Model Reduction of Genetic-Metabolic Networks via Time Scale Separation

Abstract

Model reduction techniques often prove indispensable in the analysis of physical and biological phenomena. A succesful reduction technique can substantially simplify a model while retaining all of its pertinent features. In metabolic networks, metabolites evolve on much shorter time scales than the catalytic enzymes. In this chapter, we exploit this discrepancy to justify the reduction via time scale separation of a class of models of metabolic networks under genetic regulation . We formalise the concept of a metabolic network and employ Tikhonov’s Theorem for singularly perturbed systems. We demonstrate the applicability of our result by using it to address a problem in metabolic engineering: the genetic control of branched metabolic pathways. We conclude by providing guidelines on how to generalise our result to larger classes of networks.

Appendices

Appendix

In the appendices we assume that the reader has some familiarity with non-linear systems theory. Specifically, we assume that the reader is comfortable with the various notions of stability of equilibria, Lyapunov functions and the existence and uniqueness results. If not, we refer the reader to the excellent text [8].

We begin by presenting Tikhonov’s Theorem over finite time intervals and some related results. Next, we discuss the notion of converging input converging state systems. Lastly, we employ the previous two to prove Lemmas 1 and 2 and Theorem 1.

Throughout the appendices we use ||$\cdot$|| to denote any vector norm.

A: Tikhonov’s Theorem

As discussed in the main text, a method for dimensionality reduction of non-linear systems is time scale separation. This is applicable to systems whose state variables exhibit large differences in the ‘speed’ of their time responses. Core to time scale separation is the following result first proved by Tikhonov 60 years ago [21, 22]. The version of it presented here is not the original version by Tikhonov, but instead the version published by Vasil’eva in 1963, which we find easier to work with.

Theorem 2

[9, 23] Let $f\text {:}\, {\mathbb {R}}^n \times {\mathbb {R}}^m\times \rightarrow {\mathbb {R}}^n$ and $g\text {:}\, {\mathbb {R}}^n \times {\mathbb {R}}^m\rightarrow {\mathbb {R}}^m$ both be smooth functions. Consider the system

$$\begin{aligned} \varepsilon&\dot{z}(t)=f(x(t),z(t)),\quad z(0) = z_0, \quad z\in \mathbb {R}^n,\end{aligned}$$

(7.19a)

$$\begin{aligned}&\dot{x}(t)=g(x(t),z(t)),\quad x(0) = x_0, \quad x\in \mathbb {R}^m, \end{aligned}$$

(7.19b)

where $\varepsilon > 0$. Assume for all $t\in [0,T]$ where $T\in {\mathbb {R}}_{\ge 0}$ that (7.19) has the unique solutions $x(t), z(t)$. Consider the following conditions:

• 1. There exists a unique function $\phi (\cdot )$ such that $g(\bar{x}(t),\phi (\bar{x}(t))=0$ for all $t\in [0,T]$ where $\bar{x}(t)$ denotes the unique solution over $[0,T]$ of the reduced system $\dot{\bar{x}}=g(\bar{x},\phi (\bar{x})),\;x(0) = x_0$.

• 2. Consider the ‘boundary layer’ system

  $$\begin{aligned} \frac{d\hat{z}}{d\tau }(\tau ) = f(x_0,\hat{z}(\tau )+\phi (x_0)). \end{aligned}$$

  (7.20)

  Assume that the equilibrium$\hat{z}=0$ of (7.20) is globally asymptotically stable, uniformly in $x_0$.

• 3. The eigenvalues of $\left[ \frac{\partial f}{\partial z}(\cdot )\right]$ evaluated along $\bar{x}(t)$, $\bar{z}(t)$, have real parts smaller than a fixed negative number, i.e.,

  $$\begin{aligned} \mathrm{{Re}}\left( \lambda _i\left( \left[ \frac{\partial f}{\partial z}\right] (\bar{x}(t),\bar{z}(t))\right) \right) \le -c,\quad c\in \mathbb {R}_{>0},\quad \forall i,\quad \forall t \ge 0. \end{aligned}$$

  where $\mathrm{{Re}}(a)$ denotes the real part of $a\in \mathbb {C}$ and $\lambda _i(A)$ denotes the ith eigenvalue of $A\in \mathbb {R}^{n\times n}$.

If the three conditions above are satisfied, then relations (7.21) and (7.22) hold for all $t\in [0,T]$ and there exists a time $t_1 \ge 0$, $O(\varepsilon \ln (1/\varepsilon ))$, such that (7.23) holds for all $t\in [t_1,T]$.

$$\begin{aligned} x(t) = \bar{x}(t) + O(\varepsilon ). \end{aligned}$$

(7.21)

$$\begin{aligned} z(t) = \bar{z}(t) + \hat{z}(t) + O(\varepsilon ). \end{aligned}$$

(7.22)

$$\begin{aligned} z(t) = \bar{z}(t) + O(\varepsilon ). \end{aligned}$$

(7.23)

The Theorem’s first condition ensures that there exists a well defined reduced model. The second condition verifies that, initially, the trajectory of the complete system rapidly converges to the one of the reduced system. The third condition guarantees that after the initial transient dies out the trajectory of the complete system remains close to the that of the reduced system. It is worth mentioning, that the above is Tikhonov’s theorem restricted to the special case when the systems are time invariant and (7.19a) has a unique root. For an excellent treatment of Tikhonov’s theorem (including its most general form) and its applications in control theory see [9].

In verifying the theorem’s last two conditions the following two lemmas will be useful.

Lemma 3

[22] Consider the boundary system (7.20). Assume that $f$ and the root$\phi$ are continuous functions and that $x_0\in \fancyscript{P}$ where $\fancyscript{P}$ is a compact subset of $\mathbb {R}^m$. Suppose that for all $x_0\in \fancyscript{P}$, the origin of (7.20) is globally asymptotically stable. Then the origin of (7.20) is globally asymptotically stable, uniformly in $x_0$.

Lemma 4

Consider $f(\cdot )$ in (7.19). Let $A$ be a compact subset of $\,\mathbb {R}^{n+m}$ and suppose that

$$\begin{aligned} \mathrm{{Re}}\left( \lambda _i\left( \left[ \frac{\partial f}{\partial z}\right] (x,z)\right) \right) <0,\quad \forall i,\quad \forall \begin{bmatrix}x,z\end{bmatrix}^T\in A. \end{aligned}$$

Then, there exists a $c\in \mathbb {R}_{>0}$ such that

$$\begin{aligned} \mathrm{{Re}}\left( \lambda _i\left( \left[ \frac{\partial f}{\partial z}\right] (x,z)\right) \right) \le -c,\quad \forall \begin{bmatrix}x\\ z\end{bmatrix}^T\in A. \end{aligned}$$

Proof

First, we show that

$$\begin{aligned} \lambda ^*(x,z) := \max _i \left( \lambda _i\left( \left[ \frac{\partial f}{\partial z}\right] (x,z)\right) \right) , \end{aligned}$$

(7.24)

that is, the maximum real part of the eigenvalues of the Jacobian, is a continuous function of $x$ and $z$.

The eigenvalues are the roots of the characteristic polynomial of the Jacobian (i.e., the solutions to $\det \left( \lambda I-\left[ \frac{\partial f}{\partial z}\right] (x,z)\right) =0$ where $\lambda \in \mathbb {C}$). The roots of a polynomial depend continuously on the coefficients of a polynomial. The coefficients of the characteristic polynomial of the Jacobian above depend continuously of the entries of the Jacobian. The entries of the Jacobian are continuous functions of $x$ and $z$. The composition of two continuous functions is also a continuous function. Thus, the eigenvalues of the Jacobian are continuous functions of $x$ and $z$. Thus, (7.24) is a continuous function of $x$ and $z$.

The supremum of a continuous function over a compact set is achieved by an element in the set. This fact and the lemma’s premise imply that $\sup _{[x,z]^T\in A}\lambda ^*(x,z) < 0$ which completes the proof.$\square$

B: Converging Input Converging State Systems

In Appendix C, we need to prove that the unique equilibrium of the network with the enzyme concentrations fixed in time (system (7.8)) is globally asymptotically stable (GAS). To accomplish this we exploit the acyclycity of the network to break system (7.8) down into $n$ one dimensional subsystems and study how they interact. To this end, we introduce the notions of converging input bounded state (CIBS) and converging input converging state (CICS) systems. These were original presented in [20] and relate to other more well known concepts such as input to state stable (ISS) systems.

Definition 1

We say that $u(\cdot )$ is an input if it is a continuous function that maps from $\mathbb {R}_{\ge 0}$ to $\mathbb {R}^m$.

Now, consider the non-autonomous system

$$\begin{aligned} \dot{x}(t) = f(x(t),u(t)), \end{aligned}$$

(7.25)

where $f(\cdot )$ is continuous, $x\in \mathbb {R}^n$ and $u(\cdot )$ is an input. In addition, consider the same system with ‘zero input’

$$\begin{aligned} \dot{x}(t) = f(x(t),0). \end{aligned}$$

(7.26)

Definition 2

System (7.25) is said to be converging input bounded state (CIBS) if for any input $u(\cdot )$ such that $u(t)\rightarrow 0$ as $t\rightarrow +\infty$ and for any initial conditions $x_0\in \mathbb {R}^n$, the solution exists for all $t\ge 0$ and is bounded.

Definition 3

System (7.25) is said to be converging input converging state (CICS) if for any input $u(\cdot )$ such that $u(t)\rightarrow 0$ as $t\rightarrow +\infty$ and for any initial conditions $x_0\in \mathbb {R}^n$, the solution exists for all $t\ge 0$ and converges to $0$ as time tends to infinity.

Lemma 5

Assume that for any input, $x(t)$ exists for all $t\ge 0$. Let $V\text {:}\,\mathbb {R}^n\rightarrow \mathbb {R}$ be a continuously differentiable, bounded from below and radially unbounded (i.e., $||x||\rightarrow +\infty \Rightarrow V(x)\rightarrow +\infty )$ function. If there exists constants $\alpha >0$ and $\beta >0$ such that

$$\begin{aligned} \dot{V}(x) = \frac{\partial V}{\partial x}f(x,u) \le 0 \quad \forall \quad (x,u)\in \mathbb {R}^{n+m}\mathrm{{:}}\,\quad ||x||\ge \beta , \,||u||\le \alpha , \end{aligned}$$

then system (7.25) is CIBS.

Proof

We prove by contradiction. Assume that the premise of the lemma is satisfied and that there exists a $u(t)$ such that $||u(t)||\rightarrow 0$ as $t\rightarrow +\infty$ but $x(t)$ is unbounded. By our premise, $x(t)$ is defined for all $t\ge 0$. Thus, there does not exist a finite escape time, i.e., there does not exists a time $T\ge 0$ such that $||x(t)||\rightarrow +\infty$ as $t\rightarrow T$. Thus, the fact that $x(t)$ is unbounded implies that $||x(t)||\rightarrow +\infty$ as $t\rightarrow +\infty$.

Now, $||u(t)||\rightarrow 0$ as $t\rightarrow +\infty$ implies that there exists a $t_1\ge 0$ such that $\forall t \ge t_1$, $||u(t)||\le \alpha$. In addition, $||x(t)||\rightarrow +\infty$ as $t\rightarrow +\infty$ implies that there exists a $t_2\ge 0$ such that $\forall t \ge t_2$, $||x(t)||\ge \beta$. Let $t_3:= \max \{t_1,t_2\}$. Thus, $\forall t\ge t_3$, $\dot{V}(x(t))\le 0$ which implies that $\forall t\ge t_3$, $V(x(t))\le V(x(t_3))$. This implies that $x(t)$ does not tend to +$\infty$ as t tends to +$\infty$. We have reached a contradiction.$\square$

Theorem 3

[20] If $0$ is a GAS equilibrium of (7.26) then CIBS and CICS are equivalent for (7.25).

Theorem 4

[20] Consider the cascade formed by system (7.25) and the autonomous system $\dot{y}=g(y)$,

$$\begin{aligned}&\dot{x}=f(x,y),\end{aligned}$$

(7.27a)

$$\begin{aligned}&\dot{y}=g(y), \end{aligned}$$

(7.27b)

where $g$ is continuous, $y\in \mathbb {R}^m$. Assume the origin of (7.27b) is GAS and that (7.25) is CICS. Then the origin of (7.27) is GAS.

C: Proof of the Main Results

We begin by demonstrating a series of results regarding the metabolic model when enzymes are kept at a fixed value. In other words, up to and including the proof of Lemma 1 we neglect the enzyme dynamics (7.1b) and assume $e(t) \equiv e$, where $e\in \mathbb {R}^m_{>0}$ is a constant such that Conditions 1–3 hold. In Sect. 7.3.1, we argued that if Conditions 1–3 are satisfied, the metabolic network (7.8) has a unique equilibrium$\bar{s}$.

We now establish global asymptotic stability of the equilibrium. To do this, instead of studying the behaviour of the whole network in one go, we examine the behaviour of individual metabolites, or individual subsystems first, and then using these we establish the stability property for the whole network. We call

$$\begin{aligned} \dot{x}(t) = f_1\left( x(t),e\right) ,\quad x(0)=x_0\in \mathbb {R}_{\ge 0} \end{aligned}$$

the 1st subsystem where $f_1$ is defined as in (7.8). Similarly, we call

$$\begin{aligned} \dot{x}(t) = f_i\left( w(t),x(t),e\right) ,\quad x(0)=x_0\in \mathbb {R}_{\ge 0} \end{aligned}$$

the ith subsystem ¹ where $w\text {:}\,\mathbb {R}_{\ge 0}\rightarrow \mathbb {R}^{i-1}_{\ge 0}$ plays the role of an input and $f_i$ is defined as in (7.8) for $i=2,\dots ,n$. Note that, given that the domain of $f_i$, with $i=2,\dots ,n$, is $\mathbb {R}^{i-1}_{\ge 0}\times \mathbb {R}_{\ge 0}\times \mathbb {R}_{\ge 0}$ (the reaction rates, $v_{j\rightarrow i}$ are only defined for non-negative arguments), it is important that the range of $w$ is $\mathbb {R}^{i-1}_{\ge 0}$ instead of $\mathbb {R}^{i-1}$. For this reason, if we want to employ the CICS machinery introduced in Appendix 2, we first must alter slightly our definition of an input $u(\cdot )$ (Definition 1, Appendix B).

Definition 4

We say that $u(\cdot )$ is an input to the system $\dot{x} = f(x,u)$, $f\text {:}\,A\times B\rightarrow \mathbb {R}^n$ where $A\times B\subset \mathbb {R}^n\times \mathbb {R}^m$, if it is a continuous function that maps from $\mathbb {R}_{\ge 0}$ to $B$.

It can be shown, in a similar manner as in Appendix B and [20], that Lemma 5 and Theorems 3 and 4 hold if one replaces the original definition of an input (Definition 1) with the one above (Definition 4) and $x_0\in \mathbb {R}^n$ with $x_0\in A$.

Returning to our original problem, it is convenient to introduce the change of coordinates $z := x-\bar{s}$ and $u(\cdot ) := w(\cdot )-\bar{s}^i$ where $\bar{s}^i:=\begin{bmatrix}\bar{s}_1&\cdots&\bar{s}_{i-1}\end{bmatrix}^T$ for $i=2,\ldots ,n$. Then, we can re-write the 1st subsystem as

$$\begin{aligned} \dot{z}(t) = f_1\left( z(t)+\bar{s}_1,e\right) ,\quad z(0)=z_0\in [-\bar{s}_1,+\infty ). \end{aligned}$$

and the ith subsystem

$$\begin{aligned} \dot{z}(t) = f_i\left( u(t)+\bar{s}^i,z(t)+\bar{s}_i,e\right) ,\quad z(0)=z_0\in [-\bar{s}_i,+\infty ). \end{aligned}$$

(7.28)

for $i=2,\ldots ,n$. In addition, from now onwards we will say an input $u(\cdot )$ meaning an input to the ith subsystem (7.28) in the sense of Definition 4.

Proposition 1

For any input given $u(\cdot )$, then the $i$th subsystem has a unique, continuous solution $z(t)\in [-s_i,+\infty )$ for all $t\ge 0$.

Proof

Each component of $f(\cdot )$ is a linear combination of globally Lipschitz continuous functions (Assumptions 1 and 2), hence $f(\cdot )$ is globally Lipschitz continuous as well. This and the definition of $u(\cdot )$ (which implies that it is a continuous function of $t$), ensure that the $i$th subsystem, $\dot{z}=f_i\left( u(t)+\bar{s}^i,z(t)+\bar{s}_i,e\right)$, satisfies the usual conditions for global existence of solutions of time varying systems. Hence the $i$th subsystem has a unique, continuous solution $z(t)$ that exists for all $t\ge 0$. Then, due to the positive definiteness of the $g_is$ and $h_is$

$$\begin{aligned} z= -\bar{s}_1 \Rightarrow \dot{z} = f_1\left( 0,e\right) = I_1 - h_1(0,e) = I_1\ge 0 \end{aligned}$$

which proves that $z(t)\in [-\bar{s}_1,+\infty )$ for all $t\ge 0$ were $z(t)$ is the solution of the $1^{st}$ subsystem, and

$$\begin{aligned} z= -\bar{s}_i \Rightarrow \dot{z}&= f_i\left( u(t)+\bar{s}^i,0,e\right) \nonumber \\&= g_i\left( u(t)+\bar{s}^i,e\right) - h_i(0,e) = g_i\left( u(t)+\bar{s}^i,e\right) \ge 0 \end{aligned}$$

which proves that $z(t)\in [-\bar{s}_i,+\infty )$ for all $t\ge 0$ were $z(t)$ is the solution of the $i$th subsystem, $i=2,\ldots ,n$.$\square$

Proposition 1 is important for two reasons. First, it allows us to regard the state space of $i$th subsystem, (7.28), to be $[-\bar{s}_i,+\infty )$ instead of $\mathbb {R}$. This makes sense, we are only interested in non-negative concentrations of the metabolites. Second, it shows that the vector containing the state of the first $i-1$ subsystems is input to the $i$th subsystem, in the sense of Definition 4.

Proposition 2

The $i$th subsystem is CIBS, for any $i=2,\ldots ,n$.

Proof

Let $V:\mathbb {R}_{\ge 0}\rightarrow \mathbb {R}_{\ge 0}$ be defined as

$$\begin{aligned} V(z) := \frac{1}{2}z^2 \Rightarrow \dot{V}(z) = \frac{\partial V}{\partial z} \dot{z}= z\dot{z}=z\left( g_i\left( u+\bar{s}^i,e\right) -h_i\left( z+\bar{s}_i,e\right) \right) . \end{aligned}$$

By Condition 3, $\hat{h}_i(e)>g_i(\bar{s}^i,e)$ thus $\hat{h}_i(e)\ge g_i(\bar{s}^i,e) +\delta _1$, for some $\delta _1>0$. In addition, by continuity and monotonicity of $g_i$ (monotonicity in each of its arguments), there exists a sufficiently small $\alpha >0$ such that

$$\begin{aligned} g_i\left( \alpha 1\!\!\!\!1 +\bar{s}^i,e\right) -g_i\left( \bar{s}^i,e\right) \le \frac{\delta _1}{2}, \end{aligned}$$

where $1\!\!\!\!1 :=\begin{bmatrix}1&\dots&1\end{bmatrix}^T$. In addition,

$$\begin{aligned} ||u||_\infty \le \alpha \Rightarrow g_i\left( u+\bar{s}^i,e\right) \le g_i\left( \alpha 1\!\!\!\!1 +\bar{s}^i,e\right) \le g_i\left( \bar{s}^i,e\right) + \frac{\delta _1}{2} \le \hat{h}_i(e) - \frac{\delta _1}{2}. \end{aligned}$$

Hence, we have

$$\begin{aligned} ||u||_\infty \le \alpha \Rightarrow g_i\left( u+\bar{s}^i,e\right) -h_i\left( z+\bar{s}_i,e\right) \le \hat{h}_i(e)- \frac{\delta _1}{2}-h_i(z+\bar{s}_i,e). \end{aligned}$$

Because $h_i(z+\bar{s}_i,e)\rightarrow \hat{h}_i(e)$ from below as $z\rightarrow +\infty$ we can always find a $\beta _1$ such that $z \ge \beta _1 \Rightarrow \hat{h}_i(e)-h_i(z+\bar{s}_i,e) \le \delta _2$ for any given $\delta _2>0$. In addition, because $z\in [-\bar{s}_i,+\infty )$, $||z||>\bar{s}_i$ implies $||z||=z$. Hence, choosing $\delta _2\le \frac{\delta _1}{2}$ and defining $\beta := \max \{\beta _1,\bar{s}_i+\varepsilon \}$, where $\varepsilon >0$, we have

$$\begin{aligned} u,z\text {:}\,||u||_\infty \le \alpha , ||z|| \ge \beta \Rightarrow \dot{V}(z) \le z(\hat{h}_i(e)-h(\hat{s}_i,e)-\frac{\delta _1}{2}) \le z(\delta _2- \frac{\delta _1}{2})\le 0 \end{aligned}$$

Then, applying Lemma 5 completes the proof.$\square$

Proposition 3

The origin of $i$th subsystem with zero input (i.e., $u(t)\equiv 0$) is a globally asymptotically stable equilibrium, for any $i=1,\ldots ,n$.

Proof

We use the Lyapunov function

$$\begin{aligned} V(z) := \frac{1}{2}z^2 \Rightarrow \dot{V}(z)&= \frac{\partial V}{\partial z} \dot{z}= zf_i(\bar{s}^i,z(t)+\bar{s}_i,e)\nonumber \\&= z(g_i(\bar{s}_1,\dots ,\bar{s}_{i-1},e)-h_i(z+\bar{s}_i,e)). \end{aligned}$$

By the definition of $\bar{s}$, we have that $g_i(\bar{s}_1,\dots ,\bar{s}_{i-1},e)=h_i(\bar{s}_i,e)$. So

$$\begin{aligned} \dot{V}(z) = z(h_i(\bar{s}_i,e)-h_i(z+\bar{s}_i,e)). \end{aligned}$$

Due to the strict monotonicity of $h_i$, $z$ and $(h_i(\bar{s}_i,e)-h_i(z+\bar{s}_i,e))$ have opposite signs and are both equal to zero if and only if $z=0$. Hence, applying Lyapunov’s Direct Method completes the proof.$\square$

Proposition 4

The $i$th subsystem is CICS, for any $i=1,\ldots ,n$.

Proof

This follows directly from Propositions 2 and 3 and Theorem 3.$\square$

With these preliminary results in mind, we are now ready to prove Lemma 1.

Proof

(Lemma 1) $\square$As previously pointed out, the solution to the first subsystem is an input to the second subsystem, in the sense of Definition 4. Consider the cascade obtained by setting the input of the 2nd subsystem to the state of the 1st subsystem,

$$\begin{aligned} \dot{z}_1(t)&= f_1\left( z_1(t)+\bar{s}_1,e\right) ,\nonumber \\ \dot{z}_2(t)&= f_2\left( z_1(t)+\bar{s}_1,z_2(t)+\bar{s}_2,e\right) . \end{aligned}$$

Propositions 3 (i.e., the origin of the 1st subsystem is a GAS equilibrium) and 4 (i.e., the 2nd subsystem is CICS) and Theorem 4 (i.e., that the origin of the interconnection of an autonomous system which has a GAS equilibrium at the origin and a CICS system has a GAS equilibrium) imply that the origin of the above cascade is GAS. Then, by induction, we see that the origin of the system obtained by iteratively cascading the $i$th subsystem with the cascade formed by the previous $i-1$ subsystems is a GAS equilibrium. In other words, the origin of

$$\begin{aligned} \dot{z} = f(z+\bar{s},e) \end{aligned}$$

is a GAS equilibrium, which completes the proof.$\square$

Proposition 5

Let $A$ denote the subset of $\,\mathbb {R}^m_{>0}$ whose elements are such that Condition 3 holds. There exists a unique function $\phi \text {:}\,A\rightarrow \mathbb {R}^n_{\ge 0}$ such that $f(\phi (e),e)=0$ for all $e\in A$. Furthermore, this function is continuously differentiable and globally Lipschitz continuous.

Proof

Existence and uniqueness of $\phi$ follows from our discussion in Sect. 7.3.1 of the main text regarding the existence and uniqueness of an equilibrium if the enzymes are constant. Each component of $f(\cdot )$ is a linear combination of continuously differentiable and globally Lipschitz continuous functions (Assumptions 1 and 2). Thus, $f(\cdot )$ is continuously differentiable and globally Lipschitz continuous or, equivalently its partial derivatives exists everywhere, are continuous and bounded. The fact that $f(\phi (e),e)=0$ for all $e\in A$ implies that the total derivative of $f(\cdot )$ along $\begin{bmatrix}\phi (e)&e\end{bmatrix}^T$ is also equal to zero, i.e., $f'(\phi (e),e)=0$ for all $e\in A$. The total derivative of a function exists and is continuous if and only if the partial derivatives of the function exist and are continuous. Hence,

$$\begin{aligned} \frac{\partial f}{\partial \phi }(\phi (e),e)\frac{\partial \phi }{\partial e}(e) + \frac{\partial f}{\partial e}(\phi (e),e)=0 \end{aligned}$$

which implies that

$$\begin{aligned} \frac{\partial \phi }{\partial e}(e) = -\left( \frac{\partial f}{\partial \phi }(\phi (e),e)\right) ^{-1}\frac{\partial f}{\partial e}(\phi (e),e). \end{aligned}$$

By Condition 1, $v_{j\rightarrow i} \equiv 0$ if $i<j$. Hence, $i<j \Rightarrow \frac{\partial f_i}{\partial \phi _j}(\phi (e),e)=\frac{\partial v_{j\rightarrow i}}{\partial \phi _j}(\phi _j(e),e)= 0$. Thus, $\frac{\partial f}{\partial \phi }(\phi (e),e)$ is lower triangular. Furthermore, by Condition 2, $h_i$ is strictly increasing, hence

$$\begin{aligned} \frac{\partial f_i}{\partial \phi _i}(\phi (e),e) = -\frac{\partial h_i}{\partial \phi _i}(\phi _i(e),e) < 0. \end{aligned}$$

Thus, $\left( \frac{\partial f}{\partial \phi }(\phi (e),e)\right) ^{-1}$ exists for all $e\in A$. Hence, $\frac{\partial \phi }{\partial e}(e)$ exists for all $e\in A$. Furthermore, $\frac{\partial \phi }{\partial e}(e)$ is continuous and bounded which shows that $\phi$ is continuously differentiable and globally Lipschitz continuous.$\square$

We are now in a position to prove Lemma 2 and Theorem 1.

Proof

(Lemma 2) The existence and uniqueness of $s(t)$ and $e(t)$ follow from our assumption that $f(\cdot )$ and $g(\cdot )$ are smooth and globally Lipschitz continuous (Assumptions 1–3). The existence and uniqueness of $\phi (\cdot )$ is proven in Proposition 5. The domain of $\phi (\cdot )$ is $A$. Thus, (7.14) is well-defined if and only if $\bar{e}(t)$ remains in $A$. This is ensured by the premise, $B\subseteq A$ is forward invariant with respect to (7.14) and $e_0\in B$. In addition, $g(\cdot )$ and $\phi (\cdot )$ are globally Lipschitz continuous (Assumption 3, Proposition 5, respectively), which implies that (7.14) satisfies the usual conditions for global existence and uniqueness solutions.$\square$

Proof

(Theorem 1) The proof is an application of Tikhonov’s Theorem on finite time intervals (Theorem 2). The existence and uniqueness of $\phi (\cdot )$ satisfies the first condition in the premise of Theorem 2 which requires that the metabolite dynamics, $f(s,e)$, has a unique root.

The second condition of Tikhonov’s Theorem is that $z=\phi (e_0)$ is a globally asymptotically stable equilibrium, uniformly in $e_0$, of the boundary layer system $\dot{z}=f(z,e_0)$. Lemma 1 shows that for any given $e_0\in B\subseteq A$, $\phi (e_0)$ is a globally asymptotically stable equilibrium of $\dot{z}=f(z,e_0)$. The fact that $B$ is compact combined with the previous statement form the premise of Lemma 3. Then, Lemma 3 establishes the desired result, i.e., that the equilibrium $z=\phi (e_0)$ is a globally asymptotically stable, uniformly in $e_0$.

Proposition 5 shows that $\phi (\cdot )$ is continuous. Because $\bar{e}(t)\in B$ for all time, and $B$ is a compact set, $\bar{s}(t)=\phi (\bar{e}(t))$ must also be confined to some compact set. In the proof of Proposition 5 we established that for any given $e\in B\subseteq A$, the eigenvalues of the Jacobian of the boundary layer system evaluated at $[\phi (e),e]^T$, $\frac{\partial f}{\partial \phi }(\phi (e),e)$, have negative real parts. The previous two statements form the premise of Lemma 4 which shows that the eigenvalues of the Jacobian of the boundary layer system, evaluated along $\begin{bmatrix}\phi (\bar{e}(t))&\bar{e}(t)\end{bmatrix}^T$ have real parts smaller than a certain negative real number, i.e., that the third condition of Tikhonov’s theorem is satisfied. $\square$