# Metabolism in states of maximal enzyme efficiency

Andreas Kremling, Wolfram Liebermeister, Elad Noor, Meike T. Wortel

► To cite this version:

Andreas Kremling, Wolfram Liebermeister, Elad Noor, Meike T. Wortel. Metabolism in states of maximal enzyme efficiency. The Economic Cell Collective. Economic Principles in Cell Biology, 2023, 10.5281/zenodo.8164468. [https://doi.org/10.5281/zenodo.8164468](https://doi.org/10.5281/zenodo.8164468)

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# Optimization of metabolic states

Andreas Kremling, Wolfram Liebermeister, Elad Noor, Meike Wortel

► To cite this version:

Andreas Kremling, Wolfram Liebermeister, Elad Noor, Meike Wortel. Optimization of metabolic states. The Economic Cell Collective. Economic Principles in Cell Biology, The Economic Cell Collective, 2023, 10.5281/zenodo.8164468. [https://hal.inrae.fr/hal-04172128v2](https://hal.inrae.fr/hal-04172128v2)

HAL Id: hal-04172128

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Submitted on 6 Aug 2025

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archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

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recherche français ou étrangers, des laboratoires
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# Metabolism in states of maximal enzyme efficiency

Andreas Kremling¹ ID, Wolfram Liebermeister² ID, Elad Noor³ ID and Meike T. Wortel⁴ ID

¹ Systems Biotechnology, School of Engineering and Design, Technical University of Munich, Germany; ² Université Paris-Saclay, INRAE, MalAGE, 78350 Jouy-en-Josas, France; ³ Department of Plant and Environmental Sciences, Weizmann Institute of Science, 76100 Rehovot, Israel; ⁴ Molecular Biology and Microbial Food Safety, Swammerdam Institute for Life Sciences, University of Amsterdam, the Netherlands

## Abstract

Enzyme-efficient states are metabolic states that realize a given flux objective at a minimal enzyme cost. Enzyme-efficient states can be hypothesized to be used under growth rate optimization regimes, as biomass production rate per enzyme can be converted to into cell growth rate. In models without further constraints, enzyme-efficient states are Elementary Flux Modes (EFMs). This allows for a algorithm to find enzyme-efficient states: Enumerate the EFMs, calculate the minimal enzyme cost per EFM, and choose the one with the lowest enzyme investment. This algorithm allows for finding of enzyme efficient states for larger models then can be optimized by 'brute force', but still need to be small enough to enumerate the EFMs. Such optimization has lead to the insights on the effect of changing external nutrient conditions: As growth conditions are changing, the optimal flux profile either changes continuously (and metabolite and enzyme concentrations change continuously as well) when the same EFM remains optimal, or fluxes change discontinuously together with metabolite and enzyme concentrations when a different EFM becomes optimal.

**Contributions:** This chapter was planned and written by the authors and initially discussed with J. Zanghellini. The chapter was reviewed by D.S. Tourigny and H. Dourado and discussed with S. Müller.

**Keywords:** Elementary Flux Mode (EFM), Enzyme Cost Minimization

**To cite this chapter:** A. Kremling, W. Liebermeister, E. Noor, M.T. Wortel. Metabolism in states of maximal enzyme efficiency. Version July 2023. The Economic Cell Collective. Economic Principles in Cell Biology (2023). No commercial publisher | Authors open access book.

The authors are listed in alphabetical order.

<figure><img src="image_2.png" /></figure>

This is a chapter from the free textbook "Economic Principles in Cell Biology".
Free download from [principlescellphysiology.org/book-economic-principles/](https://principlescellphysiology.org/book-economic-principles/).
Lecture slides for this chapter are available on the website.

© 2023 The Economic Cell Collective.

Licensed under Creative Commons License CC-BY-SA 4.0.

An authors open access book. No publisher has been paid.

<figure><img src="image_3.png" /></figure>
# Optimization of metabolic states

Andreas Kremling¹ ID, Wolfram Liebermeister² ID, Elad Noor³ ID and Meike T. Wortel⁴ ID

¹ Systems Biotechnology, School of Engineering and Design, Technical University of Munich, Germany; ² Université Paris-Saclay, INRAE, MaIAGE, 78350 Jouy-en-Josas, France; ³ Department of Plant and Environmental Sciences, Weizmann Institute of Science, 76100 Rehovot, Israel; ⁴ Swammerdam Institute for Life Sciences, University of Amsterdam, Amsterdam, the Netherlands

## Abstract

Cells in a well-mixed and nutrient-rich environment can be expected to experience selection on their growth rate. Such cells optimize their metabolic state to achieve a high growth rate. Metabolic states that lead to a high growth rate are states that realize a maximal biomass production rate at a minimal enzyme cost. Metabolic states that optimize a specific flux at minimal enzyme cost are called enzyme-efficient metabolic states and in this chapter we refer to them as optimal metabolic states. The calculation of optimal metabolic states is facilitated by the result that, in models without further constraints, the flux distributions in enzyme-efficient states are Elementary Flux Modes (EFMs). This result allows for an algorithm to find enzyme-efficient states by: 1) Enumerating the EFMs, 2) calculating the minimal enzyme cost per EFM, and 3) choosing the one with the lowest enzyme investment. This algorithm finds optimal metabolic states for larger models which cannot be optimized by 'brute force', but are still small enough to enumerate the EFMs. Finding optimal metabolic states uncovered the effect of changing external nutrient conditions: As growth conditions are changing, the optimal flux profile either changes continuously (and metabolite and enzyme concentrations change continuously as well) when the same EFM remains optimal, or fluxes change discontinuously together with metabolite and enzyme concentrations when a different EFM becomes optimal.

**Contributions:** This chapter was drafted and written by Andreas Kremling, Wolfram Liebermeister, Elad Noor, and Meike Wortel. It was initially discussed with Jürgen Zanghellini and later reviewed by David S. Tourigny and Hugo Dourado and discussed with Stefan Müller.

**Keywords:** elementary flux mode (EFM) - enzyme cost minimization - enzyme-efficient state

**To cite this chapter:** A. Kremling, W. Liebermeister, E. Noor, M.T. Wortel. Optimization of metabolic states (Version July 2025). doi: 10.5281/zenodo.8164468. Chapter from: The Economic Cell Collective (2025). Economic Principles in Cell Biology. No commercial publisher | Online open access book | doi: 10.5281/zenodo.8156386

The authors are listed in alphabetical order.

<figure><img src="image_2.png" /></figure>

This is a chapter from the open textbook "Economic Principles in Cell Biology".
Free download from [principlescellphysiology.org/book-economic-principles/](http://principlescellphysiology.org/book-economic-principles/).
Lecture slides for this chapter are available on the website.

© 2025 The Economic Cell Collective.

Licensed under Creative Commons License CC-BY-SA 4.0.
An online open access book. No publisher has been paid.
doi: 10.5281/zenodo.8156386

<figure><img src="image_3.png" /></figure>
## Chapter highlights

* Enzyme-efficient states are metabolic states that realize a given flux objective at a minimal enzyme cost.

* In models without further constraints, enzyme-efficient states are Elementary Flux Modes (EFMs).

* Elementary Flux Modes can be used to find enzyme-efficient states in networks that would be too large to optimize metabolic states "by brute force".

* Biomass per enzyme efficiency can be converted to into cell growth rate.

* As growth conditions are changing, the flux profile either changes continuously (and metabolite and enzyme concentrations as well) or fluxes change discontinuously together with metabolite and enzyme concentrations.

## 7.1. Introduction

In a simple economic picture of cells, we assume that cells adjust their metabolic state in each environment to obtain a maximal fitness advantage. This may be impossible in reality, but it remains an interesting question what this best metabolic state would look like, according to our knowledge of cells. So what is the best metabolic state overall (comprising metabolic fluxes, metabolite concentrations and enzyme levels)? What pathways should a cell use, which enzymes should be induced or repressed, and how should this change in a new environment? To answer this, we need to remember that all metabolic variables (fluxes, metabolite levels, enzyme levels, and enzyme efficiencies) depend on each other. Physically, fluxes depend on metabolite concentrations through kinetics and enzyme regulation (e.g. competitive inhibition) and metabolites are produced and consumed by the fluxes until a steady state is reached. Hence, if we think in terms of cellular economics (treating enzymes as control variables), then all metabolic variables must be optimized together.

In the previous chapters we saw some ways to predict optimal metabolic fluxes, metabolite concentrations and enzyme levels separately: in Flux Balance Analysis (FBA, Chapter 5 in [1]), we optimized fluxes (while ignoring kinetics or assuming constant enzyme efficiencies); in Enzyme Costs Minimization (Chapter 6 in [1]), in contrast, fluxes were fixed and given and concentrations (and enzyme efficiencies) were optimized. Each of these methods is based on a strong assumption: FBA requires measured flux ranges and/or apparent catalytic rates and assumes enzyme saturation effects can be neglected, while enzyme cost minimization requires a given flux distribution. But what if we don't know any of the variables in advance? How can we predict all of them from first principles?

Before thinking about this, let us briefly step back: what do we actually mean by an "optimal state"? What quantity should be maximized in metabolism? There could be very different aims (e.g. production in biotechnology, vs offspring and survival in a wild-type cell). However, in both cases an important aim is cell growth – or at least, avoiding strong growth deficits. Below we will see that cell growth depends, to a good approximation, on biomass/enzyme efficiency, that is, biomass production per total enzyme invested. Hence, whenever fast growth is important, cells should maximize this efficiency.

Thus, we will consider the following optimality problem: maximize biomass/enzyme efficiency, defined as the production flux per invested enzyme with respect to all metabolic variables (metabolites, enzymes and fluxes) and under all constraints (steady state, enzyme kinetics, etc). Solutions to this problem are considered optimal states.

## 7.2. The enzyme-efficient metabolic states are Elementary Flux Modes

The optimization problem in this chapter is to reach maximal objective flux(es) for minimal enzyme investment. The biological interpretation is that this would lead to the highest growth rate, because it optimizes the ratio between gains (flux(es)) and costs (enzymes). When we solve this optimization problem with mathematical tools, it is convenient to either find the minimal enzyme investment for a certain flux, or the maximum flux for a fixed enzyme investment. Although one could think of different biological explanations for those two ways to state the optimization problem,
## Chapter overview

* Optimal metabolic states in this chapter refer to enzyme-efficient states, which are metabolic states that realize a given objective flux at a minimal enzyme cost.

* In models without further flux constraints, flux distributions of enzyme-efficient states are Elementary Flux Modes (EFMs).

* Elementary Flux Modes can be used to find enzyme-efficient states in networks that would be too large to optimize metabolic states "by brute force".

* Biomass per enzyme efficiency can be converted to cell growth rate by simple approximate formulae.

* The Elementary Flux Mode that is realized in an enzyme-efficient state depends on the external conditions.

* As growth conditions change, either the flux profile changes continuously (together with metabolite and enzyme concentrations), or fluxes change discontinuously, implying jumps also in metabolite and enzyme concentrations.

## 7.1. Introduction

In a simple economic picture of cells, we assume that cells adjust their metabolic state in each environment to obtain a maximal fitness advantage. This may be impossible in reality, but it remains an interesting question what this best metabolic state would look like, according to our knowledge of cells. So what is the best metabolic state overall (comprising metabolic fluxes, metabolite concentrations and enzyme levels)? What pathways should a cell use, which enzymes should be induced or repressed, and how should this change in a new environment? To answer these questions, we need to remember that all metabolic variables (fluxes, metabolite levels, enzyme levels, and enzyme efficiencies) depend on each other. Physically, fluxes depend on metabolite concentrations through kinetics and enzyme regulation (e.g. competitive inhibition) and metabolites are produced and consumed by the fluxes until a steady state is reached. Hence, if we think in terms of cellular economics (treating enzymes as control variables), then all metabolic variables must be optimized together.

In the previous chapters we saw some ways to predict optimal metabolic fluxes, metabolite concentrations and enzyme levels separately: in Flux Balance Analysis (FBA, Chapter 5 in [1]), we optimized fluxes by maximizing an objective function (typically biomass) while in Enzyme Cost Minimization [2, 3] (Chapter 6 in [1]) metabolite concentrations were optimized by minimizing cost (or, equivalently, maximizing the enzyme efficiencies). Each of these methods is based on a strong assumption: FBA requires measured flux ranges and/or apparent catalytic rates and assumes enzyme saturation effects can be neglected, while enzyme cost minimization requires a given flux distribution. But what if we don't know any of the variables in advance? How can we predict all of them from first principles?

Before thinking about this, let us briefly step back: what do we actually mean by an "optimal state"? What quantity should be maximized in metabolism? There could be very different aims (e.g. production in biotechnology, versus number of offspring and survival in a wild-type cell). However, in both cases an important aim is cell growth – or at least, avoiding strong growth deficits. Below we will see that cell growth depends, to a good approximation, on biomass/enzyme efficiency, that is, biomass production per total enzyme invested. Hence, whenever fast growth is important, cells should maximize this efficiency.

In conclusion, we will consider the following optimality problem: maximize biomass/enzyme efficiency, defined as the production flux per invested enzyme with respect to all metabolic variables (metabolites, enzymes and fluxes) and under all constraints (steady state, enzyme kinetics, etc.). Solutions to this problem are considered optimal states.
mathematically they are equivalent. For the outline of the proof that optimal states are Elementary Flux Modes, it is convenient to fix the objective flux to an arbitrary value (we choose 1) and then minimize the enzyme investment. This leads to the following optimization problem:

$$
\begin{array}{ll}
\underset{v,e,c}{\text{minimize}} & \displaystyle\sum_{i=1}^{r} h_i e_i \qquad (7.1) \\
\text{subject to:} & \mathbf{N} \cdot \mathbf{v} = \mathbf{0} \qquad \text{steady state} \\
& \forall i : v_i = e_i f_i(\mathbf{c}) \qquad \text{enzyme kinetics} \\
& \mathbf{e}, \mathbf{c} \ge 0 \qquad \text{positive concentrations} \\
& v_r = 1 \qquad \text{fixed objective flux} \\
& \mathbf{c} \le c_{\max} \qquad \text{metabolite bounds}
\end{array}
$$

where $\forall i$ means for all reactions $i$, $r$ is the number of reactions (with the last the objective), $h_i$ are the weights, and bold
face items denote vectors. This optimization problem states that by adjusting the fluxes ($v$), metabolite concentrations
($c$) and enzyme concentrations ($e$), the cost (sum of the costs ($h_i e_i$) for every reaction) is minimized, while keeping the
objective flux constant. The weights ($h_i$) can be thought of as the size or production costs of the enzymes (measured
in e.g. molecular weight or gene length), which might be different for different enzymes. We need certain constraints
to be met for this solution: 1) the metabolic network needs to be in steady state to avoid built-up of intermediates, 2) enzyme kinetics: the flux of each reaction ($v_i$) has to be equal to the enzyme concentration ($e_i$) times a metabolite dependent term ($f_i(c)$), 3) all enzyme metabolite concentrations have to be positive, and 4) the objective flux is equal to
1. Optionally, we can add bounds on the metabolite levels, which is mostly necessary with irreversible kinetics. Reversible kinetics usually lead to bounded metabolite levels, because very high concentrations of products will inhibit the reaction forming the products.

In this section we will explain why the optimal state is reached at an Elementary Flux Mode (EFM). One important starting point is that, as we have seen before in Chapter 4 in [1], convex optimization problems with only positivity or equality constraints (no other inequalities) lead to an optimal solution at an extreme points of the feasible solution space, and those extreme points are Elementary Flux Modes (EFMs). However, the optimization problem (7.1) is not convex, mainly due to the hyperbolic dependency of reaction rates on the metabolite concentrations ($f_i(c)$ is usually not linear). There are several ways to proof that the solution of this optimization problem is an EFM, of which some are outlined in the papers by Wortel et al. [2] and Müller et al. [3]. In this book we will outline the proof that a solution that would not be an EFM, will lead to a contradiction.

Assume we have some optimal state that is not an EFM. Any optimal solution is associated with a set of fluxes, and enzyme and metabolite concentrations. Now we set the metabolite concentrations to the concentrations of the assumed optimal state. Then all metabolite dependent terms ($f_i(c)$) become constants and we return to a convex problem. As explained in Chapter 9 in [1] and Figure 7.1, the optimum (now in terms of enzyme concentrations and fluxes) of this problem is a flux distribution that is an EFM. But that is a contradiction: we starting with assuming the optimal state from which we took the set of enzyme concentrations was not an EFM. We conclude that the optimal state has to be an EFM.

7.3. Illustration with an example network

To illustrate the proof, we study a simple network that we have seen previously in Chapter 5 in [1] (Figure 7.2), in which
we will now include enzyme kinetics. We will use the factorized rate law as in Chapter 6 in [1], but then generalized for
$n_s$ substrates and $n_p$ products (also compare Eq. (3.15 in [1]) in Chapter 3 in [1]):
## 7.2. Enzyme-efficient metabolic states use elementary flux modes

The optimization problem in this chapter is to reach maximal objective flux with minimal enzyme investment. The biological interpretation is that this would lead to the highest growth rate, because it optimizes the ratio between gains (fluxes) and costs (enzymes). When we solve this optimization problem with mathematical tools, it is convenient to either find the minimal enzyme investment for a certain flux, or the maximum flux for a fixed enzyme investment. Although one could think of different biological explanations for those two ways to state the optimization problem, mathematically they are equivalent. For the outline of the proof that optimal states are elementary flux modes, it is convenient to fix the objective flux to an arbitrary value (we choose 1) and then minimize the enzyme investment. This leads to the following optimization problem over the fluxes (v), enzymes levels (e) and internal metabolite concentrations (s):

$$
\begin{array}{ll}
\underset{\mathbf{v},\mathbf{e},\mathbf{s}}{\text{minimize}} & \displaystyle\sum_{i=1}^{r} h_i e_i \qquad (7.1) \\
\text{subject to:} & \mathbf{N} \cdot \mathbf{v} = \mathbf{0} \qquad \text{steady state} \\
& \forall i : v_i = e_i \quad \kappa_i(\mathbf{s}) \qquad \text{enzyme kinetics} \\
& \mathbf{e}, \mathbf{s} \ge \mathbf{0} \qquad \text{positive concentrations} \\
& v_r = 1 \qquad \text{fixed objective flux} \\
& \mathbf{s} \le \mathbf{s}_{\text{max}} \qquad \text{metabolite bounds}
\end{array}
$$

where $h_i$ are the weights, $\mathbf{N}$ is the stoichiometry matrix, $i$ is the index of the reactions (ranging from 1 to $r$), with the last reaction (with index $r$) representing the objective. This optimization problem states that by adjusting the fluxes (v), metabolite concentrations (s) and enzyme concentrations (e), the total cost (sum of costs – $h_i e_i$ – for every reaction) is minimized, while keeping the objective flux constant (any arbitrary constant can be chosen, here we chose 1). The weights ($h_i$) can be thought of as the size or production costs of the enzymes (measured, for example, in molecular weight or gene length) We require certain constraints: (i) the metabolic network needs to be in steady state to avoid built-up of intermediates, (ii) enzyme kinetics – the flux of each reaction ($v_i$) has to be equal to the enzyme concentration ($e_i$) times a metabolite dependent (e.g., saturation) term ($\kappa_i(s)$), (iii) all enzyme and metabolite concentrations have to be positive, (iv) the objective flux is equal to 1, and (v) the metabolite concentrations are within their given bounds. The latter constraint is optional and is mostly necessary when dealing with irreversible kinetics. Reversible kinetics usually lead to bounded metabolite levels because very high concentrations of products inhibit the reaction that forms the products.

In this section, we will explain why the optimal state is reached at an Elementary Flux Mode (EFM). One important starting point is that, as we have seen before in Chapter 4 in [1], convex optimization problems with only positivity or equality constraints (no other inequalities) lead to an optimal solution at an extreme point of the feasible solution space, and those extreme points are Elementary Flux Modes. However, the optimization problem (7.1) is not convex, mainly due to the hyperbolic dependence of reaction rates on the concentrations of metabolites ($\kappa_i(s)$ is usually not linear in the internal metabolite concentrations).

There are several ways to prove that the solution of this optimization problem is an EFM, of which some are outlined in the papers by Wortel et al. [4] and Müller et al. [5]. Here we will outline a proof by contradiction: assuming a solution to the optimization problem that is not an EFM and showing that this leads to a contradiction.

**Theorem 1.** The flux distribution that maximizes an objective flux over the total enzyme cost in a metabolic network without additional constraints is an Elementary Flux Mode.

*Proof.* Assume we have some optimal state where the flux distribution is not an EFM. Any optimal solution is
<figure><img src="image_4.png" /><figcaption>Figure 7.1: Translation from flux to enzyme space retains EFMs as extreme rays. The top left panel shows the feasible flux space with the steady state constraints, all fluxes postive (using splitting of fluxes, as explained in the text, if necessary) and a fixed objective flux. The extreme points here are points where one flux becomes 0 and are Elementary Flux Modes (see Chapter 5 in [1]). Here we show that when we have assumed metabolite concentrations, such as when we keep them at an optimal solution, we get a linear transformation and the extreme rays are maintained. Different metabolite levels, for example solutions to different environmental conditions, can lead to different transformations and therefore different optima (mimimal total enzyme), but those are always located at an EFM.</figcaption></figure>

$$v = e \cdot k_{\text{cat}}^{+} \cdot \frac{\prod_{n_s}^{j=1} s_j / K_{S,j}}{1 + \prod_{n_p}^{k=1} p_k / K_{P,k} + \prod_{n_s}^{j=1} s_j / K_{S,j}} \cdot (1 - e^{\Delta G'_{r}/RT}) \quad (7.2)$$

See Box 7.4 for all detailed rate laws of the example networks. We can simplify this equation by combining the forward catalytic constant, the thermodynamic efficiency factor, the saturation efficiency factor, and the regulation efficiency factor (if that exists) in a function f(c), which only depends on the metabolites, and not on the enzyme concentrations. We will below write f for f(c).

$$v_i = e_i \cdot f_i \quad (7.3)$$

Now we take $v_{BM} = 1$ and optimize all fluxes, enzymes concentrations and metabolite concentrations to minimize the enzyme costs, while satisfying the constraints posed in Equations (7.1), for different levels of external glucose and standard levels of the other external metabolites. We see that for different external glucose concentrations, we get different optimal fluxes, enzyme levels and metabolite levels (Table 7.1).

We notice that the total enzyme needed for the biomass flux of one decreases with the increasing glucose levels, as we expect. Also, the optimal level of internal glucose increases with increasing external glucose. Moreover, the fluxes of the solutions do follow an EFM (see Figure 7.2b). We can now reformulate the problem for only the flux and enzyme levels, while keeping the metabolite levels as they are in the table. With the metabolite levels in the first row of the table, we
<figure><img src="image_4.png" /><figcaption>Figure 7.1: Translation from flux to enzyme space retains EFM as extreme rays – The top left panel shows the feasible flux space with the steady state constraints, all fluxes positive (using splitting of fluxes, as explained in the text, if necessary) and a fixed objective flux. The extreme points here are points where one flux becomes 0 and are elementary flux modes (see Chapter 5 in [1]). Here we show that when we have assumed metabolite concentrations, such as when we keep them at an optimal solution, we get a linear transformation and the extreme rays are maintained. Different metabolite levels, for example solutions to different environmental conditions, can lead to different transformations and therefore different optima (minimal total enzyme), but those are always located at an EFM.</figcaption></figure>

associated with a set of fluxes, enzyme concentrations and metabolite concentrations. Now we set the metabolite concentrations to the concentrations of the assumed optimal state. Then all metabolite-dependent terms ($κ_i(s)$) become constants, and we return to a convex problem. As explained in Chapter 10 in [1] and Figure 7.1, the optimum of this problem (now in terms of enzyme concentrations and fluxes) is a flux distribution that is an EFM. But this contradicts our initial assumption that the optimal state from which we took the set of metabolite concentrations was not an EFM. The proof by contradiction shows that the optimal state must be an EFM. ☐

## 7.3. Enzyme-efficient states in an example network

To illustrate the proof, we study a simple network representing growth on glucose and pyruvate that we have seen previously in Chapter 5 in [1] (Figure 7.2). We use G and P for glucose and pyruvate in the equations, we use the subscript ex when a metabolite is extracellular and square brackets to denote a concentration. For the use in this chapter, we add enzyme kinetics to this network. We will use the factorized rate law as in Chapter 6 in [1], but then
<table><caption>Table 7.1: Outcomes of the optimization of the example network with standard kinetics, parameter values and external concentrations (see Box 7.4) for varying levels of [G<sub>ex</sub>].</caption><thead><tr><th>[G<sub>ex</sub>]</th><th>E<sub>tot</sub></th><th>v<sub>0</sub></th><th>v<sub>1</sub></th><th>v<sub>2</sub></th><th>v<sub>3</sub></th><th>v<sub>4</sub></th><th>v<sub>BM</sub></th><th>e<sub>0</sub></th><th>e<sub>1</sub></th><th>e<sub>2</sub></th><th>e<sub>3</sub></th><th>e<sub>4</sub></th><th>e<sub>BM</sub></th><th>[G]</th><th>[P]</th><th>[ATP]</th><th>[ADP]</th></tr></thead><tbody><tr><td>0.01</td><td>156.2</td><td>5</td><td>5</td><td>0</td><td>9</td><td>0</td><td>1</td><td>54.4</td><td>4.4</td><td>0</td><td>94.4</td><td>0</td><td>2.9</td><td>0.08</td><td>15.14</td><td>0.05</td><td>20.09</td></tr><tr><td>0.1</td><td>91.3</td><td>50</td><td>50</td><td>99</td><td>0</td><td>0</td><td>1</td><td>61.3</td><td>11.3</td><td>14.2</td><td>0</td><td>0</td><td>4.4</td><td>0.13</td><td>4.55</td><td>0.11</td><td>20.09</td></tr><tr><td>1</td><td>36.2</td><td>50</td><td>50</td><td>99</td><td>0</td><td>0</td><td>1</td><td>13.0</td><td>8.0</td><td>12.5</td><td>0</td><td>0</td><td>2.7</td><td>0.60</td><td>7.65</td><td>0.11</td><td>20.09</td></tr></tbody></table>

can linearly relate the enzyme and flux levels (with the factors $f_i$), and thereby the extreme rays of the enzyme and flux space will be equal and EFMs, as pointed out above (see also Chapter 5 in [1], Figure 7.1). Optimization in this space will lead to the optimal flux distributions following an EFM. If the fluxes in the table would not follow an EFM, this will lead to an inconsistency; if there is an optimum, this should still be the outcome if we keep some variables in the optimum constant and optimize over the rest of the variables.

We point out two important aspects, using the network (Figure 7.2) as an example. First, it is convenient to split reversible reactions, such that fluxes are always positive. In this case that means that the reversible reaction from P to P<sub>1</sub> is split into the forward reaction v<sub>2</sub> and the reverse reaction v<sub>4</sub>, which both can have only positive flux. This splitting makes sure that EFMs are the extreme rays of the flux space (see Chapter 5 in [1]). This splitting is purely a mathematical conveniency, we still assume this to be one reaction in the biological sense, and therefore the kinetic equations of both the forward and the backward reaction will be exactly the same. Only, depending on in which direction the flux goes, either one of the reactions will be positive. Any solution with both reactions positive is infeasible, but minimizing enzyme levels will never lead to such a solution, therefore we do not need to set additional constraints. Second, the feasibility of EFMs can depend on external concentrations. In this network the biomass reaction (v<sub>BM</sub>) is the objective flux and there are three EFMs leading to the production of biomass: EFM1 consisting of v<sub>0</sub>, v<sub>1</sub>, v<sub>2</sub> and v<sub>BM</sub>, EFM2 consisting of v<sub>0</sub>, v<sub>1</sub>, v<sub>3</sub> and v<sub>BM</sub> and EFM3 consisting of v<sub>4</sub>, v<sub>3</sub> and v<sub>BM</sub>. However, if P<sub>1</sub> is absent in the environment, the uptake flux v<sub>4</sub> will always be 0 and therefore EFM3 will not be feasible.

## 7.4. Computation of the optimal state

We can now use the result that states of maximal enzyme efficiency are reached at an Elementary Flux Mode to calculate these states. We can follow the following steps to calculate these states:

1. Enumerate the Elementary Flux Modes

2. Calculate the minimal enzyme for each EFM scaled to an objective flux of 1

3. Compare the EFMs and select the one with minimal enzyme demands

Step 2 is a convex optimization problem as we have seen in Chapter 6 in [1]. Step 1 is possible for relatively large networks, although usually not for genome scale metabolic networks. The method of these three steps is called Enzyme Flux Cost Minimization, because it is similar to Enzyme Cost Minimization, but while that is focussed on fixes fluxes, Enzyme Flux Cost Minimization simultaneously finds the optimal fluxes. In this section we will show the method on the example network of Figure 7.2.

First, we describe the network with the Stoichiometric matrix (N) and the concentration vector (c):
Box 7.A Kinetics of the example network

The detailed kinetic equations for the example model (Figure 7.2) using the factorized rate law (see Equation (7.2) and Chapters 3 in [1] and 6 in [1]) are:

$$
\begin{align}
v_0 &= e_0 \cdot k_{\text{cat},0}^+ \cdot \frac{[\text{Gex}]/K_{\text{Gex}}}{1 + [\text{G}]/K_{\text{G}} + [\text{Gex}]/K_{\text{Gex}}} \cdot (1 - e^{\Delta_r G'_{0}/RT}) \nonumber \\
v_1 &= e_1 \cdot k_{\text{cat},1}^+ \cdot \frac{([\text{G}]/K_{\text{G}})([\text{ADP}]/K_{\text{ADP}})}{1 + ([\text{P}]/K_{\text{P}})([\text{P}]/K_{\text{P}})([\text{ATP}]/K_{\text{ATP}}) + ([\text{G}]/K_{\text{G}})([\text{ADP}]/K_{\text{ADP}})} \cdot (1 - e^{\Delta_r G'_{1}/RT}) \nonumber \\
v_2 &= e_2 \cdot k_{\text{cat},2}^+ \cdot \frac{[\text{P}]/K_{\text{P}}}{1 + [\text{Pex}]/K_{\text{Pex}} + [\text{P}]/K_{\text{P}}} \cdot (1 - e^{\Delta_r G'_{2}/RT}) \nonumber \\
v_3 &= e_3 \cdot k_{\text{cat},3}^+ \cdot \frac{([\text{P}]/K_{\text{P}})([\text{ADP}]/K_{\text{ADP}})([\text{O}_2]/K_{\text{O}_2})}{1 + ([\text{CO}_2]/K_{\text{CO}_2})([\text{ATP}]/K_{\text{ATP}}) + ([\text{P}]/K_{\text{P}})([\text{ADP}]/K_{\text{ADP}})([\text{O}_2]/K_{\text{O}_2})} \cdot (1 - e^{\Delta_r G'_{3}/RT}) \tag{7.3} \\
v_4 &= e_4 \cdot k_{\text{cat},4}^+ \cdot \frac{[\text{Pex}]/K_{\text{Pex}}}{1 + [\text{Pex}]/K_{\text{Pex}} + [\text{P}]/K_{\text{P}}} \cdot (1 - e^{\Delta_r G'_{4}/RT}) \nonumber \\
v_{\text{BM}} &= e_{\text{BM}} \cdot k_{\text{cat,BM}}^+ \cdot \frac{([\text{P}]/K_{\text{P}})([\text{ATP}]/K_{\text{ATP}})}{1 + ([\text{BM}]/K_{\text{BM}})([\text{ADP}]/K_{\text{ADP}}) + ([\text{P}]/K_{\text{P}})([\text{ATP}]/K_{\text{ATP}})} \cdot (1 - e^{\Delta_r G'_{5}/RT}) \nonumber
\end{align}
$$

Note that P is a product twice in $v_1$, as $v_1$ produces 2P. Note that $v_2$ and $v_4$ are the same reaction, but defined in the opposite direction. The standard set of parameters we used for the toy model is all $k_{\text{cat},i}^+ = 10 \, s^{-1}$ except $k_{\text{cat},3}^+ = 0.1 \, s^{-1}$, all $\Delta_r G'_{i}/RT = -440$ and all $K_{\text{M}} = 1$ mM. For the external metabolites $[\text{Pex}] = 1$ mM, $[\text{Gex}] = 0.05$ mM, $[\text{O}_2] = 0.1$ mM, $[\text{BM}] = 1$ mM and $[\text{CO}_2] = 10$ mM unless mentioned otherwise.

generalized for $n_s$ substrates and $n_p$ products (also compare Eq. (3.10 in [1]) in Chapter 3 in [1]):

$$v = e \cdot k_{\text{cat}}^{+} \cdot \frac{\prod_{n_s}^{j=1} s_j / K_{S,j}}{1 + \prod_{n_p}^{k=1} p_k / K_{P,k} + \prod_{n_s}^{j=1} s_j / K_{S,j}} \cdot (1 - e^{\Delta_r G' / RT}) \quad (7.2)$$

See Box 7.A for all detailed rate laws of the example network. We can simplify this equation by combining the forward catalytic constant, the thermodynamic efficiency factor, the saturation efficiency factor, and the regulation efficiency factor (if that exists) in a function $\kappa(s)$, which only depends on the metabolites, and not on the enzyme concentrations. We will below write $\kappa$ for $\kappa(s)$.

$$v_i = e_i \cdot \kappa_i \quad (7.4)$$

Now we take $v_{\text{BM}} = 1$ and optimize all fluxes, enzyme concentrations and metabolite concentrations to minimize the enzyme costs ($e_{\text{tot}} = \sum_i e_i$), while satisfying the constraints posed in Equations (7.1), for different levels of external glucose and standard levels of the other external metabolites. We see that for different concentrations of external glucose, lead to different optimal fluxes, enzyme levels and metabolite levels (Table 7.1).

The table shows that the total enzyme needed for a biomass flux of one decreases with increasing glucose levels, as we expect. In addition, the optimal level of internal glucose increases with increasing external glucose. This is

<table><caption>Table 7.1: Outcomes of the optimization of the example network with standard kinetics, parameter values and external concentrations (see Box 7.A) for varying levels of $[G_{\text{ex}}]$.</caption><thead><tr><th>[G<sub>ex</sub>]</th><th>e<sub>tot</sub></th><th>v<sub>0</sub></th><th>v<sub>1</sub></th><th>v<sub>2</sub></th><th>v<sub>3</sub></th><th>v<sub>4</sub></th><th>v<sub>BM</sub></th><th>e<sub>0</sub></th><th>e<sub>1</sub></th><th>e<sub>2</sub></th><th>e<sub>3</sub></th><th>e<sub>4</sub></th><th>e<sub>BM</sub></th><th>[G]</th><th>[P]</th><th>[ATP]</th><th>[ADP]</th></tr></thead><tbody><tr><td>0.01</td><td>156.2</td><td>5</td><td>5</td><td>0</td><td>9</td><td>0</td><td>1</td><td>54.4</td><td>4.4</td><td>0</td><td>94.4</td><td>0</td><td>2.9</td><td>0.08</td><td>15.14</td><td>0.05</td><td>20.09</td></tr><tr><td>0.1</td><td>91.3</td><td>50</td><td>50</td><td>99</td><td>0</td><td>0</td><td>1</td><td>61.3</td><td>11.3</td><td>14.2</td><td>0</td><td>0</td><td>4.4</td><td>0.13</td><td>4.55</td><td>0.11</td><td>20.09</td></tr><tr><td>1</td><td>36.2</td><td>50</td><td>50</td><td>99</td><td>0</td><td>0</td><td>1</td><td>13.0</td><td>8.0</td><td>12.5</td><td>0</td><td>0</td><td>2.7</td><td>0.60</td><td>7.65</td><td>0.11</td><td>20.09</td></tr></tbody></table>
<figure><img src="image_5.png" /><figcaption>Figure 7.2: Maximal efficiency states in example model. (a) Example network from Chapter 5 in [1] with added stoichiometry. (b) Three elementary flux modes of this network. (c) Calculated enzyme investment needed for a biomass flux of 1. There is a single concentration of $G_{ex}$ for which the optimal EFM switches from EFM2 to EFM1. (d) Specific fluxes (flux divided by total enzyme) associated with the optimal EFM for different levels of $G_{ex}$. The rates show a discontinuity when the optimal EFM switches from EFM1 to EFM2.</figcaption></figure>

$$ 
\begin{aligned}
\mathbf{N} &= \begin{pmatrix}
1 & -1 & 0 & 0 & 0 & 0 \\
0 & 2 & -1 & -1 & 1 & -1 \\
0 & 2 & 0 & 10 & 0 & -100 \\
0 & -2 & 0 & -10 & 0 & 100
\end{pmatrix} \\
\mathbf{c} &= \begin{pmatrix}
[G] \\
[P] \\
[ATP] \\
[ADP]
\end{pmatrix}
\end{aligned}
\qquad (7.4) 
$$
because a higher external glucose allows for a higher internal glucose while still maintaining a steady glucose influx, and a higher internal glucose allows fewer enzymes to drive further metabolism. Moreover, the fluxes of the solutions follow an EFM (see Figure 7.2b).

We can now reformulate the problem for only the flux and enzyme levels while keeping the metabolite levels as they are in the table. With the metabolite levels in the first row of the table, we can linearly relate the enzyme and flux levels (with the factors $κ_i$ that have become constants now we have set the internal metabolite concentrations), and thus the extreme rays of the enzyme and flux space will be equal and EFMs, as pointed out above (see also Chapter 5 in [1] and Figure 7.1). Optimization in this space will lead to the optimal flux distributions following an EFM (see Box 7.B for the detailed calculations). As fixing part of the optimal solution should lead to the same optimal solution, this required the flux distribution of the optimization over all variables to follow an EFM, as was indeed the case.

We point out two important aspects, using the network (Figure 7.2) as an example. First, it is convenient to split reversible reactions such that fluxes are always positive. In this case, that means that the reversible reaction from P to P<sub>ex</sub> is split into the forward reaction v<sub>2</sub> and the reverse reaction v<sub>4</sub>, both of which can have only positive flux. This splitting makes sure that EFMs are the extreme rays of the flux space (see Chapter 5 in [1]). This splitting is purely a mathematical convenience; we still assume this to be one reaction in the biological sense, and therefore the kinetic equations of both the forward and the backward reactions will be exactly the same. Depending on in which direction the flux goes, either one of the reactions will be positive and the other zero. Any solution with both reactions positive is infeasible, but minimizing enzyme levels will never lead to such a solution; therefore we do not need to set additional constrains to avoid this. Second, the feasibility of EFMs can depend on external concentrations. In this network, the biomass reaction (v<sub>BM</sub>) is the objective flux and there are three EFMs leading to the production of biomass: EFM1 consisting of v<sub>0</sub>, v<sub>1</sub>, v<sub>2</sub> and v<sub>BM</sub>, EFM2 consisting of v<sub>0</sub>, v<sub>1</sub>, v<sub>3</sub> and v<sub>BM</sub> and EFM3 consisting of v<sub>4</sub>, v<sub>3</sub> and v<sub>BM</sub>. However, if P<sub>ex</sub> is absent in the environment, the uptake flux v<sub>4</sub> will always be 0 and therefore EFM3 will not be feasible.

## 7.4. Calculation of optimal states

We can now use the result that states of maximal enzyme efficiency are reached at an elementary flux mode to calculate optimal states in a metabolic network using the following steps:

1. Enumerate the elementary flux modes that include the objective flux

2. Calculate the minimal enzyme for each EFM scaled to an objective flux of 1

3. Compare the EFMs and select the one with minimal enzyme demands

Step 1 is possible for relatively large networks, although usually not for genome scale metabolic networks. Step 2 is a convex optimization problem as we have seen in Chapter 6 in [1] and Step 3 is straightforward. These three steps together are called Enzyme Flux Cost Minimization, because it is similar to Enzyme Cost Minimization, but while that is focused on fixed fluxes, Enzyme Flux Cost Minimization simultaneously finds the optimal fluxes, enzyme and metabolite levels. In this section we will show the method on the example network of Figure 7.2.

First, we describe the network with the stoichiometric matrix (N) and the concentration vector (s):

$$N = \begin{pmatrix} 1 & -1 & 0 & 0 & 0 & 0 \\ 0 & 2 & -1 & -1 & 1 & -1 \\ 0 & 2 & 0 & 10 & 0 & -100 \\ 0 & -2 & 0 & -10 & 0 & 100 \end{pmatrix}, \quad s = \begin{pmatrix} [G] \\ [P] \\ [ATP] \\ [ADP] \end{pmatrix} \qquad (7.6)$$

And with the stoichiometric matrix we can describe the steady state constraints:
And with the stoichiometric matrix we can describe the steady state constraints:

$$ \dot{\mathbf{c}} = \mathbf{N}\mathbf{v} = \begin{pmatrix} 1 & -1 & 0 & 0 & 0 & 0 \\ 0 & 2 & -1 & -1 & 1 & -1 \\ 0 & 2 & 0 & 10 & 0 & -100 \\ 0 & -2 & 0 & -10 & 0 & 100 \end{pmatrix} \begin{pmatrix} v_0 \\ v_1 \\ v_2 \\ v_3 \\ v_4 \\ v_{BM} \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{pmatrix} \quad (7.5) $$

Now we find the EFM5 (for example with EFMtool [4]). It is easily cheched that the following EFM5 are in the nullspace of the Stoichiometric matrix:

$$ \text{EFM1} = \begin{pmatrix} 5 \\ 5 \\ 0 \\ 9 \\ 0 \\ 1 \end{pmatrix}, \quad \text{EFM2} = \begin{pmatrix} 50 \\ 50 \\ 99 \\ 0 \\ 0 \\ 1 \end{pmatrix}, \quad \text{EFM3} = \begin{pmatrix} 0 \\ 0 \\ 0 \\ 10 \\ 11 \\ 1 \end{pmatrix} \quad (7.6) $$

The next step is to do the convex optimization over the metabolite levels for all three EFM5. Therefore we convert the enzyme levels to ratio of the flux and the function $f(c)$, using Equation 7.3. By taking the sum of those we make a function for the total enzyme as a function of fluxes, metabolite concentrations and parameters. We use the standard parameters (Box 7.4) and for each EFM input the fluxes. We are then left with a convex optimization over the metabolite levels, an Enzyme Cost Minimization problem as in Chapter 6 in [1]. For $[G_{ex}] = 0.05$ we obtain a total enzyme of 111.1 for EFM1 and of 146.3 for EFM2. That means that for this conditions we will conclude that EFM1 is optimal, and we obtain the metabolite concentrations from the optimization of $[G] = 0.08$, $[P] = 3.93$, $[ATP] = 0.11$ and $[ADP] = 20.09$. We can next use the rate equations to calculate the enzyme levels from the fluxes and metabolite levels, using the values for the parameters and external concentrations.

We can repeat this procedure for different levels of external concentrations and see that the optimal EFM can change depending on the external concentration (Figure 7.2c). When the optimum shifts to using a different EFM, there is a discontinuity in the fluxes at the external metabolite concentration (Figure 7.2d). Many cells show shifts in metabolic strategies depending on the external conditions and with Enzyme Flux Cost Minimization those shift could be explained.

In conclusion, we have now found the metabolic state of maximum enzyme efficiency. Even though we obtained the enzyme concentrations last, this is how the cell controls the system. When the cells makes enzymes in the concentrations we calculated, a steady state will set in with the fluxes and metabolite levels that will lead to this optimal state.

### Box 7.A: Kinetics of the example network

The detailed kinetic equations for the example model (Figure 7.2) using the factorized rate law (see Equation (7.2) and Chapters 3 in [1] and 6 in [1]) are:
Box 7.B Optimal metabolic states in the example network

We minimize the enzyme investment for $v_{BM} = 1$ with $[P_{ex}] = 0$ (and therefore $v_4 = 0$ and EFM3 is not feasible) for the network in Figure 7.2 (the optimization problem in Equation (7.1)). Assuming all $h_i = 1$, the objective function $\sum_{i=1}^{r} h_i e_i = e_0 + e_1 + e_2 + e_3 + e_{BM}$. The constraints $v_{BM} = 1$ and $e, s \ge 0$ in Eq. (7.1) are straightforward. The steady state of all internal metabolites (G, P, ADP and ATP) leads to the following equalities (the steady states of ADP and ATP lead to the same equality):

$$
\begin{align*}
\text{Steady state ATP} &\implies 100 v_{BM} = 2 v_1 + 10 v_3 \\
\text{Steady state P} &\implies 2 v_1 + v_4 = v_2 + v_3 + v_{BM} \\
\text{Steady state G} &\implies v_0 = v_1
\end{align*}
$$

Substituting $v_{BM} = 1$ and $v_4 = 0$ and solving this set of linear equations, we can write all fluxes as functions of $v_2$: $v_0 = v_1 = 5 + \frac{5}{11}v_2$ and $v_3 = 9 - \frac{1}{11}v_2$ (there is only one independent flux in this system). This means we can draw the feasible flux space on the $v_2$ line and we can express the objective function in terms of $v_2$:

$$
\begin{align}
\sum_{i=1}^{r} h_i e_i &= e_0 + e_1 + e_2 + e_3 + e_{BM} \nonumber \\
&= v_0/\kappa_0 + v_1/\kappa_1 + v_2/\kappa_2 + v_3/\kappa_3 + v_{BM}/\kappa_{BM} \nonumber \\
&= (5 + 5/11v_2)/\kappa_0 + (5 + 5/11v_2)/\kappa_1 + v_2/\kappa_2 + (9 - 1/11v_2)/\kappa_3 + 1/\kappa_{BM} \tag{7.5} \\
&= \underbrace{(5/\kappa_0 + 5/\kappa_1 + 9/\kappa_3 + 1/\kappa_{BM})}_{\alpha} + \underbrace{(5/(11\kappa_0) + 5/(11\kappa_1) + 1/\kappa_2 - 1/(11\kappa_3))}_{\beta} v_2 \nonumber \\
&= \alpha + \beta v_2 \nonumber
\end{align}
$$

The kinetic functions ($\kappa_i$) depend on several parameters (external metabolite levels [$G_{ex}$], $O_2$, [CO$_2$] and [P$_{ex}$], catalytic constants, Michaelis constants and Gibbs free energies) and the variables [G], [P], [ATP] and [ADP]. That means that once we have a set of internal metabolite concentrations $s$, the enzyme levels in the objective function can be written as a constant times the flux: $e_i = v_i/\kappa_i$, with $\kappa_i$ a constant. For a set of parameters, $\alpha$ and $\beta$ are positive or negative depending on the choice of $s$. It is clear that when we minimize this objective function by adjusting $v_2$, we will always have an optimum at $v_2 = 0$ (when $\beta$ is positive) or $v_2 = 99$ (when $\beta$ is negative). $v_2 = 99$ is the maximum of $v_2$ because then $v_3 = 9 - \frac{1}{11}v_2 = 0$, and higher values of $v_2$ would lead to negative values for $v_3$.

In conclusion, the optimum cannot be at a value of $0 < v_2 < 99$. If there would be an optimum with $0 < v_2 < 99$, we can determine $s$ and calculate whether $\beta > 0$ to find a lower objective value at $v_2 = 0$ or $v_2 = 99$, contradicting that we started with an optimum. Only if $\beta = 0$ there is a range of optima, but this requires very precise parameter values. $v_2 = 0$ and $v_2 = 99$ correspond to EFMs of this network (Figure 7.2).

$$
\frac{\mathrm{d}}{\mathrm{d}t} \mathbf{s} = \mathbf{N} \mathbf{v} = \begin{pmatrix} 1 & -1 & 0 & 0 & 0 & 0 \\ 0 & 2 & -1 & -1 & 1 & -1 \\ 0 & 2 & 0 & 10 & 0 & -100 \\ 0 & -2 & 0 & -10 & 0 & 100 \end{pmatrix} \begin{pmatrix} v_0 \\ v_1 \\ v_2 \\ v_3 \\ v_4 \\ v_{BM} \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{pmatrix} \quad (7.7)
$$

Now we find the EFMs (for example with EFMtool [6]). It can easily be checked that the following EFMs (denoted
<figure><img src="image_6.png" /><figcaption>Figure 7.3: Translation of enzyme-specific biomass rate to growth rate. A Both from experimental data and a cell-optimization point of view, the ribosomal fraction of the proteome increases with the growth rate, while the metabolic fraction decreases. B This leads to a hyperbolic dependency of the growth rate on the biomass production rate.</figcaption></figure>

$$
\begin{align}
v_0 &= e_0 \cdot k_{\text{cat},0}^+ \cdot \frac{[G_{\text{ex}}]/K_{G_{\text{ex}}}}{1 + [G]/K_G + [G_{\text{ex}}]/K_{G_{\text{ex}}}} \cdot \left(1 - e^{\Delta G'_{r,0}/RT}\right) \nonumber \\
v_1 &= e_1 \cdot k_{\text{cat},1}^+ \cdot \frac{([G]/K_G)([\text{ADP}]/K_{\text{ADP}})}{1 + ([P]/K_P)([P]/K_P)([\text{ATP}]/K_{\text{ATP}}) + ([G]/K_G)([\text{ADP}]/K_{\text{ADP}})} \cdot \left(1 - e^{\Delta G'_{r,1}/RT}\right) \nonumber \\
v_2 &= e_2 \cdot k_{\text{cat},2}^+ \cdot \frac{[P]/K_P}{1 + [P_1]/K_{P_1} + [P]/K_P} \cdot \left(1 - e^{\Delta G'_{r,2}/RT}\right) \nonumber \\
v_3 &= e_3 \cdot k_{\text{cat},3}^+ \cdot \frac{([P]/K_P)([\text{ADP}]/K_{\text{ADP}})([0_2]/K_{0_2})}{1 + ([C_{0_2}]/K_{C_{0_2}})([\text{ATP}]/K_{\text{ATP}}) + ([P]/K_P)([\text{ADP}]/K_{\text{ADP}})([0_2]/K_{0_2})} \cdot \left(1 - e^{\Delta G'_{r,3}/RT}\right) \tag{7.7} \\
v_4 &= e_4 \cdot k_{\text{cat},4}^+ \cdot \frac{[P_1]/K_{P_1}}{1 + [P_1]/K_{P_1} + [P]/K_P} \cdot \left(1 - e^{\Delta G'_{r,4}/RT}\right) \nonumber \\
v_{\text{BM}} &= e_{\text{BM}} \cdot k_{\text{cat,BM}}^+ \cdot \frac{([P]/K_P)([\text{ATP}]/K_{\text{ATP}})}{1 + ([\text{BM}]/K_{\text{BM}})([\text{ADP}]/K_{\text{ADP}}) + ([P]/K_P)(\text{ATP}/K_{\text{ATP}})} \cdot \left(1 - e^{\Delta G'_{r,0}/RT}\right) \nonumber
\end{align}
$$

Note that P is a product twice in $v_1$, as $v_1$ produces 2P. Note that $v_2$ and $v_4$ have a very similar rate equation, but in the different direction, and both should only be used in the positive direction.

The standard set of parameters we used for the toy model is all $k_{cat,i} = 10$ except $k_{cat,3} = 0.1$, all $\Delta G'_{r,i} = -1000$, $R = 8.3$, $T = 293$ and all $K_M = 1$. For the external metabolites $[P_1] = 1$, $[G_{ex}] = 0.05$, $[0_2] = 0.1$, $[BM] = 1$ and $[C_{0_2}] = 10$ unless mentioned otherwise.

## 7.5. Translating enzyme efficiency into cell growth rate

In the section above, we learned how to optimize metabolic states for a maximal overall enzyme efficiency. Why is this quantity relevant? One reason is that overall enzyme efficiency, according to some simple reasoning, determines the cell growth rates. If microbes compete by growing fast, their fitness is largely determined by their momentary growth rate in their respective environment. In such environments, the biomass/enzyme efficiency will be under selection, which makes it one of the important objective functions in this book. If higher enzyme efficiency means higher growth rate, and if we have a conversion formula for this, we can put "growth rate" instead of "overall enzyme efficiency" on the axes of our plots.

Enzyme-efficient metabolic states allow us to compute specific biomass production rates, i.e. the rate of biomass production per metabolic enzyme invested. If biomass consisted only of enzymes, the ratio "enzyme production rate / enzyme concentration" would give us directly the growth rate. However, biomass does not only consist of metabolic enzymes, but includes ribosomal enzymes, RNA, DNA, lipids, and other compounds. Therefore we need a formula for converting biomass/enzyme efficiency into cellular growth rate.

Mathematically, a cell's growth rate is given by $\mu = v_{BM}/c_{BM}$, where $v_{BM}$ is the biomass production rate (biomass produced per cell volume and time) and $c_{BM}$ is the biomass amount per cell volume. If a cell consisted only of metabolic
<figure><img src="image_5.png" /><figcaption>Figure 7.2: States of maximal efficiency in an example model – (A) Example network from Chapter 5 in [1] with added stoichiometry. (B) Three elementary flux modes of this network. (C) Calculated enzyme investment needed for a biomass flux of 1. At a very low concentration of extracellular glucose ([Gex]), EFM3 has the lowest cost. But as we move along the x-axis, at around [Gex] = 0.02 there is a switch to EFM1 and later, at around [Gex] = 0.07, EFM2 becomes the one with the lowest cost. (D) Specific fluxes (flux divided by total enzyme) associated with the optimal EFM for different levels of Gex. Note that v1 is not shown as it is always equal to v0. The rates show a discontinuity when there is a switch from one optimal EFM to another.</figcaption></figure>

by vectors $f^{(i)}$ are in the nullspace of the stoichiometric matrix:

$$f^{(1)} = \begin{pmatrix} 5 \\ 5 \\ 0 \\ 9 \\ 0 \\ 1 \end{pmatrix}, \quad f^{(2)} = \begin{pmatrix} 50 \\ 50 \\ 99 \\ 0 \\ 0 \\ 1 \end{pmatrix}, \quad f^{(3)} = \begin{pmatrix} 0 \\ 0 \\ 0 \\ 10 \\ 11 \\ 1 \end{pmatrix} \qquad (7.8)$$
<figure><img src="image_7.png" /><figcaption>Figure 7.4: Optimal growth rates of the two EFMs for different levels of the external metabolite G<sub>ex</sub> (using Equation 7.8)</figcaption></figure>

enzymes (more precisely, of the enzymes described in our model) the biomass/enzyme efficiency $K_{BM} = v_{BM}/h_{enz}$ would directly describe the cellular growth rate. Since that is not the case, we need to convert between $h_{enz}$ and $c_{BM}$. The metabolic protein fraction decreases with the growth rate, leading to a hyperbolic dependency of the growth rate on the biomass production rate (Figure 7.3). We may use the empirical approximation $h_{enz}/c_{BM} = f_{prot}(a - b \mu)$, where $f_{prot} = 0.5$ is the fraction of protein mass within the cell dry mass and the parameters $a = 0.27$ and $b = 0.2$ h were fitted to describe the metabolic enzyme fraction in proteomics data, assuming a linear dependence on growth rate [5]. This yields the conversion formula (see also [6]):

$$ \mu = \frac{a f_{prot} v_{BM}}{h_{enz} + b f_{prot} v_{BM}}. \qquad (7.8) $$

This formula has been used to convert the minimal enzyme cost per biomass flux for different external concentrations in the toy model (Figure 7.2c) to the maximal growth for each EFM (Figure 7.4).

## 7.6. Conclusions

In this chapter we considered the metabolic network of a cell - and enzyme levels, metabolite concentrations, and fluxes as the state variables - and studied its maximally efficient states. Finding such states can be difficult because fluxes, metabolite concentrations, and enzyme levels are tightly coupled: metabolite concentrations determine enzyme efficiencies, enzyme efficiencies determine optimal enzyme levels, and enzyme levels determine fluxes and metabolite concentrations, which in turn determine enzyme efficiencies. To find an optimal state, all variables need to be optimized at the same time, which is a non-linear optimality problem with (possibly) many local optima. In small toy models, solutions can be found numerically, but for large detailed models, the computational effort becomes enormous. Instead of simplifying the problem (as in the previous chapters) we here used the insight that (in models without extra flux bounds) the optimal solutions must be EFMs. Biologically, it is enzyme levels rather than fluxes that are regulated directly (by transcriptional regulation), while metabolite concentrations and fluxes respond dynamically. But from a functional point of view, we may see this differently: we may first think of a task (a flux to be realized) and then ask how this flux can be performed optimally.
<figure><img src="image_6.png" /><figcaption>Figure 7.3: Translation of enzyme-specific biomass rate to growth rate – (A) Both from experimental data and a cell-optimization point of view, the ribosomal fraction of the proteome increases with the growth rate, while the metabolic fraction decreases. (B) This leads to a hyperbolic dependency of the growth rate on the biomass production rate per amount of enzymes.</figcaption></figure>

The next step is to perform the convex optimization over the metabolite levels for each one of the three EFMs. Therefore, we express the enzyme levels as a ratio of the flux and the function $f(s)$, using Equation 7.4. Summing over all enzymes, we get a function for the total enzyme cost (level) as a function of fluxes, metabolite concentrations and parameters:

$$e_{\text{tot}} = \sum_{i} e_{i} = \sum_{i} \frac{v_{i}}{\kappa_{i}(s)}. \qquad (7.9)$$

We use the standard parameters (Box 7.A) and replace $v_i$ by the values given by each EFM. We are then left with a convex optimization over the metabolite levels, an Enzyme Cost Minimization problem as in Chapter 6 in [1]. For $[G_{ex}] = 0.05$ we obtain a total enzyme of 111.1 for EFM1, of 146.3 for EFM2 and 136.5 for EFM3. That means that for these conditions we will conclude that EFM1 is optimal. From the optimization we obtain the metabolite concentrations: $[G] = 0.08$, $[P] = 3.93$, $[ATP] = 0.11$ and $[ADP] = 20.09$ (that the internal glucose concentration is higher than the external is because we described the transport with regular enzyme kinetics instead of transporter enzyme kinetics, which would have been more realistic). We can next use the rate equations to calculate the enzyme levels from the fluxes and metabolite levels, using the values for the parameters and external concentrations.

We can repeat this procedure for different levels of external concentrations and see that the optimal EFM can change depending on the external concentration (Figure 7.2c). When the optimum shifts to using a different EFM, there is a discontinuity in the fluxes at the external metabolite concentration (Figure 7.2d). Many cells show shifts in metabolic strategies depending on the external conditions and Enzyme Flux Cost Minimization is one way of explaining those shifts.

Above, Enzyme Cost Flux Minimization was used to find the metabolic state with the maximum enzyme efficiency. Although in our calculation we obtain the enzyme concentrations last, it is by enzyme concentrations that cells actually control metabolism. If cells produce enzymes at the concentrations we calculated and reach a steady state, this state will realize the fluxes and metabolite levels that lead to our optimal state.

## 7.5. Translating enzyme efficiency into cell growth rate

In the section above, we learned how to optimize metabolic states for a maximal overall enzyme efficiency. Why is this quantity relevant? One reason is that overall enzyme efficiency, according to some simple reasoning, determines the cell's growth rate. If microbes compete by growing fast, their fitness is largely determined by their momentary growth rate in their respective environment. In such environments, the biomass/enzyme efficiency will be under selection, which makes it one of the important objective functions in this book. If higher enzyme efficiency means higher growth rate, and if we have a conversion formula for this, we can plot the growth rate of the different EFMs instead of overall enzyme efficiency.
Recommended readings

M.T. Wortel, H. Peters, J. Hulshof, B. Teusink, and F.J. Bruggeman. Metabolic states with maximal specific rate carry flux through an elementary flux mode. FEBS Journal, 281(6):1547-1555, 2014.

S. Müller, G. Regensburger, and R. Steuer. Enzyme allocation problems in kinetic metabolic networks: Optimal solutions are elementary flux modes. Journal of Theoretical Biology, 347:182-190, 2014.
<figure><img src="image_7.png" /><figcaption>Figure 7.4: Optimal growth rates of the two EFMs for different levels of the external metabolite Gex, computed using Equation (7.10) from the enzyme demands (at a unit biomass production rate) shown in Figure 7.2 (C).</figcaption></figure>

Enzyme-efficient metabolic states allow us to compute specific biomass production rates, i.e. the rate of biomass production per metabolic enzyme invested. If biomass consisted only of enzymes, the ratio "enzyme production rate per total enzyme demand" would give us directly the growth rate. However, biomass does not only consist of metabolic enzymes, but includes ribosomal enzymes, RNA, DNA, lipids, and other compounds. Therefore we need a formula for converting biomass/enzyme efficiency into cellular growth rate.

Mathematically, a cell's growth rate is given by $\mu = v_{BM}/s_{BM}$, where $v_{BM}$ is the biomass production rate (biomass produced per cell volume and time) and $s_{BM}$ is the biomass amount per cell volume. If a cell contained nothing but metabolic enzymes (more precisely, the enzymes described in our model), the biomass/enzyme efficiency $\kappa_{BM} = v_{BM}/h_{enz}$ would directly describe the cellular growth rate. Since that is not the case, we need to convert $h_{enz}$ to $s_{BM}$. The metabolic protein fraction decreases with the growth rate, leading to a hyperbolic dependency of the growth rate on the biomass production rate (Figure 7.3). We may use the empirical approximation $h_{enz}/s_{BM} = \kappa_{prot}(a - b \mu)$, where $\kappa_{prot} = 0.5$ is the fraction of protein mass within the cell dry mass and the parameters $a = 0.27$ and $b = 0.2$ h were fitted to describe the metabolic enzyme fraction in proteomics data, assuming a linear dependence on growth rate [7]. This yields the conversion formula (see also [8]):

$$ \mu = \frac{a \ \kappa_{prot} \ v_{BM}}{h_{enz} + b \ \kappa_{prot} \ v_{BM}}. \qquad (7.10) $$

This formula has been used to convert the minimal enzyme cost per biomass flux for different external concentrations in the toy model (Figure 7.2c) to the maximal growth for each EFM (Figure 7.4).

## 7.6. Application to central metabolism in E. coli bacteria

In the previous sections, we saw that finding enzyme-efficient metabolic states can be done by iterating through all possible EFMs and performing the enzyme cost minimization on each one. We demonstrated it on a toy model comprising only 3 EFMs. In Wortel et al. [8], this method was scaled up and applied to a more realistic model covering the central metabolic network, as shown in Figure 7.5A. For this larger network, there are 1566 biomass-generating EFMs. Each reaction is assigned to a single enzyme along with its molecular weight, $k_{cat}^{+}$, $K_{M}$, and $\Delta_{r}G^{\circ}$, and follows the generalized factorized rate law as in Equation (7.2). These parameters are listed in Appendix section 7.8, and the full procedure for obtaining them is described in Wortel et al. [8], along with other model parameters.
Bibliography

[1] The Economic Cell Collective, editor. *Economic Principles in Cell Biology*. Free online book, 2023.

[2] M.T. Wortel, H. Peters, J. Hulshof, B. Teusink, and F.J. Bruggeman. Metabolic states with maximal specific rate carry flux through an elementary flux mode. *FEBS Journal*, 281(6):1547–1555, 2014.

[3] S. Müller, G. Regensburger, and R. Steuer. Enzyme allocation problems in kinetic metabolic networks: Optimal solutions are elementary flux modes. *Journal of Theoretical Biology*, 347:182–190, 2014.

[4] Marco Terzer and Jörg Stelling. Large-scale computation of elementary flux modes with bit pattern trees. *Bioinformatics*, 24(19):2229–2235, 08 2008. ISSN 1367-4803. doi: 10.1093/bioinformatics/btn401.

[5] Matthew Scott, Carl W Gunderson, Eduard M Mateescu, Zhongge Zhang, and Terence Hwa. Interdependence of cell growth and gene expression: origins and consequences. *Science*, 330(6007):1099–1102, 2010. doi: 10.1126/science.1192588.

[6] Meike T Wortel, Elad Noor, Michael Ferris, Frank J Bruggeman, and Wolfram Liebermeister. Metabolic enzyme cost explains variable trade-offs between microbial growth rate and yield. *PLoS Comput. Biol.*, 14(2):e1006010, February 2018. doi: 10.1371/journal.pcbi.1006010.
<figure><img src="image_8.png" /><figcaption>Figure 7.5: Model of central metabolism in *E. coli* bacteria. (A) The metabolic network of the *E. coli* model used by Wortel et al. [8]. Note that only for the purpose of visualization, the network shown here has been condensed by lumping consecutive reactions that are fully coupled (e.g., the reactions between DHAP and PEP are now represented by a single arrow). Furthermore, some groups of metabolites have been merged to a single node: H6P – representing the hexose phosphates G6P, F6P, and FBP; T3P – representing the triose phosphates G3P and DHAP; P5P – representing the pentose phosphates R5P, X5P, and Ru5P. The metabolites that are direct substrates of the biomass reaction are marked in bold. (B) A Venn diagram showing statistics of biomass-producing EFMs in the model and their reliance on oxygen.</figcaption></figure>

First, Wortel et al. [8] wanted to study the effect of environmental conditions on the growth rate of *E. coli*, and see whether the model would be able to recapitulate empirical phenomena. The external glucose concentration was set to 100 mM and oxygen levels were varied between 1 µ and 10 mM. They selected 4 flux modes as representatives (the EFMs *max-gr*, *ana-lac*, and *aero-ace*, and *exp*, which is based on experimentally measured fluxes; the flux distributions are shown in Figure 7.6), and calculated their predicted growth rates in each condition, using Equation (7.10). The results are shown in Figure 7.7. When focusing on a single flux mode, one can see that as the oxygen level increases so does the growth rate. The increase saturates at some point, which depends on the flux modes and on the kinetic parameters in the model. Indeed, it has long been known that growth rate dependence on a limiting substrate concentration has this specific shape – a relationship generally called the *Monod curve*.

In this specific example, it is interesting to see the Monod curves of the different EFMs, and try to understand the
<figure><img src="image_9.png" /></figure>

<figure><img src="image_10.png" /></figure>

<figure><img src="image_11.png" /></figure>

<figure><img src="image_12.png" /></figure>

Figure 7.6: The four flux modes chosen for drawing the Monod curves in Figure 7.7: (A) *max-gr* – the EFM with the highest growth rate under the standard conditions chosen in the study, (B) *aero-ace* – an EFM which mixes between respiration and acetate fermentation, (C) *ana-lac* – an EFM that does not require oxygen (i.e. anaerobic) and uses lactate fermentation, and (D) *exp* – which is not an elementary flux mode, but rather one based on the measured flux distribution for *E. coli* growing on minimal media and glucose. The active reactions are highlighted in color (with the flux direction indicated by the arrowhead). The magnitude of each flux is not shown here but can be found in the Supplementary section of Wortel et al. [8]. The biomass reaction is not shown here due to space limitations, but is always active.

differences. First, the EFM called *ana-lac* (red curve), is a flat line. This makes sense because cells that use this EFM do not utilize the oxidative phosphorylation system and therefore do not require oxygen at all for growth. *max-gr*,
<figure><img src="image_13.png" /><figcaption>Figure 7.7: Monod curves (cell growth rate as a function of oxygen level) computed using the model shown in Figure 7.5 – Each curve was computed using one of the EFMs and the associated (oxygen-dependent) enzyme demands. The ana-lac strategy (anaerobic growth with lactate secretion) does not use oxygen, therefore it curve is flat.</figcaption></figure>

on the other hand, is very sensitive to the level of oxygen mainly because of the high flux going through oxidative phosphorylation. It is also the EFM with the highest growth rate in standard oxygen levels (0.21 mM), even when taking all the other ~1500 EFMs into account (not shown here).

Instead of screening only external oxygen levels, we can also screen several model parameters and compute "winning EFMs", their enzyme demands, and the resulting growth rates for our parameter combination. By screening glucose and oxygen concentrations, we obtain the Monod landscape shown in Figure 7.8 (A). Just like in Figure 7.7, there are distinct parameter regions in which optimal growth is reached by specific EFMs. While the *max-gr* EFM remains best when glucose and oxygen levels are high, at low oxygen levels we see a large number of different EFMs, one of them *ana-lac* (see the EFM phase diagram in Figure 7.8 (B)). More results for this model (fluxes plotted in the EFM phase diagram and in flux space, as well as ideal and real enzyme costs for all EFMs), are shown in Appendix Section 7.9.

## 7.7. Concluding remarks

In this chapter we considered the metabolic network of a cell - and enzyme levels, metabolite concentrations, and fluxes as the state variables - and studied its maximally efficient states. Finding such states can be difficult because fluxes, metabolite concentrations, and enzyme levels are tightly coupled: metabolite concentrations determine enzyme efficiencies, enzyme efficiencies determine optimal enzyme levels, and enzyme levels determine fluxes and metabolite concentrations, which in turn determine enzyme efficiencies. To find an optimal state, all variables need to be optimized at the same time, which is a non-linear optimality problem with (possibly) many local optima. In small toy models, solutions can be found numerically, but for large detailed models, the computational effort becomes enormous. Instead of simplifying the problem (as in the previous chapters) we here used the insight that (in models without extra flux bounds) the optimal solutions must be EFMs.

## Recommended readings

* M.T. Wortel, H. Peters, J. Hulshof, B. Teusink, and F.J. Bruggeman. Metabolic states with maximal specific rate carry flux through an elementary flux mode. FEBS Journal, 281(6):15471555, 2014.
(A)

<figure><img src="image_14.png" /></figure>

(B)

<figure><img src="image_15.png" /></figure>

Figure 7.8: Monod landscape – (A) Similar to the 1-dimensional Monod curve (Figure 7.7), the graphics shows the cell growth rate as a function of external glucose and oxygen concentrations, predicted from the *E. coli* model in Figure 7.5. The growth rate of the “winning” EFM – i.e. the one with the highest growth rate under the glucose and oxygen levels matching the *x* and *y* values – determines the height of each point. Each color represents the region in which a certain EFM is the “winning” one. (B) EFM phase diagram. The same plot as in (A), seen from above. The “winning EFMs” form a sort of phase diagram. At the boundary between every two regions, the two EFMs lead to the same growth rate (similar to the intersections between curves in Figure 7.7). The EFMs from Figure 7.6 are marked by their names. Note that the colors in this figure do not match the previous colors marking these select EFMs.

○ S. Müller, G. Regensburger, and R. Steuer. Enzyme allocation problems in kinetic metabolic networks: Optimal solutions are elementary flux modes. Journal of Theoretical Biology, 347:182190, 2014.

○ M.T. Wortel, E. Noor, M. Ferris, F.J. Bruggeman, and W. Liebermeister. Metabolic enzyme cost explains variable trade-offs between microbial growth rate and yield. PLoS Computational Biology, 14 (2):e1006010, 2018

## Problems

Computer exercises for this chapter can be found on the book website.

### Problem 7.1 Effect of oxygen concentration

Consider the model in Figure 7.2. What would be the qualitative effect of a change in oxygen concentration on the enzyme cost of the three EFMs and on the choice of the optimal strategy?

### Problem 7.2 Effect of external metabolites

Consider the model in Figure 7.2 under standard conditions (Box 7.A and [$G_{ex}$] = 1, such that EFM2 is optimal, and EFM1 second best (remember that the higher the enzyme cost, the less optimal the EFM). What might happen when we gradually increase the concentration [$P_{ex}$]? What is the qualitative effect on the enzyme cost of the three EFMs?

### Problem 7.3 States of maximal growth rate

Consider the following small toy network:
<figure><img src="image_16.png" /></figure>

We want to optimize the specific pathway flux for the production of P (which is $v_3/e_{tot}$, where we assume all enzymes to have equal costs: i.e. $e_{tot} = e_1 + e_2 + e_3$) at steady state. We assume mass-action kinetics, meaning the rate is the enzyme concentration times the forward rate constant times the substrate minus the backward rate constant times the product: $v = e(k^+s - k^-p)$. Unless mentioned otherwise, we use the values $s_2 = 10$, $k_1^+ = 2$, $k_1^- = 1$, $k_2^+ = 3$, $k_2^- = 1$, $k_3^+ = 1$, $k_3^- = 0.1$, $p = 0$ (concentrations are denoted by lower case letters).

(a) Write out the rate equations for all three rates in terms of the parameters and the concentrations.

(b) Give an expression of the total enzyme concentration in terms of fluxes and the metabolite concentrations $s_1$ and $x$.

(c) Find the concentration of X for which the specific flux $v_3/e_{tot}$ is maximal for $e_1 = 0$ and $s_2 = 10$, and also give the corresponding value of $v_3$. HINT: Is is easiest to set $e_{tot} = 1$ and maximize $v_3$, replacing $e_3$ using the equation for the total enzyme cost and the steady state assumption.

(d) Find the concentration of X for which the specific flux $v_3/e_T$ is maximal for $e_2 = 0$ and $s_1 = 10$, and also give the corresponding value of $v_3/e_{tot}$.

(e) Find the concentration of X for which the enzyme cost is minimal for $e_1 = e_2$ and $s_1 = s_2 = 10$, and also give the corresponding value of $v_3/e_{tot}$.

(f) What was the best distribution of enzymes from the three options above for $s_1 = 10$?

(g) Find the concentration of X for which the enzyme cost is minimal for $e_1 = 0$ and $s_1 = 50$, and also give the corresponding value of $v_3/e_{tot}$.

(h) Find the concentration of X for which the enzyme cost is minimal for $e_2 = 0$ and $s_1 = 50$, and also give the corresponding value of $v_3/e_{tot}$.

(i) Find the concentration of X for which the enzyme cost is minimal for $e_1 = e_2$ and $s_1 = 50$, and also give the corresponding value of $v_3/e_{tot}$.

(j) What was the best distribution of enzymes from the three options above for $s_1 = 50$?

(k) Interpret the results from this problem in light of the proof shown in this chapter about the optimal specific flux being attained at an EFM.

# Appendix sections

## 7.8. A model of central metabolism in *Escherichia coli*
<table><caption>Table 7.2: Stoichiometry of biomass reaction – R70</caption><thead><tr><th>Metabolite name</th><th>Biomass stoichiometric coefficient</th></tr></thead><tbody><tr><td>AcCoA</td><td>-41</td></tr><tr><td>ADP</td><td>547</td></tr><tr><td>2-oxoglutarate</td><td>-14</td></tr><tr><td>ATP</td><td>-547</td></tr><tr><td>H<sub>2</sub>O</td><td>-547</td></tr><tr><td>P<sub>i</sub></td><td>547</td></tr><tr><td>CO<sub>2</sub></td><td>2</td></tr><tr><td>CoA</td><td>41</td></tr><tr><td>DHAP</td><td>-5</td></tr><tr><td>G6P</td><td>-4</td></tr><tr><td>NAD<sup>+</sup></td><td>178</td></tr><tr><td>NADH</td><td>-178</td></tr><tr><td>NH<sub>3</sub></td><td>-139</td></tr><tr><td>2-oxoglutarateAcetate</td><td>-24</td></tr><tr><td>PEP</td><td>-32</td></tr><tr><td>Pyruvate</td><td>-38</td></tr><tr><td>E4P</td><td>-5</td></tr><tr><td>R5P</td><td>-13</td></tr></tbody></table>

<table><caption>Table 7.3: Glycolysis</caption><thead><tr><th>Reaction ID</th><th>EC number</th><th>Reaction name</th><th>Formula</th></tr></thead><tbody><tr><td>R1</td><td>2.7.1.69</td><td>pts</td><td>Glucose + PEP ⇌ G6P + Pyruvate</td></tr><tr><td>R2r</td><td>5.3.1.9</td><td>pgi</td><td>G6P ⇌ F6P</td></tr><tr><td>R3</td><td>2.7.1.11</td><td>pfk</td><td>F6P + ATP ⇌ FBP + ADP</td></tr><tr><td>R4</td><td>3.1.3.11</td><td>fbp</td><td>FBP + H<sub>2</sub>O ⇌ F6P + P<sub>i</sub></td></tr><tr><td>R5r</td><td>4.1.2.13</td><td>ald</td><td>FBP ⇌ DHAP + G3P</td></tr><tr><td>R6r</td><td>5.3.1.1</td><td>tim</td><td>G3P ⇌ DHAP</td></tr><tr><td>R7ra</td><td>1.2.1.12</td><td>gap</td><td>G3P + NAD<sup>+</sup> + P<sub>i</sub> ⇌ BPG + NADH</td></tr><tr><td>R7rb</td><td>2.7.2.3</td><td>pgk</td><td>BPG + ADP ⇌ 3PG + ATP</td></tr><tr><td>R7rc</td><td>5.4.2.11 / 5.4.2.12</td><td>pgm</td><td>3PG ⇌ 2PG</td></tr><tr><td>R8r</td><td>4.2.1.11</td><td>pgh</td><td>2PG ⇌ PEP</td></tr><tr><td>R9</td><td>2.7.1.40</td><td>pyk</td><td>PEP + ADP ⇌ Pyruvate + ATP</td></tr><tr><td>RR9</td><td>2.7.9.2</td><td>pps</td><td>Pyruvate + 2 ATP ⇌ PEP + 2 ADP + P<sub>i</sub></td></tr></tbody></table>

<table><caption>Table 7.4: Pentose Phosphate Pathway</caption><thead><tr><th>Reaction ID</th><th>EC number</th><th>Reaction name</th><th>Formula</th></tr></thead><tbody><tr><td>R10a</td><td>1.1.1.49</td><td>zwf</td><td>G6P + NAD<sup>+</sup> ⇌ 6PGL + NADH</td></tr><tr><td>R10b</td><td>3.1.1.31</td><td>glh</td><td>6PGL ⇌ 6PGC</td></tr><tr><td>R10c</td><td>1.1.1.44</td><td>pgd</td><td>6PGC + NAD<sup>+</sup> ⇌ NADH + CO<sub>2</sub> + Ru5P</td></tr><tr><td>R11r</td><td>5.1.3.1</td><td>rpe</td><td>Ru5P ⇌ X5P</td></tr><tr><td>R12r</td><td>5.3.1.6</td><td>rpi</td><td>Ru5P ⇌ R5P</td></tr><tr><td>R13r</td><td>2.2.1.1</td><td>txt1</td><td>R5P + X5P ⇌ S7P + G3P</td></tr><tr><td>R14r</td><td>2.2.1.2</td><td>tal</td><td>G3P + S7P ⇌ E4P + F6P</td></tr><tr><td>R15r</td><td>2.2.1.1</td><td>txt2</td><td>E4P + X5P ⇌ G3P + F6P</td></tr><tr><td>R60</td><td>4.2.1.12</td><td>edd</td><td>6PGC ⇌ KDPG</td></tr><tr><td>R61r</td><td>4.1.2.14</td><td>eda</td><td>KDPG ⇌ G3P + Pyruvate</td></tr></tbody></table>

<table><caption>Table 7.5: TCA Cycle</caption><thead><tr><th>Reaction ID</th><th>EC number</th><th>Reaction name</th><th>Formula</th></tr></thead><tbody><tr><td>R20</td><td>2.3.1.54</td><td>pfl</td><td>Pyruvate + CoA ⇌ AcCoA + Formate</td></tr><tr><td>R21</td><td>1.2.4.1 / 2.3.1.12</td><td>pdh</td><td>Pyruvate + NAD<sup>+</sup> + CoA ⇌ AcCoA + CO<sub>2</sub> + NADH</td></tr><tr><td>R22</td><td>2.3.3.1</td><td>csn</td><td>2-oxoglutarateacetate + AcCoA ⇌ Citrate + CoA</td></tr><tr><td>R23r</td><td>4.2.1.3</td><td>acn</td><td>Citrate ⇌ iso-Citrate</td></tr><tr><td>R24</td><td>1.1.1.41</td><td>icd</td><td>iso-Citrate + NAD<sup>+</sup> ⇌ 2-oxoglutarate + NADH + CO<sub>2</sub></td></tr><tr><td>R25</td><td>1.2.4.2</td><td>kgd</td><td>2-oxoglutarate + NAD<sup>+</sup> + CoA ⇌ NADH + Succinateinyl-CoA + CO<sub>2</sub></td></tr><tr><td>R26r</td><td>6.2.1.5</td><td>scs</td><td>Succinateinyl-CoA + ADP + P<sub>i</sub> ⇌ Succinateinate + ATP + CoA</td></tr><tr><td>R27</td><td>1.3.5.1</td><td>sdh</td><td>Succinateinate + ADP + O<sub>2</sub>[e] + P<sub>i</sub> ⇌ Fumarate + ATP</td></tr><tr><td>R27b</td><td>1.3.5.4</td><td>frd</td><td>Fumarate + NADH ⇌ Succinateinate + NAD<sup>+</sup></td></tr><tr><td>R28r</td><td>4.2.1.2</td><td>fum</td><td>Fumarate ⇌ Malate</td></tr><tr><td>R29r</td><td>1.1.1.37</td><td>mdh</td><td>Malate + NAD<sup>+</sup> ⇌ 2-oxoglutarateacetate + NADH</td></tr></tbody></table>
<table><caption>Table 7.6: Anaplerotic Reactions</caption><thead><tr><th>Reaction ID</th><th>EC number</th><th>Reaction name</th><th>Formula</th></tr></thead><tbody><tr><td>R40</td><td>4.1.1.31</td><td>ppc</td><td>PEP + CO<sub>2</sub> ⇌ 2-oxoglutarateacetate + P<sub>i</sub></td></tr><tr><td>R41</td><td>1.1.1.38</td><td>me</td><td>Malate + NAD<sup>+</sup> ⇌ Pyruvate + NADH + CO<sub>2</sub></td></tr><tr><td>R42</td><td>4.1.1.49</td><td>ppck</td><td>2-oxoglutarateacetate + ATP ⇌ PEP + ADP + CO<sub>2</sub></td></tr></tbody></table>

<table><caption>Table 7.7: Redox-associated reactions</caption><thead><tr><th>Reaction ID</th><th>EC number</th><th>Reaction name</th><th>Formula</th></tr></thead><tbody><tr><td>R53r</td><td>1.1.1.27</td><td>ldh</td><td>Pyruvate + NADH ⇌ Lactate + NAD<sup>+</sup></td></tr><tr><td>R54ra</td><td>1.2.1.10</td><td>ada</td><td>AcCoA + NADH ⇌ Acetaldehyde + NAD<sup>+</sup> + CoA</td></tr><tr><td>R54rb</td><td>1.1.1.1</td><td>adh</td><td>Acetaldehyde + NADH ⇌ ETOH + NAD<sup>+</sup></td></tr><tr><td>R55a</td><td>2.3.1.8</td><td>pta</td><td>AcCoA + P<sub>i</sub> ⇌ Acetyl-P + CoA</td></tr><tr><td>R55b</td><td>2.7.2.1</td><td>ack</td><td>Acetyl-P + ADP ⇌ Acetate + ATP</td></tr></tbody></table>

<table><caption>Table 7.8: Oxidative phosphorylation</caption><thead><tr><th>Reaction ID</th><th>Reaction name</th><th>Formula</th></tr></thead><tbody><tr><td>R80</td><td>oxphos</td><td>NADH + 2 ADP + 0.5 O<sub>2</sub>[e] + 2 P<sub>i</sub> ⇌ NAD<sup>+</sup> + 2 ATP + 3 H<sub>2</sub>O</td></tr><tr><td>R82</td><td>atpmain</td><td>ATP + H<sub>2</sub>O ⇌ ADP + P<sub>i</sub> + ATP<sub>main</sub></td></tr></tbody></table>

<table><caption>Table 7.9: Membrane Transport Reactions</caption><thead><tr><th>Reaction ID</th><th>Reaction name</th><th>Formula</th></tr></thead><tbody><tr><td>R90</td><td>ex<sub>etoh</sub></td><td>ETOH ⇌ ETOH[e]</td></tr><tr><td>R91</td><td>ex<sub>ace</sub></td><td>Acetate ⇌ Acetate[e]</td></tr><tr><td>R93</td><td>ex<sub>NH<sub>3</sub></sub></td><td>NH<sub>3</sub>[e] ⇌ NH<sub>3</sub></td></tr><tr><td>R94</td><td>ex<sub>lac</sub></td><td>Lactate ⇌ Lactate[e]</td></tr><tr><td>R95</td><td>ex<sub>suc</sub></td><td>Succinateinate ⇌ Succinateinate[e]</td></tr><tr><td>R96</td><td>ex<sub>for</sub></td><td>Formate ⇌ Formate[e]</td></tr><tr><td>R97r</td><td>ex<sub>CO<sub>2</sub></sub></td><td>CO<sub>2</sub> ⇌ CO<sub>2</sub>[e]</td></tr></tbody></table>
<table><caption>Table 7.10: Kinetic parameters associated with reactions</caption><thead><tr><th>Reaction ID</th><th>k<sup>+</sup><sub>cat</sub> [1/s]</th><th>K<sub>eq</sub> [unitless]</th><th>Enzyme molecular weight [Da]</th></tr></thead><tbody><tr><td>R1</td><td>100</td><td>N/A</td><td>2.6·10<sup>5</sup></td></tr><tr><td>R10a</td><td>240</td><td>N/A</td><td>5.6·10<sup>4</sup></td></tr><tr><td>R10b</td><td>410</td><td>N/A</td><td>3.6·10<sup>4</sup></td></tr><tr><td>R10c</td><td>110</td><td>N/A</td><td>1.0·10<sup>5</sup></td></tr><tr><td>R11r</td><td>130</td><td>2.3</td><td>2.5·10<sup>4</sup></td></tr><tr><td>R12r</td><td>1400</td><td>2.3</td><td>1.9·10<sup>4</sup></td></tr><tr><td>R13r</td><td>46</td><td>3.7</td><td>7.3·10<sup>4</sup></td></tr><tr><td>R14r</td><td>17</td><td>0.9</td><td>3.5·10<sup>4</sup></td></tr><tr><td>R15r</td><td>75</td><td>38</td><td>7.3·10<sup>4</sup></td></tr><tr><td>R20</td><td>4800</td><td>N/A</td><td>8.5·10<sup>4</sup></td></tr><tr><td>R21</td><td>38</td><td>N/A</td><td>2.8·10<sup>5</sup></td></tr><tr><td>R22</td><td>360</td><td>N/A</td><td>9.6·10<sup>4</sup></td></tr><tr><td>R23r</td><td>33</td><td>0.074</td><td>9.6·10<sup>4</sup></td></tr><tr><td>R24</td><td>110</td><td>N/A</td><td>4.6·10<sup>4</sup></td></tr><tr><td>R25</td><td>150</td><td>N/A</td><td>1.2·10<sup>6</sup></td></tr><tr><td>R26r</td><td>89</td><td>0.52</td><td>7.1·10<sup>4</sup></td></tr><tr><td>R27</td><td>78</td><td>N/A</td><td>7.9·10<sup>5</sup></td></tr><tr><td>R27b</td><td>180</td><td>N/A</td><td>1.8·10<sup>5</sup></td></tr><tr><td>R28r</td><td>280</td><td>4.7</td><td>6.0·10<sup>4</sup></td></tr><tr><td>R29r</td><td>210</td><td>6.1·10<sup>-5</sup></td><td>3.2·10<sup>4</sup></td></tr><tr><td>R2r</td><td>320</td><td>0.51</td><td>6.2·10<sup>4</sup></td></tr><tr><td>R3</td><td>110</td><td>N/A</td><td>1.4·10<sup>5</sup></td></tr><tr><td>R4</td><td>25</td><td>N/A</td><td>3.7·10<sup>4</sup></td></tr><tr><td>R40</td><td>120</td><td>N/A</td><td>2.0·10<sup>5</sup></td></tr><tr><td>R41</td><td>76</td><td>N/A</td><td>6.3·10<sup>4</sup></td></tr><tr><td>R42</td><td>51</td><td>N/A</td><td>6.0·10<sup>4</sup></td></tr><tr><td>R53r</td><td>140</td><td>2.1·10<sup>4</sup></td><td>3.7·10<sup>4</sup></td></tr><tr><td>R54ra</td><td>0.35</td><td>2.3·10<sup>-3</sup></td><td>9.6·10<sup>4</sup></td></tr><tr><td>R54rb</td><td>320</td><td>2.8·10<sup>3</sup></td><td>9.6·10<sup>4</sup></td></tr><tr><td>R55a</td><td>91</td><td>N/A</td><td>7.7·10<sup>4</sup></td></tr><tr><td>R55b</td><td>59</td><td>N/A</td><td>4.3·10<sup>4</sup></td></tr><tr><td>R5r</td><td>8.0</td><td>3.0·10<sup>-4</sup></td><td>3.9·10<sup>4</sup></td></tr><tr><td>R60</td><td>250</td><td>N/A</td><td>6.5·10<sup>4</sup></td></tr><tr><td>R61r</td><td>80</td><td>9.6·10<sup>-3</sup></td><td>2.2·10<sup>4</sup></td></tr><tr><td>R6r</td><td>7800</td><td>11</td><td>5.4·10<sup>4</sup></td></tr><tr><td>R70</td><td>99</td><td>N/A</td><td>6.0·10<sup>4</sup></td></tr><tr><td>R7ra</td><td>230</td><td>0.088</td><td>3.6·10<sup>4</sup></td></tr><tr><td>R7rb</td><td>390</td><td>730</td><td>4.1·10<sup>4</sup></td></tr><tr><td>R7rc</td><td>53</td><td>0.16</td><td>2.9·10<sup>4</sup></td></tr><tr><td>R80</td><td>4.0·10<sup>6</sup></td><td>N/A</td><td>9.1·10<sup>5</sup></td></tr><tr><td>R82</td><td>180</td><td>N/A</td><td>6.0·10<sup>4</sup></td></tr><tr><td>R8r</td><td>210</td><td>3.5</td><td>4.6·10<sup>4</sup></td></tr><tr><td>R9</td><td>510</td><td>N/A</td><td>5.0·10<sup>4</sup></td></tr><tr><td>R90</td><td>100</td><td>N/A</td><td>N/A</td></tr><tr><td>R91</td><td>100</td><td>N/A</td><td>5.9·10<sup>4</sup></td></tr><tr><td>R93</td><td>100</td><td>N/A</td><td>4.5·10<sup>4</sup></td></tr><tr><td>R94</td><td>100</td><td>N/A</td><td>5.9·10<sup>4</sup></td></tr><tr><td>R95</td><td>100</td><td>N/A</td><td>4.5·10<sup>4</sup></td></tr><tr><td>R96</td><td>100</td><td>N/A</td><td>3.1·10<sup>4</sup></td></tr><tr><td>R97r</td><td>100</td><td>N/A</td><td>N/A</td></tr><tr><td>RR9</td><td>13</td><td>N/A</td><td>8.7·10<sup>4</sup></td></tr></tbody></table>
<table><thead><tr><th>Reaction ID</th><th>Metabolite name</th><th>K<sub>M</sub> [mM]</th></tr></thead><tbody><tr><td>R1</td><td>G6P</td><td>0.102</td></tr><tr><td>R1</td><td>Glucose</td><td>0.116</td></tr><tr><td>R1</td><td>PEP</td><td>0.0983</td></tr><tr><td>R1</td><td>Pyruvate</td><td>0.102</td></tr><tr><td>R10a</td><td>G6P</td><td>0.314</td></tr><tr><td>R10a</td><td>6PGL</td><td>0.129</td></tr><tr><td>R10a</td><td>NAD<sup>+</sup></td><td>0.863</td></tr><tr><td>R10a</td><td>NADH</td><td>0.129</td></tr><tr><td>R10b</td><td>6PGL</td><td>0.168</td></tr><tr><td>R10b</td><td>6PGC</td><td>0.0594</td></tr><tr><td>R10c</td><td>CO<sub>2</sub></td><td>0.0626</td></tr><tr><td>R10c</td><td>6PGC</td><td>0.101</td></tr><tr><td>R10c</td><td>Ru5P</td><td>0.0626</td></tr><tr><td>R10c</td><td>NAD<sup>+</sup></td><td>0.0591</td></tr><tr><td>R10c</td><td>NADH</td><td>0.0626</td></tr><tr><td>R11r</td><td>Ru5P</td><td>0.0878</td></tr><tr><td>R11r</td><td>X5P</td><td>0.114</td></tr><tr><td>R12r</td><td>R5P</td><td>1.25</td></tr><tr><td>R12r</td><td>Ru5P</td><td>0.558</td></tr><tr><td>R13r</td><td>G3P</td><td>1.23</td></tr><tr><td>R13r</td><td>R5P</td><td>0.972</td></tr><tr><td>R13r</td><td>S7P</td><td>2.11</td></tr><tr><td>R13r</td><td>X5P</td><td>0.157</td></tr><tr><td>R14r</td><td>E4P</td><td>0.175</td></tr><tr><td>R14r</td><td>F6P</td><td>0.888</td></tr><tr><td>R14r</td><td>G3P</td><td>0.578</td></tr><tr><td>R14r</td><td>S7P</td><td>0.206</td></tr><tr><td>R15r</td><td>E4P</td><td>0.0934</td></tr><tr><td>R15r</td><td>F6P</td><td>0.737</td></tr><tr><td>R15r</td><td>G3P</td><td>1.27</td></tr><tr><td>R15r</td><td>X5P</td><td>0.152</td></tr><tr><td>R20</td><td>AcCoA</td><td>0.0352</td></tr><tr><td>R20</td><td>CoA</td><td>0.0168</td></tr><tr><td>R20</td><td>Formate</td><td>6.35</td></tr><tr><td>R20</td><td>Pyruvate</td><td>2.18</td></tr><tr><td>R21</td><td>AcCoA</td><td>0.159</td></tr><tr><td>R21</td><td>CO<sub>2</sub></td><td>0.159</td></tr><tr><td>R21</td><td>CoA</td><td>0.0629</td></tr><tr><td>R21</td><td>Pyruvate</td><td>0.291</td></tr><tr><td>R21</td><td>NAD<sup>+</sup></td><td>0.0629</td></tr><tr><td>R21</td><td>NADH</td><td>0.159</td></tr><tr><td>R22</td><td>AcCoA</td><td>0.0867</td></tr><tr><td>R22</td><td>Citrate</td><td>0.0756</td></tr><tr><td>R22</td><td>CoA</td><td>0.0756</td></tr><tr><td>R22</td><td>2-oxoglutarate</td><td>0.0287</td></tr><tr><td>R23r</td><td>Citrate</td><td>3.49</td></tr><tr><td>R23r</td><td>iso-Citrate</td><td>2.42</td></tr><tr><td>R24</td><td>2-oxoglutarate</td><td>0.483</td></tr><tr><td>R24</td><td>CO<sub>2</sub></td><td>2.02</td></tr><tr><td>R24</td><td>iso-Citrate</td><td>0.0227</td></tr></tbody></table>

<table><thead><tr><th>Reaction ID</th><th>Metabolite name</th><th>K<sub>M</sub> [mM]</th></tr></thead><tbody><tr><td>R24</td><td>NAD<sup>+</sup></td><td>1.06</td></tr><tr><td>R24</td><td>NADH</td><td>0.0119</td></tr><tr><td>R25</td><td>2-oxoglutarate</td><td>0.0670</td></tr><tr><td>R25</td><td>CO<sub>2</sub></td><td>0.108</td></tr><tr><td>R25</td><td>CoA</td><td>0.0927</td></tr><tr><td>R25</td><td>Succinyl-CoA</td><td>0.108</td></tr><tr><td>R25</td><td>NAD<sup>+</sup></td><td>0.0927</td></tr><tr><td>R25</td><td>NADH</td><td>0.108</td></tr><tr><td>R26r</td><td>CoA</td><td>0.00731</td></tr><tr><td>R26r</td><td>Succinate</td><td>0.237</td></tr><tr><td>R26r</td><td>Succinyl-CoA</td><td>0.0105</td></tr><tr><td>R26r</td><td>ADP</td><td>0.0560</td></tr><tr><td>R26r</td><td>ATP</td><td>0.0812</td></tr><tr><td>R27</td><td>Fumarate</td><td>0.0812</td></tr><tr><td>R27</td><td>O<sub>2</sub>[e]</td><td>0.371</td></tr><tr><td>R27</td><td>Succinate</td><td>0.0756</td></tr><tr><td>R27</td><td>ADP</td><td>0.371</td></tr><tr><td>R27</td><td>ATP</td><td>0.0270</td></tr><tr><td>R27b</td><td>Fumarate</td><td>0.0201</td></tr><tr><td>R27b</td><td>Succinate</td><td>0.205</td></tr><tr><td>R27b</td><td>NAD<sup>+</sup></td><td>0.0431</td></tr><tr><td>R27b</td><td>NADH</td><td>0.232</td></tr><tr><td>R28r</td><td>Fumarate</td><td>0.314</td></tr><tr><td>R28r</td><td>Malate</td><td>0.615</td></tr><tr><td>R29r</td><td>Malate</td><td>3.19</td></tr><tr><td>R29r</td><td>2-oxoglutarate</td><td>0.0283</td></tr><tr><td>R29r</td><td>NAD<sup>+</sup></td><td>0.460</td></tr><tr><td>R29r</td><td>NADH</td><td>0.0321</td></tr><tr><td>R2r</td><td>F6P</td><td>0.162</td></tr><tr><td>R2r</td><td>G6P</td><td>0.273</td></tr><tr><td>R3</td><td>F6P</td><td>0.116</td></tr><tr><td>R3</td><td>FBP</td><td>0.113</td></tr><tr><td>R3</td><td>ADP</td><td>0.113</td></tr><tr><td>R3</td><td>ATP</td><td>0.141</td></tr><tr><td>R4</td><td>F6P</td><td>0.171</td></tr><tr><td>R4</td><td>FBP</td><td>0.0161</td></tr><tr><td>R40</td><td>CO<sub>2</sub></td><td>0.115</td></tr><tr><td>R40</td><td>2-oxoglutarate</td><td>0.0426</td></tr><tr><td>R40</td><td>PEP</td><td>0.364</td></tr><tr><td>R41</td><td>CO<sub>2</sub></td><td>0.0885</td></tr><tr><td>R41</td><td>Malate</td><td>0.361</td></tr><tr><td>R41</td><td>Pyruvate</td><td>0.0885</td></tr><tr><td>R41</td><td>NAD<sup>+</sup></td><td>0.0691</td></tr><tr><td>R41</td><td>NADH</td><td>0.0885</td></tr><tr><td>R42</td><td>CO<sub>2</sub></td><td>5.21</td></tr><tr><td>R42</td><td>2-oxoglutarate</td><td>0.571</td></tr><tr><td>R42</td><td>PEP</td><td>0.0643</td></tr><tr><td>R42</td><td>ADP</td><td>0.0484</td></tr><tr><td>R42</td><td>ATP</td><td>0.0750</td></tr><tr><td>R53r</td><td>LACTATE</td><td>0.517</td></tr></tbody></table>

Michaelis constants – part I
<table><thead><tr><th>Reaction ID</th><th>Metabolite name</th><th>K<sub>M</sub> [mM]</th></tr></thead><tbody><tr><td>R53r</td><td>Pyruvate</td><td>0.0193</td></tr><tr><td>R53r</td><td>NAD<sup>+</sup></td><td>0.517</td></tr><tr><td>R53r</td><td>NADH</td><td>0.0193</td></tr><tr><td>R54ra</td><td>AcCoA</td><td>0.0242</td></tr><tr><td>R54ra</td><td>Acetaldehyde</td><td>1.80</td></tr><tr><td>R54ra</td><td>CoA</td><td>0.00786</td></tr><tr><td>R54ra</td><td>NAD<sup>+</sup></td><td>0.0415</td></tr><tr><td>R54ra</td><td>NADH</td><td>0.113</td></tr><tr><td>R54rb</td><td>Acetaldehyde</td><td>0.0593</td></tr><tr><td>R54rb</td><td>ETOH</td><td>5.49</td></tr><tr><td>R54rb</td><td>NAD<sup>+</sup></td><td>0.169</td></tr><tr><td>R54rb</td><td>NADH</td><td>0.0593</td></tr><tr><td>R55a</td><td>AcCoA</td><td>0.0424</td></tr><tr><td>R55a</td><td>Acetyl-P</td><td>0.313</td></tr><tr><td>R55a</td><td>CoA</td><td>0.0860</td></tr><tr><td>R55b</td><td>Acetate</td><td>3.44</td></tr><tr><td>R55b</td><td>Acetyl-P</td><td>0.154</td></tr><tr><td>R55b</td><td>ADP</td><td>0.402</td></tr><tr><td>R55b</td><td>ATP</td><td>0.0714</td></tr><tr><td>R5r</td><td>DHAP</td><td>0.0782</td></tr><tr><td>R5r</td><td>FBP</td><td>0.204</td></tr><tr><td>R5r</td><td>G3P</td><td>0.0782</td></tr><tr><td>R60</td><td>6PGC</td><td>0.0434</td></tr><tr><td>R60</td><td>KDPG</td><td>0.150</td></tr><tr><td>R61r</td><td>G3P</td><td>0.00146</td></tr><tr><td>R61r</td><td>KDPG</td><td>0.561</td></tr><tr><td>R61r</td><td>Pyruvate</td><td>0.00146</td></tr><tr><td>R6r</td><td>DHAP</td><td>0.0750</td></tr><tr><td>R6r</td><td>G3P</td><td>0.745</td></tr><tr><td>R70</td><td>AcCoA</td><td>0.462</td></tr><tr><td>R70</td><td>2-oxoglutarate</td><td>0.352</td></tr><tr><td>R70</td><td>BIOMASS</td><td>0.0998</td></tr><tr><td>R70</td><td>CO<sub>2</sub></td><td>0.0996</td></tr><tr><td>R70</td><td>CoA</td><td>0.891</td></tr><tr><td>R70</td><td>E4P</td><td>0.0144</td></tr><tr><td>R70</td><td>G6P</td><td>4.31</td></tr><tr><td>R70</td><td>NH<sub>3</sub></td><td>0.0151</td></tr><tr><td>R70</td><td>2-oxoglutarate</td><td>0.00672</td></tr><tr><td>R70</td><td>PEP</td><td>0.169</td></tr><tr><td>R70</td><td>Pyruvate</td><td>0.319</td></tr><tr><td>R70</td><td>R5P</td><td>0.881</td></tr><tr><td>R70</td><td>ADP</td><td>0.0293</td></tr><tr><td>R70</td><td>ATP</td><td>0.342</td></tr><tr><td>R70</td><td>NAD<sup>+</sup></td><td>1.43</td></tr></tbody></table>

<table><thead><tr><th>Reaction ID</th><th>Metabolite name</th><th>K<sub>M</sub> [mM]</th></tr></thead><tbody><tr><td>R70</td><td>NADH</td><td>0.0913</td></tr><tr><td>R7ra</td><td>DPG</td><td>0.0576</td></tr><tr><td>R7ra</td><td>G3P</td><td>0.687</td></tr><tr><td>R7ra</td><td>NAD<sup>+</sup></td><td>0.0558</td></tr><tr><td>R7ra</td><td>NADH</td><td>0.0576</td></tr><tr><td>R7rb</td><td>DPG</td><td>0.0426</td></tr><tr><td>R7rb</td><td>3PG</td><td>0.235</td></tr><tr><td>R7rb</td><td>ADP</td><td>0.0426</td></tr><tr><td>R7rb</td><td>ATP</td><td>0.235</td></tr><tr><td>R7rc</td><td>3PG</td><td>0.132</td></tr><tr><td>R7rc</td><td>2PG</td><td>0.0755</td></tr><tr><td>R80</td><td>O<sub>2</sub>[e]</td><td>0.116</td></tr><tr><td>R80</td><td>ADP</td><td>0.136</td></tr><tr><td>R80</td><td>ATP</td><td>0.0737</td></tr><tr><td>R80</td><td>NAD<sup>+</sup></td><td>0.0859</td></tr><tr><td>R80</td><td>NADH</td><td>0.116</td></tr><tr><td>R82</td><td>ATP<sub>main</sub></td><td>0.130</td></tr><tr><td>R82</td><td>ADP</td><td>0.130</td></tr><tr><td>R82</td><td>ATP</td><td>0.0769</td></tr><tr><td>R8r</td><td>PEP</td><td>0.131</td></tr><tr><td>R8r</td><td>2PG</td><td>0.108</td></tr><tr><td>R9</td><td>PEP</td><td>0.291</td></tr><tr><td>R9</td><td>Pyruvate</td><td>0.0476</td></tr><tr><td>R9</td><td>ADP</td><td>0.218</td></tr><tr><td>R9</td><td>ATP</td><td>8.45</td></tr><tr><td>R90</td><td>ETOH</td><td>0.100</td></tr><tr><td>R90</td><td>ETOH[e]</td><td>0.100</td></tr><tr><td>R91</td><td>Acetate</td><td>0.100</td></tr><tr><td>R91</td><td>Acetate[e]</td><td>0.100</td></tr><tr><td>R93</td><td>NH<sub>3</sub></td><td>0.0999</td></tr><tr><td>R93</td><td>NH<sub>3</sub>[e]</td><td>0.100</td></tr><tr><td>R94</td><td>Lactate</td><td>0.100</td></tr><tr><td>R94</td><td>Lactate[e]</td><td>0.100</td></tr><tr><td>R95</td><td>Succinate</td><td>0.100</td></tr><tr><td>R95</td><td>Succinate[e]</td><td>0.100</td></tr><tr><td>R96</td><td>Formate</td><td>0.0999</td></tr><tr><td>R96</td><td>Formate[e]</td><td>0.100</td></tr><tr><td>R97r</td><td>CO<sub>2</sub></td><td>0.0999</td></tr><tr><td>R97r</td><td>CO<sub>2</sub>[e]</td><td>0.100</td></tr><tr><td>RR9</td><td>PEP</td><td>0.0934</td></tr><tr><td>RR9</td><td>Pyruvate</td><td>0.0864</td></tr><tr><td>RR9</td><td>ADP</td><td>0.0873</td></tr><tr><td>RR9</td><td>ATP</td><td>0.0350</td></tr></tbody></table>

Michaelis constants – part II
<figure><img src="image_17.png" /></figure>

Figure 7.9: Metabolic strategies in the *E. coli* model, depending on external glucose and oxygen concentrations. In the EFM phase diagram. Each region represents the winning EFM as explained in Figure 7.8. Here, the colors represent the flux in one specific reaction based on the winning EFM in that region. (A) The lactate secretion flux is strikingly equal to 0 in most regions. The only conditions where lactate is secreted is at low oxygen and medium/high glucose concentrations. (B) The biomass yield is, in general, high if and only if lactate is not secreted. This makes sense because the carbon coming from the glucose is often the limiting nutrient for growth, and there is a trade-off between using it for biomass versus fermentation products such as lactate. Interestingly, the region with high glucose and high oxygen levels (upper right quadrant) is occupied by an EFM that doesn't achieve the highest possible yield (i.e. *max-gr*). In low glucose and high oxygen, or in medium oxygen levels, the winning EFMs are the ones with relatively higher biomass yields.

## 7.9. More results for the *E. coli* central metabolism model

This section contains additional results for the *E. coli* central metabolism model from Chapter 7 in [1], in particular, fluxes plotted in the EFM phase diagram and in flux space, as well as ideal and real enzyme costs for all EFMs.

Each point in the Monod landscape in Figure 7.8 (A) corresponds to a state of the model, and the calculations that lead to the growth rate and the "winning EFM" shown yield a full description of this state, including all fluxes, metabolite concentrations and enzyme levels. These data can be explored and visualized in many ways. For illustration, Figure 7.10 shows a variant of Figure 7.8 (A) in the horizontal axes do not describe the external concentrations of glucose and oxygen, but their uptake rates, and the vertical axes shows the biomass production rate. Since uptake rates are a function of external concentrations, and the growth rate directly depends on the biomass production rate, we might have expected that this yields the same picture, just a bit stretched along each of the axes. However, the picture looks very different: instead of forming a continuous surface, the points now fall on disconnected rays, apparently with one ray for each colored region of the surface. In fact, when looking at the picture closely, we can see that each of the winning EFMs gives rise to exactly one ray. But this, after all, is logical. In the new plot, all axes refer to reaction rates, and for each EFM all rates come in fixed ratios, giving rise to a ray. So, if our solutions are EFMs, this picture cannot be continuous – in line with the fact that, in the original Monod landscape 7.8 (A), when moving from one region to the other one, one would notice a discrete jump of the reaction rates. But why is the new picture not a continuous surface, if uptake rates depends smoothly on external metabolite concentrations? In fact, they do not only depend on these concentrations, but also on resource allocation to the transporter. If this allocation shows a discrete jump (again, when moving from one region to another one), then also the rate shows a jump. The comparison between the two plots shows us what we gain by considering enzyme kinetics as compared to a pure stoichiometric model. With the biomass rate as a proxy for cell growth, each EFM defines fixed ratios between this growth rate and each of the metabolic fluxes, including the uptake rates. When continuously scaling an EFM, the glucose uptake,
<figure><img src="image_18.png" /></figure>

Figure 7.10: Proportional scaling of fluxes within each EFM – In the diagram with glucose uptake, oxygen uptake, and biomass production rate on the axes, each colored line corresponds to one EFM, and shows the possible combinations of fluxes obtained from the model behind Figure 7.8 (which also shares the EFM colors with this figure). Importantly, here the *x* and *y* axes represent uptake rates and not substrate concentrations. Therefore, as expected, each EFM yields a straight line (because of the proportional scaling of different fluxes for each EFM). Since – according to our reasoning – optimal flux distributions must be EFMs, only these combinations of fluxes are actually possible. When glucose and oxygen concentrations are varied smoothly in Figure 7.8, the corresponding movement in this plot would be along the lines and sometimes, jumps between different lines (when the system moves from one region to another one in Figure 7.8).

oxygen uptake, and biomass production will scale proportionally. So the rays in Figure 7.10 reflect what we can know about possible metabolic fluxes based on network structure alone; but to get to the Monod landscape, as a function of concentrations, we had to use kinetic information and an extra principle of economical enzyme usage.

The phase diagram of “winning EFMs” can also be used to visualize other (optimized) quantities as functions of glucose and oxygen concentrations. Figure 7.9 shows as example the (biomass-specific) lactate secretion (showing that also a number of other winning EFMs, apart from *ana-lac*, secrete lactate) and the biomass yield on glucose.

Finally, a statistics over all EFMs shows that the range of possible enzyme demands per biomass production rate is quite large: as shown in Figure 7.11, they vary over more than two orders of magnitude, making some EFMs a hundred-fold more enzyme-expensive than others. The same plot also shows how enzyme costs depend on the fact that enzymes do not operate at their full capacity (reaching their $k_{cat}$ value), but at best at the enzyme efficiencies predicted by enzyme cost minimization. For our *E. coli* model and the aerobic glucose conditions studied, if all enzymes could operate at their $k_{cat}$ values, this would decrease to overall enzyme demand by a factor of at least 1.4, or maximally 4.7, depending on the EFM in question. But still, in this case, for determining enzyme costs the choice of the right EFM (even assuming "ideal" enzymes) is much more important than considering the actual, "non-ideal" way in which enzymes operate. But this may not always hold: under low-oxygen conditions, the enzyme demands of some EFMs may increase much more drastically.

# Solutions to problems

## Problem 7.1 (Effect of oxygen concentration)

The oxygen concentration affects only the rate $v_3$, and an increase in oxygen increases this rate for the same enzyme concentration. Since EFM2 does not contain $v_3$, the enzyme cost of this EFM will not change. EFM1 and EFM3
<figure><img src="image_19.png" /></figure>

Figure 7.11: Ideal and real enzyme costs of elementary flux modes – For each EFM (shown as a cyan dot), the ideal enzyme cost per biomass production rate (i.e. assuming that all the enzymes are saturated) is compared to the actual cost (calculated using Enzyme Cost Minimization, assuming standard aerobic glucose conditions). The costs span a wide range from the most enzyme-efficient EFMs on the lower left to the least enzyme-efficient ones on the upper right. For different EFMs, the ratio of actual and ideal costs varies between 1.4 and 4.7. Here, the EFM with the minimal actual cost is among the top 5 in terms of ideal cost.

benefit from an increase of the oxygen concentration, because they will have to invest less enzyme in $v_3$ to obtain the same rate. Therefore, those EFMs can become more beneficial and the optimal EFM could shift to one of those EFMs.

## Problem 7.2 (Effect of external metabolites)

Increasing $[P_{ex}]$ will benefit EFM3 and decrease the benefit of EFM2 (because for EFM2 $[P_{ex}]$ will inhibit reaction $v_2$ and therefore more enzyme is needed for reaction $v_2$ and the enzyme cost of EFM2 will increase. Qualitatively, with increasing $[P_{ex}]$, EFM2 might become more expensive, and either EFM1 or EFM3 will become beneficial, or both at different $P_{ex}$ concentrations, depending on the kinetics of the reactions.

## Problem 7.3 (States of maximal growth rate)

(a) $v_1 = e_1(k_1^+ s_1 - k_1^- X)$, $v_2 = e_2(k_2^+ s_2 - k_2^- x)$ and $v_3 = e_3(k_3^+ x - k_3^- p)$

(b) $e_{tot} = \frac{v_1}{(2s_1-x)} + \frac{v_2}{(30-x)} + \frac{v_3}{x}$

(c) When $e_1 = 0$, $v_1$ is also 0 and to achieve steady state $v_2 = v_3$ and using the rate equations and filling in the parameters we get $e_{tot} = \frac{v_3}{(30-x)} + \frac{v_3}{x}$. We now set $e_{tot} = 1$ to obtain $v_3 = (1 - \frac{v_3}{30-x})x$ which we can rewrite to $v_3 = \frac{x}{1 + \frac{x}{30-x}}$. We can take the derivative to $x$ and set it equal to 0 to find the optimum, which leads to $x = 15$ and $v_3/e_{tot} = \frac{15}{2}$. Note that in this specific case we did not need to set $e_{tot} = 1$ and could have maximized $v_3/e_{tot}$ directly, but in general this does not always work. In the rest of the answers we assume we set $e_{tot} = 1$ and therefore $v_3 = v_3/e_{tot}$.

(d) $e_2 = 0$ implies $v_2 = 0$ and filling in $s_1 = 10$ leads to $v_3 = (1 - \frac{v_3}{20-x})x$, which can be rewritten to $v_3 = \frac{x}{1 + \frac{x}{20-x}}$. This is optimal when $x = 10$ and $v_3/e_{tot} = 5$.

(e) $e_1 = e_2$ implies $\frac{v_1}{20-x} = \frac{v_2}{30-x}$. From the steady state we know that $v_2 = v_3 - v_1$. Filling this in and solving for $v_1$ leads to $v_1 = v_3 \frac{x-20}{x-50}$. From the total enzyme and by replacing $e_1$ by $e_2$ we get $e_3 = 1 - 2 \frac{v_1}{20-x}$. Putting this in the equation for $v_3$ and using the previous equality to replace $v_1$ leads to: $v_3 = (1 + \frac{v_3}{x-25})x$. Solving this for $v_3$ gives $v_3 = x - \frac{x^2}{25}$, which is optimal for $x = 12.5$ with $v_3/e_{tot} = \frac{25}{4}$.

(f) It was optimal to invest all enzyme in $e_2$ and none in $e_1$, because that lead to the highest specific flux $v_3$ (namely $v_3 = 7.5$).

(g) Since $v_1 = 0$, $S_1$ is not involved in any reaction and the solution is the same as above, $x = 15$ and $v_3/e_{tot} = \frac{15}{2}$.
(h) Similar calculations as above but now with $s_1 = 50$ lead to $v_3 = \frac{x}{1+\frac{x}{100-x}}$, which is optimal when $x = 50$ and gives $v_3/e_{tot} = 25$.

(i) Similar calculations as above but now with $s_1 = 50$ lead to $v_3 = x - \frac{x^2}{65}$. $v_3/e_{tot}$ is maximal at $x = 32.5$ and takes the value 16.25.

(j) Now $s_1$ increased the optimal strategy would be to invest all enzymes in $e_1$, and have $e_2 = 0$, because that leads to the highest specific flux of $v_3$, namely $v_3/e_{tot} = 25$.

(k) The two EFMs in this pathway that produce $P$ are $v_1 = v_3$ with $v_2 = 0$ and $v_2 = v_3$ with $v_1 = 0$. In the problem we saw when we optimize the specific flux, we always obtained one of those EFMs as the best solution, from the options that we tested. Therefore, these results are in agreement with the proof outlined in this chapter.
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