# Optimal cell behavior in time

Hidde de Jong, Dafni Giannari, Diego A Oyarzún, Steffen Waldherr, Agustín G Yabo

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## ► To cite this version:

Hidde de Jong, Dafni Giannari, Diego A Oyarzún, Steffen Waldherr, Agustín G Yabo. Optimal cell behavior in time. Economic Principles in Cell Biology, Zenodo, 2023, 10.5281/zenodo.8157066. [https://doi.org/10.5281/zenodo.8157066](https://doi.org/10.5281/zenodo.8157066)

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Optimal cell behavior in time

Dafni Giannari, Hidde de Jong, Diego A Oyarzún, Steffen Waldherr, Agustín G. Yabo

► To cite this version:

Dafni Giannari, Hidde de Jong, Diego A Oyarzún, Steffen Waldherr, Agustín G. Yabo. Optimal cell behavior in time. Economic Principles in Cell Biology, , pp.153-172, 2023, 10.5281/zenodo.8157066. hal-04383880

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Submitted on 5 Apr 2024

HAL is a multi-disciplinary open access
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.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
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publics ou privés.

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# Optimal cell behavior in time

Hidde de Jong¹ ID, Dafni Giannari² ID, Diego A. Oyarzún³,⁴ ID, Steffen Waldherr⁵ ID, and Agustín G. Yabo⁶ ID

¹ Université Grenoble Alpes, Inria, France; ² Department of Chemical Engineering and Applied Chemistry, University of Toronto, Toronto, Canada; ³ School of Informatics, The University of Edinburgh, EH8 9AB, Edinburgh, United Kingdom; ⁴ School of Biological Sciences, The University of Edinburgh, EH9 3JH, Edinburgh, United Kingdom; ⁵ Department of Functional and Evolutionary Ecology, University of Vienna, 1030 Vienna, Austria; ⁶ MISTEA, Université Montpellier, INRAE, Institut Agro, Montpellier, France

## Abstract

Microorganisms live in continually changing environments, which require them to develop adaptation strategies. These strategies have been profitably studied under the assumption that microorganisms have evolved to optimize one or several aspects of their adaptive response. The mathematical formalization of this assumption leads to dynamic optimization problems that can be solved by means of techniques from optimal control theory. The chapter discusses three example problems: dynamic optimization of enzyme expression in metabolic pathways, dynamic optimization of coarse-grained models of cellular growth, and dynamic flux balance analysis. The results obtained for these problems illustrate the interest of studying adaptation strategies from the perspective of dynamic optimization, and the strengths and weaknesses of this approach.

**Contributions:** This chapter was planned and outlined by Hidde de Jong and Steffen Waldherr (chapter editors). The sections were written Diego Oyarzún (dynamic optimization of enzyme expression in metabolic pathways), Hidde de Jong and Agustín Yabo (dynamic optimization of coarse-grained models of cellular growth), and Dafni Giannari and Steffen Waldherr (dynamic flux balance analysis). Agustín Yabo performed the numerical optimization studies for the examples discussed in the chapter. Andreas Kremling, Wolfram Liebermeister, and Felipe Scott provided comments on the chapter draft. Hidde de Jong acknowledges support from the Agence National de la Recherche (MAXIMIC, ANR-17-CE40-0024-01).

**To cite this chapter:** H. de Jong, D. Giannari, D.A. Oyarzún, S. Waldherr, and A.G. Yabo. Optimal cell behavior in time. 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_3.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_4.png" /></figure>
# Optimal cell behavior in time

Hidde de Jong¹ ID, Dafni Giannari² ID, Diego A. Oyarzún³,⁴ ID, Steffen Waldherr⁵ ID, and Agustín G. Yabo⁶ ID

¹ Université Grenoble Alpes, Inria, France; ² Department of Chemical Engineering and Applied Chemistry, University of Toronto, Toronto, Canada; ³ School of Informatics, The University of Edinburgh, EH8 9AB, Edinburgh, United Kingdom; ⁴ School of Biological Sciences, The University of Edinburgh, EH9 3JH, Edinburgh, United Kingdom; ⁵ Department of Functional and Evolutionary Ecology, University of Vienna, 1030 Vienna, Austria; ⁶ MISTEA, Université Montpellier, INRAE, Institut Agro, Montpellier, France

## Abstract

Microorganisms live in continually changing environments, which require them to develop adaptation strategies. These strategies have been profitably studied under the assumption that microorganisms have evolved to optimize one or several aspects of their adaptive response. The mathematical formalization of this assumption leads to dynamic optimization problems that can be solved by means of techniques from optimal control theory. The chapter discusses three example problems: dynamic optimization of enzyme expression in metabolic pathways, dynamic optimization of coarse-grained models of cellular growth, and dynamic flux balance analysis. The results obtained for these problems illustrate the interest of studying adaptation strategies from the perspective of dynamic optimization, and the strengths and weaknesses of this approach.

**Contributions:** This chapter was planned and outlined by Hidde de Jong and Steffen Waldherr (chapter editors). The sections were written Diego Oyarzún (dynamic optimization of enzyme expression in metabolic pathways), Hidde de Jong and Agustín Yabo (dynamic optimization of coarse-grained models of cellular growth), and Dafni Giannari and Steffen Waldherr (dynamic flux balance analysis). Agustín Yabo performed the numerical optimization studies for the examples discussed in the chapter. Andreas Kremling, Wolfram Liebermeister, and Felipe Scott provided comments on the chapter draft. Hidde de Jong acknowledges support from the Agence National de la Recherche (MAXIMIC, ANR-17-CE40-0024-01).

**To cite this chapter:** H. de Jong, D. Giannari, D.A. Oyarzún, S. Waldherr, and A.G. Yabo. Optimal cell behavior in time. 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>
## Chapter highlights

* Microorganisms live in continually changing environments, which require them to develop adaptation strategies.

* These strategies have been profitably studied under the assumption that microorganisms have evolved to optimize one or several aspects of their adaptive response.

* The mathematical formalization of this assumption leads to dynamic optimization problems that can be solved by means of techniques from optimal control theory.

* The chapter discusses three example problems: dynamic optimization of enzyme expression in metabolic pathways, dynamic optimization of coarse-grained models of cellular growth, and dynamic flux balance analysis.

* The results obtained for these problems illustrate the interest of studying adaptation strategies from the perspective of dynamic optimization, and the strengths and weaknesses of this approach.

### 10.1. Introduction

The study of microorganisms in the laboratory has often focused on the creation of stable conditions enabling balanced, reproducible growth of the population. Such conditions are almost never found in nature. Microorganisms live in continually changing environments in which nutrients are only intermittently available and in which the cells are submitted to a variety of other temporally varying stresses (acidity, heat and cold, drought, ...). In order to survive in these conditions, microorganisms have developed a range of molecular mechanisms to detect changes in the environment, or signals announcing such changes, and to adapt their functioning accordingly.

A well-studied example of the dynamic response of bacteria to changes in their environment is the phenomenon of diauxic growth, discovered by Jacques Monod ([1] (see also Chapter 8 in [2]). When *Escherichia coli* is grown in a medium containing a mixture of two carbon sources, e.g., glucose and lactose, the cells generally first deplete the carbon source supporting the highest growth rate (glucose) before starting to assimilate the other carbon source (lactose). A variety of mechanisms are involved in this switch from a preferred to a secondary carbon source, including the release of the repression of enzymes necessary for lactose utilization, the release of the inhibition of lactose transporters, and the global regulation of a large number of other genes [3, 4].

In many situations, the precise functioning of the molecular mechanisms regulating the adaptation of microbial physiology to changes in the environment is not or only qualitatively understood. This precludes their inclusion in quantitative models that accurately predict the dynamic response of the cell in a variety of conditions. The lack of mechanistic, quantitative information can be bypassed by making appropriate assumptions about the regulatory systems, in particular that the latter have evolved under the selection pressure of the environment to optimize the response to external perturbations. More precisely, it is assumed that microorganisms have developed mechanisms that allocate limiting resources (proteins, fluxes, ...) to cellular processes so as to maximize or minimize some objective function, or combination of objective functions, over the time-interval of environment changes.

The use of an optimality assumption to make up for missing or incomplete information was already exploited with success in Chapter 5 in [2] of this book. The difference with those approaches is that here we are interested in cases where the optimality criterion is defined over an interval of time rather than at steady state, and thus we need to consider dynamic instead of static optimization. Moreover, some methods take into account that cells may vary the allocation of limiting resources to cellular processes over the time interval in which the environmental changes occur, instead of only considering a constant response in a stable environment. This generalization of the problem enormously increases its complexity. It may also lead to nontrivial dynamical effects that are not found in the case of static optimization, such as the accumulation of resource buffers to anticipate future changes in the environment [5].

The classical argument motivating the optimality assumption in the case of microorganisms is that mutants of genes coding for enzymes in central metabolism often have a lower growth rate than the wild-type strain, where growth rate
## Chapter highlights

* Microorganisms live in continually changing environments, which require them to develop adaptation strategies.

* These strategies have been profitably studied under the assumption that microorganisms have evolved to optimize one or several aspects of their adaptive response.

* The mathematical formalization of this assumption leads to dynamic optimization problems that can be solved by means of techniques from optimal control theory.

* The chapter discusses three example problems: dynamic optimization of enzyme expression in metabolic pathways, dynamic optimization of coarse-grained models of cellular growth, and dynamic flux balance analysis.

* The results obtained for these problems illustrate the interest of studying adaptation strategies from the perspective of dynamic optimization, and the strengths and weaknesses of this approach.

### 10.1. Introduction

The study of microorganisms in the laboratory has often focused on the creation of stable conditions enabling balanced, reproducible growth of the population. Such conditions are almost never found in nature. Microorganisms live in continually changing environments in which nutrients are only intermittently available and in which the cells are submitted to a variety of other temporally varying stresses (acidity, heat and cold, drought, ...). In order to survive in these conditions, microorganisms have developed a range of molecular mechanisms to detect changes in the environment, or signals announcing such changes, and to adapt their functioning accordingly.

A well-studied example of the dynamic response of bacteria to changes in their environment is the phenomenon of diauxic growth, discovered by Jacques Monod ([1] (see also Chapter 8 in [2]). When *Escherichia coli* is grown in a medium containing a mixture of two carbon sources, e.g., glucose and lactose, the cells generally first deplete the carbon source supporting the highest growth rate (glucose) before starting to assimilate the other carbon source (lactose). A variety of mechanisms are involved in this switch from a preferred to a secondary carbon source, including the release of the repression of enzymes necessary for lactose utilization, the release of the inhibition of lactose transporters, and the global regulation of a large number of other genes [3, 4].

In many situations, the precise functioning of the molecular mechanisms regulating the adaptation of microbial physiology to changes in the environment is not or only qualitatively understood. This precludes their inclusion in quantitative models that accurately predict the dynamic response of the cell in a variety of conditions. The lack of mechanistic, quantitative information can be bypassed by making appropriate assumptions about the regulatory systems, in particular that the latter have evolved under the selection pressure of the environment to optimize the response to external perturbations. More precisely, it is assumed that microorganisms have developed mechanisms that allocate limiting resources (proteins, fluxes, ...) to cellular processes so as to maximize or minimize some objective function, or combination of objective functions, over the time-interval of environment changes.

The use of an optimality assumption to make up for missing or incomplete information was already exploited with success in Chapter 5 in [2] of this book. The difference with those approaches is that here we are interested in cases where the optimality criterion is defined over an interval of time rather than at steady state, and thus we need to consider dynamic instead of static optimization. Moreover, some methods take into account that cells may vary the allocation of limiting resources to cellular processes over the time interval in which the environmental changes occur, instead of only considering a constant response in a stable environment. This generalization of the problem enormously increases its complexity. It may also lead to nontrivial dynamical effects that are not found in the case of static optimization, such as the accumulation of resource buffers to anticipate future changes in the environment [5].

The classical argument motivating the optimality assumption in the case of microorganisms is that mutants of genes coding for enzymes in central metabolism often have a lower growth rate than the wild-type strain, where growth rate
is interpreted as indicating fitness [6]. This argument, however, derives from observations of balanced growth in a
stable environment. Is there any evidence that, in the case of changing environments, microorganisms have evolved to
perform dynamic optimization? Some circumstantial evidence is provided by the observed capacity of microorganisms to
anticipate changes in their environment. For example, when moving along the digestive tract, *E. coli* cells are exposed
first to lactose and then to maltose, thus requiring the ability to switch from growth on lactose to growth on maltose
(reminiscent of diauxic growth in the laboratory) [7]. Interestingly, reporter gene studies found that the enzymes required
for maltose assimilation are expressed at a much higher level in the presence than in the absence of lactose, in otherwise
identical conditions [8]. This suggests a specific effect of the presence of lactose on the expression of maltose enzymes,
preparing the cells for the expected future availability of maltose. This and other examples of anticipatory behavior are
not conclusive in themselves, but they suggest that dynamic optimization is a plausible working hypothesis that may be
useful in practice.

The aim of this chapter is to show how microbial physiology can be studied by means of dynamic optimization, by
combining a specific objective function, or combination of objective functions, with models of different scope and
granularity, while taking into account a number of biophysical and biochemical constraints. We first provide a general
definition of dynamic optimization problems in the mathematical framework of optimal control. We then instantiate
this general definition for three types of biological problems, each giving rise to a specific class of models. In particular,
we discuss (i) dynamic optimization of enzyme expression in metabolic pathways, (ii) dynamic optimization of resource
allocation in coarse-grained models of cellular growth, and (iii) dynamic flux balance analysis (dFBA) of metabolic
networks. Across the different examples, the scope of the models varies from metabolic pathways (i) to metabolic
networks (iii) to the entire cell (ii). The increase in scope is sometimes traded against a lower granularity of the
description of cellular process (ii). Some of the models provide a kinetic description of the rates of the individual
reactions (i and ii), whereas other models only provide constraints on the reaction rates (iii). In every case, different
objective functions are tried, for example the minimization of the time to produce a given compound or the maximization
of the amount of biomass produced.

For each of the biological problems and corresponding models considered, we give the precise definition of the modeling
formalism and the optimization problem, a small example as an illustration, a discussion of the solution of the problem,
and a brief description of more realistic applications and the insights they have given into the functioning of cellular
networks. The chapter does not give a detailed explanation of the mathematical methods that are used for solving
different classes of optimal control problems, because this would require knowledge of specialized mathematical concepts
with which the average reader of the book may not be familiar. Moreover, these methods have been the subject
of dedicated textbooks [9, 10]. Rather, we focus on the definition of the dynamic optimization problems and the
interpretation of the solutions returned by available numerical solvers of optimal control problems.

10.2. Mathematical formalization of dynamic optimization problems

The models of cellular processes considered in this chapter have the form of systems of ordinary differential equations (ODEs) (Chapter 3 in [2]). Dynamic optimization problems for such systems take the form of so-called optimal control problems, which have their roots in physics and engineering [9, 10].

Let $\mathbf{x}(t)$ be the (time-varying) state of the dynamical system, typically concentrations of (intracellular or extracellular) metabolites or proteins, and let $\mathbf{f}(\cdot)$ describe the (linear or nonlinear) dynamics of the state. $\mathbf{u}(t)$ denotes the (time-varying) control variables, e.g., fluxes allocated to specific reactions or protein fractions allocated to specific enzymes. The time-points $0$ and $T > 0$ indicate the bounds of the interval over which the behavior of the system is optimized, with respect to an objective function $J$. The behavior of the system, given the control exerted by $\mathbf{u}(t)$, is subject to constraints $c_1(\cdot)$ and $c_2(\cdot)$ on the admissible control inputs at specific time-points $t$ or over the entire time-interval $[t_0, t_e]$, respectively. The constraints express physical limitations, such as the intracellular density of molecular constituents (Chapter 2 in [2]), or biochemical limitations, such as the maximum protein synthesis rate. Combining the above
is interpreted as indicating fitness [6]. This argument, however, derives from observations of balanced growth in a
stable environment. Is there any evidence that, in the case of changing environments, microorganisms have evolved to
perform dynamic optimization? Some circumstantial evidence is provided by the observed capacity of microorganisms to
anticipate changes in their environment. For example, when moving along the digestive tract, *E. coli* cells are exposed
first to lactose and then to maltose, thus requiring the ability to switch from growth on lactose to growth on maltose
(reminiscent of diauxic growth in the laboratory) [7]. Interestingly, reporter gene studies found that the enzymes required
for maltose assimilation are expressed at a much higher level in the presence than in the absence of lactose, in otherwise
identical conditions [8]. This suggests a specific effect of the presence of lactose on the expression of maltose enzymes,
preparing the cells for the expected future availability of maltose. This and other examples of anticipatory behavior are
not conclusive in themselves, but they suggest that dynamic optimization is a plausible working hypothesis that may be
useful in practice.

The aim of this chapter is to show how microbial physiology can be studied by means of dynamic optimization, by
combining a specific objective function, or combination of objective functions, with models of different scope and
granularity, while taking into account a number of biophysical and biochemical constraints. We first provide a general
definition of dynamic optimization problems in the mathematical framework of optimal control. We then instantiate
this general definition for three types of biological problems, each giving rise to a specific class of models. In particular,
we discuss (i) dynamic optimization of enzyme expression in metabolic pathways, (ii) dynamic optimization of resource
allocation in coarse-grained models of cellular growth, and (iii) dynamic flux balance analysis (dFBA) of metabolic
networks. Across the different examples, the scope of the models varies from metabolic pathways (i) to metabolic
networks (iii) to the entire cell (ii). The increase in scope is sometimes traded against a lower granularity of the
description of cellular process (ii). Some of the models provide a kinetic description of the rates of the individual
reactions (i and ii), whereas other models only provide constraints on the reaction rates (iii). In every case, different
objective functions are tried, for example the minimization of the time to produce a given compound or the maximization
of the amount of biomass produced.

For each of the biological problems and corresponding models considered, we give the precise definition of the modeling
formalism and the optimization problem, a small example as an illustration, a discussion of the solution of the problem,
and a brief description of more realistic applications and the insights they have given into the functioning of cellular
networks. The chapter does not give a detailed explanation of the mathematical methods that are used for solving
different classes of optimal control problems, because this would require knowledge of specialized mathematical concepts
with which the average reader of the book may not be familiar. Moreover, these methods have been the subject
of dedicated textbooks [9, 10]. Rather, we focus on the definition of the dynamic optimization problems and the
interpretation of the solutions returned by available numerical solvers of optimal control problems.

10.2. Mathematical formalization of dynamic optimization problems

The models of cellular processes considered in this chapter have the form of systems of ordinary differential equations (ODEs) (Chapter 3 in [2]). Dynamic optimization problems for such systems take the form of so-called optimal control problems, which have their roots in physics and engineering [9, 10].

Let $\mathbf{x}(t)$ be the (time-varying) state of the dynamical system, typically concentrations of (intracellular or extracellular) metabolites or proteins, and let $\mathbf{f}(\cdot)$ describe the (linear or nonlinear) dynamics of the state. $\mathbf{u}(t)$ denotes the (time-varying) control variables, e.g., fluxes allocated to specific reactions or protein fractions allocated to specific enzymes. The time-points $0$ and $T > 0$ indicate the bounds of the interval over which the behavior of the system is optimized, with respect to an objective function $J$. The behavior of the system, given the control exerted by $\mathbf{u}(t)$, is subject to constraints $c_1(\cdot)$ and $c_2(\cdot)$ on the admissible control inputs at specific time-points $t$ or over the entire time-interval $[t_0, t_e]$, respectively. The constraints express physical limitations, such as the intracellular density of molecular constituents (Chapter 2 in [2]), or biochemical limitations, such as the maximum protein synthesis rate. Combining the above
elements, we obtain the following definition of dynamic optimization problems:

$$ \max_{u \in U} J(\mathbf{x}(t), \mathbf{u}(t), 0, T), \qquad (10.1) $$

such that

$$ \frac{d\mathbf{x}}{dt} = \mathbf{f}(\mathbf{x}(t), \mathbf{u}(t)), \quad \mathbf{x}(0) = \mathbf{x}_0, \qquad (10.2) $$

$$ 0 \ge \mathbf{c}_1(\mathbf{x}(t), \mathbf{u}(t)), \qquad (10.3) $$

$$ 0 \ge \mathbf{c}_2(\mathbf{x}(0), \mathbf{x}(T)). \qquad (10.4) $$

In summary, the problem consists in finding controls that, given the dynamics of the system, maximize the objective function and satisfy the constraints [11].

The above definition makes no specific assumptions about the dynamics of the system under consideration. Given that we deal with biochemical reaction systems, the dynamics can be refined to

$$ \frac{d\mathbf{x}}{dt} = \mathbf{N}\mathbf{v}(\mathbf{x}(t), \mathbf{u}(t)) - \mu(t)\mathbf{x}(t), \quad \mathbf{x}(0) = \mathbf{x}_0, \qquad (10.5) $$

where $\mathbf{N}$ represents the stoichiometry matrix and $\mu$ is the (time-varying) growth rate. The principles of describing the structure of biochemical reactions systems by means of a stoichiometry matrix were described in Chapter 3 in [2] above.

The problem definition assumes that there is only a single objective function to be optimized. This may not be appropriate, since microorganisms seem to optimize several criteria in parallel, for example growth rate and survival under stress [12]. In many situations, it is therefore more appropriate to generalize the above problem to the case where $\mathbf{J}(\cdot, \cdot)$ represents a vector of $n$ objective functions $\mathbf{J} = [J_1, \dots, J_n]$. Thus generalized, the problem does not usually have a single solution, but rather an infinite set of solutions located on a so-called Pareto surface [13]. Solutions on the Pareto surface have the property that every alternative solution improving the performance with respect to some objective necessarily degrades the performance with respect to at least one of the other objectives. In the problems developed in the sections below, we principally consider optimality in the case of a single, possibly composite objective.

Many methods for solving optimal control problems (10.1)-(10.4) exist. While some optimal control problems can be solved analytically, most of the problems considered in the examples below require numerical approximations to be solved. All examples developed in the sections below have been solved by means of freely available solvers.

## 10.3. Dynamic optimization of enzyme expression in metabolic pathways

A number of experimental works suggest that metabolic regulation encodes temporal patterns in enzyme expression that may be beneficial for cell physiology [14, 15]. Since the timing of gene expression can directly control resource expenditure, several authors have attempted to rationalize such patterns as solutions of optimal control problems defined as in (10.1)-(10.4). The general idea is to optimize the temporal evolution of enzyme concentrations using objective functions that are representative of cellular goals. This provides a rationale to reverse-engineer optimality principles that underlie the expression patterns observed in experiments. In this section, we briefly describe results obtained for unbranched metabolic pathways, the basic building blocks of the metabolic networks of the cell.

Dynamic optimization of enzymatic concentrations was first considered by Klipp and co-workers [16]. The problem under study was the minimal-time activation of an unbranched network from an “off” state, where only the precursor is present, to a state where all substrate has been converted into product. To this end, the authors considered an
elements, we obtain the following definition of dynamic optimization problems:

$$ \max_{u \in U} J(\mathbf{x}(t), \mathbf{u}(t), 0, T), \qquad (10.1) $$

such that

$$ \frac{d\mathbf{x}}{dt} = \mathbf{f}(\mathbf{x}(t), \mathbf{u}(t)), \quad \mathbf{x}(0) = \mathbf{x}_0, \qquad (10.2) $$

$$ 0 \ge \mathbf{c}_1(\mathbf{x}(t), \mathbf{u}(t)), \qquad (10.3) $$

$$ 0 \ge \mathbf{c}_2(\mathbf{x}(0), \mathbf{x}(T)). \qquad (10.4) $$

In summary, the problem consists in finding controls that, given the dynamics of the system, maximize the objective function and satisfy the constraints [11].

The above definition makes no specific assumptions about the dynamics of the system under consideration. Given that we deal with biochemical reaction systems, the dynamics can be refined to

$$ \frac{d\mathbf{x}}{dt} = \mathbf{N}\mathbf{v}(\mathbf{x}(t), \mathbf{u}(t)) - \mu(t)\mathbf{x}(t), \quad \mathbf{x}(0) = \mathbf{x}_0, \qquad (10.5) $$

where $\mathbf{N}$ represents the stoichiometry matrix and $\mu$ is the (time-varying) growth rate. The principles of describing the structure of biochemical reactions systems by means of a stoichiometry matrix were described in Chapter 3 in [2] above.

The problem definition assumes that there is only a single objective function to be optimized. This may not be appropriate, since microorganisms seem to optimize several criteria in parallel, for example growth rate and survival under stress [12]. In many situations, it is therefore more appropriate to generalize the above problem to the case where $\mathbf{J}(\cdot, \cdot)$ represents a vector of $n$ objective functions $\mathbf{J} = [J_1, \dots, J_n]$. Thus generalized, the problem does not usually have a single solution, but rather an infinite set of solutions located on a so-called Pareto surface [13]. Solutions on the Pareto surface have the property that every alternative solution improving the performance with respect to some objective necessarily degrades the performance with respect to at least one of the other objectives. In the problems developed in the sections below, we principally consider optimality in the case of a single, possibly composite objective.

Many methods for solving optimal control problems (10.1)-(10.4) exist. While some optimal control problems can be solved analytically, most of the problems considered in the examples below require numerical approximations to be solved. All examples developed in the sections below have been solved by means of freely available solvers.

## 10.3. Dynamic optimization of enzyme expression in metabolic pathways

A number of experimental works suggest that metabolic regulation encodes temporal patterns in enzyme expression that may be beneficial for cell physiology [14, 15]. Since the timing of gene expression can directly control resource expenditure, several authors have attempted to rationalize such patterns as solutions of optimal control problems defined as in (10.1)-(10.4). The general idea is to optimize the temporal evolution of enzyme concentrations using objective functions that are representative of cellular goals. This provides a rationale to reverse-engineer optimality principles that underlie the expression patterns observed in experiments. In this section, we briefly describe results obtained for unbranched metabolic pathways, the basic building blocks of the metabolic networks of the cell.

Dynamic optimization of enzymatic concentrations was first considered by Klipp and co-workers [16]. The problem under study was the minimal-time activation of an unbranched network from an “off” state, where only the precursor is present, to a state where all substrate has been converted into product. To this end, the authors considered an
unbranched pathway with n enzymes and (n + 1) metabolites:

$$
\begin{align}
\frac{dx_0}{dt} &= -k_1 e_1 x_0, \nonumber \\
\frac{dx_i}{dt} &= k_i e_i x_{i-1} - k_{i+1} e_{i+1} x_i, \tag{10.6} \\
\frac{dx_n}{dt} &= k_n e_n x_{n-1}, \nonumber
\end{align}
$$

with a given initial condition $x_0(0) \neq 0$ and $x_i(0) = 0$ for $i = 1, 2, \dots, n$, and where all enzymatic reactions are assumed to follow mass-action kinetics with rate constant $k_i$. To model the "off" state prior to pathway activation, the initial conditions can be set to $x_0(0) = s$, where s is the concentration of precursor at $t = 0$, and $x_i(0) = e_i(0) = 0$ for all $i = 1, \dots, n$. The goal was to determine a vector of optimal enzyme concentrations $e(t)$ that solve the following problem:

$$
\mathbf{e}^*(t) = \arg \min_{e \in U} \frac{1}{s} \int_{0}^{\infty} (s - x_n(t)) \, dt, \quad (10.7)
$$

subject to the dynamic model in (10.6) and constraint set U as in (10.1) defined by a limited overall enzyme abundance
over the optimization horizon:

$$
\sum_{i=1}^{n} e_i(t) = e_{\text{tot}}, \tag{10.8}
$$

where $e_{\text{tot}}$ is a constant amount of total enzyme concentration. The objective function in (10.7) is called the *transition time* of the pathway and quantifies the time needed to convert all precursor into product. Note that the optimization problem (10.6)-(10.8) falls within the general class of problems defined by (10.1)-(10.4).

Numerical solutions of the optimization problem reveal that the enzyme profiles have a temporal sequence that matches the order in which the enzymes appear in the pathway. Crucially, such pattern resembles the “just-in-time” strategies widely studied in operations research [17], whereby costly resources are deployed only when needed in a production line. In the context of cellular metabolism, such a strategy implies that minimal time activation tends to express biosynthetic enzymes only when their substrates have been built up to sufficiently high concentrations, and thus avoid wasteful protein expression.

The first experimental demonstration of the just-in-time principle was presented by Zaslaver and colleagues [14]. This work employed luminescent and fluorescent reporters to measure the temporal adaptation of *Escherichia coli* upon withdrawal of amino acids from the growth media. Clear just-in-time patterns of enzyme expression were found in the serine, methionine and arginine biosynthetic pathways. To better understand such patterns, the authors studied a model for an unbranched pathway with three enzymatic steps and Michaelis-Menten kinetics:

$$
\frac{dx_i}{dt} = k_{\text{cat},i} e_i \frac{x_{i-1}}{x_{i-1} + K_{\text{M},i}} - k_{\text{cat},i+1} e_{i+1} \frac{x_i}{x_i + K_{\text{M},i}} - \mu x_i, \quad i = 1, \dots, 3, \quad (10.9)
$$

with given initial conditions $x_1(0) \neq 0$, $x_2(0) = x_3(0) = 0$, and where $(k_{\text{cat},i}, K_{\text{M},i})$ are the enzyme turnover rate and Michaelis-Menten constants of each enzyme, respectively. The precursor concentration $x_0$ is assumed to be constant. The model also includes a dilution term that accounts for dilution by cell growth at rate $\mu$. In contrast to previous works, this model also includes an explicit description of enzyme expression:

$$
\frac{de_i}{dt} = \frac{\beta_i}{1 + r/\kappa_i} - \mu e_i, \quad i = 1, \dots, 3, \tag{10.10}
$$

where the first term is a lumped model of enzyme expression controlled by a time-varying (active) repressor concentration
$r(t)$, with maximal expression rate $\beta_i$, and $\kappa_i$ being the concentration of (active) repressor required for half-maximal
expression. Moreover, since bacterial amino acid pathways are often subject to end-product feedback, the model assumed
unbranched pathway with n enzymes and (n + 1) metabolites:

$$
\begin{align}
\frac{dx_0}{dt} &= -k_1 e_1 x_0, \nonumber \\
\frac{dx_i}{dt} &= k_i e_i x_{i-1} - k_{i+1} e_{i+1} x_i, \tag{10.6} \\
\frac{dx_n}{dt} &= k_n e_n x_{n-1}, \nonumber
\end{align}
$$

with a given initial condition $x_0(0) \neq 0$ and $x_i(0) = 0$ for $i = 1, 2, \dots, n$, and where all enzymatic reactions are assumed to follow mass-action kinetics with rate constant $k_i$. To model the "off" state prior to pathway activation, the initial conditions can be set to $x_0(0) = s$, where s is the concentration of precursor at $t = 0$, and $x_i(0) = e_i(0) = 0$ for all $i = 1, \dots, n$. The goal was to determine a vector of optimal enzyme concentrations $e(t)$ that solve the following problem:

$$
\mathbf{e}^*(t) = \arg \min_{e \in U} \frac{1}{s} \int_0^\infty (s - x_n(t)) \, dt, \quad (10.7)
$$

subject to the dynamic model in (10.6) and constraint set U as in (10.1) defined by a limited overall enzyme abundance
over the optimization horizon:

$$
\sum_{i=1}^{n} e_i(t) = e_{\text{tot}}, \tag{10.8}
$$

where $e_{\text{tot}}$ is a constant amount of total enzyme concentration. The objective function in (10.7) is called the *transition time* of the pathway and quantifies the time needed to convert all precursor into product. Note that the optimization problem (10.6)-(10.8) falls within the general class of problems defined by (10.1)-(10.4).

Numerical solutions of the optimization problem reveal that the enzyme profiles have a temporal sequence that matches the order in which the enzymes appear in the pathway. Crucially, such pattern resembles the “just-in-time” strategies widely studied in operations research [17], whereby costly resources are deployed only when needed in a production line. In the context of cellular metabolism, such a strategy implies that minimal time activation tends to express biosynthetic enzymes only when their substrates have been built up to sufficiently high concentrations, and thus avoid wasteful protein expression.

The first experimental demonstration of the just-in-time principle was presented by Zaslaver and colleagues [14]. This work employed luminescent and fluorescent reporters to measure the temporal adaptation of *Escherichia coli* upon withdrawal of amino acids from the growth media. Clear just-in-time patterns of enzyme expression were found in the serine, methionine and arginine biosynthetic pathways. To better understand such patterns, the authors studied a model for an unbranched pathway with three enzymatic steps and Michaelis-Menten kinetics:

$$
\frac{dx_i}{dt} = k_{\text{cat},i} e_i \frac{x_{i-1}}{x_{i-1} + K_{\text{M},i}} - k_{\text{cat},i+1} e_{i+1} \frac{x_i}{x_i + K_{\text{M},i}} - \mu x_i, \quad i = 1, \dots, 3, \quad (10.9)
$$

with given initial conditions $x_1(0) \neq 0$, $x_2(0) = x_3(0) = 0$, and where $(k_{\text{cat},i}, K_{\text{M},i})$ are the enzyme turnover rate and Michaelis-Menten constants of each enzyme, respectively. The precursor concentration $x_0$ is assumed to be constant. The model also includes a dilution term that accounts for dilution by cell growth at rate $\mu$. In contrast to previous works, this model also includes an explicit description of enzyme expression:

$$
\frac{de_i}{dt} = \frac{\beta_i}{1 + r/\kappa_i} - \mu e_i, \quad i = 1, \dots, 3, \tag{10.10}
$$

where the first term is a lumped model of enzyme expression controlled by a time-varying (active) repressor concentration
$r(t)$, with maximal expression rate $\beta_i$, and $\kappa_i$ being the concentration of (active) repressor required for half-maximal
expression. Moreover, since bacterial amino acid pathways are often subject to end-product feedback, the model assumed
that the repressor gets activated by the pathway product:

$$r(t) = r_T(t) \frac{x_3(t)}{K_r + x_3(t)}, \qquad (10.11)$$

where $r_T(t)$ denotes the total (active and inactive) repressor concentration. The model also included negative autoregulation of the repressor itself:

$$\frac{dr_T}{dt} = \frac{\beta_0}{1 + r/\kappa_0} - \mu r_T, \qquad (10.12)$$

where $\beta_0$ and $\kappa_0$ define the strength of autoregulation similarly as in the lumped model for enzyme expression in (10.10).

The authors constructed an optimization problem so as to study the relation between optimality, and the strength of the regulatory parameters $k = (k_1, k_2, k_3)$ and $\beta = (\beta_1, \beta_2, \beta_3)$. To this end, they defined the optimization problem

$$\min_{k, \beta} a \cdot \sum_{i=1}^{3} \underbrace{\int_{0}^{T} \frac{\beta_0}{1+r(t)/\kappa_0} dt}_{\text{total amount of repressor}} + \underbrace{\int_{0}^{T} |F-F_{\text{goal}}| dt}_{\text{deviation from steady state}}, \qquad (10.13)$$

where $a$ is a scalar weight accounting for the protein costs, $T$ is the optimization horizon, and $F$ is the rate of product synthesis:

$$F = k_{\text{cat},3} e_3 \frac{x_2}{x_2 + K_{\text{M},3}}. \qquad (10.14)$$

In problem (10.13), the constant $F_{\text{goal}}$ is a prescribed production flux that the pathway should achieve at steady state. Minimization of the objective in (10.13) accounts for the activation of the pathway from an “off” state until it reaches a prescribed flux $F_{\text{goal}}$. This formulation differs from the previous example [16] in two important ways. First, it accounts for cellular resources in the objective function itself. The first term of the objective quantifies the total amount of repressor produced through the optimization horizon, and thus relates to the amount of cellular resources required to activate the pathway. Second, the decision variables are the regulatory parameters, not the temporal profiles of the molecular species. Therefore, strictly speaking, this is not an optimal control problem but rather a static optimization problem subject to dynamic constraints encapsulated by the pathway ODE model. Through numerical solutions for different values of the protein cost weight $a$ and optimization horizon $T$, the authors determined conditions under which the optimal solutions showed two features of the just-in-time property, namely:

$$\tau_1 < \tau_2 < \tau_3, \quad \max_t e_1 > \max_t e_2 > \max_t e_3, \qquad (10.15)$$

where $\tau_i$ is the response time, i.e. the time to reach 50% of maximal concentration, and $\max_t e_i$ is the peak concentration of each enzyme. This theoretical model was designed to mimic the architecture of gene regulation in such pathways, whereby the end product commonly represses the expression of upstream enzymes, and thus gave both experimental and computational evidence that just-in-time patterns may be the result of optimality principles underlying the regulation of metabolic pathways.

Further experimental evidence of temporal patterns in enzyme expression have been found in other pathways [18] and organisms [15], and number of subsequent works have explored their optimality in more detail; we refer the reader to the review in [19] for a detailed discussion on such approaches. Oyarzún and colleagues [20], in particular, gave the first mathematical proof that just-in-time dynamics are a general property in models of unbranched metabolic pathways. Using a cost-benefit objective function that balances the speed of response against the cost of expressing pathway enzymes, they showed that the just-in-time patterns emerge in pathways of arbitrary length and with minimal assumptions on the enzyme kinetics. Specifically, they considered a general model for an unbranched pathway with $n + 1$ reactions:

$$\frac{dx_i}{dt} = g_{i-1}(x_{i-1}) e_{i-1} - g_i(x_i) e_i, \quad i = 1, \dots, n, \qquad (10.16)$$
that the repressor gets activated by the pathway product:

$$r(t) = r_T(t) \frac{x_3(t)}{K_r + x_3(t)}, \qquad (10.11)$$

where $r_T(t)$ denotes the total (active and inactive) repressor concentration. The model also included negative autoregulation of the repressor itself:

$$\frac{dr_T}{dt} = \frac{\beta_0}{1 + r/\kappa_0} - \mu r_T, \qquad (10.12)$$

where $\beta_0$ and $\kappa_0$ define the strength of autoregulation similarly as in the lumped model for enzyme expression in (10.10).

The authors constructed an optimization problem so as to study the relation between optimality, and the strength of the regulatory parameters $k = (k_1, k_2, k_3)$ and $\beta = (\beta_1, \beta_2, \beta_3)$. To this end, they defined the optimization problem

$$\min_{k, \beta} a \cdot \sum_{i=1}^{3} \underbrace{\int_{0}^{T} \frac{\beta_0}{1+r(t)/\kappa_0} dt}_{\text{total amount of repressor}} + \underbrace{\int_{0}^{T} |F-F_{\text{goal}}| dt}_{\text{deviation from steady state}}, \qquad (10.13)$$

where $a$ is a scalar weight accounting for the protein costs, $T$ is the optimization horizon, and $F$ is the rate of product synthesis:

$$F = k_{\text{cat},3} e_3 \frac{x_2}{x_2 + K_{\text{M},3}}. \qquad (10.14)$$

In problem (10.13), the constant $F_{\text{goal}}$ is a prescribed production flux that the pathway should achieve at steady state. Minimization of the objective in (10.13) accounts for the activation of the pathway from an “off” state until it reaches a prescribed flux $F_{\text{goal}}$. This formulation differs from the previous example [16] in two important ways. First, it accounts for cellular resources in the objective function itself. The first term of the objective quantifies the total amount of repressor produced through the optimization horizon, and thus relates to the amount of cellular resources required to activate the pathway. Second, the decision variables are the regulatory parameters, not the temporal profiles of the molecular species. Therefore, strictly speaking, this is not an optimal control problem but rather a static optimization problem subject to dynamic constraints encapsulated by the pathway ODE model. Through numerical solutions for different values of the protein cost weight $a$ and optimization horizon $T$, the authors determined conditions under which the optimal solutions showed two features of the just-in-time property, namely:

$$\tau_1 < \tau_2 < \tau_3, \quad \max_t e_1 > \max_t e_2 > \max_t e_3, \qquad (10.15)$$

where $\tau_i$ is the response time, i.e. the time to reach 50% of maximal concentration, and $\max_t e_i$ is the peak concentration of each enzyme. This theoretical model was designed to mimic the architecture of gene regulation in such pathways, whereby the end product commonly represses the expression of upstream enzymes, and thus gave both experimental and computational evidence that just-in-time patterns may be the result of optimality principles underlying the regulation of metabolic pathways.

Further experimental evidence of temporal patterns in enzyme expression have been found in other pathways [18] and organisms [15], and number of subsequent works have explored their optimality in more detail; we refer the reader to the review in [19] for a detailed discussion on such approaches. Oyarzún and colleagues [20], in particular, gave the first mathematical proof that just-in-time dynamics are a general property in models of unbranched metabolic pathways. Using a cost-benefit objective function that balances the speed of response against the cost of expressing pathway enzymes, they showed that the just-in-time patterns emerge in pathways of arbitrary length and with minimal assumptions on the enzyme kinetics. Specifically, they considered a general model for an unbranched pathway with $n + 1$ reactions:

$$\frac{dx_i}{dt} = g_{i-1}(x_{i-1}) e_{i-1} - g_i(x_i) e_i, \quad i = 1, \dots, n, \qquad (10.16)$$
with initial conditions $x_i(0) = 0$ for $i = 1, 2, \dots, n$, and the precursor $x_0$ assumed to be at a constant concentration. The functions $g_i$ represent a general kinetic turnover rate satisfying the following conditions:

$$ 
\begin{aligned}
g_i(0) &= 0, \\
\frac{\partial g_i(x_i)}{\partial x_i} &> 0.
\end{aligned} 
\qquad (10.17) $$

The above assumptions are generally satisfied by most enzyme kinetic functions, as catalytic rates are typically a monotonic function of the substrate concentration. In particular, the assumptions in (10.17) are met by common kinetics such as mass-action, Michaelis-Menten and Hill equations. The optimization problem considered in [20] corresponds to a free final-time optimal control problem:

$$ e^*(t) = \arg \min_{e \in U} \int_0^T (1 + \alpha' e(t)) dt, \qquad (10.18) $$

where $e(t)$ is the vector of enzyme concentration, $\alpha$ is an $(n+1)$-dimensional vector of tuneable weights, $T$ is a free optimization horizon, and $U$ is a constraint set as in (10.1). The first term in the objective function (10.18) accounts for the total time taken to activate the pathway from the “off” state up to a steady state flux, while the second term weighs the cost of pathway activation. To account for limited availability of cellular resources, the authors also included a temporal constraint on the enzyme concentrations:

$$ \sum_{i=0}^{n} e_i(t) \le e_{\text{tot}}, \qquad (10.19) $$

which is a relaxation of the constraint originally employed by Klipp et al in (10.8), as well as a terminal constraint of the form:

$$ e_i(t) = \frac{F_{\text{goal}}}{g_i(x_i(T))}, \quad \text{for } t \ge T, \qquad (10.20) $$

where $F_{\text{goal}}$ is a (constant) target pathway flux, similar as in (10.13). The terminal constraint ensures that the pathway reaches a steady state at the final time $T$. Using Pontryagin's Minimum Principle [10], the authors showed that the optimal enzyme concentrations follow a bang-bang temporal profile that matches the order in which they act on the pathway. This result was shown to be independent of the weight $\alpha$, the number of enzymatic steps, and valid for a wide range of enzyme kinetics satisfying the assumptions in (10.17), thus extending the original finding in [16] to a larger class of pathways. Figure 10.1 shows a numerical example of the optimal activation pattern obtained for an unbranched metabolic pathway of length three.

In this section we have reviewed some optimal control approaches for the optimization of unbranched metabolic pathways. While differing in their formulations and solution strategies, these approaches provide substantial computational evidence that some temporal patterns observed in metabolic dynamics can be understood as the solution of an optimal control problem. In the next section we focus on approaches that go beyond individual pathways and include additional components and processes of the cellular machinery.

## 10.4. Dynamic optimization of resource allocation in coarse-grained models of cellular growth

In the previous section, we considered models that were essentially limited to metabolic pathways. The optimization problems were formulated in terms of the allocation of enzymes to the different reactions in the pathway. In this section, we generalize the perspective by increasing the scope of the models from metabolism to protein synthesis and growth. The optimization problems concern the allocation of resources to the synthesis of enzymes catalyzing different metabolic reactions, but also to the synthesis of ribosomes in charge of the production of proteins. Growth is explicitly defined in terms of the increase of protein mass, and leads to growth dilution of all cellular components. The models are very
with initial conditions $x_i(0) = 0$ for $i = 1, 2, \dots, n$, and the precursor $x_0$ assumed to be at a constant concentration. The functions $g_i$ represent a general kinetic turnover rate satisfying the following conditions:

$$ 
\begin{aligned}
g_i(0) &= 0, \\
\frac{\partial g_i(x_i)}{\partial x_i} &> 0.
\end{aligned} 
\qquad (10.17) $$

The above assumptions are generally satisfied by most enzyme kinetic functions, as catalytic rates are typically a monotonic function of the substrate concentration. In particular, the assumptions in (10.17) are met by common kinetics such as mass-action, Michaelis-Menten and Hill equations. The optimization problem considered in [20] corresponds to a free final-time optimal control problem:

$$ e^*(t) = \arg \min_{e \in U} \int_{0}^{T} (1 + \alpha' e(t)) dt, \qquad (10.18) $$

where $e(t)$ is the vector of enzyme concentration, $\alpha$ is an $(n+1)$-dimensional vector of tuneable weights, $T$ is a free optimization horizon, and $U$ is a constraint set as in (10.1). The first term in the objective function (10.18) accounts for the total time taken to activate the pathway from the “off” state up to a steady state flux, while the second term weighs the cost of pathway activation. To account for limited availability of cellular resources, the authors also included a temporal constraint on the enzyme concentrations:

$$ \sum_{i=0}^{n} e_i(t) \le e_{\text{tot}}, \qquad (10.19) $$

which is a relaxation of the constraint originally employed by Klipp et al in (10.8), as well as a terminal constraint of the form:

$$ e_i(t) = \frac{F_{\text{goal}}}{g_i(x_i(T))}, \quad \text{for } t \ge T, \qquad (10.20) $$

where $F_{\text{goal}}$ is a (constant) target pathway flux, similar as in (10.13). The terminal constraint ensures that the pathway reaches a steady state at the final time $T$. Using Pontryagin's Minimum Principle [10], the authors showed that the optimal enzyme concentrations follow a bang-bang temporal profile that matches the order in which they act on the pathway. This result was shown to be independent of the weight $\alpha$, the number of enzymatic steps, and valid for a wide range of enzyme kinetics satisfying the assumptions in (10.17), thus extending the original finding in [16] to a larger class of pathways. Figure 10.1 shows a numerical example of the optimal activation pattern obtained for an unbranched metabolic pathway of length three.

In this section we have reviewed some optimal control approaches for the optimization of unbranched metabolic pathways. While differing in their formulations and solution strategies, these approaches provide substantial computational evidence that some temporal patterns observed in metabolic dynamics can be understood as the solution of an optimal control problem. In the next section we focus on approaches that go beyond individual pathways and include additional components and processes of the cellular machinery.

## 10.4. Dynamic optimization of resource allocation in coarse-grained models of cellular growth

In the previous section, we considered models that were essentially limited to metabolic pathways. The optimization problems were formulated in terms of the allocation of enzymes to the different reactions in the pathway. In this section, we generalize the perspective by increasing the scope of the models from metabolism to protein synthesis and growth. The optimization problems concern the allocation of resources to the synthesis of enzymes catalyzing different metabolic reactions, but also to the synthesis of ribosomes in charge of the production of proteins. Growth is explicitly defined in terms of the increase of protein mass, and leads to growth dilution of all cellular components. The models are very
A

$$x_0 \rightarrow x_1 \rightarrow x_2 \rightarrow x_3$$

B

<figure><img src="image_5.png" /><figcaption>Figure 10.1: Example of optimal enzyme expression in an unbranched metabolic pathway. A. A simple scheme of the metabolic pathway. B. Time evolution of the optimal enzyme expression $u_i$ and metabolite concentration $x_i$. For the simulations, the functions $g_i$ are Michaelis-Menten with constants $k = (1, 2, 4, 3)$ s$^{-1}$, $K = 1$ mM, $V = 0.2$ mM s$^{-1}$ and $x_0 = 5$ mM. Enzymatic weights are set to $\alpha_i = 1$ mM$^{-1}$ s and maximum enzyme availability $E_{tot} = 1$ mM. Resulting activation times are $t_0 = 1.59$ s, $t_1 = 2.2$ s and $t_f = 2.55$ s.</figcaption></figure>

similar to those considered in Chapter 8 in [2], but the optimization problems are dynamic rather than static. That is, instead of searching an allocation of cellular resources to the synthesis of different classes of proteins that is optimal at steady state, during balanced growth, we are interested in finding a time-varying resource allocation strategy optimizing an objective defined over an interval of time, e.g., during a transition between two states of balanced growth.

We consider the class of models with dynamics given by Eq. 10.5, where the input $\mathbf{u}$ is interpreted as the (time-varying) resource allocation strategy. Among the cellular components $\mathbf{x}$, we distinguish between metabolites and proteins, with concentrations $\mathbf{c}$ and $\mathbf{p}$, respectively. Accordingly, the concentration vector can be written as $\mathbf{x} = [\mathbf{c}, \mathbf{p}]^T$. We also distinguish between enzymatic reactions and protein synthesis reactions. While the former have metabolites as substrates and products, the latter convert metabolites (amino acids) into proteins. An enzymatic reaction $i$ has the following reaction rate function:

$$v_i(t) = k_i p_j(t) h_i(c), \qquad (10.21)$$

where $k_i$ is a catalytic constant, $p_j$ the concentration of protein $j$, and $h_i$ a function describing enzyme saturation. Enzyme saturation is determined by the substrates, products, and activators/inhibitors of the reaction. Typical rate functions $v_i$ follow mass-action kinetics or (ir)reversible Michaelis-Menten kinetics. The synthesis of protein $i$ is associated with the reaction-rate function

$$v_i(t) = u_i(t) v_R(t), \qquad (10.22)$$

where $v_R$ is the total protein synthesis rate defined by

$$v_R(t) = k_R p_R(t) h_R(c(t)), \qquad (10.23)$$
A

$$x_0 \rightarrow x_1 \rightarrow x_2 \rightarrow x_3$$

B

<figure><img src="image_4.png" /><figcaption>Figure 10.1: Example of optimal enzyme expression in an unbranched metabolic pathway. A. A simple scheme of the metabolic pathway. B. Time evolution of the optimal enzyme expression $u_i$ and metabolite concentration $x_i$. For the simulations, the functions $g_i$ are Michaelis-Menten with constants $k = (1, 2, 4, 3)$ s$^{-1}$, $K = 1$ mM, $V = 0.2$ mM s$^{-1}$ and $x_0 = 5$ mM. Enzymatic weights are set to $\alpha_i = 1$ mM$^{-1}$ s and maximum enzyme availability $E_{tot} = 1$ mM. Resulting activation times are $t_0 = 1.59$ s, $t_1 = 2.2$ s and $t_f = 2.55$ s.</figcaption></figure>

similar to those considered in Chapter 8 in [2], but the optimization problems are dynamic rather than static. That is, instead of searching an allocation of cellular resources to the synthesis of different classes of proteins that is optimal at steady state, during balanced growth, we are interested in finding a time-varying resource allocation strategy optimizing an objective defined over an interval of time, e.g., during a transition between two states of balanced growth.

We consider the class of models with dynamics given by Eq. 10.5, where the input $\mathbf{u}$ is interpreted as the (time-varying) resource allocation strategy. Among the cellular components $\mathbf{x}$, we distinguish between metabolites and proteins, with concentrations $\mathbf{c}$ and $\mathbf{p}$, respectively. Accordingly, the concentration vector can be written as $\mathbf{x} = [\mathbf{c}, \mathbf{p}]^T$. We also distinguish between enzymatic reactions and protein synthesis reactions. While the former have metabolites as substrates and products, the latter convert metabolites (amino acids) into proteins. An enzymatic reaction $i$ has the following reaction rate function:

$$v_i(t) = k_i p_j(t) h_i(c), \qquad (10.21)$$

where $k_i$ is a catalytic constant, $p_j$ the concentration of protein $j$, and $h_i$ a function describing enzyme saturation. Enzyme saturation is determined by the substrates, products, and activators/inhibitors of the reaction. Typical rate functions $v_i$ follow mass-action kinetics or (ir)reversible Michaelis-Menten kinetics. The synthesis of protein $i$ is associated with the reaction-rate function

$$v_i(t) = u_i(t) v_R(t), \qquad (10.22)$$

where $v_R$ is the total protein synthesis rate defined by

$$v_R(t) = k_R p_R(t) h_R(c(t)), \qquad (10.23)$$
with $k_R$ the maximum protein synthesis rate, $p_R$ the concentration of ribosomes, and $h_R$ a function describing the saturation of ribosomes by their substrate, that is, amino acids (or more precisely, tRNAs charged with amino acids). The function $u_i$ in Eq. 10.22 is a time-varying resource allocation function, describing the fraction of the total protein synthesis rate allocated to the synthesis of protein $i$. The fractions are non-negative and sum to 1, that is, for every time $t$,

$$ \sum_{i} u_{i}(t) = 1, \quad \text{and} \quad u_{i}(t) \ge 0, \quad \text{for all } i. \qquad (10.24) $$

In most models, the biomass of a growing cell population is equated with the mass of proteins, the most abundant cellular component (Chapter 2 in [2]). Under the further assumption that the biomass density is constant, it follows that the total protein concentration $p_{tot}$ must be constant, where

$$ p_{tot} = \sum_{i} p_{i}(t), \qquad (10.25) $$

with the index $i$ running over all proteins. Moreover, the growth rate reduces to the relative (or specific) increase of the protein mass, which leads to

$$ \mu(t) = \frac{v_R(t)}{p_{tot}} = \frac{k_R p_R(t) h_R(c(t))}{p_{tot}}. \qquad (10.26) $$

The above model couples metabolism, protein synthesis, and growth in a single formalism, in the spirit of the small resource allocation models discussed in Chapter 8.

Figure 10.2 gives an example of a resource allocation model, describing a simple self-replicatory microbial system [21, 22] inspired by the model of Scott et al. [23] (see Chapter ?? in [2] related models). The model divides the proteome into three categories: ribosomes, enzymes, and housekeeping proteins, with concentrations $p_Q$, $p_R$, and $p_M$, respectively. In addition to the three categories of protein, we add a metabolite representing the precursors for protein synthesis, with concentration $c$. The precursors are produced from nutrients in the environment at a rate $v_M$, a macroreaction catalyzed by the enzymes. Protein synthesis occurs at a rate $v_R$, catalyzed by the ribosomes. The resource allocation functions $u_Q$, $u_R$, and $u_M$ determine the fraction of the protein synthesis rate assigned to each of the three protein categories, where $u_Q$ is assumed to be a constant, growth-rate-independent fraction. The rate equations for the metabolic and protein synthesis reactions follow irreversible Michaelis-Menten kinetics, where the substrate concentration in the medium is assumed saturating.

The resource allocation functions in the model are not explicitly specified by regulatory mechanisms, but assumed to follow a dynamic pattern optimizing an objective criterion. In many cases, the objective criterion is based on the hypothesis that microorganisms have evolved to maximize growth, that is, the accumulation of biomass. While this hypothesis can be criticized on theoretical and empirical reasons, it is a reasonable choice in well-mixed environments and provides an interesting baseline in other environments. In the model framework considered here, this gives rise to the following objective function:

$$ \max_{u \in U} J(x(t), u(t), 0, T) = \int_{0}^{T} k_{R} p_{R}(t) h_{R}(c(t)) dt, \qquad (10.27) $$

where like in the general case of Eq. 10.1, $U$ denotes the set of admissible profiles for the resource allocation functions $u$. Note that the maximization of growth over the time-interval $[0, T]$ amounts to taking the integral of the instantaneous growth rate over that time-interval, defined by Eq. 10.26. This objective does not generally reduce to maximizing the instantaneous growth rate at every time-point of this interval.

The question can be asked, for the microbial self-replicator in Figure 10.2, how the cells redistribute their resources over the different protein categories after a change in environment, in particular a shift of the cells from a poor to a
with $k_R$ the maximum protein synthesis rate, $p_R$ the concentration of ribosomes, and $h_R$ a function describing the saturation of ribosomes by their substrate, that is, amino acids (or more precisely, tRNAs charged with amino acids). The function $u_i$ in Eq. 10.22 is a time-varying resource allocation function, describing the fraction of the total protein synthesis rate allocated to the synthesis of protein $i$. The fractions are non-negative and sum to 1, that is, for every time $t$,

$$ \sum_{i} u_{i}(t) = 1, \quad \text{and} \quad u_{i}(t) \ge 0, \quad \text{for all } i. \qquad (10.24) $$

In most models, the biomass of a growing cell population is equated with the mass of proteins, the most abundant cellular component (Chapter 2 in [2]). Under the further assumption that the biomass density is constant, it follows that the total protein concentration $p_{tot}$ must be constant, where

$$ p_{tot} = \sum_{i} p_{i}(t), \qquad (10.25) $$

with the index $i$ running over all proteins. Moreover, the growth rate reduces to the relative (or specific) increase of the protein mass, which leads to

$$ \mu(t) = \frac{v_R(t)}{p_{tot}} = \frac{k_R p_R(t) h_R(c(t))}{p_{tot}}. \qquad (10.26) $$

The above model couples metabolism, protein synthesis, and growth in a single formalism, in the spirit of the small resource allocation models discussed in Chapter 8.

Figure 10.2 gives an example of a resource allocation model, describing a simple self-replicatory microbial system [21, 22] inspired by the model of Scott et al. [23] (see Chapter ?? in [2] related models). The model divides the proteome into three categories: ribosomes, enzymes, and housekeeping proteins, with concentrations $p_Q$, $p_R$, and $p_M$, respectively. In addition to the three categories of protein, we add a metabolite representing the precursors for protein synthesis, with concentration $c$. The precursors are produced from nutrients in the environment at a rate $v_M$, a macroreaction catalyzed by the enzymes. Protein synthesis occurs at a rate $v_R$, catalyzed by the ribosomes. The resource allocation functions $u_Q$, $u_R$, and $u_M$ determine the fraction of the protein synthesis rate assigned to each of the three protein categories, where $u_Q$ is assumed to be a constant, growth-rate-independent fraction. The rate equations for the metabolic and protein synthesis reactions follow irreversible Michaelis-Menten kinetics, where the substrate concentration in the medium is assumed saturating.

The resource allocation functions in the model are not explicitly specified by regulatory mechanisms, but assumed to follow a dynamic pattern optimizing an objective criterion. In many cases, the objective criterion is based on the hypothesis that microorganisms have evolved to maximize growth, that is, the accumulation of biomass. While this hypothesis can be criticized on theoretical and empirical reasons, it is a reasonable choice in well-mixed environments and provides an interesting baseline in other environments. In the model framework considered here, this gives rise to the following objective function:

$$ \max_{u \in U} J(x(t), u(t), 0, T) = \int_{0}^{T} k_{R} p_{R}(t) h_{R}(c(t)) dt, \qquad (10.27) $$

where like in the general case of Eq. 10.1, $U$ denotes the set of admissible profiles for the resource allocation functions $u$. Note that the maximization of growth over the time-interval $[0, T]$ amounts to taking the integral of the instantaneous growth rate over that time-interval, defined by Eq. 10.26. This objective does not generally reduce to maximizing the instantaneous growth rate at every time-point of this interval.

The question can be asked, for the microbial self-replicator in Figure 10.2, how the cells redistribute their resources over the different protein categories after a change in environment, in particular a shift of the cells from a poor to a
A

<figure><img src="image_6.png" /></figure>

B

### Dynamical system

$$ \dot{c} = v_M - \mu c $$

$$ \dot{p}_Q = u_Q v_R - \mu p_Q $$

$$ \dot{p}_R = u_R v_R - \mu p_R $$

$$ \dot{p}_M = u_M v_R - \mu p_M $$

C

<figure><img src="image_7.png" /></figure>

### Synthesis rates

$$ v_M = k_M p_M $$

$$ v_R = k_R p_R \frac{c}{c + K_R} $$

$$ \mu = \frac{v_R}{P_Q + P_R + P_M} $$

$$ u_Q + u_R + u_M = 1 $$

D

<figure><img src="image_8.png" /></figure>

### Objective function

$$ J(u) = \int_{0}^{T} \mu(t) dt $$

### Optimal control

$$ u_{\text{opt}} = \arg \max_{u \in \mathcal{U}} J(u) $$

Figure 10.2: Example of optimal resource allocation strategy in a coarse-grained model of microbial growth. A. Representation of simple self-replicator model of microbial growth. B. Model and optimization problem for the self-replicator shown in panel A, as discussed in the text. C. Optimal dynamic resource allocation strategy, in terms of the fraction of resources attributed to ribosome synthesis ($u_R$). D. Time-varying protein mass fractions corresponding to the optimal solution shown in panel C. The parameter values used for the simulation are $k_M = 0.5$, $k_R = 1$, $K_R = 0.5$ and $u_Q = 0.6$.

rich carbon source. In the case of E. coli, for example, such as shift might involve a change from minimal medium with acetate to minimal medium with glucose. Given that E. coli grows faster on glucose than on acetate, and that a higher growth rate requires an increased proportion of resources to be allocated to ribosomes according to the growth law (Chapter 10 in [2]), one expects $u_R$ to increase after the shift. Since $u_Q$ is assumed constant, and the resource allocation functions must sum to 1 at every time-point, this overall increase of $u_R$ must be balanced by a decrease of $u_M$. These expectations concern resource allocation before the shift (balanced growth on acetate) and a long time after the shift (balanced growth on glucose), but the growth law provides no information on the pattern of adaptation immediately after the shift.

In order to investigate the optimal adaptation pattern of $u_R$ immediately after the growth transition, we solve the dynamic optimization problem specified in Figure 10.2. For the simple example considered here, the optimal solution can
<figure><img src="image_5.png" /></figure>

**Dynamical system**

$$ \dot{c} = v_M - \mu c $$

$$ \dot{p}_Q = u_Q v_R - \mu p_Q $$

$$ \dot{p}_R = u_R v_R - \mu p_R $$

$$ \dot{p}_M = u_M v_R - \mu p_M $$

**Synthesis rates**

$$ v_M = k_M p_M $$

$$ v_R = k_R p_R \frac{c}{c + K_R} $$

$$ \mu = \frac{v_R}{P_Q + P_R + P_M} $$

$$ u_Q + u_R + u_M = 1 $$

**Objective function**

$$ J(u) = \int_{0}^{T} \mu(t) dt $$

**Optimal control**

$$ u_{\text{opt}} = \arg \max_{u \in \mathcal{U}} J(u) $$

<caption>Figure 10.2: Example of optimal resource allocation strategy in a coarse-grained model of microbial growth. **A.** Representation of simple self-replicator model of microbial growth. **B.** Model and optimization problem for the self-replicator shown in panel A, as discussed in the text. **C.** Optimal dynamic resource allocation strategy, in terms of the fraction of resources attributed to ribosome synthesis ($u_R$). **D.** Time-varying protein mass fractions corresponding to the optimal solution shown in panel C. The parameter values used for the simulation are $k_M = 0.5$, $k_R = 1$, $K_R = 0.5$ and $u_Q = 0.6$.</caption>

rich carbon source. In the case of E. coli, for example, such as shift might involve a change from minimal medium with acetate to minimal medium with glucose. Given that E. coli grows faster on glucose than on acetate, and that a higher growth rate requires an increased proportion of resources to be allocated to ribosomes according to the growth law (Chapter 10 in [2]), one expects $u_R$ to increase after the shift. Since $u_Q$ is assumed constant, and the resource allocation functions must sum to 1 at every time-point, this overall increase of $u_R$ must be balanced by a decrease of $u_M$. These expectations concern resource allocation before the shift (balanced growth on acetate) and a long time after the shift (balanced growth on glucose), but the growth law provides no information on the pattern of adaptation immediately after the shift.

In order to investigate the optimal adaptation pattern of $u_R$ immediately after the growth transition, we solve the dynamic optimization problem specified in Figure 10.2. For the simple example considered here, the optimal solution can
be characterized analytically [24, 21, 22]. This is not possible for more complicated examples, however, which require
the optimal solution to be constructed numerically, using one of the tools discussed in Appendix ]. Figure 10.2C-D show
a typical solution for parameter values estimated from experimental data [22]. Starting from a low value of uR during
balanced growth on acetate, the optimal resource allocation scheme consists of a sequence of switches between uR = 1
(maximal ribosome synthesis) and uR = 0 (no ribosome synthesis), until an intermediate value of uR for balanced
growth on glucose is attained. The value of uR during balanced growth on glucose is higher than that for balanced
growth on acetate, as expected from the growth law.

The sequence of on-off switches followed by the intermediate steady-state value is called a bang-bang-singular solution in
optimal control theory [21, 22]. The solution reflects a dynamic trade-off between the two different functions contributing
to growth: metabolism and protein synthesis. When, due to growth dilution, the ribosome concentration falls to a level
that is limiting for maximal protein synthesis, the synthesis of ribosomal proteins is switched on (uR = 1), leading to
an increase of the ribosome concentration. Switching on the synthesis of ribosomal proteins causes the synthesis of
metabolic enzymes to be switched off. When, due to growth dilution, the concentration of metabolic enzymes next falls
to a level that the precursors produced by the latter become limiting, the synthesis of metabolic enzymes is switched on
(uR = 0) to replenish the precursor pool, and so on.

Optimal solutions with a similar bang-bang pattern were already encountered in the previous section. They also occur
in a model with a more detailed description of different precursor (amino acid) synthesis pathways under the objective
under the minimal time of adaptation after a shift from a medium supplemented with amino acid to a medium lacking
amino acids [25]. In a different type of problem, the development of intestinal crypts, the minimal time to mature crypts
was found to depend on the on-off control of the proliferation of stem and non-stem cells [26]. There is no convincing
experimental evidence that the adaptation of ribosomal synthesis after a nutrient upshift from a poor to a carbon source
follows a bang-bang singular pattern. The interpretation of proteomics data after a nutrient upshift in E. coli shows
that the simple upregulation of ribosomal resource allocation to the steady-state value for growth on the rich nutrient
captures the ribosomal protein expression data well [27].

This example serves to emphasize that, while the optimality assumption may lead to thought-provoking predictions, these
need to be confronted with experimental data. In case the optimal solutions do not agree with the data, several revisions
of the problem could be considered. While growth optimization was chosen as the objective criterion in the example
of Figure 10.2, there is evidence that during balanced growth, microorganisms find a trade-off between maximizing
growth rate in a given environment and minimizing necessary adjustments to other environments [12]. The problem
could therefore be generalized to a multi-criteria optimization problem. An example is the analysis of a model similar to
that considered here under the objectives of biomass maximization and minimal adaptation time after a nutrient shift
[28]. The formulation of the optimization problem in Figure 10.2 does not put any constraints on valid optimal resource
allocation strategies, except that the individual functions uᵢ components need to sum to 1 (Eq. 10.24). Bearing in
mind that the regulatory mechanisms underlying a resource allocation come with a cost, and need to respect certain
physical constraints, the predicted resource allocation strategy may not be feasible. When such constraints are taken
into account, the optimal solution may no longer be bang-bang singular, but resemble the observed adaptation pattern
[29, 22].

In summary, the dynamical optimization approach for studying microbial growth presented here provides a way to test the
consequences of hypothesized objective functions in combination with simple resource allocation models. The predictions
can be confronted with experimental data, but may also inform the redesign of microbial strains for metabolic engineering
purposes [30] (Box 10.A).

**Box 10.A: Time-dependent optimization for biotechnological applications and process design**

As explained in Section 10.2, time-dependent optimization problems are defined by an objective function, expressing
the criterion that microorganisms presumably optimize. In the case of microorganisms growing in natural conditions,
the choice of a particular objective function is difficult to make and several functions may qualify. For example,
be characterized analytically [24, 21, 22]. This is not possible for more complicated examples, however, which require
the optimal solution to be constructed numerically, using one of the tools discussed in Appendix ]. Figure 10.2C-D show
a typical solution for parameter values estimated from experimental data [22]. Starting from a low value of $u_R$ during
balanced growth on acetate, the optimal resource allocation scheme consists of a sequence of switches between $u_R = 1$
(maximal ribosome synthesis) and $u_R = 0$ (no ribosome synthesis), until an intermediate value of $u_R$ for balanced
growth on glucose is attained. The value of $u_R$ during balanced growth on glucose is higher than that for balanced
growth on acetate, as expected from the growth law.

The sequence of on-off switches followed by the intermediate steady-state value is called a bang-bang-singular solution in
optimal control theory [21, 22]. The solution reflects a dynamic trade-off between the two different functions contributing
to growth: metabolism and protein synthesis. When, due to growth dilution, the ribosome concentration falls to a level
that is limiting for maximal protein synthesis, the synthesis of ribosomal proteins is switched on ($u_R = 1$), leading to
an increase of the ribosome concentration. Switching on the synthesis of ribosomal proteins causes the synthesis of
metabolic enzymes to be switched off. When, due to growth dilution, the concentration of metabolic enzymes next falls
to a level that the precursors produced by the latter become limiting, the synthesis of metabolic enzymes is switched on
($u_R = 0$) to replenish the precursor pool, and so on.

Optimal solutions with a similar bang-bang pattern were already encountered in the previous section. They also occur
in a model with a more detailed description of different precursor (amino acid) synthesis pathways under the objective
under the minimal time of adaptation after a shift from a medium supplemented with amino acid to a medium lacking
amino acids [25]. In a different type of problem, the development of intestinal crypts, the minimal time to mature crypts
was found to depend on the on-off control of the proliferation of stem and non-stem cells [26]. There is no convincing
experimental evidence that the adaptation of ribosomal synthesis after a nutrient upshift from a poor to a carbon source
follows a bang-bang singular pattern. The interpretation of proteomics data after a nutrient upshift in *E. coli* shows
that the simple upregulation of ribosomal resource allocation to the steady-state value for growth on the rich nutrient
captures the ribosomal protein expression data well [27].

This example serves to emphasize that, while the optimality assumption may lead to thought-provoking predictions, these
need to be confronted with experimental data. In case the optimal solutions do not agree with the data, several revisions
of the problem could be considered. While growth optimization was chosen as the objective criterion in the example
of Figure 10.2, there is evidence that during balanced growth, microorganisms find a trade-off between maximizing
growth rate in a given environment and minimizing necessary adjustments to other environments [12]. The problem
could therefore be generalized to a multi-criteria optimization problem. An example is the analysis of a model similar to
that considered here under the objectives of biomass maximization and minimal adaptation time after a nutrient shift
[28]. The formulation of the optimization problem in Figure 10.2 does not put any constraints on valid optimal resource
allocation strategies, except that the individual functions $u_i$ components need to sum to 1 (Eq. 10.24). Bearing in
mind that the regulatory mechanisms underlying a resource allocation come with a cost, and need to respect certain
physical constraints, the predicted resource allocation strategy may not be feasible. When such constraints are taken
into account, the optimal solution may no longer be bang-bang singular, but resemble the observed adaptation pattern
[29, 22].

In summary, the dynamical optimization approach for studying microbial growth presented here provides a way to test the
consequences of hypothesized objective functions in combination with simple resource allocation models. The predictions
can be confronted with experimental data, but may also inform the redesign of microbial strains for metabolic engineering
purposes [30] (Box 10.A).

Box 10.A: Time-dependent optimization for biotechnological applications and process design

As explained in Section 10.2, time-dependent optimization problems are defined by an objective function, expressing
the criterion that microorganisms presumably optimize. In the case of microorganisms growing in natural conditions,
the choice of a particular objective function is difficult to make and several functions may qualify. For example,
microorganisms could be assumed to maximize their biomass over a given interval of time or minimize the time to adapt to their new environment after a change in growth conditions. The choice may be somewhat arbitrary and in many cases it makes sense to consider a multi-criteria optimization problem. Even more fundamentally, the idea that microorganisms have evolved to the point that they optimize one or several objective functions, is controversial.

In bioengineering applications, however, the formulation of an objective function is less problematic. In this context, the objective function is not assumed to have evolved through natural selection, but is rather stipulated by the metabolic engineer in an *a-priori* manner, in agreement with a practical objective. Possible objective functions are the maximal amount of fermentation product that can be obtained from a given amount of substrate (maximal yield) or the minimal time to produce a given amount of fermentation product (maximal productivity). The use of optimal control methods for process design in bioengineering is well-known [31, 32]. Most of these methods, however, treat microbial growth as a black box and do not provide much detail about the underlying cellular processes, contrary to the formalisms discussed in this chapter. Opening up the black box of microbial growth allows the use of control variables that go beyond standard process parameters of the bioreactor and represent directed perturbations of specific cellular processes.

One example is the use of coarse-grained models of microbial growth for the design of optimal operating conditions for the so-called growth switch [30]. The growth switch is a synthetic regulatory circuit allowing growth of *E. coli* to be arrested in order to passively reorient the resources thus becoming available towards the production of a metabolite of interest [33]. The maximal production of this metabolite from a given amount of substrate, within a given interval of time, was formulated as an optimal control problem. Its solution showed that the optimal solution consists of two phases: a first phase of maximal biomass production followed by a second phase of maximal product synthesis [30]. The conclusion that this two-phase procedure is optimal corresponds well with established practice in biotechnology [34]. Very similar conclusions were attained in related work by Jeanne et al. [35].

## 10.5. Dynamic flux balance analysis (dFBA) of metabolic networks

Dynamic Flux Balance Analysis (dFBA) is an extension of Flux Balance Analysis (FBA) as described in Chapter 5 in [2], that can simulate the interactions between the metabolism of an organism and its dynamic environment. In contrast to the constant, steady-state flux solutions that are generated by classical FBA, dFBA yields flux solutions that may dynamically depend on concentrations of extracellular metabolites, such as sugars or other carbon sources, dissolved oxygen, or secreted waste metabolites. Applying these fluxes to the concentration balance of extracellular metabolites also permits to capture dynamic changes in these concentrations due to the cells' metabolic activity, and track the resulting overall biomass growth. It is noted that a basic assumption of dFBA is that organisms rapidly reach intracellular steady state in response to extracellular perturbations, and on the long run no metabolite can accumulate or deplete.

In general, a dFBA model comprises three main parts as demonstrated in Figure 10.3: the dynamic equations, in the form of differential equations, for biomass and extracellular metabolites, constraints on the fluxes as in the FBA model, and an optimization objective that determines how to choose the optimal fluxes.

We first consider the dynamic equations used for dFBA.

The biomass dynamics are given by

$$ \dot{X} = \mu X, \qquad (10.28) $$

where *X* denotes the biomass concentration, typically measured as dry mass in g/L, and *μ* denotes the growth rate, typically measured in 1/h. In principle, this equation follows the equations for balanced growth. However, instead of using simple models, like a Monod equation for the growth rate, the growth rate is taken from the value of the biomass reaction in an FBA model (check in Chapter 5 in [2]!).
microorganisms could be assumed to maximize their biomass over a given interval of time or minimize the time to adapt to their new environment after a change in growth conditions. The choice may be somewhat arbitrary and in many cases it makes sense to consider a multi-criteria optimization problem. Even more fundamentally, the idea that microorganisms have evolved to the point that they optimize one or several objective functions, is controversial.

In bioengineering applications, however, the formulation of an objective function is less problematic. In this context, the objective function is not assumed to have evolved through natural selection, but is rather stipulated by the metabolic engineer in an *a-priori* manner, in agreement with a practical objective. Possible objective functions are the maximal amount of fermentation product that can be obtained from a given amount of substrate (maximal yield) or the minimal time to produce a given amount of fermentation product (maximal productivity). The use of optimal control methods for process design in bioengineering is well-known [31, 32]. Most of these methods, however, treat microbial growth as a black box and do not provide much detail about the underlying cellular processes, contrary to the formalisms discussed in this chapter. Opening up the black box of microbial growth allows the use of control variables that go beyond standard process parameters of the bioreactor and represent directed perturbations of specific cellular processes.

One example is the use of coarse-grained models of microbial growth for the design of optimal operating conditions for the so-called growth switch [30]. The growth switch is a synthetic regulatory circuit allowing growth of E. coli to be arrested in order to passively reorient the resources thus becoming available towards the production of a metabolite of interest [33]. The maximal production of this metabolite from a given amount of substrate, within a given interval of time, was formulated as an optimal control problem. Its solution showed that the optimal solution consists of two phases: a first phase of maximal biomass production followed by a second phase of maximal product synthesis [30]. The conclusion that this two-phase procedure is optimal corresponds well with established practice in biotechnology [34]. Very similar conclusions were attained in related work by Jeanne et al. [35].

## 10.5. Dynamic flux balance analysis (dFBA) of metabolic networks

Dynamic Flux Balance Analysis (dFBA) is an extension of Flux Balance Analysis (FBA) as described in Chapter 5 in [2], that can simulate the interactions between the metabolism of an organism and its dynamic environment. In contrast to the constant, steady-state flux solutions that are generated by classical FBA, dFBA yields flux solutions that may dynamically depend on concentrations of extracellular metabolites, such as sugars or other carbon sources, dissolved oxygen, or secreted waste metabolites. Applying these fluxes to the concentration balance of extracellular metabolites also permits to capture dynamic changes in these concentrations due to the cells' metabolic activity, and track the resulting overall biomass growth. It is noted that a basic assumption of dFBA is that organisms rapidly reach intracellular steady state in response to extracellular perturbations, and on the long run no metabolite can accumulate or deplete.

In general, a dFBA model comprises three main parts as demonstrated in Figure 10.3: the dynamic equations, in the form of differential equations, for biomass and extracellular metabolites, constraints on the fluxes as in the FBA model, and an optimization objective that determines how to choose the optimal fluxes.

We first consider the dynamic equations used for dFBA.

The biomass dynamics are given by

$$ \dot{X} = \mu X, \qquad (10.28) $$

where X denotes the biomass concentration, typically measured as dry mass in g/L, and μ denotes the growth rate, typically measured in 1/h. In principle, this equation follows the equations for balanced growth. However, instead of using simple models, like a Monod equation for the growth rate, the growth rate is taken from the value of the biomass reaction in an FBA model (check in Chapter 5 in [2]!).
Denoting the concentrations of the extracellular metabolites that are modelled dynamically as the vector $c$, the dynamics for these metabolites can be formulated as the differential equation

$$ \dot{c} = S_{exch}vX. \qquad (10.29) $$

Here, $v$ is the flux vector for the complete metabolic network, including uptake and production reactions for exchange metabolites, and $S_{exch}$ is the stoichiometric matrix that links these reaction fluxes to the metabolite concentrations which are balanced dynamically. Multiplication with the biomass $X$ is necessary, since the flux values in the FBA model are determined relative to biomass, whereas the concentrations $c$ of the dynamic metabolites are relative to the system volume. The equations are not yet closed, because the fluxes $v$ (including the growth rate $\mu$ as one element of the flux vector) still need to be determined by optimization.

As constraints, two types of constraints are used in dFBA models. A flux balance constraint as in steady state FBA models is applied to the concentrations of all metabolites that are not dynamically balanced in (10.29), e.g., intracellular metabolites. This steady state constraint is given by

$$ S_{int}v = 0, \qquad (10.30) $$

where $S_{int}$ is the stoichiometric matrix that links reaction fluxes in the vector $v$ to the steady state metabolites. Further, upper and lower bounds need to be put on the individual reaction fluxes. In contrast to classical FBA, where these bounds are constant, in dFBA flux bounds can depend on concentrations of metabolites in the vector $c$. This is mostly applied to uptake reactions for nutrients, and often as Michelis-Menten kinetics. For example, if $c_i$ is the concentration of a sugar substrate, and $v_i$ is the uptake reaction for this substrate (conventionally negative in FBA models), bounds of the form

$$ -\frac{V_{i,max}c_i}{K_M + c_i} \le v_i \le 0 \qquad (10.31) $$

would be used, where $V_{i,max}$ and $K_M$ are the common parameters of the Michealis-Menten kinetics (Chapter 3 in [2] [MET]).

In recent years, dFBA is increasingly applied for the simulation of dynamic biological systems, especially due to the promising use of GSMMs for interpreting cell physiology and evolution, as well as for guiding metabolic engineering and bioprocess design and optimization [36, 37, 38]. The dFBA applications based on GSMMs include the bacteria *Escherichia coli* [39, 40, 41, 42, 43, 44, 45, 46] and *Lactococcus lactis* [47], as well as the yeast species *Saccharomyces cerevisiae* [48, 49, 50, 51, 52, 40, 43, 53, 54, 46] and *Scheffersomyces (Pichia) stipites* [43]. However, the majority of dFBA applications use small-scale metabolic models, most of which include less than 100 reactions. Such applications include models of bacteria, like *Escherichia coli* [55, 56, 57, 43, 58, 59, 60, 61], *Corynebacterium glutamicum* [62, 42, 63, 45] and *Bordetella pertussis* [64], models of yeast for wine fermentation [65] and *Saccharomyces cerevisiae* [66, 54, 67, 68], but also plant and animal models, such as a model for the photosynthetic metabolism of C3 plants [69], a four-tissue (leaf, root, seed, and stem) model of the core metabolism of *Arabidopsis thaliana* [70], a model for fatty acid metabolism and lipid accumulation in rat hepatocytes [71], and a model for energy metabolism in myocardial cells [72].

It is noted that most of the dFBA applications for microorganisms simulate microbial fermentations under batch or fed-batch conditions. Since dFBA can be used for the analysis, control and optimization of biochemical processes, many dFBA applications focus on either dynamic metabolic engineering or optimal control of bioreactors, or both simultaneously. Dynamic metabolic engineering studies can predict the effect of strain gene insertion and deletion on the dynamic behavior and productivity of a bioprocess [48, 39, 49, 50], while optimal control of batch or fed-batch operation of bioreactors is important for the production of desired chemicals [48, 68]. Finally, dFBA has also been expanded for the study of microbial communities, where each microorganism is represented by an LP that is solved independently [73, 74, 75]. Co-culture simulations with dFBA can predict possible consortia compositions, as well as metabolic engineering approaches to improve the productivity of the consortia, but they are out of the scope of this
Denoting the concentrations of the extracellular metabolites that are modelled dynamically as the vector $c$, the dynamics for these metabolites can be formulated as the differential equation

$$ \dot{c} = S_{exch}vX. \qquad (10.29) $$

Here, $v$ is the flux vector for the complete metabolic network, including uptake and production reactions for exchange metabolites, and $S_{exch}$ is the stoichiometric matrix that links these reaction fluxes to the metabolite concentrations which are balanced dynamically. Multiplication with the biomass $X$ is necessary, since the flux values in the FBA model are determined relative to biomass, whereas the concentrations $c$ of the dynamic metabolites are relative to the system volume. The equations are not yet closed, because the fluxes $v$ (including the growth rate $\mu$ as one element of the flux vector) still need to be determined by optimization.

As constraints, two types of constraints are used in dFBA models. A flux balance constraint as in steady state FBA models is applied to the concentrations of all metabolites that are not dynamically balanced in (10.29), e.g., intracellular metabolites. This steady state constraint is given by

$$ S_{int}v = 0, \qquad (10.30) $$

where $S_{int}$ is the stoichiometric matrix that links reaction fluxes in the vector $v$ to the steady state metabolites. Further, upper and lower bounds need to be put on the individual reaction fluxes. In contrast to classical FBA, where these bounds are constant, in dFBA flux bounds can depend on concentrations of metabolites in the vector $c$. This is mostly applied to uptake reactions for nutrients, and often as Michelis-Menten kinetics. For example, if $c_i$ is the concentration of a sugar substrate, and $v_i$ is the uptake reaction for this substrate (conventionally negative in FBA models), bounds of the form

$$ -\frac{V_{i,max}c_i}{K_M + c_i} \le v_i \le 0 \qquad (10.31) $$

would be used, where $V_{i,max}$ and $K_M$ are the common parameters of the Michealis-Menten kinetics (Chapter 3 in [2] [MET]).

In recent years, dFBA is increasingly applied for the simulation of dynamic biological systems, especially due to the promising use of GSMMs for interpreting cell physiology and evolution, as well as for guiding metabolic engineering and bioprocess design and optimization [36, 37, 38]. The dFBA applications based on GSMMs include the bacteria *Escherichia coli* [39, 40, 41, 42, 43, 44, 45, 46] and *Lactococcus lactis* [47], as well as the yeast species *Saccharomyces cerevisiae* [48, 49, 50, 51, 52, 40, 43, 53, 54, 46] and *Scheffersomyces (Pichia) stipites* [43]. However, the majority of dFBA applications use small-scale metabolic models, most of which include less than 100 reactions. Such applications include models of bacteria, like *Escherichia coli* [55, 56, 57, 43, 58, 59, 60, 61], *Corynebacterium glutamicum* [62, 42, 63, 45] and *Bordetella pertussis* [64], models of yeast for wine fermentation [65] and *Saccharomyces cerevisiae* [66, 54, 67, 68], but also plant and animal models, such as a model for the photosynthetic metabolism of C3 plants [69], a four-tissue (leaf, root, seed, and stem) model of the core metabolism of *Arabidopsis thaliana* [70], a model for fatty acid metabolism and lipid accumulation in rat hepatocytes [71], and a model for energy metabolism in myocardial cells [72].

It is noted that most of the dFBA applications for microorganisms simulate microbial fermentations under batch or fed-batch conditions. Since dFBA can be used for the analysis, control and optimization of biochemical processes, many dFBA applications focus on either dynamic metabolic engineering or optimal control of bioreactors, or both simultaneously. Dynamic metabolic engineering studies can predict the effect of strain gene insertion and deletion on the dynamic behavior and productivity of a bioprocess [48, 39, 49, 50], while optimal control of batch or fed-batch operation of bioreactors is important for the production of desired chemicals [48, 68]. Finally, dFBA has also been expanded for the study of microbial communities, where each microorganism is represented by an LP that is solved independently [73, 74, 75]. Co-culture simulations with dFBA can predict possible consortia compositions, as well as metabolic engineering approaches to improve the productivity of the consortia, but they are out of the scope of this
<figure><img src="image_9.png" /><figcaption>Figure 10.3: Schematic representation of dFBA. As in FBA, the intracellular environment in dFBA is represented by a linear programming (LP) optimization problem that describes the metabolism of the microorganism based on its genome-scale metabolic model (GSMM). FBA assumes that all intracellular metabolite concentrations remain constant while the cells optimally distribute their metabolic fluxes to maximize their growth rate and hence, an LP can calculate the growth rate, as well as the intracellular and exchange fluxes of the GSMM. The calculated growth rate and exchange fluxes can be used to update the extracellular environment. The extracellular environment in dFBA is represented by ordinary differential equations (ODEs) that describe the mass balance equations for biomass and metabolites found outside of the cell. Moreover, the intracellular GSMM and the extracellular mass balance equations can be linked through kinetic rules for substrate uptake, like the Michaelis-Menten equations, that can raise concentration-dependent constraints for exchange fluxes and predict growth rate dependencies on substrate concentrations.</figcaption></figure>

chapter.

Coming to the mathematical formulation of dFBA models, dFBA is an optimization problem coupled with a system of ordinary differential equations, that can be solved with the help of various mathematical and numerical techniques. Even though dFBA was first introduced in 1994 [55], it was not formalized until 2002 [56]. The existing formalized solution approaches that are going to be discussed here involve the static optimization approach (SOA), the dynamic optimization approach (DOA), and the direct approach (DA). More recently, reformulation approaches and surrogate models for the optimization problem have also been proposed in order to ease the computational complexity of dFBA simulations. This complexity arises from several characteristics of dFBA. More specifically, the solution of dFBA problems faces challenges in terms of:

1. problem size and scalability: As the size of the metabolic network increases, the computational cost increases. For this reason, simulations that involve large genome-scale metabolic models or multispecies microbial communities are limited.

2. stiffness: The stiff behavior of dFBA has been observed in many cases, such as the simulation of the diauxic growth in E. coli [56].

3. nonlinearity: The presence of nonlinear constraints or objective functions can significantly increase the computa-tional cost.

4. feasibility: The intracellular optimization problem can become infeasible and lead to failure of the integration of the extracellular ODEs.

5. differentiability: The optimal value of the intracellular optimization problem may not be continuously differentiable, which poses an obstacle when dFBA is used for optimal control or parameter estimation.

6. non-unique solutions: The solution of the intracellular optimization problem is usually not unique which can make
<figure><img src="image_6.png" /><figcaption>Figure 10.3: Schematic representation of dFBA. As in FBA, the intracellular environment in dFBA is represented by a linear programming (LP) optimization problem that describes the metabolism of the microorganism based on its genome-scale metabolic model (GSMM). FBA assumes that all intracellular metabolite concentrations remain constant while the cells optimally distribute their metabolic fluxes to maximize their growth rate and hence, an LP can calculate the growth rate, as well as the intracellular and exchange fluxes of the GSMM. The calculated growth rate and exchange fluxes can be used to update the extracellular environment. The extracellular environment in dFBA is represented by ordinary differential equations (ODEs) that describe the mass balance equations for biomass and metabolites found outside of the cell. Moreover, the intracellular GSMM and the extracellular mass balance equations can be linked through kinetic rules for substrate uptake, like the Michaelis-Menten equations, that can raise concentration-dependent constraints for exchange fluxes and predict growth rate dependencies on substrate concentrations.</figcaption></figure>

chapter.

Coming to the mathematical formulation of dFBA models, dFBA is an optimization problem coupled with a system of ordinary differential equations, that can be solved with the help of various mathematical and numerical techniques. Even though dFBA was first introduced in 1994 [55], it was not formalized until 2002 [56]. The existing formalized solution approaches that are going to be discussed here involve the static optimization approach (SOA), the dynamic optimization approach (DOA), and the direct approach (DA). More recently, reformulation approaches and surrogate models for the optimization problem have also been proposed in order to ease the computational complexity of dFBA simulations. This complexity arises from several characteristics of dFBA. More specifically, the solution of dFBA problems faces challenges in terms of:

1. problem size and scalability: As the size of the metabolic network increases, the computational cost increases. For this reason, simulations that involve large genome-scale metabolic models or multispecies microbial communities are limited.

2. stiffness: The stiff behavior of dFBA has been observed in many cases, such as the simulation of the diauxic growth in E. coli [56].

3. nonlinearity: The presence of nonlinear constraints or objective functions can significantly increase the computa-tional cost.

4. feasibility: The intracellular optimization problem can become infeasible and lead to failure of the integration of the extracellular ODEs.

5. differentiability: The optimal value of the intracellular optimization problem may not be continuously differentiable, which poses an obstacle when dFBA is used for optimal control or parameter estimation.

6. non-unique solutions: The solution of the intracellular optimization problem is usually not unique which can make
fluxes unrealistically “jump” between different optimal solutions.

**Static Optimization Approach (SOA)** divides the total time horizon of the dFBA simulation into several smaller time intervals. The optimization problem is solved to obtain the flux distribution at the beginning of each time interval, and then the ODEs are integrated over the time interval with this fixed flux distribution. The dynamics calculated from this time step are used to constrain the optimization problem solved at the beginning of the next time interval, and the process is repeated until the end of the simulation time is reached. SOA can be implemented easily with the use of an Euler scheme for integrating the system and a suitable existing LP solver for solving the FBA at each time step. SOA is also implemented in the constraint-based reconstruction and analysis (COBRA) toolbox for MATLAB [76] which can perform dFBA simulations. Since its implementation is relatively simple, SOA has been widely used in studies for the diauxic [56, 58, 77], aerobic and fermentative [55, 78, 79, 39] growth of *E. coli*, for *S. cerevisiae* fermentations [66, 50, 51, 52], as well as for the growth of other bacterial [47] and plant organisms [70]. Many of these applications include larger-scale or genome-scale metabolic networks, due to the scalability of SOA. However, the main drawback is that SOA is inefficient and can become computationally expensive because it has to solve the optimization problem at each time step. This can be challenging for most dFBA problems which are stiff and require small time steps to ensure accuracy, convergence, and stability of the solution.

**Dynamic Optimization Approach (DOA)** follows closely the general dynamic optimization framework described in Section 10.2: an objective function that depends on the dynamic states of the system over the complete time horizon of interest is formulated, and the dynamics (10.28)–(10.29) and algebraic constraints (10.30)–(10.31) are added as optimization constraints. In other words, DOA discretizes the total time horizon of the dFBA simulation, and then transforms the dynamic optimization problem into a non-linear programming (NLP) problem, which is solved once by simultaneously optimizing over the entire time of the simulation. In this way, DOA obtains the time profiles of fluxes and metabolite concentrations in the system, and allows the formulation of a dynamic objective function, which could provide useful information about the design of genetically modified metabolic networks or the maximization of bioprocess productivity. Because of this characteristic, DOA is often used in dynamic metabolic engineering, parameter estimation and optimal control applications. For example, DOA has been used for simulating the diauxic growth of *E. coli* [56, 80, 81], as well as the growth of engineered *E. coli* strains on glucose [82, 83], and the growth of various eukaryotic organisms, such as *S. cerevisiae* [67, 84], plant [69] and animal [71, 72] cells under genetic and environmental perturbations. On the downside, even though the optimization problem does not need to be repeatedly solved like in SOA, the single NLP of DOA can become easily intractable, as its dimension increases with the fineness of time discretization. Additionally, DOA has been mainly limited to small-scale metabolic networks, since it cannot be easily applied to genome-scale metabolic networks due to the large number of variables and constraints that are introduced in the NLP as the size of the network increases.

**Direct Approach (DA)** has been formulated more recently than SOA and DOA, and directly includes the LP solver for the FBA in the right-hand side evaluator function of the ODEs. In this way, it can take advantage of existing ODE integrators with adaptive step size and error control that can reduce the number of integration steps and provide better solution accuracy compared to SOA. DA has been implemented in the ORCA toolbox [85], which complements the constraint-based reconstruction and analysis (COBRA) toolbox for MATLAB [76]. Furthermore, DA has been used for studying the diauxic [86], aerobic and anaerobic [57, 87] growth of wild type and engineered *E. coli* strains, the aerobic growth of *Corynebacterium glutamicum* on glucose and xylose in biorefinery simulations [63], as well as the aerobic and anaerobic growth of wild type and engineered *S. cerevisiae* strains [48, 49, 51]. Some of these applications involve dynamic metabolic engineering for product maximization, and many of them include genome-scale metabolic networks, since DA is relatively easily scalable like SOA.

However, DA requires the LP to be resolved at least once, every time the right-hand side of the ODEs is evaluated [88]. This can make DA computationally demanding, especially for larger metabolic networks. Another major challenge is that
fluxes unrealistically “jump” between different optimal solutions.

**Static Optimization Approach (SOA)** divides the total time horizon of the dFBA simulation into several smaller time intervals. The optimization problem is solved to obtain the flux distribution at the beginning of each time interval, and then the ODEs are integrated over the time interval with this fixed flux distribution. The dynamics calculated from this time step are used to constrain the optimization problem solved at the beginning of the next time interval, and the process is repeated until the end of the simulation time is reached. SOA can be implemented easily with the use of an Euler scheme for integrating the system and a suitable existing LP solver for solving the FBA at each time step. SOA is also implemented in the constraint-based reconstruction and analysis (COBRA) toolbox for MATLAB [76] which can perform dFBA simulations. Since its implementation is relatively simple, SOA has been widely used in studies for the diauxic [56, 58, 77], aerobic and fermentative [55, 78, 79, 39] growth of *E. coli*, for *S. cerevisiae* fermentations [66, 50, 51, 52], as well as for the growth of other bacterial [47] and plant organisms [70]. Many of these applications include larger-scale or genome-scale metabolic networks, due to the scalability of SOA. However, the main drawback is that SOA is inefficient and can become computationally expensive because it has to solve the optimization problem at each time step. This can be challenging for most dFBA problems which are stiff and require small time steps to ensure accuracy, convergence, and stability of the solution.

**Dynamic Optimization Approach (DOA)** follows closely the general dynamic optimization framework described in Section 10.2: an objective function that depends on the dynamic states of the system over the complete time horizon of interest is formulated, and the dynamics (10.28)–(10.29) and algebraic constraints (10.30)–(10.31) are added as optimization constraints. In other words, DOA discretizes the total time horizon of the dFBA simulation, and then transforms the dynamic optimization problem into a non-linear programming (NLP) problem, which is solved once by simultaneously optimizing over the entire time of the simulation. In this way, DOA obtains the time profiles of fluxes and metabolite concentrations in the system, and allows the formulation of a dynamic objective function, which could provide useful information about the design of genetically modified metabolic networks or the maximization of bioprocess productivity. Because of this characteristic, DOA is often used in dynamic metabolic engineering, parameter estimation and optimal control applications. For example, DOA has been used for simulating the diauxic growth of *E. coli* [56, 80, 81], as well as the growth of engineered *E. coli* strains on glucose [82, 83], and the growth of various eukaryotic organisms, such as *S. cerevisiae* [67, 84], plant [69] and animal [71, 72] cells under genetic and environmental perturbations. On the downside, even though the optimization problem does not need to be repeatedly solved like in SOA, the single NLP of DOA can become easily intractable, as its dimension increases with the fineness of time discretization. Additionally, DOA has been mainly limited to small-scale metabolic networks, since it cannot be easily applied to genome-scale metabolic networks due to the large number of variables and constraints that are introduced in the NLP as the size of the network increases.

**Direct Approach (DA)** has been formulated more recently than SOA and DOA, and directly includes the LP solver for the FBA in the right-hand side evaluator function of the ODEs. In this way, it can take advantage of existing ODE integrators with adaptive step size and error control that can reduce the number of integration steps and provide better solution accuracy compared to SOA. DA has been implemented in the ORCA toolbox [85], which complements the constraint-based reconstruction and analysis (COBRA) toolbox for MATLAB [76]. Furthermore, DA has been used for studying the diauxic [86], aerobic and anaerobic [57, 87] growth of wild type and engineered *E. coli* strains, the aerobic growth of *Corynebacterium glutamicum* on glucose and xylose in biorefinery simulations [63], as well as the aerobic and anaerobic growth of wild type and engineered *S. cerevisiae* strains [48, 49, 51]. Some of these applications involve dynamic metabolic engineering for product maximization, and many of them include genome-scale metabolic networks, since DA is relatively easily scalable like SOA.

However, DA requires the LP to be resolved at least once, every time the right-hand side of the ODEs is evaluated [88]. This can make DA computationally demanding, especially for larger metabolic networks. Another major challenge is that
when evaluating the right-hand side of the ODEs close to the boundary of feasibility, the LP can become infeasible and make the dFBA simulation fail. The LP can become infeasible either because it is really infeasible and the simulation should be terminated, or because the ODE integrator becomes unable to evaluate the right-hand side of the ODEs and the simulation is discontinued, or erroneous death phase messages are being displayed. The latter can happen as dFBA simulations involve discrete events that correspond to switches in the active set of the LP solution. More specifically, different bases for the optimal solution of the LP can emerge at each time step. Moreover, at the points of change of the active set, the dFBA model is not differentiable, since the optimal value of the LP as a function of the right-hand side of the constraints is not continuously differentiable. This is a problem because the first and second derivatives of the model must be computed when dFBA is used for optimal control or parameter estimation applications. Finally, another drawback emerges due to the primal multiplicity of the LP. As it is well-known, FBA is formulated as an underdetermined problem and therefore, the LP does not have a unique solution [89]. Non-unique optimal reaction fluxes can lead different ODE integrators to different results.

## Mathematical details 10.B: dFBAlab

In order to address some of the computational challenges of dFBA, Höffner, Harwood, and Barton proposed a simulator for dFBA, which was initially coded in FORTRAN [90], but gained popularity when implemented in MATLAB with the name Dynamic Flux Balance Analysis laboratory (DFBAlab) [75], and more recently in Python [91]. It is noted that the DFBAlab is compatible with the COBRA toolbox [76]. Based on this dFBA simulator, it is not necessary to resolve the LP each time the right-hand side of the ODEs is evaluated and consequently, the solution process becomes faster. This is possible because the FBA solution at an initial time can be used to compute future optimal solutions by detecting changes in the active set [45] or by computing the optimal basis [92] of the FBA. Unfortunately, such formulations need to continuously monitor the active set of the LP, which increases with the size of the metabolic network, or need to choose a basis for the optimal solution that is most likely to remain optimal as the simulation proceeds [92]. The latter is challenging since the optimal basis can be non-unique even for a unique optimal solution. Nevertheless, DFBAlab manages to reduce the number of times that the LP is resolved, and also avoids obtaining infeasible LPs and numerical failure by using the LP feasibility problem and the Karush-Kuhn-Tucker (KKT) optimality conditions of the FBA problem (see below) [40, 42, 45]. In addition, the differentiability problem could be solved with the help of non-smooth analysis which provides optimality conditions in terms of sub-gradients or generalized gradients, for convex and non-convex functions respectively [41]. Furthermore, to tackle the issue of primal multiplicity of the FBA problem, DFBAlab performs lexicographic optimization [40, 41, 42, 45].

## 10.6. Concluding remarks

One approach for understanding the response of microorganisms to changes in their environment is to assume that this response has been optimized by evolution. That is, the regulatory mechanisms controlling the response optimize an objective, or a trade-off between competing objectives, subject to a variety of physical and biochemical constraints. This approach gives rise to dynamic optimization problems (10.1)-(10.4) that can be solved by techniques from optimal control theory. Three examples of such problems were considered in this chapter: dynamic optimization of enzyme expression in metabolic pathways, dynamic optimization of coarse-grained models of cellular growth, and dynamic flux balance analysis. This does not exhaust the range of possible problems that can be considered. One example is the combination of the resource allocation perspective with dynamic flux balance analysis [80].

Some of the predictions obtained by means of dynamic optimization seem to be supported by available experimental data, such as the time-ordering of enzyme expression in a linear pathway. Other predictions cannot currently be tested or may not be consistent with the available experimental data, such as the dynamical ribosomal protein synthesis pattern. The contradiction between a predicted optimal response and the observations is interesting, because it indicates that some of the assumptions underlying the problem need to be revised. The model may not account for all relevant processes taking
when evaluating the right-hand side of the ODEs close to the boundary of feasibility, the LP can become infeasible and make the dFBA simulation fail. The LP can become infeasible either because it is really infeasible and the simulation should be terminated, or because the ODE integrator becomes unable to evaluate the right-hand side of the ODEs and the simulation is discontinued, or erroneous death phase messages are being displayed. The latter can happen as dFBA simulations involve discrete events that correspond to switches in the active set of the LP solution. More specifically, different bases for the optimal solution of the LP can emerge at each time step. Moreover, at the points of change of the active set, the dFBA model is not differentiable, since the optimal value of the LP as a function of the right-hand side of the constraints is not continuously differentiable. This is a problem because the first and second derivatives of the model must be computed when dFBA is used for optimal control or parameter estimation applications. Finally, another drawback emerges due to the primal multiplicity of the LP. As it is well-known, FBA is formulated as an underdetermined problem and therefore, the LP does not have a unique solution [89]. Non-unique optimal reaction fluxes can lead different ODE integrators to different results.

## Mathematical details 10.B: dFBAlab

In order to address some of the computational challenges of dFBA, Höffner, Harwood, and Barton proposed a simulator for dFBA, which was initially coded in FORTRAN [90], but gained popularity when implemented in MATLAB with the name Dynamic Flux Balance Analysis laboratory (DFBAlab) [75], and more recently in Python [91]. It is noted that the DFBAlab is compatible with the COBRA toolbox [76]. Based on this dFBA simulator, it is not necessary to resolve the LP each time the right-hand side of the ODEs is evaluated and consequently, the solution process becomes faster. This is possible because the FBA solution at an initial time can be used to compute future optimal solutions by detecting changes in the active set [45] or by computing the optimal basis [92] of the FBA. Unfortunately, such formulations need to continuously monitor the active set of the LP, which increases with the size of the metabolic network, or need to choose a basis for the optimal solution that is most likely to remain optimal as the simulation proceeds [92]. The latter is challenging since the optimal basis can be non-unique even for a unique optimal solution. Nevertheless, DFBAlab manages to reduce the number of times that the LP is resolved, and also avoids obtaining infeasible LPs and numerical failure by using the LP feasibility problem and the Karush-Kuhn-Tucker (KKT) optimality conditions of the FBA problem (see below) [40, 42, 45]. In addition, the differentiability problem could be solved with the help of non-smooth analysis which provides optimality conditions in terms of sub-gradients or generalized gradients, for convex and non-convex functions respectively [41]. Furthermore, to tackle the issue of primal multiplicity of the FBA problem, DFBAlab performs lexicographic optimization [40, 41, 42, 45].

## 10.6. Concluding remarks

One approach for understanding the response of microorganisms to changes in their environment is to assume that this response has been optimized by evolution. That is, the regulatory mechanisms controlling the response optimize an objective, or a trade-off between competing objectives, subject to a variety of physical and biochemical constraints. This approach gives rise to dynamic optimization problems (10.1)-(10.4) that can be solved by techniques from optimal control theory. Three examples of such problems were considered in this chapter: dynamic optimization of enzyme expression in metabolic pathways, dynamic optimization of coarse-grained models of cellular growth, and dynamic flux balance analysis. This does not exhaust the range of possible problems that can be considered. One example is the combination of the resource allocation perspective with dynamic flux balance analysis [80].

Some of the predictions obtained by means of dynamic optimization seem to be supported by available experimental data, such as the time-ordering of enzyme expression in a linear pathway. Other predictions cannot currently be tested or may not be consistent with the available experimental data, such as the dynamical ribosomal protein synthesis pattern. The contradiction between a predicted optimal response and the observations is interesting, because it indicates that some of the assumptions underlying the problem need to be revised. The model may not account for all relevant processes taking
Mathematical details 10.C: Lexicographic Optimization

Lexicographic or hierarchical optimization involves the solution of a series of LPs with auxiliary objectives ranked in priority order. The use of auxiliary objectives reduces the feasible space and leads to a unique optimal solution, while the auxiliary objectives can have specific biological meanings and can be selected based on prior knowledge about the organism [93]. Some of the most used auxiliary objectives are based on the assumption that evolution leads to the exclusion of inefficient pathways so that cells can biosynthesize the smaller possible number of enzymes. Examples of such auxiliary objectives include the minimization of enzyme cost [94], the minimization of the total reaction flux [95], and the minimization of the number of active reactions [96]. However, it has been shown that such objectives may not be suitable for some engineered cells [64]. In general, it is not trivial to find a series of auxiliary objectives that are consistent with experimental data, assure uniqueness and preserve continuity of the optimal solution. In some cases, even when all auxiliary objectives have been used, hierarchical optimization cannot ensure the uniqueness of the dFBA solution. Apart from the use of auxiliary objectives, auxiliary rules or auxiliary parameters have also been proposed to address the primal multiplicity of FBA. For example, geometric methods have been proposed to identify a unique distribution of reaction fluxes for FBA [97], even though there is no biological evidence to justify such methods.

place in the cell, important constraints may have been ignored, or the objective may not capture the actual processes taking place.

The solution of a dynamic optimization problem may be different from the concatenation of the solutions of repeated static optimization problems defined over short, consecutive time intervals making up the time horizon. For instance, the optimal pattern of resource allocation over a time horizon may involve the accumulation of a reserve of unused resources that, while being wasteful in the short run, is beneficial when the whole time interval of interest is considered. One example was given in the introduction of this chapter, concerning the expression of maltose enzymes in the presence of lactose [8]. Another example is the accumulation of glycogen in cyanobacteria during daylight, providing the energetic resources for maintenance metabolism in the night time [5].

The analysis of the growth of microorganisms using dynamic optimization critically depends on the choice of an appropriate objective function. In the examples discussed above, optimization was performed with respect to a single objective, e.g., the maximal accumulation of biomass over a time interval or the minimal time to deplete a given amount of substrate. It is plausible, however, that microorganisms have evolved under the necessity to satisfy several objectives simultaneously. This can be taken into account by formulating a weighted sum of the different objective functions, such as the simultaneous minimization of throughput time and investment in enzymes in Eq. 10.18. Another approach is to generalize the optimization problem to a multi-objective optimization problem, with sets of Pareto optimal solutions, each providing a trade-off between mutually conflicting objectives. One example of such a multi-objective optimization problem is given by a generalization of the dynamic optimization of enzyme expression in metabolic pathways in Section 10.3, with the double objective of minimizing the time to consume a given amount of substrate and minimizing the concentration of (possibly toxic) intermediate metabolites [99].

Instead of making *a-priori* assumptions about the objectives presumably optimized by microorganisms, one could try to infer the latter from the experimental data. This inverse optimization approach leverages the large amounts of time-course data on the dynamic response of microorganisms to environmental perturbations that have accumulated in the past decade. Inverse optimization requires the solution of complex inverse optimal control problems that have been little explored until now [11].
Mathematical details 10.C: Lexicographic Optimization

Lexicographic or hierarchical optimization involves the solution of a series of LPs with auxiliary objectives ranked
in priority order. The use of auxiliary objectives reduces the feasible space and leads to a unique optimal solution,
while the auxiliary objectives can have specific biological meanings and can be selected based on prior knowledge
about the organism [93]. Some of the most used auxiliary objectives are based on the assumption that evolution
leads to the exclusion of inefficient pathways so that cells can biosynthesize the smaller possible number of enzymes.
Examples of such auxiliary objectives include the minimization of enzyme cost [94], the minimization of the total
reaction flux [95], and the minimization of the number of active reactions [96]. However, it has been shown that
such objectives may not be suitable for some engineered cells [64]. In general, it is not trivial to find a series
of auxiliary objectives that are consistent with experimental data, assure uniqueness and preserve continuity of
the optimal solution. In some cases, even when all auxiliary objectives have been used, hierarchical optimization
cannot ensure the uniqueness of the dFBA solution. Apart from the use of auxiliary objectives, auxiliary rules or
auxiliary parameters have also been proposed to address the primal multiplicity of FBA. For example, geometric
methods have been proposed to identify a unique distribution of reaction fluxes for FBA [97], even though there is
no biological evidence to justify such methods.

place in the cell, important constraints may have been ignored, or the objective may not capture the actual processes
taking place.

The solution of a dynamic optimization problem may be different from the concatenation of the solutions of repeated
static optimization problems defined over short, consecutive time intervals making up the time horizon. For instance, the
optimal pattern of resource allocation over a time horizon may involve the accumulation of a reserve of unused resources
that, while being wasteful in the short run, is beneficial when the whole time interval of interest is considered. One
example was given in the introduction of this chapter, concerning the expression of maltose enzymes in the presence of
lactose [8]. Another example is the accumulation of glycogen in cyanobacteria during daylight, providing the energetic
resources for maintenance metabolism in the night time [5].

The analysis of the growth of microorganisms using dynamic optimization critically depends on the choice of an ap-
propriate objective function. In the examples discussed above, optimization was performed with respect to a single
objective, e.g., the maximal accumulation of biomass over a time interval or the minimal time to deplete a given amount
of substrate. It is plausible, however, that microorganisms have evolved under the necessity to satisfy several objectives
simultaneously. This can be taken into account by formulating a weighted sum of the different objective functions, such
as the simultaneous minimization of throughput time and investment in enzymes in Eq. 10.18. Another approach is to
generalize the optimization problem to a multi-objective optimization problem, with sets of Pareto optimal solutions,
each providing a trade-off between mutually conflicting objectives. One example of such a multi-objective optimization
problem is given by a generalization of the dynamic optimization of enzyme expression in metabolic pathways in Sec-
tion 10.3, with the double objective of minimizing the time to consume a given amount of substrate and minimizing the
concentration of (possibly toxic) intermediate metabolites [99].

Instead of making *a-priori* assumptions about the objectives presumably optimized by microorganisms, one could try to
infer the latter from the experimental data. This inverse optimization approach leverages the large amounts of time-
course data on the dynamic response of microorganisms to environmental perturbations that have accumulated in the
past decade. Inverse optimization requires the solution of complex inverse optimal control problems that have been little
explored until now [11].
<figure><img src="image_10.png" /></figure>

B

Exchange constraints

$$-10.5 \frac{\text{mmol}}{\text{gDW h}} \frac{c_G}{2.7 \frac{\text{mg}}{\text{L}} + c_G} \le v_G \le 0$$

$$-30 \frac{\text{mmol}}{\text{gDW h}} \frac{c_{O_2}}{20 \frac{\text{mg}}{\text{L}} + c_{O_2}} \le v_{O_2} \le 0$$

$$0 \le v_E$$

$$-30 \frac{\text{mmol}}{\text{gDW h}} \frac{c_A}{100 \frac{\text{mg}}{\text{L}} + c_A} \le v_A$$

C

Dynamic equations

$$\dot{X} = \mu X$$

$$\dot{c}_{O_2} = v_{O_2} m_{O_2} X + k_L a \left(10 \frac{\text{mg}}{\text{L}} - c_{O_2}\right)$$

$$\dot{c}_G = v_G m_G X$$

$$\dot{c}_E = v_E m_E X$$

$$\dot{c}_A = v_A m_A X$$

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

E

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

F

<figure><img src="image_13.png" /><figcaption>Figure 10.4: Example result of a DFBA simulation for the E. coli core model [98] performed with DFBAlab. **A:** Illustration of dynamic metabolites and relevant exchange fluxes. Extracellular metabolites use mass concentration (g/L), exchange fluxes are in molar amount per dry biomass and time (mmol/gDW h). **B:** Concentration-dependent constraints applied to the exchange fluxes during the DFBA simulation. **C:** Differential equation model for biomass concentration X and metabolite mass concentrations. Growth rate μ and exchange fluxes v* (in red) are optimal values from the underlying FBA model. mO₂, mG, mE, mA are molar masses of O₂, glucose, ethanol, and acetate, respectively. kLa = 8.5 h⁻¹ is the volumetric mass transfer coefficient for oxygen. **D:** Simulation results for concentrations of biomass and metabolites. We can observe four growth phases: aerobic growth on glucose with production of acetate, anaerobic growth on glucose with production of ethanol and acetate, aerobic growth on acetate, and a stationary phase. **E:** Simulation result for oxygen concentration in the liquid medium. Oxygen is depleted in the second growth phase due to mass transfer limitations, but replenishes at the start of the third phase. **F:** Penalty function time course. Increases of the penalty function indicate periods where the underlying FBA model is infeasible. Here, this occurs when the ATP maintenance constraint cannot be satisfied due to a lack of substrates, and happens in this simulation during a brief period where the switch from glucose to acetate as a substrate takes place, since oxygen needs to be replenished first, and in the stationary phase.</figcaption></figure>
<figure><img src="image_7.png" /></figure>

B

Exchange constraints

$$
\begin{array}{l}
-10.5 \frac{\text{mmol}}{\text{gDW h}} \frac{c_G}{2.7 \frac{\text{mg}}{\text{L}} + c_G} \le v_G \le 0 \\
-30 \frac{\text{mmol}}{\text{gDW h}} \frac{c_{O_2}}{20 \frac{\text{mg}}{\text{L}} + c_{O_2}} \le v_{O_2} \le 0 \\
\qquad 0 \le v_E \\
-30 \frac{\text{mmol}}{\text{gDW h}} \frac{c_A}{100 \frac{\text{mg}}{\text{L}} + c_A} \le v_A
\end{array}
$$

C

Dynamic equations

$$
\dot{X} = \mu X
$$

$$
\dot{c}_{O_2} = v_{O_2} m_{O_2} X + k_L a \left(10 \frac{\text{mg}}{\text{L}} - c_{O_2}\right)
$$

$$
\dot{c}_G = v_G m_G X
$$

$$
\dot{c}_E = v_E m_E X
$$

$$
\dot{c}_A = v_A m_A X
$$

<figure><img src="image_8.png" /></figure>

E

<figure><img src="image_9.png" /></figure>

F

<figure><img src="image_10.png" /><figcaption>Figure 10.4: Example result of a DFBA simulation for the E. coli core model [98] performed with DFBAlab. **A:** Illustration of dynamic metabolites and relevant exchange fluxes. Extracellular metabolites use mass concentration (g/L), exchange fluxes are in molar amount per dry biomass and time (mmol/gDW h). **B:** Concentration-dependent constraints applied to the exchange fluxes during the DFBA simulation. **C:** Differential equation model for biomass concentration X and metabolite mass concentrations. Growth rate μ and exchange fluxes v* (in red) are optimal values from the underlying FBA model. mO2, mG, mE, mA are molar masses of O2, glucose, ethanol, and acetate, respectively. kLa = 8.5 h-1 is the volumetric mass transfer coefficient for oxygen. **D:** Simulation results for concentrations of biomass and metabolites. We can observe four growth phases: aerobic growth on glucose with production of acetate, anaerobic growth on glucose with production of ethanol and acetate, aerobic growth on acetate, and a stationary phase. **E:** Simulation result for oxygen concentration in the liquid medium. Oxygen is depleted in the second growth phase due to mass transfer limitations, but replenishes at the start of the third phase. **F:** Penalty function time course. Increases of the penalty function indicate periods where the underlying FBA model is infeasible. Here, this occurs when the ATP maintenance constraint cannot be satisfied due to a lack of substrates, and happens in this simulation during a brief period where the switch from glucose to acetate as a substrate takes place, since oxygen needs to be replenished first, and in the stationary phase.</figcaption></figure>
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[69] Ruoyu Luo, Haibin Wei, Lin Ye, Kankan Wang, Fan Chen, Lijun Luo, Lei Liu, Yuanyuan Li, M. James C. Crabbe, Li Jin, Yixue Li, and Yang Zhong. Photosynthetic metabolism of C3 plants shows highly cooperative regulation under changing environments: A systems biological analysis. *Proceedings of the National Academy of Sciences*, 106(3):847-852, January 2009. ISSN 0027-8424, 1091-6490. doi: 10.1073/pnas.0810731105.

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