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Large resource allocation models of cells

Hugo Dourado, Anne Goelzer, Pranas Grigaitis, Wolfram Liebermeister, Elad Noor

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Hugo Dourado, Anne Goelzer, Pranas Grigaitis, Wolfram Liebermeister, Elad Noor. Large resource allocation models of cells. The Economic Cell Collective. Economic Principles in Cell Biology, 2023, 10.5281/zenodo.8156647. http://hal-inrae.fr/hal-04172144

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# Resource allocation in complex cell models

Hugo Dourado, Anne Goelzer, Pranas Grigaitis, Wolfram Liebermeister, Elad Noor

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Hugo Dourado, Anne Goelzer, Pranas Grigaitis, Wolfram Liebermeister, Elad Noor. Resource allocation in complex cell models. The Economic Cell Collective. Economic Principles in Cell Biology, The Economic Cell Collective, 2023, 10.5281/zenodo.8156647. [https://doi.org/10.5281/zenodo.8156647](https://doi.org/10.5281/zenodo.8156647)

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# Large resource allocation models of cells

Hugo Dourado¹ ID, Anne Goelzer² ID, Pranas Grigaitis³ ID, Wolfram Liebermeister² ID, and Elad Noor⁴ ID

¹ Institute for Computer Science and Department of Biology, Heinrich-Heine Universität, Düsseldorf, Germany; ² Université Paris-Saclay, INRAE, MalAGE, 78350 Jouy-en-Josas, France; ³ Systems Biology Lab, Amsterdam Institute of Molecular and Life Sciences (AIMMS), Vrije Universiteit Amsterdam, Amsterdam, the Netherlands; ⁴ Department of Plant and Environmental Sciences, Weizmann Institute of Science, 76100 Rehovot, Israel

## Abstract

Resource balance analysis models are cell models based on three basic constraints formulated at genome-scale: stationary fluxes (balancing production and consumption fluxes, uptake and excretion fluxes, as well as compound dilution by cell growth); flux coupling constraints relating fluxes to the amounts of catalyzing enzymes (or other machines); and density constraints, limiting molecule amounts in cell compartments, or molecule concentrations. These constraints narrow down the solution space predicted by FBA towards more physiological solutions. Large resource allocation models build on the same principles, and have been implemented as different variations (RBA models in a narrow sense, ME-models, and pc-models).

**Keywords:** Cell model, Resource allocation, Constraint, Biomass

**Contributions:** This chapter was planned and outlined by the authors after discussions with O. Bodeit, M. Lercher, and R. Steuer, and has been reviewed by S. Waldherr and F. Moro.

**To cite this chapter:** H. Dourado, A. Goelzer, P. Grigaitis, W. Liebermeister, and E. Noor. Large resource allocation models of cells. 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>
# Resource allocation in complex cell models

Hugo Dourado¹ ID, Anne Goelzer² ID, Pranas Grigaitis³ ID, Wolfram Liebermeister² ID, and Elad Noor⁴ ID

¹ Institute for Computer Science and Department of Biology, Heinrich-Heine Universität, Düsseldorf, Germany; ² Université Paris-Saclay, INRAE, MalAGE, 78350 Jouy-en-Josas, France; ³ Systems Biology Lab, Amsterdam Institute of Molecular and Life Sciences (AIMMS), Vrije Universiteit Amsterdam, Amsterdam, the Netherlands; ⁴ Department of Plant and Environmental Sciences, Weizmann Institute of Science, 76100 Rehovot, Israel

## Abstract

Whole-cell resource allocation models on a genomic scale combine a detailed, FBA-like description of metabolism with a model of macromolecule synthesis, formulated as linear constraint-based problems. They are based on three basic constraints formulated at genome-scale: stationary fluxes (balancing production and consumption fluxes, uptake and excretion fluxes, as well as compound dilution by cell growth); flux coupling constraints relating fluxes to the amounts of catalyzing enzymes (or other machines); and density constraints, limiting molecule amounts in cell compartments, or molecule concentrations. Compared to FBA models, which only describe metabolism, these constraints narrow down the solution space towards more physiological solutions. Large resource allocation models build on the same principles, and have been implemented as different variations (RBA models, ME-models, and pc-models).

**Keywords:** cell model - resource allocation - constraint - biomass

**Contributions:** This chapter was drafted and written by Anne Goelzer, Pranas Grigaitis, Wolfram Liebermeister with the help of Hugo Dourado and Elad Noor and after initial discussions with Oliver Bodeit, Martin Lercher, and Ralf Steuer. It has been reviewed by Stefan Waldherr and Francesco Moro.

**To cite this chapter:** H. Dourado, A. Goelzer, P. Grigaitis, W. Liebermeister, and E. Noor. Resource allocation in complex cell models (Version July 2025). doi: 10.5281/zenodo.8156647. Chapter from: The Economic Cell Collective (2025). Economic Principles in Cell Biology. No commercial publisher | Online open access book | doi: 10.5281/zenodo.8156386

The authors are listed in alphabetical order.

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

This is a chapter from the open textbook "Economic Principles in Cell Biology".

Free download from principlescellphysiology.org/book-economic-principles/.

Lecture slides for this chapter are available on the website.

© 2025 The Economic Cell Collective.

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

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

doi: 10.5281/zenodo.8156386

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

* Resource balance analysis models are cell models based on three basic constraints formulated at genome-scale: stationary fluxes (balancing production and consumption fluxes, uptake and excretion fluxes, as well as compound dilution by cell growth); flux coupling constraints relating fluxes to the amounts of catalyzing enzymes (or other machines); and density constraints, limiting molecule amounts in cell compartments, or molecule concentrations.

* These constraints narrow down the solution space predicted by FBA towards more physiological solutions

* Large resource allocation models build on the same principles, and have been implemented as different variations (RBA models in a narrow sense, ME-models, and pc-models).

## 9.1. Introduction: Quantifying cell investments at genome-scale

### 9.1.1. Can we build detailed models of entire growing cells?

In the previous chapter on growing cells, we derived possible growth rates and corresponding cell compositions and metabolic strategies based on resource allocation ideas. We asked what amounts of cellular machines will allow a cell to self-replicate, and possibly, to self-replicate at a maximal possible speed. However, the models described were rather simple, with reactions coarse-grained into proteome sectors. Following the same idea, would it be possible to model an entire cell by a one-to-one mapping of all of its reactions, including realistic kinetics and find the configuration for this entire system that would allow the cell to optimally self-replicate? Such models would in principle be capable of describing the detailed resource allocation into each reaction in the cell: they would extend genome-scale models of metabolism (such as in FBA) to models covering all cell subsystems, at the same time accounting for all important constraints on growth, in particular realistic kinetics.

So why don't we directly describe metabolism and macromolecule production in one single, detailed model? If we do that and attempt to find the optimal state, we quickly see that the optimality problems for genome-size models are hard to solve, and maybe unsolvable with current means. The reason is simple: due to non-linearities in the rate laws (where rates are typically dependent on substrates and machines, and are therefore at least quadratic), the optimality problem may have multiple local optima. Since the space for optimization is very high-dimensional (with the number of dimensions on the order of the number of compounds and reactions considered) and non-convex (because certain regions of the initially convex solution space are excluded by thermodynamic constraints), the global optimum may be hard (or in practice, impossible) to find.

We can also see this more formally. While kinetic self-replicator models could be formulated, the resulting problems would be non-convex [1, 2, 3] and potentially hard to solve. How do we know this? In the optimal solution, the flux distribution must be an elementary flux mode [2]. Since no better method is known, to find the globally optimal metabolic fluxes, protein and metabolite abundances all elementary flux modes must be enumerated. For large-scale metabolic networks this is hard, and in fact this problem is computationally \#P-complete.[4], which prevents its resolution for large-scale models. However, recent attempts of integration of thermodynamics constraints in EFM computation drastically reduced their number [5], which may provide significant help in future developments to solve large self-replicator models.

So in order to model a cell in high resolution (including individual metabolic reactions and their enzymes), what can we do? In previous chapters, we approached this problem from two directions. Flux Balance Analysis (FBA) models, as described in Chapter 5 in [6], capture metabolic fluxes in detail, but ignore the rest of the cell (such as macromolecule production; and also the fact that all cellular compounds are diluted by cell growth). Self-replicator models, as described in Chapter 8 in [6], describe a growing cell directly and consider whole-cell aspects such as dilution by growth. However, they contain only very few reactions. Larger cell processes are lumped into a few global variables, ignoring the details. This makes it impossible to model the effects of detailed changes (e.g. the fate of individual metabolites, the inhibition of single proteins, or deletions of single genes).
## Chapter overview

* Whole-cell resource allocation models on a genomic scale combine a detailed, FBA-like description of metabolism with a model of macromolecule synthesis, formulated as linear constraint-based problems.

* Resource allocation models of cells can be built based on three basic constraints: stationary fluxes (balancing production and consumption fluxes, uptake and excretion fluxes, as well as compound dilution by cell growth); catalytic constraints relating fluxes to the amounts of catalyzing enzymes (or other machines); and density constraints, limiting molecule amounts in cell compartments, or molecule concentrations.

* Large resource allocation models build on the same principles, and have been implemented as different variations (RBA models, ME-models, and pc-models).

* These constraints narrow down the solution space predicted by FBA towards more physiological solutions

# 10.1. Detailed resource allocation models of cells

In the previous chapters, we have seen two principal approaches to modeling production processes in cells. To keep the number of variables low, but with intention of well-parametrizing the model, one can construct small, coarse-grained models of growing cells (Chapter 8 in [1]). On the contrary, Flux Balance Analysis (FBA) models can accommodate a very large number of variables (Chapter 5 in [1]), making them an excellent choice to model metabolic networks at genome-scale.

Small, coarse-grained models are very suitable for investigating base principles of life. Likely the best example to illustrate this is the work of Douwe Molenaar and co. [2], where a self-replicator model was used to propose that the low-yield, or substrate-inefficient ("wasteful") metabolic strategies are adopted as a consequence of these pathways being more efficient in terms of protein use, compared to the high-yield pathways. In other terms, the growth output of the "wasteful" strategy per unit protein is higher than the "efficient" one. Thus we now believe that fermentation of glucose, often called under an umbrella term "overflow metabolism", will take place in many organisms if the substrate in their environments is abundant enough.

However, the chemistry of life is extremely diverse, and even such a familiar concept as fermentation can become complicated. Take three representatives of the tree of life: a bacterium *Escherichia coli*, budding yeast *Saccharomyces cerevisiae*, and mammalian, say, human cells. All three exhibit overflow metabolism – even when enough oxygen is available in the environment – yet the underlying biochemistry tells us that *E. coli* ferments glucose into acetic acid, *S. cerevisiae* – into ethanol, and human cells – into lactate. Bringing more contrasts on the table, there might be extreme differences in a single taxon already: some yeasts, for instance, will never produce ethanol when oxygen is present; some of them have lost the ability to do respiration at all over the course of evolution. This might sound like playing a trivia game, but in many cases, meaningful modeling of complex biological systems requires both taking and making biochemical insight. Therefore, when we aim not only to uncover the underlying principles, but also to learn biochemistry, more detailed models such as FBA model have an upper hand.

Yet we already know from previous chapters that the predictions of canonical FBA models are limited to substrate-efficient metabolic states. Continuing with the example of the overflow metabolism, FBA models would predict *E. coli* or *S. cerevisiae* to respire on minimal medium with glucose as the main carbon source – regardless of the maximal flux of glucose into the cell. Thus the prediction of substrate-inefficient metabolism using FBA over the years used to rely on introducing additional, mainly empirical (e.g. maximal oxygen uptake), constraints onto the system [3]. Moreover, we can impose only linear constraints in FBA models, and this greatly reduces our options.

Overall, we often seek to take the advantageous points of both "schools of modeling", however, this is where we need to start doing compromises. In an ideal world, the self-replicator models from Chapter 8 in [1] would be much more detailed, and would be extended with explicit kinetics and thermodynamic constraints to obtain a detailed cell model.
So why don't we just combine the two approaches? In Chapter 6 in [6] we already did this, to an extent: we considered a metabolic (FBA-like) model to compute biomass/enzyme productivity, and then used a formula derived from bacterial growth laws (see Chapter 8 in [6]) to convert biomass/enzyme productivity into cell growth rate. But there were still some disadvantages, most importantly, that the biomass composition has to be known. Biological facts such as biomass composition should not just come as empirical knowledge, but should be explained *ab initio*. Subdividing a system into metabolism and other systems (with "fixed interfaces", like a predefined biomass composition) makes it impossible to understand details of complex behavior. For example, if some proteins are needed to retrieve iron via metabolism and other proteins (or possibly, the same proteins) also contain iron, this will create a complicated protein allocation response to changing iron supply. Iron depletion is expected lead to an upregulation of the first type of proteins, and a downregulation of the second type. However, FBA can only capture the first effect, because its biomass function is fixed.

### 9.1.2. Linearization: replacing kinetics by catalytic constraints

So how can large-scale models be made tractable? If we radically linearize all formulae, then instead of a biconvex or convex/concave problem, we obtain a linear problem (a bit like FBA); more precisely, a system of linear equalities and inequalities that define a set of feasible states. This set is a polytope, and linear optimality problems on this set can be solved easily. More specifically, to model metabolism in a growing cell, we need to consider dilution in the growing cell volume. If compounds are described by concentration, dilution can be effectively modeled by "consuming reactions", with a flux given by $v^{\text{dil}} = \mu c$, the compound concentration multiplied by the growth rate. By adding these hypothetical dilution reactions to the metabolic network, we obtain a new stationarity condition $N v = \mu c$ that connects the vectors of fluxes and compound concentrations, and in which the growth rate $\mu$ appears as a parameter. For each choice of the parameter $\mu$, we can ask whether a feasible steady growth state – i.e. a feasible combination of $v$ and $c$ exists, Furthermore, the feasible combinations $(\mu, v, c)$ form a convex set, with possible solutions $(v, c)$ for low values of $\mu$ and no solutions above a critical value $\mu^{\text{max}}$. Finding this critical value – the maximal possible growth rate for our model – as well as the corresponding optimal fluxes $v$ and compound concentrations $c$ is relatively easy, and can be done by solving a series of Linear Programming problems (checking for potential solutions $(v, c)$ for different values of $\mu$).

The main downside of this approach is that all relationships between models variables need to be linearized. This concerns, most importantly, all catalyzed processes, in which we assume a linear dependence between catalyzed flux and catalyst (enzyme or machine) concentration, but ignore the dependence on the concentrations of substrates, products, cofactors, or additional regulators. What does this mean in practice? As we know from Chapter 3 in [6], typical enzymatic rate laws have the form $v = e k(c)$: the rate $v$ is proportional to enzyme level $e$ and enzyme efficiency $k$, which is given by a kinetic rate law $k(c)$, a nonlinear function of the metabolite concentrations. Depending on the context, $k$ is also called catalytic rate or apparent $k_{\text{cat}}$. The kinetic rate laws $k(c)$ have typical shapes, as described in Chapter 3 in [6]. To linearize the expression for $v$, while keeping the dependence on $e$, we need to replace $k$ by a fixed number, a model parameter. If the metabolite concentrations were known (experimentally, or from kinetic models under optimality assumptions, see Chapter 6 in [6]), the value of $k$ could be computed. Otherwise, it can also be determined experimentally, by measuring $v$ and $e$ and setting $k = v/e$. Obviously, in reality, neither $c$ nor $k$ will be fixed and given, but for our linearized model, we need to assume this. This holds both for metabolic reactions (with enzymes as catalysts) and for macromolecular reactions (with molecular machines as catalysts). Under this assumption, we can replace all kinetic constraints by two linear constraints on the enzyme to fulfill its function as $-e k' \le v \le e k$, where coefficients $k$ and $k'$ are the approximation coefficients of the enzyme kinetics in the forward and backward direction respectively. These constraints, which replace the kinetic rate laws, are called capacity constraints and resemble the linear flux/enzyme relation in FBA with molecular crowding. Finally, there is one important exception: like in satFBA, some kinetic rate laws can remain, namely those in which the metabolite concentrations themselves are not variables (or disregarded completely), but are given as model parameters. This concerns, in particular, all transport reactions whose substrate is an extracellular compound with a known concentration in the growth medium. Nonlinearities can be kept as long as they do not depend on decision variables (even if the nonlinear problem can still sometimes be solved for a
However, the number of variables would increase tremendously, and non-linear optimization is very inefficient already past even small systems. On the contrary, we could try to advance on existing FBA-type models by introducing the concepts of protein economy (Chapter 7 in [1]) at genome-scale, as well as self-replication. Following our best understanding, these, again, would constitute non-linear relationships (e.g. enzyme kinetics), yet large-scale non-linear programming is not a viable option either. Thus simplifications are necessary to keep linearity (and convexity) to solve optimization problems for large-scale models.

So can we make large-scale models tractable? If we linearize all formulae, then instead of a biconvex or convex/concave problem, we obtain a linear problem (a bit like FBA); more precisely, a system of linear equalities and inequalities that define a set of feasible states. This set is a polytope, and linear optimality problems on this set can be solved easily. More specifically, to model metabolism in a growing cell, we need to consider dilution of metabolites in the growing cell volume, or simply - the growth rate $\mu$ of the cell.

### 10.1.1. Replacing enzyme kinetics by linear catalytic constraints

To obtain large, detailed cell model that we can actually solve, all relationships between models variables have to be linearized. This concerns, most importantly, all catalyzed processes: we assume a linear dependence between a catalyzed flux and its catalyst (enzyme or machine) concentration, but ignore the dependence on the concentrations of substrates, products, cofactors, or additional regulators. What does this mean in practice? As we know from Chapter 3 in [1], typical enzymatic rate laws have the form $v = e k(s)$: the rate $v$ is proportional to enzyme level $e$ and enzyme efficiency $k$, which is given by a kinetic rate law $k(s)$, a nonlinear function of the metabolite concentrations. Depending on the context, $k$ is also called apparent $k_{cat}$. The kinetic rate laws $k(s)$ have typical shapes, as described in Chapter 3 in [1].

To linearize the expression for $v$, while keeping the dependence on $e$, we need to replace the ratio $k = v/e$ by a fixed number, and so $k$ becomes a model parameter. If the metabolite concentrations were known (experimentally, or from kinetic models under optimality assumptions, see Chapter 6 in [1]), the value of $k$ could be computed. Otherwise, it can also be determined experimentally, by measuring $v$ and $e$ and setting $k = v/e$ [4], which is feasible for a limited number of enzymes, however. Obviously, in reality, neither $v$ nor $k$ will be fixed and given, but for our linearized model, we need to assume this. This holds both for metabolic reactions (with enzymes as catalysts) and for macromolecular reactions (with molecular machines as catalysts).

Under this assumption, we can replace all kinetic constraints by two linear constraints on the enzyme. If we consider coefficients $k$ and $k'$ to approximate enzyme kinetics in the forward and backward direction, respectively, the flux the enzyme $e$ catalyzes should satisfy $-e k' \le v \le e k$. We set $k' = 0$ for irreversible reactions, and, for simplicity reasons, we usually assume $k = k'$ for reversible reactions, unless kinetic measurements are available that suggest otherwise. This relationship can be formulated as enzyme capacity constraints in order to replace the kinetic rate laws in the FBA model. By writing down such constraints for each enzyme in the model, we can couple the metabolic fluxes with the demand for enzymes, needed to operate these fluxes.

By linearizing all formulae as described above, it is possible to build very large models, describing resource allocation on genome scale. What we commonly refer to as "resource allocation models" therefore formalize the mathematical relationships defining the interactions and allocation of resources between the cellular processes to describe optimal resource allocation using constraint-based models. All these relationships take the form of linear, growth-rate dependent equalities and inequalities, and, when linearized, form a convex feasibility problem [5, 6, 7].

### 10.1.2. Overview of existing FBA extensions

By itself, the idea of constraining metabolic models to represent limited metabolic capacity of cells is not new. There are two ways to approach this budgeting problem. The first possibility is "protein budgeting", where a fixed amount of protein needs to be partitioned in the optimal manner (e.g. to maximize growth). The second, alternative method
few nonlinearities if convexity is kept)

### 9.1.3. Towards fine-grained resource allocation models

Some FBA variants account for extra empirical constraints on the total concentration of metabolic enzymes (FBA with molecular crowding, or FBAwMC), or on proteome sectors (Constrained-Allocation FBA, or CAFBA). While these can predict metabolic states more reliably, the empirical constraints come as model assumptions and thus cannot be understood by the models themselves. In these models, the primary assumption is that the cell phenotype is obtained by genetic regulations, and the main goal and utility of genetic regulation can be interpreted as ways of saving resources. Thus in many cases when we predict cell phenotype maximizing growth, we find predictions in good agreement with the experimental observations.

To obtain an ideal detailed cell model, the self-replicator models from the previous chapter would have to be extended with explicit kinetics and thermodynamic constraints. Since we know that such problems are hard to solve, simplifications are currently necessary to keep linearity (and convexity) to solve optimization problems for large-scale models.

## 9.2. Additional constraints in resource allocation models

The canonical FBA/genome-scale metabolic models (GEMs), as described above (and in previous chapters), hold a number of assumptions which allow metabolic states irregardless of the corresponding resource allocation pattern. Therefore, we need to impose sets of additional constraints on metabolic networks. The fine-grained models build on the genome-scale metabolic models to encompass all the reactions that potentially could happen in a metabolic network, and different extensions of GEMs were proposed to predict optimal resource allocation in different microorganisms [7].

Resource allocation models describe optimal resource allocation in constraint-based, genome-scale whole-cell models and formalizes the mathematical relationships defining the interactions and allocation of resources between the cellular processes. All these relationships take the form of linear, growth-rate dependent equalities and inequalities, and form a convex feasibility problem [8, 9, 10]. We will now expand on the three types of constraints used in resource allocation models; note that the description is not exhaustive and peculiarities may vary among different formulations.

### Box 9.A: Current frameworks for cell modelling with protein constraints

Constraining metabolic models to represent limited metabolic capacity of cells is not a new idea. There are two ways to approach this budgeting problem: "protein budgeting", where a fixed amount of protein needs to be partitioned in the optimal manner (maximizing growth), and "resource budgeting", where models include both the protein budgeting and the descriptions of demands for protein synthesis. However, "protein budgeting" problems assume that investments in protein production follow the budget, and not vice versa. Therefore, resource allocation models extend and embed the ideas of proteome partitioning beyond frameworks like CAFBA [11] and GECKO [12], or representing metabolic capacity limitations beyond FBAwMC [13].

Currently, there are three main implementations of large-scale resource allocation models: Resource Balance Analysis (RBA) [14], Models of Metabolism and Macromolecular Expression (ME-models) [15] and proteome-constrained models (pc-models) [16]. All these implementations are formalized as LP feasibility problems at fixed growth rate.

Originally, ME-models were considered as an extension of M-models, by including predictions for mRNA, protein, and ribosome levels. Importantly, they do not consider density constraints that, for instance, RBA does. Therefore, limitations on the capacity of exchange fluxes (as in FBA) are necessary to obtain a solution.
is "resource budgeting", where models include both the protein budgeting and the descriptions of demands for protein synthesis. However, "protein budgeting" problems assume that investments in protein production follow the budget, and not vice versa.

Some enzyme-constrained variants of FBA account for empirical constraints on the total concentration of metabolic enzymes (FBA with molecular crowding, or FBAwMC [8]), or on proteome sectors (Constrained-Allocation FBA, or CAFBA [9]). While these types of models can predict metabolic states more reliably, the empirical constraints come as model assumptions and thus cannot be understood by the models themselves. In these models, the primary assumption is that the cell phenotype is obtained by genetic regulations, and the main goal and utility of genetic regulation can be interpreted as ways of saving resources. Thus in many cases when we predict cell phenotype maximizing growth, we find predictions in good agreement with the experimental observations. Therefore, resource allocation models extend and embed the ideas of proteome partitioning beyond frameworks like CAFBA and GECKO [10], or representing metabolic capacity limitations beyond FBAwMC.

Currently, there are three main implementations of large-scale resource allocation models: Resource Balance Analysis (RBA) [11], Models of Metabolism and Macromolecular Expression (ME-models) [12] and proteome-constrained models (pc-models) [13]. All these implementations are formalized as LP feasibility problems at a fixed growth rate, where the growth rate can then be maximized in an additional optimization loop. Originally, ME-models were considered as an extension of M-models, by including predictions for mRNA, protein, and ribosome levels. Importantly, they do not consider density constraints that, for instance, RBA does. Therefore, limitations on the capacity of exchange fluxes (as in FBA) are necessary to obtain a solution.

### 10.1.3. Why maximize the growth rate?

Under the assumption of the balanced growth, the copy number of each cell component is doubled between two consecutive cell divisions. If metabolites are described by their concentration, we can think of cell growth as dilution by which the concentrations of all compounds would go down if their amounts remain the same. For a given compounds, dilution by growth can be effectively modeled of every metabolite by a "consuming reaction", with a flux given by $v_{dil} = \mu s$, the compound concentration multiplied by the growth rate. By adding these hypothetical dilution reactions to the metabolic network, we obtain a new stationarity condition $N v = \mu s$ that connects the vectors of fluxes and compound concentrations, and in which the growth rate $\mu$ appears as a parameter. For each choice of the parameter $\mu$, we can ask whether a feasible steady growth state – i.e. a feasible combination of $v$ and $s$ exists. Furthermore, the feasible combinations $(\mu, v, s)$ form a convex set, with possible solutions $(v, s)$ for low values of $\mu$ and no solutions above a critical value $\mu_{max}$, the maximal possible growth rate for our model. Finding this critical value as well as the corresponding optimal fluxes $v$ and compound concentrations $s$ is relatively easy, and can be done by bisection: solving a series of Linear Programming problems (checking for potential solutions $(v, s)$ for different values of $\mu$).

## 10.2. The basic constraints in resource allocation models

As mentioned above, fine-grained resource allocation models build on genome-scale metabolic models (GEMs) to encompass all the reactions that can be employed in a metabolic network. The technical advance, when constructing such models, is to impose sets of additional constraints onto GEMs to couple the metabolic fluxes with investment into metabolic pathways (production of enzymes). To the date, different implementations of this concept were proposed to predict optimal resource allocation in different microorganisms [14].

Although the precise formulations vary, resource allocation models build on three principal types of constraints (Figure 10.1):

1. Mass-conservation constraints
9.2.1. Types of constraints

The general description of these constraints in fact is the same as for small, coarse-grained self-replicator models, only
the number of individual constraints increases. Although the precise formulations vary, resource allocation models build
on three principal types of constraints:

* Mass-conservation constraints

* Flux coupling constraints

* Compartment capacity constraints

Alongside these three major classes there is another set of constraints, which we could call "environment" constraints - these correspond to, e.g. the composition of growth medium, biomass composition at a given growth rate $\mu$, etc. They are implemented by setting target values for amounts and/or fluxes defining a viable cell in a given (or several) environmental conditions, but they are not structural constraints. These constraints usually are added ad hoc and do not need to bear any functional meaning per se.

The *mass-conservation* constraints define the metabolic network (stoichiometry and relation between fluxes). The initial building blocks of these extended models are GEMs, and thus the metabolic network stoichiometry is already there; what remains to be defined are the protein turnover processes. We consider 4 types of protein turnover reactions in fine-grained resource allocation models: protein synthesis, folding, degradation and dilution-by-growth. So, for every protein present in such a model, we add these four reactions: two of them, translation and degradation, include the stoichiometry of amino acids needed for its translation and released during degradation based on the protein sequence. Protein folding and dilution are modeled by the same types of reactions for different proteins (a series of two steps, "unfolded" → "folded" → ∅).

Next, the *flux coupling* constraints couple the metabolic fluxes with protein usage: usually, the usage scales with the catalytic turnover value $k_{\text{cat}}$ of the enzyme. In this step we have to collect the kinetic information (in most cases, $k_{\text{cat}}$ values), which are used as model input. We establish the coupling between fluxes and protein synthesis by setting $v = k_{\text{cat}} e \eta$, where $e$ is the enzyme concentration and $0 < \eta \le 1$ is an efficiency term summarizing the effects of reaction thermodynamics, enzyme saturation, and possibly small-molecule regulation. Coupling constraints are introduced to couple both (i) the metabolic reactions with enzyme usage (as described above) and (ii) protein turnover reactions with the respective macromolecular machinery (e.g. sum demand of ribosomes for protein translation, $v_{\text{translation}} = [\text{Ribosome}] \times k_{\text{cat,ribosome}}$). The sheer number of the kinetic parameters needed for formulating the coupling constraints in the fine-grained models requires the modeller to consider different assumptions and simplifications when building and parametrizing these models, as briefly discussed below.

The number of processes described in a fine-grained manner directly translates to the number of reactions and metabolites in the model. For instance, transcription is modelled explicitly in the ME-models [15]. The modellers’ decision is key here: under assumption that transcription and translation form a linear pathway with fixed scaling factors (i.e. there is a fixed ratio of peptides translated per mRNA transcribed), the flux through mRNA translation reaction can be computed post-optimization based on the flux through the protein translation reaction. Explicit modelling of transcription would require describing processes of mRNA transcription, processing, export from nucleus, and then cytosolic degradation after the mRNA is translated – for each of the transcripts, with precise stoichiometry and a new set of coupling constraints.

The next issue is kinetic parametrization of these fine-grained models. We currently can use only very simplified kinetics in the models (flux coupling $v = k_{\text{cat}} e \eta$), and simplify such factors as enzyme saturation and thermodynamic driving force into a single value of factor $\eta$. Two approaches are used to deal with this, as a large fraction of parameters are not even available. First, condition-dependent kinetic parameters ("apparent catalytic constants", $k_{\text{app}}$) are fitted from experimental (mostly quantitative proteomics) data (setting $k_{\text{eff}} = k_{\text{cat}} \alpha$, where $0 < \alpha \le 1$) with a value $\alpha$ chosen to match predicted enzyme abundance and experimental measurements. Otherwise, for the enzymes with measured $k_{\text{cat}}$ values, we can assume that enzymes work at their maximal rate, i.e. the saturation function $\eta = 1$. Then the model computes the *minimal* protein requirement to sustain the flux through the metabolic reactions. The comparison of
(2) Flux coupling constraints

(3) Compartment capacity, or protein density, constraints

The general description of these constraints in fact is the same as for small, coarse-grained self-replicator models, only the number of individual constraints increases. Moreover, every of the constraints described can be split into a number of constraints, considering only a subset of fluxes in the model (e.g., fluxes taking place in a certain cell compartment).

Alongside these three major types of constraints, there is another set of constraints, which we may call "environment" constraints - these correspond to, e.g. the composition of growth medium, biomass composition at a given growth rate $\mu$, etc. They are implemented by setting target values for amounts and/or fluxes defining a viable cell in a given (or several) environmental conditions, but they are not structural constraints. These constraints usually are added *ad hoc* and do not need to bear any functional meaning *per se*. We will now expand on the three types of constraints used in resource allocation models; note that the description is not exhaustive and peculiarities may vary among different formulations.

### 10.2.1. Steady-state and mass-conservation constraints

*Mass-conservation* constraints define the metabolic network (stoichiometry and relation between fluxes). The initial building blocks of these extended models are GEMs, and thus the metabolic network stoichiometry is already there; what remains to be defined are the protein turnover processes. We consider four types of protein turnover reactions in fine-grained resource allocation models: protein synthesis, folding, degradation and dilution-by-growth. So, for every protein present in such a model, we add these four reactions: two of them, translation and degradation, include the stoichiometry of amino acids needed for its translation and released during degradation based on the protein sequence. The reactions which represent either protein folding modeled as the conversion of the "unfolded" protein species into the "folded" ones, and the dilution-by-growth is modeled as a sink for the "folded" protein species ("folded" $\rightarrow \emptyset$).

### 10.2.2. Catalytic constraints

Next, the flux *coupling* constraints couple the metabolic fluxes with protein usage: usually, the usage scales with the catalytic turnover value $k_{\text{cat}}$ of the enzyme. In this step we have to collect the kinetic information (in most cases, $k_{\text{cat}}$ values), which are used as model parameters. We establish the coupling between fluxes and protein synthesis by setting $v = k_{\text{cat}} e \eta$, where $e$ is the enzyme concentration and $0 < \eta \le 1$ is an efficiency term summarizing the effects of reaction thermodynamics, enzyme saturation, and possibly small-molecule regulation. The value for $\eta$ can be either assumed or fitted from experimental data, and when $\eta = 1$, the enzyme is considered to operate at its maximal rate. Coupling constraints are introduced to couple both (i) the metabolic reactions with enzyme usage (as described above) and (ii) protein turnover reactions with the respective macromolecular machinery (e.g. sum demand of ribosomes for protein translation, $v_{\text{translation}} = [\text{Ribosome}] \times k_{\text{cat,ribosome}}$). The sheer number of the kinetic parameters needed for formulating the coupling constraints in the fine-grained models requires the modeler to consider different assumptions and simplifications when building and parameterizing these models, as briefly discussed below.

The number of processes described in a fine-grained manner directly translates to the number of reactions and metabolites in the model. For instance, transcription is modelled explicitly in the ME-models [12]. The modeler's decision is key here: under assumption that transcription and translation form a linear pathway with fixed scaling factors (i.e. there is a fixed ratio of peptides translated per mRNA transcribed), the flux through mRNA translation reaction can be computed post-optimization based on the flux through the protein translation reaction. Explicit modelling of transcription would require describing processes of mRNA transcription, processing, export from nucleus, and then cytosolic degradation after the mRNA is translated – for each of the transcripts, with precise stoichiometry and a new set of coupling constraints.
minimal predicted vs. observed protein abundance can represent the "apparent saturation", or "overcapacity" of enzymes.
For instance, it is common in yeast *S. cerevisiae* that the flux and not protein expression varies across conditions, and
the relationship between predicted and measured expression can suggest the nature of the observed protein expression
[17].

The final layer of information in the fine-grained resource allocation models is a set of capacity constraints. These con-
straints describe the [upper] limit of cellular process(es), e.g. maximal protein capacity of a compartment. The capacity
constraints are formulated as weighted sums of protein abundance, and usual weighing multipliers are proportional to the
molecular weight of the protein. Usually, the capacity constraints are expressed in terms of mass, area, and volume of
the compartment (e.g. "what is the maximal mass the mitochondrial proteins can take up in gDW of cells?"). Based on
the biological interpretation of the constraints, we formulate the weighing multipliers to represent either of the metrics
(mass/area/volume) that every protein occupies.

The capacity constraints can be both *equality* and *inequality* constraints: more frequent are the latter (usually defining the "upper limit" of, e.g. amount of protein targeted to mitochondria). However, some cell properties should be described through equality constraints: one of these is the protein density of biomass, defining the "target" protein translation per gram dry cell biomass.

Box 9.B: Protein abundance versus concentration in fine-grained resource allocation models

Here we would like to include a relevant note for interpretation of the output of the fine-grained resource allocation
models. Both the classical FBA and these extensions do not consider "metabolite concentration" as a concept:
optimization variables are all *fluxes*. Frameworks discussed in this chapter model protein synthesis from amino acids
and energy equivalents explicitly, with a typical flux dimension of *mmol gDW⁻¹ h⁻¹* (as for any other *fluxes*).
To compute the amount of protein that has to be produced in the steady-state growth, we should consider the
flux balance for the protein *e*: *v*<sub>synthesis,e</sub> = *v*<sub>degradation,e</sub> + *v*<sub>dilution,e</sub>, or, rewritten with the respective parameters,
*v*<sub>synthesis,e</sub> = (*k*<sub>deg,e</sub> + *μ*) *e*. Here, *k*<sub>deg,e</sub> is the degradation rate for the protein *e*, and *μ* is the specific growth
(= dilution-by-growth) rate. The [*e*] in the rewritten equation holds dimension of mmol gDW<sup>-1</sup>, which is protein
*abundance*, rather than *concentration*.

The predicted amount of protein in cells can be compared to experimental measurements in two ways. First option is to convert abundance to concentration using the relationship between the cell volume and dry weight (e.g. *V*<sub>gDW</sub> = 1.7 *mL gDW*<sup>-1</sup> in *Saccharomyces cerevisiae*, [18]). Alternatively, proteome mass fractions are a popular unit in label-free mass spectrometry-based protein quantification, a popular method in quantitative microbiology. Respectively, predicted proteome mass fractions can be inferred by converting protein abundance in *mmol* to *g*, and scaling to the protein content in dry cell biomass. Here, it is important to consider the conversion factors (protein content in dry biomass). Fast-growing bacteria, e.g. *E. coli*, maintains rather constant protein content in dry weight across growth rates (ca. 0.55 (*g* protein) *gDW*<sup>-1</sup>) [19, 20]. On the contrary, the protein content is known to vary in *S. cerevisiae* as a function of growth rate [18].

9.2.2. Interpretation and outcomes of the additional constraints

For cells growing in exponential phase at a growth rate $\mu$, the constraints listed above formalize different structural
constraints:

(I) the metabolic network has to produce all metabolic precursors necessary for biomass production and mass conservation must hold for all intracellular molecule species - i.e. intracellular metabolites and molecular machines.

(II) the capacity of each type of molecular machine must be sufficient to ensure its function, i.e. to catalyze chemical conversions at a sufficient rate;

(III) the intracellular density of compartments and the occupancy of membranes must not exceed the defined limits.

Metabolism and protein production depend on each other in both ways!
<figure><img src="image_4.png" /></figure>

<figure><img src="image_5.png" /><figcaption>Figure 10.1: Overview of biological components and mathematical constraints in large-scale resource allocation models – A Resource Balance Analysis (RBA) model is shown as an example. (A) Typically, an RBA model describes metabolisms and macromolecule production in a growing cell (yellow blocks). Precursors from metabolism are needed to produce macromolecules, and some macromolecules serve as enzymes to catalyze metabolic reactions. In addition, macromolecules are diluted and are localized in cell compartments. (B) Sets of mathematical constraints. The variables and processes described by an RBA model must satisfy a number of constraints, include mass-balance constraints (between production, degradation, and dilution of compounds); capacity constraints (relating process velocities to the concentrations of catalysts); density constraints (on the total amount of compounds in a cell compartment); and possibly empirical physiological constraints on any types of "target variables", to ensure realistic models.</figcaption></figure>

The next issue is kinetic parametrization of these fine-grained models. We currently can use only very simplified kinetics in the models (flux coupling $v = k_{\text{cat}} e \eta$), and simplify such factors as enzyme saturation and thermodynamic driving force into a single value of factor $\eta$. Two approaches are used to deal with this, as a large fraction of parameters are not even available. First, condition-dependent kinetic parameters ("apparent catalytic constants", $k_{\text{app}}$) are fitted from experimental (mostly quantitative proteomics) data (setting $k_{\text{eff}} = k_{\text{cat}} \alpha$, where $0 < \alpha \le 1$) with a value $\alpha$ chosen to match predicted enzyme abundance and experimental measurements. Otherwise, for the enzymes with measured $k_{\text{cat}}$ values, we can assume that enzymes work at their maximal rate, i.e. the saturation function $\eta = 1$. Then the model computes the minimal protein requirement to sustain the flux through the metabolic reactions. The comparison of minimal predicted vs. observed protein abundance can represent the "apparent saturation", or "overcapacity" of enzymes. For instance, it is common in yeast S. cerevisiae that the flux and not protein expression varies across conditions, and the relationship between predicted and measured expression can suggest the nature of the observed protein expression [15].
<figure><img src="image_5.png" /><figcaption>Figure 9.1: Resource balance analysis model: overview of biological components and mathematical constraints. (a) Typically, an RBA model describes metabolisms and macromolecule production in a growing cell (yellow blocks). Precursors from metabolism are needed to produce macromolecules, and some macromolecules serve as enzymes to catalyze metabolic reactions. In addition, macromolecules are diluted and are localized in cell compartments. (b) Mathematical constraints. The variables and processes described by an RBA model must satisfy a number of constraints, include mass-balance constraints (between production, degradation, and dilution of compounds); capacity constraints (relating process velocities to the concentrations of catalysts); density constraints (on the total amount of compounds in a cell compartment); and possibly empirical physiological constraints on any types of "target variables", to ensure realistic models.</figcaption></figure>

* Possible states are defined by constraints:

(i) stationarity, including dilution

(ii) kinetics/capacity constraints

(iii) density constraints

(iv) other physiological bounds ("targets")

* Select an optimal state by an assumed objective (e.g. maximal growth; protein production; in chemostat, growth at a given growth rate and at minimal external substrate level)

* A change in one cell component may lead to small perturbations or global rearrangements

* Relaxing one constraint can lead to changes in the entire system!

## 9.3. Resource balance analysis

Resource Balance Analysis (RBA) has been developed as (and is considered) a flexible and generic modeling framework which describes the functioning of an organism using the most relevant set of linear equality and inequality constraints. Importantly, RBA does not refer to a *specific* unique set of constraints, but it provides the framework in which such constraints can be taken into account. Here we distinguish the general RBA framework from a specific type of RBA model, i.e. the model of a specific organism, which contains by definition a set of specific constraints. The general structure of such models, as well as the typical constraints, are shown in Figure 9.1.
Box 10.A Protein abundance versus concentration in resource allocation models

Here we would like to include a relevant note for interpretation of the output of the fine-grained resource allocation models. Both the classical FBA and these extensions do not consider "metabolite concentration" as a concept: optimization variables are all *fluxes*. Frameworks discussed in this chapter model protein synthesis from amino acids and energy equivalents explicitly, with a typical flux dimension of *mmol gDW⁻¹ h⁻¹* (as for any other fluxes). To compute the amount of protein that has to be produced in the steady-state growth, we should consider the flux balance for the protein *e*: *v*<sub>synthesis,e</sub> = *v*<sub>degradation,e</sub> + *v*<sub>dilution,e</sub>, or, rewritten with the respective parameters, *v*<sub>synthesis,e</sub> = (*v*<sub>deg,e</sub> + μ) *e*. Here, *k*<sub>deg,e</sub> is the degradation rate for the protein *e*, and μ is the specific growth (= dilution-by-growth) rate. The [*e*] in the rewritten equation holds dimension of mmol gDW⁻¹, which is protein *abundance*, rather than *concentration*.

The predicted amount of protein in cells can be compared to experimental measurements in two ways. First option is to convert abundance to concentration using the relationship between the cell volume and dry weight (e.g. V<sub>gDW</sub> = 1.7 mL gDW<sup>-1</sup> in *Saccharomyces cerevisiae*, [16]). Alternatively, proteome mass fractions are a popular unit in label-free mass spectrometry-based protein quantification, a popular method in quantitative microbiology. Respectively, predicted proteome mass fractions can be inferred by converting protein abundance in *mmol* to *g*, and scaling to the protein content in dry cell biomass. Here, it is important to consider the conversion factors (protein content in dry biomass). *E. coli* maintains rather constant protein content in dry weight across growth rates (ca. 0.55 (*g* protein) *gDW*<sup>-1</sup>) [17, 18]. On the contrary, the protein content is known to vary in *S. cerevisiae* as a function of growth rate [16].

10.2.3. Protein density constraints

The final layer of information in our resource allocation models is a set of protein density constraints in each cell.
These constraints put an upper limit on the amounts machines driving the cellular processes, e.g. a maximal protein
capacity of a compartment. These constraints are formulated as weighted sums of protein abundance, usually with
weights proportional to the proteins' molecular weight. Usually, the density constraints are expressed in terms of
(usually maximal) mass, area, and volume of the compartment (e.g. "what is the maximal mass the mitochondrial
proteins can take up in gDW of cells?"). Based on the biological interpretation of the constraints, we formulate the
weighing multipliers to represent either of the metrics (mass/area/volume) that every protein occupies.

The capacity constraints can be both *equality* and *inequality* constraints: more frequent are the latter (usually
defining the "upper limit" of, e.g. amount of protein targeted to mitochondria). However, some cell properties should
be described through equality constraints: one of these is the protein density of biomass, defining the "target" protein
translation per gram dry cell biomass.

10.2.4. Interpreting the consequences of the additional constraints

We have briefly discussed what types of additional constraints need to be implemented to extend FBA models to account for cellular resource allocation, and now let us recap on what these sets of rules mean in biological terms. The constraints described above shall couple the metabolic fluxes with the production of enzymes that operate these functions, so the model has to produce amino acids and generate ATP in order to use them for protein translation. Moreover, the enzyme demand will be coupled with the production of the macromolecular machines required to produce, fold, and degrade these enzymes (ribosomes, chaperones, and proteases, respectively), requiring the same building blocks (see Chapter 2 in [1]). These constraints therefore formalize a self-replicating molecular system in balanced growth subject to different structural constraints:

1. The metabolic network has to produce all metabolic precursors necessary for biomass production and mass conservation must hold for all intracellular molecule species - i.e. intracellular metabolites and molecular machines.

2. The capacity of each type of molecular machine must be sufficient to ensure its function, i.e. to catalyze chemical conversions at a sufficient rate;
### 9.3.1. Mathematical description

**Notation.** Below $A^T$ refers to the transpose of the matrix $A$. $\mathbb{R}_{>0}^n \triangleq \{x \in \mathbb{R}^n | x_i > 0 \text{ for all } i \in \{1, \dots, n\}\}$, $\mathbb{R}_{>0} \triangleq \mathbb{R}_{>0}^1$, $\mathbb{R}_{\ge 0}^n \triangleq \{x \in \mathbb{R}^n | x_i \ge 0 \text{ for all } i \in \{1, \dots, n\}\}$ and $\mathbb{R}_{\ge 0} \triangleq \mathbb{R}_{\ge 0}^1$.

In a standard RBA model, we consider balanced growth, that is, the average state of a cell in a cell bacterial population growing exponentially at the specific (constant) growth rate $\mu \ge 0$, i.e. the amount of produced biomass per biomass per cell per unit of time. Our simulated average cell is composed of different molecule species:

(i) $n_y$ types of molecular machines, which can be subdivided further into $n_e$ enzymes and transporters involved in the metabolic network $E \triangleq (E_1, \dots, E_{n_e})$ at the concentrations $e \triangleq (e_1, \dots, e_{n_e})^T$ and metabolic fluxes $\nu \triangleq (\nu_1, \dots, \nu_{n_e})^T$; and $n_m$ macromolecular machines $M \triangleq (M_1, \dots, M_{n_m})$ involved in non-metabolic cellular processes, such as the translation apparatus, at the concentrations $m \triangleq (m_1, \dots, m_{n_m})^T$;

(ii) $n_p$ proteins $P \triangleq \{P_1, \dots, P_{n_p}\}$ belonging to unspecified cellular processes. $p \triangleq (p_1, \dots, p_{n_p})^T$ denotes the set of concentrations of $P$;

(iii) $n_s$ intracellular and mass-balanced metabolites $S \triangleq (S_1, \dots, S_{n_s})$. Within the set $S$, we distinguish a subset $B \triangleq (B_1, \dots, B_{n_b})$ of abundant metabolites which have fixed growth-independent concentrations $\bar{b} \triangleq (\bar{b}_1, \dots, \bar{b}_{n_b})^T$ (and usually coincide with biomass macro-components such as DNA, cell wall or plasmic membrane). We also consider a set of extracellular metabolites $S_{\text{ext}} \triangleq (S_{\text{ext},1}, \dots, S_{\text{ext},n_{\text{ext}}})$ of concentrations $s_{\text{ext}} \triangleq (s_{\text{ext},1}, \dots, s_{\text{ext},n_{\text{ext}}})^T$ that are not mass-balanced.

Finally, let us introduce the vector $\mathbf{y}^T \triangleq (\mathbf{e}^T, \mathbf{m}^T)$ of concentrations of molecular machines of size $n_y$. Typical units of concentrations $e$, $m$ and $p$ are in millimoles per gram of cell dry weight, and fluxes $\nu$ in millimoles per gram of cell dry weight per unit of time.

For a given cell growth rate $\mu \ge 0$, the RBA optimization problem (named $\mathcal{P}_{\text{rba}}(\mu)$) can be formalized mathematically as follows.

For a fixed vector of concentrations $\mathbf{p} \in \mathbb{R}_{>0}^{N n_p}$ and the given growth rate $\mu \ge 0$,

$$
\begin{array}{ll}
\text{find possible cell states} & \mathbf{y} \in \mathbb{R}_{\ge 0}^{n_y}, \nu \in \mathbb{R}^{n_e}, \\
\text{subject to} & \\
(C_1) & -\Omega \nu + \mu(\mathbf{C}_Y^S \mathbf{y} + \mathbf{C}_B^S \bar{\mathbf{b}} + \mathbf{C}_P^S \mathbf{p}) = 0 \\
(C_{2a}) & \mu(\mathbf{C}_Y^M \mathbf{y} + \mathbf{C}_P^M \mathbf{p}) - \mathbf{K}_T \mathbf{y} \le 0 \\
(C_{2b}) & -\mathbf{K}_E' \mathbf{y} \le \nu \le \mathbf{K}_E \mathbf{y} \\
(C_3) & \mathbf{C}_Y^D \mathbf{y} + \mathbf{C}_P^D \mathbf{p} - \bar{\mathbf{d}} \le 0
\end{array}
$$

where all the inequalities are defined componentwise and:

* $\Omega$ is the stoichiometry matrix of the metabolic network of size $n_s \times n_e$, where $\Omega_{ij}$ corresponds to the stoichiometry of metabolite $S_i$ in the j-th enzymatic reaction;

* $\mathbf{C}_Y^S$ (resp. $\mathbf{C}_P^S$) is an $n_s$ $n_y$ (resp. $n_s$ $n_p$) matrix where each coefficient $\mathbf{C}_{Y_{ij}}^S$ corresponds to the number of metabolite $S_i$ consumed (or produced) for the synthesis of one machine $Y_j$ (resp. $P_j$); $\mathbf{C}_{Y_{ij}}^S$ is then positive, negative or null if $S_i$ is produced, consumed or not involved in the synthesis of one machine $Y_j$ (resp. $P_j$);

* $\mathbf{C}_B^S$ is an $n_s \times n_b$ matrix in which each coefficient $\mathbf{C}_{B_{ij}}^S$ corresponds to a metabolite $S_i$ consumed (or produced) for the synthesis of one $B_j$;

* $\mathbf{K}_T$ ($\mathbf{K}_E$ and $\mathbf{K}_E'$, respectively) are matrices of size $n_m \times n_y$ ($n_e \times n_y$, respectively) in which each coefficient $k_{T_i}$ ($k_{E_i}$ and $k_{E_i'}$, respectively) is positive and corresponds to the efficiency of molecular machine $M_i$, i.e. the rate of the process per amount of the catalyzing molecular machine, (the efficiency of the enzyme $E_i$ in forward and
3. The intracellular density of compartments and the occupancy of membranes must not exceed the defined limits.

As highlighted before, the biological interpretation of the additional constraints discussed above is rather universal for different implementations of resource allocation models, with minor deviations in terminology and/or formulation. To illustrate how resource allocation models are built from conventional GEMs, and how the respective models are formalized in mathematical terms, in the following we will consider one of the popular formulations of resource allocation models in more depth.

## 10.3. Resource Balance Analysis: model construction and simulation

Resource Balance Analysis (RBA) is a flexible and generic modeling framework that describes the functioning of an organism using linear equality and inequality constraints, as described in general terms in Section 10.2. As a consequence, an RBA model includes all known metabolic reactions coupled to relevant cell processes with major protein investments (production of biomass precursors; including, but not limited to protein translation, protein folding, protein transmembrane transport, and protein degradation). Where applicable, circumstantial information can be included into the model to establish the dependency of enzyme activity on metal ions, vitamins, and/or cofactors. Which metabolic reactions and cell processes are regarded as relevant may vary between organisms and is a modeler's choice.

### 10.3.1. Building a draft RBA model

The software package RBApy [19] contains all the routines needed to build and simulate RBA models. In order to build a new RBA model, it takes as an input a genome-scale metabolic network in SBML format [20], together with additional information to formulate all the constraints described above. Different types of biological data are needed to build an RBA model for a given type of cell:

* Amino acid sequences for metabolic enzymes and macromolecular machines (e.g. ribosomes and chaperones),

* If applicable, stoichiometry of known cofactors (e.g. metal ions),

* Efficiencies of metabolic enzymes,

* Molecular weights and localization of proteins (for density constraints),

* Any empirical constraints on concentrations or fluxes ("targets", see previous section).

The software first extends the input GEM with a description of protein production and dilution in the cell. To do so, it extracts information from the input files on (i) protein sequences and cofactors, (ii) the subunit stoichiometry of protein complexes, and (iii) protein localization (using information from public databases such as UniProt). Using this information, reactions corresponding to protein synthesis, folding, degradation, and dilution by growth are added automatically. Finally, the software maps enzymes to the reactions they catalyze and to the proteins they consist of. The output of the routine is a draft (uncalibrated) RBA model.

### 10.3.2. Mathematical description of a RBA problem

**Notation.** Below $A^T$ refers to the transpose of the matrix $A$. $\mathbb{R}_{>0}^n \triangleq \{x \in \mathbb{R}^n | x_i > 0 \text{ for all } i \in \{1, \dots, n\}\}$, $\mathbb{R}_{>0} \triangleq \mathbb{R}_{>0}^1$, $\mathbb{R}_{\ge 0}^n \triangleq \{x \in \mathbb{R}^n | x_i \ge 0 \text{ for all } i \in \{1, \dots, n\}\}$ and $\mathbb{R}_{\ge 0} \triangleq \mathbb{R}_{\ge 0}^1$.

In a standard RBA model, we consider balanced growth (see Chapter ?? in [1]), that is, the average state of a cell in a cell bacterial population growing exponentially at the specific (constant) growth rate $\mu \ge 0$, i.e. the amount of produced biomass per biomass per cell per unit of time. Our simulated average cell is composed of different molecule species:

1. $n_y$ types of molecular machines, which can be subdivided further into $n_e$ enzymes and transporters involved
backward sense, respectively);

* $C_Y^M$ (resp. $C_P^M$) is an $n_m \times n_y$ (resp. $n_m \times n_p$) matrix in which each coefficient $C_{Yij}^M$ typically corresponds to the length in amino acids of the machine $Y_j$ (resp. $P_j$). In some cases (for instance for the constraints on protein chaperoning), the length in amino acids can be multiplied by a coefficient, such as the fraction of the whole proteome that necessitates chaperoning;

* $\bar{d}$ is a vector of size $n_c$, where $n_c$ is the number of compartments (compartment membrane and/or compartment interior for which density constraints are considered. $\bar{d}^i$ is the density of molecular entities within the volume or surface area. Densities are typically expressed as a number of amino-acid residues by volume or surface area.

* $C_Y^D$ (resp. $C_P^D$) is an $n_c \times N_y$ (resp. $n_c \times n_p$) matrix in which each coefficient $C_{Yij}^D$ corresponds to the density of one machine $Y_j$ (resp. $P_j$) in the compartment $i$. By construction, we have one unique localization per machine.

At given $\mu$, $P_{rba}(\mu)$ is a feasibility optimization problem, where constraints ($C_1-C_3$) define the feasibility domain. The feasibility domain can be empty or non-empty. If there exists a solution $(\mathbf{y}, \nu)$ to $P_{rba}(\mu)$ -i.e. the feasibility domain is non-empty-, then there exists a feasible resource distribution compatible with the given growth rate. In other words, the cell can grow at this growth rate value.

**Remarks:**

1. In practice, the vector $\bar{b}$ contains non-zero values only for the concentrations of macro-components such as DNA, cell wall, and lipid membranes, and for a few set of metabolites. These values are usually extracted from the biomass formation reaction used in FBA models (see Chapter 5 in [6]).

2. To model reversible enzymes, we introduced two diagonal matrices containing the enzyme efficiencies, i.e. $K_E$ and $K'_E$, describing the capacity constraints of enzymes in both directions. If an enzyme $E_i$ is considered irreversible, $k'_{E_i}$ is set to 0.

3. In [9, 21], an RBA model was built for *Bacillus subtilis*. It integrates two macromolecular processes in constraint $C_{2a}$, the translation and chaperoning of proteins, and two density constraints, the limitation of the cytosolic density and of the membrane occupancy. An RBA model can be refined by integrating for instance other cellular processes and molecular machines, such as the transcription machinery, the protein secretion apparatus (see [21, 22]), or molecule turnover [23], as well as other types of constraints.

**How to incorporate the medium composition.** We represent the medium composition by extracellular concentrations, which determine the efficiencies of metabolic transporters via Michaelis-Menten like rate laws (as nonlinear $k(c)$ functions; see section 9.1.2). For an extracellular nutrient $S_{ext,i}$ with concentration $s_{ext,i} \ge 0$, the efficiency of the corresponding metabolic transporter(s) is given by $k_E(s_{ext,i}) = \frac{k_{cat}s_{ext,i}}{K_m+s_{ext,i}}$, with parameters $k_{cat}$ and $K_m$ for the turnover number and the affinity of the transporter, respectively.

**Convexity.** For given growth rate and medium composition, all equalities and inequalities in our RBA problem $P_{rba}(\mu)$ is linear in the decision variables $(\mathbf{y}, \nu)$ and is proven to be convex [8, 10]. By construction, the feasibility domain of $P_{rba}(\mu)$ corresponds to the set of all possible phenotypes of the cell at a growth rate $\mu \ge 0$.

**Parsimonious resource allocation.** The principle of parsimonious resource allocation between cellular processes coincides with principles of optimal performance, for example, that a cell phenotype should maximize growth rate. Importantly, for an RBA problem with given parameters, there exists a maximal growth rate $\mu^* \ge 0$, such that for any $\mu$, $P_{rba}(\mu)$ is feasible if and only if $\mu \le \mu^*$ [8, 10]. For a given medium composition, the maximal growth rate $\mu^*$ can be computed by using a bisection algorithm, in which a series of LP problems are solved to narrow down the exact growth rate at which the problem becomes infeasible. Together with the maximal feasible growth rate one obtains the optimal cell configuration maximizing growth $(\mu^*, \mathbf{y}^*, \nu^*)$.

**Exploration of the feasibility domain.** It was proven that the feasibility domain contracts with increasing growth rate [8, 10]. In the same vein as Flux Variability Analysis ([24], see Chapter 5 in [6]), the feasibility domain can be explored at optimal $(\mu^*)$ or sub-optimal $(\mu \le \mu^*)$ growth rates. For one decision variable $y_i$ (resp. $\nu_i$), two LP problems are solved, where $(i)$ constraints $C_1, C_2$ and $C_3$ remain unchanged; $(ii)$ the decision variable $y_i$ (resp. $\nu_i$) is maximized (LP 1) and minimized (LP 2). This operation is repeated for each decision variable to obtain *in fine* the feasibility domain of all
in the metabolic network $\mathbb{E} \triangleq (E_1, \dots, E_{n_e})$ at the concentrations $\mathbf{e} \triangleq (e_1, \dots, e_{n_e})^T$ and metabolic fluxes $\mathbf{v} \triangleq (v_1, \dots, v_{n_e})^T$; and $n_m$ macromolecular machines $\mathbb{M} \triangleq (M_1, \dots, M_{n_m})$ involved in non-metabolic cellular processes, such as the translation apparatus, at the concentrations $\mathbf{m} \triangleq (m_1, \dots, m_{n_m})^T$;

2. $n_p$ proteins $\mathbb{P} \triangleq \{P_1, \dots, P_{n_p}\}$ belonging to unspecified cellular processes. $\mathbf{p} \triangleq (p_1, \dots, p_{n_p})^T$ denotes the set of concentrations of $\mathbb{P}$;

3. $n_s$ intracellular and mass-balanced metabolites $\mathbb{S} \triangleq (S_1, \dots, S_{n_s})$. Within the set $\mathbb{S}$, we distinguish a subset $\mathbb{B} \triangleq (B_1, \dots, B_{n_b})$ of abundant metabolites which have fixed growth-independent concentrations $\bar{\mathbf{b}} \triangleq (\bar{b}_1, \dots, \bar{b}_{n_b})^T$ (and usually coincide with biomass macro-components such as DNA, cell wall or plasmic membrane). We also consider a set of extracellular metabolites $\mathbb{S}_{\text{ext}} \triangleq (S_{\text{ext},1}, \dots, S_{\text{ext},n_{\text{ext}}})$ of concentrations $\mathbf{s}_{\text{ext}} \triangleq (s_{\text{ext},1}, \dots, s_{\text{ext},n_{\text{ext}}})^T$ that are not mass-balanced.

Finally, let us introduce the vector $\mathbf{y}^T \triangleq (\mathbf{e}^T, \mathbf{m}^T)$ of concentrations of molecular machines of size $n_y$. Typical units of concentrations $\mathbf{e}$, $\mathbf{m}$ and $\mathbf{p}$ are in millimoles per gram of cell dry weight, and fluxes $\mathbf{v}$ in millimoles per gram of cell dry weight per unit of time.

For a given cell growth rate $\mu \ge 0$, the RBA optimization problem (named $\mathcal{P}_{\text{rba}}(\mu)$) can be formalized mathematically as follows. For a fixed vector of concentrations $\mathbf{p} \in \mathbb{R}_{>0}^{N n_p}$ and the given growth rate $\mu \ge 0$,

$$
\begin{array}{ll}
\text{find possible cell states} & \mathbf{y} \in \mathbb{R}_{\ge 0}^{n_y}, \mathbf{v} \in \mathbb{R}^{n_e}, \\
\text{subject to} & \\
(C_1) & -\Omega \mathbf{v} + \mu(\mathbf{C}_Y^S \mathbf{y} + \mathbf{C}_B^S \bar{\mathbf{b}} + \mathbf{C}_P^S \mathbf{p}) = 0 \\
(C_{2a}) & \mu(\mathbf{C}_Y^M \mathbf{y} + \mathbf{C}_P^M \mathbf{p}) - \mathbf{K}_T \mathbf{y} \le 0 \\
(C_{2b}) & -\mathbf{K}'_E \mathbf{y} \le \mathbf{v} \le \mathbf{K}_E \mathbf{y} \\
(C_3) & \mathbf{C}_Y^D \mathbf{y} + \mathbf{C}_P^D \mathbf{p} - \bar{\mathbf{d}} \le 0
\end{array}
$$

where all the inequalities are defined component-wise and:

○ $\Omega$ is the stoichiometry matrix of the metabolic network of size $n_s \times n_e$, where $\Omega_{ij}$ corresponds to the stoichiometry of metabolite $S_i$ in the j-th enzymatic reaction;

○ $\mathbf{C}_Y^S$ (resp. $\mathbf{C}_P^S$) is an $n_s$ $n_y$ (resp. $n_s$ $n_p$) matrix where each coefficient $\mathbf{C}_{Y_{ij}}^S$ corresponds to the number of metabolite $S_i$ consumed (or produced) for the synthesis of one machine $Y_j$ (resp. $P_j$); $\mathbf{C}_{Y_{ij}}^S$ is then positive, negative or null if $S_i$ is produced, consumed or not involved in the synthesis of one machine $Y_j$ (resp. $P_j$);

○ $\mathbf{C}_B^S$ is an $n_s \times n_b$ matrix in which each coefficient $\mathbf{C}_{B_{ij}}^S$ corresponds to a metabolite $S_i$ consumed (or produced) for the synthesis of one $B_j$;

○ $\mathbf{K}_T$ ($\mathbf{K}_E$ and $\mathbf{K}'_E$, respectively) are matrices of size $n_m \times n_y$ ($n_e \times n_y$, respectively) in which each coefficient $k_{T_i}$ ($k_{E_i}$ and $k'_{E_i}$, respectively) is positive and corresponds to the efficiency of molecular machine $M_i$, i.e. the rate of the process per amount of the catalyzing molecular machine, (the efficiency of the enzyme $E_i$ in forward and backward sense, respectively);

○ $\mathbf{C}_Y^M$ (resp. $\mathbf{C}_P^M$) is an $n_m \times n_y$ (resp. $n_m \times n_p$) matrix in which each coefficient $\mathbf{C}_{Y_{ij}}^M$ typically corresponds to the length in amino acids of the machine $Y_j$ (resp. $P_j$). In some cases (for instance for the constraints on protein chaperoning), the length in amino acids can be multiplied by a coefficient, such as the fraction of the whole proteome that necessitates chaperoning;

○ $\bar{\mathbf{d}}$ is a vector of size $n_c$, where $n_c$ is the number of compartments (compartment membrane and/or compartment interior for which density constraints are considered. $\bar{\mathbf{d}}^i$ is the density of molecular entities within the volume or surface area. Densities are typically expressed as a number of amino-acid residues by volume or surface area.

○ $\mathbf{C}_Y^D$ (resp. $\mathbf{C}_P^D$) is an $n_c \times N_y$ (resp. $n_c \times n_p$) matrix in which each coefficient $\mathbf{C}_{Y_{ij}}^D$ corresponds to the density of one machine $Y_j$ (resp. $P_j$) in the compartment $i$. By construction, we have one unique localization per machine.
decision variables. In practice, at the optimum, the cell configuration $(\mu^*, y^*, \nu^*)$ is often unique. Indeed, non-unique solutions will exist if two alternative metabolic pathways have exactly the same cost in resources. Since all enzymes have different amino acid sequences, use different cofactors, are differently localized, etc, this is highly unlikely. A caricatural example of a model with non-unique solutions would be one in which an enzyme pool is arbitrarily split into two, and the two new "enzyme species" are given different names, although they are physically exactly the same.

### 9.3.2. How to build and calibrate RBA models?

For a given organism, an RBA model can include all known metabolic reactions and all (potentially relevant) cell processes with major protein investments (including in particular metabolism, transport, ribosomes, chaperones). Which metabolic reactions and cell processes are regarded as relevant may vary between organisms and is a modeling choice. RBA integrates the description of the stoichiometry of the metabolic network, the composition of some macromolecular components as DNA, mRNA, lipids, cell wall components, but also a detailed description of molecular machines including the precise protein and nucleic sequences, their dependency on metallic ions, vitamins, cofactors for activity, and their cellular localization.

**Requirement for RBA model creation.** Several pieces of information, that is, several types of biological data, are needed to build an RBA model for an organism:

* Metabolic network and reaction stoichiometry

* Mapping from reactions to enzymes and further to proteins

* Protein amino acid sequences and cofactors (e.g. metal ions)

* Efficiencies of metabolic enzymes,

* Types of cellular machines (e.g. ribosomes and chaperones), their efficiencies and cofactor demands

* Molecule sizes and localisations (for density constraints)

* Other constraints on concentrations or fluxes ("targets") defining a viable cell

In practice, RBA models can be built and simulated using a software called RBApy [22]. It takes as input (i) a genome-scale metabolic network in SBML format [25] and having a high-quality mapping between genes and reactions; (ii) the description of molecular machines involved in macromolecular processes (e.g ribosomes); (iii) the NCBI taxon of the organism of interest. Then it automatically extracts information on protein sequences and cofactors, subunit structures, localization from public repositories such as Uniprot, maps enzymes to the reactions they catalyze and to the proteins they consist of, and builds a draft uncalibrated RBA model.

**Calibration of RBA model parameters.** An RBA model may contain a high number of model parameters. How can we obtain them? Remember that the main parameters to be estimated are the abundance of housekeeping proteins (p), densities (d̄) and efficiencies of molecular machines ($K_E, K'_E, K_T)^1$. As we learned in Chapter 3 in [6], the rate of an enzymatic reaction, $v = e k_{app}$, depends on the enzyme's efficiency or "apparent catalytic rate", given by $k_{app} = f(c) = k_{cat}^+ \cdot \eta^{\text{rev}}(c) \cdot \eta^{\text{sat}}(c) < k_{cat}^+$. The $k_{app}$ values are always below the $k_{cat}$ value, but may vary from state to state depending on metabolite concentrations. Since internal metabolite concentrations c are unknown and difficult to measure at genome-scale, we cannot estimate $k_{app}$ from the explicit kinetic law $f(c)$. We need to obtain these $k_{app}$ parameters empirically, for example by measuring the flux v and the protein abundance e in one condition and taking their ratio. Hence, for a given environmental condition, RBA parameters can be estimated using quantitative proteomics [26] for p and d̄, and a combination of quantitative proteomics and fluxomics (if available) as in [21] or FBA to estimate the flux distribution as in [22] for efficiency parameters. To account for variable enzyme efficiencies, one may make the simplifying assumption that enzyme efficiencies depend mostly on growth rate. By estimating the enzyme efficiencies at different growth rates and interpolating between them, one obtains empirical relationships between them and the growth rate [21] that can be further used in $P_{rba}(\mu)$. For instance, several estimates of enzymatic efficiencies obtained in several contrasted growth conditions will provide a relationship such as $K_E(\mu)$ instead of a constant $K_E$ value.

<footnote n="1">¹d̄ is deduced from the biomass reaction of the genome-scale metabolic network of the organism</footnote>
For given growth rate and medium composition, all equalities and inequalities in our RBA problem $P_{rba}(\mu)$ is linear in the decision variables $(\mathbf{y}, \mathbf{v})$ and is proven to be convex [5, 7]. At given $\mu$, $P_{rba}(\mu)$ is a feasibility optimization problem, where constraints $(C_1-C_3)$ define the feasibility domain. The feasibility domain can be empty or non-empty. If there exists a solution $(\mathbf{y}, \mathbf{v})$ to $P_{rba}(\mu)$ -i.e. the feasibility domain is non-empty-, then there exists a feasible resource distribution compatible with the given growth rate. In other words, the cell can grow at this growth rate value. By construction, the feasibility domain of $P_{rba}(\mu)$ corresponds to the set of all possible phenotypes of the cell at a growth rate $\mu \ge 0$.

We conclude this with some remarks:

1. In practice, the vector $\bar{\mathbf{b}}$ contains non-zero values only for the concentrations of macro-components such as DNA, cell wall, and lipid membranes, and for a few set of metabolites. These values are usually extracted from the biomass formation reaction used in FBA models (see Chapter 5 in [1]).

2. To model reversible enzymes, we introduced two diagonal matrices containing the enzyme efficiencies, i.e. $K_E$ and $K'_E$, describing the capacity constraints of enzymes in both directions. If an enzyme $E_i$ is considered irreversible, $k'_{E_i}$ is set to 0.

3. In [6, 21], an RBA model was built for *Bacillus subtilis*. It integrates two macromolecular processes in constraint $C_{2a}$, the translation and chaperoning of proteins, and two density constraints, the limitation of the cytosolic density and of the membrane occupancy. An RBA model can be refined by integrating for instance other cellular processes and molecular machines, such as the transcription machinery, the protein secretion apparatus (see [21, 19]), or molecule turnover [22], as well as other types of constraints.

### 10.3.3. Simulation and analysis of RBA models

**How to incorporate the medium composition.** We represent the medium composition in two aspects, namely (i) qualitatively, by allowing exchange of the medium metabolites in the model ($UB_{Exchange,n} > 0$). Note that some metabolites, although not explicitly represented by the growth media, should also adhere to this rule (e.g. oxygen, water, and protons). The (ii) quantitative composition of the growth medium is determined by extracellular concentrations, which, in turn, dictate the efficiencies of metabolic transporters via Michaelis-Menten-like rate laws (as nonlinear $k(c)$ functions; see section 10.1.3). For an extracellular nutrient $S_{ext,i}$ with concentration $s_{ext,i} \ge 0$, the efficiency of the corresponding metabolic transporter(s) is given by $k_E(s_{ext,i}) = \frac{k_{cat}s_{ext,i}}{K_M+s_{ext,i}}$, with parameters $k_{cat}$ and $K_M$ for the turnover number and the affinity of the transporter, respectively.

**Obtaining the RBA solution for a given parameter set.** For an RBA problem with given parameters, there exists a maximal growth rate $\mu^* \ge 0$, such that for any $\mu$, $P_{rba}(\mu)$ is feasible if and only if $\mu \le \mu^*$ [5, 7]. For a given medium composition, the maximal growth rate $\mu^*$ can be computed by using a bisection algorithm, in which a series of LP problems are solved to narrow down the exact growth rate at which the problem becomes infeasible. A real-life example would be simulating growth in glucose-limited chemostat cultures under different dilution rates $D$. With increasing $D$, the glucose availability increases, and a set of $n$ different glucose uptake rates $q_{Glc}(q_{Glc,1}, q_{Glc,2}, \dots, q_{Glc,n})$ can be subjected to an RBA model to obtain a set of optimal metabolic states $(\mu_1^*, \mu_2^*, \dots, \mu_n^*)$.

Together with the maximal feasible growth rate one obtains the optimal cell configuration maximizing growth $(\mu^*, \mathbf{y}^*, \mathbf{v}^*)$. The principle of optimal performance, in this case, that a cell phenotype should maximize growth rate, in fact, coincides with the principle of parsimonious resource allocation between cellular processes.

**Exploration of the feasibility domain.** Although RBA models inherently reduce the solution space due to principle of parsimonious resource allocation, the solutions obtained might still contain considerable flux variability. In the same vein as Flux Variability Analysis ([23], see Chapter 5 in [1]), the feasibility domain can be explored at optimal $(\mu^*)$ or sub-optimal $(\mu \le \mu^*)$ growth rates. For one decision variable $y_i$ (resp. $v_i$), two LP problems are solved, where (i) constraints $C_1, C_2$ and $C_3$ remain unchanged; (ii) the decision variable $y_i$ (resp. $v_i$) is maximized (LP 1) and minimized (LP 2). This operation is repeated for each decision variable to obtain *in fine* the feasibility domain
### 9.3.3. Examples of RBA applications

Despite its assumptions and simplifications, RBA is able to predict realistic complex adaptations (e.g. some proteins contain iron; under iron starvation, the cell may *increase* the import of iron, but also *avoid* using proteins that contain iron, and the pathways in which they operate), even in complex growth conditions [21, 27]. These effects cannot be captured by standard CBM methods in an autonomous way, i.e. without the addition of specific constraints on fluxes. Thus altogether, RBA provides a reasonable compromise between numerical tractability, cell phenotype prediction accuracy and model complexity.

## 9.4. Prediction of variable biomass composition

Cell models describe, among other things, what a cell is composed of (see Chapter 2 in [6]). In such models, cell composition may either serve as input information (e.g. putting a certain amount of lipids as a constraint) or model results (to be validated by data). Cell composition may be described broadly by molecule classes (e.g. the overall amounts of lipids, protein, DNA, RNA, and so on), or more precisely for individual molecule types (individual lipids, proteins, mRNA species, and so on), or even in terms of atomic composition (which in turn gives clues about the amounts of molecule classes). Another measure of biomass composition is the amounts of precursors that are made in metabolism in order to produce macromolecules. In FBA, specifically, "biomass" refers to the proportions of these precursor amounts (including cofactors), assuming that the total production may change, but the proportions stay fixed.

Cell composition may greatly vary between cell types, and also for a given cell type between conditions (for instance, because of the different ribosome content of a cell at different growth rates). This variable composition often poses a challenge for models. Just like the uptake rates, the varying biomass composition reflects complex global rearrangements of resources, and choices between metabolic strategies. In classical FBA, uptake rates and biomass composition need to be predefined; if they are not known from experiments, predictions may not be reliable (see Chapter 5 in [6]).

A main achievement of RBA models is that the biomass composition is not fixed and given as input information as in FBA, but is predicted by the model itself. The model predicts the optimal macromolecule composition of the cell and – unlike in small self-replicator models (see the models in Chapter 8 in [6]) – this prediction is very detailed: it follows from the predicted proteome and may comprise details like trace elements (e.g. iron, which appears in some of the proteins) and whose contribution to the overall biomass will vary with the expression level of those proteins. How does this work? In RBA models, in contrast to FBA, uptake rates and biomass composition are usually not predefined (although this is possible, if required, and may be used to fit models to data), but obtained as model predictions from resource allocation ideas. For example, in RBA, the proportions of biomass precursors would reflect the types and amounts of macromolecules (to be made from these precursors) in the optimal state, which in turn (among other things) reflect the amounts of enzymes needed to make the precursors.

### Recommended readings

* Website rba.inrae.fr for further details on RBA

* A recent review on large-scale resource allocation models: K. de Becker et al. "Using resource constraints derived from genomic and proteomic data in metabolic network models" *Curr Opin Syst Biol* 2022, 29:100400

## Problems

### Problem 9.1 The role of metabolite concentrations

* What if the metabolite content of the cell has been underestimated? Assume that the amount of small metabolites in cells is currently underestimated. What problems in model predictions would arise from the fact? In what way
of all decision variables.

It was proven that the feasibility domain becomes smaller with increasing growth rate [5, 7], so it might be worthwhile to probe the solution space at slow-growth regimes. In practice, at the optimum, the cell configuration ($\mu^*, y^*, v^*$) is often unique. Indeed, non-unique solutions will exist if two alternative metabolic pathways have exactly the same cost in resources. Since all enzymes have different amino acid sequences, use different cofactors, are differently localized, etc, this is highly unlikely. A caricatural example of a model with non-unique solutions would be one in which an enzyme pool is arbitrarily split into two, and the two new "enzyme species" are given different names, although they are physically exactly the same.

### 10.3.4. Calibration of model parameters

An RBA model may contain a high number of model parameters. First, the global parameters to be estimated are related to cell composition: (i) the concentrations of bulk biomass components $\bar{b}$, which is usually deduced from the biomass reaction of the genome-scale metabolic network of the organism. Using quantitative proteomics data [24], one can infer (ii) the protein densities in different compartments ($\bar{d}$), and (iii) the abundance of housekeeping (unspecified) proteins ($\bar{p}$).

The next set of parameters we need to collect concerns the efficiencies of molecular machines ($K_E, K'_E, K_T$). As we learned in Chapters 2 in [1] and 3 in [1], the rate of an enzymatic reaction $v$ depends on the enzyme efficiency or "apparent catalytic rate", given by $v = ek_{app}$, with $k_{app} = \kappa(c) = k_{cat}^+ \cdot \eta^{\text{rev}}(c) \cdot \eta^{\text{sat}}(c) < k_{cat}^+$. The $k_{app}$ values are always below the $k_{cat}$ value, but may vary from state to state depending on metabolite concentrations. Since internal metabolite concentrations $s$ are unknown and difficult to measure at genome-scale, we cannot estimate $k_{app}$ from the explicit kinetic law $\kappa(c)$. We need to obtain these $k_{app}$ parameters empirically, for example by measuring the flux $v$ and the protein abundance $e$ in one condition and taking their ratio.

Hence, for a given environmental condition, efficiency parameters can be estimated using quantitative proteomics in combination with fluxomics [21] or FBA to estimate the flux distribution [19]. To account for variable enzyme efficiencies, one may make the simplifying assumption that enzyme efficiencies depend mostly on growth rate. By estimating the enzyme efficiencies at different growth rates and interpolating between them, one obtains empirical relationships between efficiency and the growth rate [21] to be used in $P_{rba}(\mu)$. For instance, several estimates of enzymatic efficiencies obtained in contrasting growth conditions will provide a relationship $K_E(\mu)$ instead of a constant $K_E$ value.

### 10.3.5. Enzyme efficiencies: use of -omics data-informed $k_{app}$ vs. naïve $k_{cat}$ values

The three most popular formalisms of fine-grained resource allocation models, RBA [11], ME-models [12], and pc-models [13], are variations on the same theme, implementing the four major constraints discussed in Section 10.2. Thus most of the ideas, concepts, and constraints are equivalent (or at least highly similar) in their biological interpretation. Most of the differences arise from the approach taken towards parametrization of these models, and consequently, interpretation of model output. Here we will discuss an example where implementations differ significantly.

In resource allocation models, two types of constraints define the proteome capacity at given growth rate $\mu$, the protein density vector $\bar{b}$, and the fraction of housekeeping proteins $\bar{p}$ in the proteome. The remaining proteome space is to be distributed among the proteins that are explicitly defined in the model. The RBA formalism requires to formulate the function $k_{app}(\mu)$ (or $K_E(\mu)$ in the RBA problem, Section 10.3.2) for every protein in the model using -omics data (see Section 10.3.3), and the fraction of the "housekeeping" proteins in the proteome is determined from data for each simulation.

Conversely, the formulation of pc-models [13] allows more flexibility to the "unspecified" protein $UP$, represented by
would predictions (by FBA or other methods) be distorted?

* In what way would a cell, in reality, profit from a lower small metabolite content? Can we assume that the ratio between small metabolites and proteins is optimized? Describe possible aspects of this compromise! For inspiration, see [28].
a single artificial protein of average size and amino acid composition. Instead of setting a fixed amount allocated to **p** which changes across conditions, one can determine the minimal fraction of this protein in proteome *UP*<sub>min</sub>, and formulate the demand to produce *UP* as an inequality constraint *UP* ≥ *UP*<sub>min</sub>. Interestingly, in *Saccharomyces cerevisiae*, the proteome mass fraction occupied by non-metabolic proteins is relatively constant under different glucose-limited conditions, as determined by quantitative proteomics data (see [13], Fig. S1 for a plot).

This inequality constraint can be interpreted as the upper limit of available protein space, i.e., under fixed protein
density $\mathbf{y} + \mathbf{p} = \text{const.}$, the proteome not occupied by $\mathbf{y} \stackrel{\Delta}{=} \mathbf{e} + \mathbf{m}$ is allocated to $\mathbf{p}$. Since now the model can
distribute the proteome among explicitly-defined vs. unspecified protein freely, the procedure of fitting $k_{\text{app}}$ values
is no longer a prerequisite. Using $k_{\text{cat}}$ values, collected from literature/databases/own experimental measurements,
rather than apparent $k_{\text{app}}$ values, has consequences both for predictions and the data use: first, the model prediction
on the protein use is the "demand" of the enzyme and is strictly coupled to the flux through the enzyme (equivalent
to the ECM1 layer of enzyme costs in the *Enzyme Cost Minimization* method, Chapter 6 in [1]). Second, the
condition-dependent quantitative proteomics data can be used as validation dataset for model predictions instead
[25], as the predicted protein abundance is not dependent on these datasets.

Using less data for parameter fitting and redirecting these data-rich datasets towards validation of model prediction
strengthens the argument for using resource allocation models for learning new biology, and already has real-life
examples. For instance, the discrepancies in predicted vs. observed levels of glycolytic enzymes at glucose-scarce
conditions in [13] inspired the same team to revisit the question whether the high levels of glycolytic enzymes repre-
sent the optimal expression given very low thermodynamic driving force and undersaturation of glycolytic enzymes.
Comparing predictions of *Enzyme Cost Minimization* models with the results of the pc-model and experimental data,
[15] proposed that *S. cerevisiae* expresses genuine excess of glycolytic enzymes in glucose-limited conditions, meant
to amply consume any glucose as soon as it appears in the environment.

10.4. Biomass composition as a constraint or as a prediction

Cell models describe, among other things, what a cell is composed of (see Chapter 2 in [1]). In FBA, specifically, biomass refers to the proportions of different molecule classes (e.g. lipids, protein, DNA, RNA, cofactors) in 1 gram dry weight of cells, and biomass composition needs to be defined prior to optimization. Since, at least for FBA models of microbes, biomass production usually is the optimization objective, the literature frequently refers to the mathematical description of cell composition as "biomass objective function" (BOF). In most cases, it is assumed that the proportions of biomass constituents are fixed, only the total production (flux through BOF) changes.

For the predictions of FBA models to be reliable, a realistic BOF is a must (see Chapter 5 in [1]). Therefore, there
is a sustained effort to experimentally determine biomass composition, even for *E. coli* [26]; for more details on the
usual experimental measurement methods, see the box in Chapter 2 in [1]. In case supporting data are available,
the cell composition in the BOF may be described in a more fine-grained manner for individual molecule types (e.g.,
individual lipids, proteins, mRNA species, etc), or even in terms of atomic composition (which in turn gives clues
about the amounts of molecule classes). So, overall, biomass composition acts as a global, and one of the most
stringent, constraint on the predicted solution space in FBA-based models.

However, cell composition may greatly vary not only between (micro-)organisms, or different cell types within the
same organism, but also for the same organism/cell type across different conditions. The budding yeast *S. cerevisiae*,
for instance, exhibits rather linear relationships between the proportions of bulk biomass constituents as a function of
growth rate in glucose-limited cultures [27]. This variable composition often poses a challenge for models: just like
the uptake rates, the varying biomass composition reflects complex global rearrangements of resources (for instance,
different ribosome content at different growth rates [28] leads to changes in RNA-to-protein ratio in the cells), and
choices between metabolic strategies (e.g. depletion of storage carbohydrates in glucose-fermenting *S. cerevisiae*
[16]).
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[10] Anne Goelzer, Vincent Fromion, and Gérard Scorletti. Cell design in bacteria as a convex optimization problem. *Automatica*, 47(6):1210-1218, 2011. doi: 10.1016/j.automatica.2011.02.038.

[11] Matteo Mori, Terence Hwa, Olivier C Martin, Andrea De Martino, and Enzo Marinari. Constrained allocation flux balance analysis. *PLoS computational biology*, 12(6):e1004913, 2016. doi: 10.1371/journal.pcbi.1004913.

[12] Benjamín J Sánchez, Cheng Zhang, Avlant Nilsson, Petri-Jaan Lahtvee, Eduard J Kerkhoven, and Jens Nielsen. Improving the phenotype predictions of a yeast genome-scale metabolic model by incorporating enzymatic constraints. *Molecular systems biology*, 13(8):935, 2017. doi: 10.15252/msb.20167411.

[13] Alexei Vazquez, Qasim K Beg, Marcio A DeMenezes, Jason Ernst, Ziv Bar-Joseph, Albert-László Barabási, László G Boros, and Zoltán N Oltvai. Impact of the solvent capacity constraint on e. coli metabolism. *BMC systems biology*, 2(1):1-10, 2008. doi: 10.1186/1752-0509-2-7.

[14] Anne Goelzer, Vincent Fromion, and Gérard Scorletti. Cell design in bacteria as a convex optimization problem. *Automatica*, 47(6):1210-1218, 2011. doi: 10.1016/j.automatica.2011.02.038.

[15] Edward J O'Brien, Joshua A Lerman, Roger L Chang, Daniel R Hyduke, and Bernhard Ø Palsson. Genome-scale models of metabolism and gene expression extend and refine growth phenotype prediction. *Molecular systems biology*, 9(1):693, 2013. doi: 10.1038/msb.2013.52.
A main advance of resource allocation models, compared to conventional FBA models, is that only a part of the biomass composition is given as input information just like in FBA (b in RBA). The proteome composition, on the contrary, becomes a genuine prediction of the optimization procedure. Unlike small self-replicator models (see the models in Chapter 8 in [1]), this prediction is very detailed, as the predicted proteome composition is represented by the sum of individual protein abundances. Moreover, if proteins require trace elements or cofactors (e.g. iron in iron-containing proteins) for function, the demand and contribution to the overall biomass of these metabolites will also be predicted by the model (as it will vary with the expression level of those proteins).

In theory, the abundance of biomass constituents other than proteome could be formulated in the way they become predictions of the resource allocation models, rather than hardcoded inputs. Following the idea implemented in the small, coarse-grained models of [2], one could set relationships between, e.g., protein density in the cells and production of lipids (in [2], the biological interpretation was to maintain the surface area-to-volume ratio constant). Currently this is not widely accepted as a standard practice, and, as we can see from the example above, requires comprehensive experimental evidence, which, by itself, could be interpreted still as "input to the model".

## 10.5. Concluding remarks

In this chapter we considered whole-cell resource allocation models that couple metabolic networks with a description of the macromolecular machinery that is required to operate them. Compared to FBA models, these models contain a large number of additional reactions, metabolites, constraints, and model parameters, and, overall, offer a fine-grained representation of cellular economy. Many of the kinetic parameters cannot be accurately measured for individual enzymes, and/or are condition-dependent. The quantitative nature of the predictions of resource allocation models (and the most cellular decisions/phenotype shifts), however, is largely due to global constraints: for instance, when the protein density $g gDW^{-1}$ in a compartment reaches its upper limit (=that compartment is fully packed with protein), the cells switch from fully-respiratory to respiro-fermentative growth (see [12] for *E. coli*, or [13] for *S. cerevisiae*). Unlike the kinetic parameters in single reactions, which are rather uncertain, these "global", cell-wide constraints are based on more trustworthy evidence.

Thus these models still retain a reasonable compromise concerning numerical tractability and model complexity, and can accurately predict complex adaptations, which cannot be captured by GEMs in an autonomous way, i.e. without the addition of empirical constraints on fluxes. A successful use case of using resource allocation models is dissecting iron economy, using RBA models: some proteins require iron for their function, and the cell growth can become iron-limited in some conditions. The RBA model was used to predict cell behavior under iron starvation, and the predictions suggested couple of scenarios, (i) the cell may increase the import of iron, but also (ii) avoid using proteins that contain iron (and the pathways in which they operate) [21, 29].

As with the biomass composition, another aspect of resource allocation models (and FBA-based models in general) with some duality in its interpretation is the objective function. Although its validity has been always debated since conception, maximization of instantaneous growth rate as the optimization objective has shown incredible success in predicting microbial physiology. The current approach we apply for resource allocation models still remains the FBA-based assumption that the desired cell phenotypes are the ones maximizing instantaneous growth rate $\mu$. This time, however, the $\mu$ is also a model variable, so we have to apply bisection to obtain the optimal solution for each parameter set we use in resource allocation models.

It is becoming more and more evident that many cell phenotypes (and microbial species!), which we try to predict, do not actually maximize instantaneous growth rate. For instance, most experimental research on microbial physiology has been focused on carbon-limited (C-limited) cultures, especially the yeast work in Delft, the Netherlands (see [27, 16] for examples). It seems that the principle of growth rate maximization works very well in C-limited case, and the success of resource allocation models to quantitatively capture these phenotypes [12, 21, 13] affirms this assumption. But is C-limitation descriptive of natural environments? Let us continue the argument with yeasts as
[16] Ibrahim E Elsemman, Angelica Rodriguez Prado, Pranas Grigaitis, Manuel Garcia Albornoz, Victoria Harman, Stephen W Holman, Johan van Heerden, Frank J Bruggeman, Mark MM Bisschops, Nikolaus Sonnenschein, et al. Whole-cell modeling in yeast predicts compartment-specific proteome constraints that drive metabolic strategies. *Nature Communications*, 13(1):1–12, 2022. doi: 10.1038/s41467-022-28467-6.

[17] Pranas Grigaitis and Bas Teusink. An excess of glycolytic enzymes under glucose-limited conditions may enable saccharomyces cerevisiae to adapt to nutrient availability. *FEBS letters*, 2022.

[18] André B. Canelas, Cor Ras, Angela ten Pierick, Walter M. van Gulik, and Joseph J. Heijnen. An in vivo data-driven framework for classification and quantification of enzyme kinetics and determination of apparent thermodynamic data. *Metabolic Engineering*, 13(3):294–306, 2011. doi: 10.1016/j.ymben.2011.02.005.

[19] E Martinez-Salas, JA Martin, and M Vicente. Relationship of escherichia coli density to growth rate and cell age. *Journal of bacteriology*, 147(1):97–100, 1981. doi: 10.1128/jb.147.1.97\-100.1981.

[20] Steven B Zimmerman and Stefan O Trach. Estimation of macromolecule concentrations and excluded volume effects for the cytoplasm of escherichia coli. *Journal of molecular biology*, 222(3):599–620, 1991. doi: 10.1016/0022-2836(91)90499-v.

[21] Anne Goelzer, Jan Muntel, Victor Chubukov, Matthieu Jules, Eric Prestel, Rolf Nölker, Mahendra Mariadassou, Stéphane Aymerich, Michael Hecker, Philippe Noirot, et al. Quantitative prediction of genome-wide resource allocation in bacteria. *Metabolic engineering*, 32:232–243, 2015. doi: 10.1016/j.ymben.2015.10.003.

[22] Ana Bulović, Stephan Fischer, Marc Dinh, Felipe Golib, Wolfram Liebermeister, Christian Poirier, Laurent Tournier, Edda Klipp, Vincent Fromion, and Anne Goelzer. Automated generation of bacterial resource allocation models. *Metabolic engineering*, 55:12–22, 2019. doi: 10.1016/j.ymben.2019.06.001.

[23] Anne Goelzer and Vincent Fromion. Rba for eukaryotic cells: foundations and theoretical developments. *bioRxiv*, page 750182, 2019. doi: 10.1101/750182.

[24] Radhakrishnan Mahadevan and Chrisophe H Schilling. The effects of alternate optimal solutions in constraint-based genome-scale metabolic models. *Metabolic engineering*, 5(4):264–276, 2003. doi: 10.1016/j.ymben.2003.09.002.

[25] Michael Hucka, Andrew Finney, Herbert M Sauro, Hamid Bolouri, John C Doyle, Hiroaki Kitano, Adam P Arkin, Benjamin J Bornstein, Dennis Bray, Athel Cornish-Bowden, et al. The systems biology markup language (sbml): a medium for representation and exchange of biochemical network models. *Bioinformatics*, 19(4):524–531, 2003. doi: 10.1093/bioinformatics/btg015.

[26] Mélisande Blein-Nicolas and Michel Zivy. Thousand and one ways to quantify and compare protein abundances in label-free bottom-up proteomics. *Biochimica et Biophysica Acta (BBA)-Proteins and Proteomics*, 1864(8):883–895, 2016. doi: 10.1016/j.bbapap.2016.02.019.

[27] Laurent Tournier, Anne Goelzer, and Vincent Fromion. Optimal resource allocation enables mathematical exploration of microbial metabolic configurations. *Journal of mathematical biology*, 75(6):1349–1380, 2017. doi: 10.1007/s00285-017-1118-5.

[28] Naama Tepper, Elad Noor, Daniel Amador-Noguez, Hulda S Haraldsdóttir, Ron Milo, Josh Rabinowitz, Wolfram Liebermeister, and Tomer Shlomi. Steady-State metabolite concentrations reflect a balance between maximizing enzyme efficiency and minimizing total metabolite load. *PLoS One*, 8(9):e75370, September 2013. doi: 10.1371/journal.pone.0075370.
an example.

Yeasts in the wild, for instance, are often subjected to feast-famine cycles in terms of carbon availability, and one
could argue that in the famine phase of the cycle, these yeasts should act as if they were glucose-limited. Yet the
current opinion in the yeast ecology seems to see feast-famine cycles as a continuous, although reduced, supply
of carbon, and steer towards embracing a higher role of nitrogen (N) limitation in natural environments instead.
Currently, our understanding of N-limited growth is not very comprehensive, and N-limitation is also a case where the
instantaneous growth rate maximization breaks down: the pc-models of S. cerevisiae cannot quantitatively capture
the cell behavior under N-limited conditions.

So the selection of a suitable optimization objective can be a choice followed by huge success, but also, the optimal
solution might end up contradicting the existing knowledge. How can we try to mitigate that? One huge advance
of resource allocation models is that at any condition, the available solution space is greatly reduced, compared to
conventional FBA. We can argue that we have introduced a whole new set, a whole new type of constraints into the
model by accepting assumptions stemming from the metabolism-molecular machinery coupling. In theory, we should
be able to reason further regarding any additional (even empirical/ad hoc) constraints and/or additional optimization
objectives which would bring our model predictions closer to observed biology. Just remember: fitting models is not a
sin; but nontransparent/reckless fitting is! After all, modeling is an art, and there is no one cookbook that represents
the ground truth: we should be free to explore the secrets of biology, as unrealistic as our assumptions are at times.

A final remark on modeling being an art. In this book, we have explored several types of cell models of different
size, detail, and assumptions behind. This whole hierarchy and diversity of different implementations and formalisms
might seem overcomplicated and unnecessary, but it is a mere reflection that "one size does not fit all". Hence, and
we invite (future) modelers to be creative, mix, match, and tailor different models (and modeling types) to cover
more and more biological knowledge. The compromise between fine-grained but linear modeling vs. complex kinetics
that materialized into resource allocation models is an inspiring example of how one can push bounds of different
methods to create mathematical representation of our mental pictures of cells.

Recommended readings

**RBA website** Website rba.inrae.fr for further details on RBA. Under Tools, you can find example models and
Jupyter notebooks for running them.

**Review article on large-scale resource allocation models** K. de Becker et al. "Using resource constraints derived from genomic and proteomic data in metabolic network models" *Curr Opin Syst Biol* 2022, 29:100400

Problems

**Problem 10.1 The role of metabolite concentrations**

The available cell space for proteins depends on the assumed space occupied by small metabolites.

(a) What if the metabolite content of the cell has been underestimated? Assume that the amount of small metabolites in cells is currently underestimated. What problems in model predictions would arise from the fact? In what way would predictions (by FBA or other methods) be distorted?

(b) In what way would a cell, in reality, profit from a lower small metabolite content? Can we assume that the ratio between small metabolites and proteins is optimized? Describe possible aspects of this compromise! For inspiration, see [30].
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[24] Mélisande Blein-Nicolas and Michel Zivy. Thousand and one ways to quantify and compare protein abundances in label-free bottom-up proteomics. *Biochimica et Biophysica Acta (BBA)-Proteins and Proteomics*, 1864(8): 883-895, 2016. doi: 10.1016/j.bkapap.2016.02.019.

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[29] Laurent Tournier, Anne Goelzer, and Vincent Fromion. Optimal resource allocation enables mathematical exploration of microbial metabolic configurations. *Journal of mathematical biology*, 75(6):1349-1380, 2017. doi: 10.1007/s00285-017-1118-5.

[30] Naama Tepper, Elad Noor, Daniel Amador-Noguez, Hulda S Haraldsdóttir, Ron Milo, Josh Rabinowitz, Wolfram Liebermeister, and Tomer Shlomi. Steady-State metabolite concentrations reflect a balance between maximizing enzyme efficiency and minimizing total metabolite load. *PLoS One*, 8(9):e75370, September 2013. doi: 10.1371/journal.pone.0075370.