∗ = lim Wi (t) when it exists. Having low equilibrium t→∞ power is equivalent to saying that the consumed energy does grow slowly and, hence, that the metabolic cost of the reactions is reduced.
S6.2
Main result
We then have the following result: Proposition S6.1. Assume that the positive equilibrium point of the closed-loop network is asymptotically stable. Then, the equilibrium power consumption of the controller network is given by We∗ = (κr + κm + κc )ν + κa βp p∗ > `∗ (µ)
(S6.3)
where `∗ (µ) := κa βp p∗ , the so-called constitutive limit or static limit, is a decreasing function of µ. Proof. The proof of the expression (S6.3) follows from simple substitutions. To show that `∗ (µ) is decreasing, let us prove that p∗ (µ) is a decreasing function of µ. To this aim, let us rewrite the polynomial (S2.5) as the multivariate polynomial Z(p, µ) = −σ1 µp2 − µσ2 p + γp (αn − δn µ)
(S6.4)
where σ1 = αn δb /αb > 0 and σ2 = δn + θγp > 0. Clearly, we have that Z(p∗ (µ), µ) = 0 for any µ ∈ (0, ρ). The implicit function theorem yields dp =− dµ
dZ dp
−1
dZ σ1 p2 + σ2 p + γp δn =− 0.
24
7 16
(S9.3)
1/2
Proof. Using the change of variables X := αb be written as
∗ Pmax =
and Pe∗ (µρ ) = hence
and Y := (δb ργp )1/2 , the above performance levels can V X 4Y 4 X 2 (X + Y )4
i2 p V h −(Y 2 + X 2 ) + (Y 2 − X 2 )2 + 16X 2 Y 2 2 64X
i2 h p (X + Y )4 −(Y 2 + X 2 ) + (Y 2 − X 2 )2 + 16X 2 Y 2 Pe∗ (µρ ) G(X, Y ) := = . ∗ Pmax 64X 4 Y 4
(S9.4)
(S9.5)
(S9.6)
Note that
(r + 1)4 (r2 − (r4 + 14r2 + 1)1/2 + 1)2 (S9.7) 64r4 so the function is constant on all lines in the plane (X, Y ) going through the origin and G(X, X) = 1; i.e. the error is 0 on the line X = Y . The function G(X, Y ) is symmetric since G(X, Y ) = G(Y, X). The derivative of the function g(r) can be shown to only vanish at r = 1, which is obviously a global maximum. Hence, the function is monotonically decreasing as r increases. Taking now the limits as r goes to 0 and infinity we get that G(X, rX) = g(r) :=
lim g(r) = lim g(r) =
r→0
S9.2
r→∞
9 ≈ 0.5625. 16
(S9.8)
Approximate robustness
Interestingly, since µρ only depends on ρ, then in the uncertain case where ρ ∈ [ρ− , ρ+ ], we simply have to implement the value ρ− /4 (see Theorem S8.1). In this case, we will simply have at most a deterioration of 43.75% of the best worst-case performance given by (S8.5).
S9.3
Numerical results
Using the numerical values of Table 1, we find that r = 0.7125 which implies that ρ < 0.0005 < 0.01, meaning that we lose much less than 1% of the performance.
S10
Discussion about model modifications
We discuss here on the impact of model variations on the conclusions of the paper.
S10.1
Capturing the dynamics of the RNA polymerase
As the dynamics of the RNA polymerase has been neglected in the model (S1.5), it seems important to briefly analyze the impact of these additional dynamics on the closed-loop system and its performance. We assume here that free RNA polymerase is abundant in the cell and cannot be sequestered away from the σ-factors we are using in the circuit. Under this assumption, we can then neglect the competition between all the σ-factors present in the cell for the RNA polymerase. In this regard, the controller model is given by z˙1 = ν − ηz1 z2 z˙2 = θbi − ηz1 z2 − ξz2 + ζz3 (S10.1) z˙3 = ξz2 − ζz3 u = kz3 where z3 denotes the concentration of the holoenzyme complex consisting of a RNA polymerase molecule and a σ-factor molecule bound together. The dynamics of the free RNA polymerase is neglected since it 25
is assumed that it is abundant. In this regard, its concentration (assumed then to be relatively constant) is incorporated in the rate constant ξ. Note also that the control input is now the holoenzyme complex as it is the molecule that will bind the promoter region of the pump genes. We can clearly see that, at equilibrium, we have that ξz2∗ − ζz3∗ = 0 and, hence, ν − θb∗i = 0. In this regard, the integral action is preserved and all the conclusions of the paper remain valid. Only the stability conditions will be different but since the additional dynamics are stable, no dramatic changes in the theoretical stability results are expected.
S10.2
Capturing the unbinding of the σ-factors
Even though the dissociation constant of the σ-factors and anti σ-factors is small (approximately 0.01nM [3, 15]), it is still not zero. Hence, it seems interesting to study the effect of the dissociation of the complexes consisting of σ-factors and anti σ-factors. In this case, the model of the controller is given by z˙1 z˙2 z˙3 u
= = = =
ν − ηz1 z2 + ξz3 θbi − ηz1 z2 + ξz3 ηz1 z2 − ξz3 − ζz3 kz2
(S10.2)
where z3 denotes the complex formed by a pair of σ- and anti σ-factors which is subject to some degradation with rate ζ (which is necessary for stability otherwise the states of the controller will grow without bound as no degradation will be present). In this case, again, the integral action is preserved and all the conclusions of the paper remain valid. Only the stability conditions will be different but since the additional dynamics are stable, no dramatic changes in the theoretical stability results are expected.
S10.3
Capturing the dilution of the σ-factors
We argue here that the degradation of the σ-factors and the anti σ-factors destroys the integral action and will lead to a steady-state error. Note that degradation has a stabilizing effect and, hence, the closed-loop network will be necessarily stable if the closed-loop network controlled with the antithetic integral controller is stable. In this case, the model of the controller is given by z˙1 = gν − ηz1 z2 − ζz1 z˙2 = gθbi − ηz1 z2 − ζz2 u = kz2
(S10.3)
where ζ is the degradation/dilution rate of the σ-factor and the anti σ-factor molecules and g is an additional gain. In this case, we have the following equilibrium relationship g(ν − θb∗i ) − ζ(z1∗ − z2∗ ) = 0.
(S10.4)
We now prove by contradiction that tracking is generally not achieved when degradation/dilution is present. To this aim, assume that tracking is achieved (i.e. ν − θb∗i = 0). This then implies that z1∗ = z2∗ = z ∗ and, thus, that η(z ∗ )2 + ζz ∗ − ν = 0. The only positive solution then only depends on the controller parameters η, ζ and ν, and so is the value of the control input u∗ = kz2∗ . Consequently, the value of the control input is independent of the parameters of the biofuel model and, hence, tracking cannot be achieved since a necessary condition for tracking is that the control input depends on (some of) the parameters of the system we have to control. Hence, tracking is not achieved, yielding a contradiction. The consequence of that is that a steady-state error will arise, the impact of which on the equilibrium productivity can be analyzed using the sensitivity analysis developed in Section S7. Note, however, that for the current set of parameters, this degradation is pretty low. Finally, it is interesting to point out that the error can be reduced by using a high annihilation constant η and a high-gain g. Indeed, in the limit η → ∞, the equilibrium point satisfies g(ν − θb∗i ) + ζz2∗ = 0.
(S10.5)
We can see now that in the limit g → ∞, we will have that b∗i = ν/θ, showing then that dilution can be compensated by considering a high-annihilation constant η and a high-gain g. 26
S10.4
Effect of the saturation of protein pump production
Let us now assume that the activation of pump genes saturates and we model this as u=u ¯
kz2 , γu + kz2
(S10.6)
where u ¯ is the saturating value for the rate of production of pump proteins and γu is the saturation threshold. The effect of the saturation is summarized in the following result: Proposition S10.1. Assume that u is constant. Then, the equilibrium value bi (u) for the intracellular biofuel concentration is given by bi (u) =
αn βp2 αb γp (u + βp γp )(αn δb u + βp αb δn )
(S10.7)
and is a monotonically decreasing function of u. If 0 < u < u ¯, then we get that the equilibrium values for bi are necessarily contained in the interval ! αn βp2 αb γp ,ρ . (S10.8) E := (¯ u + βp γp )(αn δb u ¯ + βp αb δn ) Moreover, if µopt ∈ E, then the set-point for which the equilibrium productivity is maximum is µopt . Otherwise, a suitable one is given by αn βp2 αb γp + (¯ u + βp γp )(αn δb u ¯ + βp αb δn ) where is any sufficiently small positive real number. Proof. Assume that u is constant. Hence, the equilibrium value for p is given by p(u) = u/βp . Substituting this expression in the dynamics of n and bi yields the system of linear equations αn βp γp αn δn n(u) (S10.9) = u + βp γ p βp αb −δb u bi (u) 0 and we get that bi (u) =
αn βp2 αb γp . (u + βp γp )(αn δb u + βp αb δn )
(S10.10)
Notably, bi (u) is monotonically decreasing and we have that bi (0) = ρ and bi (u) → 0 as u → ∞. If we have the restriction u < u ¯ then we have that the equilibrium values for bi are necessarily contained in the interval E; i.e. bi (¯ u) < bi (u) < ρ. This proves the first part of the result. To prove the second part, it is enough to remark that if the set-point µopt is achievable; i.e. µopt ∈ E, then there is no problem. On the other hand, if the set-point is not achievable, then from the fact that the productivity curve is decreasing over µ ∈ (µopt , ρ), then we need to pick the smallest value for the set-point to maximize the equilibrium productivity. However, since this is a supremum (the maximum is attained for the value u ¯ for the control input, which is not achievable), we need to pick a slightly larger value. This proves the second part of the result. Finally, it is worth mentioning that the impact of this saturation on the stability of the closed-loop network will be rather minor.
S10.5
Effect of the saturation of the measurement reaction
Assume here that the propensity function of the measurement reaction is not linear anymore but saturates and takes the form θbi /(γb +bi ) where γb is the saturation threshold. In this case, the controller expression becomes z˙1 = ν − ηz1 z2 θbi − ηz1 z2 z˙2 = (S10.11) γ b + bi u = kz2 . 27
Hence, the following expression holds at equilibrium µ=
ν b∗i = θ γb + b∗i
ρ ν µ0 , for some . In this regard, choosing := γb + ρ θ γb + µ0 µ0 ∈ (0, ρ), leads to b∗i = µ0 . Hence, a suitable set-point for b∗i can be enforced provided that we can µ0 which may be a problem when γb is poorly characterized. In this regard, a compute the ratio γb + µ0 steady-state error will likely occur in this case because of a poor implementation of the set-point value – an issue that was addressed using a sensitivity analysis in Section S7. The impact of this saturation on the stability of the closed-loop network will also be pretty minor as in the previous cases. and can only have a solution if 0 < µ
