kinetic rate model and is commonly expressed as:

$$v = \frac{s \cdot E_{tot} \cdot k_{cat}}{K_{M} + s}$$

where $E_{tot}$ is equal to C and represents total enzyme concentration, $k_{cat}$ is equal to $k_3$ and is known as the maximal catalytic rate of an enzyme, and $K_M$ is equal to $(k_2 + k_3)/k_1$ and is known as the Michaelis-Menten coefficient of the enzyme. Plotting this rate against increasing substrate concentration would show that the rate is a 'saturating function' of S, i.e. the rate approaches a threshold point - given by $v_{max} = E_{tot} \cdot k_3$ as substrate increases. The enzymatic nature of the reaction introduces a limiting factor on the reaction rate! This saddle point is actually a underpinning point for some of the constraint-based methods discussed in this book.

### A.1.2. Derivation of a two substrate, irreversible rate equation

See Exercise 3.2

### A.1.3. Derivation of the single substrate, reversible rate equation

We now return to the reaction scheme we considered in the above section:

$$S + E \underset{k_2}{\overset{k_1}{\rightleftharpoons}} ES \underset{k_4}{\overset{k_3}{\rightleftharpoons}} EP \underset{k_6}{\overset{k_5}{\rightleftharpoons}} P + E.$$

The corresponding ODE system, written only for the key variables ES, EP, and P, is as follows:

$$\frac{des}{dt} = e \cdot s \cdot k_1 + ep \cdot k_4 - es \cdot (k_2 + k_3)$$

$$\frac{dep}{dt} = e \cdot p \cdot k_6 + es \cdot k_3 - ep \cdot (k_4 + k_5)$$

$$\frac{dp}{dt} = ep \cdot k_5 - e \cdot p \cdot k_6$$

As above, we will now introduce the assumptions of (1) total enzyme being conserved, and (2) the quasi steady-state, but this time for both of the enzyme-substrate and enzyme-product complexes. We will denote total enzyme concentration as C, as before, and use these two assumptions to express es and ep in terms of each other, and the other variables. Let us first proceed with es;

$$\frac{des}{dt} = 0 = e \cdot s \cdot k_1 + ep \cdot k_4 - es \cdot (k_2 + k_3)$$

$$es \cdot (k_2 + k_3) = (C - es - ep) \cdot s \cdot k_1 + ep \cdot k_4$$

$$es \cdot (k_2 + k_3 + s \cdot k_1) = (C - ep) \cdot s \cdot k_1 + ep \cdot k_4$$

$$es = \frac{C \cdot s \cdot k_1 + ep \cdot (k_4 - s \cdot k_1)}{(k_2 + k_3 + s \cdot k_1)}$$

We carry the same derivation for ep;

$$\frac{dep}{dt} = 0 = e \cdot p \cdot k_6 + es \cdot k_3 - ep \cdot (k_4 + k_5)$$

$$ep \cdot (k_4 + k_5) = (C - es - ep) \cdot p \cdot k_6 + es \cdot k_3$$

$$ep \cdot (k_4 + k_5 + p \cdot k_6) = (C - es) \cdot p \cdot k_6 + es \cdot k_3$$

$$ep = \frac{C \cdot p \cdot k_6 + es \cdot (k_3 - p \cdot k_6)}{(k_4 + k_5 + p \cdot k_6)}$$
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We see that we have a symmetry in the expressions for es and ep, in that the two expressions can be derived from each other by a replacement of variables ($k_1, k_4, k_2, s$) → ($k_6, k_3, k_5, p$). Keeping this symmetry in mind, we now attempt to eliminate one of the complexes from the equation for the other:

$$ep \cdot (k_4 + k_5 + p \cdot k_6) = C \cdot p \cdot k_6 + es \cdot (k_3 - p \cdot k_6)$$

$$ep \cdot (k_4 + k_5 + p \cdot k_6) = C \cdot p \cdot k_6 + \frac{C \cdot s \cdot k_1 + ep \cdot (k_4 - s \cdot k_1)}{(k_2 + k_3 + s \cdot k_1)} \cdot (k_3 - p \cdot k_6)$$

$$ep \cdot (k_4 + k_5 + p \cdot k_6) = C \cdot p \cdot k_6 + \frac{C \cdot s \cdot k_1 k_3 - C \cdot s \cdot k_1 \cdot p \cdot k_6 + ep \cdot (k_4 - s \cdot k_1) \cdot (k_3 - p \cdot k_6)}{(k_2 + k_3 + s \cdot k_1)}$$

$$ep \cdot (k_4 + k_5 + p \cdot k_6) \cdot (k_2 + k_3 + s \cdot k_1) = C \cdot p \cdot k_6 \cdot (k_2 + k_3 + s \cdot k_1) + C \cdot s \cdot k_1 k_3 - C \cdot s \cdot k_1 \cdot p \cdot k_6 + \\ ep \cdot (k_4 - s \cdot k_1) \cdot (k_3 - p \cdot k_6)$$

$$ep \cdot (k_4 + k_5 + p \cdot k_6) \cdot (k_2 + k_3 + s \cdot k_1) = C \cdot p \cdot k_6 k_2 + C \cdot p \cdot k_6 k_3 + C \cdot s \cdot k_1 k_3 + ep \cdot (k_4 - s \cdot k_1) \cdot (k_3 - p \cdot k_6)$$

$$ep \cdot ((k_4 + k_5 + p \cdot k_6) \cdot (k_2 + k_3 + s \cdot k_1) - (k_4 - s \cdot k_1) \cdot (k_3 - p \cdot k_6)) = C \cdot p \cdot k_6 k_2 + C \cdot p \cdot k_6 k_3 + C \cdot s \cdot k_1 k_3$$

$$ep = \frac{C \cdot p \cdot k_6 \cdot (k_2 + k_3) + C \cdot s \cdot k_1 k_3}{(k_4 + k_5 + p \cdot k_6) \cdot (k_2 + k_3 + s \cdot k_1) - (k_4 - s \cdot k_1) \cdot (k_3 - p \cdot k_6)}$$

$$ep = \frac{C \cdot p \cdot (k_6 k_2 + k_6 k_3) + C \cdot s \cdot k_1 k_3}{(k_4 k_2 + k_5 k_2 + k_5 k_3 + s \cdot k_1 k_3 + s \cdot k_1 k_4 + s \cdot k_1 k_5 + p \cdot k_6 k_2 + p \cdot k_6 k_3 + p \cdot k_6 k_4)}$$

Note that, in the above equation set, we have dropped the dot notation from multiplication of parameters for simplicity of expression. Based on the above argument of symmetry, or by following the same steps for “es”, we can show that we will have a similar expression with different parameters in the numerator:

$$es = \frac{C \cdot s \cdot (k_1 k_5 + k_1 k_4) + C \cdot p \cdot k_6 k_4}{(k_4 k_2 + k_5 k_2 + k_5 k_3 + s \cdot k_1 k_3 + s \cdot k_1 k_4 + s \cdot k_1 k_5 + p \cdot k_6 k_2 + p \cdot k_6 k_3 + p \cdot k_6 k_4)}$$

With these expressions for es and ep at hand, we can now derive an expression for e:

$$e = C - es - ep$$

$$e = C - \frac{C \cdot s \cdot (k_1 k_5 + k_1 k_4) + C \cdot p \cdot k_6 k_4}{(k_4 k_2 + k_5 k_2 + k_5 k_3 + s \cdot k_1 k_3 + s \cdot k_1 k_4 + s \cdot k_1 k_5 + p \cdot k_6 k_2 + p \cdot k_6 k_3 + p \cdot k_6 k_4)} - \frac{C \cdot p \cdot (k_6 k_2 + k_6 k_3) + C \cdot s \cdot k_1 k_3}{(k_4 k_2 + k_5 k_2 + k_5 k_3 + s \cdot k_1 k_3 + s \cdot k_1 k_4 + s \cdot k_1 k_5 + p \cdot k_6 k_2 + p \cdot k_6 k_3 + p \cdot k_6 k_4)}$$

$$e = C - C \cdot \frac{s \cdot (k_1 k_3 + k_1 k_5 + k_1 k_4) + p \cdot (k_6 k_2 + k_6 k_3 + k_6 k_4)}{(k_4 k_2 + k_5 k_2 + k_5 k_3 + s \cdot k_1 k_3 + s \cdot k_1 k_4 + s \cdot k_1 k_5 + p \cdot k_6 k_2 + p \cdot k_6 k_3 + p \cdot k_6 k_4)}$$

$$e = C \cdot \frac{k_3 k_5 + k_2 k_5 + k_2 k_4}{(k_4 k_2 + k_5 k_2 + k_5 k_3 + s \cdot k_1 k_3 + s \cdot k_1 k_4 + s \cdot k_1 k_5 + p \cdot k_6 k_2 + p \cdot k_6 k_3 + p \cdot k_6 k_4)}$$

We are now ready to substitute all these expressions into the ODE for the product, so to obtain our rate equation:

$$\frac{dp}{dt} = \frac{C \cdot p \cdot (k_6 k_2 + k_6 k_3) + C \cdot s \cdot k_1 k_3}{(k_4 k_2 + k_5 k_2 + k_5 k_3 + s \cdot k_1 k_3 + s \cdot k_1 k_4 + s \cdot k_1 k_5 + p \cdot k_6 k_2 + p \cdot k_6 k_3 + p \cdot k_6 k_4)} \cdot k_5 - C \cdot \frac{k_3 k_5 + k_2 k_5 + k_2 k_4}{(k_4 k_2 + k_5 k_2 + k_5 k_3 + s \cdot k_1 k_3 + s \cdot k_1 k_4 + s \cdot k_1 k_5 + p \cdot k_6 k_2 + p \cdot k_6 k_3 + p \cdot k_6 k_4)} \cdot p \cdot k_6$$

$$\frac{dp}{dt} = C \cdot \frac{s \cdot k_1 k_3 k_5 - p \cdot k_2 k_4 k_6}{(k_4 k_2 + k_5 k_2 + k_5 k_3 + s \cdot k_1 k_3 + s \cdot k_1 k_4 + s \cdot k_1 k_5 + p \cdot k_6 k_2 + p \cdot k_6 k_3 + p \cdot k_6 k_4)}$$
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We can somewhat simplify this expression by defining the following composite rate constants:

$$K_S = \frac{k_3 k_5 + k_2 k_5 + k_2 k_4}{k_1 \cdot (k_3 + k_4 + k_5)}$$

$$K_P = \frac{k_3 k_5 + k_2 k_5 + k_2 k_4}{k_6 \cdot (k_2 + k_3 + k_4)}$$

$$k_{\text{cat}}^{+} = \frac{k_3 k_5}{k_3 + k_4 + k_5}$$

$$k_{\text{cat}}^{-} = \frac{k_2 k_4}{k_2 + k_3 + k_4}$$

and substituting them into the rate expression from above, to get:

$$\frac{dp}{dt} = v = C \cdot \frac{k_{\text{cat}}^{+}}{K_S} \cdot \frac{s - p \cdot \frac{k_{\text{cat}}^{-}}{K_P}}{1 + \frac{p}{K_P} + \frac{s}{K_S}}$$

This reaction rate is referred to as the Haldane kinetic rate equation, named after Jack Burden Sanderson Haldane (5 November 1892 – 1 December 1964). It can be re-expressed by recognizing the fact that the fraction entering as a multiplier for the product concentration is actually equivalent to the equilibrium constant of the reaction scheme drawn above, at the beginning of this section, when we assume the reaction proceeding in the forward direction, i.e. towards product formation:

$$\frac{\frac{k_{\text{cat}}^{-}}{K_P}}{\frac{k_{\text{cat}}^{+}}{K_S}} = \frac{k_2 k_4 k_6}{k_1 k_3 k_5} = 1/K_{\text{eq}}$$

This allows us to re-express the Haldane rate equation as:

$$v = C \cdot k_{\text{cat}}^{+} \cdot \frac{s/K_{\text{S}}}{1 + \frac{p}{K_{\text{P}}} + \frac{s}{K_{\text{S}}}} \cdot \left(1 - \frac{p/s}{K_{\text{eq}}}\right)$$

This re-arranged expression is interesting because we can recognize that the last term is related to the thermodynamic Gibbs free energy of the reaction, allowing us to finally derive:

$$v = C \cdot k_{\text{cat}}^{+} \cdot \frac{s/K_{\text{S}}}{1 + p/K_{\text{P}} + s/K_{\text{S}}} \cdot \left(1 - e^{\Delta G'/RT}\right)$$

where $\Delta_r G'$ is the Gibbs free energy of reaction for given substrate and product levels, considering forward direction, and R and T stand for the gas constant and temperature respectively. This rate equation shows that forward reaction rate will be independent of thermodynamics, when the reaction free energy is highly negative (i.e. thermodynamically highly favored), but the reaction rate will decrease as Gibbs free energy gets close to zero.

A second, faster derivation of this rate equation is found by noting that the ODEs for $\frac{des}{dt}$ and $\frac{dep}{dt}$ are linear in e, es and ep, and can therefore be solved with linear matrix algebra. One may write:

$$\begin{pmatrix} sk_1 & -(k_2 + k_3) & k_4 \\ pk_6 & k_3 & -(k_4 + k_5) \\ 1 & 1 & 1 \end{pmatrix} \begin{pmatrix} e \\ es \\ ep \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \\ C \end{pmatrix}, \qquad (A.3)$$

where the first two rows of the matrix correspond to $\frac{des}{dt} = 0$ and $\frac{dep}{dt} = 0$, and the last row represents conservation of total enzyme concentration. The equilibrium concentrations of e, es and ep are then found by left-multiplying both sides of the equation by the inverse of this matrix. The obtained results are the same as given above.
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### A.1.4. Derivation of two substrate, reversible rate equation for simultaneous binding of substrates

The two-substrate case is described by the following reaction scheme:

$$S_1 + S_2 + E \underset{k_2}{\overset{k_1}{\rightleftharpoons}} ES_1S_2 \underset{k_4}{\overset{k_3}{\rightleftharpoons}} EP_1P_2 \underset{k_6}{\overset{k_5}{\rightleftharpoons}} P_1 + P_2 + E,$$

Where we assume that binding and unbinding of the substrates and products occurs simultaneously. Proceeding as above we let $e, es_1s_2, ep_1p_2, s_1, s_2, p_1$ and $p_2$ denote the concentrations of $E, ES_1S_2, EP_1P_2, S_1, S_2, P_1$ and $P_2$ respectively. The differential equations for $es_1s_2, ep_1p_2$ and $p_1 + p_2$ are:

$$\frac{des_1s_2}{dt} = e \cdot s_1 \cdot s_2 \cdot k_1 + ep_1p_2 \cdot k_4 - es_1s_2 \cdot (k_2 + k_3)$$

$$\frac{dep_1p_2}{dt} = e \cdot p_1 \cdot p_2 \cdot k_6 + es_1s_2 \cdot k_3 - ep_1p_2 \cdot (k_4 + k_5)$$

$$\frac{d(p_1 + p_2)}{dt} = ep_1p_2 \cdot k_5 - e \cdot p_1 \cdot p_2 \cdot k_6.$$

Proceeding as in the single substrate case, we note that the ODEs for $\frac{des_1s_2}{dt}$ and $\frac{dep_1p_2}{dt}$ are linear in $e, es_1s_2$ and $ep_1p_2$, and that the total enzyme concentration $e + es_1s_2 + ep_1p_2$ is constant, denoted $C$.

$$\begin{pmatrix} s_1s_2k_1 & -(k_2+k_3) & k_4 \\ p_1p_2k_6 & k_3 & -(k_4+k_5) \\ 1 & 1 & 1 \end{pmatrix} \begin{pmatrix} e \\ es_1s_2 \\ ep_1p_2 \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \\ C \end{pmatrix}. \qquad (A.4)$$

We therefore see that the results for the two-substrate case are the same as for the single substrate case, with $s$ replaced by $s_1s_2$ and $p$ replaced by $p_1p_2$. This result is dependent on the assumption that binding/unbinding of substrates/products occurs simultaneously.

## A.2. Example Metabolic Models

### A.2.1. A simple model illustrating product activation

This model demonstrates that allosteric regulation of an enzymatic reaction by its product can create a bistable system. In this simple example, we consider enzymatic production of a metabolite (labelled 'x') and its non-enzymatic consumption. It is assumed that the metabolite allosterically regulates the enzyme that produces it. The listing uses the Antimony format [624] which can be easily converted into SBML [625]. An online converter can be found at https://sys-bio.github.io/makesbml/

1. // The following model admits three steady-states at:

2. // x = 0.325, x = 1.671, and x = 0.873

3. // The first reaction step '-> x' uses a rate law that models

4. // positive feedback via the product x. The constant 0.2

5. // is to ensure that the lower steady-state is non-zero.

6. // The statement 'ext Xo' indicates that the species Xo is fixed.

7. ext Xo

8. Xo -> x; (vo*x^n)/(1 + x^n) + 0.2

10. x ->; k1*x

11. k1 = 0.65

13. n = 4; vo = 1
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14 x = 0

<caption>Listing A.1: Model illustrating bistability</caption>

1 # Equivalent model as a differential equation in python:

2 def ode (x, t):

3 vo = 1

4 n = 4

5 k1 = 0.65

6 return [((vo*x**n)/(1 + x**n) + 0.2) - k1*x]

<caption>Listing A.2: Equivalent model as a differential equation in python</caption>
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# Appendix B

## Solutions to problems

### B.1. What makes up a cell

**Problem 1 (Intuition for biological numbers)**

Check the results at http://bionumbers.hms.harvard.edu, http://book.bionumbers.org/ or https://doi.org/10.1016/j.cell.2010.06.019.

**Problem 2 (Proteins per cell - estimate one)**

$$ \frac{\text{Proteins}}{\text{mL}} = 0.2 \frac{\text{g}}{\text{mL}} \cdot 6 \cdot 10^{23} \frac{1}{\text{mol}} \cdot \frac{1}{40000} \frac{\text{mol}}{\text{g}} = 3 \cdot 10^{18} \frac{1}{\text{mL}} $$

$$ \frac{\text{Proteins}}{\mu\text{m}^3} = 3 \cdot 10^{18} \frac{1}{\text{mL}} \cdot 10^{-12} \frac{\text{mL}}{\mu\text{m}^3} = 3 \cdot 10^6 \frac{1}{\mu\text{m}^3} $$

$$ \frac{\text{Proteins}}{\text{cell}} \approx \begin{cases} 3 \cdot 10^6 & \text{E. coli} \\ 2 \cdot 10^8 & \text{S. cerevisiae} \\ 9 \cdot 10^9 & \text{mammalian cells} \end{cases} $$

**Problem 3 (Proteins/ribosomes per cell - estimate two)**

$$ 10^9 \cdot 0.3/25 \approx 1.2 \cdot 10^7 \text{ proteins} $$

$$ 10^9 \cdot 0.3/3400 \approx 88000 \text{ ribosomes} $$

**Problem 4 (Buoyant cell density)**

$$ \text{Density} = 1 \cdot 0.7 + 1.3 \cdot 0.18 + 1.7 \cdot 0.08 + 1 \cdot 0.03 + 1.5 \cdot 0.01 = 1.115 $$

**Problem 5 (Concentrations enzymes and substrates)**

Concentration of one molecule per E. coli cell:

$$ \frac{1}{\mu\text{m}^3} \cdot 10^{15} \frac{\mu\text{m}^3}{\text{L}} \cdot \frac{1}{6 \cdot 10^{23}} \frac{\text{mol}}{1} = 1.7 \cdot 10^{-9} \frac{\text{mol}}{\text{L}} $$

Enzyme mass: 40 000 g mol⁻¹ (BNID 105861)

Glucose mass: 180 g mol⁻¹
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$$40000/180 = 222$$

Assuming that the mass of the free enzyme and the mass of the substrate should be equal for optimal flux, we need approximately 222 molecules of substrate (glucose) per one molecule of free enzyme. This corresponds to the following concentrations:

Enzyme: ~ 1.7 nmol L⁻¹

Substrate: ~ 370 nmol L⁻¹

### Problem 6 (Cell size in different dimensions)

<table><thead><tr><th></th><th>Cell 1</th><th>Cell 2</th><th>Ratio</th></tr></thead><tbody><tr><td>Diameter</td><td>1</td><td>2</td><td>2</td></tr><tr><td>Surface area</td><td>3.14</td><td>12.57</td><td>4</td></tr><tr><td>Volume</td><td>0.52</td><td>4.19</td><td>8</td></tr><tr><td>SA/V</td><td>6</td><td>3</td><td>0.5</td></tr></tbody></table>

As the cell diameter increases, the volume grows faster than the surface area, leading to a decrease in the surface area-to-volume ratio (SA/V). This decrease can impact nutrient uptake as there is less relative surface area available for membrane transporters. Therefore, larger cells with lower SA/V ratios may face challenges in obtaining enough nutrients to meet their metabolic demands.

### Problem 7 (Alien lifeforms)

We do not know the answer to this question. These are ideas from ChatGPT (https://openai.com, GPT-3.5, version 2023) – take them with a grain of salt:

When imagining alien lifeforms, it is difficult to make definitive statements about their nature since we have no direct knowledge of extraterrestrial life. However, we can explore some possibilities and consider the principles of biology and physics as we currently understand them.

1. **Composition of Cells:** While cells are the fundamental building blocks of life on Earth, it is not certain that extraterrestrial life would be composed of cells. Cells are a product of evolution on our planet and are highly efficient structures for carrying out biological processes. However, alternative forms of life may exist with entirely different structures and organization. For example, hypothetical non-cellular life forms based on self-replicating molecules or other complex arrangements have been proposed.

2. **Different Features of Alien Cells:** a. **Membranes:** While cells on Earth have lipid bilayer membranes, it is conceivable that alien life forms might have different membrane compositions. For example, they could have membranes composed of different molecules or organized in alternative ways.

b. **Genetic Material:** On Earth, DNA and RNA are the primary carriers of genetic information. However, it is possible that alternative genetic molecules or information storage systems could exist in alien life. These could have different base pairs or use different coding mechanisms.

c. **Metabolism:** Alien life could have different metabolic processes compared to Earth organisms. They might utilize different energy sources, employ unique enzymatic reactions, or even rely on completely novel biochemical pathways.

d. **Size and Structure:** Cells on Earth exhibit a wide range of sizes, from microscopic bacteria to the largest known cells in organisms like ostrich eggs. It is conceivable that alien cells could differ significantly in size and overall structure, depending on the specific conditions and evolutionary paths of their respective environments.

3. **Features Dictated by Physics:** Certain fundamental principles of physics are likely to impose constraints on the functioning and structure of any kind of cell, including potential alien cells. These features include:

a. **Biochemistry:** Regardless of the specific molecular composition, alien cells would need a biochemistry that allows
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for the storage and utilization of energy, the replication and expression of genetic information, and the maintenance of internal equilibrium.

b. Thermodynamics: The laws of thermodynamics, such as energy conservation and entropy increase, are universal physical principles. Any living system, including alien cells, would need to adhere to these principles to maintain their internal processes.

c. Water: Water is a highly abundant molecule and a fundamental solvent for life on Earth. It provides a medium for biochemical reactions and allows for efficient transport of molecules within cells. It is possible that water or another suitable liquid would be essential for alien life, but alternative solvents cannot be ruled out entirely.

While these considerations provide a starting point for thinking about alien life, the possibilities are vast, and it is challenging to predict the specific characteristics of extraterrestrial organisms. Future discoveries and investigations in astrobiology will help refine our understanding of life beyond Earth.

## B.2. Metabolism

### Problem 1

(a)

$$E + S_1 + S_2 \underset{k_2}{\overset{k_1}{\rightleftharpoons}} ES_1S_2 \xrightarrow{k_3} E + P_1 + P_2 \quad (B.1)$$

(b)

$$\frac{dp}{dt} = k_3 \frac{s_1 s_2 C}{s_1 s_2 + \frac{k_2 + k_3}{k_1}}, \quad (B.2)$$

where $p = [P_1 + P_2]$ and $C = [E] + [ES_1S_2]$.

### Problem 2

(a)

$$E + S_1 + S_2 \underset{k_2}{\overset{k_1}{\rightleftharpoons}} ES_1S_2 \xrightarrow{k_3} E + P_1 + P_2 \quad (B.3)$$

(b)

$$\frac{dp}{dt} = k_3 \frac{C(s_1 s_2 - \frac{k_2 k_4}{k_1 k_3} p)}{s_1 s_2 + \frac{k_4}{k_1} p + \frac{k_2 + k_3}{k_1}} \quad (B.4)$$

where $p = [P_1 + P_2]$ and $C = [E] + [ES_1S_2]$.

### Problem 3

(a)

$$E + S_1 \underset{k_2}{\overset{k_1}{\rightleftharpoons}} ES_1 \\ ES_1 + S_2 \underset{k_4}{\overset{k_3}{\rightleftharpoons}} ES_1S_2 \xrightarrow{k_5} E + P_1 + P_2 \quad (B.5)$$

(b)

$$\frac{dp}{dt} = k_5 \frac{s_1 s_2 C}{s_1 s_2 + s_1 \frac{k_4 + k_5}{k_3} + s_2 \frac{k_5}{k_3} + \frac{k_2}{k_1 k_3} (k_1 + k_5)}, \quad (B.6)$$

where $p = [P_1 + P_2]$ and $C = [E] + [ES_1] + [ES_1S_2]$.

### Problem 4
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(a)

$$ 
\begin{aligned}
& E + S_1 \underset{k_2}{\overset{k_1}{\rightleftharpoons}} ES_1 \\
& ES_1 + S_2 \underset{k_4}{\overset{k_3}{\rightleftharpoons}} ES_1 S_2 \\
& E + S_2 \underset{k_6}{\overset{k_5}{\rightleftharpoons}} ES_2 \\
& ES_2 + S_1 \underset{k_8}{\overset{k_7}{\rightleftharpoons}} ES_1 S_2 \\
& ES_1 S_2 \underset{k_9}{\overset{k_8}{\rightleftharpoons}} E + P_1 + P_2
\end{aligned} 
\qquad (B.7) $$

(b)

$$ \frac{dp}{dt} = k_9 \frac{C s_1 s_2 (k_1 k_3 (k_6 + k_7 s_1) + k_5 k_7 (k_2 + k_3 s_2))}{s_1 A(s_1) + s_2 B(s_2) + s_1 s_2 C(s_1, s_2) + D} \qquad (B.8) $$

where $p = [P_1 + P_2]$, $C = [E] + [ES_1] + [ES_2] + [ES_1S_2]$, and

$$ A(s_1) = k_1 k_6 (k_4 + k_8 + k_9) + k_7 (k_0 + k_4) (k_2 + k_1 s_1) $$

$$ B(s_2) = k_2 k_5 (k_4 + k_8 + k_9) + k_3 (k_0 + k_8) (k_6 + k_5 s_2) $$

$$ C(s_1, s_2) = k_1 k_3 (k_6 + k_8 + k_7 s_1) + k_5 k_7 (k_2 + k_4 + k_3 s_2) + k_3 k_7 k_9 $$

$$ D = k_2 k_6 (k_4 + k_8 + k_9) $$

## B.3. The space of metabolic flux distributions

**Problem 4.4** Solution (EFMs with forward fluxes only):

<figure><img src="image_99.png" /></figure>

EFMs with forward and backward fluxes:
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<figure><img src="image_100.png" /></figure>

B.4. Rating metabolic pathways by enzyme efficiency

**Problem 6.9 (Haldane kinetic rate law)** First, we add the constraint on the total enzyme concentration ($[E] + [ES] + [EP] = E_{tot}$) and rewrite the ODE system in matrix notation:

$$ \begin{pmatrix} 1 \\ [S]k_1 \\ [P]k_6 \\ -[S]k_1 - [P]k_6 \end{pmatrix} \begin{pmatrix} 1 \\ -(k_2 + k_3) \\ k_3 \\ k_2 \end{pmatrix} \begin{pmatrix} 1 \\ k_4 \\ -(k_4 + k_5) \\ k_5 \end{pmatrix} \begin{pmatrix} [E] \\ [ES] \\ [EP] \end{pmatrix} = \begin{pmatrix} [E_0] \\ 0 \\ 0 \\ 0 \end{pmatrix}. \quad (B.9) $$

Note that the last row is linearly dependent on the two previous ones (it is minus their sum). Therefore, we can drop it from the system without losing information. Then, we will find explicit expressions for $[E]$, $[ES]$, and $[EP]$ by using Gaussian elimination – a process of eliminating off-diagonal values in the matrix until we reach the identity matrix, while at the same time applying the same operations to the vector on the right-hand side of the equality.

Step 1, elimination the off-diagonal elements on the first column (subtracting the first row times $[S]k_1$ from the 2nd row and the first row times $[P]k_6$ from the 3rd row)

$$ \begin{pmatrix} 1 & 1 & 1 \\ 0 & -(k_2 + k_3) - [S]k_1 & k_4 - [S]k_1 \\ 0 & k_3 - [P]k_6 & -(k_4 + k_5) - [P]k_6 \end{pmatrix} \begin{pmatrix} [E] \\ [ES] \\ [EP] \end{pmatrix} = [E_{tot}] \begin{pmatrix} 1 \\ -[S]k_1 \\ -[P]k_6 \end{pmatrix}. $$
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Step 2, dividing the second row by $-(k_2 + k_3 + [S]k_1)$ to have 1 on the diagonal:

$$ \begin{pmatrix} 1 & 1 & 1 \\ 0 & 1 & \frac{[S]k_1-k_4}{k_2+k_3+[S]k_1} \\ 0 & k_3-[P]k_6 & -(k_4+k_5)-[P]k_6 \end{pmatrix} \begin{pmatrix} [E] \\ [ES] \\ [EP] \end{pmatrix} = [E_{\text{tot}}] \begin{pmatrix} 1 & \frac{[S]k_1}{k_2+k_3+[S]k_1} \\ \frac{[S]k_1}{k_2+k_3+[S]k_1} & -[P]k_6 \end{pmatrix}. $$

Step 3, subtracting the second row from the 1st, and again from the 3rd (after multiplying by $k_3 - [P]k_6$):

$$ \begin{pmatrix} 1 & 0 & 1-\frac{[S]k_1-k_4}{k_2+k_3+[S]k_1} \\ 0 & 1 & \frac{[S]k_1-k_4}{k_2+k_3+[S]k_1} \\ 0 & 0 & -(k_4+k_5)-[P]k_6-\frac{([S]k_1-k_4)(k_3-[P]k_6)}{k_2+k_3+[S]k_1} \end{pmatrix} \begin{pmatrix} [E] \\ [ES] \\ [EP] \end{pmatrix} = [E_{\text{tot}}] \begin{pmatrix} 1-\frac{[S]k_1}{k_2+k_3+[S]k_1} & \frac{[S]k_1}{k_2+k_3+[S]k_1} \\ \frac{[S]k_1}{k_2+k_3+[S]k_1} & -[P]k_6-\frac{[S]k_1(k_3-[P]k_6)}{k_2+k_3+[S]k_1} \end{pmatrix}. $$

which after simplifying becomes:

$$ \begin{pmatrix} 1 & 0 & \frac{k_2+k_3+k_4}{k_2+k_3+[S]k_1} \\ 0 & 1 & \frac{[S]k_1-k_4}{k_2+k_3+[S]k_1} \\ 0 & 0 & -\frac{[S]k_1(k_3+k_4+k_5)+[P]k_6(k_2+k_3+k_4)+k_2k_4+k_2k_5+k_3k_5}{k_2+k_3+[S]k_1} \end{pmatrix} \begin{pmatrix} [E] \\ [ES] \\ [EP] \end{pmatrix} = [E_{\text{tot}}] \begin{pmatrix} \frac{k_2+k_3}{k_2+k_3+[S]k_1} & \frac{[S]k_1}{k_2+k_3+[S]k_1} \\ \frac{[S]k_1}{k_2+k_3+[S]k_1} & -\frac{[P]k_6k_2+[P]k_6k_3+[S]k_1k_3}{k_2+k_3+[S]k_1} \end{pmatrix}. $$

and we normalize the last row to have 1 on the diagonal:

$$ \begin{pmatrix} 1 & 0 & \frac{k_2+k_3+k_4}{k_2+k_3+[S]k_1} \\ 0 & 1 & \frac{[S]k_1-k_4}{k_2+k_3+[S]k_1} \\ 0 & 0 & 1 \end{pmatrix} \begin{pmatrix} [E] \\ [ES] \\ [EP] \end{pmatrix} = [E_{\text{tot}}] \begin{pmatrix} \frac{k_2+k_3}{k_2+k_3+[S]k_1} & \frac{[S]k_1}{k_2+k_3+[S]k_1} \\ \frac{[P]k_6k_2+[P]k_6k_3+[S]k_1k_3}{[S]k_1(k_3+k_4+k_5)+[P]k_6(k_2+k_3+k_4)+k_2k_4+k_2k_5+k_3k_5} & \frac{[S]k_1(k_3+k_4+k_5)+[P]k_6(k_2+k_3+k_4)+k_2k_4+k_2k_5+k_3k_5}{[S]k_1(k_3+k_4+k_5)+[P]k_6(k_2+k_3+k_4)+k_2k_4+k_2k_5+k_3k_5} \end{pmatrix}. $$

Step 4, we eliminate the off-diagonal values of the third column using the 3rd row:

$$ \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix} \begin{pmatrix} [E] \\ [ES] \\ [EP] \end{pmatrix} = [E_{\text{tot}}] \begin{pmatrix} \frac{k_2+k_3}{k_2+k_3+[S]k_1} & -\frac{k_2+k_3+k_4}{k_2+k_3+[S]k_1} \cdot \frac{[P]k_6k_2+[P]k_6k_3+[S]k_1k_3}{[S]k_1(k_3+k_4+k_5)+[P]k_6(k_2+k_3+k_4)+k_2k_4+k_2k_5+k_3k_5} \\ \frac{[S]k_1}{k_2+k_3+[S]k_1} & -\frac{[S]k_1-k_4}{k_2+k_3+[S]k_1} \cdot \frac{[P]k_6k_2+[P]k_6k_3+[S]k_1k_3}{[S]k_1(k_3+k_4+k_5)+[P]k_6(k_2+k_3+k_4)+k_2k_4+k_2k_5+k_3k_5} \\ \frac{[S]k_1}{k_2+k_3+[S]k_1} & \frac{[P]k_6k_2+[P]k_6k_3+[S]k_1k_3}{[S]k_1(k_3+k_4+k_5)+[P]k_6(k_2+k_3+k_4)+k_2k_4+k_2k_5+k_3k_5} \end{pmatrix} $$

Simplifying the expressions on the right-hand side is a lengthy process (which we do not show here) and in the end we get:

$$ \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix} \begin{pmatrix} [E] \\ [ES] \\ [EP] \end{pmatrix} = [E_{\text{tot}}] \begin{pmatrix} \frac{k_2k_4+k_2k_5+k_3k_5}{[S]k_1(k_3+k_4+k_5)+[P]k_6(k_2+k_3+k_4)+k_2k_4+k_2k_5+k_3k_5} \\ \frac{[P]k_6k_2+[P]k_6k_3+[S]k_1k_5}{[S]k_1(k_3+k_4+k_5)+[P]k_6(k_2+k_3+k_4)+k_2k_4+k_2k_5+k_3k_5} \\ \frac{[P]k_6k_2+[P]k_6k_3+[S]k_1k_3}{[S]k_1(k_3+k_4+k_5)+[P]k_6(k_2+k_3+k_4)+k_2k_4+k_2k_5+k_3k_5} \end{pmatrix} $$

Therefore,

$$ [E] = [E_{\text{tot}}] \frac{k_2k_4 + k_2k_5 + k_3k_5}{[S]k_1(k_3 + k_4 + k_5) + [P]k_6(k_2 + k_3 + k_4) + k_2k_4 + k_2k_5 + k_3k_5} \quad (B.10) $$

$$ [ES] = [E_{\text{tot}}] \frac{[P]k_4k_6 + [S]k_1k_4 + [S]k_1k_5}{[S]k_1(k_3 + k_4 + k_5) + [P]k_6(k_2 + k_3 + k_4) + k_2k_4 + k_2k_5 + k_3k_5} \quad (B.11) $$

$$ [EP] = [E_{\text{tot}}] \frac{[P]k_2k_6 + [P]k_3k_6 + [S]k_1k_3}{[S]k_1(k_3 + k_4 + k_5) + [P]k_6(k_2 + k_3 + k_4) + k_2k_4 + k_2k_5 + k_3k_5} \quad (B.12) $$
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Index

adder, 231

allometry, 261

anabolism, 136

apparent kcat value, 164

autocatalysis, 151

autocatalytic cores, 153

bacterial persistence, 214

balanced growth, 186

bang-bang solution, 182, 186

batch growth, 133

Bayesian inference, 203

bet-hedging, 214

biomass composition, 173

biomass reaction, 56, 86

bioprocesses, 188

Boltzmann distribution, 200

bow-tie structure, 25

carbohydrates, 17

carbon catabolite repression, 133, 140

catabolism, 136

cell composition, 15, 173

cell cycle, 229, 235

cell density, 30

cell division, 229

cell size, 22

cell surface, 138

cell volume, 138

chemostat, 134

CHR algorithm, 73

chromosome, 236

coarse-grained model, 134, 183

concurrent-cycles scenario, 240

constraint, 250

control, 231

convection, 256

conversion cone, 71

convexity, 106

correlation, 230

cost of oxygen transport (COT), 264

coupling, 245

decision making, 215

DFBAlab, 191

diauxic growth, 133, 177

diffusion, 255

Dirac's δ-distribution, 199

direct approach (DA), 190

diversity, 196

dynamic flux balance analysis (dFBA), 187

dynamic optimization, 178, 179, 183, 186, 190

dynamical autocatalysis, 154

dynamical system, 35

EFM phase diagram, 130

EFM weight polyhedron, 65

electron flow, 38

elementary conversion mode, 55

elementary conversion modes, 71

elementary flux mode, 80, 120

elementary flux modes, 57, 59

elementary vector, 71

Embden-Meyerhof-Parnas pathway, 96

end-product feedback, 181

Entner-Doudoroff pathway, 97

Entropy, 210

entropy, 221

enzyme, 42

enzyme catalysis, 44

enzyme cost minimization, 106

enzyme demand, 99, 100

enzyme efficiency, 29, 120

equilibrium constant, 44

evolutionary stability, 146

expanding economy model, 155

exponential growth, 135

extensive quantity, 137

factorized rate law, 101, 102

fermentation, 141

finance, 219

fine-grained model, 134

fitness-information relationship, 208

flux, 137
flux balance analysis, 84

flux cone, 65, 79

flux coupling constraint, 166

flux distribution, 55

flux polyhedron, 65

flux sampling, 55, 73

flux space, 79

flux variability analysis, 91

flux vector, 58, 59, 61, 79, 83

formose, 156

gambling, 219

gene dosage compensation, 147

geometric mean, 218

Gibbs free energy, 43

glycolysis, 111

Gordan's theorem, 152

growth law, 19, 134, 145, 186

growth laws, 159

growth rate, 126, 136, 144, 184, 188, 218

growth rate maximization, 141

hazard, 232

hazard rate, 235

hierarchical optimization, 191

homeostasis, 234

information, 221

information theory, 207

inhomogeneous constraints, 65

intensive quantity, 137

irreversibility constraints, 65

Jacques Monod, 133

just-in-time principle, 180

Kelly, 219

kinetic rate law, 137

Leontief model, 156

lexicographic optimization, 191

linear programming (LP), 84, 189

linear response, 235

linearization, 165

lipids, 17

lung, 251

macromolecules, 16

maintenance energy, 134

mass action rate law, 42, 44

mass action ratio, 41

mass balance, 56, 136

mass conservation, 136

mass-conservation constraint, 166

Max-Min Driving Force method (MDF), 104

maximum a posteriori inference, 202

maximum entropy principle, 203

maximum likelihood inference, 202

mechanistic assumption, 140

metabolic engineering, 112, 188

metabolic enzyme, 26

metabolic network, 55

metabolic pathway, 35, 96, 180

metabolic regime, 263

methionine and arginine biosynthetic pathways, 180

Michaelis constant, 27

Michaelis-Menten rate law, 45, 46

microbial community, 189

minimal cut set, 55, 73

Monod curve, 128

Monod landscape, 129

Monod law, 134, 145

morphogenesis, 252

multi-criteria optimization, 179, 187, 193

multiple optima in constraint-based models, 196

non-enzymatic reaction, 44

non-equilibrium thermodynamics, 254

non-linear programming (NLP), 190

non-uniqueness (of optima), 91

nucleic acid, 16

nutrient upshift, 186

objective function, 86, 179

objective vs subjective uncertainty, 198

optimal control theory, 178, 179

optimality, 213

optimization, 178, 249

ordinary differential equation (ODE), 50, 179

overflow metabolism, 141, 142

Pareto, 223

Pareto optimality, 179

Pathway Specific Activity (PSA), 112

phase plane analysis, 90

physical constraint, 31

polymerization, 29

polytope, 83, 91

Pontryagin's minimum principle, 182

probability density, 198

production function, 157

protein, 16

protein density constraint, 168

proteome, 184

proteome sector, 137

proteome space, 30
quasi-steady-state assumption, 136

redox reaction, 38

regulation of metabolic pathways, 182

replication-segregation cycle, 236

resource allocation, 183

resource balance analysis (RBA), 169

respiration, 141

ribosome, 183

risk, 214

self-replication, 135, 184

shadow price, 90

Shannon, 219

single-cell data, 230

single-cell methods, 196

sizer, 231

small molecules, 16

static optimization approach (SOA), 190

statistical inference, 202

steady state, 50, 56

stem cell, 186

stochasticity, 213

stoichiometric coefficient, 85

stoichiometric matrix, 50, 58, 179, 188

stoichiometry, 57

synthetic gene circuit, 146

system equation, 50

thermodynamic constraint, 67

thermodynamic efficiency, 102

thermodynamic equilibrium, 43

timer, 231

transcription, 145

transition time, 180

translation, 145

turnover number, 27

universal constructor, 153

vertex, 91
List of boxes

<table><tr><td>Economics 1.A: A cell and a construction firm as black boxes</td><td>12</td></tr><tr><td>Economics 1.B: A cell to a construction firm as growing systems</td><td>13</td></tr><tr><td>Economics 1.C: Allocation of workforce</td><td>13</td></tr><tr><td>General 2.A: Macromolecular machines</td><td>18</td></tr><tr><td>General 2.B: Orders of magnitude</td><td>22</td></tr><tr><td>Experiments 2.C: Quantification of biomass composition</td><td>24</td></tr><tr><td>Experiments 2.D: Examples of biomass quantification methods</td><td>25</td></tr><tr><td>Economics 2.E: Comparing the cell to a bakery</td><td>26</td></tr><tr><td>Economics 3.A: Metabolism as an assembly line</td><td>37</td></tr><tr><td>Physics 3.B: The redox ladder in metabolism</td><td>40</td></tr><tr><td>Physics 3.C: Cyclic reaction motifs</td><td>41</td></tr><tr><td>Physics 3.D: Mass action law for chemical reactions</td><td>43</td></tr><tr><td>Mathematics 4.A: Generators of a polyhedral cone</td><td>59</td></tr><tr><td>Mathematics 4.B: Equivalent Definitions of Elementary Flux Modes (EFMs)</td><td>62</td></tr><tr><td>Economics 5.A: Optimization and economic planning in the Soviet Union</td><td>84</td></tr><tr><td>Philosophy 5.B: Qualities of a model</td><td>92</td></tr><tr><td>Philosophy 6.A: What do we mean by a "pathway"?</td><td>96</td></tr><tr><td>Mathematics 6.B: Integer cuts for iterating all possible pathway variants</td><td>98</td></tr><tr><td>Mathematics 6.C: Factorized rate laws and enzyme cost function</td><td>101</td></tr><tr><td>Economics 6.D: The push for fast growth</td><td>113</td></tr><tr><td>Economics 6.E: Two central assumptions: homogeneity and stationarity</td><td>115</td></tr><tr><td>General 7.A: Kinetics of the example network</td><td>123</td></tr><tr><td>General 7.B: Optimal metabolic states in the example network</td><td>124</td></tr><tr><td>Mathematics 9.A: Gordan's theorem</td><td>152</td></tr><tr><td>Economics 9.B: Production functions</td><td>157</td></tr><tr><td>General 10.A: Protein abundance versus concentration in resource allocation models</td><td>168</td></tr><tr><td>General 11.A: Temporal optimization for biotechnological applications and process design</td><td>187</td></tr><tr><td>Mathematics 11.B: DFBAlab</td><td>191</td></tr><tr><td>Mathematics 11.C: Lexicographic Optimization</td><td>191</td></tr><tr><td>Experiments 12.A: Quantitative methods for single-cell analysis</td><td>196</td></tr><tr><td>Mathematics 12.B: Dirac's δ-distribution</td><td>199</td></tr><tr><td>Mathematics 12.C: Gaussian model</td><td>203</td></tr><tr><td>Mathematics 12.D: Inference in metabolic network modeling</td><td>204</td></tr><tr><td>Economics 12.E: Maximum Entropy economic equilibrium</td><td>205</td></tr><tr><td>Mathematics 13.A: Arithmetic versus geometric mean and logarithmic utility</td><td>218</td></tr><tr><td>Mathematics 13.B: Kelly's optimal betting strategy</td><td>219</td></tr><tr><td>Mathematics 13.C: The value and cost of information for growing populations</td><td>221</td></tr><tr><td>Mathematics 14.A: Probability distribution of (un)divided cells</td><td>233</td></tr><tr><td>Mathematics 14.B: Linear formalism and adder plots</td><td>238</td></tr><tr><td>Mathematics 14.C: Comparing cell-cycle sub-periods models with data</td><td>239</td></tr><tr><td>Mathematics 14.D: The concurrent-cycles framework</td><td>241</td></tr><tr><td>Mathematics 14.E: A mathematical model</td><td>243</td></tr><tr><td>Philosophy 15.A: The origin of shape?</td><td>253</td></tr><tr><td>Physics 15.B: Energy conversion in muscles</td><td>254</td></tr></table>
<table><tr><td>Mathematics 15.C: Conditions for the numerical simulations</td><td>257</td></tr><tr><td>Physics 15.D: The cost of oxygen transport</td><td>264</td></tr><tr><td>Physics 15.E: The cost of oxygen transport as a function of speed</td><td>265</td></tr></table>
# Mathematical symbols

## Metabolic models

<table><thead><tr><th>Quantity</th><th>Symbol</th><th>Dimensionality</th><th>Unit</th></tr></thead><tbody><tr><td colspan="4">State variables</td></tr><tr><td>Metabolic flux</td><td>v</td><td>concentration/time</td><td>mmol/L/sec</td></tr><tr><td>Metabolite concentration</td><td>s</td><td>concentration</td><td>mmol L<sup>-1</sup></td></tr><tr><td>Enzyme level</td><td>e</td><td>concentration</td><td>mmol L<sup>-1</sup></td></tr><tr><td colspan="4">Rate laws and enzyme kinetics</td></tr><tr><td>Reaction rate</td><td>ν</td><td>concentration/time</td><td>mmol/L/sec</td></tr><tr><td>Maximal velocity</td><td>ν<sup>max</sup></td><td>concentration/time</td><td>mmol/L/sec</td></tr><tr><td>Catalytic rate (enzyme efficiency)</td><td>κ</td><td>1/time</td><td>mmol/L/sec</td></tr><tr><td>Apparent catalytic constant</td><td>k<sub>app</sub></td><td>1/time</td><td>1/sec</td></tr><tr><td colspan="4">Kinetic constants</td></tr><tr><td>General rate constant</td><td>k</td><td>1/time</td><td>1/sec</td></tr><tr><td>Catalytic constant (irrev. reactions)</td><td>k<sub>cat</sub></td><td>1/time</td><td>1/sec</td></tr><tr><td>Forward catalytic constant</td><td>k<sub>cat</sub><sup>+</sup></td><td>1/time</td><td>1/sec</td></tr><tr><td>Backward catalytic constant</td><td>k<sub>cat</sub><sup>-</sup></td><td>1/time</td><td>1/sec</td></tr><tr><td>Michaelis-Menten constant</td><td>K<sub>M</sub></td><td>concentration</td><td>mmol L<sup>-1</sup></td></tr><tr><td>Inhibition constant</td><td>K<sub>I</sub></td><td>concentration</td><td>mmol L<sup>-1</sup></td></tr><tr><td>Activation constant</td><td>K<sub>A</sub></td><td>concentration</td><td>mmol L<sup>-1</sup></td></tr><tr><td colspan="4">Flux constraints</td></tr><tr><td>Flux polytope</td><td>P</td><td>-</td><td>-</td></tr><tr><td>Full stoichiometric matrix</td><td>N = (n<sub>ij</sub>)</td><td>dimensionless</td><td>1</td></tr><tr><td>Internal stoichiometric matrix</td><td>N<sub>int</sub> = (n<sub>ij</sub>)</td><td>dimensionless</td><td>1</td></tr><tr><td>Elementary flux mode</td><td>f</td><td>-</td><td>-</td></tr><tr><td>Flux (as an objective)</td><td>J</td><td>concentration/time</td><td>mmol/L/sec</td></tr><tr><td colspan="4">Thermodynamics</td></tr><tr><td>Equilibrium constant</td><td>K<sub>eq</sub></td><td>-</td><td>-</td></tr><tr><td>Reaction Gibbs free energy</td><td>Δ<sub>r</sub>G'</td><td>energy/mol</td><td>kJ mol<sup>-1</sup></td></tr><tr><td>Reaction standard Gibbs free energy</td><td>Δ<sub>r</sub>G'</td><td>energy/mol</td><td>kJ mol<sup>-1</sup></td></tr><tr><td>Thermodynamic driving force</td><td>θ</td><td>dimensionless</td><td>1</td></tr><tr><td colspan="4">Enzyme cost and efficiency</td></tr><tr><td>Reversibility efficiency term</td><td>η<sup>for</sup></td><td>dimensionless</td><td>1</td></tr><tr><td>Saturation efficiency term</td><td>η<sup>sat</sup></td><td>dimensionless</td><td>1</td></tr><tr><td>Regulation efficiency term</td><td>η<sup>reg</sup></td><td>dimensionless</td><td>1</td></tr><tr><td>Enzyme molecular mass</td><td>m</td><td>mass/mol</td><td>Dalton = g mol<sup>-1</sup></td></tr><tr><td colspan="4">Cell physiology</td></tr><tr><td>(Specific) cell growth rate</td><td>μ</td><td>1/time</td><td>1/h</td></tr><tr><td>Total enzyme level</td><td>e<sub>tot</sub></td><td>concentration</td><td>mmol L<sup>-1</sup></td></tr><tr><td>Yield on substrate</td><td>Y</td><td>-</td><td>-</td></tr></tbody></table>
## Physical units

<table><thead><tr><td>Unit</td><td>Symbol</td></tr></thead><tbody><tr><td>second</td><td>s</td></tr><tr><td>meter</td><td>m</td></tr><tr><td>liter</td><td>L</td></tr><tr><td>Joule</td><td>J</td></tr><tr><td>Kelvin</td><td>K</td></tr><tr><td>mol</td><td>mol</td></tr></tbody></table>

## Physical variables and constants

<table><thead><tr><td>Symbol</td><td>Name</td><td>Value</td><td>Unit</td></tr></thead><tbody><tr><td>t</td><td>Time</td><td></td><td>s or h</td></tr><tr><td>T</td><td>Temperature</td><td></td><td>K</td></tr><tr><td>R</td><td>Boltzmann's gas constant</td><td>≈ 8.31</td><td>J K<sup>-1</sup> mol<sup>-1</sup></td></tr><tr><td>N<sub>A</sub></td><td>Avogadro number</td><td>≈ 6.022 · 10<sup>23</sup></td><td>mol<sup>-1</sup></td></tr></tbody></table>

## Mathematical symbols

<table><thead><tr><td>Name</td><td>Symbol</td></tr></thead><tbody><tr><td>real numbers</td><td>R</td></tr><tr><td>nonnegative integers</td><td>N</td></tr><tr><td>integer numbers</td><td>Z</td></tr><tr><td>Euler's constant</td><td>e or exp(·)</td></tr><tr><td>imaginary unit</td><td>i</td></tr><tr><td>differential (as in d/dt)</td><td>d</td></tr><tr><td>kernel of a matrix</td><td>ker(·)</td></tr><tr><td>support of a function</td><td>supp(·)</td></tr><tr><td>sign of a vector</td><td>sign(·)</td></tr></tbody></table>
# List of abbreviations

## General

WT Wildtype (non-mutant, non-modified cell strain)

## Biochemical compounds

ADP Adenosine diphosphate

ATP Adenosine triphosphate

DNA Deoxyribonucleic acid

RNA Ribonucleic acid

rRNA ribosomal RNA

tRNA transfer RNA

mRNA messenger RNA

## Biochemical pathways

ED Entner-Doudoroff pathway (a version of glycolysis)

EMP Embden-Meyerhof-Parnas pathway (a version of glycolysis)

## Modeling methods

CAFBA Constrained Allocation Flux Balance Analysis

DA Direct Approach

DFBA Dynamic Flux Balance Analysis

DOA Dynamic Optimization Approach

EFM Elementary Flux Mode

FBA Flux Balance Analysis

MDF Max-min Driving Force

MILP Mixed-Integer Linear Problem

LP Linear Programming

ODE Ordinary Differential Equation

NLP Non-Linear Programming

PSA Pathway Specific Activity

(Q)SSA (quasi-) Steady-State Assumption

RBA Resource Balance Analysis

SOA Static Optimization Approach

## Physical quantities

SA/V Surface Area/Volume ratio

OD Optical Density
# Author contributions

**Preface.** The preface was written by Wolfram Liebermeister, with parts written by Paul Ross and feedback from Orkun Soyer.

**Overview of the book.** The overview chapter was drafted by W. Liebermeister, edited by Paul Ross and Michela Pauletti, and reviewed by Diana Széliová. The graphics were designed and made by Michela Pauletti.

**1. The cell as a factory.** This chapter was written by Ohad Golan with the help of Wolfram Liebermeister and Elad Noor.

**2. An inventory of cell components.** This chapter was drafted and written by Pranas Grigaitis and Diana Széliová, with the help of Wolfram Liebermeister and Elad Noor. Feedback was provided by Samira van den Bogaard and Ohad Golan. Michela Pauletti drew many of the figures in this chapter.

**3. Cell metabolism.** The chapter was originally drafted and written by Orkun Soyer, Hadrien Delattre, and Robert West. Later, sections were added by Wolfram Liebermeister, Elad Noor, and Herbert Sauro. We thank Francesco Moro, Markus Arthur Köbis, and Maarten Droste for comments on an earlier version of this chapter. We acknowledge funding from the UK's Biotechnology and Biological Sciences Research Council (BBSRC) grant ID BB/T010150/1 and from the Gordon and Betty Moore Foundation grant ID GBMF9200.

**4. Metabolic flux distributions.** This chapter was originally drafted and written by Daan de Groot, David Tourigny and Felipe Scott. Stefan Müller worked substantially on the sections about elementary flux modes, David Ruckerbauer and Maxime Mahout added sections on Minimal Cut Sets and on the enumeration of EFMs under extra constraints. Wolfram Liebermeister wrote parts of the introduction. Exercises were added by Felipe Scott and Andreas Kremling. Feedback was provided by Diana Széliová, Samira van den Bogaard, Maarten Droste, Elad Noor, and Herbert Sauro.

**5. Optimization of metabolic fluxes.** This chapter was originally drafted and written by Daan de Groot, David Tourigny and Felipe Scott. Newer sections were added by Stefan Müller and Hester Chapman. Exercises were added by Felipe Scott. Jan Philipp Dapprich wrote the Box on Leonid Kantorovich. Feedback was provided by Diana Széliová, Samira van den Bogaard, Maarten Droste, Wolfram Liebermeister, Elad Noor, and Herbert Sauro.

**6. The enzyme cost of metabolic fluxes.** This chapter was drafted and written by Wolfram Liebermeister and Elad Noor and has been reviewed by Avi Flamholz, Daan de Groot, Maarten Droste, and Ohad Golan.

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

**8. Principles of cell growth.** Ohad Golan conceived and drafted the chapter. Ohad Golan, Hidde de Jong, and Andrea Weiße wrote sections 1.1-1.3. All authors provided critical feedback and helped shape the chapter. Hidde de Jong acknowledges support from the Agence National de la Recherche (MAXIMIC, ANR-17-CE40-0024-01).

**9. Universal features of autocatalytic systems.** This book chapter is mainly based on these two recent articles: Anjan Roy et al., A unifying auto-catalytic network-based framework for bacterial growth laws, Proc. Natl. Acad. Sci. U.S.A. 118(33):e2107829118 (2021) and A. Blokhuis et al., Universal motifs and the diversity of autocatalytic systems, Proc. Natl. Acad. Sci. U.S.A., 117, 25230 (2020). We thank R. Pugatch for comments.

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

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

12. **Diversity of metabolic fluxes in a cell population.** This chapter was drafted and written by Andrea De Martino and Marcelo Rivas-Astroza, who are indebted with Daan de Groot for many useful comments and suggestions, as well as with Daniele De Martino, Wolfram Liebermeister, Elad Noor and David Tourigny for discussions. Michela Pauletti redrew all figures.

13. **Cells in the face of uncertainty.** The chapter was written by D. Lacoste, O. Rivoire, and D. Tourigny. We thank Daan de Groot for reviewing this chapter.

14. **Strategies for cell cycle control.** This chapter was drafted and written by Marco Cosentino Lagomarsino, with the help of the other authors. All the authors wrote the introduction and discussion sections (14.1 and 14.7). Mattia Corigliano drafted section 14.5, Marco Cosentino Lagomarsino drafted sections 14.2 and 14.6, Jacopo Grilli drafted section 14.3, Gabriele Micali drafted section 14.4. Michela Pauletti redrew most of the figures. We thank Andrea De Martino for proof correction and feedback on our manuscript.

15. **Economy of organ form and function.** Cyril Karamaoun, Benjamin Mauroy and Frédérique Noël wrote the major content of the chapter and contributed equally to this work. Boxes were contributed by Christophe Goupil and Eric Herbert, and further sections were provided by Paul Ross.
# Reading recommendations

Cellular economics - and systems biology more generally - builds on knowledge from different disciplines and on a history of ideas in biology and beyond. Below you will find a number of books, articles, and online resources that provide background information. For readings specific to individual chapters, please see the "Recommended readings" sections at the end of each chapter.

## Recommended textbooks

### Books on Biochemistry and cell biology:

**Cornish-Bowden, Athel. *Fundamentals of Enzyme Kinetics*;** Wiley Blackwell, 2012 (4th Edition). A detailed presentation of enzyme kinetics that also considers the broader implications of the field for areas such as systems biology and bioinformatics. The book devotes particular attention to the areas of multi-enzyme complexes and cooperativity.

**Alberts Bruce, Heald Rebecca, et al. *Molecular Biology of the Cell*;** Garland Science, 2022 (6th Edition). A comprehensive textbook of cell biology which covers expression and transmission of genetic information, internal organization of cells, and behavior of cells in multicellular organisms. The book also presents experimental methods used to investigate cells and understand their behavior. Free access online (5th Edition): archive.org/details/MolecularBiologyOfTheCell5th.

**Swanson, Michele, S., Joyce, Elizabeth, A. and Horak, Rachel, E.A. *Microbe*;** ASM Press, 2022 (3rd Edition). An introductory textbook to microbiology that incorporates both fundamental principles and accessible case studies to illustrate microbiology's social and environmental relevance.

**Milo, Ron and Phillips, Rob et. al. *Cell Biology By The Numbers*;** Taylor & Francis, 2016. The book addresses the relative paucity of quantitative information about biological systems. The authors propose ways to quantify biological processes and objects and develop a "sense" for applicable scales and sizes. Free access online: book.bionumbers.org.

**Phillips, Rob, Kondev, Jane, et al. *Physical Biology of the Cell*;** Garland Science, 2012 (2nd Edition). This textbook presents molecular and cell biology through the lens of physical biology. Biological phenomena are treated as coherent systems founded on physical principles. The overarching topic is that quantitative biological intuition builds on applying few fundamental physical models, and this logic can be used to address a wide range of biological problems.

**Harold, Franklin M. *The Way of the Cell: Molecules, Organisms, and the Order of Life*;** Oxford University Press, 2003. The text offers connections between physics, chemistry, and biology and how insights gained from integrative approach help to understand the processes and principles that make the creation of living organisms from inanimate chemicals possible.

### Books on Biothermodynamics and biological physics:

**Haynie, Donald T. *Biological Thermodynamics*;** Cambridge University Press, 2008 (2nd Edition). An introduction to bioenergetics focused on the physical nature of energy transformation in living cells. The book covers relevant concepts of physics, like the laws of thermodynamics, Gibbs free energy, statistical thermodynamics, binding equilibria and reaction kinetics.

**Nicholls, David, G and Fergueso, Scott J. *Bioenergetics*;** Academic Press, 2013 (4th Edition). A textbook on cellular energy flows and intracellular thermodynamics. The book takes recent advances in chemiosmotic energy conversion into account; the book also covers the role that mitochondria play in the life and death of the cell.

**Nelson, Philip. *Biological Physics*;** Chiliagon Science, 2020. The book covers biophysics, bioengineering and nanotechnology and highlights recent advances in molecular motors, self-assembly, and single-molecule
manipulation. It also describes practical applications, and contains exercises designed to develop model-
ing skills of computer programming in languages such as MATLAB and Python.

Books on Systems Biology:

Klipp, Edda, Liebermeister, Wolfram, et. al. *Systems biology, A Textbook*; Wiley-Blackwell, 2016 (2nd Edition). An overview of basic concepts in systems biology supplemented with illustrative examples and practical case studies. The book presents an integrative approach to living systems combining biology, computer science, and engineering.

Voit, Eberhard. *A First Course in Systems Biology*; Garland Science, 2018 (2nd Edition). A moderately advanced introduction to systems biology. The book prioritizes conceptualization and development of computational models and the ways theoretical work enhances understanding of diverse biological systems.

Sauro, Herbert. *Systems Biology: Introduction to Pathway Modeling*; Ambrosius Publishing, 2020. The book covers fundamentals of biochemical modeling, including a review on differential equations and stochastic models. The book highlights the increasingly prominent role of computer models in modern systems and synthetic biology. Book includes hands-on modeling exercises to illustrate computational models methods for designing, building, simulating models; performing stability analysis, and parameter estimation.

Palsson, Bernhard. *Systems Biology: Simulation of Dynamic Network States*; Cambridge University Press, 2011. An introduction to the mass action stoichiometric simulation (MASS) approach to transform stoichiometric reconstructions into dynamic models making use of metabolomic and fluxomic data. The MASS approach aims at depicting integrated processes that depend on a precise accounting of small molecules and proteins.

Alon, Uri. *An Introduction to Systems Biology: Design Principles of Biological Circuits*; Chapman & Hall, 2006. The book presents design principles of biological systems such as stability, robustness, and optimal design, and how they can be applied to deepen the understanding of living cells. The book provides a mathematical framework that can be used to better understand and design biological circuits by highlighting the recurring circuit elements that make up biological networks.

Szallasi, Zoltan, Periwal, Vipul, and Stelling, Joerg. *System Modeling in Cellular Biology: From Concepts to Nuts and Bolts*; MIT Press, 2010. An overview of systems modeling in cell biology. The book incorporates concepts from biology, computer science, mathematics, statistics, physics, and biochemistry and considers how they can be integrated to study biological systems. The book also covers multiple modeling paradigms and discusses their suitability for modeling different biological systems.

Savageau, Michael, A. *Biochemical Systems Analysis: A Study of Function and Design in Molecular Biology*; CreateSpace Independent Publishing Platform, 2010. A primer to identifying and defining biological design principles for systems like complex biochemical pathways, intricate gene regulation circuits, network interactions within the immune system, plasticity of neural networks, and the pattern formations of cellular networks. The book addresses integration of multiple viewpoints, the relation between the behavior of intact systems and their molecular components; unifying design principles that give meaning for vast diversity of alternative molecular designs; higher-level theory and quantitative prediction.

Books on Metabolic control:

Heinrich, Reinhart and Schuster, Stefan. *The Regulation of Cellular Systems*; Springer, 1996. The book covers the mathematical analysis of enzymatic systems such as stoichiometric analysis, enzyme kinetics, dynamical simulation, metabolic control analysis, and evolutionary optimization.

Fell, David. *Understanding The Control of Metabolism*; Frontiers in Metabolism Ser. No. 2; Ashgate Publishing Company, 1997. The book offers a comprehensive theory of metabolic regulation, metabolic control analysis, that encompasses the philosophical underpinnings of regulation and control concepts. It also reviews and summarizes experimental methods that add a new dimension to the study of metabolism
and its regulation in complete multicellular organisms, isolated tissues, organs and individual cells.

**Sauro, Herbert.** *Systems Biology: An Introduction to Metabolic Control Analysis*; Ambrosius Publishing, 2018. This book provides an introduction to the field of Systems Biology and demonstrates how computational models can be used to understand the dynamics of complex biological systems and how the individual components that comprise those systems interact to produce complex behaviors. The book details the fundamentals of modeling biological pathways, such as metabolic, signaling, and gene regulatory networks and instruction in the application of tools that can create those models.

**Books on Mathematics:**

**Slivanus, P. Thompson and Gardner, Martin.** *Calculus Made Easy*; St. Martin's Press, 1998 (revised/expanded edition). An introduction to Calculus for beginners and those with no previous background. Essentials of differentiation and integration are covered: the power rule, product rule, and chain rule for differentiation, as well as basic techniques of integration. The book takes a different approach to calculus by emphasizing an intuitive understanding of calculus rather than insisting on a rigorous formalism. The applications-oriented approach highlights the applicability of those concepts to real-world cases. Free pdf at calculusmadeeasy.org.

**Johnston, Nathaniel.** *Introduction to Linear and Matrix Algebra*; Springer, 2021. The book introduces key concepts of linear algebra, such as vectors, matrices, systems of linear equations. The book primarily covers vector spaces and their subspaces, and linear transformations between vector spaces. The book is full of examples in which linear algebra is used in fields such as computer science, physics, and engineering, e.g., least-squares solutions, Markov chains, and cryptography.

**Strang, Gilbert.** *Introduction to Linear Algebra*; Wellesley-Cambridge Press, 2023 (6th Edition). The book covers theory and applications of vector algebra and geometry, systems of linear equations, vector spaces and subspaces, orthogonality, determinants, and eigenvalues and eigenvectors, and linear statistical analysis. Later chapters concern contemporary applications and computational issues in linear algebra.

**Strang, Gilbert.** *Calculus*; Wellesley-Cambridge Press, 1991 (2nd Edition). This book is a thorough review of both single-variable and multivariable calculus. The book makes use of visual methodologies (diagrams) and illustrative examples to aid understanding and enhance the learning experience. Examples highlight the application of calculus to a variety of domains such as physics and engineering and economics.

**Lay, David, Lay, Steven, and McDonald, Judi.** *Linear Algebra and Its Applications*; Pearson, 2014 (5th Edition). The book provides a basic introduction to linear algebra, illustrated by a broad selection of case studies. The emphasis is on developing a conceptual view of the subject and an understanding of why, and not just how, certain methods are used. Problems and cases presented demonstrate how linear algebra can be applied to various fields.

**Strogatz, Steven.** *Nonlinear dynamics and chaos: With applications to physics, biology, chemistry, and engineering*; CRC Press, 2000. An exploration of concepts related to dynamical systems and nonlinear behaviors that exhibit unpredictable or chaotic behavior and cannot be easily predicted by simple equations. The book covers the basics of dynamical systems, characterizing the properties of non-linear systems and the points of equilibrium. The book also describes temporal dynamics of systems. Finally, it considers fractals and unique properties of chaotic systems with practical examples.

**Books on Information theory:**

**Cover, Thomas A. and Thomas, Joy A.** *Elements of Information Theory*; Wiley Series in Telecommunications and Signal Processing; Wiley-Interscience, 2006 (2nd Edition). The book considers the theoretical underpinnings and practical application of information theory. It also provides a thorough review of information theory's constituent elements: entropy, data compression, channel capacity, rate distortion, network information theory, and hypothesis testing.

**Bialek, William.** *Searching for Principles*; Princeton University Press, 2012. The book explores the inter-
section of theoretical physics and biology and argues that the fundamental principles of physics can be applied to living systems and used to better understand their behavior. The book advocates for an inter-disciplinary approach that integrates insights from physics, biology, and mathematics to develop a deeper understanding of how living organisms behave and function.

**Book on Economics:**

**Core ECON.** *Economics for a Changing World.* A free textbook offering a new approach to teaching and learning economics, supported by an extensive collection of online resources. Written by a global collective of economists. All content is freely available at www.core-econ.org.

**Classical books, relevant to cell biology and the "economy of the cell"**

Brasier, Martin. *Secret Chambers: The Inside Story of Cells and Complex Life*; Oxford University Press, 2012. This book draws on insights from paleontology, biology, and history of science in its exploration of the evolution of complex life on Earth. It focuses specifically on how cells developed the capacity for organization and complexity using an investigation into the origins of eukaryotic cells, the fundamental elements of all complex life forms, as illustration.

Zimmer, Carl. *Microcosm: E. coli and the New Science of Life*; Knopf Doubleday, 2009. This book describes the role *E. coli* has played in defining the history of biology - from the discovery of DNA to more recent advances in biotechnology. The book highlights the bacterium's malleability, mutability, and survivability. Methods which exploited *E. coli*'s inherent malleability have modified and repurposed the bacterium to produce a range of beneficial products from drugs to fuels. Because it mutates in near real-time, *E. coli* has also been a source of insight into the process of evolution. Detailed examination of its genome has revealed a record of billions of years of evolutionary history. Finally, the book highlights the numerous strategies *E. coli* uses to survive, from practicing chemical warfare to building microbial cities.

Monod, Jacques. *Chance and Necessity: An Essay on the Natural Philosophy of Modern Biology*; Vintage Books, 1972. Monod argues that life should be understood as a result of natural processes that do not adhere to a predetermined purpose or follow a predefined design: it arose by chance and that was then conditioned by natural selection. Hence the book's central thesis is that the observed evolution of life and its diversity is the product of "Chance" - random mutations in genetic material and "Necessity" the deterministic laws of physics and chemistry that govern these mutations. Together, "Chance" and "Necessity" explain the evolution and diversity of life.

Prigogine, Ilya and Stengers, Isabelle. *Order Out Of Chaos*; Bantam, 1984. The text reconciles order with chaos and synthesizes concepts of time and chance to form a lens through which thermodynamics can be appreciated and its attendant laws can be better understood. In doing so, it offers a provocative view of the universe and novel insights into humankind's position within it.

Harold, Franklin M. *In Search of Cell History: The Evolution of Life's Building Blocks*; University of Chicago Press, 2014. The book presents research into the history of the cell and the debate conclusions from that research have generated while describing the evolution of cellular organization, the origin of complex cells, and the incorporation of symbiotic organelles.

Lyons, Sherrie L. *From Cells to organisms: re-envisioning cell theory*; University of Toronto Press, 2020. The book integrates the history of science with fundamental biological concepts and provides context that draws on the nature of scientific practice and rise of ideas. The book shows how discoveries, debates, and scientific processes regarding understanding of cells and organisms shaped modern biology. The focus on controversy highlights the iterative nature of science, showing how scientific knowledge evolves through argument, testing, and reinterpretation of evidence. The book connects past and present by describing how cell theory laid the foundation for the development of fields such as molecular biology, biochemistry, and genetics.

Niklas, Karl J. *Plant Allometry: The Scaling of Form and Process*; University of Chicago Press, 1994. The author seeks to apply allometry to studies of plant evolution, morphology, physiology, and reproduction
and shed light on the relationship between organ size and plant form and physiology.

**Rosen, Robert.** *Optimality Principles in Biology*; Springer, 1967. The text concentrates on conceptual and theoretical aspects of cost functions and the decisive role they play in the achievement of optimality in biological systems. It focuses in particular on the relationship between extremization and stationarity, and the significance of the necessary conditions for stationarity.

**Rosen, Robert.** *Life Itself: A Comprehensive Inquiry Into the Nature, Origin, and Fabrication of Life*; Columbia University Press, New York, 1991. A treatise on complex biological systems that calls into question "reductionism", the belief that all complex systems can be broken down (reduced) to their constituent parts. The goal of this approach is to effect more rigorous analysis.

**Schrödinger, Erwin.** *What is life? With Mind and Matter and Autobiographical Sketches*; Canto Classics (Cambridge University Press reprint), 2012. The book investigates the nature of life from the physics perspective and speculates on the age-old question of the relationship between mind and matter. Throughout the book, the author attempts to reconcile biology and quantum physics, in order to demonstrate how seemingly chaotic processes of living organisms in fact obey the laws of physics while maintaining order and stability. Some of the hypotheses articulated in the book laid the conceptual foundation for molecular biology, e.g. the supposition that the key to life lies in molecular structures that store and transmit information.

**Cornish-Bowden, Athel.** *The Pursuit of Perfection: Aspects of Biochemical Evolution*; Oxford University Press, 2004. The book explores the interplay of biochemistry and evolutionary biology, the two being implicitly connected and that each can only be understood fully in the context of the other. The book also argues for a more balanced approach to science: in light of Nature's complexity and unpredictability, perfect models are unattainable; we should strike a balance between accuracy and practical application. Diversity of biological systems is a prime example of this. The book examines how natural selection optimizes biochemical processes and enzymes but stops short of achieving absolute perfection due to constraints like energy efficiency, environmental fluctuations, and genetic variation.

**West, Geoffrey.** *Scale: The Universal Laws of Growth, Innovation, Sustainability, and the Pace of Life in Organisms, Cities, Economies, and Companies*; Weidenfeld & Nicolson, 2017. This book presents the theory of scale that is grounded in the realization that universal mathematical laws govern complex systems in nature, society, and economics. These so-called scaling laws reflect underlying efficiencies and constraints based on how energy, resources, or information are distributed across networks. The book draws from biology, physics, and complexity theory to explain the mathematical principles that govern the scaling of both living systems and social organizations: cities, companies, and organisms.

**Dawkins, Richard.** *The Blind Watchmaker: Why the Evidence of Evolution Reveals a Universe Without Design*; Norton & Company, 1986. Dawkins makes the case for the theory of evolution through natural selection and in doing so refutes the intervention of the divine (metaphorically referred to as the "Watchmaker"). The book highlights the differences between human design and planning and the way natural selection works. Drawing a distinction with the original metaphor for the presence of the divine, the book concludes that the evolutionary process, on further investigation, reveals itself to be more similar to that of a blind watchmaker.

## Classical articles that shaped the thinking in our field

**Jacob, François.** "Evolution and Tinkering"; Science, volume 196, issue 4295, 1977. The article elucidates the concept of evolution by viewing it through the lens of "tinkering". It proposes that consideration of the cumulative effects of history on the evolution of life leads to an alternative account of the patterns that characterize the history of life on earth.

**Gould, Stephen Jay, Lewontin, Richard.** *The spandrels of San Marco and the panglossian paradigm: a critique of the adaptationist programme*; Proceedings of the Royal Society B: Biological Sciences, 205(1161), 581-598, 1979. This paper is a critique of the adaptationist view of evolution that the authors illustrate by drawing an zanalogy to an architectural feature called the spandrel. The adaptationist school holds that most biological features adaptations and that these adaptations are the result of evolutionary pressures
that have favored their development because they offer some advantage to the organism in its environ-
ment. Countering the adaptationist view, the authors of the critique posit that just as the spandrel, a
triangle-shaped space between arches in structures like the Basilica of San Marco, is a byproduct of the
arches and domes that are the Basilica's constituent elements but can be artistically decorated, making
them look intentional many biological traits may be byproducts of other structural necessities, not direct
adaptations. In fact, these traits may not serve any specific adaptive purpose but have come into being as
a result of constraints, like the need to maintain structural integrity, or as side effects of other adaptations.

Lazebnik, Yuri. *Can a biologist fix a radio?— Or, what I learned while studying apoptosis*; Cancer Cell, 3, 179-182, 2002. The paper offers a critique of the way systems biology, in particular, approaches the study of living organisms through the analogy of the way an engineer approaches the diagnosis and repair of a broken radio. Engineers typically understand systems holistically while the approach of a biologist is characteristically reductionist. The reductionist approach to a malfunctioning radio is to isolate and count radio's components without fully understanding how they interact with one another as parts of a larger, integrated system. The paper argues that biologists should follow the example of engineers in taking a more integrative and systems-based approach: an understanding of how an entire system functions is the key to identifying and diagnosing problems. The approach the paper advocates takes the form of improved models and frameworks that enable a better understanding of the complex interactions that characterize biological systems.

**Newsholme, Philip.** *Mapping life's reactions: a brief history of metabolic pathways and their regulation;*
The Biochemist Volume 31, Issue 3, 2009. The article provides a historical review of discovery of metabolic
functions and assembling them into metabolic pathways. This period begins in the early 20th century with
a highlight of contributions made by German chemists such as Meyerhof and Krebs. The paper charts
advances in the understanding of metabolic regulation and the refinement of mathematically-grounded
methods for accurately and adequately modeling metabolic pathways from the post-war era. Finally, it ex-
tends the account into the current time with a description of the advent of metabolomics and the evolution
of systems biology.

Recommended online videos and podcasts

Milo, Ron. *Lectures on "Cell biology by the numbers"* The lectures, accompanying the book "Cell biology by the numbers", invite viewers to think about cells quantitatively. Ron Milo shows that knowing a few important numbers (or looking them up in the BioNumbers database) and using them for rough estimates gives an intuitive sense for how cells function!

TEDx talk “A sixth sense for understanding our cells”: www.youtube.com/watch?v=JC7WnzM2Lsc

Full BioNumbers lectures: www.youtube.com/channel/UChCzuzoZp5NheAPWH5YoGWw

Lercher, Martin J., and Yanai, Itai. *Night Science*. Series of editorials in *Genome Biology* (Springer Nature) 2019-2022, and podcast 2021-ongoing. The title, "Night Science", draws on a term coined by the biologist Francois Jacob to characterize the significant parts of scientific research that occur "behind the scenes". This iteration on the original concept charts the decline of innovation and creative thinking in Science in favor of a focus on near-term results and financial gain. It also offers prescriptions for restoring the creative spirit to science and revitalizing innovation.

Website: night-science.org

Materials: www.biomedcentral.com/collections/night-science

Podcast on Spotify: open.spotify.com/show/6berzd2rX6rpJQ6CPnbOtl?si=2176cbb795f24c6a

**Khan Academy.** *Free Math Courses*. Khan Academy offers free math courses, including advanced topics like differential equations, linear algebra, and multivariable calculus. These courses feature video tutorials, practice problems, and step-by-step solutions, making complex topics like systems of equations, vector spaces, and partial derivatives easier to grasp. The lessons are self-paced and adaptable.

Website: www.khanacademy.org
# Appendix A

# Appendix “Cell metabolism”

## A.1 Dynamics of metabolism: experimental evidence and model-based explanations

The high-level of connectivity among reactions, together with the plurality of molecular level mechanisms that can arise in enzyme-mediated reactions, gives metabolic systems the capacity to display rich dynamic behaviors [61, 90]. Here, we highlight some of the illustrative experimental observations on these metabolic dynamics, and their possible model-based explanations.

### A.1.1 Flux switching / regulation

We have introduced above the redox-based, electron flow view of metabolism. A common electron donor in metabolism is glucose, while a common electron acceptor is oxygen together with the associated, membrane-bound electron transport chain (ETC). The ensuing metabolic pathway linking glucose oxidation to oxygen reduction is termed as 'respiration', resulting in formation of CO₂ (from full oxidation of glucose) and water (from reduction of oxygen, cf. Fig. ??). However, it is possible for cell metabolism to stop the sequential oxidation of glucose (or other sugars) at an intermediate level. In this case, the ensuing metabolism is termed as 'fermentation' due to production of partially oxidized carbon compounds such as acetate and ethanol (Fig. A.1) [51].

One of the earliest observation on metabolic dynamics is a shift from pure respiration into fermentation or respiro-fermentation with changing conditions. This shift, known as contre-Pasteur, Warburg, or, Crabtree effect, is described initially in yeast and mammalian cells, especially cancerous cells. The respiration to fermentation shift happens under lack of electron acceptors or with increasing growth rate. While a shift into fermentative pathways due to lack of electron acceptors can be intuitively understood as the only route to sustain electron flow, a similar shift due to increased carbon availability or growth rate are non-intuitive as they occur under the continued presence of strong electron acceptors such as oxygen.

A dominant concept to explain the switch to respiro-fermentation has been the idea of 'overflow metabolism'. It postulates that this switch should be seen as an overflow, arising due to limitations in respiration not being sufficient in sustaining metabolic fluxes in the face of increasing substrate availability. The dynamic regulation and origin of this respiro-fermentation switch is still a focus of significant systems biology research. Hence, this topic is discussed further in other chapters of this book with several alternative models presented for its underlying causes.

It must also be noted that, while respiro-fermentation switch is commonly referred to as 'overflow metabolism' (due to excretion of fermentation products such as acetate, lactate and ethanol), the phenomenon of overflow, i.e. excretion of energy rich compounds is not limited to fermentation. Excretion of amino acids and vitamins seem particularly common, and it is not clear in these cases what type of metabolic flux switching happens or how it happens.
<figure><img src="image_150.png" /></figure>

Figure A.1: Respiration-fermentation switch – Cartoon representation of the respiration and fermentation pathways. There is an observed switch of metabolic fluxes between these pathways, from full respiration to fermentation or a combination of respiration and fermentation, as glucose levels (and consequently growth rate) increases. It is postulated that this relates to a limitation in the respiration and ETC system, but the molecular basis of the switch is not fully clear with several, equally plausible hypotheses postulated.

**Flux switching / regulation - flux sensors, branch point dynamics and dynamical flux regulation** How can we understand cells switching their metabolic fluxes with changing external or internal conditions. As discussed above, one possibility is that cells alter the expression levels of their various enzymes, so to achieve a re-distribution of fluxes (given that enzyme levels are directly involved in the determination of fluxes, see Eq. (3.8)). This kind of enzyme-level regulation can be mediated through regulation of transcription factors by specific signaling molecules, including metabolites. The latter case is explored in models of central metabolism, and it was shown that fructose-1,6-bisphosphate could act as a "flux sensor", conveying information about the glycolytic flux onto key transcription factors regulating glycolytic enzymes.

While regulation of enzyme levels can alter flux levels, this type of regulation can be made more sensitive if the coupling between enzyme and flux levels can be made more nonlinear. It has been shown that such nonlinear coupling of flux and enzyme levels can arise at branching points in metabolism [59]. In particular, branch points (as a metabolic motif) have been shown to give rise to ultrasensitivity - a system dynamics feature that describes the situation when a given input to a system results in more than a proportional change in its output. When enzymes at the two branches of a branching point have highly differing affinities for the substrate (i.e. different $K_M$ values, see Eq. (3.8)), then alterations of the maximal rate of one enzyme with higher affinity to the substrate can result in a nonlinear effect on the flux into the other branch of the branching point [59]. Thus, branching points can be one structural motif that can result in switch-like, nonlinear flux changes within metabolism.

It is clear that changing of enzyme levels can regulate fluxes, and can do so in an abrupt, switch-like fashion through structural motifs such as branching points. However, regulation of enzyme levels via transcription factors is found to not capture all observed flux changes in experiments. This suggests that cells might be able to regulate fluxes by other means as well. Recently, one such possible mechanisms is proposed to be the co-substrate pools [65]. For example, in a metabolic branch point, where the two branches involve different co-substrates, regulating the pool sizes of those two co-substrates can induce flux switching at the branch point.

### A.1.2 Bistability

Bistability is introduced above, and refers to a dynamical system having three steady state, two of which are dynamically stable and can be attained by the system. When bistable systems exist in cell metabolism, their combination with population level variance (i.e. noise) in enzyme levels or activity can lead to bimodal
distribution of metabolic fluxes (i.e phenotypes) in isogenic population of cells. In this context, it is notable
that significant level of variance is seen in several metabolic parameters, including sugar uptake, ATP
levels, and expression levels of the enzymes involved in glycolysis and the TCA cycle.

Bistability in metabolic responses is experimentally implicated in the context of respiration to fermentation switch, and when carbon metabolism is initiated on glucose or when it switches from glucose to other carbon sources. In particular, the latter studies found sub-populations, within isogenic populations (i.e. no mutations), that show different metabolic responses to changing conditions. Experiments with isotope labeled carbon indicated that these sub-populations emerged at the time of the shift in carbon source is induced, i.e. in response to changing conditions, and in a manner dependent on the concentration of the new carbon source. This suggests that the metabolic system implements bistable dynamics, such that changes in external glucose concentrations can lead some cells to shift to a new metabolic steady-state flux distribution, while others remain at their original steady state.

**Bistability - negative feedback via substrate inhibition** There have been many theoretical studies indicat-
ing the possibility of bistability within simple enzymatic reaction systems. Bistability is shown to be possible
even in a single enzymatic reaction, involving allosteric regulation, or in a system of few coupled enzymatic
reactions. A particular 'reaction motif' that has been studied extensively is a two-enzyme cyclic reaction
system, where a substrate is converted into a product and then back again, with both forward and back-
ward reactions usually involving different enzymes (see Fig. A.2). It is common, in these models, that the
enzyme catalyzing the forward reaction is assumed to be regulated by substrate inhibition, or by substrate
inhibition coupled with product activation. This motif is found in several locations within metabolism, par-
ticularly around dehydrogenases, such as lactate dehydrogenase, and kinase/phosphatase pairs, such as
those involved around fructose-6-phosphate, that can convert different metabolites back and forth, using
the NAD+/NADH or ADP/ATP pairs as reaction partners.

These theoretical findings are supported by several *in vitro* re-constitution experiments that confirmed bistability experimentally using enzyme preparations of pyruvate kinase, lactate dehydrogenase, and isocitrate dehydrogenase enzymes and their corresponding partners resulting in cyclic reaction schemes.

It is notable that many of these models incorporated negative feedback via empirical alteration of a Michaelis-Menten type reaction rate model (i.e. one of the approaches we mentioned in the paragraph above on allosteric rate models, see 3.12). This raises the question about the actual biochemical mechanisms that can lead to bistability in a enzyme-mediated reaction model. In a recent study, it was shown that the presence of multiple enzyme-substrate complexes, as would be the case in an enzyme with multiple substrate binding sites, creates a potential in the reaction system for bistability [80] (see Fig. A.2). Thus, multi-site enzymes could be points of multistability generation in metabolic systems and any larger models featuring such enzymes or inherently including feedback regulation can demonstrate bistability.

A.1.3 Oscillations

Sustained and damped oscillations are common dynamics in nonlinear systems and can arise from a com-
bination of positive and negative feedbacks [86]. In metabolic systems both types of oscillations are seen
*in vivo* or *in situ*, with cell extracts, where concentrations of all observed metabolites are found to oscillate
over time. In the case of experiments involving cell extracts, these metabolite oscillations had a period
ranging from few to tens of minutes. In these experiments, oscillations are verified not to be due to arti-
ficial changes in ATP dynamics arising in the cell extract preparations, and oscillations could be entrained
by controlled glucose additions. This shows that there is an inherent ability for oscillatory dynamics in the
underpinning enzymatic reaction system. This ability is suggested to be linked to the enzyme phosphofruc-
tokinase (PFK), which catalyzes the phosphorylation of fructose-6-phosphate into fructose-diphosphate in
the glycolysis pathway and is allosterically regulated.

*In vivo*, oscillatory dynamics are observed to occur within the central carbon pathways and displaying a phase of tens of minutes up to several hours. Metabolic oscillations were demonstrated at single cell level and are found to be autonomous of, but coupled with, the cell cycle oscillations. Additional studies across cell populations found that cells can synchronize metabolic oscillations under some conditions,