Global dynamics of the chemostat with overflow metabolism

Fast growing E. coli cells, in glucose-aerobic conditions, excrete fermentation by-products such as acetate. This phenomenon is known as overflow metabolism and has been observed in a diverse range of microorganisms. In this paper, we study a chemostat model subject to overflow metabolism: if the substrate uptake rate (or the specific growth rate) is above a threshold rate (different from zero), then secretion of a by-product happens. We assume that the presence of the by-product has an inhibitory effect on the growth of the microorganism. The model is described by a non-smooth differential system of dimension three. We prove the existence of at most one equilibrium (or steady-state) with presence of microorganism, which is globally stable. We use these results to discuss the performance of chemostat-type systems to produce biomass or recombinant proteins.


Introduction
Escherichia coli (E. coli) is a bacterium that is naturally found in the intestine of humans and other mammals. This bacterium has been a preferred choice for largescale production of recombinant proteins 1 such as insulin, GFP (green fluorescent protein), or the human growth hormone (Baeshen et al. 2015;Huang et al. 2012). For high density cultivation of E. coli, glucose is generally the preferred and most 1 Recombinant proteins are proteins that are artificially made through the recombinant DNA technology. Jean-Luc Gouzé jean-luc.gouze@inria.fr common carbon and energy source (Bren et al. 2016), since this is inexpensive and readily utilizable. To harvest energy from glucose, E. coli combines two different metabolic strategies, aerobic respiration, which needs oxygen, and fermentation, which does not need oxygen Gerosa et al. (2015). Respiration is more energy-efficient than fermentation, nevertheless, in fast growing cells, some energy is also obtained by fermentation. This seemingly wasteful strategy in which cells use fermentation instead of respiration, even in the presence of oxygen, is known as overflow metabolism (Basan et al. 2015). This phenomenon is not only limited to E. coli, but to a diverse range of microorganisms (Vazquez 2017). For example, in yeasts, overflow metabolism is known as Crabtree effect (Dashko et al. 2014), and in cancer cells it is known as Warburg effect (Kim and Dang 2006).
Overflow metabolism results in the secretion of fermentation by-products, such as acetate in E. coli cultures or ethanol in yeast cultures, which accumulation can have an inhibitory effect on cells growth. For example, glucose uptake is inhibited in E. coli and yeast cultures in presence of acetate (Luli and Strohl 1990) and ethanol (Liu and Wu 2008) respectively. Moreover, the formation of these by-products constitutes a diversion of carbon that might have contributed to biomass or protein synthesis. Thus, overflow metabolism can pose a major problem in large-scale production of biomass or recombinant proteins (Eiteman and Altman 2006;Hensing et al. 1995).
Cultivation of E. coli, yeasts, and other microorganisms can be done in a chemostat. The chemostat, introduced in the 1950s independently by Monod (1950) and Novick and Szilard (1950), is a perfectly mixed reactor, permanently fed with a nutrient rich medium and simultaneously emptied so that the culture volume is kept constant. Using the chemostat is a way to maintain indefinitely a non-zero growth rate, and therefore to study the organisms under various constant growth rates. The classical chemostat model describes the dynamics of a single population with growth limited by a single nutrient. We refer the reader to Smith and Waltman (1995) and Ajbar and Alhumaizi (2011) for the theory of the chemostat and for different variations of the classical chemostat model. Yano and Koga (1973), Xiu et al. (1998), Heßeler et al. (2006), and Harvey et al. (2014) studied the dynamics of chemostat models with the production of a toxic by-product. Xiu et al. (1998) described the production of the byproduct as a consequence of overflow metabolism. However, in all these works the authors assume that the secretion of the by-product occurs at any growth rate while experimental evidence shows that by-product secretion does not take place at low growth rates Basan et al. (2015).
In this paper, we study the long-term behavior of a chemostat model accounting for the following features of overflow metabolism: -secretion of a by-product when the substrate uptake rate is above a threshold; -biomass loss due to secretion of the by-product; -inhibition of substrate uptake in presence of the by-product.
The model is mainly inspired by the recently proposed model by Mauri et al. (2020) that describes the growth of an E. coli culture producing a recombinant protein. In contrast to our model, Mauri et al. (2020) consider growth on the by-product (acetate) and an additional variable describing the dynamics of a recombinant protein concentration. E. coli consume acetate only after the glucose (substrate) is totally consumed, phenomenon known as carbon catabolite repression (Wolfe 2005). Thus, given the continuous supply of substrate in chemostats, we neglect the consumption of the byproduct in our model. Note that carbon catabolite repression is also observed in yeasts (Gancedo 1998). With respect to the recombinant protein concentration, in the discussion section we show that our results can be easily extended when considering the dynamics of a recombinant protein.
The chemostat model with overflow metabolism is described by an autonomous system of ordinary differential equations. Using a conservation principle, the model can be reduced to a planar system. Thus, we study the dynamics of the planar system by finding appropriate invariant sets and using results on cooperative systems (Smith 2008). To extend our results to the original model, we use the well known Theorem of Butler-McGehee Smith and Waltman (1995). This technique requires the stability of equilibria, which may be difficult to obtain due to the non-smoothness of the byproduct excretion rate function (overflow metabolism). This situation is treated with classical results of the theory of differential equations such as the comparison method (Coppel 1965).
This paper is organized as follows. In Sect. 2, we describe the chemostat model and the main hypotheses. Sections 3 and 4 are devoted to the mathematical analysis of the model. In Sect. 3, we characterize the existence of equilibria and their local stability. In Sect. 4, we present the results on the global behavior of the model. The main result is given in this section (Theorem 1). In the last section, Sect. 5, we begin presenting a brief summary of our mathematical results. Then, we finish the paper with a discussion on the steady state production of biomass and recombinant proteins in chemostat-type systems.

Chemostat model
We consider a chemostat (see Fig. 1) with a single population of microorganisms whose concentration is denoted by x. This population grows at a specific growth rate μ(·). The specific growth rate considers the carbon gain by substrate uptake and the carbon loss due to metabolic overflow i.e.
where r S is the substrate uptake rate, r of is the metabolic overflow rate (or by-product formation rate), and Y S , Y R are yield coefficients. Following Basan et al. (2015), when  By-product excretion rate (r of ) as a function of the substrate uptake rate (r S ) r S is higher than a threshold rate r S0 , then the excretion of by-product occurs at a rate proportional to the difference between r S and r S0 i.e. r of = f (r S ) with f defined as (see Fig. 2): with k > 0. The substrate uptake rate r S is a function of the substrate bulk concentration (S) and the overflow metabolism by-product (R) i.e. r S = r S (S, R). We assume that r S is continuously differentiable for all S, R ≥ 0 and that: Thus, the by-product R has an inhibitory effect on the substrate uptake rate. An example for r S is given by (see Mauri et al. 2020): where r S,max is the maximal substrate uptake rate, K S is a half saturation constant, and K i,R is an inhibition constant. The chemostat is fed at a rate F > 0 with a substrate concentration S in . The dilution rate is defined as D = F/V , with V the volume of the culture. Mass balance equations lead to: Model (4) is that of a standard chemostat with a single species with growth limited by a single substrate, with the added feature that a by-product is produced as a consequence of overflow metabolism.
Throughout the paper we assume: This assumption implies that the growth rate function μ is strictly increasing in S and strictly decreasing in R. This follows directly from noting that: Assumption (4) is satisfied by the parameters given by Mauri et al. (2020). We also assume that In the long-term operation with presence of microorganisms, the substrate concentration in the medium cannot be higher than S in . Then, if r S (S in , 0) ≤ r S0 , overflow metabolism is not possible in the long-term, and the study of the dynamics of (4) is reduced to that of a classical chemostat model. Recalling the definition of μ and combining (4) and (5), we have the following inequality: This inequality allows us to consider dilution rates between Y S r S0 and μ(S in , 0). As we will show in the next sections, in the long-term operation, only when Y S r S0 < D < μ(S in , 0) there is presence of the by-product in the culture.
As expected, the domain of biological interest, that is R 3 The conservation principle for chemostats is satisfied by the variable We can rapidly verify that In view of the definition of W we have Since W (t) → Y S S in as t → ∞, we conclude that (4) is dissipative i.e. solutions of (4) are attracted by the bounded set [0,

Existence of steady states and local stability
Equation (4) admits at most two equilibria. A trivial equilibrium corresponds to the absence of microorganisms. It is given by and it always exists. The other possible equilibrium is characterized by the presence of microorganisms. The presence of the by-product depends on the dilution rate. The following proposition formally characterizes the existence of this equilibrium.
Proposition 1 (Existence of the non-trivial equilibrium)

then (4) has no equilibrium with presence of microorganisms.
Proof Assume that D ≤ Y S r S0 . In this case, any positive steady state of (4) has no by-product. Indeed, by contradiction, if (x * , S * , R * ) is a positive steady state of (4) with R * > 0, then r of (S * , R * ) = D R * /x * > 0 (from the third equation in (4)). Thus, from the first equation in (4) we obtain: which contradicts the fact that (x * , S * , R * ) is a positive steady state. Hence, any positive steady state of (4) has the form (x * , S * , 0). As in a classical chemostat model (note that S −→ μ(S, 0) is strictly increasing), (4) admits a unique positive steady state if μ(S in , 0) > D, and has no positive steady states if μ(S in , 0) ≤ D. Now assume that D > Y S r S0 . In this case, the by-product is present in any positive equilibrium of (4). Indeed, by contradiction, if (x * , S * , 0) is a positive steady state of (4), then r of (S * , 0) = 0. But we have Y S r S (S * , 0) = D > Y S r S0 , which implies r S (S * , 0) > r S0 , and hence r of (S * , 0) > 0, which is a contradiction. Then, we look for positive steady states (x * , S * , R * ) with R * > 0. If R * > 0, then r of (S * , R * ) > 0. Thus, we study the following system of equations: From the two first equations in (10), we obtain that: Combining the three equations in (10), we obtain that x + Y S S + Y R R = Y S S in (conservation principle, see (6)). Combining this equation with (10), we obtain: Combining (10) and (11) with the first equation in (10), we obtain the following equation for x: Since f is strictly decreasing, It remains to prove the inequalities in (a). Assume that μ(S in , 0) > D and let (x * , S * , R * ) be the unique positive steady state of (4). If D ≤ Y S r S0 , then R * = 0, hence r of (S * , 0) = 0 (from the third equation in (4)). This implies that The following result shows that the equilibrium with presence of microorganisms is locally stable when D = Y S r S0 .
Proposition 2 (Local stability of E * ) Assume that D < μ(S in , 0) and let E * be the non-trivial equilibrium given by Proposition 1. If D = Y S r S0 , then E * is locally stable.
Proof If D > Y S r S0 , according to Proposition 1, R * > 0 and r of (S * , R * ) > 0. Thus, we can study the local stability of E * in the following system: The Jacobian matrix associated with (15) and evaluated at E * is: It is clear that one eigenvalue of J is −D. The other two eigenvalues are those of the matrix: We note that: Thus, it is easy to verify that T r(J 1 ) < 0 and that det(J 1 ) > αr S0 x ∂r S ∂ S > 0. This implies that both eigenvalues of J 1 have negative real part. Thus, E * is locally stable.
If 0 < D < Y S r S0 , according to Proposition 1, R * = 0 and r of (S * , 0) = 0. Thus, we can study the local stability of E * in the following system: The Jacobian matrix associated with (17) and evaluated at E * is: As in the previous case, one eigenvalue of J 2 is −D. The other two eigenvalues are those of the matrix: .
It is clear that T r(J 2 ) < 0 and det(J 2 ) > 0. Hence, both eigenvalues of J 2 have negative real part. Thus, E * is locally stable.

Global behavior and main result
In this section, we aim to prove that if (4) admits an equilibrium with presence of microorganisms, which is unique according to Proposition 1, then any solution to (4) approaches it asymptotically, provided a positive initial population. The first result in this section shows the existence of two positively invariant sets, which will be repeatedly used in this section.

Lemma 1 (Positively invariant sets)
(a) The set Proof We have that the variable V := x + Y S S satisfies the following differential equation: The proof of (a) follows from the fact that dV dt V =Y S S in ≤ 0 and R 3 + is positively invariant. For (b), let us consider the variable y = r S (S, R). Then we have: Since Ω 1 is positively invariant, it is enough to show that dy Since ∂r S (S,R) This completes the proof.
The following result shows that if there is no equilibrium with presence of microorganisms (i.e. μ(S in , 0) ≤ D), then the population goes to extinction.
Proof Let (x,S,R) be a solution of (4), and letṼ =x + Y SS . We have that We can easily verify that : (Smith and Waltman 1995). Then, applying Theorem B.1 from Appendix B in Smith and Waltman (1995), we conclude thatx Again, due to the cooperativity of (19) we conclude that (x,V ) approaches an equilibrium asymptotically. Since the unique equilibrium of (19) is (0, Y S S in ), we conclude that (x,V ) approaches (0, Y S S in ) asymptotically. From (20),x approaches 0 asymptotically. Now, noting thatS(t) ≤Ṽ (t)/Y S for all t ≥ 0, we have that Let V be the unique solution of: satisfying V (0) =Ṽ (0). Thus, by a comparison theorem argument, we conclude that 0)x(t) → 0 as t → ∞, we can apply Theorem 1.2 in Thieme (1992) and conclude that V approaches Y S S in asymptotically. Now, since V (t) ≤Ṽ (t) ≤V (t) for all t ≥ 0, we conclude thatṼ approaches Y S S in asymptotically. Finally, consider the variableW =x + Y SS + Y RR . In view of (6), W converges to Y S S in . Consequently,R =W −Ṽ Y R converges to 0, and the proof is complete.
In view of (6), the solutions of (4) approach the hyperplane: The set Ω is positively invariant with respect to (4). This implies that the dynamics of solutions starting in Ω correspond to that of a two-dimensional system. The following two results describe the dynamics of any solution starting in Ω.
The following result describes the global behavior of solutions of (4) starting on Ω.
According to Proposition 1, if D = Y S r S0 , then the positive equilibrium E * = (x * , S * , 0) satisfies r S (S * , 0) = r S0 and the function r of is not differentiable at (S * , 0). This poses a problem for the study of the local stability of E * , and consequently for the application of classical arguments (e.g. Butler-McGehe Theorem) to extend Proposition 4 to any initial condition. The following result considers this particular case.
Proposition 5 Assume that μ(S in , 0) > D and let E * be given by Proposition 1. If D = Y S r S0 , then E * is stable.

Proof Let ξ(t) = (x(t), S(t), R(t)
) be a solution of (4) with x(0) > 0 and S(0), R(0) ≥ 0, and let Ω 1 and Ω 2 be the sets defined in Lemma 1. Consider the following sets: Given sets A, B ∈ {Ω 1 , Ω 2 , Ω 2 }, we will say that ξ moves from A to B, if there are t ≥ 0 and τ > 0 such that ξ(t) ∈ A for all t ∈ (t − τ, t ), ξ(t ) ∈ A ∪ B, and ξ(t) ∈ B − A for all t ∈ (t , t + τ ). This means that if ξ moves from A to B, then there is a time when ξ is in A and then later is in B but not in A. Since Ω 1 and Ω 2 are positively invariant (see Lemma 1), ξ can only move from Ω 1 to Ω 2 , from Ω 1 to Ω 2 , or from Ω 2 to Ω 2 . Hence, ξ has one of the following global behaviors: ξ starts on Ω 1 and moves either to Ω 2 or to Ω 2 , (c) ξ starts on Ω 2 and moves to Ω 2 , (d) ξ starts on Ω 1 , then moves to Ω 2 , and then to Ω 2 .
Let > 0 be given. We have to prove the existence of a δ > 0 such that in any situation listed above, if ||ξ(0) − E * || < δ then ||ξ(t) − E * || < for all t ≥ 0. We only give the proof in the situation d) because the proof in the other situations is almost the same. Thus, let us assume the existence of t 1 , t 2 > 0 such that t 1 < t 2 and ξ(t) ∈ Ω 1 for all t ∈ [0, t 1 ), ξ(t) ∈ Ω 2 for all t ∈ [t 1 , t 2 ), and ξ(t) ∈ Ω 2 for all t ≥ t 2 . For all t ≥ t 2 , ξ(t) can be seen as a solution of (14). In such a case, we can study the Jacobian matrix of (14) evaluated at E * (as done in the proof of Proposition 2) to conclude the existence of δ 2 > 0 such that ||ξ(t) − E * || < for all t ≥ t 2 provided ||ξ(t 2 ) − E * || < δ 2 . Now for all t ∈ [t 1 , t 2 ), ξ(t) can be seen as a solution of (17). In such a case, we can study the Jacobian matrix of (17) evaluated at E * to conclude the existence of δ 1 > 0 such that ||ξ(t) − E * || < δ 2 /2 for all t ∈ [t 1 , t 2 ) provided ||ξ(t 1 ) − E * || < δ 1 . Finally, for all t ∈ [0, t 1 ), consider the variables V = x + Y S S and W = x + Y S S + Y R R. It is clear that: Using the definition of Ω 1 and (28) we obtain: Again, using the definition of Ω 1 and (7), we obtain that We note that g is strictly increasing in v and strictly decreasing in  (29) and (30), we conclude that |x(t) − x * | < , Choosing an appropriate , and writing S and R in terms of x, V , and W , we can find Since ξ is continuous, we conclude that ||ξ(t) − E * || < for all t ≥ 0 provided ||ξ(t) − E * || < δ( ).
Proof Part (b) follows directly from Proposition 3. For (a), let (x, S, R) be a solution of (4) with x(0) > 0, S(0), R(0) ≥ 0. Let us write P = (x(0), S(0), R(0)). In view of (7), we have that ω(P) ⊂ Ω, where ω(P) denotes the ω-limit set of P and Ω is defined in (23). From Proposition 4, the ω-limit set of any trajectory passing through Ω is either E 0 or E * . Consequently, The Jacobian matrix associated with (4) and evaluated at E 0 is: It is clear that J has two negative eigenvalues and one positive eigenvalue. Let Ω 0 be the two-dimensional subspace spanned by the eigenvectors corresponding to the negative eigenvalues i.e. Ω 0 := {0} × R 2 + . It is clear that Ω 0 is positively invariant and that any solution starting on Ω 0 approaches E 0 asymptotically. Since Ω 0 is a manifold trivially tangent to Ω 0 at 0, we conclude that Ω 0 is the stable (global) manifold of (4) at E 0 (see Chapter 2.7 in Perko 2013). Since P / ∈ Ω 0 , we have that ω(P) = {E 0 }. Now, let us assume that E 0 ∈ ω(P). According to the Theorem of Butler-McGehee (see for example page 12 in Smith and Waltman 1995), ω(P) intersects Ω 0 in a point other than E 0 . The (whole) trajectory of that point, say (0, S 0 , R 0 ), is given by It is clear that γ is unbounded (as t → −∞). Consequently, ω(P) contains an unbounded trajectory. However, ω(P) is a bounded set because the solutions to (4) are ultimately bounded (see Lemma 3.1.2 in Hale 2010). This contradiction implies that E 0 cannot be in ω(P). Hence, from (31), we conclude that E * ∈ ω(P). From Propositions 2 and 5 we have that E * is stable, hence ω(P) = {E * }. This completes the proof.

Summary of our mathematical results: survival, extinction, and stability
The chemostat with overflow metabolism, described by (4), admits at most two equilibria. An extinction equilibrium, denoted by E 0 = (0, S in , 0), that corresponds to the absence of microorganisms and always exists. The other possible equilibrium, denoted by E * = (x * , S * , R * ), is characterized by the presence of microorganism i.e. x * > 0. Our main result (Theorem 1), states that if E * exists, then any solution to (4) with a positive initial population approaches (asymptotically) E * . That is, given a solution (x(t), S(t), R(t)) of (4) with x(0) > 0, we have that On the other hand, the non-existence of E * implies that any solution to (4) approaches the extinction equilibrium asymptotically: meaning that lim t→∞ x(t) = 0. Proposition 1 in Sect. 3 gives necessary and sufficient conditions for the existence of E * . Indeed, E * exists if and only if μ(S in , 0) > D. The survival of microorganisms (existence of E * ) does not ensure the presence of the overflow metabolism by-product in the medium. According to Proposition 1, R * > 0 if and only if Y S r S0 < D < μ(S in , 0).
Overflow metabolism, and the consequent presence of a by-product, does not generate multistability. That is, if E * exists, there are no solutions with positive initial population converging to E 0 . Xiu et al. (1998) observed the multiplicity of stable steady states. However, apart from taking r S0 = 0, they assume that excess of substrate inhibits the growth rate. Thus, the existence of multiple steady states is due to substrate inhibition and not to overflow metabolism.

Acetate formation and productivity in E. coli cultures
In E. coli cultures, the by-product corresponds to acetate. According to Proposition 1, the presence of acetate in the non-trivial equilibrium E * depends on the dilution rate. This is illustrated in Fig. 3a. Indeed, in presence of bacteria, R * > 0 if and only if D > r S0 Y S . This relation between the acetate steady state concentration and the dilution rate has been observed experimentally by El-Mansi and Holms (1989). This may suggest an optimal operation of the chemostat at dilution rates lower than Y S r S0 to avoid the presence of acetate in the culture. Indeed, different authors have shown that preventing acetate formation in fed-batch leads to higher density cultures (Korz et al. 1995;Babu et al. 2000). To evaluate this strategy in chemostat cultures, let us consider the (steady state) productivity defined as P * = Dx * , with x * the steady state concentration of bacteria at the dilution rate D. P * quantifies the biomass that is produced per unit of time at steady state. To determine P * numerically, let us assume that r S is given by (2), and consider the parameters estimated by Mauri et al. (2020) (see Table 1). Figure 3b shows that the steady state productivity is maximal at a value of the dilution rate higher than Y S r S0 (continuous line). This suggest that preventing acetate formation is not a good strategy in chemostat cultures, in contrast to fed-batch cultures. The veracity of this observation depends on the choice of parameters. For instance, for low values of K i,R (strong inhibition), the maximal productivity is reached at D = Y S r S0 (see dashed line in Fig. 3).
As shown in Fig. 3b (continuous line), maximal productivity of the system is accompanied by the secretion of acetate. A natural strategy to increase this maximal productivity is removing acetate from the culture during fermentation. This can be done with a dialysis reactor (Nakano et al. 1997), or with macroporous ion-exchange resins (Huang et al. 2012). However, these methods tend to remove nutrients that are necessary for cell growth. A promising alternative consists in introducing an additional E. coli strain (a cleaner), which has been metabolically engineered to consume acetate. Thus, two different E. coli populations coexist in the culture: one producing biomass, and one reducing the presence of acetate. Experimental results have shown an increase of the productivity with this strategy (Bernstein et al. 2012). A few math-  Fig. 3 Acetate concentration (a) and productivity (b) evaluated at steady state for different dilution rates. The function r S is taken as in (2). The continuous line is obtained with the parameters from  (Heßeler et al. 2006;Harvey et al. 2014). However, as mentioned in the introduction, the authors assume that overflow metabolism always occur (i.e. r S0 = 0). Thus, our results give a basis to understand the dynamics of such microbial communities when r S0 > 0.

Recombinant protein production
Following Mauri et al. (2020), and using the notation of this paper, the dynamics of a recombinant protein, which concentration is denoted by H , follows from: Maximal (with respect to D) protein productivity as a function of Y H . The function r S is taken as in (2) and the parameters are taken from Table 1 Here, Y H is the protein yield coefficient representing the carbon diversion to protein production. Let (H , x, S, R) be a solution of (32) with x(0) > 0, H (0), S(0), R(0) ≥ 0. The dynamics of (x, S, R) is independent of H and can be described by Theorem 1. 2 Indeed, if (1 − Y H )μ(S in , 0) > D, then there is x * > 0 such that lim t→∞ x(t) = x * . Now, it is easy to verify that the variable y := Y H 1−Y H x − H satisfies dy dt = −Dy. Therefore, lim t→∞ y(t) = 0, which implies that lim t→∞ H (t) = Y H 1−Y H x * . Thus, we define the steady state protein productivity as: Note that the value of x * depends on the values of Y H and D and that P * H only exists if 0 < D < (1 − Y H )μ(S in , 0). These results allow to illustrate the impact of Y H on the protein productivity. If Y H = 0, there is no production of H , and consequently P * H = 0. On the other hand, if Y H approaches 1, it can be shown that P * H approaches 0. Indeed, using the restriction over D we obtain P * H < μ(S in , 0)Y H x * , where it is clear that lim Y H →1 x * = 0. 3 This shows the existence of an intermediate value of Y H maximizing P * H . Now, for each value of Y H ∈ [0, 1) we compute the maximal productivity with respect to the dilution rate i.e. max{P ; 0 < D ≤ (1−Y H )μ(S in , 0)}. These results are depicted in Fig. 4. We observe that the optimal value of Y H is 0.505, suggesting that protein productivity is maximal (0.373 g L −1 d −1 ) when 50% of the absorbed substrate, that is not excreted in form of acetate, is diverted into protein production.