Economic Principles in Cell Biology

Agustín Gabriel Yabo, Andrea de Martino, Andrea Weisse, Andreas

Kremling, Anne Goelzer, Benjamin Mauroy, Christophe Goupil, Cyril

Karamaoun, Daan de Groot, Dafni Giannari, et al.

► To cite this version:

Agustín Gabriel Yabo, Andrea de Martino, Andrea Weisse, Andreas Kremling, Anne Goelzer, et al. "Economic Principles in Cell Biology. The Economic Cell Collective", pp.1-285, 2023, 10.5281/zen-odo.8156386. [hal-04172118](https://hal.inrae.fr/hal-04172118)

HAL Id: hal-04172118

https://hal.inrae.fr/hal-04172118

Submitted on 27 Jul 2023

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

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

<figure><img src="image_1.png" /></figure>
# Economic Principles in Cell Biology

Agustín G. Yabo, Andrea de Martino, Andrea Weisse, Andreas Kremling, Anne Goelzer, Benjamin Mauroy, Christophe Goupil, Cyril Karamaoun, Daan de Groot, Dafni Giannari, et al.

► To cite this version:

Agustín G. Yabo, Andrea de Martino, Andrea Weisse, Andreas Kremling, Anne Goelzer, et al. "Economic Principles in Cell Biology. The Economic Cell Collective", pp.1-285, 2025, 10.5281/zen-odo.8156386. [https://hal.inrae.fr/hal-04172118v2](https://hal.inrae.fr/hal-04172118v2)

HAL Id: hal-04172118

https://hal.inrae.fr/hal-04172118v2

Submitted on 29 Jun 2025

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

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

<figure><img src="image_1.png" /></figure>
# Economic Principles in Cell Biology

## The Economic Cell Collective

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

Edition July 2023

<figure><img src="image_3.png" /></figure>
ECONOMIC
PRINCIPLES IN
CELL BIOLOGY

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

the economic cell collective

july 2025 edition
<figure><img src="image_4.png" /></figure>

Some of the authors at the Paris Summer School 2023, where material from this book was presented. Other authors presented their chapters online. Lecture slides can be found at [principlescellphysiology.org/book-economic-principles/](https://principlescellphysiology.org/book-economic-principles/).

## About this book

Living cells, shaped by billions of years of evolution, have developed many ways to adapt to their environment, e.g. by regulation of gene expression. But the rules of physics and chemistry enforce certain boundaries on what cells can achieve and how they can allocate their own resources. The goal of this book is to uncover some of these governing principles, linking them to fundamental questions in cell physiology.

The book gives an overview of established approaches to "cellular economics", from descriptions of simple metabolic systems to cell growth, variability, and dynamic behavior. It is written as a free and open textbook to which anyone can contribute. The authors of the current edition (July 2023), forming the Economic Cell Collective, are Agustín G. Yabo, Andrea De Martino, Andrea Weiße, Andreas Kremling, Anne Goelzer, Benjamin Mauroy, Christophe Goupil, Cyril Karamaoun, Daan de Groot, Dafni Giannari, David Lacoste, David Tourigny, Diana Széliová, Diego A. Oyarzún, Elad Noor, Elena Pascual Garcia, Éric Herbert, Felipe Scott, Frédérique Noël, Gabriele Micali, Hadrien Delattre, Herbert Sauro, Hidde de Jong, Hollie J. Hindley, Hugo Dourado, Jacopo Grilli, Marcelo Rivas-Astroza, Marco Cosentino Lagomarsino, Markus Köbis, Mattia Corigliano, Meike Wortel, Ohad Golan, Olivier Rivoire, Orkun S. Soyer, Pranas Grigaitis, Robert West, Steffen Waldherr, and Wolfram Liebermeister, but many others also contributed to this book.

Chapter authors are listed in alphabetical order.

The book and computer exercises can be downloaded from [principlescellphysiology.org/book-economic-principles/](https://principlescellphysiology.org/book-economic-principles/).

<figure><img src="image_5.png" /></figure>

© 2023 The Economic Cell Collective

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

An authors open access book. No publisher has been paid.
<figure><img src="image_3.png" /></figure>
# Contents

Preface 1

1 The cell as a factory 3

2 What makes up a cell 7

2.1 The components of a cell 7

2.1.1 Cell composition and structures 7

2.1.2 Biological molecules 8

2.2 Cell composition in numbers 10

2.2.1 Variability of biomass composition 12

2.2.2 Biomass composition is not uniform 13

2.3 Cell size 15

2.4 Cell density 16

2.5 The physical constraints of cell growth 16

2.6 Macromolecule synthesis and the resources needed 17

2.6.1 Precursors of macromolecules 17

2.6.2 Catalysts needed for macromolecule synthesis 18

2.6.3 Physical proteome space 21

2.7 Concluding remarks 22

3 The dynamics of metabolic systems 25

3.1 Conceptualizing cell metabolism as a dynamical system 25

3.1.1 Metabolism as a collection of pathways 26

3.1.2 Metabolism - coarse grained views 27

3.1.3 Keeping flows in a system of interconnected fluxes 30

3.1.4 Metabolic system and recurring motifs 30

3.2 Dynamics and regulation of metabolism 31

3.2.1 Biochemical reactions and thermodynamics 31

3.2.2 Stoichiometric matrix and ordinary differential equations 33
## About this book

Living cells, shaped by billions of years of evolution, have developed many ways to adapt to their environment in order to thrive and to maximize their chance for survival. Cell metabolism is shaped by the rules of physics and chemistry, which enforce boundaries on what cells can achieve. But within these boundaries, we can still ask how cells should allocate their resources. By looking at cells through this economic lens, we may uncover some general design principles that characterize life. The goal of this book is to describe some of these governing principles, linking them to fundamental questions in cell physiology. The book gives an overview of established approaches and topics in cellular economics, from descriptions of simple metabolic systems to cell growth, variability, and dynamic behavior. It is written as a free and open textbook to which anyone can contribute.

The authors of the current edition (July 2025) and further main contributors, forming the Economic Cell Collective, are Agustín G. Yabo, Andrea De Martino, Andrea Weiße, Andreas Kremling, Anne Goelzer, Barnabé Ledoux, Benjamin Mauroy, Christophe Goupil, Cyril Karamaoun, Daan de Groot, Dafni Gian-nari, David Lacoste, David Ruckerbauer, David Tourigny, Diana Széliová, Diego A. Oyarzún, Elad Noor, Elena Pascual Garcia, Éric Herbert, Felipe Scott, Frédérique Noël, Gabriele Micali, Hadrien Delattre, Herbert Sauro, Hester Chapman, Hidde de Jong, Hollie J. Hindley, Hugo Dourado, Jacopo Grilli, Jan-Philipp Dapprich, Marcelo Rivas-Astroza, Marco Cosentino Lagomarsino, Markus Köbis, Mattia Corigliano, Maxime Mahout, Meike Wortel, Michela Pauletti, Milena Milovanović, Miriam Neubauer, Ohad Golan, Olivier Rivoire, Orkun S. Soyer, Paul Ross, Pranas Grigaitis, Robert West, Stefan Müller, Steffen Waldherr, and Wolfram Lieber-meister. Aside from these authors, many others contributed to this book.

The authors of each chapter are listed in alphabetical order.

Graphics by Michela Pauletti and Elad Noor based on designs by all authors.

Layout and cover design by Elad Noor and Wolfram Liebermeister.

Typeset by the authors using LaTeX.

The book, lecture slides, lecture videos, and computer exercises are available at principlescellphysiology.org/book-economic-principles/.

<figure><img src="image_4.png" /></figure>

<figure><img src="image_5.png" /></figure>

© 2025 The Economic Cell Collective

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

An authors open access book. No publisher has been paid.
<table><tbody><tr><td>3.2.3</td><td>Dynamic steady-state</td><td>33</td></tr><tr><td>3.2.4</td><td>Multiple steady-states and oscillations</td><td>34</td></tr><tr><td>3.2.5</td><td>Regulation of fluxes</td><td>35</td></tr><tr><td>3.3</td><td>Toolbox for modelling dynamics of metabolism</td><td>35</td></tr><tr><td>3.3.1</td><td>Enzymes - a brief note</td><td>36</td></tr><tr><td>3.3.2</td><td>Modeling reaction fluxes - reaction rate models</td><td>37</td></tr><tr><td>3.4</td><td>Dynamics of metabolism: Examples of experimental evidence and model-based explanations</td><td>42</td></tr><tr><td>3.4.1</td><td>Flux switching / regulation</td><td>43</td></tr><tr><td>3.4.2</td><td>Bistability</td><td>44</td></tr><tr><td>3.4.3</td><td>Oscillations</td><td>46</td></tr><tr><td>3.5</td><td>Concluding remarks</td><td>46</td></tr><tr><td>4</td><td>The space of metabolic flux distributions</td><td>49</td></tr><tr><td>4.1</td><td>Modeling metabolic fluxes in cells</td><td>49</td></tr><tr><td>4.2</td><td>The flux cone</td><td>51</td></tr><tr><td>4.2.1</td><td>Mass-balance constraints</td><td>51</td></tr><tr><td>4.2.2</td><td>Elementary flux modes</td><td>53</td></tr><tr><td>4.2.3</td><td>Irreversibility constraints</td><td>55</td></tr><tr><td>4.2.4</td><td>Practical uses of elementary flux modes</td><td>56</td></tr><tr><td>4.2.5</td><td>Computational challenges for elementary flux mode analysis</td><td>57</td></tr><tr><td>4.3</td><td>Additional constraints and flux polyhedra</td><td>57</td></tr><tr><td>4.3.1</td><td>Inhomogeneous linear flux constraints</td><td>57</td></tr><tr><td>4.3.2</td><td>Thermodynamic constraints</td><td>60</td></tr><tr><td>4.4</td><td>Alternative methods for flux space exploration</td><td>61</td></tr><tr><td>4.4.1</td><td>Elementary conversion modes</td><td>61</td></tr><tr><td>4.4.2</td><td>Flux sampling</td><td>63</td></tr><tr><td>4.4.3</td><td>Minimal cut sets</td><td>63</td></tr><tr><td>4.5</td><td>Conclusions of our flux cone explorations</td><td>64</td></tr><tr><td>5</td><td>Solutions of constraint-based metabolic models</td><td>69</td></tr><tr><td>5.1</td><td>Can we use an optimality assumption to predict metabolic behavior?</td><td>69</td></tr><tr><td>5.2</td><td>Metabolic models based on linear optimization problems</td><td>70</td></tr><tr><td>5.2.1</td><td>Types of linear objective functions used in FBA</td><td>71</td></tr><tr><td>5.3</td><td>Optimal metabolism in terms of elementary flux modes</td><td>73</td></tr><tr><td>5.4</td><td>Phenotypic phase plane analysis</td><td>75</td></tr><tr><td>5.5</td><td>Non-uniqueness of the optimal metabolic state</td><td>76</td></tr></tbody></table>
# Contents

<table><tr><td>Preface</td><td>1</td></tr><tr><td>Overview</td><td>5</td></tr><tr><td>Getting started</td><td>5</td></tr><tr><td>A guide to the book</td><td>6</td></tr><tr><td>Where to find more information</td><td>8</td></tr><tr><td>1 The cell as a factory</td><td>11</td></tr><tr><td>2 An inventory of cell components</td><td>15</td></tr><tr><td>2.1 Describing and counting cellular components</td><td>15</td></tr><tr><td>2.2 The components of a cell</td><td>16</td></tr><tr><td>2.2.1 Elemental composition of the cell</td><td>16</td></tr><tr><td>2.2.2 Biological molecules</td><td>16</td></tr><tr><td>2.3 Cell organization and size</td><td>17</td></tr><tr><td>2.3.1 Membrane-bound structures of the cell</td><td>17</td></tr><tr><td>2.3.2 Cell size</td><td>18</td></tr><tr><td>2.3.3 Variation of single-cell sizes and shapes</td><td>19</td></tr><tr><td>2.4 Cell composition in numbers</td><td>19</td></tr><tr><td>2.4.1 Variability of biomass composition</td><td>21</td></tr><tr><td>2.4.2 Biomass composition varies within and between cells</td><td>23</td></tr><tr><td>2.5 Macromolecule synthesis and the resources needed</td><td>25</td></tr><tr><td>2.5.1 Precursors of macromolecules</td><td>25</td></tr><tr><td>2.5.2 Catalysts needed for macromolecule synthesis</td><td>26</td></tr><tr><td>2.5.3 Physical proteome space</td><td>30</td></tr><tr><td>2.6 Physicochemical considerations about cells</td><td>30</td></tr><tr><td>2.6.1 Cell density</td><td>30</td></tr><tr><td>2.6.2 Physical constraints of cell growth</td><td>31</td></tr><tr><td>2.7 Concluding remarks</td><td>31</td></tr><tr><td>3 Cell metabolism</td><td>35</td></tr><tr><td>3.1 Conceptualizing cell metabolism as a dynamical system</td><td>35</td></tr><tr><td>3.1.1 Metabolism as a collection of pathways</td><td>35</td></tr><tr><td>3.1.2 Coarse grained views of metabolism</td><td>38</td></tr><tr><td>3.1.3 Keeping flows in a system of interconnected fluxes</td><td>38</td></tr><tr><td>3.1.4 Metabolic system and recurring motifs</td><td>39</td></tr><tr><td>3.2 Reaction thermodynamics and enzyme kinetics</td><td>39</td></tr><tr><td>3.2.1 Biochemical reactions and thermodynamics</td><td>40</td></tr><tr><td>3.2.2 Enzymes as catalysts of biochemical reactions</td><td>42</td></tr></table>
<table>
  <tbody>
    <tr>
      <td>5.5.1 Flux Variability Analysis</td>
      <td>77</td>
    </tr>
    <tr>
      <td>5.6 Limitations of constraint-based metabolic models</td>
      <td>77</td>
    </tr>
    <tr>
      <td>5.7 Conclusions of our flux cone optimizations</td>
      <td>78</td>
    </tr>
    <tr>
      <td><b>6 Rating metabolic pathways by enzyme efficiency</b></td>
      <td><b>81</b></td>
    </tr>
    <tr>
      <td>6.1 What guides evolution to select one pathway over another?</td>
      <td>81</td>
    </tr>
    <tr>
      <td>6.2 Pathway efficiency - some basic notions and thoughts</td>
      <td>85</td>
    </tr>
    <tr>
      <td>6.3 The role of thermodynamics</td>
      <td>88</td>
    </tr>
    <tr>
      <td>6.3.1 Kinetics and driving forces</td>
      <td>88</td>
    </tr>
    <tr>
      <td>6.3.2 Small driving forces should be avoided</td>
      <td>89</td>
    </tr>
    <tr>
      <td>6.3.3 Max-min Driving Force method</td>
      <td>90</td>
    </tr>
    <tr>
      <td>6.3.4 The roles of thermodynamics for metabolic states</td>
      <td>92</td>
    </tr>
    <tr>
      <td>6.4 Enzyme cost minimization</td>
      <td>92</td>
    </tr>
    <tr>
      <td>6.4.1 Enzyme cost minimization</td>
      <td>92</td>
    </tr>
    <tr>
      <td>6.4.2 Enzyme cost landscape of a metabolic pathway</td>
      <td>92</td>
    </tr>
    <tr>
      <td>6.4.3 Enzyme cost as a function of metabolite profiles</td>
      <td>94</td>
    </tr>
    <tr>
      <td>6.4.4 General lessons from Enzyme Cost Minimization</td>
      <td>95</td>
    </tr>
    <tr>
      <td>6.5 Comparison of alternative pathways</td>
      <td>96</td>
    </tr>
    <tr>
      <td>6.5.1 A tale of two glycolyses</td>
      <td>96</td>
    </tr>
    <tr>
      <td>6.5.2 Metabolic engineering</td>
      <td>97</td>
    </tr>
    <tr>
      <td>6.5.3 Predicting concentrations</td>
      <td>98</td>
    </tr>
    <tr>
      <td>6.6 Can we now predict which pathways will be selected by evolution?</td>
      <td>99</td>
    </tr>
    <tr>
      <td><b>7 Metabolism in states of maximal enzyme efficiency</b></td>
      <td><b>103</b></td>
    </tr>
    <tr>
      <td>7.1 Introduction</td>
      <td>103</td>
    </tr>
    <tr>
      <td>7.2 The enzyme-efficient metabolic states are Elementary Flux Modes</td>
      <td>104</td>
    </tr>
    <tr>
      <td>7.3 Illustration with an example network</td>
      <td>105</td>
    </tr>
    <tr>
      <td>7.4 Computation of the optimal state</td>
      <td>106</td>
    </tr>
    <tr>
      <td>7.5 Translating enzyme efficiency into cell growth rate</td>
      <td>109</td>
    </tr>
    <tr>
      <td>7.6 Conclusions</td>
      <td>110</td>
    </tr>
    <tr>
      <td><b>8 Models of growing cells</b></td>
      <td><b>113</b></td>
    </tr>
    <tr>
      <td>8.1 Introduction</td>
      <td>113</td>
    </tr>
    <tr>
      <td>8.2 Fundamental modeling assumptions of microbial growth</td>
      <td>115</td>
    </tr>
    <tr>
      <td>8.2.1 Conservation of mass and quasi-steady-state assumption</td>
      <td>116</td>
    </tr>
    <tr>
      <td>8.2.2 Proteome allocation assumption</td>
      <td>117</td>
    </tr>
    <tr>
      <td>8.2.3 Mathematical description of reaction fluxes</td>
      <td>117</td>
    </tr>
  </tbody>
</table>
<table>
  <tbody>
    <tr>
      <td>3.2.3 Modeling reaction fluxes - reaction rate models</td>
      <td>43</td>
    </tr>
    <tr>
      <td>3.3 Dynamics and regulation of metabolism</td>
      <td>49</td>
    </tr>
    <tr>
      <td>3.3.1 Stoichiometric matrix and ordinary differential equations</td>
      <td>50</td>
    </tr>
    <tr>
      <td>3.3.2 Dynamic steady state</td>
      <td>50</td>
    </tr>
    <tr>
      <td>3.3.3 Multiple steady-states and oscillations</td>
      <td>52</td>
    </tr>
    <tr>
      <td>3.4 Concluding remarks</td>
      <td>52</td>
    </tr>
    <tr>
      <td>4 Metabolic flux distributions</td>
      <td>55</td>
    </tr>
    <tr>
      <td>4.1 Modeling metabolic fluxes in cells</td>
      <td>55</td>
    </tr>
    <tr>
      <td>4.2 The flux cone</td>
      <td>57</td>
    </tr>
    <tr>
      <td>4.2.1 Mass-balance constraints</td>
      <td>57</td>
    </tr>
    <tr>
      <td>4.2.2 Irreversibility constraints</td>
      <td>58</td>
    </tr>
    <tr>
      <td>4.2.3 The flux cone</td>
      <td>58</td>
    </tr>
    <tr>
      <td>4.3 Elementary flux modes</td>
      <td>59</td>
    </tr>
    <tr>
      <td>4.3.1 Practical relevance of EFMs</td>
      <td>61</td>
    </tr>
    <tr>
      <td>4.3.2 Reaction splitting for EFM computation</td>
      <td>63</td>
    </tr>
    <tr>
      <td>4.4 Extra constraints and flux polyhedra</td>
      <td>64</td>
    </tr>
    <tr>
      <td>4.4.1 Inhomogeneous linear flux constraints</td>
      <td>64</td>
    </tr>
    <tr>
      <td>4.4.2 From the flux polyhedron to the EFM weight polyhedron</td>
      <td>65</td>
    </tr>
    <tr>
      <td>4.4.3 Thermodynamic constraints</td>
      <td>67</td>
    </tr>
    <tr>
      <td>4.4.4 Computational challenges for EFM analysis</td>
      <td>67</td>
    </tr>
    <tr>
      <td>4.4.5 Reducing combinatorics of EFMs computation</td>
      <td>68</td>
    </tr>
    <tr>
      <td>4.5 Alternative methods for flux space exploration</td>
      <td>71</td>
    </tr>
    <tr>
      <td>4.5.1 Elementary conversion modes</td>
      <td>71</td>
    </tr>
    <tr>
      <td>4.5.2 Flux sampling</td>
      <td>73</td>
    </tr>
    <tr>
      <td>4.5.3 Minimal cut sets</td>
      <td>73</td>
    </tr>
    <tr>
      <td>4.6 Concluding remarks</td>
      <td>74</td>
    </tr>
    <tr>
      <td>5 Optimization of metabolic fluxes</td>
      <td>79</td>
    </tr>
    <tr>
      <td>5.1 Can optimality principles help us predict metabolic behavior?</td>
      <td>79</td>
    </tr>
    <tr>
      <td>5.2 Metabolic modeling based on linear optimization</td>
      <td>80</td>
    </tr>
    <tr>
      <td>5.2.1 Linear programming problems</td>
      <td>80</td>
    </tr>
    <tr>
      <td>5.2.2 Metabolic modeling via linear programs aka Flux balance analysis</td>
      <td>81</td>
    </tr>
    <tr>
      <td>5.2.3 Optimization problems for metabolic fluxes</td>
      <td>83</td>
    </tr>
    <tr>
      <td>5.3 Choice of objective functions in Flux Balance Analysis</td>
      <td>86</td>
    </tr>
    <tr>
      <td>5.4 Enzyme-constrained FBA</td>
      <td>87</td>
    </tr>
    <tr>
      <td>5.4.1 Optimal metabolism in terms of elementary flux modes</td>
      <td>88</td>
    </tr>
    <tr>
      <td>5.5 Multi-objective flux analysis and flux variability</td>
      <td>89</td>
    </tr>
    <tr>
      <td>5.5.1 Phenotypic phase plane analysis</td>
      <td>89</td>
    </tr>
    <tr>
      <td>5.5.2 Non-uniqueness of optimal metabolic states: possible reasons</td>
      <td>91</td>
    </tr>
    <tr>
      <td>5.5.3 Flux Variability Analysis</td>
      <td>91</td>
    </tr>
  </tbody>
</table>
<table>
  <tbody>
    <tr>
      <td>8.2.4</td>
      <td>Volume and surface area assumptions</td>
      <td>118</td>
    </tr>
    <tr>
      <td>8.3</td>
      <td>Derivation of growth laws from basic modeling assumptions</td>
      <td>118</td>
    </tr>
    <tr>
      <td><b>9</b></td>
      <td><b>Large resource allocation models of cells</b></td>
      <td><b>123</b></td>
    </tr>
    <tr>
      <td>9.1</td>
      <td>Introduction: Quantifying cell investments at genome-scale</td>
      <td>123</td>
    </tr>
    <tr>
      <td>9.1.1</td>
      <td>Can we build detailed models of entire growing cells?</td>
      <td>123</td>
    </tr>
    <tr>
      <td>9.1.2</td>
      <td>Linearization: replacing kinetics by catalytic constraints</td>
      <td>124</td>
    </tr>
    <tr>
      <td>9.1.3</td>
      <td>Towards fine-grained resource allocation models</td>
      <td>125</td>
    </tr>
    <tr>
      <td>9.2</td>
      <td>Additional constraints in resource allocation models</td>
      <td>125</td>
    </tr>
    <tr>
      <td>9.2.1</td>
      <td>Types of constraints</td>
      <td>126</td>
    </tr>
    <tr>
      <td>9.2.2</td>
      <td>Interpretation and outcomes of the additional constraints</td>
      <td>128</td>
    </tr>
    <tr>
      <td>9.3</td>
      <td>Resource balance analysis</td>
      <td>128</td>
    </tr>
    <tr>
      <td>9.3.1</td>
      <td>Mathematical description</td>
      <td>128</td>
    </tr>
    <tr>
      <td>9.3.2</td>
      <td>How to build and calibrate RBA models?</td>
      <td>131</td>
    </tr>
    <tr>
      <td>9.3.3</td>
      <td>Examples of RBA applications</td>
      <td>132</td>
    </tr>
    <tr>
      <td>9.4</td>
      <td>Prediction of variable biomass composition</td>
      <td>132</td>
    </tr>
    <tr>
      <td><b>10</b></td>
      <td><b>Optimal cell behavior in time</b></td>
      <td><b>135</b></td>
    </tr>
    <tr>
      <td>10.1</td>
      <td>Introduction</td>
      <td>135</td>
    </tr>
    <tr>
      <td>10.2</td>
      <td>Mathematical formalization of dynamic optimization problems</td>
      <td>137</td>
    </tr>
    <tr>
      <td>10.3</td>
      <td>Enzyme expression in metabolic pathways</td>
      <td>138</td>
    </tr>
    <tr>
      <td>10.4</td>
      <td>Coarse-grained models of cellular growth</td>
      <td>141</td>
    </tr>
    <tr>
      <td>10.5</td>
      <td>Dynamic flux balance analysis (dFBA) of metabolic networks</td>
      <td>145</td>
    </tr>
    <tr>
      <td>10.6</td>
      <td>Concluding remarks</td>
      <td>149</td>
    </tr>
    <tr>
      <td><b>11</b></td>
      <td><b>Metabolic diversity in cell populations: probability densities over the flux polytope</b></td>
      <td><b>153</b></td>
    </tr>
    <tr>
      <td>11.1</td>
      <td>Introduction</td>
      <td>153</td>
    </tr>
    <tr>
      <td>11.2</td>
      <td>Sources of variability and uncertainty in metabolism</td>
      <td>156</td>
    </tr>
    <tr>
      <td>11.3</td>
      <td>Probability densities over the flux polytope</td>
      <td>156</td>
    </tr>
    <tr>
      <td>11.4</td>
      <td>Representing heterogeneity and uncertainty</td>
      <td>160</td>
    </tr>
    <tr>
      <td>11.4.1</td>
      <td>ML, MAP and Bayesian inference</td>
      <td>160</td>
    </tr>
    <tr>
      <td>11.4.2</td>
      <td>MaxEnt inference</td>
      <td>163</td>
    </tr>
    <tr>
      <td>11.5</td>
      <td>Representing optimal populations</td>
      <td>166</td>
    </tr>
    <tr>
      <td>11.6</td>
      <td>Discussion</td>
      <td>169</td>
    </tr>
    <tr>
      <td><b>12</b></td>
      <td><b>Control of cell division</b></td>
      <td><b>171</b></td>
    </tr>
    <tr>
      <td>12.1</td>
      <td>Introduction: the decision to divide illustrated through single-cell <i>E. coli</i> data.</td>
      <td>171</td>
    </tr>
  </tbody>
</table>
<table>
  <tbody>
    <tr>
      <td>5.6</td>
      <td>Concluding remarks</td>
      <td>92</td>
    </tr>
    <tr>
      <td>6</td>
      <td>The enzyme cost of metabolic fluxes</td>
      <td>95</td>
    </tr>
    <tr>
      <td>6.1</td>
      <td>What guides evolution to select one pathway over another?</td>
      <td>95</td>
    </tr>
    <tr>
      <td>6.2</td>
      <td>Pathway efficiency - some notions and thoughts</td>
      <td>98</td>
    </tr>
    <tr>
      <td>6.3</td>
      <td>The role of thermodynamics</td>
      <td>101</td>
    </tr>
    <tr>
      <td>6.3.1</td>
      <td>Enzyme kinetics and driving forces</td>
      <td>102</td>
    </tr>
    <tr>
      <td>6.3.2</td>
      <td>Driving forces should not be too small</td>
      <td>102</td>
    </tr>
    <tr>
      <td>6.3.3</td>
      <td>Max-Min driving force method</td>
      <td>104</td>
    </tr>
    <tr>
      <td>6.3.4</td>
      <td>The role of thermodynamics for metabolic states</td>
      <td>106</td>
    </tr>
    <tr>
      <td>6.4</td>
      <td>Enzyme cost minimization</td>
      <td>106</td>
    </tr>
    <tr>
      <td>6.4.1</td>
      <td>Enzyme cost minimization</td>
      <td>106</td>
    </tr>
    <tr>
      <td>6.4.2</td>
      <td>Enzyme cost landscape of a metabolic pathway</td>
      <td>106</td>
    </tr>
    <tr>
      <td>6.4.3</td>
      <td>Enzyme cost as a function of metabolite concentrations</td>
      <td>107</td>
    </tr>
    <tr>
      <td>6.4.4</td>
      <td>General lessons from Enzyme Cost Minimization</td>
      <td>109</td>
    </tr>
    <tr>
      <td>6.5</td>
      <td>Comparison of alternative pathways</td>
      <td>110</td>
    </tr>
    <tr>
      <td>6.5.1</td>
      <td>A tale of two glycolyses</td>
      <td>110</td>
    </tr>
    <tr>
      <td>6.5.2</td>
      <td>Metabolic engineering</td>
      <td>112</td>
    </tr>
    <tr>
      <td>6.5.3</td>
      <td>Predicting the metabolite concentrations</td>
      <td>112</td>
    </tr>
    <tr>
      <td>6.6</td>
      <td>Concluding remarks</td>
      <td>114</td>
    </tr>
    <tr>
      <td>7</td>
      <td>Optimization of metabolic states</td>
      <td>119</td>
    </tr>
    <tr>
      <td>7.1</td>
      <td>Introduction</td>
      <td>119</td>
    </tr>
    <tr>
      <td>7.2</td>
      <td>Enzyme-efficient metabolic states use elementary flux modes</td>
      <td>120</td>
    </tr>
    <tr>
      <td>7.3</td>
      <td>Enzyme-efficient states in an example network</td>
      <td>121</td>
    </tr>
    <tr>
      <td>7.4</td>
      <td>Calculation of optimal states</td>
      <td>123</td>
    </tr>
    <tr>
      <td>7.5</td>
      <td>Translating enzyme efficiency into cell growth rate</td>
      <td>126</td>
    </tr>
    <tr>
      <td>7.6</td>
      <td>Application to central metabolism in E. coli bacteria</td>
      <td>127</td>
    </tr>
    <tr>
      <td>7.7</td>
      <td>Concluding remarks</td>
      <td>130</td>
    </tr>
    <tr>
      <td>8</td>
      <td>Principles of cell growth</td>
      <td>133</td>
    </tr>
    <tr>
      <td>8.1</td>
      <td>Introduction</td>
      <td>133</td>
    </tr>
    <tr>
      <td>8.2</td>
      <td>Fundamental modeling assumptions of microbial growth</td>
      <td>135</td>
    </tr>
    <tr>
      <td>8.2.1</td>
      <td>Conservation of mass and quasi-steady-state assumption</td>
      <td>136</td>
    </tr>
    <tr>
      <td>8.2.2</td>
      <td>Proteome allocation assumption</td>
      <td>137</td>
    </tr>
    <tr>
      <td>8.2.3</td>
      <td>Mathematical description of reaction fluxes</td>
      <td>137</td>
    </tr>
    <tr>
      <td>8.2.4</td>
      <td>Volume and surface area assumptions</td>
      <td>138</td>
    </tr>
    <tr>
      <td>8.3</td>
      <td>Growth laws derived from basic modeling assumptions</td>
      <td>138</td>
    </tr>
    <tr>
      <td>8.4</td>
      <td>Mechanistic links between cellular trade-offs, gene expression, and growth</td>
      <td>143</td>
    </tr>
    <tr>
      <td>8.4.1</td>
      <td>Model definitions</td>
      <td>144</td>
    </tr>
    <tr>
      <td>8.4.2</td>
      <td>Model predictions</td>
      <td>145</td>
    </tr>
  </tbody>
</table>
<table><tbody><tr><td>12.2 Hazard rate approach to cell division</td><td>173</td></tr><tr><td>12.3 Cell-division control as discrete-time linear response process</td><td>177</td></tr><tr><td>12.4 Coordination of cell division with different cell-cycle processes</td><td>179</td></tr><tr><td>12.5 Protein sectors and cell division</td><td>185</td></tr><tr><td>12.6 Control of cell division across species and kingdoms</td><td>190</td></tr><tr><td>12.7 Concluding remarks</td><td>192</td></tr><tr><td><b>13 Cell behavior in the face of uncertainty</b></td><td><b>195</b></td></tr><tr><td>13.1 Strategies to cope with uncertainty: a financial analogy</td><td>196</td></tr><tr><td>13.2 Modeling cells growing in uncertain environments</td><td>197</td></tr><tr><td>13.3 Optimization in uncertain environments</td><td>199</td></tr><tr><td>13.3.1 Long-term versus short term optimization</td><td>199</td></tr><tr><td>13.3.2 Trade-offs at intermediate time scales</td><td>201</td></tr><tr><td>13.4 Strategies in correlated environments</td><td>205</td></tr><tr><td>13.5 Perspectives</td><td>207</td></tr><tr><td><b>14 Economy of organ shapes and function</b></td><td><b>211</b></td></tr><tr><td>14.1 Optimization of organs and systems</td><td>211</td></tr><tr><td>14.1.1 Organs and constraints</td><td>211</td></tr><tr><td>14.1.2 The example of the lung</td><td>213</td></tr><tr><td>14.2 The lung as a model organ for optimization under constraints</td><td>217</td></tr><tr><td>14.2.1 Lung morphology, a complex structure</td><td>217</td></tr><tr><td>14.2.2 Lung dynamics: where physics enters the play</td><td>218</td></tr><tr><td>14.2.3 The energy expenditure or the cost of breathing</td><td>220</td></tr><tr><td>14.3 Allometric scaling laws for respiration and ventilation</td><td>222</td></tr><tr><td>14.3.1 The emergence of scaling relations in nature</td><td>222</td></tr><tr><td>14.3.2 A brief history of allometry</td><td>223</td></tr><tr><td>14.3.3 Allometry: a mechanistic approach</td><td>224</td></tr><tr><td>14.3.4 Allometric relations for the respiratory system</td><td>225</td></tr><tr><td>14.4 Conclusion and perspectives</td><td>229</td></tr><tr><td><b>Bibliography</b></td><td><b>231</b></td></tr><tr><td><b>A Metabolism</b></td><td><b>273</b></td></tr><tr><td>A.1 Derivation of enzymatic reaction rate models</td><td>273</td></tr><tr><td>A.1.1 Derivation of the single substrate, irreversible rate equation</td><td>273</td></tr><tr><td>A.1.2 Derivation of a two substrate, irreversible rate equation</td><td>275</td></tr></tbody></table>
<table>
  <tbody>
    <tr>
      <td>8.4.3 Applications of the model</td>
      <td>146</td>
    </tr>
    <tr>
      <td>8.5 Concluding remarks</td>
      <td>147</td>
    </tr>
    <tr>
      <td>9 Universal features of autocatalytic systems</td>
      <td>151</td>
    </tr>
    <tr>
      <td>9.1 Stoichiometric versus dynamical autocatalysis</td>
      <td>151</td>
    </tr>
    <tr>
      <td>9.1.1 Stoichiometric autocatalysis</td>
      <td>151</td>
    </tr>
    <tr>
      <td>9.1.2 Dynamical autocatalysis</td>
      <td>154</td>
    </tr>
    <tr>
      <td>9.2 von Neumann's model of an expanding economy</td>
      <td>155</td>
    </tr>
    <tr>
      <td>9.2.1 Leontief's production function</td>
      <td>156</td>
    </tr>
    <tr>
      <td>9.2.2 Liebig's law</td>
      <td>158</td>
    </tr>
    <tr>
      <td>9.2.3 Application to metabolism</td>
      <td>158</td>
    </tr>
    <tr>
      <td>9.3 Concluding remarks</td>
      <td>160</td>
    </tr>
    <tr>
      <td>10 Resource allocation in complex cell models</td>
      <td>163</td>
    </tr>
    <tr>
      <td>10.1 Detailed resource allocation models of cells</td>
      <td>163</td>
    </tr>
    <tr>
      <td>10.1.1 Replacing enzyme kinetics by linear catalytic constraints</td>
      <td>164</td>
    </tr>
    <tr>
      <td>10.1.2 Overview of existing FBA extensions</td>
      <td>165</td>
    </tr>
    <tr>
      <td>10.1.3 Why maximize the growth rate?</td>
      <td>165</td>
    </tr>
    <tr>
      <td>10.2 The basic constraints in resource allocation models</td>
      <td>166</td>
    </tr>
    <tr>
      <td>10.2.1 Steady-state and mass-conservation constraints</td>
      <td>166</td>
    </tr>
    <tr>
      <td>10.2.2 Catalytic constraints</td>
      <td>166</td>
    </tr>
    <tr>
      <td>10.2.3 Protein density constraints</td>
      <td>168</td>
    </tr>
    <tr>
      <td>10.2.4 Interpreting the consequences of the additional constraints</td>
      <td>168</td>
    </tr>
    <tr>
      <td>10.3 Resource Balance Analysis: model construction and simulation</td>
      <td>169</td>
    </tr>
    <tr>
      <td>10.3.1 Building a draft RBA model</td>
      <td>169</td>
    </tr>
    <tr>
      <td>10.3.2 Mathematical description of a RBA problem</td>
      <td>169</td>
    </tr>
    <tr>
      <td>10.3.3 Simulation and analysis of RBA models</td>
      <td>171</td>
    </tr>
    <tr>
      <td>10.3.4 Calibration of model parameters</td>
      <td>172</td>
    </tr>
    <tr>
      <td>10.3.5 Enzyme efficiencies: use of -omics data-informed k<sub>app</sub> vs. naïve k<sub>cat</sub> values</td>
      <td>172</td>
    </tr>
    <tr>
      <td>10.4 Biomass composition as a constraint or as a prediction</td>
      <td>173</td>
    </tr>
    <tr>
      <td>10.5 Concluding remarks</td>
      <td>174</td>
    </tr>
    <tr>
      <td>11 Optimal cell behavior in time</td>
      <td>177</td>
    </tr>
    <tr>
      <td>11.1 Introduction</td>
      <td>177</td>
    </tr>
    <tr>
      <td>11.2 Mathematical form of dynamic optimization problems</td>
      <td>179</td>
    </tr>
    <tr>
      <td>11.3 Enzyme expression in metabolic pathways</td>
      <td>180</td>
    </tr>
    <tr>
      <td>11.4 Coarse-grained models of cellular growth</td>
      <td>183</td>
    </tr>
    <tr>
      <td>11.5 Dynamic flux balance analysis (dFBA) of metabolic networks</td>
      <td>187</td>
    </tr>
    <tr>
      <td>11.6 Concluding remarks</td>
      <td>191</td>
    </tr>
    <tr>
      <td>12 Diversity of metabolic fluxes in a cell population</td>
      <td>195</td>
    </tr>
  </tbody>
</table>
<table><tbody><tr><td>A.1.3 Derivation of the single substrate, reversible rate equation</td><td>275</td></tr><tr><td>A.1.4 Derivation of two substrate, reversible rate equation for simultaneous binding of substrates</td><td>278</td></tr><tr><td>A.2 Example Metabolic Models</td><td>278</td></tr><tr><td>A.2.1 A simple model illustrating product activation</td><td>278</td></tr><tr><td><b>B Solutions to problems</b></td><td><b>281</b></td></tr><tr><td>B.1 What makes up a cell</td><td>281</td></tr><tr><td>B.2 Metabolism</td><td>283</td></tr><tr><td>B.3 The space of metabolic flux distributions</td><td>284</td></tr><tr><td>B.4 Rating metabolic pathways by enzyme efficiency</td><td>285</td></tr></tbody></table>
<table>
  <tbody>
    <tr>
      <td>12.1 Introduction</td>
      <td>195</td>
    </tr>
    <tr>
      <td>12.2 Sources of variability and uncertainty in metabolism</td>
      <td>197</td>
    </tr>
    <tr>
      <td>12.3 Probability densities over the flux polytope</td>
      <td>198</td>
    </tr>
    <tr>
      <td>12.4 Representing heterogeneity and uncertainty</td>
      <td>202</td>
    </tr>
    <tr>
      <td>12.4.1 Maximum Likelihood, Maximum a Posteriori and Bayesian inference</td>
      <td>202</td>
    </tr>
    <tr>
      <td>12.4.2 MaxEnt inference</td>
      <td>203</td>
    </tr>
    <tr>
      <td>12.5 Representing optimal populations</td>
      <td>207</td>
    </tr>
    <tr>
      <td>12.6 Concluding remarks</td>
      <td>210</td>
    </tr>
    <tr>
      <td>13 Cells in the face of uncertainty</td>
      <td>213</td>
    </tr>
    <tr>
      <td>13.1 Introduction</td>
      <td>213</td>
    </tr>
    <tr>
      <td>13.2 Strategies to cope with uncertainty: a financial analogy</td>
      <td>214</td>
    </tr>
    <tr>
      <td>13.3 Modeling cells growing in uncertain environments</td>
      <td>215</td>
    </tr>
    <tr>
      <td>13.4 Optimization in uncertain environments</td>
      <td>217</td>
    </tr>
    <tr>
      <td>13.4.1 Long-term versus short-term optimization</td>
      <td>217</td>
    </tr>
    <tr>
      <td>13.4.2 Trade-offs at intermediate time scales</td>
      <td>218</td>
    </tr>
    <tr>
      <td>13.5 Strategies in correlated environments</td>
      <td>223</td>
    </tr>
    <tr>
      <td>13.6 Concluding remarks</td>
      <td>224</td>
    </tr>
    <tr>
      <td>14 Strategies for cell cycle control</td>
      <td>229</td>
    </tr>
    <tr>
      <td>14.1 Introduction: the decision to divide illustrated through single-cell E. coli data.</td>
      <td>229</td>
    </tr>
    <tr>
      <td>14.2 Hazard rate approach to cell division</td>
      <td>231</td>
    </tr>
    <tr>
      <td>14.3 Cell-division control as a discrete-time linear response process</td>
      <td>234</td>
    </tr>
    <tr>
      <td>14.4 Coordination of cell division with different cell-cycle processes</td>
      <td>235</td>
    </tr>
    <tr>
      <td>14.5 Protein sectors and cell division</td>
      <td>241</td>
    </tr>
    <tr>
      <td>14.6 Control of cell division across species and kingdoms</td>
      <td>242</td>
    </tr>
    <tr>
      <td>14.7 Concluding remarks</td>
      <td>245</td>
    </tr>
    <tr>
      <td>15 Economy of organ form and function</td>
      <td>249</td>
    </tr>
    <tr>
      <td>15.1 Optimization of organs and systems</td>
      <td>249</td>
    </tr>
    <tr>
      <td>15.1.1 Organs and constraints</td>
      <td>249</td>
    </tr>
    <tr>
      <td>15.1.2 Energy conversion in living systems</td>
      <td>251</td>
    </tr>
    <tr>
      <td>15.1.3 The example of the lung</td>
      <td>251</td>
    </tr>
    <tr>
      <td>15.2 The lung as a model organ for optimization under constraints</td>
      <td>254</td>
    </tr>
    <tr>
      <td>15.2.1 Lung morphology, a complex structure</td>
      <td>255</td>
    </tr>
    <tr>
      <td>15.2.2 Lung dynamics: where physics enters the play</td>
      <td>255</td>
    </tr>
    <tr>
      <td>15.2.3 The energy expenditure or the cost of breathing</td>
      <td>257</td>
    </tr>
    <tr>
      <td>15.3 Allometric scaling laws for respiration and ventilation</td>
      <td>259</td>
    </tr>
    <tr>
      <td>15.3.1 The emergence of scaling relations in nature</td>
      <td>259</td>
    </tr>
    <tr>
      <td>15.3.2 A brief history of allometry</td>
      <td>259</td>
    </tr>
    <tr>
      <td>15.3.3 Allometry: a mechanistic approach</td>
      <td>261</td>
    </tr>
  </tbody>
</table>
# Preface

Wolfram Liebermeister

How can a cell maintain itself as a living being? Living cells, shaped by billions of years of evolution, have developed many ways to adapt to their environment, e.g. by regulation of gene expression. But the rules of physics and chemistry enforce certain boundaries on what cells can achieve and how they can allocate their own resources. Shaped by evolution, cells "do certain things right", and computational models of cells often assume that this "doing something right" can be described by evoking optimality principles. The goal of this book is to uncover some of these governing principles. While biological optimality is often contested for good reasons, theories based on economic principles can explain many observations (about cell growth or the usage of cellular resources) much better than purely mechanistic models. Methods such as Flux Balance Analysis are well established, but the idea of resource allocation is gaining ground, and metaphors like "currency metabolites" or "energy budget" are common in cell biology, optimality principles are often applied ad hoc, and a coherent picture - in which many single observations or models would have their place - is still missing. This book - a free and open textbook to which anyone is invited to contribute - gives an overview of established approaches to "cellular economics", from descriptions of simple metabolic systems to cell growth, variability, and dynamic behavior.

Compared to non-living matter, living organisms have some very specific abilities. How can a tiny cell maintain itself, whereas a cloud fades away? How can it grow and divide, how can it make copies of itself? Or in other words, what does it take to be alive? There is no special "life force"; what makes matter alive is its microscopic structure or molecular organization. Living matter follows the laws of physics. However, to understand life, physics alone is not enough! On the one hand, living beings are complex on many levels of organization, from biomolecules to cells, the body, population, and ecosystem. Each of these levels follows its own laws, but in some cases a change on the lowest level - a point mutation - may change the fate of a population. On the other hand, living systems do not just exist as they are, but have been shaped by billions of years of evolution. This is also why some of their features - look like they were perfectly engineered. Since we do not know - and certainly cannot always consider - evolution in its entirety, we often use "optimality" as a shortcut. To explain a biological feature, like the shape of dolphin, instead of telling all the story of dolphin evolution, and how changes in shape appeared, and some were conserved, we simply say: this is the shape that functions best, and apparently evolution, by mutation and selection, converged to this shape.

In this book we mostly focus on microbes, and how they function internally: what compounds they need to produce, and how, in order to live and self-replicate. We can describe this on three different levels. Level 1, the "inventory" of a cell, in a molecular view, consists of molecules and biochemical reactions, which form a complex chemical network. Level 2, the dynamics of molecule concentrations, is determined by physical laws like the conservation of mass and by specific biochemical regulation mechanisms, for example molecular recognition. But there is also a third level, concerning the function (or possibly, optimality) of these dynamics, for which economic metaphors are appropriate: given a limited "protein budget", what biochemical pathways should a cell prioritize to thrive, grow, and survive? In this book we focus on the third layer, the "economy of the cell", which in fact encompasses the previous two.

What do we mean by the "economy of the cell"? Economic theory is of course vast, and only a tiny bit of it has made its way to biology so far. In this book, by "economy" we mostly mean resource allocation and scheduling problems: what is the best allocation of protein resources in a bacterial cell (see the graphics above)? How should photosynthetic bacteria
<table>
  <tbody>
    <tr>
      <td>15.3.4 Allometric relations for the respiratory system</td>
      <td>261</td>
    </tr>
    <tr>
      <td>15.4 Concluding remarks</td>
      <td>265</td>
    </tr>
    <tr>
      <td>Bibliography</td>
      <td>269</td>
    </tr>
    <tr>
      <td>Index</td>
      <td>303</td>
    </tr>
    <tr>
      <td>List of boxes</td>
      <td>308</td>
    </tr>
    <tr>
      <td>Further information</td>
      <td>310</td>
    </tr>
    <tr>
      <td>Mathematical symbols</td>
      <td>310</td>
    </tr>
    <tr>
      <td>List of abbreviations</td>
      <td>312</td>
    </tr>
    <tr>
      <td>Author contributions</td>
      <td>313</td>
    </tr>
    <tr>
      <td>Reading recommendations</td>
      <td>315</td>
    </tr>
    <tr>
      <td>A Appendix "Cell metabolism"</td>
      <td>321</td>
    </tr>
    <tr>
      <td>A.1 Dynamics of metabolism: experimental evidence and model-based explanations</td>
      <td>321</td>
    </tr>
    <tr>
      <td>A.1.1 Flux switching / regulation</td>
      <td>321</td>
    </tr>
    <tr>
      <td>A.1.2 Bistability</td>
      <td>322</td>
    </tr>
    <tr>
      <td>A.1.3 Oscillations</td>
      <td>323</td>
    </tr>
    <tr>
      <td>A.2 Derivation of enzymatic rate laws</td>
      <td>325</td>
    </tr>
    <tr>
      <td>A.2.1 Derivation of the single substrate, irreversible rate law</td>
      <td>325</td>
    </tr>
    <tr>
      <td>A.2.2 Derivation of a two substrate, irreversible rate law</td>
      <td>326</td>
    </tr>
    <tr>
      <td>A.2.3 Derivation of the single substrate, reversible rate law</td>
      <td>326</td>
    </tr>
    <tr>
      <td>A.2.4 Derivation of two-substrate, reversible rate law with simultaneous binding</td>
      <td>329</td>
    </tr>
    <tr>
      <td>A.3 A simple model illustrating product activation</td>
      <td>330</td>
    </tr>
    <tr>
      <td>B Appendix "Optimization of metabolic states"</td>
      <td>331</td>
    </tr>
    <tr>
      <td>B.1 A model of central metabolism in Escherichia coli</td>
      <td>331</td>
    </tr>
    <tr>
      <td>B.2 More results for the E. coli central metabolism model</td>
      <td>336</td>
    </tr>
    <tr>
      <td>C Appendix "Strategies for cell cycle control"</td>
      <td>339</td>
    </tr>
    <tr>
      <td>C.1 Equations for birth size</td>
      <td>339</td>
    </tr>
    <tr>
      <td>C.2 Growth laws</td>
      <td>340</td>
    </tr>
    <tr>
      <td>D Solutions to problems</td>
      <td>343</td>
    </tr>
    <tr>
      <td>D.1 Chapter "The inventory of a cell"</td>
      <td>343</td>
    </tr>
    <tr>
      <td>D.2 Chapter "Cell metabolism"</td>
      <td>345</td>
    </tr>
    <tr>
      <td>D.3 Chapter "Metabolic flux distributions"</td>
      <td>346</td>
    </tr>
    <tr>
      <td>D.4 Chapter "The enzyme cost of metabolic fluxes"</td>
      <td>346</td>
    </tr>
    <tr>
      <td>D.5 Chapter "Optimization of metabolic states"</td>
      <td>347</td>
    </tr>
    <tr>
      <td>D.6 Chapter "Universal features of autocatalytic systems"</td>
      <td>348</td>
    </tr>
    <tr>
      <td>D.7 Chapter "Cells in the face of uncertainty"</td>
      <td>350</td>
    </tr>
  </tbody>
</table>
<figure><img src="image_6.png" /></figure>

Figure 1: Protein abundances in the yeast Saccharomyces cerevisiae. Measured amounts of different sorts of proteins are shown as areas, proteins of related functions are arranged into larger regions, shown by colors. Why does the cell invest such a large fraction of their protein budget into the glycolysis pathway? Such economic questions are central in this book.

adjust these investments during the day-night cycle? Our answers to such questions, also in this book, are often based on an underlying assumption of optimality. But often we simply consider all the constraints under which a cell needs to act, and figure out what cellular behaviors are possible.

As we look at cells from the perspective of resource allocation, we will neglect other aspects: we will rarely talk about regulation (e.g. the mechanisms for regulation of gene expression), and even more rarely about gene or protein sequences. Instead, we assume that certain mechanisms are in place in the cell, and that molecules encoded by sequences exist, and either ask why (that is, for what functional reason) they are the way they are, or what the cell can do with them to perform certain tasks. This often means that we assume a mechanistic system with possible “choices” (among flux profiles, expression levels, enzyme parameters, etc), and ask, first, what choices exist (considering all the constraints) and, second, how profitable these choices are for the cell (assuming certain objectives). While we are hardly concerned with genetics, we are certainly interested in how optimality may arise from evolution - to connect the two, we need to think about fitness (how long-term fitness can be defined, and how it gives rise to “momentary” or “local” optimization objectives in a given part of the cell).

The topic of this book has emerged from the field of systems biology, and accordingly we address students and researchers in related fields, with a background in biology, physics, engineering, or math. For readers without a biological background, we recommend the book "Cell biology by the numbers" (book.bionumbers.org), which takes the reader on a journey through various aspects of cell biology.

Finally, why did we choose to write this textbook as a collaborative, open book? Publishing with a commercial publisher has a number of downsides, most of them reflecting a clash of interests between publishers, authors, and readers. We wish to write this book as a community for the community. Many colleagues were and are involved, and we would be glad to welcome you as part of the team!
# Chapter 1

## The cell as a factory

Ohad Golan

The term “metabolism” is usually used to describe the chemical reactions that occur within biological organisms to produce and transform molecules needed to sustain life. While this definition is useful, it gives no scientific or mathematical ground for the analysis of metabolic systems. Here we consider metabolic systems in a much broader sense, and in order to provide a logical framework for the analysis of metabolic systems, we begin with a more formal definition that also covers systems outside biology. Metabolic system: “A well defined system that takes up nutrients and uses them to sustain itself”. This definition can be represented by a simple chemical equation:

$$ \text{nutrients} \rightarrow \text{metabolic system} + \text{waste products} $$

The process is carried out by the metabolic system itself - a point we will expand on later. The waste products are typical leftovers of the reaction in case such products exist. The most obvious example of a metabolic system is a biological system that takes up substances from its environment and assimilates them to reproduce its own components (often summarized as “biomass”). The chemical equation of metabolism for biological systems is:

$$ c_1 \text{ sugar} + c_2 \text{ oxygen} + c_3 \text{ ammonia} \rightarrow \text{biomass} + \text{waste products} $$

The equation describes all the nutrients, including sugar, oxygen and ammonia, that are necessary to sustain a biological system. Other molecules such as certain metals and phosphate are also necessary for the reaction to occur but we neglect them for the sake of brevity. The typical waste products are CO2, H2O and other possible chemicals secreted by the system.
# Preface

Wolfram Liebermeister

How can a cell maintain itself as a living being? Living cells, shaped by billions of years of evolution, have developed many ways to adapt to their environment, for example, by regulation of gene expression. But the rules of physics and chemistry enforce certain boundaries on what cells can achieve and how they can allocate their own resources. Shaped by evolution, cells "do certain things right", and computational models of cells often assume that this "doing something right" can be described by evoking optimality principles. The goal of this book is to uncover some of these governing principles. Although biological optimality is often contested for good reasons, theories based on economic principles can explain many observations (about cell growth or the usage of cellular resources) much better than purely mechanistic models. Methods such as Flux Balance Analysis are well established, but the idea of resource allocation is gaining ground, and metaphors like "currency metabolites" or "energy budget" are common in cell biology. Optimality principles are often applied ad hoc, and a coherent picture in which many single observations or models would have their place is still missing. This book - a free and open textbook to which anyone is invited to contribute - gives an overview of established approaches to "cellular economics", from descriptions of simple metabolic systems to cell growth, variability, and dynamic behavior.

Compared to non-living matter, living organisms have some very specific abilities. How can a tiny cell maintain itself, while a cloud fades away? How can it grow and divide, how can it make copies of itself? Or in other words, what does it take to be alive? There is no special "life force"; what makes matter alive is its microscopic structure or molecular organization. Living matter follows the laws of physics. However, to understand life, physics alone is not enough! On the one hand, living beings are complex at many levels of organization, from biomolecules to cells, body, population, and ecosystem. Each of these levels follows its own laws, but in some cases, a change on the lowest level, a point mutation, may change the fate of a population. On the other hand, living systems do not just exist as they are, but have been shaped by billions of years of evolution. This is also why some of their features look like they were perfectly engineered. Since we do not know – and certainly cannot always consider – evolution in its entirety, we often use "optimality" as a shortcut. To explain a biological feature, like the shape of dolphins, we might tell all the story of dolphin evolution and how changes in shape appeared and some were conserved. But instead, we may simply say: this is the shape that functions best, and apparently evolution, by mutation and selection, converged to this shape.

In this book, we mostly focus on microbes, and how they function internally: what compounds they need to produce, and how, in order to live and self-replicate. We can describe this at three different levels. Level 1, the 'inventory' of a cell, from a molecular point of view, consists of molecules and biochemical reactions, which form a complex chemical network. Level 2, the dynamics of molecule concentrations, is determined by physical laws like the conservation of mass and by specific biochemical regulation mechanisms, for example molecular recognition. But there is also a third level, concerning the function (or possibly optimality) of these dynamics, for which economic metaphors are appropriate. Given a limited "protein budget", what biochemical pathways should a cell prioritize to thrive, grow, and survive? In this book, we focus on the third layer, the "economy of the cell", which, in fact, encompasses the previous two.

What do we mean by the "economy of the cell"? Economic theory is, of course, vast and only a small bit of it
Economic analogy 1.A

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

Comparison of black box description of microbial growth to that of an economic system.

Another, less typical, example of a metabolic system in this general sense is an economic firm supplying a product. In this example we will consider for simplicity a firm that builds houses, but any kind of product can be equally used. Such a firm takes in land and different construction materials, these would be equivalent to the nutrients, and by the use of the labor force, which would be equivalent to the proteins, uses them to build houses. The houses are then sold in order to sustain and increase the value of the firm, just as the biological cell sustains itself. The chemical equation of metabolism for a construction firm is:

$$c_1 \text{ land} + c_2 \text{ construction materials} \rightarrow \text{value} + c_3 \text{ waste}$$

In this book we focus on the analysis of biological metabolic systems. However, given that economic systems fall under the same definition of a metabolic system, we will use them as analogies to simplify explanations. Whenever an analogy to economical systems is presented in this book, it will be displayed in an "Economic analogy" box such as the one above.

Many metabolic systems use a strategy of reproduction to sustain itself. That is, the nutrients are used to make more of the metabolic system and not only maintain it. This means that the output of the metabolic process is more of the metabolic system. This creates a system that when unlimited resources are available grows exponentially - the metabolic system takes in nutrients which it uses to replicate, the output of the process is also the metabolic system which takes in more nutrients and also replicates. Metabolism includes all the processes that take place in order to carry out the chemical equation of metabolism - that is, everything that happens inside the black box described above. The most fundamental model of a metabolic system is one that takes nutrients from the environment, breaks them down into building blocks, and uses these building blocks to sustain itself. In biological systems, these processes are termed catabolism and anabolism. In catabolism the cell takes up carbon and nitrogen sources from the environment and uses them to synthesize the necessary building blocks: amino acids, nucleic acids and fatty acids. In the anabolic process, the building blocks are used to form biomass which includes the functioning systems of the cells - proteins, DNA strands and the membrane. Each process is catalyzed by a specific set of enzymes. These enzymes that catalyze the reactions are actually the metabolic system itself. When the cell grows it makes more enzymes to catalyze more reactions - this is the reproduction process which leads to exponential growth.
<figure><img src="image_6.png" /><figcaption>Figure 1: Protein abundances in the yeast Saccharomyces cerevisiae. Measured amounts of different sorts of proteins are shown as areas, proteins of related functions are arranged into larger regions, shown by colors. Why does the cell invest such a large fraction of their protein budget into the glycolysis pathway? Such "economic" questions are central in this book.</figcaption></figure>

has made its way into biology so far. In this book, by "economy" we mean primarily resource allocation and scheduling problems: What is the best allocation of protein resources in a bacterial cell (see the graphic above)? How should photosynthetic bacteria adjust these investments during the day-night cycle? Our answers to such questions, also in this book, are often based on an underlying assumption of optimality. But often we simply consider all the constraints under which a cell needs to act and figure out what cellular behaviors are possible.

As we look at cells from the perspective of resource allocation, we will neglect other aspects: we will rarely talk about regulation (e.g. the mechanisms for regulation of gene expression), and even more rarely about gene or protein sequences. Instead, we assume that certain mechanisms are in place in the cell, and that molecules encoded by sequences exist, and either ask why (that is, for what functional reason) they are the way the are, or what the cell can do with them to perform certain tasks. This often means that we assume a mechanistic system with possible 'choices' (among flux profiles, expression levels, enzyme parameters, etc.) and ask, first, what choices exist (considering all the constraints) and, second, how profitable these choices are for the cell (assuming certain objectives). While we are hardly concerned with genetics, we are certainly interested in how optimality may arise from evolution - to connect the two, we need to think about fitness (how long-term fitness can be defined and how it gives rise to "momentary" or "local" optimization objectives in a given part of the cell).

The source of inspiration for the book and the questions (discussions) that motivated the investigation of the various mechanisms the cell uses to allocate resources in the most efficient way possible were a series of events in formal settings such as an annual summer workshop, the monthly online Forum "Economic principles in cell physiology", and more informal hackathons. The development of the book is an endeavor that is truly global in scope, drawing on the expertise and integrating the contributions of scientists who were members of a global network (formed a global community) representing research institutions located in more than a dozen countries on three continents. Those who contributed to the creation of the book recognize that the success they achieved in bringing it to a satisfactory conclusion is due, in no small part, to the support of the institutions with which they are affiliated and are grateful to INRAE, the Learning Planet Institute Paris, and the home institutions of all other authors (as well as the taxpayers who finance these institutions) who encouraged the creation of the book by providing its authors and contributors with the time and space necessary to sustain its development and achieve its completion.
## Economic analogy 1.B

In an analogy to an economic system of a construction firm, the catabolic process would correspond to the purchase and transfer of the construction materials to the construction site and the anabolic processes would correspond to the construction of the house, the catalytic enzymes would correspond to the workers carrying out the transfer of the materials and construction process. The growth process in bacteria is analogous to the growth of the firm - when the construction of the house is complete, the house is then sold to increase the value of the firm. The increased value enables the firm to hire more work force and construct more houses.

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

**Analogy of the catabolic and anabolic processes in biological cell to that of an economic system, in this case a construction firm.**

The metabolic system controls the allocation of the available resources. When coordinating the process, the metabolic system decides between different strategies of how to best use the resources. For example, the cell decides how much of the available enzymes to allocate to the catabolic process and much to the anabolic process. When making these decisions, the cell takes into account different physical constraints. Examples of these physical constraints are: a limited physical volume to maintain and carry out the metabolic processes, a limited surface area that constrains the ability to take up nutrients or limiting thermodynamic constraints on the activity of the enzymes. There is no one best strategy that is always utilized - different organisms decide on different strategies based on the living conditions. This decision process is carried out by many mechanisms in the cell with the main information processing core of the cell being the DNA. The decisions carried out by the cell are based on the evolutionary process the metabolic system has gone through during its existence. A description of cell information processing and how it is carried out is brought in appendix A.

## Economic analogy 1.C

In an analogy to the economic system of a construction company, the manager of the company faces the decision of how to allocate his workforce, how many of his workers to assign to bring in materials from the factory and how many of his workers to assign to the construction process. In a similar way to the biological system, there are different limiting constraints such as a difficult topographic constructions site or limited available resources. Unlike the biological cell though, in which the decision making is embedded by the evolutionary processes, here the decision is made by the manager of the construction site.

So far, we describe the most fundamental metabolic system. This is a coarse-grained description in which the cell catabolizes the nutrients into one type of precursor and does not take into account all the processes that take place in catabolism and anabolism. In a biological metabolic system, the cell requires multiple different types of precursors such as amino acids, nucleic acids and fatty acids. To create all the different precursors, the cell takes in nutrients from the environment and through a set of chemical reactions, turns the nutrients into the precursors that are necessary for the
Finally, why did we choose to write this textbook as a collaborative, open book? Publishing with a com-
mercial publisher has several downsides, most of which reflect a clash of interests between publishers,
authors, and readers. We wish to write this book as a community for the community. Many colleagues
were and are involved, and we would be glad to welcome you as part of the team! If you would like to
join for writing, reviewing chapters, designing graphics, or discussing new ideas, please have a look at our
website and get in touch.
<figure><img src="image_9.png" /><figcaption>Figure 1.1: In the metabolic process the cell takes in available nutrients and through a set of biochemical reactions, turns them into precursors necessary for growth. The chemical reactions are carried out by proteins in the cell.</figcaption></figure>

cell to sustain itself. Each chemical reaction in the metabolic process is carried out by proteins. The different precursors can be produced through different sets of chemical reactions known as metabolic pathways and the different chemicals in the metabolic pathways are known as metabolites. The cell decides which metabolic pathway to activate by producing the necessary enzymes. In an analogy to the economic system of a construction company, each chemical reaction is one process carried out by a worker – for example, the assembly of the frame of the house requires a carpenter while the next step in the construction pathway is to place the foundation in the correct location which is done by another worker. The workers are analogous to enzymes and the different parts necessary for construction are the metabolites.

Many metabolic pathways have overlapping metabolite reactants and products. Some of the key parameters that describe metabolic pathways are the enzymatic catalytic rates. These parameters describe the rate at which the enzymes turn metabolites and at what concentration of reactants they saturate. In the analogy to the construction firm, the enzymatic parameters are parameters that describe the rate of work of each worker. Given that each metabolic pathway is made of a series of chemical reactions, each with different catalytic rates, the different enzymes of each pathway must be coordinated perfectly to avoid any excess buildup of metabolites - just like in a factory assembly line all the workers must be coordinated together to avoid buildup of an intermediate.

In order to make sense of the complex network of metabolic reactions, different mathematical models were developed. The models take into account the known experimental data of the different reactions and compile them together to predict the overall response of the system in different growth conditions.

The metabolic models described above describe biological systems that are disconnected from the environment except for some artificial supply of nutrients. In natural ecological systems, different organisms exist together under a limited supply of nutrients. They compete or cooperate to best utilize the limited available resources. All organisms try to improve their chances for survival according to the laws of evolution. In such a setting, the metabolism of organisms living in an ecological system is directly dependent on the other organisms that coexist with them. In an analogy to an economic system, this would be a competition between different companies for the same possible clientele. Some companies would compete against each other while others cooperate to improve their profit.
# Chapter 2

## What makes up a cell

Pranas Grigaitis and Diana Széliová

### Chapter highlights

* Cells use the same building blocks to give rise to a high number of molecular species

* There are many parallel processes happening in cells, with similar precursors

* Composition of cells is environment-dependent

* Different time-scales give rise to coordination of processes

### Describing and counting cellular components

Cells contain a diverse spectrum of molecules, needed to create two cells out of one (as Rudolf Virchow proposed, *omnis cellula e cellula*, all cells come from cells). These molecules come in different sizes and properties, and therefore create a demand for a cell to keep these components in different places (*spatial* organization) with different patterns of use (*temporal* organization), and book-keep their quantities. Cell composition directly influences the function of the cell: thus we observe different cellular make-up in different organisms or even in different cells of the same organism. Both historical research and the latest advancements in instrumentation allow us to characterize the constituents of cells in more and more depth. Thus in this chapter, we will do a census of cellular components: we will discuss what molecules make up a cell, what they are derived from, how to measure these components in the lab and we will briefly consider allocation of resources, directed to synthesize individual cellular components.

#### 2.1. The components of a cell

##### 2.1.1. Cell composition and structures

Although living matter comes in different shapes and sizes, over 99% of the cellular mass can be described by only a handful of chemical elements. 6 most abundant elements form the famous CHNOPS notation: carbon (C), hydrogen (H), nitrogen (N), oxygen (O), phosphorus (P), and sulfur (S). Taken together, these 6 elements encompass the vast majority of the mass, namely, ca. 97.5% in budding yeast *Saccharomyces cerevisiae* [1]. Living cells also contain minute amounts of different metal ions, such as sodium (Na), potassium (K), iron (Fe), molybdenum (Mo) and others – usually facilitating signal transduction or supporting enzymatic catalysis.

In an extremely simplified way, cells can be looked at as bags of fluid-like material, kept together by a membrane. These "bags of things" can also contain other membrane structures inside them, forming so-called organelles. In cell biology,
# Overview

## Getting started

### What is this book about?

In biology, the "economy of the cell" has become increasingly central as a way of understanding cells. In particular, it has been used as a perspective on metabolic states, the allocation of protein resources in cells, and the interplay between production processes and cell growth. In this book, we focus on diverse biological topics of interest and, where possible, use economic analogies to show that, much like in human economics, balance and resource management are crucial for cells. The "economy of the cell" is based on, and is a part of, systems biology, a branch of biology that is typically concerned with networks, large cell biological data sets, and dynamic models.

Functional thinking – as opposed to describing cells mechanistically, as physical objects – is fundamental to biology. In biology, the notion of "function" is justified by the fact that organisms emerged from evolution – that is, as a result of mutation and selection – where completely "nonfunctional" solutions are probably being selected against. Evolution itself is an open-ended process and does not entail any simple criterion for "optimality". Since selection depends on changing environments, and since environments themselves can be shaped by organisms, there is no simple, general criterion for Darwinian fitness (except for the fact, post hoc, that a species managed to survive over a long period of time). However, if we look at the end result – an evolved species, or an evolved trait in microbes, for example, how cells allocate their resources – and assume that this species evolved in a constant environment, it is tempting (and, as we argue here, meaningful) to describe this result by optimality approaches or economic thinking. Hence, it is not by chance that some cell models bear strong resemblances with economic models.

Since the integration of systems biology and economics is long overdue—and no books currently focus on the intersection of these two fields, despite some addressing systems biology from an economic perspective—we decided to write a textbook that covers the fundamentals of cellular production processes, their regulation, and how they can be described in terms of resource allocation, costs, and benefits.

### Who is this book for?

The book focuses on the application of economic principles to cell biology, providing readers with a quantitative framework to understand how cells allocate resources, optimize processes, and make trade-offs. The topic of this book has emerged from the field of systems biology, and accordingly, we address students and researchers in related fields with a background in biology, physics, engineering, or math who want to explore this interdisciplinary field. For students, the textbook offers a structured introduction to the economic principles that govern cellular behavior, starting with basic concepts and advancing to more complex models. For researchers, it provides an overview of the current literature, helping those in related fields quickly grasp key ideas and approaches in this area of study. For readers without a biological background, we recommend the book "Cell biology by the numbers" (book.bionumbers.org), which takes the reader on a journey through various aspects of cell biology.

Our aim is to make this book accessible to as many people as possible by ensuring that the concepts are accessible to everyone, covering both beginner and advanced topics, and by offering it as a free resource.
we call cells *prokaryotic* if they do not possess these membrane structures, and *eukaryotic* if they do. The most common example to illustrate the prokaryote-eukaryote divide are two organisms: prokaryotic bacterium *Escherichia coli* and eukaryotic yeast *Saccharomyces cerevisiae*. They both are organisms, composed of a single cell (thus called *unicellular*), and they both are very small, compared to a typical human cell. However, *E. coli* does not contain any additional membrane structures except from the plasma membrane (which encompasses the cellular contents). Meanwhile, a handful of different organelles can be observed in *S. cerevisiae*. The cellular organization of these cells is shown in Figure 2.1.

Most biological membranes and membrane-based structures, including the plasma membrane itself, have multiple func-
tions (not only separating space), and are highly dynamic. Some membranes can fold into very compact structures with
extremely high surface area (endoplasmic reticulum, Golgi apparatus), occupy different volumes - from small vesicles to
large vacuoles, occupying a major fraction of the cell volume. Moreover, some molecules can form very large structures,
which might be transient (short-lived), thus capturing and defining them remains a major challenge. For these reasons,
the fine structure of cells is unclear - some findings (e.g. organelle contact sites, see [2] for a recent review) hint into
some functional organization of organelles, yet the canonical way to look at the cellular structure remains as to a "bag
of things".

A notable example of a highly specialized organelle is the mitochondrion. The mitochondrion is separated from the rest of the cell by two (outer and inner) membranes; this feature is essential for their function. In eukaryotes, mitochondria are a major hub of metabolism: they house essential biochemical pathways, such as tricarboxylic acid cycle (also known as citric acid-, or Krebs cycle), as well as the so-called *respiratory chain*, the machinery for generating energy with the use of oxygen (see Chapter 3 for more details). While the most biochemical interconversions happen inside the mitochondria (in mitochondrial *matrix*), the respiratory chain proteins are located in the inner mitochondrial membrane: these proteins create an *electrochemical gradient* across this membrane, and use it to drive the conversion of energy, stored in nutrients, into the energy the cell can use (in a form of ATP). What makes mitochondria even more interesting is that they also contain mitochondria-specific genetic information (mitochondrial DNA), which is essential for mitochondria to function inside the cell. In many organisms, the loss of mitochondrial DNA results in impaired growth (in yeasts, that is called the *petite* phenotype) [3], and some organisms cannot grow unless mitochondrial DNA is present (*petite-negative* yeasts).

2.1.2. Biological molecules

Although cells contain many different molecular *species* ("molecular identities"), we can crudely categorize them into *small molecules* and *macromolecules* based on their molecular weight and complexity. Small molecules, as the name suggests, are small chemical compounds, up to 1000 Daltons in mass (1 Dalton = 1 atomic mass unit, 1 amu), and are usually composed of a non-repeating single chemical unit (called *monomer*). Macromolecules, on the contrary, are up to several megadaltons (MDa = $10^6$ Da) in weight, and are frequently composed of multiple monomers (forming so-called *polymers*). Compounds in the cells, both macro- and small molecules, based on their chemical nature, fall into 5 big groups: proteins, nucleic acids (both macromolecules), carbohydrates (exist as both small molecules and polymers), lipids (small molecules), and cofactors/other small molecules.

Proteins are polymers, composed of amino acids. Proteins are an exceptionally diverse class of molecules: in Nature,
20 amino acids can be incorporated into proteins (so-called *proteogenic* amino acids), which, combinatorially provides
20 options for each position in the protein chain. Therefore, there is an enormous amount of possible combinations to
make a protein of a length of 100 amino acids (20¹⁰⁰, to be precise), even for a amino acid chain way shorter than
the average in *E. coli*, around 325 amino acids (*BioNumbers ID* (BNID) [4] 108986). This diversity gives rise to the
spectrum of functions proteins can do, for instance, catalysis (catalytic proteins are also called *enzymes*), transport of
molecules, keeping structural integrity of membranes, and others. Also two notable properties of proteins are that they
(1) need to acquire a specific three-dimensional structure ("to fold") in order to become functionally active, and (2)
sometimes, they also need to form complexes of the same or other proteins (called *multimers*). Protein production is a
major consumer of energy and biosynthetic intermediates in the cell, therefore, in this book we will frequently consider
<figure><img src="image_7.png" /><figcaption>Figure 2: Background knowledge and modeling in this book – Many topics in this book are presented via mathematical models. Models can capture and structure knowledge from biology, chemistry and physics in a mathematical formulation. As a "simplified replica" of reality, they highlight certain aspects of cells that we would like to describe and make them amenable to analysis. In the book, most models either describe cell metabolism (as a whole, or parts of it) or a growing cell as a whole. Aside from the basic description of steady states (in metabolism) or steady growth states (of cells), the book captures some advanced topics related to cell behavior in time, in cell communities, in uncertain environments, or aspects of spatial structure.</figcaption></figure>

The book and its individual chapters can be downloaded from our [book website](#). A new version is released every three months and since the project is still ongoing, the text will be improved edition by edition.

# A guide to the book

## Chapters overview

A main topic of this book is resource allocation in cells. Focusing on metabolism, we can ask, more specifically, about (potentially optimal) configurations of fluxes, protein concentrations, and metabolite concentrations. This question may be given in a simplified form, e.g. as a choice of fluxes under constraints (in Flux Balance Analysis models) or an allocation of a finite protein budget to cellular tasks (as in whole-cell models). But the overall aim behind this is to describe an entire growing cell. If we simplify this again by looking at parts of a cell (e.g. considering small-molecule metabolism only) or looking at "low resolution" (i.e. considering only a few global variables), this leads to different modeling approaches which we explore in this book.

The reader will learn how economic principles such as optimization, resource allocation, and trade-offs can be applied to cellular biology. The chapters are organized to guide the reader from basic concepts to more advanced applications. The book covers foundational topics first and then progresses to more specialized areas, including how to develop and analyze models that explain how cells manage resources and optimize their internal processes. By the end of the book, the readers will have a solid understanding of how economic principles can be used to analyze and model cellular behavior.

The book chapters are related to a number of larger topics, as shown in Figure 3.

1. **The functioning of cells** - After the introductory chapter 1, "The cell as a factory", you will find two chapters with background information about cells and their metabolism. Chapter 2, "An inventory of cell components" describes the main components of a cell, their functions, and their typical abundances in a cell. In a self-replicating cell, these are the components that need to be reproduced while also acting as the "materials" and "machines" that make reproduction possible. Chapter 3, "Cell metabolism", focuses on metabolic reactions and pathways and shows how chemical conversions depend on enzyme kinetics and reaction thermodynamics. Readers familiar with cell biology and metabolic models may
proteins as central players in implementing economic principles in cell physiology.

Nucleic acids are another category of macromolecules; their monomers are called nucleotides. There are two major classes of nucleic acids, RNA (ribonucleic acid) and DNA (deoxyribonucleic acid). RNA and DNA chemically have a slight, yet critical difference: the sugar, which is a part of the nucleotides, differs between RNA (ribose) and DNA (deoxyribose). The two sugars are almost the same but for one chemical group: one of the carbon atoms in ribose is connected to two another carbon atoms, a hydrogen atom, and a chemical group, called hydroxy- ($-OH$). In deoxyribose, the hydroxy-group is substituted with another hydrogen atom, hence the prefix “deoxy-” (“minus oxygen”). RNA and DNA have different functions in the cell: the primary function of DNA is to store genetic information, while RNA can work both as an intermediate agent to transfer that genetic information to protein production (messenger RNA, mRNA) or to participate in catalysis and protein production in general (e.g. transfer and ribosomal RNA, tRNA and rRNA, respectively). Outside the polymers, nucleotides can also act as energy-accumulating compounds (e.g. ATP, adenosine triphosphate) or signaling molecules (e.g. cyclic adenosine monophosphate, cAMP). In this text, we will mostly refer to the energy-storing function of the nucleotides, although other functions, such as signaling, also are essential aspects of describing cell physiology.

### Box 2.A: Macromolecular machines

An important consideration about both proteins and nucleic acids is that they are polymerized by very specialized protein- and protein-nucleic acid complexes. These molecular motors use energy (in terms of ATP equivalents) to form chains of the respective monomers. In the case of proteins, the individual amino acids are combined into a so-called *peptide chain* by a ribosome, a macromolecular complex made from proteins and RNA. The nucleic acids are synthesized by a class of enzymatic complexes, called *nucleic acid polymerases*. There are two major classes of them, specific to the nucleic acid: RNA and DNA polymerases, respectively.

Carbohydrates are another major class of biological molecules, and are important both as monomers and high molecular-weight polymers. Monomeric carbohydrates (sometimes also referred to as sugars) are mainly used as carbon and energy sources for organisms, e.g. glucose or fructose. In oligomeric form (up to 10 monomers), carbohydrate chains are essential for cellular sensing systems, to be specific, receptor-ligand binding. Finally, polymers of carbohydrates usually serve as structural components (part of peptidoglycan, major part of bacterial cell walls) or energy/carbon storage (glycogen in, e.g. yeasts and animal cells, or starch in plants).

Lipids are a vaguely-described class of compounds, which have an overarching similarity, being water-insoluble. The major function of lipids in biological cells is structural: a very abundant subclass of lipids, phospholipids, is an essential constitutent of biological membranes. As discussed in Section 2.1.1, membranes themselves have a variety of functions, which are mostly carried out by lipids (structural) or proteins (transport, sensing, signaling etc.). Some lipids can also undertake other functions, such as signaling (various sterols), or energy storage (tryglycerides, or fats).

As we see, the metabolism of biological molecules is tightly interlinked, although they exhibit major differences in their abundance, size and chemical properties. Macromolecules are present in very low concentrations, and their biosynthesis usually takes minutes. Meanwhile, the time scale of small molecule reactions is usually seconds (or fraction of), and the concentrations of small molecules are usually several magnitudes higher than these of macromolecules. Yet, despite acting at different rates and concentrations, these two types of biological molecules work in an orchestrated manner. To begin with, a number of different small molecules are required to produce both other small molecules and the macromolecules. In return, the macromolecules ensure cell integrity and growth by, among other functions, operating the reaction networks of small molecule interconversions (which we usually refer to as metabolism). Additionally, presence of some small molecules can influence the function of macromolecules, both directly (e.g. essential cofactors, needed for enzymatic reactions; enzyme activation or inhibition), and indirectly (e.g. modulation of gene expression, signaling). Therefore, a lot of different processes have to happen in parallel to ensure the operation of the cells. Having defined the major types of molecules we find in living cells, next we will discuss how abundant are different components of the cells.
<figure><img src="image_8.png" /></figure>

Figure 3: Topics of the book shown as the branches of a tree, and book chapters shown as fruit – The first chapters provide background knowledge, represented by the roots of the tree. In the following chapters we explain different modeling approaches that focus on different aspects, represented by the tree’s trunk. In the chapters on metabolism and on cell models, we assume steady (growth) states and move step by step towards more complex models (resource allocation in cells), shown as the first line of branches of the tree. Finally, we consider more specific aspects such as time, variability, and space, as higher branches of our tree. The numbers in the figure indicate chapters (for chapter titles see text).

skip these two chapters.

2. **Metabolism** - The following four chapters concern metabolic models, starting with models focusing only on metabolic fluxes (chapters 4, "Metabolic flux distributions", and 5, "Optimization of metabolic fluxes") and then continuing with models that consider enzyme kinetic rate laws to link metabolic production to enzyme demand. Chapter 6, "The enzyme cost of metabolic fluxes" assumes that (desired) metabolic fluxes are given and asks how much enzyme is needed to support them, and how metabolite concentrations should be chosen to minimize this enzyme demand. Chapter 7, "Optimization of metabolic states", combines these aspects and presents a general way to determine optimal metabolic fluxes, metabolite concentrations, and enzyme levels at the same time. At the end of these chapters, you will have learned what arrangements of fluxes and concentrations make cell metabolism maximally efficient, that is, allowing to produce a maximal amount of product at a limited enzyme capacity.

3. **Cell models** - In the following two chapters, we consider the cell as a whole. Chapter 8, "Principles of cell growth", describes what a system, the cell, needs to do in order to replicate, and what internal arrangements will lead to a maximal growth rate. Self-replication can be seen as a form of autocatalysis, which is described in detail in in Chapter 9. Chapter 10, "Resource allocation in complex cell models", shows how these general principles are applied in large cell models that describe small-molecule and macromolecule metabolism at a great level of detail.

4. **Time and uncertainty** - While the models in the previous chapters all assumed steady states, and often a simple choice of the "best state" for a cell, the following chapters explore some more possibilities and how one can describe them by models. Chapter 11, "Optimal cell behavior in time", extends the question of optimal resource allocation to optimal scheduling processes in time, where the cell needs to achieve its goal in a certain time horizon and resources can be shifted between different moments in time. The next two chapters are concerned with variability. Chapter 12, "Diversity of metabolic fluxes in a cell population", explores how cells in a population, instead of realizing the same optimal flux distribution, may realize different fluxes, creating random differences between individual cells in a given environment. On the contrary, Chapter 13, "Cells in the face of uncertainty" assumes that cells live in an unpredictable environment and need to "make bets" on how the environment will change in the future, and addresses what are the best strategies.
<figure><img src="image_10.png" /></figure>

Figure 2.1: Biomass composition and cell structure of a typical bacterial, yeast, and a mammalian cell. While the average composition is quite similar in the three groups, there are major differences in size and internal organization (especially when comparing prokaryotes with eukaryotes). Data for proteome groups (length-weighted protein abundances) was obtained from Proteomaps. Sources of composition data: bacteria [5], yeast (BNID 108200, 108196, 107234, 100261, [6]), mammalian cells (BNID 107131, 107235, 107234). Pictures of cells were created using Bioicons<fnref n="1" />.

## 2.2. Cell composition in numbers

Cells are composed of around 70% water and 30% dry mass. As mentioned in the previous section, we can describe the composition of the dry mass with the most abundant chemical elements. For example, the elemental formula for *E. coli* is CH₁.₇七O₀.₄₉N₀.₂₄ (BNID 101800) and for *S. cerevisiae* CH₁.₆₁O₀.₅₆N₀.₁₆ (BNID 101801).

However, more often, we are interested in biomass composition in terms of the main macromolecules (proteins, nucleic acids, lipids, and carbohydrates) and small molecules (metabolites, cofactors, ions). Table 2.1 summarizes an average composition of *E. coli* and *S. cerevisiae* during exponential growth, the typical molecular masses and copy numbers of the components. The most abundant component is protein, forming around half of the cell's dry mass. When we divide the proteome into functional groups, we find that the biggest fractions belong to translation, central carbon metabolism, folding, sorting and degradation, and biosynthesis. A substantial fraction belongs to proteins that are not mapped (especially in mammalian cells), illustrating that we still lack knowledge about the function of many proteins (Figure 2.1).

RNA forms 20% of dry cell mass in *E. coli*, but this number is lower in eukaryotes, such as yeast (11%) or mammalian cells (4%). While the total amount of RNA is variable in different organisms, its relative composition is similar – most of the RNA mass is formed by rRNA (80%), followed by tRNA (15%) and mRNA (5%) (BNID 100258, 100261, 106154). Lipid content is the highest in mammalian cells (13%) compared to yeast and bacteria (4-10%, BNID 111209,

<footnote n="1">¹bacterium-interior icon by Servier is licensed under CC-BY 3.0 Unported, golgi-3d-1 icon by Servier is licensed under CC-BY 3.0 Unported, mitochondria-3 icon by Servier is licensed under CC-BY 3.0 Unported, endoplasmatic-reticulum-3d-medium icon by Servier is licensed under CC-BY 3.0 Unported, endoplasmatic-reticulum-rough-3d-2 icon by Servier is licensed under CC-BY 3.0 Unported, endoplasmatic-reticulum-rough-3d icon by Servier is licensed under CC-BY 3.0 Unported, nucleus icon by Servier is licensed under CC-BY 3.0 Unported</footnote>
<figure><img src="image_9.png" /><figcaption>Figure 4: Metabolic models and levels of description – (A) Main elements of metabolic models. The example shows a linear pathway of 3 enzyme-catalyzed reactions. Following the convention in kinetic models, the “boundaries” of the model are external metabolites (marked as “ext”). In flux analysis models, the boundaries are typically not formed by metabolites, but by exchange reactions. (B) Levels of description of a metabolic model, from network structure (metabolites and reactions) to a quantitative physical description (comprising for example concentrations and fluxes) and further to a function-related “economic” description (comprising physiological constraints, costs, and benefits).</figcaption></figure>

5. **Sizes and shapes** - The last two chapters of the book are concerned with space in a broader sense. Chapter 14, "Strategies for cell size control" describes how cells choose the moment of cell division, which determines the distribution of cell sizes in cell populations. Chapter 15, on the "Economy of organ form and function", goes beyond microbiology and describes more broadly how systems in the body and their physiological usage – in this case, the lungs in mammals and the speed and depth of respiration – are shaped by their size, and how general scaling laws for shapes can give rise to similar laws for biological function and the "economics" of the system in question.

**Some words about mathematical models**

As shown in Figure 2, in this book, cells and cell behavior will be largely described with the help of mathematical models, often used in biology to gain insight into biological systems through simulations and quantitative analysis. As shown in Figure 4, our models typically describe a set of metabolites and the reactions that convert them, forming a network; we then attribute concentrations to the metabolites and chemical fluxes to the reactions and describe their dynamics; and finally, based on this dynamics, we consider "economic" questions, often in the form of optimality problems. Although different chapters will focus on different types of models (describing metabolic fluxes, compound concentrations, cell growth, or all of these aspects together) and models of different size (from simple instructive 3-variable models to models covering thousands of different cell components), all these models eventually describe different aspects of one cell and the same cell. Therefore, the different types of model are closely related and sometimes one model can be seen as a simplified form of another one. Figure 5 shows a basic scheme of a cell, where precursors produced in metabolism are converted into proteins, which then constitute the machines that catalyze metabolic reactions (as enzymes) or protein production (as ribosomes). By "zooming in" and focusing on different aspects of this scheme, we obtain the main types of models that we will encounter in this book.

**Where to find more information**

In addition to the main text, the book offers additional material.

**Background knowledge and literature.** Cellular economics - and systems biology more generally - builds on knowledge from different disciplines and on a history of ideas in biology and beyond. In the section "Reading recommendations" at the end of the book, you will find a number of books, articles, and online resources that provide background information.
<table><caption>Table 2.1: Amounts, characteristic molecular masses and copy numbers of the main biomass components for *Escherichia coli* (*E. c.*) and *Saccharomyces cerevisiae* (*S. c.*). The composition data is shown for *E. c.* with a doubling time of 40 minutes (BNID 104954) and for *S. c.* with a doubling time of 110 minutes ([8], BNID 111755). The storage carbohydrates include glycogen for *E. c.* / glycogen and trehalose for *S. c.* The structural carbohydrates include peptidoglycan and lipopolysaccharides for *E. c.* / mannan and glucan for *S. c.* Sources for molecular masses (BNID 105861, 115091, 101838, 104886, 107678, 109645, 102502, 100459); molecule copy numbers (BNID 108248, 108197, 114950).</caption><thead><tr><th rowspan="2"></th><th colspan="2">% of dry mass</th><th colspan="2">Mass per cell [fg]</th><th colspan="2">Molecular mass [Da]</th><th colspan="2">Copy number</th></tr><tr><th>E. c.</th><th>S. c.</th><th>E. c.</th><th>S. c.</th><th>E. c.</th><th>S. c.</th><th>E. c.</th><th>S. c.</th></tr></thead><tbody><tr><td>Proteins</td><td>55</td><td>51</td><td>165</td><td>7650</td><td>40000</td><td>55000</td><td>3 × 10<sup>6</sup></td><td>10<sup>8</sup></td></tr><tr><td>RNA</td><td>20</td><td>11</td><td>60</td><td>1650</td><td>10<sup>4</sup>-10<sup>6</sup></td><td>10<sup>4</sup>-10<sup>6</sup></td><td>3 × 10<sup>5</sup></td><td>4 × 10<sup>6</sup></td></tr><tr><td>DNA (chromosomal)</td><td>3</td><td>0.5</td><td>9</td><td>75</td><td>3 × 10<sup>9</sup></td><td>2.5 × 10<sup>8</sup></td><td>2</td><td>16</td></tr><tr><td>Lipids</td><td>9</td><td>6</td><td>27</td><td>900</td><td>800</td><td>800</td><td>2 × 10<sup>7</sup></td><td>10<sup>9</sup></td></tr><tr><td>Storage carbohydrates</td><td>3</td><td>0.5</td><td>9</td><td>75</td><td>10<sup>6</sup></td><td>variable</td><td>4000</td><td>—</td></tr><tr><td>Structural polymers</td><td>6</td><td>23</td><td>18</td><td>3450</td><td>variable</td><td>variable</td><td>—</td><td>—</td></tr><tr><td>Metabolites/cofactors</td><td>3</td><td>2</td><td>9</td><td>300</td><td>&lt;1000</td><td>&lt;1000</td><td>—</td><td>—</td></tr><tr><td>Other</td><td>1</td><td>6</td><td>3</td><td>900</td><td>—</td><td>—</td><td>—</td><td>—</td></tr></tbody></table>

<table><caption>Table 2.2: Useful quantities for unit conversions. Note that these are average or "rule of thumb" values. In reality, these values typically cover a broad range and depend on environmental conditions.</caption><thead><tr><th>Name</th><th>Unit</th><th>E. coli</th><th>S. cerevisiae</th><th>BNID/Reference</th></tr></thead><tbody><tr><td>Surface area/volume (SA/V)</td><td>µm<sup>-1</sup></td><td>6</td><td>1.2</td><td>calculated here</td></tr><tr><td>Dry cell mass</td><td>pg</td><td>0.3</td><td>15</td><td>104954, 108315</td></tr><tr><td>Total cell mass (with water)</td><td>pg</td><td>1</td><td>60</td><td>104954, 108315</td></tr><tr><td>Bilayer membrane thickness</td><td>nm</td><td>4</td><td>4</td><td>[5]</td></tr><tr><td>Cell size</td><td>µm</td><td>1-2</td><td>5</td><td>[5], 101796</td></tr><tr><td>Cell surface area</td><td>µm<sup>2</sup></td><td>6</td><td>70</td><td>101792, 113854</td></tr><tr><td>Cell volume</td><td>µm<sup>3</sup></td><td>1</td><td>60</td><td>101788, 101794</td></tr><tr><td>Cell density</td><td>g mL<sup>-1</sup></td><td>1.1</td><td>1.1</td><td>103875, 103876</td></tr></tbody></table>

Table 2.1). Remarkably, there are cases where engineered yeast cells accumulated up to 80 % of lipids per cell dry mass [7]. The content of storage carbohydrates varies from around 30% in yeast to 3% in bacteria (Table 2.1). In bacteria, carbohydrates are stored as the polysaccharide glycogen, while yeast cells use glycogen and the disaccharide trehalose. Yeast cells also contain structural polysaccharides, such as mannan and glucan [6]. Bacteria contain structural molecule peptidoglycan (3% of dry mass) – a polymer of sugars and amino acids, which forms bacterial cell walls. In addition, some bacteria (e.g. *E. coli*) also have lipopolysaccharides on their cell wall (3% of dry mass).

A small fraction of cell mass (2-3%) is formed by small molecules (< 1000 Da) such as metabolites and ions. This group contains thousands of different molecules with vastly different functions and concentrations. For illustration, the concentrations of the most abundant metabolites in *E. coli* range from $10^{-1}$ to $10^{-7}$ moles per cell, corresponding to a range from $10^8$ to only 100 copies per cell [5]. Possibly, there are metabolites with even lower concentrations, but these are much more difficult to quantify. Similarly, the concentrations of the most common inorganic ions ($K^+$, $Na^+$, $Mg^+$, $Ca^+$, $Cl^-$) span several orders of magnitude [5].

The quantities of biomass components are usually expressed in relation to other quantities. The most common units are copy numbers, moles, grams, or fractions which can be expressed per cell, per gram dry mass or per cell volume. Membrane components can also be expressed per surface area. Often, experimental data for these quantities is not readily available, so we need to extract it from literature. Useful sources for average or "rule of thumb" values include *BioNumbers* database [4] and the book *Cell Biology by the numbers* [5]. Some useful quantities are summarized in Table 2.2. They are organized in increasing order with respect to the dimensions (1 – mass, size, thickness; 2 – area; 3 – volume, density). Notice how the dimensions influence the numerical values. For example, while the cell size differs only about 3-fold between bacteria and yeast, the surface area differs by more than tenfold and the volume by about 60-fold. Because volume grows faster than area, the ratio of cell surface area to volume (SA/V) gets smaller and smaller as cells get bigger (see more in Section 2.3).
<figure><img src="image_10.png" /></figure>

Figure 5: Simple scheme of a cell, and common function-based (“economic”) cell models – A microbial cell, depicted here by a very simplified scheme (center top), can be viewed in various ways. Two main views on cells come from quantitative data obtained from experiments (here represented by the proteome, top right) and from the network of metabolic reactions (left), covering all (or a part) of the production and conversion processes in the cell. These conceptual pictures can be translated into mathematical models that describe (and predict) a number of cell variables. The three remaining boxes refer to three common types of resource allocation models presented in this book, each covering a different scope - from metabolism to entire cells. The formulae are explained in later chapters (N: stoichiometric matrix; v: vector of fluxes; b(v): flux benefit function; a(v): flux cost function; k_app: apparent catalytic rate of an enzyme; v(e, c): rate law of an enzymatic reaction, giving the rate as a function of enzyme level e and metabolite concentration vector c).

**Reading recommendations for individual chapters.** For readings specific to individual chapters, please see the "Recommended readings" sections at the end of each chapter.

**Boxes.** In the chapters, some specialized topics or thoughts on the side can be found in separate boxes. Most of these boxes belong to one of these categories: Economic analogies, Philosophical remarks, Physical thoughts and analogies, Mathematical details, Experimental methods in biology. The remaining boxes contain ideas that did not fit into this simple scheme. A list of all the boxes can be found at the end of the book.

**Book website.** More information about the book and the economic cell collective behind it can be found on our website https://principlescellphysiology.org/book-economic-principles/.

**Problems and computer exercises.** The problems at the end of each chapter are a mix of conceptual questions, paper-and-pencil calculation exercises, and computer exercises. Solutions to some of the problems can be found in the end of the book. More computer exercises (Jupyter notebooks) can be found on our website.
<figure><img src="image_11.png" /><figcaption>Figure 2.2: Cell volume grows exponentially with growth rate in *E. coli* (data from [13]). Growth rate was varied by changing medium composition.</figcaption></figure>

### 2.2.1. Variability of biomass composition

Table 2.1 shows biomass composition of a typical *E. coli* and *S. cerevisiae* cell – these are average values in certain environmental conditions. However, cell size, mass, and composition vary with growth rate and environmental conditions.

One of the most extensively studied relationships in literature is the correlation of growth rate with cell size. The increase of cell mass and volume with growth rate has been observed in bacteria (Figure 2.2), yeast, and mammalian cells [9, 10, 11, 12] (BNID 107948, 110191, 105103). For example, the cell mass of *E. coli* can vary fivefold – 150 to 870 fg per cell for generation times between 100 to 24 minutes [5].

Larger cell mass goes hand in hand with larger amounts of the individual biomass components. The absolute amounts of protein, RNA, and DNA increase with cell size. However, the ratios of the components do not stay the same and the relative composition changes with growth rate [9, 14].

One of the most consistent observations is that the relative amount of RNA per cell increases with a higher growth rate [9, 14, 15], (BNID 111460, 111755, 108200). On the other hand, the data for relative protein content is more variable. For example, in bacteria, protein content decreases with growth rate in some studies [14, 15] but goes up and down in another (BNID 111460); in yeast, it increases (BNID 108200, 111755). Nevertheless, when looking at RNA:protein ratio we consistently find a positive correlation with growth rate across various species of bacteria and yeast [16]. RNA:protein ratio is a measure of protein production capacity since most RNA is dedicated to protein synthesis. 80% is rRNA, which forms 2/3 of the mass of a bacterial ribosome – the molecular machine that makes proteins, and 15% is tRNA which brings new amino acids to the ribosome (for more details about ribosomes, see Section 2.6). Indeed, we also observe a correlation between ribosome content and growth rate. The increase of RNA:protein ratio and ribosome content with increasing growth rate reflect higher biosynthetic needs of faster-growing cells. To support higher growth rate, cells need to reallocate resources according to the growth demands (for example, make more ribosomes which can then make more proteins) [17, 16, 18, 19]. For more details about resource allocation and how it is modeled see Chapters 8 and 9.

Similarly to protein content, there is no clear correlation between the relative DNA and lipid content with growth rate across studies [14] (BNID 111460, 111755, 108196). The content of storage carbohydrates decreases at higher growth rates in yeast and bacteria [15] (BNID 111755, 111460).

As we have seen, biomass composition changes with growth rate, and for some components, we can describe this relationship with simple mathematical equations [17, 15, 14]. However, the growth rate is a result of environmental conditions (the amount or the quality of a carbon source, temperature, oxygen concentration, presence of inhibitors, and so on), and the same growth rate can be achieved in different ways. However, it may not lead to the same changes in cell physiology [10]. For example, modulation of growth rate by temperature rather than medium composition does not significantly alter cell size, and composition [9, 14]. The inhibition of ribosomes with an antibiotic decreases growth rate but increases the ribosome content (as opposed to reduced ribosome content in a medium with a “worse” carbon
website.

**Lectures.** All book chapters have been presented as lectures at our "Economic Principles in Cell Biology" summer schools at LPI Paris. Lecture slides are provided on our website.

**You can participate in writing this book.** You can participate in our project in many ways. If you have direct feedback for us (which may concern anything from typos to proposing new topics), please let us know via our feedback form on the website. If you would like to be directly involved (in writing, reviewing, proofreading, graphics design, or any other smaller or larger tasks), please contact us anytime. For more information, see the book website.
<figure><img src="image_12.png" /><figcaption>Figure 2.3: Two cells with the same number of molecules per cell but with different concentrations.</figcaption></figure>

source) [18].

Conversely, environmental factors can influence cell composition without affecting growth rate. This shows that cell metabolism is flexible – cells can reach the same growth rate in different ways, depending on the conditions. For example, in yeast, changes of O₂ concentration lead to changes in biomass composition while keeping the growth constant using a chemostat [20]. In mammalian cells, a change of a cultivation medium leads to significant changes in lipid composition without having a considerable effect on growth rate [12]. Genetic background (mutations or a presence of a transgene) can also affect cell characteristics without changing the growth rate [10, 20].

Interestingly, even though the total protein content is variable, the amino acid composition is roughly constant at different growth rates/conditions in bacteria, yeast, and mammalian cells and can even be predicted from a genome sequence with reasonable accuracy [20, 21, 12].

### 2.2.2. Biomass composition is not uniform

In the previous paragraphs, we considered average cells with a homogeneous composition across the cell. However, we need to keep in mind that cells have an internal structure, and the biomass components are not uniformly distributed throughout the cell (as illustrated in Figure 2.3). Even though prokaryotic cells do not have compartments separated by membranes, they have some internal organization. For example, DNA is not spread across the cytoplasm, but wrapped around proteins and packed in a compact structure called a nucleoid. Another example is the preferential localization of certain proteins on the poles in rod-shaped bacteria. Eukaryotes have compartments with distinct compositions, pH, and membrane potential. DNA is localized only in the nucleus and mitochondria, and many proteins localize only in a particular compartment. Small molecules and ions also have different concentrations in the different compartments. Often they cannot freely diffuse through membranes, but the transport is regulated and requires energy.

These differences in concentrations have implications for cellular functions. Some processes are restricted only to a particular compartment/area. For example, transcription only happens in the nucleus and mitochondria (nucleoid), and some metabolic pathways occur only in a specific compartment (e.g. tricarboxylic acid cycle in the mitochondria). Even if the same enzyme is present in several compartments, it might work at a different rate or in the opposite direction because of the different concentrations of substrates or products. In eukaryotes, certain digestive enzymes only work at low pH present in lysosomes (thus preventing a cell from digesting itself). Sometimes, consecutive enzymes in a metabolic pathway are not freely floating in a cell but form an assembly or bind to a scaffold, allowing intermediates to be channeled directly from one enzyme to another. This accelerates metabolic reactions because intermediates do not diffuse away into the bulk solution and are not consumed by competing reactions.

Finally, we need to zoom out from a single-cell (or average) view of a cell and consider the heterogeneity at the population level. This heterogeneity is often neglected, and we use a single number to describe a concentration of a molecule in a cell/compartment – an average value of the population. However, biological processes are *stochastic* (noisy), and the actual molecule numbers follow a certain distribution (Figure 2.4), which can be characterized by mean and variance. The effect of the heterogeneity becomes especially important at low copy numbers.

The heterogeneity in molecule copy numbers leads to a heterogeneity in cell phenotypes such as generation time, cell size, stress tolerance and others. Population heterogeneity can impact fitness in a positive or negative way, depending
# Chapter 1

## The cell as a factory

Ohad Golan

The term "metabolism" is usually used to describe the chemical reactions that occur within biological organisms to produce and transform molecules needed to sustain life. Although this definition is useful, it does not give scientific or mathematical ground for the analysis of metabolic systems. Here we consider metabolic systems in a much broader sense, and in order to provide a logical framework for the analysis of metabolic systems, we begin with a more formal definition that also covers systems outside biology. Metabolic system: "A well-defined system that takes up nutrients and uses them to sustain itself". This definition can be represented by a simple chemical equation:

$$ \text{nutrients} \rightarrow \text{metabolic system} + \text{waste products} $$

The process is carried out by the metabolic system itself - a point we will expand on later. The waste products are typical leftovers of the reaction in case such products exist. The most obvious example of a metabolic system is a biological system that takes up substances from its environment and assimilates them to reproduce its own components (often summarized as "biomass"). The chemical equation of metabolism for biological systems is:

$$ c_1 \text{ sugar} + c_2 \text{ oxygen} + c_3 \text{ ammonia} \rightarrow \text{biomass} + \text{waste products} $$

The equation describes all the nutrients, including sugar, oxygen, and ammonia, that are necessary to sustain a biological system. Other molecules such as certain metals and phosphate are also necessary for the reaction to occur, but we neglect them for the sake of brevity. The typical waste products are water, carbon dioxide, and other possible chemicals secreted by the system.

In this book we focus on the analysis of biological metabolic systems. However, given that economic systems fall under the same definition of a metabolic system, we will use them as analogies to simplify explanations. Whenever an analogy to economical systems is presented in this book, it will be displayed in an "Economic analogy" box such as the one above.

Many metabolic systems use a strategy of reproduction to sustain itself. That is, nutrients are used to make more of the metabolic system and not only to maintain it. This means that the output of the metabolic process is more of the metabolic system. This creates a system that, when unlimited resources are available, grows exponentially - the metabolic system takes in nutrients which it uses to replicate, the output of the process is also the metabolic system which takes in more nutrients and also replicates. Metabolism includes all the processes that take place in order to carry out the overall chemical conversion of metabolism - that is, everything that happens inside the black box described above. The most fundamental model of a metabolic system is one that takes nutrients from the environment, breaks them down into building
<figure><img src="image_13.png" /><figcaption>Figure 2.4: Number of molecules per cell in a population (for example protein or mRNA). The red line is the population mean, which is often the value we use (for modeling). However, the values at a single-cell level can differ several fold.</figcaption></figure>

on conditions. For example, when a cell population encounters an unexpected environment, a certain subpopulation might be better suited to survive. In a different environment, another subpopulation might thrive. We can view this as a microbial “bet-hedging” which increases the chances that at least some part of a population will survive the new conditions. However, when cells try to maximize growth rate, the variability in the population can decrease fitness because it decreases the average population growth rate [22]. This topic is discussed in detail in the Chapter 13.

Experimental methods 2.B: Experimental quantification of biomass composition

We can measure biomass composition at different levels of detail – from a coarse elemental or macromolecular composition of an average cell to the quantities of individual molecules in each cellular compartment.

To quantify the main chemical elements (CHNOPS), we can use devices called elemental analyzers. The main macromolecular components – the total protein, lipid, carbohydrate, DNA, and RNA content – can be quantified with simple assays such as detection with fluorescent dyes, chemical reactions that lead to color change or extraction and weighing of a component. Going into more detail typically requires more sophisticated methods such as liquid or gas chromatography (LC, GC), mass spectrometry (MS) or nuclear magnetic resonance (NMR). For example, for proteins, we can measure an average amino acid composition, and for lipids, the main lipid classes (glycerophospholipids, sphingolipids, sterols, etc.). For comprehensive reviews and protocols, see for example [23, 24, 25].

If we go down to the level of individual molecules, we enter fields of study collectively termed as *omics*, which aim to characterize and quantify certain pools of biomolecules. Omics methods typically involve high-throughput measurements of hundreds or thousands of different molecules and require a lot of resources (specialized equipment, computational resources) and expertise. The classic omics fields include genomics [26], transcriptomics [27] and proteomics [28] and study the components of the central dogma of molecular biology – DNA, RNA and proteins. Other examples include metabolomics which focuses on small metabolites [29] or fluxomics which measures metabolic fluxes (for example ¹³C metabolic flux analysis [30]).

Combinations of different omics can help us obtain other parameters that are difficult to measure. For example, turnover numbers of enzymes ($k_{cat}$) are notoriously difficult to quantify because the measurements are error-prone and low-throughput. With proteomics and fluxomics data we can calculate apparent turnover numbers ($k_{app}$) at various conditions (see Figure 2.8) and use the maximum value ($k_{app}^{max}$) as an estimate of *in vivo* $k_{cat}$ [31, 32].

The following table shows examples of biomass quantification methods:
<figure><img src="image_11.png" /></figure>

Economics analogy 1.A A cell and a construction firm as black boxes

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

<figure><img src="image_13.png" /></figure>

Another, less typical, example of a metabolic system in this general sense is an economic firm supplying a product. In this example we will consider for simplicity a firm that builds houses, but any kind of product can be equally used. Such a firm takes in land and different construction materials, these would be equivalent to the nutrients, and by the use of the labor force, which would be equivalent to the proteins, uses them to build houses. The houses are then sold to maintain and increase the value of the firm, just as the biological cell maintains itself. The chemical equation of metabolism for a construction firm is:

$$c_1 \text{ land} + c_2 \text{ construction materials} \rightarrow \text{value} + c_3 \text{ waste}$$

blocks, and uses these building blocks to sustain itself. In biological systems, these processes are termed catabolism and anabolism. In catabolism, the cell takes up carbon and nitrogen sources from the environment and uses them to synthesize the necessary building blocks: amino acids, nucleic acids, and fatty acids. In the anabolic process, the building blocks are used to form biomass which includes the functioning systems of the cells, proteins, DNA strands, and the membrane. Each process is catalyzed by a specific set of enzymes. These enzymes that catalyze the reactions are actually the metabolic system itself. When the cell grows, it makes more enzymes to catalyze more reactions - this is the reproduction process that leads to exponential growth.

The metabolic system controls the allocation of the available resources. When coordinating the process, the metabolic system decides between different strategies on how to best use the resources. For example, the cell decides how much of the available enzymes to allocate to the catabolic process and how much to the anabolic process. When making these decisions, the cell takes into account different physical constraints. Examples of these physical constraints are: a limited physical volume to maintain and carry out the metabolic processes, a limited surface area that constrains the ability to take up nutrients, or limiting thermodynamic constraints on the activity of the enzymes. There is no one best strategy that is always utilized - different organisms decide on different strategies based on the living conditions. This decision process is carried out by many mechanisms in the cell, the main information processing core of the cell being DNA. The decisions carried out by the cell are based on the evolutionary process the metabolic system has gone through during its existence. A description of cell information processing and how it is carried out is given in Appendix A.

So far, we describe the most fundamental metabolic system. This is a coarse-grained description in which the cell catabolizes nutrients into one type of precursor and does not take into account all the processes that take place in catabolism and anabolism. In a biological metabolic system, the cell requires multiple different types of precursors, such as amino acids, nucleic acids, and fatty acids. To create all the different precursors, the cell takes in nutrients from the environment and, through a set of chemical reactions, turns the nutrients into the precursors that are necessary for the cell to sustain itself. Each chemical reaction in the metabolic process is carried out by proteins. The different precursors can be produced through different sets of chemical reactions known as metabolic pathways, and the different chemicals in the metabolic pathways are known as metabolites. The cell decides which metabolic pathway to activate
<figure><img src="image_14.png" /><figcaption>Figure 2.5: Variability of cell size across organisms</figcaption></figure>

<table><thead><tr><th>Component/parameter</th><th>Examples of methods</th></tr></thead><tbody><tr><td>Cell size</td><td>microscopically</td></tr><tr><td>Dry cell mass</td><td>weighing of a defined amount of dry cells</td></tr><tr><td>Buoyant density</td><td>Percoll gradient</td></tr><tr><td>Protein</td><td>colorimetric (Bradford assay; Lowry assay)</td></tr><tr><td>Lipid</td><td>weighing of extracted and dried lipids</td></tr><tr><td>Carbohydrates</td><td>colorimetric (anthrone assay; phenol-sulphuric acid assay)</td></tr><tr><td>RNA</td><td>fluorimetric (RiboGreen), spectrophotometric</td></tr><tr><td>DNA</td><td>fluorimetric (PicoGreen, Hoechst), spectrophotometric</td></tr><tr><td>Amino acids/lipid classes</td><td>LC/MS, GC/MS</td></tr><tr><td>Genomics</td><td>next-generation sequencing (NGS) - Illumina, PacBio, Nanopore</td></tr><tr><td>Transcriptomics</td><td>NGS (RNA-seq), DNA microarrays</td></tr><tr><td>Proteomics/metabolomics</td><td>LC/MS, GC/MS, NMR</td></tr></tbody></table>

To visualize composition data, consider using Voronoi diagrams instead of the traditional pie charts or bar plots. An online tool is available at [bionic-vis.biologie.uni-greifswald.de](http://bionic-vis.biologie.uni-greifswald.de) for proteomics data, but there is also a tool that works with any type of input data [GitLab repository].

## 2.3. Cell size

There is a remarkable variability of cell sizes in nature (Figure 2.5). Figure 2.1 shows the typical sizes of bacterial, yeast and mammalian cells, which range from 1 to 15 µm. However, we can easily find more extreme values. For example, human egg cell has 100 µm (BNID 111184). The smallest known bacteria *Mycoplasma* has only 0.2 µm in diameter (BNID 104717) while the largest bacteria *Thiomargarita magnifica* can reach up to 2 cm [33] which is even more than most mammalian cells. However, this giant bacteria looks very different from typical bacteria like *E. coli* – it has hundreds of thousands of genome copies in organelle-like structures. There are exceptional cases where cells can reach even bigger sizes. The largest known single-celled organism is the alga *Caulerpa taxifolia*. It has many nuclei that are not separated by a membrane, and it reaches up to one meter [34]. Another special case is a neuron – its body has a small diameter (100 µm), but its axons can extend to more than a meter (BNID 109548).

For many organisms, cell size changes with environmental conditions. As already mentioned in Section 2.2.1 cell size varies with growth rate, and it depends on how a particular growth rate is reached. More than 60 years ago, Schaechter et al. discovered the nutrient growth law – cell volume increases exponentially with growth rate (as a result of the nutrient availability in the medium) [9]. Since then, the correlation between cell size and growth rate was also observed for other organisms [10, 11, 12] (BNID 107948, 110191, 105103). However, when the growth rate is changed by other means, for example by temperature, this relationship is not observed [9, 14]. In some cases, even the opposite is observed. For example, for a mammalian culture, it was observed that cell division stops at the end of the exponential phase, but cell volume continues to increase threefold [35].

The relationships above refer to an average volume in the population. However, size changes throughout the cell cycle
<figure><img src="image_14.png" /></figure>

Economics analogy 1.B A cell to a construction firm as growing systems

In an analogy to an economic system of a construction firm, the catabolic process would correspond to the purchase and transfer of the construction materials to the construction site, and the anabolic processes would correspond to the construction of the house; the catalytic enzymes would correspond to the workers carrying out the transfer of the materials and construction process. The growth process in bacteria is analogous to the growth of the firm - when the construction of the house is complete, the house is then sold to increase the value of the firm. The increased value enables the company to hire more workers and build more houses.

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

by producing the necessary enzymes. In an analogy to the economic system of a construction company, each chemical reaction is one process carried out by a worker – for example, the assembly of the frame of the house requires a carpenter, while the next step in the construction pathway is to place the foundation in the correct location, which is done by another worker. The workers are analogous to enzymes, and the different parts necessary for construction are the metabolites.

Many metabolic pathways have overlapping metabolite reactants and products. Some of the key parame-
ters that describe metabolic pathways are the enzyme catalytic rates. These parameters describe the rate
at which the enzymes consume and produce metabolites and at which concentration of reactants they
saturate. In the analogy to the construction firm, the enzymatic parameters are parameters that describe
the rate of work of each worker. Given that each metabolic pathway is made up of a series of chemical
reactions, each with different catalytic rates, the different enzymes of each pathway must be coordinated
perfectly to avoid any excess buildup of metabolites - just like in a factory assembly line, all the workers
must be coordinated together to avoid buildup of an intermediate.

In order to make sense of the complex network of metabolic reactions, different mathematical models
were developed. The models take into account the known experimental data for the different reactions

<figure><img src="image_16.png" /></figure>

Economics analogy 1.C Allocation of workforce

In an analogy to the economic system of a construction company, the company manager faces the decision of how to allocate his workforce, how many of his workers to assign to bring in materials from the factory and how many of his workers to assign to the construction process. In a similar way to the biological system, there are different limiting constraints, such as a difficult topographic construction site or limited available resources. Unlike the biological cell, though, in which the decision-making is embedded by the evolutionary processes, here the decision is made by the manager of the construction site.
at the level of single cells. Before cells divide, they need to increase their cell size. Otherwise they would get smaller and smaller with each division. However, they also cannot grow too much, or the average cell size would get bigger and bigger. There are various mechanisms of how cells maintain a cell size homeostasis, and they are discussed in detail in Chapter 12.

Finally, we need to discuss the importance of cellular shape. Different cell types come in different shapes, such as spheres, ovals, rods, or spirals. Differently shaped cells may have the same volume but very different surface area and surface area to volume ratio (SA/V). Spheres have the lowest possible SA/V while more complicated shapes have higher SA/V (e.g. endoplasmic reticulum). What happens to the shape when a cell changes its volume (for example, in response to environmental conditions)? For many cells, the shape remains roughly the same – for example *E. coli* always looks like a rod. As a result, SA/V decreases when cells get bigger. We see a decreasing linear relationship if we plot SA/V against growth rate. On the other hand, some cells vary their size and shape but maintain a constant condition-specific SA/V. [36]

## 2.4. Cell density

Most cellular parameters we discussed so far – cell size, mass, and composition – vary greatly with the cell type, growth rate, or conditions. However, one quantity does not show such variability – buoyant cell density. Buoyant density is the ratio of cell mass to volume, usually expressed as $g mL^{-1}$. For most organisms, prokaryotic or eukaryotic, the buoyant cell density is around 1.05-1.15 $g mL^{-1}$ [37, 5]. This range results from the fact that cells are 70% water which has a density of $1 g mL^{-1}$ and that most dry mass is formed by proteins, which have a density of 1.2-1.4 relative to water (BNID 111208, 104272, 101502). Other components range from 1 for lipids (BNID 108142) to 1.4-2 for nucleic acids (BNID 111208). To try the calculation of bacterial density, see Problem 2.4.

For many organisms (*E. coli*, the yeast *Schizosaccharomyces pombe*, Chinese hamster ovary cells, mouse cells), cell density is constant throughout the cell cycle and at different growth rates when growing exponentially. However, it was observed to increase in stationary phase for *E. coli* and *S. pombe* [37, 38]. On the other hand, the density of *S. cerevisiae* fluctuates during the cell cycle, which might be related to a different division mode. The organisms mentioned earlier divide by binary fission – cells divide in the middle and produce two (roughly) identical daughter cells. In contrast, *S. cerevisiae* divides asymmetrically - it grows a bud that breaks away and becomes a smaller daughter cell.

Nevertheless, despite the variability, the range of the observed values is relatively small and similar for most organisms, from bacteria to mammalian cells. There are special cases where cell density deviates from the characteristic values – for example, cells with very high fat content or gas bubbles have lower densities. However, assuming the density of $1.1 g mL^{-1}$ is probably a good guess unless you work with a particularly fatty or gassy cell type.

The invariability of cell density suggests that this property is highly regulated and brings us to the next question – is there an optimal density? And what are the constraints that (possibly) determine this optimum? These questions (among others) are discussed in the next section.

## 2.5. The physical constraints of cell growth

The living cells are constantly subject to a handful of so-called *physical constraints*, which are directly linked to the physics and the chemistry of life. Cells cannot override (evolve to bypass) these limits – only try to cope with them. Thus, sometimes these constraints are also called “hard” constraints. Notice that we consider the “hardness” of these constraints only in the space where conditions can still sustain life: some of these limitations could be relaxed by changing abiotic conditions, but would result in breakdown of biological systems.

One of the abiotic factors would be temperature; however, increased temperatures cause proteins to *denature* (lose their 3D-folded structure, thus functionality) and destabilize biological membranes. Although there are organisms, which live in extremely high temperatures (so-called *thermophiles*), as a rule of thumb, we usually consider the temperature above
<figure><img src="image_17.png" /><figcaption>Figure 1.1: A self-replicating cell – In the metabolic process the cell takes in available nutrients and through a set of biochemical reactions, turns them into precursors necessary for growth. The chemical reactions are carried out by proteins in the cell.</figcaption></figure>

and compile them together to predict the overall response of the system under different growth conditions.

The metabolic models described above describe biological systems that are disconnected from the environment except for some artificial supply of nutrients. In natural ecological systems, different organisms exist together under a limited supply of nutrients. They compete or cooperate to best utilize the limited available resources. All organisms try to improve their chances of survival according to the laws of evolution. In such a setting, the metabolism of organisms living in an ecological system is directly dependent on the other organisms that co-exist with them. In an analogy to an economic system, this would be a competition between different companies for the same possible clientele. Some companies would compete against each other, while others would cooperate to improve their profit.
393 K (120 °C) to be close to the limit of life. There is an organism known as *Strain 121* (*Geogemma barossii*) which can grow at 121 °C (hence the name), currently the highest temperature known [39]. Next, the suboptimal concentration of inorganic salts (osmolarity) or pH could also drive similar changes, disfavoring life. Here we will consider two prominent physical limits in life: the diffusion and density limits. These two limits describe two aspects of how molecules move in aqueous environments, in our case – living cells.

The diffusion limit describes the state where enzymatic catalysis is so specific and so fast that the reaction speed is determined only by the collisions of substrate molecules to the enzymes, which all result in conversions (i.e. no futile collisions) [40]. Usually, the number of futile collisions vary between 1 and $10^4$ per successful conversion, and thus having as little futile collisions as possible greatly enhances the overall rate of the reaction. Enzymes approaching (operating at) the diffusion limit are also called *perfect enzymes*. Currently there are no enzymes reported which are considerably “above” diffusion limit (see [40] for an in-depth discussion), suggesting the universality of the underlying constraint.

Another aspect to consider is the density, or sum concentration of molecules, of the fluid. As described in previous sections, cell cytosol contains a spectrum of different molecules at different sizes and concentrations. We normally assume that some sort of optimal cell density that maximizes fitness exists, however, the density is known to fluctuate substantially in time and across conditions [41]. One of the most prevalent properties, linked to cytosolic density, is *macromolecular crowding*. As the name suggests, it describes the concentration of biological macromolecules, mainly proteins, in cytosol (thus in bacteria, the genomic DNA also contributes to molecular crowding). For example, the macromolecular crowding is suggested to impose a limit on the protein translation [42], therefore, increased crowding would result in a growth rate decrease. The state of macromolecular crowding is relevant for the cellular function, and is proposed to be in homeostasis (reviewed in [43]): optimal macromolecular crowding corresponds to a state where crowding reduces the path proteins have to diffuse, yet does not substantially decrease the speed of diffusion. In such a way, maintaining high macromolecular crowding is suggested to maximize reaction rates in the cytosol [44].

## 2.6. Macromolecule synthesis and the resources needed

Now that we have explored the diversity of nature and abundance of biological molecules, in this and the next section we will consider the coordination of cell components in the biosynthesis of macromolecules. The overall cell growth can be called *self-replication*: a cell makes a copy of itself by synthesizing macromolecules by using molecules it either produces or takes up from the environment, all at right amounts and proportions. Three essential types of resources are needed for synthesizing the macromolecules: (1) precursors, (2) catalysts, and (3) physical space/volume for the process to happen.

As discussed earlier in the chapter, macromolecules, primarily proteins, are essential for operating metabolic networks. As synthesis of different macromolecule species competes for the same classes of resources, macromolecule synthesis can be altered to change the operational metabolic network - to switch between metabolic strategies. In different conditions, different strategies are superior in the growth they support and the best manner to allocate the limited resources will be preferred. We thus will discuss how these resources are primed and used for macromolecule synthesis, together with different considerations surrounding each type of these resources.

### 2.6.1. Precursors of macromolecules

Biosynthesis of the macromolecule precursors (e.g. amino acids, nucleotides, energy equivalents) is a major part of every metabolic network. Many microorganisms can grow on a very limited number of nutrients (in the lab context, so-called minimal media), which usually consist of a single source for carbon, nitrogen, phosphorus, and sulfur. For instance, a minimal growth medium with glucose as the sole carbon source can fully support growth: glucose enters glycolysis as the main energy harvesting route, however, some of the glycolytic intermediates serve as substrates for, e.g. amino acid, lipid, or nucleotide biosynthesis.
# Chapter 2

## An inventory of cell components

Pranas Grigaitis and Diana Széliová

### Chapter overview

* The main components of a cell are proteins, RNA, DNA, lipids and carbohydrates.

* The quantities of these components vary depending on the cell type and the cell's environment.

* These components are synthesized by enzymes and molecular machines such as ribosomes and DNA/RNA polymerases.

* There are many parallel processes happening in cells and they have to be coordinated.

* Cellular processes are constrained by factors like temperature, diffusion limits, and density.

### 2.1 Describing and counting cellular components

Cells contain a diverse spectrum of molecules, needed to create two cells out of one (as Rudolf Virchow proposed, *omnis cellula e cellula*, all cells come from cells). These molecules come in different sizes and properties and therefore create a demand for a cell to keep these components in different places (*spatial* organization) with different patterns of use (*temporal* organization), and book-keep their quantities. Cell composition directly influences cell function: thus we observe different cellular make-up in different organisms or even in different cells of the same organism.

Historical research and the latest advances in instrumentation allow us to characterize the constituents of cells in increasing depth. Today, collections of such biological numbers, like *BioNumbers* [1], store thousands of values available at your fingertips, a long way from scouting the numbers in original publications. Specialized open databases, e.g. *Human Serum Metabolome Database* [2], bring increasing amounts of measurement data available to the community.

Being able to operate basic biological numbers has multiple benefits when thinking of the cellular economy. To name a couple, first, it allows “back-of-envelope” calculations, where we aim to estimate the plausible order of magnitude of a derived value, rather than the exact value. This sort of thinking boosts interpretation of results considerably, as it allows us to rule out unrealistic outcomes. Second, computational models of cell growth (Chapter 8) usually use numbers like average cell size or protein mass as parameters. Consequently, the choice of parameters has a direct influence on the quantitative predictions. Last but not least, these simple calculations allow us to establish relationships between different components of the cells - and cells are nothing but heavily intertwined networks of molecules.

Counting molecules in a cell is as important to the cellular economy as counting different sorts of fruits and vegetables in a warehouse – and is a key ingredient in the journey towards understanding of the principles behind the cellular economy. As we unveil throughout the book, it seems that cells can be treated as “little bookkeepers under the microscope”. Thus in this chapter, we will do a census of cellular
<table><caption>Table 2.3: The estimated energetic costs (units of ATP hydrolysis) of biosynthesis of a gene, as computed by [47]. The estimates are represented as functions of the following parameters: $L_g$, gene length; $N_r$, the steady-state number of mRNAs; $L_{r,pre}$ and $L_{r,mat}$, the length of precursor and mature mRNA, respectively; $\delta_r$, the degradation rate of mRNA; $t$, division time of a cell; $N_p$, the steady-state number of protein molecules; $L_p$, length of the protein chain; $\bar{c}_{AA}$, average cost of an amino acid; $\delta_p$, the degradation rate of proteins.</caption><thead><tr><th>Expression stage</th><th>Bacteria</th><th>Eukaryotes</th></tr></thead><tbody><tr><td>DNA synthesis</td><td>101L<sub>g</sub></td><td>263L<sub>g</sub> (×2 for diploids)</td></tr><tr><td>RNA transcription</td><td>2N<sub>r</sub>L<sub>g</sub>(23 + δ<sub>r</sub>t)</td><td>N<sub>r</sub>(46 × L<sub>r,mat</sub> + 2.17 × δ<sub>r</sub>tL<sub>r,pre</sub>)</td></tr><tr><td>Protein synthesis</td><td>N<sub>p</sub>L<sub>p</sub>[(c&#773;<sub>AA</sub> - 1) + 5δ<sub>p</sub>t]</td><td></td></tr></tbody></table>

A particularly interesting fact is that metabolic networks can be described as *bow-tie* structures [45]: a large variety of nutrients can be converted into a very small number (usually counted up to 12) essential metabolic intermediates, which give rise to, again, a diverse set of molecules (for a detailed discussion, see Chapter 3). This gives two important insights into metabolic networks. First, this plasticity of the metabolic networks, allows organisms to grow in various environments, where different nutrients are available. Second, due to this organization, the biosynthesis of macromolecule precursors competes for the same starting molecules, independently from the initial nutrients.

### Economic analogy 2.C: A bakery

The diversity of metabolic intermediates/end products, stemming from small number of nutrients (e.g. minimal mineral media for yeast growth, containing glucose, ammonium, phosphate and sulphate salts), can be imagined as a bakery. Every pastry starts with a small array of ingredients (flour, water, salt, sugar, ...) and using some machinery (e.g. ovens), one ends up baking bread, pretzels, cookies, muffins etc., which are way diverse in their features, compared to the starting mixture. Likewise, by taking only a handful of compounds, cells, especially microorganisms, can synthesize most of the molecules they need to eventually replicate.

### 2.6.2. Catalysts needed for macromolecule synthesis

Many steps of the biosynthesis of macromolecules, as discussed previously, need catalysis to proceed. Therefore, another kind of investment into macromolecule synthesis is expression of necessary proteins and RNAs (in the latter case - ribosomal RNA). Expression of proteins, starting from transcription of messenger RNAs, their translation into proteins, folding, and degradation, involve many steps with energy investment (ATP hydrolysis) and consume large amounts of precursors (nucleotides, amino acids). Talking in energetic terms alone, protein expression accounts for ca. 40% of energy investments in yeast *S. cerevisiae* [46], and the investments of energy for every stage of protein expression is illustrated in Table 2.3 for typical bacterial and eukaryotic cells. This concerted action of several systems, as described above, with substantial investments at every intermediate step, means that these investments thus happen on two levels: investments in the metabolic machinery and in the machinery, producing proteins themselves. We will consider these two levels in the following.

**Metabolic enzymes.** First, metabolic enzymes need to be expressed to convert nutrients into biosynthesis precursors. Some enzymes are active only in a form of complexes, which also creates a demand to express proteins at defined ratios. Enzymes and their complexes come in different sizes and flavors, and their activity can be described (in very coarse-grained way, for more details see Chapter 3) by two kinetic aspects: the efficacy (represented by the turnover number $k_{cat}$) and substrate specificity (Michaelis constant $K_M$) of an enzyme. Importantly, these two parameters are intertwined: high substrate specificity usually comes at the cost of efficacy and vice versa. Therefore, although some enzymes tend towards extremes in terms of their specificity or efficacy, most of the enzymes land close to the average/median values of these parameters, when considering the distribution of enzyme parameters among different organisms [48] (Figure 2.6).

The metabolic networks need to work in a concerted manner, even though different enzymes need to perform different amounts of “work” (described as metabolite flux through these enzymes, $v$). Thus, even given the similarities in
components: we will discuss what molecules make up a cell, what they are derived from, how to measure these components in the lab, and we will briefly consider allocation of resources, directed to synthesize individual cellular components.

## 2.2 The components of a cell

### 2.2.1 Elemental composition of the cell

Although living matter comes in different shapes and sizes, over 99% of the cellular mass can be described by only a handful of chemical elements. 6 most abundant elements form the famous CHNOPS notation: carbon (C), hydrogen (H), nitrogen (N), oxygen (O), phosphorus (P), and sulfur (S). Taken together, these 6 elements encompass the vast majority of the mass, namely, ca. 97.5% in budding yeast *Saccharomyces cerevisiae* [3]. Living cells also contain minute amounts of different metal ions, such as sodium (Na), potassium (K), iron (Fe), molybdenum (Mo) and others – usually facilitating signal transduction or supporting enzymatic catalysis.

### 2.2.2 Biological molecules

Although cells contain many different molecular species ("molecular identities"), we can crudely categorize them into small molecules and macromolecules based on their molecular weight and complexity. Small molecules, as the name suggests, are small chemical compounds, up to 1000 Daltons in mass (1 Dalton = 1 atomic mass unit, 1 amu), and are usually composed of a non-repeating single chemical unit (called monomer). Macromolecules, on the contrary, are up to several megadaltons (MDa = $10^6$ Da) in weight, and are frequently composed of multiple monomers (forming so-called polymers). Compounds in the cells, both macro- and small molecules, based on their chemical nature, fall into 5 big groups: proteins, nucleic acids (both macromolecules), carbohydrates (exist as both small molecules and polymers), lipids (small molecules), and cofactors/other small molecules.

Proteins are polymers, composed of amino acids. Proteins are an exceptionally diverse class of molecules: in Nature, 20 amino acids can be incorporated into proteins (so-called proteogenic amino acids), which, combinatorially provides 20 options for each position in the protein chain. Therefore, there is an enormous amount of possible combinations to make a protein of a length of 100 amino acids ($20^{100}$, to be precise), even for a amino acid chain way shorter than the average in *E. coli*, around 325 amino acids (BioNumbers ID (BNID) 108986). This diversity gives rise to the spectrum of functions proteins can do, for instance, catalysis (catalytic proteins are also called enzymes), transport of molecules, keeping structural integrity of membranes, and others. Also two notable properties of proteins are that they (1) need to acquire a specific three-dimensional structure ("to fold") in order to become functionally active, and (2) sometimes, they also need to form complexes of the same or other proteins (called multimers). Protein production is a major consumer of energy and biosynthetic intermediates in the cell, therefore, in this book we will frequently consider proteins as central players in implementing economic principles in cell physiology.

Nucleic acids are another category of macromolecules; their monomers are called nucleotides. There are two major classes of nucleic acids, RNA (ribonucleic acid) and DNA (deoxyribonucleic acid). RNA and DNA chemically have a slight, yet critical difference: the sugar, which is a part of the nucleotides, differs between RNA (ribose) and DNA (deoxyribose). The two sugars are almost the same but for one chemical group: one of the carbon atoms in ribose is connected to two another carbon atoms, a hydrogen atom, and a chemical group, called hydroxy- ($-OH$). In deoxyribose, the hydroxy-group is substituted with another hydrogen atom, hence the prefix "deoxy-" ("minus oxygen"). RNA and DNA have different functions in the cell: the primary function of DNA is to store genetic information, while RNA can work both as an intermediate agent to transfer that genetic information to protein production (messenger RNA, mRNA) or to participate in catalysis and protein production in general (e.g. transfer and ribosomal RNA, tRNA and rRNA, respectively). Outside the polymers, nucleotides can also act as energy-accumulating compounds (e.g. ATP, adenosine triphosphate) or signaling molecules (e.g. cyclic adenosine monophosphate, cAMP). In this text, we will mostly refer to the energy-storing function of the nucleotides, although other functions, such as signaling, also are essential aspects of describing cell physiology.
<figure><img src="image_15.png" /><figcaption>Figure 2.6: Distributions of the $k_{cat}$ and $K_M$ values, collected for *E. coli*, yeast and human enzymes. The vertical solid line depicts the median of each distribution. Values were collected from the BRENDA database, release 2022.1 [49].</figcaption></figure>

"average" (or "moderate") enzyme properties, the expression of proteins and the abundance of their substrates span several orders of magnitude. Based on the kinetic interpretation of enzyme kinetic parameters, we can link them to either expression level of the enzyme $e$ ($e \propto \frac{v}{k_{cat}}$) or substrate concentration $s$ (usually, $0.1K_M \le s \le 10K_M$). Note that for substrate concentrations, the suggested range (order-of-magnitude difference from the $K_M$ to each side) is arbitrary, yet supported by empirical observations: enzyme kinetics set the upper bounds for the concentrations, as the benefit from high substrate concentration becomes negligible (saturation kinetics), and the lower bound is defined through high demand of enzymes (in order to sustain flux, a lot of enzyme would have to be produced).

To illustrate the diversity of enzyme turnover values $k_{cat}$ and the condition-dependent expression of enzymes (dictated by the flux $v$ these enzymes have to sustain), we can consider proteome composition of *E. coli* under two conditions: growth medium with the complete supplement of amino acids (all 20 proteogenic amino acids present in medium), in contrast to the supplement with a single amino acid missing (a "dropout" medium) (Figure 2.7). The growth of *E. coli* in a nutrient-rich medium (glucose + amino acid supplement) is indeed a very fast one (with doubling time of $\tau_{d,rich} = 21.5 \pm 0.4$ vs. $\tau_{d,minimal} = 56.3 \pm 0.5$ minutes). The omission of methionine from the amino acid supplement does increase the doubling time ($\tau_{d,-Met} = 26.5 \pm 1.1$ minutes), yet the growth rate remains high, and so is the methionine biosynthesis demand in these conditions.

Methionine is an amino acid that is energetically the most expensive to make [50], and the final enzymatic reaction in the methionine synthesis pathway is so-called *rate-limiting*, or the reaction which dictates the flux through the whole pathway. Moreover, the enzyme methionine synthase (*MetE*) is a very slow enzyme (Figure 2.7, table on the bottom), thus required at large quantities to provide enough methionine for protein synthesis at high growth. Consequently, it was observed that *MetE* alone could occupy up to ca. 7.5% of the total proteome (by mass) in medium lacking methionine, and growth on a medium, containing methionine, would reduce the proteome fraction by ca. 800-fold, to 0.009% [51]. To contrast this highly condition-dependent expression of *MetE*, we considered a protein in the lower glycolysis, called enolase *Eno*. Since both the complete, and the methionine-lacking media contained glucose as the main carbon source, the expression of glycolytic proteins, including *Eno*, was determined to be similar. A noticeable trait of *Eno* is also a high $k_{cat}$ value, ca. 3 orders-of-magnitude higher than the one for *MetE*: this allows enolase to be less expressed, although the flux through enolase is very high when cells grow fast on glucose [51].

The variable concentrations of metabolic substrates, and their relation to the enzyme parameters ($K_m$ in this case), also bring additional kinetic considerations. The above-introduced turnover value $k_{cat}$ represents the highest possible
Carbohydrates are another major class of biological molecules, and are important both as monomers and high molecular-weight polymers. Monomeric carbohydrates (sometimes also referred to as simple sugars) are mainly used as carbon and energy sources for organisms, e.g. glucose or fructose. Oligosaccharides made up of two or three linked monomers can also used as energy source and many of them are specific to certain groups of organisms (e.g. melezitose, a trisaccharide found in insect honeydew). In oligomeric form (up to 10 monomers), carbohydrate chains are essential for cellular sensing systems: proteins can be "decorated" with chains of carbohydrate monomers to be recognized by receptor molecules on the surface of the cell. Finally, polymers of carbohydrates usually serve as structural components (part of peptidoglycan, major part of bacterial cell walls) or energy/carbon storage (glycogen in, e.g. yeasts and animal cells, or starch in plants).

Lipids are a vaguely-described class of compounds, which have an overarching similarity, being water-insoluble. The major function of lipids in biological cells is structural: a very abundant subclass of lipids, phospholipids, is an essential constitutent of biological membranes. As discussed in Section 2.2.1, membranes themselves have a variety of functions, which are mostly carried out by lipids (structural) or proteins (transport, sensing, signaling etc.). Some lipids can also undertake other functions, such as signaling (various sterols), or energy storage (tryglycerides, or fats).

As we see, the metabolism of biological molecules is tightly interlinked, although they exhibit major differences in their abundance, size and chemical properties. Macromolecules are present in very low concentrations, and their biosynthesis usually takes minutes. Meanwhile, the time scale of small molecule reactions is usually seconds (or fraction of), and the concentrations of small molecules are usually several magnitudes higher than these of macromolecules. Yet, despite acting at different rates and concentrations, these two types of biological molecules work in an orchestrated manner. To begin with, a number of different small molecules are required to produce both other small molecules and the macromolecules. In return, the macromolecules ensure cell integrity and growth by, among other functions, operating the reaction networks of small molecule interconversions (which we usually refer to as metabolism). Additionally, presence of some small molecules can influence the function of macromolecules, both directly (e.g. essential cofactors, needed for enzymatic reactions; enzyme activation or inhibition), and indirectly (e.g. modulation of gene expression, signaling). Therefore, a lot of different processes have to happen in parallel to ensure the operation of the cells. Having defined the major types of molecules we find in living cells, next we will discuss how abundant are different components of the cells.

## 2.3 Cell organization and size

In an extremely simplified way, cells can be looked at as bags of fluid-like material, kept together by a membrane. These "bags of things" can also contain other membrane structures inside them, forming so-called organelles. In cell biology, we call cells *prokaryotic* if they do not possess these membrane structures, and *eukaryotic* if they do. The divide between prokaryotes and eukaryotes can be illustrated by comparing two organisms: the prokaryotic bacterium *Escherichia coli* and the eukaryotic yeast *Saccharomyces cerevisiae*. They both are organisms, composed of a single cell (thus called *unicellular*), and they both are very small, compared to a typical human cell. However, *E. coli* does not contain any additional membrane structures except from the plasma membrane (which encompasses the cellular contents). Meanwhile, a handful of different organelles can be observed in *S. cerevisiae*. The cellular organization of these cells is shown in Figure 2.2.

### 2.3.1 Membrane-bound structures of the cell

Most biological membranes and membrane-based structures, including the plasma membrane itself, have multiple functions (not only separating space), and are highly dynamic. Some membranes can fold into very compact structures with extremely high surface area (endoplasmic reticulum, Golgi apparatus), occupy different volumes - from small vesicles to large vacuoles, occupying a major fraction of the cell volume. Moreover, some molecules can form very large structures, which might be transient (short-lived), thus capturing and defining them remains a major challenge. For these reasons, the fine structure of cells is unclear - some findings (e.g. organelle contact sites, see [4] for a recent review) hint into some functional
<figure><img src="image_16.png" /><figcaption>Figure 2.7: Proteome composition of *E. coli*, grown on the growth medium with full amino acid supplement (right) or its version without amino acid methionine (left). The table on the bottom compares the abundance of two selected proteins (enolase, independent on the amino acid supplementation; methionine synthase, dependent) and their turnover numbers $k_{cat}$. Proteome composition data from [51].</figcaption></figure>

<figure><img src="image_17.png" /><figcaption>Figure 2.8: The relation between the apparent and measured turnover value($k_{app}$ and $k_{cat}$, respectively). Factors, leading to low net rate of reaction per unit of protein (e.g. low substrate concentration) lead to $k_{app}$ being significantly lower than the measured $k_{cat}$ value, latter of which corresponds to the maximal rate of the reaction.</figcaption></figure>

efficacy of the enzyme, where all substrates are accessible in concentration needed to sustain this efficacy (also called saturating concentrations). Turnover values are usually measured *in vitro*, with all the substrates highly in excess, thus deliberately minimizing many kinetic effects (enzyme saturation, reversibility of reactions etc.) that are prevalent in more physiological conditions (see Chapter 3 for details). Therefore, what we usually observe in living cells is not the enzyme efficacy in terms of the $k_{cat}$, but rather their apparent turnover value $k_{app}$ (Figure 2.8). The ratio of these values ($k_{app}/k_{cat}$) is then called the *enzyme efficiency* and can be used to infer how far away the enzyme is from its optimal working conditions. The $k_{app}$ value of an enzyme *in vivo* can be computed as follows: knowing the $k_{cat}$ value, the flux through the reaction, one can compute the minimal demand (in moles) of the enzyme to sustain that flux. Then, the $k_{app}$ value can be computed by taking the ratio of predicted minimal enzyme demand and the enzyme abundance in the cells.
Box 2.A Macromolecular machines

An important consideration about both proteins and nucleic acids is that they are polymerized by very specialized protein- and protein-nucleic acid complexes. These molecular machines use energy (in terms of ATP equivalents) to form chains of the respective monomers.

<figure><img src="image_18.png" /></figure>

For proteins, amino acids (AA) are combined into a so-called *peptide chain* in a process called translation, which is catalyzed by ribosomes – large complexes made from proteins and RNA. Nucleic acids (RNA and DNA) are synthesized from nucleotides (NT) through a process called transcription by enzyme complexes known as *nucleic acid polymerases*. There are two major classes of them, RNA and DNA polymerases, each specific to their respective nucleic acid.

organization of organelles, yet the canonical way to look at the cellular structure remains as to a "bag of things".

A notable example of a highly specialized organelle is the mitochondrion. The mitochondrion is separated from the rest of the cell by two (outer and inner) membranes; this feature is essential for their function. In eukaryotes, mitochondria are a major hub of metabolism: they house essential biochemical pathways, such as tricarboxylic acid cycle (also known as citric acid-, or Krebs cycle), as well as the so-called *respiratory chain*, the machinery for generating energy with the use of oxygen (see Chapter 3 for more details). While the most biochemical interconversions happen inside the mitochondria (in mitochondrial *matrix*), the respiratory chain proteins are located in the inner mitochondrial membrane: these proteins create an *electrochemical gradient* across this membrane, and use it to drive the conversion of energy, stored in nutrients, into the energy the cell can use (in a form of ATP). What makes mitochondria even more interesting is that they also contain mitochondria-specific genetic information (mitochondrial DNA), which is essential for mitochondria to function inside the cell. In many organisms, the loss of mitochondrial DNA results in impaired growth (in yeasts, that is called the *petite* phenotype) [5], and some organisms cannot grow unless mitochondrial DNA is present (*petite-negative* yeasts).

2.3.2 Cell size

There is a remarkable variability of cell sizes in nature (Figure 2.1). Figure 2.2 shows the typical sizes of bacterial, yeast and mammalian cells, which range from 1 to 15 µm. However, we can easily find more extreme values. For example, one of the largest cells in the human body, the egg cell, is 100 µm in diameter (BNID 111184). Bacteria are usually considered very small, in fact, the diameter of the smallest known bacterium *Mycoplasma* is only 0.2 µm (BNID 104717). Meanwhile, on the other side of the spectrum, the largest bacteria *Thiomargarita magnifica* can reach up to 2 cm [6] which is even more than most mammalian cells. However, this giant bacteria looks very different from typical bacteria like *E. coli* - it has hundreds
**Macromolecule polymerization.** Moving from the metabolic enzymes to the macromolecular synthesis machinery, the polymerization of the macromolecules (DNA replication, RNA transcription and protein translation) are catalyzed by large enzyme (and RNA, in the case of ribosomes) complexes: DNA and RNA polymerases (DNAP, RNAP) and ribosomes. Resources, needed for expressing them also significantly contribute to the total costs of macromolecule biosynthesis. For instance, the molecular weight of an intact ribosome in *E. coli* is ca. 2.3 MDa (BNID 111560), and the *E. coli* ribosome consists of 62% RNA and 38% protein (in mass %, BNID 109047). Meanwhile, eukaryal ribosomes are even larger, ca. 3.3 MDa for *S. cerevisiae* and ca. 4.3 MDa for human *H. sapiens* (BNID 111560), and have higher protein content [52]. For a comparison, the average length of a protein in *E. coli* is ca. 300 amino acids (BNID 100017) and average amino acid weight is ca. 109 Da (BNID 104877). By multiplying these numbers, the molecular mass of an average protein is ca. 32.7 kDa, roughly 70× lower than the ribosome that synthesizes this protein.

The nature of these large complexes requires an exceptional coordination of resources. The first consideration is the number of individual proteins that form these complexes: the RNA polymerases of *S. cerevisiae* contain up to 17 subunits (BNID 111568), and 79 ribosomal proteins form a fully functional ribosome [53]. Therefore, the assembly of these complexes must be fast and robust: thus cells contain a number of assembly factors for facilitating these processes. Next, the coordination also has to be temporal, especially for prokaryotes, where both messenger RNA transcription and protein translation can happen simultaneously. In *E. coli*, this is well illustrated by the 3-fold difference between elongation rates of mRNAs and proteins, ca. 62 nt s⁻¹ and 21 aa s⁻¹, respectively (BNID 103021, 107868). This coordination is essential for coordinated transcription and translation happening at the same time [54], as translation happens in steps, 3 nt each (so-called *triplets*). Even in eukaryote *S. cerevisiae* we observe a similar pattern: mRNA elongation rate of ca. 30 nt s⁻¹ (BNID 103016) and protein chain elongation rate of ca. 10.5 aa s⁻¹ [55], nearly a 3× difference. Also, the polymerization of macromolecules is very tightly connected to the metabolism: different kinds of growth limitations (limiting amounts of nutrients) were shown to create bottlenecks at different stages of protein expression [56], and the optimal regulation of these processes were selected for by the evolution [57, 58].

### 2.6.3. Physical proteome space

A final type of asset required for macromolecule synthesis is the physical volume in the cell. As the cells are, again, “bags of things”, they possess a finite volume, thus different processes compete for available proteome volume (also called “proteome space” interchangeably). A general trend across microorganisms is that ribosomes occupy larger proteome mass fraction (in the range of 10-40% total proteome) with increasing growth rate [18, 59], with an estimated maximum in *E. coli* of ca. 55% of total proteome mass [18]. Alongside ribosomes, biosynthetic pathways also occupy a substantial share of total proteome (e.g. enzymes, required for amino acid biosynthesis occupy up to 15% of the proteome space in *S. cerevisiae* [59]). Experimentally, the optimal allocation of proteome space can be challenged by, e.g. varying expression of an unneeded (gratuitous) protein. Both for *E. coli* and *S. cerevisiae* it was shown that increasing gratuitous protein expression directly affects the maximal growth rate on both minimal and rich media [60, 56], suggesting that the decrease in growth rate is not dependent on the nutrient status of the cell.

Numbers provided above were measured for cells, grown in minimal medium, and some of the costs we discussed - not only proteome space, but also precursors and enzymes - could be alleviated by growth in rich medium. Uptake of biosynthetic precursors usually is less costly than biosynthesis, as expression of a single type of transporter can substitute the need of expressing a biosynthetic pathway with tens of enzymes associated. Indeed, transfer of *S. cerevisiae* cells to a amino acid-rich growth medium resulted in an increase of growth rate, caused by increased proteome allocation to ribosomes, in place of the proteins of *de novo* amino acid biosynthesis [61]. In conclusion, the physical space that proteins can occupy is also an asset that the proteins are competing for, and thus the optimal allocation of the available space is key for the cells to grow in the most favorable way under specific conditions.
of thousands of genome copies in organelle-like structures. There are exceptional cases where cells can reach even bigger sizes. The largest known single-celled organism is the alga *Caulerpa taxifolia*. It has many nuclei that are not separated by a membrane and reaches up to one meter [7]. Another special case is a neuron – its body has a small diameter (100 µm), but its axons can extend to more than a meter (BNID 109548).

For many organisms, cell size changes with environmental conditions. As already mentioned in Section 2.4.1, the size of the cell varies with the growth rate, and depends on how a particular growth rate is reached. More than 60 years ago, Schaechter et al. discovered the nutrient growth law – cell volume increases exponentially with growth rate (as a result of the nutrient availability in the medium) [8]. Since then, the correlation between cell size and growth rate was also observed for other organisms [9, 10, 11] (BNID 107948, 110191, 105103). However, when the growth rate is changed by other means, for example by temperature, this relationship is not observed [8, 12].

### 2.3.3 Variation of single-cell sizes and shapes

The relationships above refer to an average cell volume in the population. However, at the single-cell level, size changes throughout the cell cycle. Before cells divide, they essentially need to double their size. Otherwise, they would get smaller and smaller with each division. However, they also cannot grow too much, or the average cell size would get bigger and bigger. There are various mechanisms of how cells maintain a cell size homeostasis, and they are discussed in detail in Chapter 14.

Aside from cell size, we need to consider the importance of cell shape. Different cell types come in different shapes, such as spheres, ovals, rods, or spirals. Differently shaped cells may have the same volume but very different surface area and surface area to volume ratio (SA/V). Spheres have the lowest possible SA/V while more complicated shapes have higher SA/V. What happens to the shape when a cell changes its volume (for example, in response to environmental conditions)? For many cells, the shape remains roughly the same – for example *E. coli* always looks like a rod. As a result, SA/V decreases as cells get larger. On the other hand, some cells vary their size and shape but maintain a constant condition-specific SA/V [13].

The changing SA/V ratio can impact cell biology in various ways. For example, it can change cell composition – a larger membrane area increases the lipid fraction of the cell. The SA/V ratio also affects nutrient uptake. A bigger surface area allows faster nutrient uptake per unit volume because more transporters can fit into the membrane. Additionally, cell shape influences the rate at which molecules diffuse from one end of the cell to another. Understanding these effects is relevant for some modeling approaches, especially those that consider membrane synthesis or diffusion.

## 2.4 Cell composition in numbers

Throughout this book, we will explore various ways to model cells mathematically. For that, we need to know not only what the main components are but also what are their quantities. For example, proteins are usually the most abundant constituent of the biomass, and many simplified mathematical models focus solely on proteins. However, some models include a much more detailed description of the biomass composition. Additionally, we need to know how biomass composition changes in different environments. For instance, the observation that RNA/protein ratio increases with growth rate hinted how cells reallocate

<figure><img src="image_19.png" /><figcaption>Figure 2.1: Variability of cell size across organisms</figcaption></figure>
## 2.7. Concluding remarks

In this chapter, we discussed the properties and the quantities of the main cellular components, how the composition changes in different environmental conditions, and what resources are needed for a cell to replicate itself. It may seem that we already have a vast amount of data, but a lot is still missing. Most available data comes from model organisms such as *E. coli*, *S. cerevisiae*, or humans, but the data for other organisms is still limited. Single-cell data (ideally with subcellular resolution) is also not widely available. Even though we can sequence a genome within a few hours or days, we still do not know the functions of many genes. Many experiments still need to be done, and new high-throughput experimental methods developed to fill the gaps in our knowledge.

Nevertheless, with the basic knowledge from this chapter, we can dive deeper into studying cellular economics and resource allocation with mathematical modeling. How is biomass represented in mathematical models? Often, models only focus on proteome as it is a cell's most abundant and expensive component. However, some models also include other major components (RNA, DNA, lipids, carbohydrates, cofactors, etc.). The components can be modeled at different levels of detail. For example, the cell proteome can be represented simply as a total proteome, divided into protein subgroups (e.g. metabolic, ribosomal, other), or modeled as individual proteins. Finally, there are two contrasting ways to include biomass in mathematical models. On the one hand, some models consider a fixed biomass composition based on measurements or literature (see Chapters 4 and 5). On the other hand, some models predict the biomass composition (i.e. they calculate optimal resource allocation or enumerate all possible compositions, see Chapter 9).

Apart from biomass composition, we can include other cellular properties as constraints or parameters in the models, depending on the type of a model and how detailed it is. For example, we can constrain the transcription/translation rates, enzyme turnover rates, cell surface area or volume,

In conclusion, this chapter introduced the basic building blocks of a cell, processes that make them, how they are coordinated and how they depend on environmental conditions. In the next chapters you will learn how to translate this information into mathematical models and how to use them to gain deeper knowledge of cell biology.

### Recommended readings

* Moran, U., Phillips, R., & Milo, R. (2010). SnapShot: key numbers in biology. Cell, 141(7), 1262-1262.

* Milo, R., & Phillips, R. (2015). Cell biology by the numbers. Garland Science.

* Alberts, B., et al. (2022). Molecular Biology of the Cell. WW Norton & Co.

### Problems

**Problem 2.1 Intuition for biological numbers.** Try to answer the following questions, and only then look up the results:

* What is the volume of a cell?

* What is the size of a protein?

* How many protein molecules are there in a cell?

* What is the number of genes in a genome?

* How long does it take to transcribe a gene?

* How long does it take to produce a protein molecule?

* What is the minimal doubling time of a cell?

* What other questions come to your mind?

Precise values do not matter here – think about orders of magnitude.
<figure><img src="image_20.png" /><figcaption>**Figure 2.2:** Biomass composition and cell structure of a typical bacterial, yeast, and a mammalian cell – The area of each polygon corresponds to a mass fraction of a component per cell. While the average composition is quite similar in the three groups, there are major differences in size and internal organization (especially when comparing prokaryotes with eukaryotes). Data for proteome groups (length-weighted protein abundances) was obtained from Proteomaps. Sources of composition data: bacteria [14], yeast (BNID 108200, 108196, 107234, 100261, [15]), mammalian cells (BNID 107131, 107235, 107234). Pictures of cells were created using **Bioicons**<fnref n="1" />.</figcaption></figure>

resources to grow faster, inspiring the development of mathematical models that capture this behavior.

Cells are composed of around 70% water and 30% dry mass. As mentioned previously in the chapter, we can describe the composition of the dry mass with the most abundant chemical elements. For example, the elemental formula for *E. coli* is CH<sub>1.77</sub>O<sub>0.49</sub>N<sub>0.24</sub> (BNID 101800) and for *S. cerevisiae* CH<sub>1.61</sub>O<sub>0.56</sub>N<sub>0.16</sub> (BNID 101801). However, this kind of description is not particularly useful for understanding cells because it does not capture the variety of molecules that exist in a cell.

Therefore, we are more interested in biomass composition in terms of the main macromolecules (proteins, nucleic acids, lipids, and carbohydrates) and small molecules (metabolites, cofactors, and ions). Table 2.1 summarizes an average composition of *E. coli* and *S. cerevisiae* during exponential growth, the typical molecular masses and copy numbers of the components. The most abundant component is protein, which forms around half of the dry mass of the cell. When we divide the proteome into functional groups, we find that the biggest fractions belong to translation, central carbon metabolism, folding, sorting and degradation, and biosynthesis. A substantial fraction belongs to proteins that are not mapped (especially in mammalian cells), illustrating that we still lack knowledge about the function of many proteins (Figure 2.2).

<footnote n="1">¹The icons bacterium-interior, golgi-3d-1, mitochondrium-3, endoplasmatic-reticulum-3d-medium, endoplasmatic-reticulum-rough-3d-2, endoplasmatic-reticulum-rough-3d, and nucleus by Servier are licensed under CC-BY 3.0 Unported.</footnote>
**Problem 2.2 Proteins per cell - estimate one.** How many proteins are there in a bacterial/yeast/mammalian cell [5]? Use data from the following table:

<table><tr><td>Protein mass per volume</td><td>0.2 g/ml</td></tr><tr><td>Molecular mass of a protein</td><td>40000 g/mol</td></tr><tr><td>Avogadro's number</td><td>6 · 10<sup>23</sup> 1/mol</td></tr><tr><td><i>E. coli</i> volume</td><td>1 µm<sup>3</sup></td></tr><tr><td><i>S. cerevisiae</i> volume</td><td>60 µm<sup>3</sup></td></tr><tr><td>Mammalian cell volume</td><td>3000 µm<sup>3</sup></td></tr></table>

**Problem 2.3 Proteins/ribosomes per cell - estimate two.** A typical protein has a volume of 25 $\text{nm}^3$ (BNID 101828) and a ribosome 3400 $\text{nm}^3$ (BNID 104919). Given that 70% of a cell volume is water, what is the maximum number of protein/ribosome molecules that fit into a typical *E. coli* cell (see Table 2.2)? Compare your answers with the previous problem/values in BioNumbers database.

**Problem 2.4 Buoyant cell density.** Calculate the buoyant density of a typical bacteria using the following data:

<table><thead><tr><td>Component</td><td>Density (g mL<sup>-1</sup>)</td><td>Mass fraction per cell</td></tr></thead><tbody><tr><td>Water</td><td>1</td><td>0.7</td></tr><tr><td>Proteins</td><td>1.3</td><td>0.18</td></tr><tr><td>Nucleic acids</td><td>1.7</td><td>0.08</td></tr><tr><td>Lipids</td><td>1</td><td>0.03</td></tr><tr><td>Carbohydrates</td><td>1.5</td><td>0.01</td></tr></tbody></table>

**Problem 2.5 Concentrations enzymes and substrates.** Dourado et al. suggested that there is a relationship between the concentrations of enzymes and their substrates in *E. coli*, which is a result of a constraint on the biomass density. They showed that the reaction flux is maximal when the dry mass of each substrate is equal to the dry mass of the unsaturated (free) enzyme.

What is the concentration of one enzyme per cell for *E. coli* (in $\text{mol L}^{-1}$)? What would be the optimal concentration of its substrate? Use protein mass and cell volume from Problem 2.2 and the mass of glucose as substrate

**Problem 2.6 Cell size in different dimensions.** Imagine a spherical cell that increases its diameter from 1 to 2 um. How much do the surface area, volume, and SA/V change? Think about how this could influence the import of nutrients and the diffusion across the cell.

**Problem 2.7 Alien lifeforms.** Imagine alien lifeforms. Would they be composed of cells? Why? What features of cells could be completely different? What features are so much dictated by physics that they could not be different in any type of alien cell?
RNA forms 20% of dry cell mass in *E. coli*, but this number is lower in eukaryotes, such as yeast (11%) or mammalian cells (4%). While the total amount of RNA is variable in different organisms, its relative composition is similar – most of the RNA mass is formed by rRNA (80%), followed by tRNA (15%) and mRNA (5%) (BNID 100258, 100261, 106154). Lipid content is the highest in mammalian cells (13%) compared to yeast and bacteria (4-10%, BNID 111209, Table 2.1). Remarkably, there are cases where engineered yeast cells accumulated up to 80% of lipids per cell dry mass [16]. The content of storage carbohydrates varies from around 30% in yeast to 3% in bacteria (Table 2.1). In bacteria, carbohydrates are stored as the polysaccharide glycogen, while yeast cells use glycogen and the disaccharide trehalose. Yeast cells also contain structural polysaccharides, such as mannan and glucan [15]. Bacteria contain the structural molecule peptidoglycan (3% of dry mass) – a polymer of sugars and amino acids, which forms bacterial cell walls. In addition, some bacteria (e.g. *E. coli*) also have lipopolysaccharides on their cell wall (3% of dry mass).

A small fraction of the cell mass (2-3%) is formed by small molecules (< 1000 Da) such as metabolites and ions. This group contains thousands of different molecules with vastly different functions and concentrations. For illustration, the concentrations of the most abundant metabolites in *E. coli* range from $10^{-1}$ to $10^{-7}$ moles per cell, corresponding to a range of $10^8$ to only 100 copies per cell [14]. Possibly, there are metabolites with even lower concentrations, but these are much more difficult to quantify. Similarly, the concentrations of the most common inorganic ions ($K^+$, $Na^+$, $Mg^{2+}$, $Ca^{2+}$, $Cl^-$) span several orders of magnitude [14].

<table><caption>Table 2.1: Amounts, characteristic molecular masses and copy numbers of the main biomass components for *Escherichia coli* (*E. c.*) and *Saccharomyces cerevisiae* (*S. c.*). The composition data is shown for *E. c.* with a doubling time of 40 minutes (BNID 104954) and for *S. c.* with a doubling time of 110 minutes ([17], BNID 111755). The storage carbohydrates include glycogen for *E. c.* / glycogen and trehalose for *S. c.* The structural carbohydrates include peptidoglycan and lipopolysaccharides for *E. c.* / mannan and glucan for *S. c.* Sources for molecular masses (BNID 105861, 115091, 101838, 104886, 107678, 109645, 102502, 100459); molecule copy numbers (BNID 108248, 108197, 114950).</caption><thead><tr><th rowspan="2"></th><th colspan="2">% of dry mass</th><th colspan="2">Mass per cell [fg]</th><th colspan="2">Molecular mass [Da]</th><th colspan="2">Copy number</th></tr><tr><th>E. c.</th><th>S. c.</th><th>E. c.</th><th>S. c.</th><th>E. c.</th><th>S. c.</th><th>E. c.</th><th>S. c.</th></tr></thead><tbody><tr><td>Proteins</td><td>55</td><td>51</td><td>165</td><td>7650</td><td>40000</td><td>55000</td><td>3 × 10<sup>6</sup></td><td>10<sup>8</sup></td></tr><tr><td>RNA</td><td>20</td><td>11</td><td>60</td><td>1650</td><td>10<sup>4</sup>-10<sup>6</sup></td><td>10<sup>4</sup>-10<sup>6</sup></td><td>3 × 10<sup>5</sup></td><td>4 × 10<sup>6</sup></td></tr><tr><td>DNA (chromosomal)</td><td>3</td><td>0.5</td><td>9</td><td>75</td><td>3 × 10<sup>9</sup></td><td>2.5 × 10<sup>8</sup></td><td>2</td><td>16</td></tr><tr><td>Lipids</td><td>9</td><td>6</td><td>27</td><td>900</td><td>800</td><td>800</td><td>2 × 10<sup>7</sup></td><td>10<sup>9</sup></td></tr><tr><td>Storage carbohydrates</td><td>3</td><td>0.5</td><td>9</td><td>75</td><td>10<sup>6</sup></td><td>variable</td><td>4000</td><td>-</td></tr><tr><td>Structural polymers</td><td>6</td><td>23</td><td>18</td><td>3450</td><td>variable</td><td>variable</td><td>-</td><td>-</td></tr><tr><td>Metabolites/cofactors</td><td>3</td><td>2</td><td>9</td><td>300</td><td>&lt; 1000</td><td>&lt; 1000</td><td>-</td><td>-</td></tr><tr><td>Other</td><td>1</td><td>6</td><td>3</td><td>900</td><td>-</td><td>-</td><td>-</td><td>-</td></tr></tbody></table>

### 2.4.1 Variability of biomass composition

Table 2.1 shows biomass composition of a typical *E. coli* and *S. cerevisiae* cell – these are average values in certain environmental conditions. However, cell size, mass, and composition vary with growth rate and environmental conditions. One of the most extensively studied relationships in the literature is the correlation of growth rate with cell size. The increase of cell mass and volume with growth rate has been observed in bacteria (Figure 2.3), yeast, and mammalian cells [8, 9, 10, 11] (BNID 107948, 110191, 105103). For example, the cell mass of *E. coli* can vary fivefold – 150 to 870 fg per cell for generation times between 100 and 24 minutes [14]. Larger cell mass goes hand in hand with larger amounts of individual biomass components. The absolute amounts of protein, RNA, and DNA increase with cell size. However, the ratios of the components do not stay the same and the relative composition changes with growth rate [8, 12].

One of the most consistent observations is that the relative amount of RNA per cell increases with a higher growth rate [8, 12, 18], (BNID 111460, 111755, 108200). On the other hand, the data for relative protein content is more variable. For example, in bacteria, protein content decreases with growth rate in some studies [12, 18] but goes up and down in another (BNID 111460); in yeast, it increases (BNID 108200,
Box 2.B Orders of magnitude

The quantities of biomass components are usually expressed in relation to other quantities. The most common units are copy numbers, moles, grams, or fractions which can be expressed per cell, per gram dry mass, or per cell volume. Membrane components can also be expressed per surface area. Often, experimental data for these quantities is not readily available, so we need to extract it from literature. Useful sources for average or "rule of thumb" values include *BioNumbers database* [1] and the book *Cell Biology by the numbers* [14]. Some useful quantities are summarized in the table below. They are organized in increasing order with respect to the dimensions (1 – mass, size, thickness; 2 – area; 3 – volume, density). Notice how the dimensions influence the numerical values. For example, while the cell size differs only about 3-fold between bacteria and yeast, the surface area differs by more than tenfold and the volume by about 60-fold. Because volume grows faster than area, the ratio of cell surface area to volume (SA/V) gets smaller and smaller as cells get bigger (see more in Section 2.3.2). Note that these are just "rule of thumb" values. In reality, these values typically cover a broad range and depend on environmental conditions.

<table><thead><tr><th>Name</th><th>Unit</th><th><i>E. coli</i></th><th><i>S. cerevisiae</i></th><th>BNID/Reference</th></tr></thead><tbody><tr><td>Surface area/volume (SA/V)</td><td>µm<sup>-1</sup></td><td>6</td><td>1.2</td><td>calculated here</td></tr><tr><td>Dry cell mass</td><td>pg</td><td>0.3</td><td>15</td><td>104954, 108315</td></tr><tr><td>Total cell mass (with water)</td><td>pg</td><td>1</td><td>60</td><td>104954, 108315</td></tr><tr><td>Bilayer membrane thickness</td><td>nm</td><td>4</td><td>4</td><td>[14]</td></tr><tr><td>Cell size</td><td>µm</td><td>1-2</td><td>5</td><td>[14], 101796</td></tr><tr><td>Cell surface area</td><td>µm<sup>2</sup></td><td>6</td><td>70</td><td>101792, 113854</td></tr><tr><td>Cell volume</td><td>µm<sup>3</sup></td><td>1</td><td>60</td><td>101788, 101794</td></tr><tr><td>Cell density</td><td>g mL<sup>-1</sup></td><td>1.1</td><td>1.1</td><td>103875, 103876</td></tr></tbody></table>

111755). Nevertheless, when looking at RNA/protein ratio we consistently find a positive correlation with growth rate across various species of bacteria (see Figure 2.3) and yeast [19, 20]. RNA/protein ratio is a measure of protein production capacity since most RNA is dedicated to protein synthesis. 80% is rRNA, which forms 2/3 of the mass of a bacterial ribosome – the molecular machine that makes proteins, and 15% is tRNA which brings new amino acids to the ribosome (for more details about ribosomes, see Section 2.5). Indeed, we also observe a correlation between ribosome content and growth rate. The increase of RNA/protein ratio and ribosome content with increasing growth rate reflect higher biosynthetic needs of faster-growing cells. To support higher growth rate, cells need to reallocate resources according to the growth demands (for example, make more ribosomes which can then make more proteins) [19]. For more details about resource allocation and how it is modeled see Chapters 8 and 10.

Similarly to protein content, there is no clear correlation between the relative DNA and lipid content with

<figure><img src="image_21.png" /><figcaption>Figure 2.3: Growth laws in *E. coli* – (A) Cell volume grows exponentially with growth rate (data from [21]). (B) RNA/protein ratio grows linearly with growth rate (data from [19]). In both cases, growth rate was varied by changing medium composition.</figcaption></figure>
# Chapter 3

# The dynamics of metabolic systems

Robert West, Hadrien Delattre, Elad Noor, Wolfram Liebermeister, Herbert M. Sauro, Orkun S. Soyer

## Chapter highlights

This chapter introduces cell metabolism as a 'dynamical system'. While the previous chapter has talked about the constituents of this system, i.e. enzymes, metabolites, etc., this chapter focuses on conceptual abstraction of the metabolic system as a whole and how to model its dynamics over time. The key areas introduced are:

* Conceptualizing cell metabolism as a dynamical system (section 3.1)

* Dynamics and regulation of metabolism (section 3.2)

* Toolbox for modelling dynamics of metabolism - Biochemical reaction rate equations and their derivations (section 3.3)

* Dynamics of metabolism: Examples of experimental evidence and model-based explanations (section 3.4)

* Appendix of mathematical derivations and example models (section A.1 and A.2)

* Solutions to problems (section B.2)

This chapter links to the rest of this book by introducing dynamic of metabolism and highlighting possible dynamical features as constraints or self-regulation mechanisms on metabolic fluxes. Exploring the latter possibility is challenging, requiring both theoretical and experimental efforts. The understanding of dynamics is at the forefront of the study of cell metabolism and physiology and we hope that this chapter provides a notion in the reader to explore this area of research further.

In this chapter we will switch back-and-forth between a high-level view on metabolism, considering all of it, and a more focused, low-level view focusing on modeling individual reactions or small sets of reaction systems (e.g. pathways or motifs). These two viewpoints constitute two ends of a wide spectrum, and our aim in jumping back-and-forth between them is to allow the reader to obtain the skills to model dynamics of reaction systems that make up metabolism, while at the same time to invite them to think about the overall function of the metabolic system.

## 3.1. Conceptualizing cell metabolism as a dynamical system

Cell metabolism is a dynamical process that converts available metabolites from the environment into biomass and other products. The metabolism of a typical cell involves thousands of biochemical reactions and metabolites. What would be a useful way to think about such a complex, dynamical system? We need a conceptual picture of metabolism to help us formulate more specific ideas about how it functions, how it can be manipulated, or even how it has evolved. Here, we first highlight a few such 'pictures', or ways of thinking about metabolism.
growth rate across studies [12] (BNID 111460, 111755, 108196). The content of storage carbohydrates decreases at higher growth rates in yeast and bacteria [18] (BNID 111755, 111460).

As we have seen, the composition of the biomass changes with the growth rate, and for some components, we can describe this relationship with simple mathematical equations [18, 12]. However, the growth rate is the result of environmental conditions such as the amount or quality of a carbon source, temperature, oxygen concentration, or presence of inhibitors, among others. Different conditions may lead to the same growth rate but may not result in the same changes in cell physiology [9]. For example, modulation of growth rate by temperature rather than medium composition does not significantly affect cell size and composition [8, 12]. Inhibition of ribosomes with an antibiotic decreases the growth rate but increases the ribosome content, contrasting to the nutrient law shown in Figure 2.3 [19].

In contrast, environmental factors can influence cell composition without affecting growth rate. This shows that cell metabolism is flexible – cells can reach the same growth rate in different ways, depending on the conditions. For example, in yeast, changes in O₂ concentration lead to changes in biomass composition while keeping growth constant using a chemostat [22]. In mammalian cells, a change in a culture medium leads to significant changes in lipid composition without having a considerable effect on the growth rate [11].

### 2.4.2 Biomass composition varies within and between cells

In the previous paragraphs, we considered average cells with a homogeneous composition across the cell. However, we must keep in mind that cells have an internal structure and that biomass components are not uniformly distributed throughout the cell (as illustrated in Figure 2.4). Although prokaryotic cells do not have compartments separated by membranes, they have some internal organization. For example, DNA is not spread across the cytoplasm, but wrapped around proteins and packed in a compact structure called a nucleoid. Another example is that certain proteins are preferentially localized on the poles in rod-shaped bacteria. Eukaryotes have compartments with different compositions, pH, and membrane potential. DNA is localized only in the nucleus and mitochondria, and many proteins localize only in a particular compartment. Small molecules and ions also have different concentrations in the different compartments. Often they cannot freely diffuse through membranes, but transport is regulated and requires energy.

These differences in concentrations have implications for cellular functions. Some processes are restricted only to a particular compartment/area. For example, transcription only occurs in the nucleus and mitochondria (nucleoid), and some metabolic pathways occur only in a specific compartment (e.g. tricarboxylic acid cycle in the mitochondria). Even if the same enzyme is present in several compartments, it might work at a different rate or in the opposite direction because of the different concentrations of substrates or products. In eukaryotes, certain digestive enzymes only work at low pH present in lysosomes (thus preventing a cell from digesting itself). Sometimes, consecutive enzymes in a metabolic pathway do not freely float in a cell but form an assembly or bind to a scaffold, allowing intermediates to be channeled directly from one enzyme to another. This accelerates metabolic reactions because intermediates do not diffuse away into the bulk solution and are not consumed by competing reactions.

<figure><img src="image_22.png" /></figure>

Figure 2.4: Two cells with the same numbers of molecules, but different spatial distributions. Although the average concentration is the same in both cells, the second cell has varying concentrations in different compartments.