Optimal protein production by a synthetic microbial consortium: coexistence, distribution of labor, and syntrophy

Abstract

The bacterium E. coli is widely used to produce recombinant proteins such as growth hormone and insulin. One inconvenience with E. coli cultures is the secretion of acetate through overflow metabolism. Acetate inhibits cell growth and represents a carbon diversion, which results in several negative effects on protein production. One way to overcome this problem is the use of a synthetic consortium of two different E. coli strains, one producing recombinant proteins and one reducing the acetate concentration. In this paper, we study a mathematical model of such a synthetic community in a chemostat where both strains are allowed to produce recombinant proteins. We give necessary and sufficient conditions for the existence of a coexistence equilibrium and show that it is unique. Based on this equilibrium, we define a multi-objective optimization problem for the maximization of two important bioprocess performance metrics, process yield and productivity. Solving numerically this problem, we find the best available trade-offs between the metrics. Under optimal operation of the mixed community, both strains must produce the protein of interest, and not only one (distribution instead of division of labor). Moreover, in this regime acetate secretion by one strain is necessary for the survival of the other (syntrophy). The results thus illustrate how complex multi-level dynamics shape the optimal production of recombinant proteins by synthetic microbial consortia.

Appendices

A Proofs

1.1 A.1 Change of notation

In this appendix, we present the proofs of all the statements from Sect. 3, and we present a series of technical results needed for their proofs. We begin by simplifying the notations presented in Sect. 2 and recalling the hypotheses that are necessary for the proofs.

To simplify the notation of uptake rates, we denote $r_{up,p}^g$ by $r_g$ and we define

$$\begin{aligned} r_a(A, r_g):=\left\{ \begin{array}{lcl} -k_{over} (r_g-l), &{} if &{} r_g>l, \\ k_a \frac{A}{A+k_a} d(r_g), &{} if &{} r_g \le l. \end{array} \right. \end{aligned}$$

(16)

Using property (8) and the definition of $r_{over,p}^a$, we have that

$$\begin{aligned} r_a(A,r_g(G,A))=r_{up,p}^a(G,A)+r_{over,p}^a(G,A). \end{aligned}$$

Thus, $r_a$ represents an acetate flux between producers and the medium, which can be negative or positive depending on the sign of $r_g-l$. We also define

$$\begin{aligned} \rho _a(A, r_g):=\displaystyle {k_a \frac{A}{A+K_a} d(\beta r_g)+k_{Acs} \frac{A}{A+K_{Acs}}}. \end{aligned}$$

It is important to note that $\rho _a(0, r_g)=0$ and $\rho _a(A, r_g)\ge 0$ for all $A, r_g \ge 0$, and that $A \longmapsto \rho _a(A,r_g)$ is strictly increasing. The following parameters simplify the notations of product yields:

$$\begin{aligned} \begin{array}{cc} Y_{g/p}:=(1-Y_{h/p})Y_g, &{} Y_{a/p}:=(1-Y_{h/p})Y_a,\\ Y_{g/c}:=(1-Y_{h/c})Y_g, &{} Y_{a/c}:=(1-Y_{h/c})Y_a.\\ \end{array} \end{aligned}$$

(17)

Finally, to simplify the notation of (9), we define

$$\begin{aligned} \begin{array}{ccc} \mu _p(A,r_g) &{}:=&{} Y_{g/p} r_g+ Y_{a/p} r_a(A,r_g)-k_{deg},\\ \mu _c(A,r_g) &{}:=&{} Y_{g/c} \beta r_g+ Y_{a/c} \rho _a(A,r_g)-k_{deg}.\\ \end{array} \end{aligned}$$

(18)

The system (9) can be rewritten as follows:

$$\begin{aligned} \begin{array}{lll} \displaystyle {\frac{dB_p}{dt}} &{}=&{} [\mu _p(A,r_g(G,A))-D]B_p,\\ &{}&{}\\ \displaystyle {\frac{dB_c}{dt}} &{}=&{} [\mu _c(A,r_g(G,A))-D]B_c,\\ &{}&{}\\ \displaystyle {\frac{dG}{dt}} &{} = &{} \displaystyle {D(G_{in}-G)-r_g(G,A)B_p-\beta r_g(G,A)B_c},\\ &{}&{}\\ \displaystyle {\frac{dA}{dt}} &{}=&{} \displaystyle { -DA-r_a(A, r_g(G,A))B_p-\rho _a(A, r_g(G,A))B_c},\\ \end{array} \end{aligned}$$

(19)

This new notation is a compromise between a simple notation to address the proofs and a notation that is closed to that presented in Sect. 2. Note that we have defined $\mu _p$, $\mu _c$, $r_a$, and $\rho _a$ in terms of A and $r_g$. The dilution rates defined in Propositions 1 and 2 can be rewritten as:

$$\begin{aligned} D^p{} & {} = \mu _p(0,r_g(G_{in},0)),\nonumber \\ D^c{} & {} = \mu _c(0,r_g(G_{in},0)),\nonumber \\ D^a{} & {} = Y_{g/p} l-k_{deg}.\nonumber \\ \end{aligned}$$

(20)

We also define the following constant:

$$\begin{aligned} \alpha := Y_{g/p}-k_{over}Y_{a/p}. \end{aligned}$$

(21)

We recall the hypotheses of the model in terms of the new notation. Hypothesis (11) is equivalent to

$$\begin{aligned} \alpha >0, \end{aligned}$$

(22)

and Hypothesis (12) is equivalent to

$$\begin{aligned} r_g(G_{in},0)>l. \end{aligned}$$

(23)

From now on, we assume that (22) and (23) are true.

Our main task is to prove Theorem 1, that is, to find necessary and sufficient conditions for the existence of a coexistence equilibrium of (19). Therefore, we must study the existence and uniqueness of solutions to

$$\begin{aligned} \begin{array}{lll} 0 &{}=&{} \mu _p(A,r_g(G,A))-D,\\ &{}&{}\\ 0 &{}=&{} \mu _c(A,r_g(G,A))-D,\\ &{}&{}\\ 0 &{} = &{} \displaystyle {D(G_{in}-G)-r_g(G,A)B_p-\beta r_g(G,A)B_c},\\ &{}&{}\\ 0 &{}=&{} \displaystyle { -DA-r_a(A, r_g(G,A))B_p-\rho _a(A, r_g(G,A))B_c},\\ \end{array} \end{aligned}$$

(24)

with $B_p,B_c>0$ and $G,A \ge 0$.

1.2 A.2 Results

Our first lemma establishes some properties of any non-negative solution to (24) with $B_p>0$. These properties will be repeatedly used along the appendix.

Lemma 2

Let $D>0$, and let $B_p$, $B_c$, A, and $r_g$ be such that

$$\begin{aligned} \begin{array}{lllll} \mu _p(A,r_g)-D =0,\\ -DA-r_a(A, r_g)B_p-\rho _a(A, r_g)B_c =0,\\ \end{array} \end{aligned}$$

(25)

with $B_p>0$ and $B_c, A, G, r_g \ge 0$. Let $D^a$ and $\alpha$ be given by (20) and (21), respectively, and let us define

$$\begin{aligned} \gamma :=l+\frac{1}{\alpha }(D-D^a). \end{aligned}$$

(26)

The following statements are equivalent:

• (a) $A>0$,

• (b) $r_a(A, r_g)<0$,

• (c) $r_g>l$,

• (d) $\mu _p(A, r_g)=\alpha (r_g-l)+D^a$ and $r_g> l$,

• (e) $r_g= \gamma$ and $\gamma >l$,

• (f) $D> D^a$.

Proof

If (a) holds, then $DA>0$ and $\rho _a(A,r_g)B_c \ge 0$. From the second equation in (25), we obtain that $r_a(A,r_g) B_p<0$. Since $B_p>0$, we conclude that $r_a(A,r_g)<0$, and therefore (b) holds. If (b) holds, then $r_g$ cannot be equal or lower than l, otherwise otherwise $r_a(A,r_g)$ would be non-negative. Thus, (c) holds. Now, if (c) holds, by definition, $r_a(A, r_g)=-k_{over}(r_g-l)$. Using the definition of $\mu _p$, we obtain

$$\begin{aligned} \mu _p(A, r_g)=Y_{g/p}r_g-Y_{a/p} k_{over} [r_g-l]-k_{deg}, \end{aligned}$$

which is equivalent to

$$\begin{aligned} \mu _p(A, r_g)= \alpha [r_g-l]+D^a, \end{aligned}$$

and (d) holds. If (d) holds, from the first equation in (25), we obtain that

$$\begin{aligned} \alpha [r_g-l]+D^a=D, \end{aligned}$$

from where $r_g=\gamma$, and (e) holds. Now, if (e) holds, we have that $\gamma > l$, or equivalently

$$\begin{aligned} l+\frac{1}{\alpha }(D-D^a)>l, \end{aligned}$$

from where $D>D^a$, and (f) holds. We prove the last part indirectly, that is, we prove that if (a) is false, then (f) is also false. Indeed, if $A=0$, then $\rho _a(A, r_g)=0$, and from the second equation in (25), we conclude that $r_a(A, r_g)=0$. Thus, necessarily $r_g\le l$, otherwise $r_a(A, r_g)$ is negative. Consequently, by definition of $\mu _p$,

$$\begin{aligned} \mu _p(A, r_g)= Y_{g/p} r_g-k_{deg} \le Y_{g/p} l-k_{deg} =D^a. g\end{aligned}$$

Combining the previous equation with the first equation in (25), we conclude that $D \le D^a$. This completes the proof. $\square$

The following lemma states some properties related to $D^p$.

Lemma 3

Let $\gamma$ be given by (26) and let $D^a$ and $D^p$ be given by (20). We have:

• (a) $D^p=\alpha [r_g(G_{in},0)-l]+D^a$.

• (b) $D^p>D^a$.

• (c) If $D<D^p$, then $\gamma <r_g(G_{in},0)$.

Proof

From (23), we have that $r_g(G_{in},0)>l$. Then, we have that

$$\begin{aligned} r_a(0, r_g(G_{in},0))=-k_{over}(r_g(G_{in},0)-l). \end{aligned}$$

From the previous equation and the definitions of $D^p$ (see (20)) and $\mu _p$, we obtain that

$$\begin{aligned} D^p=\alpha [r_g(G_{in},0)-l]+D^a, \end{aligned}$$

(27)

which proves (a) and implies

$$\begin{aligned} D^p-D^a=\alpha [r_g(G_{in},0)-l]>0, \end{aligned}$$

and (b) is proved. For part (c), if $D<D^p$, using (27) and the definition of $\gamma$, we have that

$$\begin{aligned} \gamma = l+\frac{1}{\alpha }(D-D^a) < l+\frac{1}{\alpha }(D^p-D^a)=l+\frac{\alpha [r_g(G_{in},0)-l]}{\alpha }=r_g(G_{in},0). \end{aligned}$$

This completes the proof. $\square$

The proof of Proposition 1 in Sect. 3 follows directly from the following proposition.

Proposition 4

Let $D^a$ and $D^p$ be given by (20). We have:

• (a) If $D^p>D$, then (19) admits a unique equilibrium of the form $E^p=(B_p^p,0,G^p,A^p)$, with $B_p^p>0$. Moreover:

  • (I) If $D > D^a$, then $A^p>0$ and $r_g(G^p,A^p)>l$.

  • (II) If $D \le D^a$, then $A^p=0$ and $r_g(G^p,0)\le l$.

• (b) If $D^p \le D$, then (19) has no equilibrium with $B_p>0$ and $B_c=0$.

Proof

The equilibria of (19) with $B_c=0$ and $B_p>0$ are given by the solutions of the following system

$$\begin{aligned} \begin{array}{lll} 0 &{}=&{} \mu _p(A, r_g(G,A))-D,\\ &{}&{}\\ 0 &{} = &{} \displaystyle {D(G_{in}-G)-r_g(G,A)B_p},\\ &{}&{}\\ 0 &{}=&{} \displaystyle { -DA-r_a(A, r_g(G,A))B_p}.\\ \end{array} \end{aligned}$$

(28)

From the equivalence among (a), (c) and (f) in Lemma 2, we have that

• (1) If $D > D^a$, then any solution of (28) satisfies $A>0$ and $r_g(G,A)>l$.

• (2) If $D \le D^a$, then any solution of (28) satisfies $A=0$ and $r_g(G,A)\le l$.

If statement (1) holds, using Lemma 2 part (d), we obtain that (28) is equivalent to

$$\begin{aligned} \begin{array}{lll} 0 &{}=&{} \alpha (r_g(G,A)-l)+D^a-D,\\ &{}&{}\\ 0 &{} = &{} \displaystyle {D(G_{in}-G)-r_g(G,A)B_p},\\ &{}&{}\\ 0 &{}=&{} \displaystyle { -DA+k_{over}[r_g(G,A)-l]B_p}.\\ \end{array} \end{aligned}$$

(29)

From the first equation in (29) (or Lemma 2 part (e)), we have that $r_g(G,A)=\gamma$, with $\gamma$ defined by (26). Thus, from the second and third equations in (29), we have

$$\begin{aligned} G=G_{in}-\beta _g B_p,\quad \beta _g=\frac{\gamma }{D}, \end{aligned}$$

(30)

$$\begin{aligned} A=\beta _a B_p,\quad \beta _a=\frac{k_{over}(\gamma -l)}{D}. \end{aligned}$$

(31)

Replacing (30) and (31) in the first equation of (29), we obtain the following equation for $B_p$:

$$\begin{aligned} \underbrace{\alpha [r_g(G_{in}-\beta _g B_p, \beta _a B_p)-l]+D^a-D}_{g(B_p)}=0 \end{aligned}$$

(32)

From the monotonicity of $r_g$, we have that g is strictly decreasing. Since $g(0)=D^p-D$ (see Lemma 3 part (a)), and $g(G_{in}/\beta _g)=D^a-D<0$, we conclude that (32) admits a unique solution $B_p^p>0$ if $D<D^p$, and has no positive solution if $D\ge D^p$. In particular, this proves (b).

If statement (2) holds, after replacing A by 0, we obtain that (28) is equivalent to

$$\begin{aligned} \begin{array}{lll} 0 &{}=&{} Y_{g/p} r_g(G,0)+Y_{a/p} r_a(0, r_g(G,0))-k_{deg}-D,\\ &{}&{}\\ 0 &{} = &{} \displaystyle {D(G_{in}-G)-r_g(G,0)B_p},\\ &{}&{}\\ 0 &{}=&{} \displaystyle { -r_a(0, r_g(G,0))B_p}.\\ \end{array} \end{aligned}$$

(33)

From the third equation in (33), we obtain that $r_a(G,A)=0$. Thus, (33) is reduced to:

$$\begin{aligned} \begin{array}{lll} 0 &{}=&{} \underbrace{Y_{g/p} r_g(G,0)-k_{deg}-D}_{h(G)},\\ &{}&{}\\ 0 &{} = &{} \displaystyle {D(G_{in}-G)-r_g(G,0)B_p}.\\ \end{array} \end{aligned}$$

(34)

From the second equation in (34), we have that $G \in [0, G_{in})$, otherwise $B_p\le 0$. Now, since $G \longmapsto h(G)$ is strictly increasing and $h(0)=-k_{def}-D<0$, (34) admits a unique positive equilibrium if and only if $h(G_{in})>0$. We have

$$\begin{aligned} \begin{array}{llll} h(G_{in}) &{}=&{} Y_{g/p} r_g(G_{in},0)-k_{deg}-D,&{}\hbox {(use (23))}\\ &{}>&{}Y_{g/p} l-k_{deg}-D, &{} \hbox {(use definition of} D^a)\\ &{}=&{} D^a-D, &{} \hbox {(apply statement (2))}\\ &{}\ge &{} 0. &{}\\ \end{array} \end{aligned}$$

Therefore, $h(G_{in})>0$ and the proof is completed. $\square$

The following proposition is equivalent to Proposition 2 in Sect. 3.

Proposition 5

(Cleaner steady state) Let $D^c$ be given by (20). We have:

• (a) If $D^c>D$, then (19) admits a unique equilibrium with $B_c>0$ and $B_p=0$, which is of the form $E^c=(0,B_c^c,G^c,0)$.

• (b) If $D^c\le D$, then (19) has no equilibrium with $B_c>0$ and $B_p=0$.

Proof

Any equilibrium with $B_c>0$ and $B_p=0$ is a solution of (replace $B_p$ by 0 in (24)):

$$\begin{aligned} \begin{array}{lll} 0 &{}=&{}\mu _c(A, r_g(G,A))-D,\\ &{}&{}\\ 0 &{} = &{} \displaystyle {D(G_{in}-G)-\beta r_g(G,A)B_c},\\ &{}&{}\\ 0 &{}=&{} \displaystyle { -DA-\rho _a(A,r_g(G,A)) B_c}.\\ \end{array} \end{aligned}$$

(35)

From the third equation in (35), we have that $A=0$, otherwise we have an equality between a positive and a negative number. Using the definition of $\mu _c$ and the fact that $\rho _a(0,r_g)=0$, (35) can be rewritten as

$$\begin{aligned} \begin{array}{lll} 0 &{}=&{}\underbrace{Y_{g/c} \beta r_g(G,0)-k_{deg}-D}_{h(G)},\\ &{}&{}\\ 0 &{} = &{} \displaystyle {D(G_{in}-G)-\beta r_g(G,0)B_c}.\\ \end{array} \end{aligned}$$

(36)

From the second equation in (36), we have that $G \in [0, G_{in})$, otherwise $B_c\le 0$. Now, since h is strictly increasing and $h(0)=-k_{deg}-D<0$, we have that (36) admits a solution, which is unique, if and only if $h(G_{in})>0$. The rest of the proof follows from noting that $h(G_{in})=D^c-D$. $\square$

The following lemma shows that if $D \le D^a$, then there is a coexistence equilibrium if and only if some parameters are perfectly balanced. Note that this lemma does not ensure the uniqueness of a coexistence equilibrium.

Lemma 4

Let $D^a$ and $D^p$ be given by (20). If $D \le D^a$, then (19) admits a coexistence equilibrium if, and only if

$$\begin{aligned} Y_{g/p} = \beta Y_{g/c}. \end{aligned}$$

Proof

Since $D\le D^a$, from the equivalence among the statements (a), (c), and (f) in Lemma 2, we have that for any solution to (24) with $B_p>0$, $A=0$ and $r_g(G,0) \le l$. Therefore, $r_a(0, r_g(G,0))=\rho _a(0, r_g(G,0))=0$ and (24) is equivalent to

$$\begin{aligned} \begin{array}{lll} 0 &{}=&{} Y_{g/p} r_g(G,0)-k_{deg}-D,\\ &{}&{}\\ 0 &{}=&{} Y_{g/c} \beta r_g(G,0)-k_{deg}-D,\\ &{}&{}\\ 0 &{} = &{} D(G_{in}-G)-r_g(G,0)B_p-\beta r_g(G,0)B_c. \end{array} \end{aligned}$$

(37)

If $Y_{g/p}\ne \beta Y_{g/c}$, then the first and second equation cannot be satisfied at the same time. Thus, there cannot be a coexistence equilbrium. On the other hand, if $Y_{g/p}= \beta Y_{g/c}$, then G is a solution of $h(G)=0$ with $h(G):=Y_{g/p} r_g(G,0)-k_{deg}-D$. Note that h is strictly increasing, $h(0)=-k_{deg}-D<0$, and

$$\begin{aligned} \begin{array}{llll} h(G_{in}) &{}=&{} Y_{g/p} r_g(G_{in},0)-k_{deg}-D,&{} \hbox {(use (23))}\\ &{}>&{}Y_{g/p} l-k_{deg}-D, &{} \hbox {(use definition of} D^a)\\ &{}=&{} D^a-D, &{} \hbox {(apply assumption on} D)\\ &{}\ge &{} 0. n&{} \end{array} \end{aligned}$$

Thus $h(G)=0$ admits a unique solution $G^* \in (0, G_{in})$. Replacing G by $G^*$ in the third equation in (37), we obtain the existence of infinity coexistence equilibria. $\square$

The following lemma shows that if the dilution rate is too high then coexistence is impossible.

Lemma 5

Let $D^p$ be given by (20). If $D\ge D^p$, then (19) has no coexistence equilibrium.

Proof

We prove this by contradiction. Let us assume that $D\ge D^p$ and that (19) admits a coexistence equilibrium $(B_p^*, B_c^*, G^*,A^*)$. Since $D^p>D^a$ (see Lemma 3 part (b)), we can use the equivalence between (d) and (f) from Lemma 2 to conclude that

$$\begin{aligned} \mu _p(A^*,r_g(G^*,A^*))=\alpha [r_g(G^*,A^*)-l]+D^a. \end{aligned}$$

(38)

From Lemma 2 part (a), we obtain that

$$\begin{aligned} D^p=\alpha [r_g(G_{in}, 0)-l]+D^a. \end{aligned}$$

(39)

From the third equation in (24), we have that $G^*<G_{in}$. Using the monotonicity of $r_{g}$, we can combine (38) and (39) to obtain $\mu _p(A^*,r_g(G^*,A^*))<D^p$. Finally, from the first equation in (24), we conclude that $D<D^p$, which contradicts our initial hypothesis ($D \ge D^p$). This completes the proof. $\square$

The following establishes some necessary conditions for coexistence.

Lemma 6

Let $D^a$, $D^p$, and $D^c$ be given by (20) and let $G^c$ be given by Proposition 5. Let us assume that $D^a<D<D^p$ and $D<D^c$. If (19) admits a coexistence equilibrium, then

$$\begin{aligned} \mu _p(0, r_g(G^c,0)) > D. \end{aligned}$$

(40)

Proof

Let us assume that (19) admits a coexistence equilibrium $(B_p^*, B_c^*,G^*,A^*)$. Since $D>D^a$, from Lemma 2 part (e), we know that $r_g(G^*,A^*)=\gamma$ with $\gamma$ defined by (26). Now, from the second equation in (24) and the definition of $G^c$ we have

$$\begin{aligned} \begin{array}{lll} \mu _c(0, \gamma )&{}=&{}D,\\ \mu _c(0, r_g(G^c, 0))&{}=&{}D, \end{array} \end{aligned}$$

which is equivalent to

$$\begin{aligned} \begin{array}{lll} Y_{g/c} \beta \gamma +Y_{a/c} \rho _a(A^*,\gamma )&{}=&{}k_{deg}+D\\ Y_{g/c} \beta r_g(G^c, 0)&{}=&{}k_{deg}+D. \end{array} \end{aligned}$$

(41)

Since $D>D^a$, from Lemma 2, we have that $A^*>0$, hence $\rho _a(A^*, \gamma )>0$. Consequently, from (41), we conclude that

$$\begin{aligned} r_g(G^c,0) > \gamma = l+\frac{1}{\alpha } (D-D^a), \end{aligned}$$

(42)

which implies

$$\begin{aligned} \alpha (r_g(G^c,0)-l)+D^a>D. \end{aligned}$$

(43)

Since $r_g(G^c,0)>l$ (see (42)), from the definitions of $r_a$ and $\mu _p$, we conclude that

$$\begin{aligned} \mu _p(0, r_g(G^c,0))=\alpha (r_g(G^c,0)-l)+D^a. \end{aligned}$$

Combining the previous equation with (43), we conclude that (40) holds and the proof is completed. $\square$

The following lemma gives upper bounds for any coexistence equilibrium.

Lemma 7

Let $D^a$ and $D^p$ be given by (20). Let us assume that $D^a< D <D^p$. Then, any solution of (24) with $B_p,B_c>0$ satisfies

$$\begin{aligned} 0<A<A^p \,\,\,\hbox {and} \,\,\, 0<G<G^p, \end{aligned}$$

(44)

with $A^p$ and $G^p$ given by Proposition 4.

Proof

Let us assume that (19) admits a solution $(B_p^*,B_c^*,G^*,A^*)$. Since $D>D^a$, we have that $\gamma = r_g(G^*, A^*)= r_g(G^p, A^p)$ (see Lemma 2). Thus, we have the following equations:

$$\begin{aligned} \begin{array}{lll} 0 &{} = &{} \displaystyle {D(G_{in}-G^p)-\gamma B_p^p},\\ &{}&{}\\ 0 &{}=&{} \displaystyle {-DA^p-r_a(A^p, \gamma )B_p^p},\\ \end{array} \end{aligned}$$

(45)

and

$$\begin{aligned} \begin{array}{lll} 0 &{} = &{} \displaystyle {D(G_{in}-G^*)-\gamma B_p^*-\beta \gamma B_c^*},\\ &{}&{}\\ 0 &{}=&{} \displaystyle { -DA^*-r_a(A^*, \gamma )B_p^*-\rho _a(A^*,\gamma )B_c^*}.\\ \end{array} \end{aligned}$$

(46)

We prove that $A^*<A^p$ by contradiction. Let us assume that $A^*\ge A^p$. Using the monotonicity of $r_g$ and the fact that $r_g(G^p,A^p)=r_g(G^*,A^*)$, we obtain that $G^p\le G^*$. From the first equation in (46) and the first equation in (45), we obtain that:

$$\begin{aligned} D(G_{in}-G^p)-\gamma B_p^p=0 < \beta \gamma B_c^* = D(G_{in}-G^*)-\gamma B_p^*, \end{aligned}$$

from where $0\le D(G^*-G^p)<\gamma (B_p^p-B_p^*)$, which implies

$$\begin{aligned} B_p^p>B_p^*. \end{aligned}$$

(47)

From the second equation in (46) and the second equation in (45), we obtain that:

$$\begin{aligned} -DA^p-r_a(A^p, \gamma )B_p^p< \rho _a(A^*, \gamma )B_c^* = -DA^*-r_a(G^*,A^*)B_p^*. \end{aligned}$$

(48)

Now, since $\gamma >l$, we have that $r_a(A^*, \gamma )=r_a(A^p, \gamma )=-k_{over}(\gamma -l)<0$. Thus, from (48), we obtain that

$$\begin{aligned} 0\le D(A^*-A^p)<k_{over}(\gamma -l)(B_p^*-B_p^p), \end{aligned}$$

which implies $B_p^*>B_p^p$. This contradicts (47). Then, $A^p> A^*$ and consequently $G^p>G^*$. $\square$

The following proposition is equivalent to Proposition 3 in Sect. 3.

Proposition 6

Let $G^c$ and $A^c$ be given by Proposition 4 and let $D^a$ and $D^p$ be given by (20). If (19) admits a coexistence equilibrium $(B_p^*, B_c^*, G^*, A^*)$, then:

• (a) $D \le D^a$ implies $A^*=0$ and $r_g(G^*,0)\le l$.

• (b) $D^a<D<D^p$ implies $0<A^*<A^p$ and $r_g(G^*,A^*)>l$.

Proof

From the equivalence among (a), (c), and (f) in Lemma 2, we obtain that immediately

• (I) $D \le D^a$ implies $A^*=0$ and $r_g(G^*,A^*)\le l$.

• (II) $D^a<D<D^p$ implies $0<A^*$ and $r_g(G^*,A^*)>l$.

And from Lemma 7, we obtain that $D^a<D<D^p$ implies $A^*<A^p$. This completes the proof. $\square$

The following result states some necessary and sufficient conditions for the coexistence of a unique coexistence equilibrium.

Lemma 8

Let $D^p$, $D^a$, and $A^p$ be given by Proposition 4 and let $\gamma$ be defined by (26). If $D^a<D<D^p$, then (9) admits a coexistence equilibrium (unique) if, and only if,

$$\begin{aligned} \mu _c(A^p, \gamma )>D\,\,\hbox {and}\,\,\mu _c(0, \gamma )<D. \end{aligned}$$

(49)

Proof

Since $D>D^a$, from Lemma 2, we know that any solution to (24) satisfies

$$\begin{aligned} r_g(G,A)=\gamma . \end{aligned}$$

(50)

Thus, from the second equation in (24), we obtain that $\phi (A):=\mu _c(A, \gamma )-D =0$. We note that (49) is equivalent to $\phi (A^p)>0$ and $\phi (0)<0$. If (19) has a coexistence equilibrium, say $(B_p^*, B_c^*, G^*, A^*)$, then $\phi (A^*)=0$. Since $\phi$ is strictly increasing and $0<A^*<A^p$ (see Lemma 7), we conclude that $\phi (A^p)>0$ and $\phi (0)<0$. Thus, (49) is a necessary condition for the existence of a coexistence equilibrium.

We prove now that (49) is also a sufficient condition, that is, if (49) holds, then (24) admits a unique solution. Since (49) holds and $\phi$ is strictly increasing, there exists a unique $A^* \in (0, A^p)$ such that $\phi (A^*)=0$.

Now, replacing A by $A^*$ in (50), we obtain the following equation for G:

$$\begin{aligned} \varphi (G):=r_g(G,A^*)-\gamma =0. \end{aligned}$$

It is clear that $\varphi$ is strictly increasing and that $\varphi (0)=-\gamma <0$. Let $G^p$ be given by Proposition 4. Since $A^p>A^*$ ( see Lemma 7), we have that $\varphi (G^p)>r_g(G^p,A^p)-\gamma$. From the definition of $G^p$ and $A^p$ (e.g. see (45)), we have that $r_g(G^p,A^p)-\gamma =0$, and hence $\varphi (G^p)>0$. Consequently, there is a unique $G^* \in (0,G^p)$ such that $\varphi (G^*)=0$. It remains to prove the existence and uniqueness of a positive solution of the following linear system for $(B_p,B_c)$ obtained from the third and fourth equations in (24):

$$\begin{aligned} \begin{array}{lll} 0 &{} = &{} \displaystyle {D(G_{in}-G^*)-\gamma B_p-\beta \gamma B_c},\\ &{}&{}\\ 0 &{}=&{} \displaystyle { -DA^*-k_{over}(\gamma -l)B_p-\rho _a(A^*, \gamma )B_c}.\\ \end{array} \end{aligned}$$

(51)

From the first equation in (51) we have that

$$\begin{aligned} B_p= \frac{\rho _a(A^*, \gamma ) B_c+D A^*}{k_{over}(\gamma -l)}. \end{aligned}$$

Combining the previous equation with the second equation in (51), we obtain:

$$\begin{aligned} \underbrace{G_{in}-G^*-A^* \frac{\gamma }{k_{over}(\gamma -l)}}_{\kappa } = \frac{\gamma }{D}\left( \beta +\frac{\rho _a(A^*,\gamma )}{k_{over}(\gamma -l)} \right) B_c. \end{aligned}$$

Since $A^p>A^*$ and $G^p>G^*$, we have that

$$\begin{aligned} \kappa > G_{in}-G^p-A^p \frac{\gamma }{k_{over}(\gamma -l)}. \end{aligned}$$

Finally, if we isolate $B_p^p$ in the second equation in (45), and we replace it in the first equation, we obtain that

$$\begin{aligned} G_{in}-G^p-A^p \frac{\gamma }{k_{over}(\gamma -l)}=0. \end{aligned}$$

This shows that $\kappa >0$. Then (51) has a positive solution. $\square$

The following result states the conditions presented in statements (b) and (c) in Theorem 1.

Proposition 7

Let $D^p$, $D^a$, and $D^c$ be given by (20) and let $\gamma$ be defined by (26). We have that:

• (a) If $D^a<D<D^p$ and $D<D^c$ then (9) admits a (unique) coexistence equilibrium, if and only if

  $$\begin{aligned} \mu _p(0, r_g(G^c,0))>D\,\,\hbox {and}\,\, \mu _c(A^p, \gamma )>D, \end{aligned}$$

  with $G^p$ and $A^p$ defined in Proposition 4, and $G^c$ defined in Proposition 5.

• (b) If $D^a<D<D^p$ and $D\ge D^c$ then (9) admits a (unique) coexistence equilibrium, if and only if

  $$\begin{aligned} \mu _c(A^p, \gamma )>D, \end{aligned}$$

  with $G^p$ and $A^p$ defined in Proposition 4.

Proof

Let us assume that $D^a<D<D^p$. From Lemmas 6 and 8, we know that (a) and (b) provide necessary conditions for the existence of a coexistence equilibrium. It remains to prove that they also provide sufficient conditions. Using Lemma 8, we know that $\mu _c(A^p, \gamma )>D$ and $\mu _c(0, \gamma )<D$ are sufficient conditions. In statements (a) and (b), it is direct to see that $\mu _c(A^p, \gamma )>D$ holds, therefore we must prove that each statement, (a) and (b), also implies $\mu _c(0, \gamma )<D$. Thus, the proof of parts (a) and (b) follows from proving that

• (I) if $D<D^c$, $D^a<D<D^p$, and $\mu _p(0,r_g(G^c,0))>D$, then $\mu _c(0, \gamma )<D$, and

• (II) if $D\ge D^c$ and $D^a<D<D^p$, then $\mu _c(0, \gamma )<D$,

respectively.

For (I), from the hypotheses, we have that $\mu _p(0,r_g(G^c,0))>D^a$. Using the definition of $\mu _p$ and $D^a$, we have

$$\begin{aligned} Y_{g/p} r_g(G^c,0)+Y_{a/p} r_a(0,r_g(G^c,0))-k_{deg} > Y_{g/p} l-k_{deg}, \end{aligned}$$

which can be rearranged as

$$\begin{aligned} Y_{g/p} [r_g(G^c,0)-l] > -Y_{a/p} r_a(0,r_g(G^c,0)). \end{aligned}$$

(52)

From the definition of $r_a$ (see (16)), we have that $r_a(0,r_g(G^c,0))$ cannot be positive (evaluate $r_a(A,r_g)$ at $A=0$). Therefore, (52) implies that $r_g(G^c,0)>l$. Using again the definition of $r_a$, we have that $r_a(0,r_g(G^c,0))=-k_{over}[r_g(G^c,0)-l]$. Hence, from the definition of $\mu _p$ (see (18)), we have

$$\begin{aligned} \mu _p(0,r_g(G^c,0))=\alpha [r_g(G^c,0)-l]+D^a, \end{aligned}$$

(53)

with $\alpha$ defined by (21). Now, since $D>D^a$, according to Proposition 4, $r_g(G^p, A^p)>l$. Thus, using Lemma 2 and the definition of $\gamma$, we have that

$$\begin{aligned} \mu _p(A^p, \gamma )=\alpha [\gamma -l]+D^a. \end{aligned}$$

(54)

By definition of $G^p$ and $A^p$, we have that $\mu _p(G^p, A^p)=D$. Thus, from the hypotheses we have that

$$\begin{aligned} \mu _p(G^c,0)>\mu _p(A^p, \gamma ). \end{aligned}$$

(55)

Combining (53), (54), and (55), we conclude that

$$\begin{aligned} r_g(G^c,0)>\gamma . \end{aligned}$$

(56)

Now, using the definition of $G^c$ (dilution rate equal to growth rate) and (56), we have that

$$\begin{aligned} D=\mu _c(0, r_g(G^c,0))=Y_{g/c} \beta r_g(G^c,0)-k_{deg}>Y_{g/c} \beta \gamma -k_{deg}= \mu _c(0,\gamma ), \end{aligned}$$

and (I) is proved.

For (II), using the definition of $D^c$ and $\mu _c$, and Lemma 3 part (c), we have

$$\begin{aligned} D^c=Y_{g/c} \beta r_g(G_{in},0)-k_{deg}> Y_{g/c} \beta \gamma -k_{deg} = \mu _c(0, \gamma ). \end{aligned}$$

Now, using the fact that $D \ge D^c$, from the previous equation, we conclude that $D>\mu _c(0,\gamma )$. This completes the proof. $\square$

B Algorithm to find the coexistence equilibrium

Let $D^a$ and $D^p$ be given by Proposition 1. From now on, we assume that $D \in (D^a, D^p)$, otherwise there is no coexistence equilibrium (see Theorem 1). The first step is to determine the equilibrium $E^p=(B_p^p,0,G^p,A^p)$ given by Proposition 1. The instructions on how to do so are dictated by the proof of Proposition 1. Indeed, $B^p_p$ is obtained as the solution of

$$\begin{aligned} g(B_p)=0, \end{aligned}$$

(57)

with g defined by

$$\begin{aligned} g(B_p)=\alpha [r_{up,p}^g(G_{in}-\beta _g B_p, \beta _a B_p)-l]+D-D^a, \end{aligned}$$

with $\alpha =(1-Y_{h/p})(Y_g-k_{over}Y_a)$, $\beta _a=\gamma /D$, $\beta _a=k_{over}(\gamma -l)/D$, and

$$\begin{aligned} \gamma =l+\frac{D-D^a}{(1-Y_{h/p})(Y_g-k_{over} Y_a)}. \end{aligned}$$

(58)

Equation (57) has a unique solution on the interval $[0,G_{in}/\beta _g]$. Moreover, $g(0)>0$ and $g(G_{in}/\beta _g)<0$, which provides an interval to look for the solution. Thus, equation (57) can be easily solved, for example, with the solver fzero in MATLAB. The values of $A^p$ and $G^p$ are obtained from

$$\begin{aligned} G^p=G_{in}-\beta _g B_p^p\,\,\hbox {and}\,\, A^p=\beta _a B_p^p. \end{aligned}$$

We also need the value of $G^c$, the glucose concentration associated with the equilibrium $E^c$ given by Proposition 2. Let $D^c$ be given by Proposition 2. If $D<D^c$, then $G^c$ is the unique solution of $(1-Y_{h/c})f_c(G,0)-k_{deg}-D=0$. This equation is easily solved explicitly. If $D\ge D^c$, we will take $G^c$ as $G_{in}$. This is useful to distinguish the cases (b) and (c) in Theorem 1 (see Remark 7).

Now, we have to determine the coexistence equilibrium. The instructions on how to do so are dictated by the proof of Lemma 8:

• 1) Determine $c_1=(1-Y_{h/p})f_p(G^c,0)-k_{deg}-D$ and $c_2=(1-Y_{h/c})f_c(G^p,A^p)-k_{deg}-D$.

• 2) If $c_1\le 0$ or $c_2 \le 0$, then there is no coexistence equilibrium. The algorithm ends. However, if $c_1$ and $c_2$ are positive, go to the next step.

• 3) Find $A^* \in [0,A^p]$ as the unique solution of $\phi (A)=0$, with $\phi$ defined by

  $$\begin{aligned} \phi (A)=(1-Y_{h/c})\left[ Y_{g} \beta \gamma +Y_a\left( k_a \frac{A}{A+K_a}d(\beta \gamma )+k_{Acs} \frac{A}{A+K_{Acs}} \right) \right] -k_{deg}-D, \end{aligned}$$

  with $\gamma$ given by (58). We have that $\phi (0)<0$ and $\phi (A^p)>0$.

• 4) Find $G^*\in [0,G_{in}]$ as the unique solution of $\varphi (G)=0$ with $\varphi (G):=(1-Y_{h/p})f_p(G,A^*)-k_{deg}-D$. We have that $\varphi (0)<0$ and $\varphi (G_{in})>0$.

• 5) Find $B_p^*$ and $B_c^*$ as the unique solution of the following linear system:

  $$\begin{aligned} \left[ \begin{array}{cc} r_{up,p}^g(G^*,A^*) &{} r_{up,c}^g(G^*,A^*)\\ r_{over,p}^a(G^*,A^*) &{} r_{up,c}^a(G^*,A^*) \end{array}\right] \left[ \begin{array}{c} B_p \\ B_c \end{array}\right] =\left[ \begin{array}{c} D(G_{in}-G^*) \\ DA^* \end{array}\right] . \end{aligned}$$

C Algorithm to solve the MOP

Problem (15) is solved numerically with the interior point algorithm implemented in the toolbox fmincon of MATLAB (Byrd et al. 1999). To use fmincon, the objective function must be continuous on the feasible region, which must be defined through equalities and non-strict inequalities. In the following, we adapt (15) to use fmincon.

Following Theorem 1, the set $\Omega$ defined in Sect. 4 can be described such that each element $v=(Y_{h/p},Y_{h/c},D)$ on $\Omega$ satisfies

$$\begin{aligned} \begin{array}{lll} D&{}< &{} D^p,\\ D&{} > &{} D^a,\\ D&{}< &{} (1-Y_{h/c})f_c(G^p,A^p)-k_{deg},\\ D&{} < &{} (1-Y_{h/p})f_p(G^c,0)-k_{deg},\\ Y_{h/p}, Y_{h/c}&{} \in &{} [0,1], \end{array} \end{aligned}$$

(59)

where $D^a$, $D^p$, $G^p$, and $A^p$ are given by Proposition 1, $D^c$ and $A^c$ are given by Proposition 2, and $G^c$ is given by (13). We define the set $\bar{\Omega }$ such that each element $v=(Y_{h/p},Y_{h/c},D)$ on $\bar{\Omega }$ satisfies

$$\begin{aligned} \begin{array}{lll} D&{} \le &{} D^p,\\ D&{} \ge &{} D^a,\\ D&{} \le &{} (1-Y_{h/c})f_c(G^p,A^p)-k_{deg},\\ D&{} \le &{} (1-Y_{h/p})f_p(G^c,0)-k_{deg},\\ Y_{h/p}, Y_{h/c}&{} \in &{} [0,1]. \end{array} \end{aligned}$$

(60)

The function $\Phi$ defined in Sect. 4 is defined on $\Omega$ and we must extend its definition on $\bar{\Omega }$. If $v \in \Omega$, we can determine the coexistence equilibrium using the algorithm presented in Appendix B, and therefore evaluate the function $\Phi$ defined in Sect. 4. For $v \in \bar{\Omega }-\Omega$, we solve the system (9)–(10) with an initial condition satisfying $B_p(0), B_c(0)>0$. We run then the model in the long-term until an equilibrium is reached. We will denote the total protein concentration associated to this equilibrium by ${\hat{H}}^*(v)$. Thus, the extension of $\Phi$ on $\bar{\Omega }$, is given by:

$$\begin{aligned} \hat{\Phi }= \left\{ \begin{array}{lcc} (D H^*(v), H^*(v)/G_{in}) &{} \text {if} &{} v \in \Omega ,\\ (D {\hat{H}}^*(v), {\hat{H}}^*(v)/G_{in}) &{} \text {if} &{} v \in \bar{\Omega }-\Omega . \end{array} \right. \end{aligned}$$

The continuity of $\hat{\Phi }$ can be observed in Figs. 3C and D. In the region of coexistence (shaded region), we have that $v\in \Omega$, while on the boundary of this interval, we have that $v\in \bar{\Omega }-\Omega$. In this situation, we observe a continuous transition between a coexistence equilibrium and a non-coexistence equilibrium as D approaches the boundary of this region.

We solve numerically the following problem instead of (15):

$$\begin{aligned} \begin{array}{ll} \max &{} \lambda \hat{\Phi }(v)+(1-\lambda ) \hat{\Phi }(v)\\ &{} Av \le b\\ &{} c(v) \le 0\\ &{} ub\le v \le up,\\ \end{array} \end{aligned}$$

(61)

where

$$\begin{aligned}{} & {} A=\left[ \begin{array}{lllll} f_p(G_{in},0)&{}0&{}1\\ -Y_g l&{} 0&{}-1 \\ \end{array}\right] , b=\left[ \begin{array}{c} f_p(G_{in},0)-k_{deg}\\ k_{deg}-Y_g l \\ \end{array}\right] , lb=\left[ \begin{array}{c} 0\\ 0 \\ 0\\ \end{array}\right] , \\{} & {} ub=\left[ \begin{array}{c} 1\\ 1 \\ f_p(G_{in},0)-k_{deg},\\ \end{array}\right] , \,\hbox {and}\,\,c(v)=\left[ \begin{array}{c} D +k_{deg}- (1-Y_{h/p}) f_p(G^c(v),0)\\ D +k_{deg}- (1-Y_{h/c}) f_c(G^p(v),A^p(v))\\ \end{array}\right] . \end{aligned}$$

D Robust choice of the down-regulation function

Let us assume that in (9) the down-regulation function d is replaced by ${\hat{d}}$, defined in (6). We then run long-term simulations of (9)–(10) until an equilibrium is reached (which is always the case).² Then, we evaluate the the process yield and the productivity of the system at the time at which steady state has been reached. Figure 7 shows the result of this experiment for 1000 different values of $(Y_{h/p},Y_{h/c},D)$. As we can see, the POF obtained when d is given by (7) represents a good approximation of the POF when d is given by (6). This shows that the choice of d in this paper is adequate to study the model proposed by Mauri et al. (2020).

Fig. 7

Scatter plot of $\Phi$ using the equilibria of model (9)–(10) with d replaced by (6). The continuous line is the POF obtained in Fig. 4