# Regulation of Eukaryote Metabolism: An Abstract Model Explaining the Warburg/Crabtree Effect

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## Abstract

**:**

## 1. Introduction

`TotemBioNet`[11,12] and

`DyMBioNet`[13]. Here, this approach is used for the first time to design and formally validate a regulation model of the cell energetic metabolism, even if a precursory model has been introduced earlier in [14]. The classical regulation view of activation and inhibition signals can be interpreted here as “substrate producers” and “substrate consumers,” to abstract the underlying mass action rules that governs metabolic fluxes in term of regulations. Moreover, our model is designed to study the shift between the two metabolic regimes (respiration and fermentation), we focus less on the stationarity of these two regimes than on the global properties under different microenvironments (such as nutrient abundance or not).

## 2. Theoretical Background, Methods and Tools

#### 2.1. Interaction Graph

#### 2.2. Dynamic Parameters

#### 2.3. Formal Validation

#### 2.4. Using Biochemical Knowledge to Induce Parameter Values

#### 2.5. Software Platforms for Validation

`TotemBioNet`[11,12] (Available now at https://gitlab.com/totembionet/totembionet, accessed on 30 May 2021)) and

`DyMBioNet`[13]. Both inherit from

`SMBioNet`[6] as they rely on intensive CTL model checking to validate the parameter settings with respect to the phenotypic knowledge (and they additionally handle fair-path CTL, another temporal logic better suited for biological knowledge of global behavior). To make a long story short,

`TotemBioNet`takes as input the description of the influences between variables, the parameter values which have been fixed by thought experiments and the global behavior expressed in CTL; then, it enumerates all the possible parametrizations of non-constrained parameters, it automates the construction of the global behavior and launches for all possible parametrizations, the verification of the global temporal properties (using intensive model checking). In the end, it lists all parametrizations that are consistent with the global temporal properties.

## 3. Metabolism Regulation and Warburg/Crabtree Effect

- Mitochondrial respiration: a slow degradation of glucose (time-consuming turnover) but efficient production yield (38 ATP per glucose molecule),
- Fermentation: a rapid degradation of glucose with an inefficient production yield (2 ATP per glucose molecule).

#### 3.1. The Main Actors of Catabolism

#### 3.2. The Main Actors of Anabolism

## 4. Graph Representation of Metabolic Regulations

#### 4.1. Biological Regulation Graphs with Multiplexes

**ATP**can reach level 0, 1 or 2, because it regulates its targets at two distinct thresholds (e.g.,

**nLBP**at level 1 via

`and`

**PPP****GLYC**at level 2 via

`), and`

**COF****FERM**is only “boolean” (level 1 if its activity affects

**NADH**, level 0 otherwise).

**FERM**to

**NADH**is direct (it simply consumes

**NADH**at level 1), while the contribution of

**ATP**to

**nLBP**is subject to conditions (for example via

`, glycolysis and nitrogen/carbon donors must also be active). Direct regulations are depicted by arrows with the threshold and a sign that indicates activation or inhibition: for example “−1” from`

**PPP****FERM**to

**NADH**. The conditional regulations that need several precursors give rise to multiplexes when we know the conditions under which the regulation can take place: for example,

`contains the formula $(GLYC\u2a7e1\phantom{\rule{3.33333pt}{0ex}}\&\phantom{\rule{3.33333pt}{0ex}}ATP\u2a7e1\phantom{\rule{3.33333pt}{0ex}}\&\phantom{\rule{3.33333pt}{0ex}}NCD\u2a7e1)$ that transcribes the necessary cooperation between $ATP$, glycolysis and nitrogen/carbon donors ($NCD$) for the production of non-lipidic biomass. By convention, “&” stands for the conjunction, “∣” stands for the disjunction, and “!” stands for the negation. Within a formula, the negation is used to denote an inhibition, for example, $!(ATP\u2a7e2)$ in the`

**PPP**`formula means that an excess of $ATP$ will participate in the inhibition of glycolysis.`

**COF****GLC**= 2 all along with the study, see Section 6.

#### 4.2. Variables and Their Biological Meaning

#### 4.2.1. Environmental Variables: **exO2**, **FA**,
**GLC** and **AA**

- ∘
- $\mathit{exO}\mathit{2}\mathit{and}\mathit{FA}\in [0,1]$ respectively abstract (external) oxygen and lipids (fatty acids) intakes. At level 1,
**FA**can participate to lipid synthesis (themultiplex, provided that**LS****ATP**is present). At level 1,**exO2**activates the in-cell oxygen**O2**. Conversely, when their value is 0, it means that the nutrient input is not sufficient to be used normally by the cell. Please note that according to the threshold approach described above, level 0 does not mean that there is no supply but simply that it does not reach a sufficient threshold. - ∘
- $\mathit{GLC}\mathit{and}\mathit{AA}\in [0,2]$ respectively abstract glucose and amino acids (derived from proteins) intakes. They are crucial nutrients for studying the Warburg/Crabtree effect [22,24], and because their action on glycolysis (
**GLYC**) or nitrogen/carbon donors (**NCD**) respectively increases with their level of presence, we need to consider an “intermediate” level. Therefore, Level 0 represents a low availability of nutrient, leading to a negligible activation of metabolic processes downstream; level 1 represents a “normal” availability sufficient for basic metabolism and level 2 represents a high level of nutrients used by the cell often inducing over-activation of glycolysis.

#### 4.2.2. Metabolites: **ATP**, **NADH**, **NCD** and **O2**

- ∘
- $\mathit{ATP}\in [0,2]$ represents the concentration ratio of $ATP/(ADP+AMP)$. It abstracts the energetic level for the cell. During anabolic reactions, $ATP$ is transformed into $ADP$ or $AMP$ to release energy. ADP and AMP are regenerated during glycolysis (aerobic glycolysis) or by mitochondrial respiration. Since there are shuttles between cytoplasm and mitochondria, the $ATP/(ADP+AMP)$, as a ratio, can be considered almost spatially homogeneous within the cell, even if individually the $ATP$ or $ADP$+$AMP$ level can vary greatly from mitochondria to the cytoplasm. Notice also that a high
**ATP**value (**ATP**= 2) might be equally viewed as a high level of $ATP$ molecules or as a low level of $ADP$+$AMP$ molecules in the cell, and conversely for**ATP**= 0. Lastly,**ATP**= 1 represents a sort of equilibrium. - ∘
- $\mathit{NADH}\in [0,1]$ similarly to
**ATP**,**NADH**represents the mean concentration ratio of $NADH/NA{D}^{+}$, $NADPH/NAD{P}^{+}$ as well as $FAD/FAD{H}_{2}$, which belong to the same electron carrier molecular group. The modeling choice of abstracting together these three different types of ratios deserves some comments. For sure, the couple $NADH/NA{D}^{+}$ is commonly used in catabolic processes whereas $NADPH/NAD{P}^{+}$ is mostly used in anabolic processes. Additionally, in Figure 3, glycolysis activates the $NADH/NA{D}^{+}$ ratio (“$+1$” plain arrow) and it does not address the same ratio as oxidative phosphorylation, which acts on $NADPH/NAD{P}^{+}$. Nevertheless, according to our very abstract level of description: (i) At the temporal resolution granularity, we consider their mean values to have the same signature as shown for example in the context of the cell cycle [25]. (ii) The thresholds between 0 and 1 are purely symbolic in the Thomas’ framework so that it is not required for the $NADH/NA{D}^{+}$ and $NADPH/NAD{P}^{+}$ thresholds to be quantitatively equal. The same reasoning applies to the $FAD/FAD{H}_{2}$ quotient. Therefore, we had no valuable reasons neither to distinguish between these three ratios nor to distinguish more than two different levels for the**NADH**targets (principle of parsimony [26]). All in all, level 0 simply represents a too low ratio to act on its targets (low $NADH$ or, equally, high $NA{D}^{+}$), contrarily to level 1. - ∘
- $\mathit{NCD}\in [0,2]$ represents the Nitrogen and Carbon Donors, useful to the cell and derived from amino acids. These elements are used to supply metabolic pathways such as
**KREBS**. At level 0**NCD**action is too low to undergo anabolic processes; at level 1 it can participate in the activation of the Pentose Phosphate Pathway (), and at level 2 it allows the production of $\alpha $-KG through glutaminolysis even if ${O}_{2}$ is absent (**PPP**) and it also participates in amino acids and lipid synthesis (**$\alpha $-KG**and**AAS**).**LS** - ∘
- $\mathit{O}\mathit{2}\in [0,1]$ represents the intracellular oxygen concentration. Once again the Thomas’ framework not being quantitative, distinguishing several thresholds for the O2 concentration can only be motivated by several targets of O2 which cannot share the same regulation thresholds. According to Figure 3, we had no valuable reason to distinguish different levels of activation for the
**O2**targets, so a present/absent state is sufficient in our model.

#### 4.2.3. Metabolic pathways: **GLYC**, **FERM**, **KREBS**
and **PHOX**

- ∘
- $\mathit{GLYC}\in [0,2]$ represents glycolysis, which degrades glucose and produces pyruvate and $ATP$ (
**GLYC**activates**ATP**) using ten chain reactions [27]. Three of these reactions are limiting and carried out using different enzymes, such as Phospho-Fructo-Kinase (PFK) which has a major role in glycolysis regulation. Level 0 represents glycolysis that does not produce enough intermediates, e.g., pyruvate, useful to other metabolic pathways (such as the Krebs cycle), nor any noticeable**ATP**. Level 1 represents a glycolytic flux sufficient to activate the related metabolic pathways (PPP). In terms of flux,can be considered to be a competitor of glycolysis. However, in terms of regulation, because the variable**PPP****GLYC**abstracts all intermediate metabolites involved in glycolysis,**GLYC**is an activator of(through one of its early metabolites), Krebs, etc., as well as fermentation in the absence of oxygen). Finally, level 2 represents a high level of activity where glycolysis can be considered to be over-functioning, compared to the needs of a healthy cell under optimal nutrient conditions. In such a case, glycolysis promotes the accumulation of pyruvate which in turn promotes the production of $\alpha $-KG and glycolysis also fosters the fermentation process if**PPP****NADH**is present, even in the presence of oxygen. - ∘
- $\mathit{FERM}\in [0,1]$ represents fermentation. This metabolic pathway becomes important when the oxygen supply is no longer sufficient. As $NADH$ production is the only target of
**FERM**in the model,**FERM**is obviously a boolean variable. - ∘
- $\mathit{KREBS}\in [0,2]$ represents the TCA cycle (tricarboxylic acid cycle) or Krebs cycle. It does not represent the reverse reactions (reductive branch of the Krebs cycle) that are implicitly taken into account within multiplexes. This central metabolic cycle allows the oxidation of groups of molecules resulting from different catabolic processes (glycolysis, $\beta $-oxidation, degradation of amino acids [28]). A sufficient flow on this pathway will allow the cell to make the oxidative phosphorylation and reduce $NA{D}^{+}$ to $NADH$ (
**KREBS**activates**NADH**). Its flux is dependent on the level of cellular oxygen but also on the quantity of precursors available [29]. Level 0 represents a low flux that does not allow one to noticeably obtain $NADH$. Level 1 represents a Krebs cycle flow capable of reducing $NA{D}^{+}$ to $NADH$ from basic catabolic processes (glycolysis and $\beta $-oxidation). This is the normal flow for healthy cells with an adapted supply of nutrients. At level 2 Krebs cycle is over-functioning and alarms the cell to lower catabolic processes, such as glycolysis (via), and promotes anabolic processes such as lipid synthesis (via**GR**).**LS** - ∘
- $\mathit{PHOX}\in [0,1]$ represents oxidative phosphorylation, a mitochondrial metabolic pathway that allows the creation of $ATP$ (
**PHOX**activates**ATP**) by consuming oxygen and $NADH$ (**PHOX**inhibits**NADH**and**O2**). Several chemical reactions allow the reduction of an oxygen molecule into a water molecule. These steps release energy, which is used to form $ATP$. This pathway depends entirely on oxygen [30].**PHOX**activates all its targets at the same level, so, it is a boolean variable.

#### 4.2.4. Biomass: **LBP** and **nLBP**

- ∘
- $\mathit{LBP}\mathit{and}\mathit{nLBP}\in [0,1]$ represents respectively the Production and storage of Lipid Biomass (all complex lipids, such as phospholipids or glycolipids) and Non-Lipidic Biomass (proteins and DNA/RNA). They are useful for component turnover and cell proliferation. Biomass production consumes energy:
**LBP**and**nLBP**inhibit**ATP**(Notice that lipid synthesis is in fact the process that consumes**ATP**, see Section 4.3.3. Technically, asis a multiplex, one must delegate the**LS****ATP**consumption to the variable**LBP**. The same applies to**nLBP**). In addition, lipid biomass production**LBP**can participate in Krebs activation through. Whatever the kind of biomass, there is either noticeable production and storage or not, so these variables are boolean.**Box**

#### 4.3. Multiplexes and Their Biological Meaning

#### 4.3.1. Multiplexes Regulating Metabolites

#### **NADH** Regulator

- □
$(NCD\u2a7e2)\phantom{\rule{3.33333pt}{0ex}}\&\phantom{\rule{3.33333pt}{0ex}}(ATP\u2a7e1)\phantom{\rule{3.33333pt}{0ex}}\&\phantom{\rule{3.33333pt}{0ex}}(NADH\u2a7e1)$ represents the processes for anabolic production of amino acids (Amino Acid Synthesis) which favor non-lipidic biomass production. They are mostly synthesized from other amino acids collected outside the cell: The multiplex**AAS**summarizes the necessary elements to produce new amino acids, such as nitrogen and carbon given off by the products of degradation of amino acids outside the cell $(NCD\u2a7e2)$, a large amount of $NADH$$(NADH\u2a7e1)$, and $ATP$ at least for some of the amino acid synthesis reactions $(ATP\u2a7e1)$. The conjunction of these three conditions yields to the formula of this multiplex [31,32]. Moreover, $NADH$ being consumed,**AAS**inhibits**AAS****NADH**: this is encoded by the “!” (negation) on the outgoing arrow fromto**AAS****NADH**.

#### 4.3.2. Multiplexes Regulating Pathways

#### **GLYC** Regulators

- □
$!(ATP\u2a7e2)\phantom{\rule{3.33333pt}{0ex}}\&\phantom{\rule{3.33333pt}{0ex}}!(NADH\u2a7e1)$ represents the cofactors needs to a correct course of glycolysis: $ADP$ and $NA{D}^{+}$ must not be limiting. This means that $\mathbf{ATP}<2$ and $\mathbf{NADH}<1$, as already explained in Section 4.2.2. Therefore, the**COF**formula properly formalizes this regulation of**COF****GLYC**.- □
$!\left[\right(KREBS\u2a7e2)\phantom{\rule{3.33333pt}{0ex}}\&\phantom{\rule{3.33333pt}{0ex}}(ATP\u2a7e1\left)\right]$ represents metabolic glycolytic flow inhibitors such as an absence of the enzyme PFK (Phospho-fructokinase) and an accumulation of citrate. Enzyme PFK of the glycolysis is allosterically inhibited by $ATP$ [33], Thus,**GR****ATP**participates to inhibition of**GLYC**via PFK. Moreover, pyruvate (the final product of glycolysis) fuels the TCA cycle and is transformed into citrate, which, if in excess, inhibits glycolysis. Reminding that the TCA cycle is abstracted by**KREBS**, an excess of citrate is produced when $KREBS\u2a7e2$, which also participates in the inhibition of glycolysis. Here, we consider that**GLYC**is inhibited when both conditions are satisfied, so**GLYC**is inhibited if $(KREBS\u2a7e2)\phantom{\rule{3.33333pt}{0ex}}\&\phantom{\rule{3.33333pt}{0ex}}(ATP\u2a7e1)$. The negation “!” at the beginning of the formula indicates the inhibition.

#### **KREBS** Regulators

- □
$(GLYC\u2a7e1)\phantom{\rule{3.33333pt}{0ex}}\&\phantom{\rule{3.33333pt}{0ex}}(O2\u2a7e1)$ represents the action of AcetylcoA as main precursor of the Krebs cycle: it directly derives from pyruvate (glycolysis product) [29], which is formalized as $GLYC\u2a7e1$. Moreover, the accumulation of Acetyl-coA coming from glycolysis activates the Krebs cycle if oxygen is present [34].**AnO**- □
$(LBP\u2a7e1)\phantom{\rule{3.33333pt}{0ex}}\&\phantom{\rule{3.33333pt}{0ex}}!(GLYC\u2a7e1)\phantom{\rule{3.33333pt}{0ex}}\&\phantom{\rule{3.33333pt}{0ex}}!(ATP\u2a7e1)$ represents $\beta $-oxidation, the fatty acid degradation pathway. It converts fatty acids to acetyl-CoA [35]. $\beta $-oxidation can be performed when the energy level of the cell is relatively low $!(ATP\u2a7e1)$, when the pool of stored lipids is large enough $(LBP\u2a7e1)$ and when glycolysis is not efficient enough to produce energy $!(GLYC\u2a7e1)$ [29]. The multiplex formula is the conjunction of these conditions.**Box**- □
$\left[\right(GLYC\u2a7e1)\&(NCD\u2a7e2\left)\phantom{\rule{3.33333pt}{0ex}}\right|\phantom{\rule{3.33333pt}{0ex}}(GLYC\u2a7e2)]\phantom{\rule{3.33333pt}{0ex}}\&\phantom{\rule{3.33333pt}{0ex}}(O2\u2a7e1)$ represents an accumulation of the $\alpha $-Ketoglutarate metabolite. This intermediate of a reaction of the Krebs cycle shows a significant flow only in the presence of oxygen $(O2\u2a7e1)$ [36]. Then, the accumulation can come from a saturation issued by glycolysis with a very high flow rate $(GLYC\u2a7e2)$. Moreover, with a lower flow rate $(GLYC\u2a7e1)$, the transformation of glutamine to $\alpha $-KG by glutaminolysis [23,37] can accumulate, provided that the amount of amino acids in the form of nitrogen and carbon is important $(NCD\u2a7e2)$.**$\alpha $-KG**

#### **FERM** Regulator

- □
$(NADH\u2a7e1)\phantom{\rule{3.33333pt}{0ex}}\&\phantom{\rule{3.33333pt}{0ex}}\left[\right(GLYC\u2a7e2\left)\phantom{\rule{3.33333pt}{0ex}}\right|\phantom{\rule{3.33333pt}{0ex}}\left(\right(GLYC\u2a7e1)\phantom{\rule{3.33333pt}{0ex}}\&\phantom{\rule{3.33333pt}{0ex}}!(O2\u2a7e1\left)\right)]$ represents an Excess of Pyruvate, which activates the fermentation flow. Fermentation always needs $NADH$ as a cofactor $(NADH\u2a7e1)$. Then, fermentation can be activated by an overproduction of pyruvate via glycolysis $(GLYC\u2a7e2)$, or if glycolysis produces pyruvate at a standard rate $(GLYC\u2a7e1)$ together with an intracellular hypoxia $!(O2\u2a7e1)$.**EP**

#### **PHOX** Regulator

- □
$(NADH\u2a7e1)\phantom{\rule{3.33333pt}{0ex}}\&\phantom{\rule{3.33333pt}{0ex}}(O2\u2a7e1)\phantom{\rule{3.33333pt}{0ex}}\&\phantom{\rule{3.33333pt}{0ex}}!(ATP\u2a7e2)$ represents the**PC****PHOX**Control. Oxidative phosphorylation requires a sufficient supply of oxygen $(O2\u2a7e1)$ as precursor of the chain. It also requires $NADH$$(NADH\u2a7e1)$. Moreover, this pathway is only activated when the energy pool of the cell is not too high [38], i.e., $!(ATP\u2a7e2)$.

#### 4.3.3. Multiplexes Regulating Biomass

#### **nLBP** Regulators

- □
$(GLYC\u2a7e1)\phantom{\rule{3.33333pt}{0ex}}\&\phantom{\rule{3.33333pt}{0ex}}(ATP\u2a7e1)\phantom{\rule{3.33333pt}{0ex}}\&\phantom{\rule{3.33333pt}{0ex}}(NCD\u2a7e1)$ represents the production of nucleotides via the Pentose Phosphate Pathway that favors non-lipidic biomass production. It produces nucleotides, thus**PPP**activates non-lipidic biomass (**PPP**). This pathway also produces $NADPH$, which is directly consumed to produce non-lipidic biomass, so that the end $NADPH$ production result is neutral for the**nLBP****NADH**ratio. Consequently, we do not put an activation arrow fromto**PPP****NADH**in the regulation graph (contrarily to). Glycolysis is required for the activation of**AAS**$(GLYC\u2a7e1)$ because Glyceraldehyde-3-phosphate (G3P) is an intermediate reaction of glycolysis and the precursor of the Pentose Phosphate Pathway. In addition, for these endergonic reactions it needs a carbon input $(NCD\u2a7e1)$ from the internalization of amino acids, as well as $ATP$$(ATP\u2a7e1)$ [39].**PPP**

`(Amino-Acid Synthesis), is obviously also an activator of`

**AAS****nLBP**. It has already been described as a

**NADH**inhibitor.

#### **LBP** regulators:

- □
$\left[\right((KREBS\u2a7e2)\phantom{\rule{3.33333pt}{0ex}}\left|\phantom{\rule{3.33333pt}{0ex}}\right(NCD\u2a7e2\left)\phantom{\rule{3.33333pt}{0ex}}\right|\phantom{\rule{3.33333pt}{0ex}}(FA\u2a7e1))\phantom{\rule{3.33333pt}{0ex}}\&\phantom{\rule{3.33333pt}{0ex}}(ATP\u2a7e1\left)\right]$ represents specifically the Synthesis of Lipids composed by fatty acids. Lipid creation uses energy $(ATP\u2a7e1)$ and fatty acids can come directly from the cellular environment $(FA\u2a7e1)$ or via the fatty acid synthesis pathway, the main precursor of which is acetyl-CoA, which is in turn provided by Krebs cycle when it is over-functioning $(KREBS\u2a7e2)$ or by glutaminolysis directly derived from amino acids degradation $(NCD\u2a7e2)$ [40].**LS**

`($\beta $-oxidation) has already been described as a`

**Box****KREBS**regulator which allows lipid degradation. It is consequently an inhibitor of

**LBP**, encoded in Figure 3 through the negation sign “!” from

`to`

**Box****LBP**.

## 5. Kinetic Parameters: How Local Reasoning Makes a Global Dynamics of the Mathematical Model Emerge

#### 5.1. Kinetic Parameters and State Transitions

**FERM**on

**NADH**,

**FERM**becomes an active resource of

**NADH**when it does not pass its threshold: there is an implicit multiplex with formula $(\mathit{FERM}\u2a7e1)$ that is the actual resource of

**NADH**. More generally, and formally, a multiplex becomes a resource of its target variable when its formula becomes true.

**GLC**to

**GLYC**, with fictitious thresholds 1 and 2 and this simulates low (

**GLC**= 0), average (

**GLC**= 1) or high (

**GLC**= 2) intakes; (ii) the same occurs with amino acids intake from

**AA**to

**NCD**; as well as (iii) the

**ATP**production by the glycolysis

**GLYC**. In the sequel,

`denotes that`

**GLC1****GLC**is at least at level 1 whereas

`denotes that`

**GLC2****GLC**reaches its highest level (and similarly for

`,`

**AA1**`,`

**AA2**`and`

**GLYC1**`).`

**GLYC2****GLC**passes threshold 2 then it also passes threshold 1, consequently only ${K}_{\mathit{GLYC},\phantom{\rule{0.166667em}{0ex}}\mathbf{GLC}\mathbf{1}\phantom{\rule{0.166667em}{0ex}}\mathbf{GLC}\mathbf{2}}$ can apply.

#### 5.2. Local Identification of Parameters for Metabolic Regulations

#### **GLYC** **Parameters**

**GLYC**: It is potentially active from its level 1 for all its targets, but, depending on external conditions, level 2 can be required. More precisely level 2 of

**GLYC**is necessary to act on three targets:

if $\mathit{O}\mathit{2}=1$ and $\mathit{NADH}\u2a7e1$,**EP**if $\mathit{NCD}<2$ and $\mathit{O}\mathit{2}=1$,**$\alpha $KG**- over-activation of
**ATP**(compared to “simple” activation performed at level 1).

`implies`

**GLC2**`as resource, thus ${K}_{\mathit{GLYC},\phantom{\rule{0.166667em}{0ex}}\mathbf{GLC}\mathbf{2}}$, ${K}_{\mathit{GLYC},\phantom{\rule{0.166667em}{0ex}}\mathbf{COF}\phantom{\rule{0.166667em}{0ex}}\mathbf{GLC}\mathbf{2}}$, ${K}_{\mathit{GLYC},\phantom{\rule{0.166667em}{0ex}}\mathbf{COF}\phantom{\rule{0.166667em}{0ex}}\mathbf{GLC}\mathbf{2}\phantom{\rule{0.166667em}{0ex}}\mathbf{GR}}$ and ${K}_{\mathit{GLYC},\phantom{\rule{0.166667em}{0ex}}\mathbf{GLYC}\mathbf{2}\phantom{\rule{0.166667em}{0ex}}\mathbf{GR}}$ reflect inconsistent sets of resources and therefore have not to be instantiated. Therefore, it remains ${2}^{4}-4=12$`

**GLC1****GLYC**parameters to estimate.

**GLC**and

**COF**(cofactors) are prerequisites for glycolysis: if one of them is missing, the glycolysis cannot occur, and it becomes unable to act on its targets. It implies that 8 parameters are equal to 0: ${K}_{\mathit{GLYC},\phantom{\rule{0.166667em}{0ex}}}$, ${K}_{\mathit{GLYC},\phantom{\rule{0.166667em}{0ex}}\mathbf{GLC}\mathbf{1}}$, ${K}_{\mathit{GLYC},\phantom{\rule{0.166667em}{0ex}}\mathbf{COF}}$, ${K}_{\mathit{GLYC},\phantom{\rule{0.166667em}{0ex}}\mathbf{GR}}$, ${K}_{\mathit{GLYC},\phantom{\rule{0.166667em}{0ex}}\mathbf{GLC}\mathbf{1}\phantom{\rule{0.166667em}{0ex}}\mathbf{GLC}\mathbf{2}}$, ${K}_{\mathit{GLYC},\phantom{\rule{0.166667em}{0ex}}\mathbf{GLC}\mathbf{1}\phantom{\rule{0.166667em}{0ex}}\mathbf{GR}}$, ${K}_{\mathit{GLYC},\phantom{\rule{0.166667em}{0ex}}\mathbf{COF}\phantom{\rule{0.166667em}{0ex}}\mathbf{GR}}$ and ${K}_{\mathit{GLYC},\phantom{\rule{0.166667em}{0ex}}\mathbf{GLC}\mathbf{1}\phantom{\rule{0.166667em}{0ex}}\mathbf{GLC}\mathbf{2}\phantom{\rule{0.166667em}{0ex}}\mathbf{GR}}$.

`reflects an inhibition of glycolysis by`

**GR****KREBS**and

**ATP**(that make PFK unavailable to glycolysis). Therefore, when

`is a resource of`

**GR****GLYC**, if cofactors and glucose are present, then the greater the level of glucose, the greater the level of glycolysis:

- When $\mathit{GLC}=2$ the functioning of glycolysis will reach a level that is sufficient to activate
even if $\mathit{O}\mathit{2}=1$ and $\mathit{NADH}\u2a7e1$, as well as**EP**if $\mathit{NCD}<2$ and $\mathit{O}\mathit{2}=1$. Thus ${K}_{\mathit{GLYC},\phantom{\rule{0.166667em}{0ex}}\mathbf{COF}\phantom{\rule{0.166667em}{0ex}}\mathbf{GLC}\mathbf{1}\phantom{\rule{0.166667em}{0ex}}\mathbf{GLC}\mathbf{2}\phantom{\rule{0.166667em}{0ex}}\mathbf{GR}}=2$.**$\alpha $KG** - When $\mathit{GLC}=1$ the functioning of glycolysis will allow the activation of its targets (if the other conditions are satisfied), but neither
if $\mathit{O}\mathit{2}=1$ and $\mathit{NADH}\u2a7e1$, nor**EP**if $\mathit{NCD}<2$ and $\mathit{O}\mathit{2}=1$. Thus ${K}_{\mathit{GLYC},\phantom{\rule{0.166667em}{0ex}}\mathbf{COF}\phantom{\rule{0.166667em}{0ex}}\mathbf{GLC}\mathbf{1}\phantom{\rule{0.166667em}{0ex}}\mathbf{GR}}=1$.**$\alpha $KG**

**GLC**) and cofactors (

`) are present, and`

**COF**`acts as an inhibitor that will reduce the effectiveness of glycolysis. In fact, the inhibition via`

**GR**`, even in the presence of high glucose, stops the “suractivation” of`

**GR****ATP**and

`when $\mathit{O}\mathit{2}=1$ and $\mathit{NADH}\u2a7e1$, thus ${K}_{\mathit{GLYC},\phantom{\rule{0.166667em}{0ex}}\mathbf{COF}\phantom{\rule{0.166667em}{0ex}}\mathbf{GLC}\mathbf{1}\phantom{\rule{0.166667em}{0ex}}\mathbf{GLC}\mathbf{2}}=1$. With the normal presence of glucose,`

**EP**`is even strong enough to stop the activation of all downstream processes through glycolysis, thus ${K}_{\mathit{GLYC},\phantom{\rule{0.166667em}{0ex}}\mathbf{COF}\phantom{\rule{0.166667em}{0ex}}\mathbf{GLC}\mathbf{1}}=0$.`

**GR**#### **KREBS** **Parameters**

**KREBS**has a dual role depending on the cell’s milieu. At level 1,

**KREBS**is a provider of

**NADH**to normal functioning of the cell. At level 2

**KREBS**is overspreading and alarms glycolysis (through the

`inhibition) to lower the production of pyruvate and to lower the intake of elements coming from amino acids to break Krebs turnover. At level 2`

**GR****KREBS**also provokes lipid production. Therefore,

**KREBS**is an inhibitor of

**GLYC**and

**NCD**, and an activator of lipid synthesis

`. See Figure 5.`

**LS****KREBS**are:

**BOX**through fatty acid degradation,

`coming from glutaminolysis, and`

**$\alpha $KG**`because acetyl-CoA is derived from pyruvate, the final product of glycolysis. Thus there are ${2}^{3}=8$ parameters to identify for`

**AnO****KREBS**.

`formula implies the one of`

**$\alpha $KG**`, reflecting the fact that both`

**AnO**`and`

**$\alpha $KG**`came from glycolysis. Moreover, the`

**AnO****BOX**formula implies

**GLYC**=0, and thus contradicts the ones of

`and`

**$\alpha $KG**`. Therefore, ${K}_{\mathit{KREBS},\phantom{\rule{0.166667em}{0ex}}\alpha \mathbf{KG}}$ , ${K}_{\mathit{KREBS},\phantom{\rule{0.166667em}{0ex}}\mathbf{BOX}\phantom{\rule{0.166667em}{0ex}}\alpha \mathbf{KG}}$ , ${K}_{\mathit{KREBS},\phantom{\rule{0.166667em}{0ex}}\mathbf{AnO}\phantom{\rule{0.166667em}{0ex}}\mathbf{BOX}\phantom{\rule{0.166667em}{0ex}}\alpha \mathbf{KG}}$ and ${K}_{\mathit{KREBS},\phantom{\rule{0.166667em}{0ex}}\mathbf{AnO}\phantom{\rule{0.166667em}{0ex}}\mathbf{BOX}}$ are useless.`

**AnO****KREBS**is available, it cannot activate any downstream target. Thus ${K}_{\mathit{KREBS},\phantom{\rule{0.166667em}{0ex}}}=0$.

**KREBS**has the full support of

`and`

**$\alpha $KG**`, it will give an alarm to catabolic processes as glycolysis or degradation of amino acids intake (`

**AnO****NCD**) to avoid overproduction of energy and it also promotes anabolic processes. Thus ${K}_{\mathit{KREBS},\phantom{\rule{0.166667em}{0ex}}\mathbf{AnO}\phantom{\rule{0.166667em}{0ex}}\alpha \mathbf{KG}}=2$.

`),`

**AnO****KREBS**produces $NADH$ without alarming the cell. Thus, ${K}_{\mathit{KREBS},\phantom{\rule{0.166667em}{0ex}}\mathbf{AnO}}=1$, and for the same reason ${K}_{\mathit{KREBS},\phantom{\rule{0.166667em}{0ex}}\mathbf{BOX}}=1$.

#### **FERM** **and** **PHOX** **Parameters**

**EP**that denotes an excess of pyruvate, and

**PC**that denotes the

**PHOX**control, see Figure 6). In both cases, if the resource is absent, then

**FERM**or

**PHOX**will become unable to act on their targets, thus ${K}_{\mathit{FERM},\phantom{\rule{0.166667em}{0ex}}}=0$ and ${K}_{\mathit{PHOX},\phantom{\rule{0.166667em}{0ex}}}=0$. Conversely, if the resource is present, they will become able to act on their targets, thus ${K}_{\mathit{FERM},\phantom{\rule{0.166667em}{0ex}}\mathbf{EP}}=1$ and ${K}_{\mathit{PHOX},\phantom{\rule{0.166667em}{0ex}}\mathbf{PC}}=1$.

#### **nLBP** **Parameters**

**nLBP**is boolean, acting on

**ATP**consumption (see Figure 7). It has 2 possible resources:

`and`

**PPP**`, whose combinations are all satisfiable, meaning that each of the 4 conjunctions of their formulas or their negations are true for at least one state of the network. Therefore, all 4 parameters are useful.`

**AAS**`or in the presence of`

**PPP**`, the non-lipidic biomass production becomes sufficiently activated to consume`

**AAS****ATP**, thus ${K}_{\mathit{nLBP},\phantom{\rule{0.166667em}{0ex}}\mathbf{PPP}}={K}_{\mathit{nLBP},\phantom{\rule{0.166667em}{0ex}}\mathbf{AAS}}={K}_{\mathit{nLBP},\phantom{\rule{0.166667em}{0ex}}\mathbf{AAS}\phantom{\rule{0.166667em}{0ex}}\mathbf{PPP}}=1$.

#### **LBP** **Parameters**

**LBP**is also boolean and it consumes

**ATP**and favors

`concomitantly. Possible resources of`

**Box****LBP**are lipid synthesis

`and`

**LS**`, and this makes the thought experiments particularly subtle: even when`

**Box**`is supposed inactive forever as a resource of`

**Box****LBP**, one must consider fully independently the possible action of

**LBP**on

`as a target. In other words, one must consider that`

**Box**`as a resource is different from`

**Box**`as a target. More generally, these thought experiments only take into account the local biochemical behavior without considering its effect on the whole network (see Figure 8).`

**Box**`and`

**LS**`are satisfiable, so that the 4 parameters are useful. Without lipidic synthesis`

**Box****LBP**lacks resources, thus ${K}_{\mathit{LBP},\phantom{\rule{0.166667em}{0ex}}}={K}_{\mathit{LBP},\phantom{\rule{0.166667em}{0ex}}\mathbf{BOX}}=0$. Conversely, with lipidic synthesis, even if

`inhibits`

**Box****LBP**, lipid biomass as precursor activates its degradation and consumes

**ATP**, thus ${K}_{\mathit{LB},\phantom{\rule{0.166667em}{0ex}}\mathbf{LS}}=1$ and a fortiori ${K}_{\mathit{LBP},\phantom{\rule{0.166667em}{0ex}}\mathbf{LS}\phantom{\rule{0.166667em}{0ex}}\mathbf{BOX}}=1$.

#### **NCD** **Parameters**

**NCD**is a provider of nitrogen and carbon coming from amino acids: In our model,

**AA**is the input provider of

**NCD**at two different levels, and

**KREBS**inhibits

**NCD**. Thus, 8 kinetic parameters drive

**NCD**. At level 1,

**NCD**allows the creation of nucleotides and DNA by providing nitrogen via

`. At level 2, it allows the creation of`

**PPP**`via glutaminolysis, as well as amino acids via the amino acids synthesis pathway`

**$\alpha $KG**`, and lipid synthesis`

**AAS**`by providing nitrogen and carbon. See Figure 9.`

**LS****AA**at level 2 implies that

**AA**at level 1 is also a resource of

**NCD**, as already explained, thus ${K}_{\mathit{NCD},\phantom{\rule{0.166667em}{0ex}}\mathbf{AA}\mathbf{2}}$ and ${K}_{\mathit{NCD},\phantom{\rule{0.166667em}{0ex}}\mathbf{AA}\mathbf{2}\phantom{\rule{0.166667em}{0ex}}\mathbf{KREBS}}$ are useless.

**KREBS**level does not reach 2:

**NCD**is fully under the control of amino acid intake

**AA**. If

**AA**=0, then without input of amino acids, no nitrogen is provided and none of the target processes downstream of

**NCD**can be activated (${K}_{\mathit{NCD},\phantom{\rule{0.166667em}{0ex}}\mathbf{KREBS}}=0$). If

**AA**=1, there is enough production of nucleotide via the pathway

`but not enough production of nitrogen and carbon to be active on amino acid`

**PPP**`or lipid`

**AAS**`synthesis, as well as`

**LS**`, thus ${K}_{\mathit{NCD},\phantom{\rule{0.166667em}{0ex}}\mathbf{AA}\mathbf{1}\phantom{\rule{0.166667em}{0ex}}\mathbf{KREBS}}=1$. Lastly if`

**$\alpha $KG****AA**= 2, then

**NCD**becomes active on

`,`

**AAS**`and`

**LS**`, thus ${K}_{\mathit{NCD},\phantom{\rule{0.166667em}{0ex}}\mathbf{AA}\mathbf{1}\phantom{\rule{0.166667em}{0ex}}\mathbf{AA}\mathbf{2}\phantom{\rule{0.166667em}{0ex}}\mathbf{KREBS}}=2$.`

**$\alpha $KG****NCD**to be active on

`,`

**PPP**`,`

**$\alpha $KG**`and`

**AAS**`, we consider that the inhibition of`

**LS****KREBS**will lower down the production speed of nitrogen and carbon, but this will not decrease the value toward which

**KREBS**asymptotically tends (technically because there is no degradation of nitrogen and carbon as such). Therefore, ${K}_{\mathit{NCD},\phantom{\rule{0.166667em}{0ex}}\mathbf{AA}\mathbf{1}}={K}_{\mathit{NCD},\phantom{\rule{0.166667em}{0ex}}\mathbf{AA}\mathbf{1}\phantom{\rule{0.166667em}{0ex}}\mathit{KREBS}}=1$ and ${K}_{\mathit{NCD},\phantom{\rule{0.166667em}{0ex}}\mathbf{AA}\mathbf{1}\phantom{\rule{0.166667em}{0ex}}\mathbf{AA}\mathbf{2}}={K}_{\mathit{NCD},\phantom{\rule{0.166667em}{0ex}}\mathbf{AA}\mathbf{1}\phantom{\rule{0.166667em}{0ex}}\mathbf{AA}\mathbf{2}\phantom{\rule{0.166667em}{0ex}}\mathbf{KREBS}}=2$.

#### **O2** **Parameters**

**O2**activator

**exO2**(oxygen supply) and

**O2**inhibitor

**PHOX**(oxidative phosphorylation which consumes oxygen) are independent of each other (see Figure 10), thus the 4 kinetic parameters for

**O2**are useful. All the targets of

**O2**,

`,`

**AnO**`,`

**$\alpha $KG**`and`

**EP**`share the same threshold:`

**PC****O2**is simply present or absent.

**O2**lack of resources, thus ${K}_{\mathit{O}\mathit{2},\phantom{\rule{0.166667em}{0ex}}}={K}_{\mathit{O}\mathit{2},\phantom{\rule{0.166667em}{0ex}}\mathbf{PHOX}}=0$. Conversely, external oxygen constantly renews

**O2**, so that ${K}_{\mathit{O}\mathit{2},\phantom{\rule{0.166667em}{0ex}}\mathbf{exO}\mathbf{2}}={K}_{\mathit{O}\mathit{2},\phantom{\rule{0.166667em}{0ex}}\mathbf{exO}\mathbf{2}\phantom{\rule{0.166667em}{0ex}}\mathbf{PHOX}}=1$.

#### **NADH** **Parameters**

**NADH**stands for the quotient $NAD\left(P\right)H\phantom{\rule{0.166667em}{0ex}}/\phantom{\rule{0.166667em}{0ex}}NAD{\left(P\right)}^{+}$, thus

**NADH**= 1 equally means presence of $NAD\left(P\right)H$ or lack of $NAD{\left(P\right)}^{+}$, and

**NADH**= 0 equally means presence of $NAD{\left(P\right)}^{+}$ or lack of $NAD\left(P\right)H$. It acts as activator for phosphorylation (

**PHOX**via

`when`

**PC****O2**is present), for fermentation (

**FERM**via

`when there is hypoxia or when glycolysis produces an excess of pyruvate) and for amino acid synthesis (when`

**EP****ATP**and

**NCD**are sufficiently present). It also acts as inhibitor for glycolysis (

**GLYC**via

`).`

**COF****NADH**has 2 source providers:

**KREBS**and

**GLYC**, and 3 consumers:

**PHOX**,

**FERM**and

`, see Figure 11. All combinations of these 5 potential resources are satisfiable, leading to 32 parameters to identify.`

**AAS****KREBS**nor

**GLYC**is present

**NADH**will obviously not activate its targets, thus ${K}_{\mathit{NADH},\phantom{\rule{0.166667em}{0ex}}}={K}_{\mathit{NADH},\phantom{\rule{0.166667em}{0ex}}\mathbf{FERM}}={K}_{\mathit{NADH},\phantom{\rule{0.166667em}{0ex}}\mathbf{PHOX}}={K}_{\mathit{NADH},\phantom{\rule{0.166667em}{0ex}}\mathbf{AAS}}={K}_{\mathit{NADH},\phantom{\rule{0.166667em}{0ex}}\mathbf{FERM}\phantom{\rule{0.166667em}{0ex}}\mathbf{PHOX}}={K}_{\mathit{NADH},\phantom{\rule{0.166667em}{0ex}}\mathbf{FERM}\phantom{\rule{0.166667em}{0ex}}\mathbf{AAS}}={K}_{\mathit{NADH},\phantom{\rule{0.166667em}{0ex}}\mathbf{AAS}\phantom{\rule{0.166667em}{0ex}}\mathbf{PHOX}}={K}_{\mathit{NADH},\phantom{\rule{0.166667em}{0ex}}\mathbf{FERM}\phantom{\rule{0.166667em}{0ex}}\mathbf{AAS}\phantom{\rule{0.166667em}{0ex}}\mathbf{PHOX}}=0$.

**NADH**quotient and thus downstream targets will finally be activated. Thus, ${K}_{\mathit{NADH},\phantom{\rule{0.166667em}{0ex}}\mathbf{GLYC}\phantom{\rule{0.166667em}{0ex}}\mathbf{FERM}\phantom{\rule{0.166667em}{0ex}}\mathbf{AAS}\phantom{\rule{0.166667em}{0ex}}\mathbf{PHOX}}={K}_{\mathit{NADH},\phantom{\rule{0.166667em}{0ex}}\mathbf{KREBS}\phantom{\rule{0.166667em}{0ex}}\mathbf{FERM}\phantom{\rule{0.166667em}{0ex}}\mathbf{AAS}\phantom{\rule{0.166667em}{0ex}}\mathbf{PHOX}}={K}_{\mathit{NADH},\phantom{\rule{0.166667em}{0ex}}\mathbf{GLYC}\phantom{\rule{0.166667em}{0ex}}\mathbf{KREBS}\phantom{\rule{0.166667em}{0ex}}\mathbf{FERM}\phantom{\rule{0.166667em}{0ex}}\mathbf{AAS}\phantom{\rule{0.166667em}{0ex}}\mathbf{PHOX}}=1$.

**NADH**, and we can set most of them owing to two hypotheses that seem sensible within our context:

- In the cell,
**PHOX**is the principal consumer of**NADH**,**KREBS**is its principal producer, and the metabolic processes they abstract are nested, therefore they balance on a long-term basis. **GLYC**is a weak producer of**NADH**,**FERM**is an average consumer, andis a weak consumer (but we do not know if**AAS****GLYC**andbalance on a long-term basis).**AAS**

**KREBS**and

**PHOX**are putted aside (they are in balance), so the situation is similar to no provider, and there is no remaining inhibitor nor activators. From the second hypothesis, it comes ${K}_{\mathit{NADH},\phantom{\rule{0.166667em}{0ex}}\mathbf{GLYC}\phantom{\rule{0.166667em}{0ex}}\mathbf{AAS}\phantom{\rule{0.166667em}{0ex}}\mathbf{PHOX}}=0$ because, here, the only producer

**GLYC**is weak and the average consumer

**FERM**is on. Taking into account the balance between

**KREBS**and

**PHOX**, the second hypothesis also implies ${K}_{\mathit{NADH},\phantom{\rule{0.166667em}{0ex}}\mathbf{GLYC}\phantom{\rule{0.166667em}{0ex}}\mathbf{KREBS}\phantom{\rule{0.166667em}{0ex}}\mathbf{AAS}}=0$ for the same reason. We can also deduce that ${K}_{\mathit{NADH},\phantom{\rule{0.166667em}{0ex}}\mathbf{FERM}\phantom{\rule{0.166667em}{0ex}}\mathbf{GLYC}\phantom{\rule{0.166667em}{0ex}}\mathbf{AAS}}=0$, as

**GLYC**is here the only activator and the principal inhibitor

**PHOX**is on. Therefore, the hypotheses require 4 parameter values to 0, and, a fortiori with less resources, we deduce 8 more parameter values: ${K}_{\mathit{NADH},\phantom{\rule{0.166667em}{0ex}}\mathbf{GLYC}\phantom{\rule{0.166667em}{0ex}}\mathbf{KREBS}}={K}_{\mathit{NADH},\phantom{\rule{0.166667em}{0ex}}\mathbf{FERM}\phantom{\rule{0.166667em}{0ex}}\mathbf{KREBS}}={K}_{\mathit{NADH},\phantom{\rule{0.166667em}{0ex}}\mathbf{KREBS}\phantom{\rule{0.166667em}{0ex}}\mathbf{AAS}}={K}_{\mathit{NADH},\phantom{\rule{0.166667em}{0ex}}\mathbf{GLYC}\phantom{\rule{0.166667em}{0ex}}\mathbf{AAS}}={K}_{\mathit{NADH},\phantom{\rule{0.166667em}{0ex}}\mathbf{GLYC}\phantom{\rule{0.166667em}{0ex}}\mathbf{PHOX}}={K}_{\mathit{NADH},\phantom{\rule{0.166667em}{0ex}}\mathbf{FERM}\phantom{\rule{0.166667em}{0ex}}\mathit{GLYC}}={K}_{\mathit{NADH},\phantom{\rule{0.166667em}{0ex}}\mathbf{KREBS}}={K}_{\mathit{NADH},\phantom{\rule{0.166667em}{0ex}}\mathbf{GLYC}}=0$.

**PHOX**is in equilibrium with

**KREBS**, there is no other inhibitor, and consequently the

**GLYC**activation makes the difference. We also obtain ${K}_{\mathit{NADH},\phantom{\rule{0.166667em}{0ex}}\mathbf{KREBS}\phantom{\rule{0.166667em}{0ex}}\mathbf{PHOX}}=1$ (where

**KREBS**is the only activator and

`and`

**AAS****FERM**are inhibitors) because the principal producer

**KREBS**, not balanced by the

**PHOX**inhibition, contributes to the $NAD\left(P\right)H\phantom{\rule{0.166667em}{0ex}}/\phantom{\rule{0.166667em}{0ex}}NAD{\left(P\right)}^{+}$ quotient more than

**FERM**and

`together, which are at most average consumers. Then, a fortiori with more resources, we deduce ${K}_{\mathit{NADH},\phantom{\rule{0.166667em}{0ex}}\mathbf{KREBS}\phantom{\rule{0.166667em}{0ex}}\mathbf{FERM}\phantom{\rule{0.166667em}{0ex}}\mathbf{PHOX}}={K}_{\mathit{NADH},\phantom{\rule{0.166667em}{0ex}}\mathbf{KREBS}\phantom{\rule{0.166667em}{0ex}}\mathbf{AAS}\phantom{\rule{0.166667em}{0ex}}\mathbf{PHOX}}={K}_{\mathit{NADH},\phantom{\rule{0.166667em}{0ex}}\mathbf{GLYC}\phantom{\rule{0.166667em}{0ex}}\mathbf{KREBS}\phantom{\rule{0.166667em}{0ex}}\mathbf{PHOX}}={K}_{\mathit{NADH},\phantom{\rule{0.166667em}{0ex}}\mathbf{FERM}\phantom{\rule{0.166667em}{0ex}}\mathbf{GLYC}\phantom{\rule{0.166667em}{0ex}}\mathbf{KREBS}\phantom{\rule{0.166667em}{0ex}}\mathbf{PHOX}}={K}_{\mathit{NADH},\phantom{\rule{0.166667em}{0ex}}\mathbf{GLYC}\phantom{\rule{0.166667em}{0ex}}\mathbf{KREBS}\phantom{\rule{0.166667em}{0ex}}\mathbf{AAS}\phantom{\rule{0.166667em}{0ex}}\mathbf{PHOX}}=1$`

**AAS****GLYC**and the consumer is

`(for the first parameter, considering the`

**AAS****KREBS**/

**PHOX**balance simplification from the first hypothesis).

**GLYC**and

`are both weak but we cannot figure out a sensible third hypothesis (that would not be too arbitrary with respect to our general context) to assert a proper comparison. Therefore, let us leave those parameters unknown and Section 7 will handle the question properly.`

**AAS**#### **ATP** **Parameters**

**ATP**stands for the quotient $ATP/ADP$, with two activity thresholds. At level 1 it participates in the amino acid, the nucleotide and the lipid syntheses (

`,`

**AAS**`and`

**PPP**`), it inhibits $\beta $-oxidation`

**LS**`, and it down-regulates glycolysis when`

**BOX****KREBS**= 2 via

`. At level 2, it additionally inhibits oxidative phosphorylation through`

**GR**`and down-regulates glycolysis once more via`

**PC**`. See in Figure 12.`

**COF****ATP**has 3 providers: two levels of glycolysis (

`and`

**GLYC1**`) as well as oxidative phosphorylation (`

**GLYC2****PHOX**). It has 2 consumers:

**LBP**and

**nLBP**(biomass productions). Of course, the resource

`implies`

**GLYC2**`, thus there are 8 useless parameters: ${K}_{\mathit{ATP},\phantom{\rule{0.166667em}{0ex}}\mathbf{GLYC}\mathbf{2}}$, ${K}_{\mathit{ATP},\phantom{\rule{0.166667em}{0ex}}\mathbf{GLYC}\mathbf{2}\phantom{\rule{0.166667em}{0ex}}\mathbf{PHOX}}$, ${K}_{\mathit{ATP},\phantom{\rule{0.166667em}{0ex}}\mathbf{GLYC}\mathbf{2}\phantom{\rule{0.166667em}{0ex}}\mathbf{LBP}}$, ${K}_{\mathit{ATP},\phantom{\rule{0.166667em}{0ex}}\mathbf{GLYC}\mathbf{2}\phantom{\rule{0.166667em}{0ex}}\mathbf{nLBP}}$, ${K}_{\mathit{ATP},\phantom{\rule{0.166667em}{0ex}}\mathbf{GLYC}\mathbf{2}\phantom{\rule{0.166667em}{0ex}}\mathbf{PHOX}\phantom{\rule{0.166667em}{0ex}}\mathbf{LBP}}$, ${K}_{\mathit{ATP},\phantom{\rule{0.166667em}{0ex}}\mathbf{GLYC}\mathbf{2}\phantom{\rule{0.166667em}{0ex}}\mathbf{PHOX}\phantom{\rule{0.166667em}{0ex}}\mathbf{nLBP}}$, ${K}_{\mathit{ATP},\phantom{\rule{0.166667em}{0ex}}\mathbf{GLYC}\mathbf{2}\phantom{\rule{0.166667em}{0ex}}\mathbf{LBP}\phantom{\rule{0.166667em}{0ex}}\mathbf{nLBP}}$ and ${K}_{\mathit{ATP},\phantom{\rule{0.166667em}{0ex}}\mathbf{GLYC}\mathbf{2}\phantom{\rule{0.166667em}{0ex}}\mathbf{PHOX}\phantom{\rule{0.166667em}{0ex}}\mathbf{LBP}\phantom{\rule{0.166667em}{0ex}}\mathbf{nLBP}}$. It remains consequently 24 parameters to identify.`

**GLYC1****PHOX**nor

**GLYC**is present

**ATP**will finally not be able to activate its targets, thus ${K}_{\mathit{ATP},\phantom{\rule{0.166667em}{0ex}}\mathbf{LBP}\phantom{\rule{0.166667em}{0ex}}\mathbf{nLBP}}={K}_{\mathit{ATP},\phantom{\rule{0.166667em}{0ex}}\mathbf{LBP}}={K}_{\mathit{ATP},\phantom{\rule{0.166667em}{0ex}}\mathbf{nLBP}}={K}_{\mathit{ATP},\phantom{\rule{0.166667em}{0ex}}}=0$.

**ATP**, the presence of any

**ATP**provider will make

**ATP**tend to its higher value where it is able to act on all its targets. Thus, ${K}_{\mathit{ATP},\phantom{\rule{0.166667em}{0ex}}\mathbf{LBP}\phantom{\rule{0.166667em}{0ex}}\mathbf{nLBP}\phantom{\rule{0.166667em}{0ex}}\mathbf{PHOX}}={K}_{\mathit{ATP},\phantom{\rule{0.166667em}{0ex}}\mathbf{GLYC}\mathbf{1}\phantom{\rule{0.166667em}{0ex}}\mathbf{LBP}\phantom{\rule{0.166667em}{0ex}}\mathbf{nLBP}\phantom{\rule{0.166667em}{0ex}}\mathbf{PHOX}}={K}_{\mathit{ATP},\phantom{\rule{0.166667em}{0ex}}\mathbf{GLYC}\mathbf{1}\phantom{\rule{0.166667em}{0ex}}\mathbf{GLYC}\mathbf{2}\phantom{\rule{0.166667em}{0ex}}\mathbf{LBP}\phantom{\rule{0.166667em}{0ex}}\mathbf{nLBP}\phantom{\rule{0.166667em}{0ex}}\mathbf{PHOX}}={K}_{\mathit{ATP},\phantom{\rule{0.166667em}{0ex}}\mathbf{GLYC}\mathbf{1}\phantom{\rule{0.166667em}{0ex}}\mathbf{LBP}\phantom{\rule{0.166667em}{0ex}}\mathbf{nLBP}}=$${K}_{\mathit{ATP},\phantom{\rule{0.166667em}{0ex}}\mathbf{GLYC}\mathbf{1}\phantom{\rule{0.166667em}{0ex}}\mathbf{GLYC}\mathbf{2}\phantom{\rule{0.166667em}{0ex}}{LBP}\phantom{\rule{0.166667em}{0ex}}\mathbf{nLBP}}=2$.

**LBP**and

**nLBP**both consume

**ATP**(i.e., when they are not resources). If

**PHOX**is present as an activator, it produces enough

**ATP**to activate anabolic pathways such as

`or`

**PPP**`, and to inhibit $\beta $-oxidation, but the production will not be powerful enough to ring overproduction retro-controls via`

**AAS**`or`

**PC**`. The same arises if glycolysis at level 2 is an activator. Indeed, in terms of metabolic pathways, the energetic balance between oxidative phosphorylation and glycolysis is widely in favor of phosphorylation, except when the glycolysis is over-speeding. For this reason,`

**COF****PHOX**and

**GLYC**at level 2 will tend to activate the same targets downstream

**ATP**when biomass production consumes

**ATP**. Thus, ${K}_{\mathit{ATP},\phantom{\rule{0.166667em}{0ex}}\mathbf{PHOX}}={K}_{\mathit{ATP},\phantom{\rule{0.166667em}{0ex}}\mathbf{GLYC}\mathbf{1}\phantom{\rule{0.166667em}{0ex}}\mathbf{PHOX}}={K}_{\mathit{ATP},\phantom{\rule{0.166667em}{0ex}}\mathbf{GLYC}\mathbf{1}\phantom{\rule{0.166667em}{0ex}}\mathbf{GLYC}\mathbf{2}}={K}_{\mathit{ATP},\phantom{\rule{0.166667em}{0ex}}\mathbf{GLYC}\mathbf{1}\phantom{\rule{0.166667em}{0ex}}\mathbf{GLYC}\mathbf{2}\phantom{\rule{0.166667em}{0ex}}\mathbf{PHOX}}=1$. When

**GLYC**=1 is the only

**ATP**producer, glycolysis is not over-speeding and there is not enough ATP produced against biomass production consumers. Thus ${K}_{\mathit{ATP},\phantom{\rule{0.166667em}{0ex}}\mathbf{GLYC}\mathbf{1}}=0$.

**ATP**parameters remain to be identified: those where at least one

**ATP**producer is present and exactly one of the two consumers

**nLBP**or

**LBP**is in action. To go further, the following hypothesis seems sensible: When lipidic biomass production is on, the

**ATP**consumption of non-lipidic biomass production will not change the limit value toward which

**ATP**is attracted. This offers 5 more parameter values: ${K}_{\mathit{ATP},\phantom{\rule{0.166667em}{0ex}}\mathbf{nLBP}\phantom{\rule{0.166667em}{0ex}}\mathbf{PHOX}}={K}_{\mathit{ATP},\phantom{\rule{0.166667em}{0ex}}\mathbf{GLYC}\mathbf{1}\phantom{\rule{0.166667em}{0ex}}\mathbf{nLBP}\phantom{\rule{0.166667em}{0ex}}\mathbf{PHOX}}={K}_{\mathit{ATP},\phantom{\rule{0.166667em}{0ex}}\mathbf{GLYC}\mathbf{1}\phantom{\rule{0.166667em}{0ex}}\mathbf{GLYC}\mathbf{2}\phantom{\rule{0.166667em}{0ex}}\mathbf{nLBP}}={K}_{\mathit{ATP},\phantom{\rule{0.166667em}{0ex}}\mathbf{GLYC}\mathbf{1}\phantom{\rule{0.166667em}{0ex}}\mathbf{GLYC}\mathbf{2}\phantom{\rule{0.166667em}{0ex}}\mathbf{nLBP}\phantom{\rule{0.166667em}{0ex}}\mathbf{PHOX}}=1$ and ${K}_{\mathit{ATP},\phantom{\rule{0.166667em}{0ex}}\mathbf{GLYC}\mathbf{1}\phantom{\rule{0.166667em}{0ex}}\mathbf{nLBP}}=0$.

**NADH**parameters, let us leave the 5 remaining

**ATP**parameters unknown and Section 7 will handle the question properly.

## 6. Validation Matrix: Encoding Global Biological Knowledge Using Temporal Logic

- The environmental conditions are defined by the four environmental variables described above (Section 4.2): presence of glucose, availability of fatty acids, etc.
- and the markers of the global functioning are the regulated variables: biomass production, the activity level of the Krebs cycle, concentration ratio $NAD\left(P\right)H/NAD{\left(P\right)}^{+}$, etc.

#### 6.1. A Logic for Asymptotic Behaviors

**ATP**or

**NADH**oscillate, and where

**NCD**is never updated. At our level of abstraction, such a path is highly improbable because it is unfair with respect to the lowering of

**NCD**. This leads us to adopt the “fair-paths” semantics: A formula containing the quantifier A is evaluated on all paths except the non “fair” ones: For a path to be taken into account, it is necessary that if a state is visited an infinite number of times, none of its outgoing events can be neglected forever. This variant of CTL, where the quantifiers A and E do not consider unfair paths, is called fair-path CTL.

- The pattern “osc” depicts oscillations of the considered marker. If one knows the highest and lowest boundaries of this oscillation, one can specify these boundaries, e.g., “osc(1,2)” indicates that the variable under consideration oscillates between 1 and 2. However, if such boundaries are unknown, “osc” simply specifies that the oscillation takes place without any constraints on the boundaries.
- The pattern “td(n)” specifies that the variable under consideration tends toward the value n. It specifies that the marker tends toward a global equilibrium.

#### 6.2. Validation Matrix for the Regulation of Metabolism

**NCD**column implicit because its asymptotic behavior is under the control of the level of

**AA**, which imposes to

**NCD**the value toward which it tends, so the

**NCD**column would be a copy (up to the “td()” pattern) of the

**AA**column. Table 2 should be regarded as a $36\times 9$ matrix of fair-path CTL formulas that formalize the known behaviors of these markers (with a tenth implicit column). When no general knowledge about the behavior of a given marker in a given context is available, the corresponding box remains empty (dark grey).

#### 6.2.1. Without Lipid Intake and without Oxygen Supply

- ⇝
**1**—**FA**= 0,**exO2**= 0,**GLC**= 0,**AA**= 0: Leads to cell death. Cells require sufficient nutrients to supply the metabolic demands for cell growth and division. When all inputs are absent, no production of any component of the cell is possible and thus each variable tends toward 0. This would lead to cell death.- ⇝
**2 & 3**—**FA**= 0,**exO2**= 0,**GLC**= 0,**AA**= (1 or 2): The cell has only amino acids as nutrient supply. There is no general knowledge to assert whether amino acid intake alone can sustain carbon-dependent metabolic activity of the cell even if there are culture media without glucose for certain cancer cell lines [20,22]. Thus, we consider that cell fate is unknown, except that with no input of oxygen, the cell becomes in hypoxia and**O2**tends toward 0.- ⇝
**4**—**FA**= 0,**exO2**= 0,**GLC**= 1,**AA**= 0: Leads to aerobic glycolysis. Without oxygen and with glucose as a unique carbon source, mitochondrial respiration cannot occur and the only metabolic pathway is fermentation. Even if ATP is consumed for cell maintenance, it at least does not tend to zero as the cell can survive with only glucose. $NA{D}^{+}$ is consumed during glycolysis and regenerated during fermentation; Glucose intake allows the production of lactate, which reduces $NADH$ to $NA{D}^{+}$, which is recycled back to $NADH$ by glycolysis, so**NADH**oscillates. By the absence of oxygen supply,**O2**tends toward 0 and mitochondrial activity is shunted, so**PHOX**tends toward 0. Lastly,**GLC**=1 denotes a glucose intake insufficient to make glycolysis reach its higher rate, so**GLYC**oscillates between 0 and 1.- ⇝
**5**—**FA**= 0,**exO2**= 0,**GLC**= 1,**AA**= 1: Leads to aerobic glycolysis. An additional supply of amino acids allows for non-lipidic biomass production, so**nLBP**does not tend toward 0. This anabolic process consumes $ATP$, and as it is also produced, $ATP$ should oscillate. In addition, as in ⇝4, the cell is in an anaerobic process, and cytosolic metabolism and mitochondrial activity act similarly on the other markers.- ⇝
**6**—**FA**= 0,**exO2**= 0,**GLC**= 1,**AA**= 2: Leads to aerobic glycolysis. A huge supply on amino acids favors the reductive phase of Krebs to provide precursors for lipid synthesis, so**LBP**does not tend toward 0. The metabolic processes are the same as ⇝5, except that a large intake of amino acids activates glutaminolysis. It creates $\alpha $-ketoglutarate that can be converted with the reductive Krebs cycle in pyruvate. This accumulation of pyruvate could also be due to high activity of**GLYC**, therefore**GLYC**could sometimes reach its highest level. Thus, we prefer to relax its oscillatory behavior (“osc” without knowledge of the boundaries instead of “osc (0,1)”).- ⇝
**7**—**FA**= 0,**exO2**= 0,**GLC**= 2,**AA**= 0): Leads to aerobic glycolysis. With high glucose intake, glycolysis can sometimes reach its highest level. So,**GLYC**could possibly oscillate from its lowest to its highest level (“osc” instead of “osc (0,1)”). Moreover, the same processes as ⇝4 are impacted, and thus the behavior of all other markers remain identical.- ⇝
**8**—**FA**= 0,**exO2**= 0,**GLC**= 2,**AA**= 1): Leads to aerobic glycolysis. Here, the catabolic activity (glycolysis, fermentation, oxidative respiration and Krebs cycle) is similar to ⇝7, so that**NADH**behaves similarly. The more precise knowledge comes from the endergonic production of non-lipidic biomass, so that**nLBP**does not tend toward 0 and**ATP**can temporarily decrease to 0, so it oscillates.- ⇝
**9**—**FA**= 0,**exO2**= 0,**GLC**= 2,**AA**= 2: Leads to aerobic glycolysis. For the same reasons as ⇝6, this context allows for the production of lipid biomass, so**LBP**does not tend toward 0. Other markers behave similarly as in ⇝8.

#### 6.2.2. Without Lipid Intake and with Oxygen Supply

- ⇝
**10**—**FA**= 0,**exO2**= 1,**GLC**= 0,**AA**= 0: Leads to cell death. The cell is now in normoxia, thus**O2**does not tend toward 0. All other markers tend toward 0 because, without glucose and amino acids entries, there is no glucose metabolism, thus no carbon and cofactors sources for anabolism, so we expect no pyruvate available for the Krebs cycle, and thus no oxidative phosphorylation; and lastly this also affects the production of ATP.- ⇝
**11 & 12**—**FA**= 0,**exO2**= 1,**GLC**= 0,**AA**= (1 or 2): No consensus phenotype. With oxygen and amino acids (glutamine) but no glucose, the general knowledge does not allow us to decide if amino acids intake is sufficient to sustain carbon dependent metabolic activity of the cell (grey boxes): Some cells survive, and some others do not. Only oxygen makes no doubt. Indeed, we prefer to take a cautious approach so as not to restrict the genericity of our study of the Warburg effect.- ⇝
**13**—**FA**= 0,**exO2**= 1,**GLC**= 1,**AA**= 0: Aerobic respiration survival. With normal intake, except amino acids and lipids, respiration can operate the processes involved in creating $ATP$:**GLYC**,**KREBS**and**PHOX**oscillate. In response,**O2**(consumed by oxidative phosphorylation) and**NADH**(produced by the Krebs cycle) also, oscillate. $ATP$ is produced but there is not enough general knowledge about its consumption to assert that**ATP**tends toward 1 or that it oscillates, so we only assert that it does not tend toward 0. Lastly, according to normal aerobic metabolism,**FERM**tend toward 0 (as fermentation is less efficient than oxidative phosphorylation).- ⇝
**14**—**FA**= 0,**exO2**= 1,**GLC**= 1,**AA**= 1: Cell culture conditions. This line can be considered to be the context representing a healthy cell. It adds amino acid inputs to ⇝13, so that non-lipid biomass can be produced (**nLBP**does not tend toward 0). Aerobic processes follow the behavior of ⇝13 but we can be more specific about**ATP**: there is now an $ATP$ consumption by biomass production, so that**ATP**oscillates.- ⇝
**15**—**FA**= 0,**exO2**= 1,**GLC**= 1,**AA**= 2: Aerobic respiration. This line is similar to ⇝14 except that the large input of amino acids enables a lipid biomass production.- ⇝
**16**—**FA**= 0,**exO2**= 1,**GLC**= 2 and**AA**= 0: Warburg/Crabtree effect.**GLYC**oscillates as in ⇝14, but the high glucose uptake provokes Warburg/Crabtree phenotype and leads to high anaerobic glycolysis, even in the presence of oxygen. From the general knowledge, we only assert that**FERM**does not tend toward 0. Glycolysis activity suffices to regenerate $NADH$: as explained in ⇝4,**NADH**oscillates. As with ⇝13**ATP**does not tend toward 0. On the opposite, oxidative phosphorylation might be present when the Warburg/Crabtree effect occurs, but for sure not constantly so**PHOX**does not tend toward 1. Therefore, oxygen could be partially consumed, but**O2**does not tend toward 0 because its consumption by oxidative phosphorylation cannot counterbalance the external intake.- ⇝
**17 & 18**—**FA**= 0,**exO2**= 1,**GLC**= 2,**AA**= (1 or 2): Warburg/Crabtree effect. The Warburg/Crabtree occurs as in ⇝16. Additionally, here, the presence of amino acids intake allows biomass productions, so that**nLBP**and**BLP**do not tend toward 0, and thus**ATP**oscillates.

#### 6.2.3. With Lipid Intake

- ⇝
**19 and 28**—**FA**= 1,**exO2**= (0 or 1),**GLC**= 0,**AA**= 0: Lead to cell death. Lipid intake alone is unable to sustain all carbon-dependent metabolic activity of the cell, so, as in ⇝1 and ⇝10 respectively, all markers tend toward 0, except**O2**for ⇝28.- ⇝
**20 & 21**—**FA**= 1,**exO2**= 0,**GLC**= 0,**AA**= (1 or 2): No consensus phenotype. As with ⇝2&3, there is no general knowledge to assert whether lipid intake alone can sustain carbon-dependent metabolic activity of the cell: we consider that the future phenotype of the cell is unknown, except for hypoxia.- ⇝
**22–27**—**FA**= 1,**exO2**= 0,**GLC**= (1 or 2),**AA**= (0, 1 or 2): Leads to aerobic glycolysis. With respect to ⇝4 to ⇝9, fatty acid intake only sustains lipidic biomass production, and consequently the $ATP$ consumption. Lipids synthesis can be done as soon as $ATP$ is available for the cell even in absence of amino acids and thus**LBP**does not tend toward 0, and the $ATP$ consumption makes**ATP**oscillate. Other markers keep the behavior already described in ⇝4 to ⇝9.- ⇝
**29 & 30**—**FA**= 1,**exO2**= 1,**GLC**= 0,**AA**= (1 or 2): No consensus phenotype. As with ⇝11&12, the general knowledge does not allow us to decide if oxygen and lipid intake are sufficient to sustain carbon-dependent metabolic activity of the cell. Only normoxia makes no doubt.- ⇝
**31**—**FA**= 1,**exO2**= 1,**GLC**= 1,**AA**= 0: Aerobic respiration survival. Compared to ⇝13, the same reasoning as ⇝22-27 applies.- ⇝
**32**—**FA**= 1,**exO2**= 1,**GLC**= 1,**AA**= 1: Compared to the cell culture conditions of ⇝14, fatty acid intake only sustains lipidic biomass production, thus**LBP**does not tend toward 0 and the other markers keep the same behavior.- ⇝
**33**—**FA**= 1,**exO2**= 1,**GLC**= 1,**AA**= 2: Aerobic respiration. With respect to ⇝32, glutaminolysis feeds Krebs (anaplerotic reactions) and this can fuel lipid synthesis through citrate export from mitochondria. This excess of citrate opens the possibility to fuel fermentation in addition to lipid production. Thus, as a precaution, we leave the behavior of**FERM**unknown.- ⇝
**34–36**—**FA**=1,**exO2**=1,**GLC**=0,**AA**=(0,1 or 2): Warburg/Crabtree effect. Compared to lines from ⇝16 to ⇝18, the same reasoning as ⇝22–27 applies:**LBP**does not tend toward 0, and**ATP**oscillates.

`TotemBioNet`can help establishing this consistency. When thought experiments did not allow the setting of all parameters, it means that the remaining possible values are all correct with respect to the used (local) biochemical knowledge. In such a case, phenotypic properties can be used to constrain the remaining parametrizations, also thank to

`TotemBioNet`, and a model is valid if the set of resulting parametrizations is not empty.

## 7. Computer-Aided Validation of the Model Dynamics

`TotemBioNet`[11,12] and

`DyMBioNet`[13]. Both inherit from

`SMBioNet`[6] as they rely on intensive model checking to validate parameter settings with respect to phenotypic knowledge (and they additionally handle fair-path CTL, as described in Section 6).

`TotemBioNet`is remarkably efficient to manage all the possible parametrizations satisfying a set of given constraints and exhaustively select the correct parametrizations. For instance,

`TotemBioNet`enumerates and checks about 100 parameter settings per second and per line of the validation matrix.

`DyMBioNet`has a more user-friendly interfaces and it additionally offers easy simulations.

`DyMBioNet`has been very useful at the beginning of the modeling process, when sensible variables, valuable interactions, and revealing behaviors were still under examination. Later,

`TotemBioNet`gave us the freedom to leave many unset parameters, so that we did not hesitate to check several local modifications of the interaction graph, different possible ranges for variables, finally leading to the model proposed in this article.

**ATP**(${K}_{\mathit{ATP},\phantom{\rule{0.166667em}{0ex}}\mathbf{LBP}\phantom{\rule{0.166667em}{0ex}}\mathbf{PHOX}}$, ${K}_{\mathit{ATP},\phantom{\rule{0.166667em}{0ex}}\mathbf{GLYC}\mathbf{1}\phantom{\rule{0.166667em}{0ex}}\mathbf{LBP}}$, ${K}_{\mathit{ATP},\phantom{\rule{0.166667em}{0ex}}\mathbf{GLYC}\mathbf{1}\phantom{\rule{0.166667em}{0ex}}\mathbf{GLYC}\mathbf{2}\phantom{\rule{0.166667em}{0ex}}\mathbf{LBP}}$, ${K}_{\mathit{ATP},\phantom{\rule{0.166667em}{0ex}}\mathbf{GLYC}\mathbf{1}\phantom{\rule{0.166667em}{0ex}}\mathbf{LBP}\phantom{\rule{0.166667em}{0ex}}\mathbf{PHOX}}$, ${K}_{\mathit{ATP},\phantom{\rule{0.166667em}{0ex}}\mathbf{GLYC}\mathbf{1}\phantom{\rule{0.166667em}{0ex}}\mathbf{GLYC}\mathbf{2}\phantom{\rule{0.166667em}{0ex}}\mathbf{LBP}\phantom{\rule{0.166667em}{0ex}}\mathbf{PHOX}}$) and 2 for

**NADH**(${K}_{\mathit{NADH},\phantom{\rule{0.166667em}{0ex}}\mathbf{FERM}\phantom{\rule{0.166667em}{0ex}}\mathbf{GLYC}\phantom{\rule{0.166667em}{0ex}}\mathbf{KREBS}}$ and ${K}_{\mathit{NADH},\phantom{\rule{0.166667em}{0ex}}\mathbf{FERM}\phantom{\rule{0.166667em}{0ex}}\mathbf{GLYC}\phantom{\rule{0.166667em}{0ex}}\mathbf{PHOX}}$).

**ATP**has 3 possible values, from 0 to 2, and

**NADH**has 2 possible values, 0 or 1. Moreover, the

**KREBS**/

**PHOX**balance hypothesis of Section 5.2 implies that ${K}_{\mathit{NADH},\phantom{\rule{0.166667em}{0ex}}\mathbf{FERM}\phantom{\rule{0.166667em}{0ex}}\mathbf{GLYC}\phantom{\rule{0.166667em}{0ex}}\mathbf{KREBS}}={K}_{\mathit{NADH},\phantom{\rule{0.166667em}{0ex}}\mathbf{FERM}\phantom{\rule{0.166667em}{0ex}}\mathbf{GLYC}\phantom{\rule{0.166667em}{0ex}}\mathbf{PHOX}}$, a constraint that

`TotemBioNet`handles easily. Thus

`TotemBioNet`had finally to explore ${3}^{5}\times {2}^{1}=486$ different parameter settings: a small number with respect to its efficiency.

`TotemBioNet`results prove that 30 parameter settings among the 486 potential ones cope with the validation matrix (The input file describing interaction graph, partially identified parameters and validation matrix is also available now at the

`TotemBioNet`repository: https://gitlab.com/totembionet/totembionet, accessed on 30 May 2021)). Therefore, the thought experiments performed in Section 5.2 are validated with respect to the behaviors inventoried in Section 6, because there exist correct values for the 7 free parameters of Table 1. If instead of 30 parameter settings

`TotemBioNet`had found 0 parameter settings, it would have invalidated Table 1.

`TotemBioNet`has found more than 1 parameter setting, leading to several potentially distinct behaviors, could raise the erroneous idea that the goal is not yet achieved. This deserves three comments.

`TotemBioNet`has found at least 1 parameter settings ($30\u2a7e1$) validates both the interaction graph of Section 4 and the 87 parameter values of Section 5.2 and this simply answers yes to the question. A common complementary question is to know if additional research is needed to obtain exactly 1 correct parameter setting. In fact, the 30 parameter settings exhibit the same answer to the question that motivated the research, consequently, the work is actually achieved with respect to the original question.

`DyMBioNet`facilitates the exploration of the 30 parameter settings via easy intensive simulations and

`TotemBioNet`can automatically fill the gaps (the grey boxes) of the validation matrix for each of the parameter settings.

`TotemBioNet`to manage exhaustively all parameter settings that cope with biological knowledge, instead of only exhibiting one proper parameter setting among others is crucial from the methodological point of view.

## 8. Conclusions

`TotemBioNet`or

`DyMBioNet`we proved the compatibility of these two mathematical descriptions (Section 7). This ensures a high level of validation of our model.

`!td(0)`may become

`osc(0-1)`for some parameter values, and

`td(1)`for some others). Such predictions would be interesting to study less generic cell types. Additionally, our definition of the Warburg/Crabtree effect only considers the ability to perform fermentation in the presence of oxygen. It could be interesting to distinguish several kinds of Warburg/Crabtree effects classified according to the concomitant aerobic respiration behavior: In [48] a quantitative interplay between glutamine, glucose and oxygen is shown decisive and this would require at least to add a supplementary threshold for our

**O2**variable and a lot of new parameters to identify.

`TotemBioNet`and

`DyMBioNet`platforms (in particular the handling of several environments [47]). It provides a solid and flexible basis to focus on different cell types, with numerous applications, for example in human therapy or bio-production. We plan to use it to study the interaction of the central carbon metabolism with other cellular subsystems, such as the cell cycle and cell proliferation, or the circadian clock. Lastly, in a long-term perspective, it could be prolific to mix flux analysis methods to such an abstract regulation model, taking into account the shift between fermentation and respiration.

## Author Contributions

## Funding

## Acknowledgments

`TotemBioNet`, the model could not have been validated. Her stimulating questions around biological and modeling aspects allowed us to deepen our modeling methodology. Déborah Boyenval, a Phd student of the I3S team, has also been a precious help for the a posteriori biological analysis of our coarse-grained model of the metabolism regulation. The authors also thank Jürgen Moll, David Lloyd, Laurent Schwartz, Anne Devin, Patrick Paumard, Jorgelindo da Veiga Moreira, Erwan Bigan and Romain Attal for fruitful discussions on cell energetic mechanisms. Finally, Sophie Vasseur and Fabienne Guillaumond, from the Centre de Recherche en Cancerologie de Marseille are thanked for suggestions of adaptation of our model in the context of cancer cell lines.

## Conflicts of Interest

## References

- Hammad, N.; Rosas-Lemus, M.; Uribe-Carvajal, S.; Rigoulet, M.; Devin, A. The Crabtree and Warburg effects: Do metabolite-induced regulations participate in their induction? Biochim. Biophys. Acta Bioenerg.
**2016**, 1857, 1139–1146. [Google Scholar] [CrossRef] [PubMed] - Molenaar, D.; van Berlo, R.; de Ridder, D.; Teusink, B. Shifts in growth strategies reflect tradeoffs in cellular economics. Mol. Syst. Biol.
**2009**, 5, 323. [Google Scholar] [CrossRef] [PubMed] - Simeonidis, E.; Murabito, E.; Smallbone, K.; Westerhoff, H.V. Why does yeast ferment? A flux balance analysis study. Biochem. Soc. Trans.
**2010**, 38, 1225–1229. [Google Scholar] [CrossRef] [PubMed] - Wortel, M.T.; Bosdriesz, E.; Teusink, B.; Bruggeman, F.J. Evolutionary pressures on microbial metabolic strategies in the chemostat. Sci. Rep.
**2016**, 6, 29503. [Google Scholar] [CrossRef] [PubMed] [Green Version] - da Veiga Moreira, J.; Hamraz, M.; Abolhassani, M.; Schwartz, L.; Jolicœur, M.; Peres, S. Metabolic therapies inhibit tumor growth in vivo and in silico. Sci. Rep.
**2019**, 9, 3153. [Google Scholar] [CrossRef] [Green Version] - Bernot, G.; Comet, J.P.; Richard, A.; Guespin, J. Application of formal methods to biological regulatory networks: Extending Thomas’ asynchronous logical approach with temporal logic. J. Theor. Biol.
**2004**, 229, 339–347. [Google Scholar] [CrossRef] [PubMed] - Bernot, G.; Comet, J.P.; Khalis, Z.; Richard, A.; Roux, O.F. A Genetically Modified Hoare Logic. Theor. Comput. Sci.
**2019**, 765, 145–157. [Google Scholar] [CrossRef] [Green Version] - Thomas, R. (Ed.) Kinetic logic: A boolean approach to the analysis of complex regulatory systems. In Lecture Notes in Biomathematics; Springer: Berlin/Heidelberg, Germany, 1979; Volume 29. [Google Scholar]
- Thomas, R. Regulatory networks seen as asynchronous automata: A logical description. J. Theor. Biol.
**1991**, 153, 1–23. [Google Scholar] [CrossRef] - Khalis, Z.; Comet, J.P.; Richard, A.; Bernot, G. The SMBioNet Method for Discovering Models of Gene Regulatory Networks. Genes Genomes Genom.
**2009**, 3, 15–22. [Google Scholar] - Boyenval, D.; Bernot, G.; Collavizza, H.; Comet, J.P. What is a cell cycle checkpoint? The TotemBioNet answer. In Proceedings of the 18th International Conference on Computational Methods in Systems Biology (CMSB), Online, 23–25 September 2020; Volume 12314, pp. 362–372. [Google Scholar]
- Gibart, L.; Bernot, G.; Collavizza, H.; Comet, J.P. TotemBioNet Enrichment Methodology: Application to the Qualitative Regulatory Network of the Cell Metabolism. In Proceedings of the 12th International Conference on Bioinformatics Models, Methods and Algorithms, Online, 11–13 February 2021. [Google Scholar]
- Khoodeeram, R. Discrete Coarse-Grained Modelling of Energy Metabolism Using Formal Approach: A Study of the Dynamics in Cell Proliferation. Ph.D. Thesis, Université Côte d’Azur, Nice, France, 2020. [Google Scholar]
- Khoodeeram, R.; Bernot, G.; Trosset, J.Y. An Ockham Razor model of energy metabolism. In Proceedings of the Thematic Research School on Advances in Systems and Synthetic Biology, Modelling Complex Biological Systems in the Context of Genomics; EDP Sciences: Les Ulis, France, 2016; pp. 81–101. ISBN 978-2-7598-2116-7. [Google Scholar]
- Thomas, R.; D’Ari, R. Biological Feedback; CRC Press: Boca Raton, FL, USA, 1990. [Google Scholar]
- Clarke, E.; Emerson, E. Design and syntheses of synchronization skeletons using branching time temporal logic. Workshop Log. Programs
**1981**, 131, 52–71. [Google Scholar] - Yin, C.; Qie, S.; Sang, N. Carbon Source Metabolism and Its Regulation in Cancer Cells. Crit. Rev. Eukaryot. Gene Expr.
**2012**, 22, 17–35. [Google Scholar] [CrossRef] [Green Version] - de Alteriis, E.; Cartenì, F.; Parascandola, P.; Serpa, J.; Mazzoleni, S. Revisiting the Crabtree/Warburg effect in a dynamic perspective: A fitness advantage against sugar-induced cell death. Cell Cycle
**2018**, 17, 688–701. [Google Scholar] [CrossRef] [Green Version] - Zhu, J.; Thompson, C.B. Metabolic regulation of cell growth and proliferation. Nat. Rev. Mol. Cell. Biol.
**2019**, 20, 436–450. [Google Scholar] [CrossRef] - Rigoulet, M.; Bouchez, C.L.; Paumard, P.; Ransac, S.; Cuvellier, S.; Duvezin-Caubet, S.; Mazat, J.P.; Devin, A. Cell energy metabolism: An update. Biochim. Biophys. Acta Bioenerg.
**2020**, 1861, 148276. [Google Scholar] [CrossRef] - Murray, D.B.; Beckmann, M.; Kitano, H. Regulation of yeast oscillatory dynamics. Proc. Natl. Acad. Sci. USA
**2007**, 104, 2241–2246. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Dang, C.V. Links between metabolism and cancer. Genes Dev.
**2012**, 26, 877–890. [Google Scholar] [CrossRef] [Green Version] - Yang, L.; Venneti, S.; Nagrath, D. Glutaminolysis: A Hallmark of Cancer Metabolism. Annu. Rev. Biomed. Eng.
**2017**, 19, 163–194. [Google Scholar] [CrossRef] [PubMed] - Liberti, M.V.; Locasale, J.W. The Warburg Effect: How Does it Benefit Cancer Cells? Trends Biochem. Sci.
**2016**, 41, 211–218. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Moreira, J.d.V.; Hamraz, M.; Abolhassani, M.; Bigan, E.; Pérès, S.; Paulevé, L.; Nogueira, M.L.; Steyaert, J.M.; Schwartz, L. The Redox Status of Cancer Cells Supports Mechanisms behind the Warburg Effect. Metabolites
**2016**, 6, 33. [Google Scholar] [CrossRef] - Sober, E. Ockham’s Razors: A User’s Manual; Cambridge University Press: Cambridge, UK, 2015. [Google Scholar] [CrossRef]
- Axelrod, B. Chapter 3—Glycolysis. In Metabolic Pathways, 3rd ed.; Greenberg, D.M., Ed.; Academic Press: Cambridge, MA, USA, 1967; pp. 112–145. [Google Scholar] [CrossRef]
- Anderson, N.M.; Mucka, P.; Kern, J.G.; Feng, H. The emerging role and targetability of the TCA cycle in cancer metabolism. Protein Cell
**2018**, 9, 216–237. [Google Scholar] [CrossRef] - Lowenstein, J.M. Chapter 4—The Tricarboxylic Acid Cycle. In Metabolic Pathways, 3rd ed.; Greenberg, D.M., Ed.; Academic Press: Cambridge, MA, USA, 1967; pp. 146–270. [Google Scholar] [CrossRef]
- Green, D.E.; MacLennan, D.H. Chapter 2—The Mitochondrial System of Enzymes. In Metabolic Pathways, 3rd ed.; Greenberg, D.M., Ed.; Academic Press: Cambridge, MA, USA, 1967; pp. 47–111. [Google Scholar] [CrossRef]
- Kumagai, H. Amino Acid Production. In The Prokaryotes: Applied Bacteriology and Biotechnology; Rosenberg, E., DeLong, E.F., Lory, S., Stackebrandt, E., Thompson, F., Eds.; Springer: Berlin/Heidelberg, Germany, 2013; pp. 169–177. [Google Scholar] [CrossRef]
- Reitzer, L. Amino Acid Synthesis. In Encyclopedia of Microbiology, 3rd ed.; Schaechter, M., Ed.; Academic Press: Oxford, UK, 2009; pp. 1–17. [Google Scholar] [CrossRef]
- Mansour, T.E. Studies on Heart Phosphofructokinase: Purification, Inhibition, and Activation. J. Biol. Chem.
**1963**, 238, 2285–2292. [Google Scholar] [CrossRef] - Shi, L.; Tu, B.P. Acetyl-CoA and the Regulation of Metabolism: Mechanisms and Consequences. Curr. Opin. Cell Biol.
**2015**, 33, 125–131. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Houten, S.M.; Wanders, R.J.A. A general introduction to the biochemistry of mitochondrial fatty acid beta-oxidation. J. Inherit. Metab. Dis.
**2010**, 33, 469–477. [Google Scholar] [CrossRef] [Green Version] - Wu, N.; Yang, M.; Gaur, U.; Xu, H.; Yao, Y.; Li, D. Alpha-Ketoglutarate: Physiological Functions and Applications. Biomol. Ther.
**2016**, 24, 1–8. [Google Scholar] [CrossRef] [PubMed] [Green Version] - McKeehan, W.L. Glutaminolysis in Animal Cells. In Carbohydrate Metabolism in Cultured Cells; Morgan, M.J., Ed.; Springer: Berlin/Heidelberg, Germany, 1986; pp. 111–150. [Google Scholar] [CrossRef]
- Wilson, D.F. Oxidative phosphorylation: Regulation and role in cellular and tissue metabolism. J. Physiol.
**2017**, 595, 7023–7038. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Gregg, X.T.; Prchal, J.T. Chapter 44—Red Blood Cell Enzymopathies. In Hematology, 7th ed.; Hoffman, R., Benz, E.J., Silberstein, L.E., Heslop, H.E., Weitz, J.I., Anastasi, J., Salama, M.E., Abutalib, S.A., Eds.; Elsevier: Amsterdam, The Netherlands, 2018; pp. 616–625. [Google Scholar] [CrossRef]
- Stein, O.; Stein, Y. Lipid Synthesis, Intracellular Transport, Storage, and Secretion. J. Cell Biol.
**1967**, 33, 319–339. [Google Scholar] [CrossRef] [Green Version] - Nelson, D.L.; Cox, M.M. Lehninger Principles of Biochemistry, 6th ed.; W.H. Freeman and Company: New York, NY, USA, 2012. [Google Scholar]
- Koolman, J.; Röhm, K.H. Color Atlas of Biochemistry, 3rd ed.; Revised and Updated Edition; Thieme Publishing Group: Stuttgart, Germany, 2012. [Google Scholar]
- Berg, J.M.; Tymoczko, J.L.; Stryer, L.; Berg, J.M.; Tymoczko, J.L.; Stryer, L. Biochemistry, 5th ed.; W.H. Freeman and Company: New York, NY, USA, 2002. [Google Scholar]
- Goldfeder, J.; Kugler, H. BRE:IN—A Backend for Reasoning About Interaction Networks with Temporal Logic. In Computational Methods in Systems Biology; Lecture Notes in Computer, Science; Bortolussi, L., Sanguinetti, G., Eds.; Springer International Publishing: Cham, Switzerland, 2019; pp. 289–295. [Google Scholar] [CrossRef]
- Richard, A. Fair Paths in CTL. Available online: https://gitlab.com/totembionet/totembionet (accessed on 30 May 2021).
- Cimatti, A.; Clarke, E.; Giunchiglia, E.; Giunchiglia, F.; Pistore, M.; Roveri, M.; Sebastiani, R.; Tacchella, A. NuSMV 2: An OpenSource Tool for Symbolic Model Checking. In Proceedings of the CAV 2002: Computer Aided Verification, Copenhagen, Denmark, Copenhagen, Denmark, 27–31 July 2002; pp. 359–364. [Google Scholar]
- Gibart, L.; Collavizza, H.; Comet, J.P. Greening R. Thomas’ Framework with Environment Variables: A Divide and Conquer Approach. In Proceedings of the 19th International Conference on Computational Methods in Systems Biology (CMSB), Bordeaux, France, 22–24 September 2021. [Google Scholar]
- Damiani, C.; Colombo, R.; Gaglio, D.; Mastroianni, F.; Pescini, D.; Westerhoff, H.V.; Mauri, G.; Vanoni, M.; Alberghina, L. A metabolic core model elucidates how enhanced utilization of glucose and glutamine, with enhanced glutamine-dependent lactate production, promotes cancer cell growth: The WarburQ effect. PLoS Comput. Biol.
**2017**, 13, e1005758. [Google Scholar] [CrossRef] [Green Version]

**Figure 1.**Toy example. (

**Left**): an influence graph made of a positive loop $(+,+)$. (

**Center**): the parameters to be identified. (

**Right**): a sensible global behavior, associated with the parameter values: ${K}_{x}=0$, ${K}_{x,y}=1$, ${K}_{y}=0$, ${K}_{y,x}=1$.

**Figure 2.**Anothertoy example. (

**Left**): A partial view of an influence graph where y inhibits itself and is activated by x. (

**Right**): the four parameters to be identified.

**Figure 3.**Influence graph of the regulation of cell metabolism. Circles represent variables whereas rectangles represent coordinated regulations (multiplexes). Blue entities refer to biomass, yellow ones to metabolites and red ones to metabolic pathways. Plain lines show regulation targets. Dashed lines denote regulation sources (mathematically useless, as they can be deduced from multiplex formulas). Within logical formulas, “!” stands for negation, “&” for conjunction and “|” for disjunction.

**Table 1.**Kinetic parameter values from Section 5.2. Only 7 parameter values remain free: ${K}_{\mathit{ATP},\phantom{\rule{0.166667em}{0ex}}\mathbf{LBP}\phantom{\rule{0.166667em}{0ex}}\mathbf{PHOX}}$, ${K}_{\mathit{ATP},\phantom{\rule{0.166667em}{0ex}}\mathbf{GLYC}\mathbf{1}\phantom{\rule{0.166667em}{0ex}}\mathbf{LBP}}$, ${K}_{\mathit{ATP},\phantom{\rule{0.166667em}{0ex}}\mathbf{GLYC}\mathbf{1}\phantom{\rule{0.166667em}{0ex}}\mathbf{GLYC}\mathbf{2}\phantom{\rule{0.166667em}{0ex}}\mathbf{LBP}}$, ${K}_{\mathit{ATP},\phantom{\rule{0.166667em}{0ex}}\mathbf{GLYC}\mathbf{1}\phantom{\rule{0.166667em}{0ex}}\mathbf{LBP}\phantom{\rule{0.166667em}{0ex}}\mathbf{PHOX}}$, ${K}_{\mathit{ATP},\phantom{\rule{0.166667em}{0ex}}\mathbf{GLYC}\mathbf{1}\phantom{\rule{0.166667em}{0ex}}\mathbf{GLYC}\mathbf{2}\phantom{\rule{0.166667em}{0ex}}\mathbf{LBP}\phantom{\rule{0.166667em}{0ex}}\mathbf{PHOX}}$, ${K}_{\mathit{NADH},\phantom{\rule{0.166667em}{0ex}}\mathbf{FERM}\phantom{\rule{0.166667em}{0ex}}\mathbf{GLYC}\phantom{\rule{0.166667em}{0ex}}\mathbf{PHOX}}$ and ${K}_{\mathit{NADH},\phantom{\rule{0.166667em}{0ex}}\mathbf{FERM}\phantom{\rule{0.166667em}{0ex}}\mathbf{GLYC}\phantom{\rule{0.166667em}{0ex}}\mathbf{KREBS}}$.

# Parameters for ATP | # Parameters for O2 | # Parameters for NADH |
---|---|---|

${K}_{\mathit{ATP}}=0$ | ${K}_{\mathit{O}\mathit{2}}=0$ | ${K}_{\mathit{NADH}}=0$ |

${K}_{\mathit{ATP},\phantom{\rule{0.166667em}{0ex}}\mathbf{LBP}}=0$ | ${K}_{\mathit{O}\mathit{2},\phantom{\rule{0.166667em}{0ex}}\mathbf{PHOX}}=0$ | ${K}_{\mathit{NADH},\phantom{\rule{0.166667em}{0ex}}\mathbf{FERM}}=0$ |

${K}_{\mathit{ATP},\phantom{\rule{0.166667em}{0ex}}\mathbf{nLBP}}=0$ | ${K}_{\mathit{O}\mathit{2},\phantom{\rule{0.166667em}{0ex}}\mathbf{exO}\mathbf{2}}=1$ | ${K}_{\mathit{NADH},\phantom{\rule{0.166667em}{0ex}}\mathbf{PHOX}}=0$ |

${K}_{\mathit{ATP},\phantom{\rule{0.166667em}{0ex}}\mathbf{LBP}\phantom{\rule{0.166667em}{0ex}}\mathbf{nLBP}}=0$ | ${K}_{\mathit{O}\mathit{2},\phantom{\rule{0.166667em}{0ex}}\mathbf{exO}\mathbf{2}\phantom{\rule{0.166667em}{0ex}}\mathbf{PHOX}}=1$ | ${K}_{\mathit{NADH},\phantom{\rule{0.166667em}{0ex}}\mathbf{AAS}}=0$ |

${K}_{\mathit{ATP},\phantom{\rule{0.166667em}{0ex}}\mathbf{PHOX}}=1$ | ${K}_{\mathit{NADH},\phantom{\rule{0.166667em}{0ex}}\mathbf{FERM}\phantom{\rule{0.166667em}{0ex}}\mathbf{PHOX}}=0$ | |

${K}_{\mathit{ATP},\phantom{\rule{0.166667em}{0ex}}\mathbf{nLBP}\phantom{\rule{0.166667em}{0ex}}\mathbf{PHOX}}=1$ | # Parameters for GLYC | ${K}_{\mathit{NADH},\phantom{\rule{0.166667em}{0ex}}\mathbf{FERM}\phantom{\rule{0.166667em}{0ex}}\mathbf{AAS}}=0$ |

${K}_{\mathit{ATP},\phantom{\rule{0.166667em}{0ex}}\mathbf{LBP}\phantom{\rule{0.166667em}{0ex}}\mathbf{nLBP}\phantom{\rule{0.166667em}{0ex}}\mathbf{PHOX}}=2$ | ${K}_{\mathit{GLYC}}=0$ | ${K}_{\mathit{NADH},\phantom{\rule{0.166667em}{0ex}}\mathbf{AAS}\phantom{\rule{0.166667em}{0ex}}\mathbf{PHOX}}=0$ |

${K}_{\mathit{ATP},\phantom{\rule{0.166667em}{0ex}}\mathbf{GLYC}\mathbf{1}}=0$ | ${K}_{\mathit{GLYC},\phantom{\rule{0.166667em}{0ex}}\mathbf{GR}}=0$ | ${K}_{\mathit{NADH},\phantom{\rule{0.166667em}{0ex}}\mathbf{FERM}\phantom{\rule{0.166667em}{0ex}}\mathbf{AAS}\phantom{\rule{0.166667em}{0ex}}\mathbf{PHOX}}=0$ |

${K}_{\mathit{ATP},\phantom{\rule{0.166667em}{0ex}}\mathbf{GLYC}\mathbf{1}\phantom{\rule{0.166667em}{0ex}}\mathbf{nLBP}}=0$ | ${K}_{\mathit{GLYC},\phantom{\rule{0.166667em}{0ex}}\mathbf{GLC}\mathbf{1}}=0$ | ${K}_{\mathit{NADH},\phantom{\rule{0.166667em}{0ex}}\mathbf{FERM}\phantom{\rule{0.166667em}{0ex}}\mathbf{KREBS}\phantom{\rule{0.166667em}{0ex}}\mathbf{AAS}}=0$ |

${K}_{\mathit{ATP},\phantom{\rule{0.166667em}{0ex}}\mathbf{GLYC}\mathbf{1}\phantom{\rule{0.166667em}{0ex}}\mathbf{LBP}\phantom{\rule{0.166667em}{0ex}}\mathbf{nLBP}}=2$ | ${K}_{\mathit{GLYC},\phantom{\rule{0.166667em}{0ex}}\mathbf{GLC}\mathbf{1}\phantom{\rule{0.166667em}{0ex}}\mathbf{GR}}=0$ | ${K}_{\mathit{NADH},\phantom{\rule{0.166667em}{0ex}}\mathbf{GLYC}\phantom{\rule{0.166667em}{0ex}}\mathbf{AAS}\phantom{\rule{0.166667em}{0ex}}\mathbf{PHOX}}=0$ |

${K}_{\mathit{ATP},\phantom{\rule{0.166667em}{0ex}}\mathbf{GLYC}\mathbf{1}\phantom{\rule{0.166667em}{0ex}}\mathbf{GLYC}\mathbf{2}}=1$ | ${K}_{\mathit{GLYC},\phantom{\rule{0.166667em}{0ex}}\mathbf{GLC}\mathbf{1}\phantom{\rule{0.166667em}{0ex}}\mathbf{GLC}\mathbf{2}}=0$ | ${K}_{\mathit{NADH},\phantom{\rule{0.166667em}{0ex}}\mathbf{GLYC}\phantom{\rule{0.166667em}{0ex}}\mathbf{FERM}\phantom{\rule{0.166667em}{0ex}}\mathbf{AAS}\phantom{\rule{0.166667em}{0ex}}\mathbf{PHOX}}=1$ |

${K}_{\mathit{ATP},\phantom{\rule{0.166667em}{0ex}}\mathbf{GLYC}\mathbf{1}\phantom{\rule{0.166667em}{0ex}}\mathbf{GLYC}\mathbf{2}\phantom{\rule{0.166667em}{0ex}}\mathbf{nLBP}}=1$ | ${K}_{\mathit{GLYC},\phantom{\rule{0.166667em}{0ex}}\mathbf{GLC}\mathbf{1}\phantom{\rule{0.166667em}{0ex}}\mathbf{GLC}\mathbf{2}\phantom{\rule{0.166667em}{0ex}}\mathbf{GR}}=0$ | ${K}_{\mathit{NADH},\phantom{\rule{0.166667em}{0ex}}\mathbf{KREBS}\phantom{\rule{0.166667em}{0ex}}\mathbf{FERM}\phantom{\rule{0.166667em}{0ex}}\mathbf{AAS}\phantom{\rule{0.166667em}{0ex}}\mathbf{PHOX}}=1$ |

${K}_{\mathit{ATP},\phantom{\rule{0.166667em}{0ex}}\mathbf{GLYC}\mathbf{1}\phantom{\rule{0.166667em}{0ex}}\mathbf{GLYC}\mathbf{2}\phantom{\rule{0.166667em}{0ex}}\mathbf{LBP}\phantom{\rule{0.166667em}{0ex}}\mathbf{nLBP}}=2$ | ${K}_{\mathit{GLYC},\phantom{\rule{0.166667em}{0ex}}\mathbf{COF}}=0$ | ${K}_{\mathit{NADH},\phantom{\rule{0.166667em}{0ex}}\mathbf{GLYC}\phantom{\rule{0.166667em}{0ex}}\mathbf{KREBS}\phantom{\rule{0.166667em}{0ex}}\mathbf{FERM}\phantom{\rule{0.166667em}{0ex}}\mathbf{AAS}\phantom{\rule{0.166667em}{0ex}}\mathbf{PHOX}}=1$ |

${K}_{\mathit{ATP},\phantom{\rule{0.166667em}{0ex}}\mathbf{GLYC}\mathbf{1}\phantom{\rule{0.166667em}{0ex}}\mathbf{PHOX}}=1$ | ${K}_{\mathit{GLYC},\phantom{\rule{0.166667em}{0ex}}\mathbf{COF}\phantom{\rule{0.166667em}{0ex}}\mathbf{GR}}=0$ | ${K}_{\mathit{NADH},\phantom{\rule{0.166667em}{0ex}}\mathbf{GLYC}\phantom{\rule{0.166667em}{0ex}}\mathbf{KREBS}\phantom{\rule{0.166667em}{0ex}}\mathbf{AAS}}=0$ |

${K}_{\mathit{ATP},\phantom{\rule{0.166667em}{0ex}}\mathbf{GLYC}\mathbf{1}\phantom{\rule{0.166667em}{0ex}}\mathbf{nLBP}\phantom{\rule{0.166667em}{0ex}}\mathbf{PHOX}}=1$ | ${K}_{\mathit{GLYC},\phantom{\rule{0.166667em}{0ex}}\mathbf{COF}\phantom{\rule{0.166667em}{0ex}}\mathbf{GLC}\mathbf{1}}=0$ | ${K}_{\mathit{NADH},\phantom{\rule{0.166667em}{0ex}}\mathbf{FERM}\phantom{\rule{0.166667em}{0ex}}\mathbf{GLYC}\phantom{\rule{0.166667em}{0ex}}\mathbf{AAS}}=0$ |

${K}_{\mathit{ATP},\phantom{\rule{0.166667em}{0ex}}\mathbf{GLYC}\mathbf{1}\phantom{\rule{0.166667em}{0ex}}\mathbf{LBP}\phantom{\rule{0.166667em}{0ex}}\mathbf{nLBP}\phantom{\rule{0.166667em}{0ex}}\mathbf{PHOX}}=2$ | ${K}_{\mathit{GLYC},\phantom{\rule{0.166667em}{0ex}}\mathbf{COF}\phantom{\rule{0.166667em}{0ex}}\mathbf{GLC}\mathbf{1}\phantom{\rule{0.166667em}{0ex}}\mathbf{GR}}=1$ | ${K}_{\mathit{NADH},\phantom{\rule{0.166667em}{0ex}}\mathbf{GLYC}\phantom{\rule{0.166667em}{0ex}}\mathbf{KREBS}}=0$ |

${K}_{\mathit{ATP},\phantom{\rule{0.166667em}{0ex}}\mathbf{GLYC}\mathbf{1}\phantom{\rule{0.166667em}{0ex}}\mathbf{GLYC}\mathbf{2}\phantom{\rule{0.166667em}{0ex}}\mathbf{PHOX}}=1$ | ${K}_{\mathit{GLYC},\phantom{\rule{0.166667em}{0ex}}\mathbf{COF}\phantom{\rule{0.166667em}{0ex}}\mathbf{GLC}\mathbf{1}\phantom{\rule{0.166667em}{0ex}}\mathbf{GLC}\mathbf{2}}=1$ | ${K}_{\mathit{NADH},\phantom{\rule{0.166667em}{0ex}}\mathbf{FERM}\phantom{\rule{0.166667em}{0ex}}\mathbf{KREBS}}=0$ |

${K}_{\mathit{ATP},\phantom{\rule{0.166667em}{0ex}}\mathbf{GLYC}\mathbf{1}\phantom{\rule{0.166667em}{0ex}}\mathbf{GLYC}\mathbf{2}\phantom{\rule{0.166667em}{0ex}}\mathbf{nLBP}\phantom{\rule{0.166667em}{0ex}}\mathbf{PHOX}}=1$ | ${K}_{\mathit{GLYC},\phantom{\rule{0.166667em}{0ex}}\mathbf{COF}\phantom{\rule{0.166667em}{0ex}}\mathbf{GLC}\mathbf{1}\phantom{\rule{0.166667em}{0ex}}\mathbf{GLC}\mathbf{2}\phantom{\rule{0.166667em}{0ex}}\mathbf{GR}}=2$ | ${K}_{\mathit{NADH},\phantom{\rule{0.166667em}{0ex}}\mathbf{KREBS}\phantom{\rule{0.166667em}{0ex}}\mathbf{AAS}}=0$ |

${K}_{\mathit{ATP},\phantom{\rule{0.166667em}{0ex}}\mathbf{GLYC}\mathbf{1}\phantom{\rule{0.166667em}{0ex}}\mathbf{GLYC}\mathbf{2}\phantom{\rule{0.166667em}{0ex}}\mathbf{LBP}\phantom{\rule{0.166667em}{0ex}}\mathbf{nLBP}\phantom{\rule{0.166667em}{0ex}}\mathbf{PHOX}}=2$ | ${K}_{\mathit{NADH},\phantom{\rule{0.166667em}{0ex}}\mathbf{GLYC}\phantom{\rule{0.166667em}{0ex}}\mathbf{AAS}}=0$ | |

# Parameters for nLBP | ${K}_{\mathit{NADH},\phantom{\rule{0.166667em}{0ex}}\mathbf{GLYC}\phantom{\rule{0.166667em}{0ex}}\mathbf{PHOX}}=0$ | |

# Parameters for LBP | ${K}_{\mathit{nLBP}}=0$ | ${K}_{\mathit{NADH},\phantom{\rule{0.166667em}{0ex}}\mathbf{FERM}\phantom{\rule{0.166667em}{0ex}}\mathbf{GLYC}}=0$ |

${K}_{\mathit{LBP}}=0$ | ${K}_{\mathit{nLBP},\phantom{\rule{0.166667em}{0ex}}\mathbf{PPP}}=1$ | ${K}_{\mathit{NADH},\phantom{\rule{0.166667em}{0ex}}\mathbf{KREBS}}=0$ |

${K}_{\mathit{LBP},\phantom{\rule{0.166667em}{0ex}}\mathbf{LS}}=1$ | ${K}_{\mathit{nLBP},\phantom{\rule{0.166667em}{0ex}}\mathbf{AAS}}=1$ | ${K}_{\mathit{NADH},\phantom{\rule{0.166667em}{0ex}}\mathbf{GLYC}}=0$ |

${K}_{\mathit{LBP},\phantom{\rule{0.166667em}{0ex}}\mathbf{BOX}}=0$ | ${K}_{\mathit{nLBP},\phantom{\rule{0.166667em}{0ex}}\mathbf{AAS}\phantom{\rule{0.166667em}{0ex}}\mathbf{PPP}}=1$ | ${K}_{\mathit{NADH},\phantom{\rule{0.166667em}{0ex}}\mathbf{FERM}\phantom{\rule{0.166667em}{0ex}}\mathbf{GLYC}\phantom{\rule{0.166667em}{0ex}}\mathbf{KREBS}\phantom{\rule{0.166667em}{0ex}}\mathbf{AAS}}=1$ |

${K}_{\mathit{LBP},\phantom{\rule{0.166667em}{0ex}}\mathbf{LS}\phantom{\rule{0.166667em}{0ex}}\mathbf{BOX}}=1$ | ${K}_{\mathit{NADH},\phantom{\rule{0.166667em}{0ex}}\mathbf{KREBS}\phantom{\rule{0.166667em}{0ex}}\mathbf{PHOX}}=1$ | |

# Parameters for KREBS | ${K}_{\mathit{NADH},\phantom{\rule{0.166667em}{0ex}}\mathbf{KREBS}\phantom{\rule{0.166667em}{0ex}}\mathbf{FERM}\phantom{\rule{0.166667em}{0ex}}\mathbf{PHOX}}=1$ | |

# Parameters for NCD | ${K}_{\mathit{KREBS}}=0$ | ${K}_{\mathit{NADH},\phantom{\rule{0.166667em}{0ex}}\mathbf{KREBS}\phantom{\rule{0.166667em}{0ex}}\mathbf{AAS}\phantom{\rule{0.166667em}{0ex}}\mathbf{PHOX}}=1$ |

${K}_{\mathit{NCD}}=0$ | ${K}_{\mathit{KREBS},\phantom{\rule{0.166667em}{0ex}}\mathbf{AnO}}=1$ | ${K}_{\mathit{NADH},\phantom{\rule{0.166667em}{0ex}}\mathbf{GLYC}\phantom{\rule{0.166667em}{0ex}}\mathbf{KREBS}\phantom{\rule{0.166667em}{0ex}}\mathbf{PHOX}}=1$ |

${K}_{\mathit{NCD},\phantom{\rule{0.166667em}{0ex}}\mathbf{KREBS}}=0$ | ${K}_{\mathit{KREBS},\phantom{\rule{0.166667em}{0ex}}\mathbf{AnO}\phantom{\rule{0.166667em}{0ex}}\alpha -\mathbf{KG}}=2$ | ${K}_{\mathit{NADH},\phantom{\rule{0.166667em}{0ex}}\mathbf{FERM}\phantom{\rule{0.166667em}{0ex}}\mathbf{GLYC}\phantom{\rule{0.166667em}{0ex}}\mathbf{KREBS}\phantom{\rule{0.166667em}{0ex}}\mathbf{PHOX}}=1$ |

${K}_{\mathit{NCD},\phantom{\rule{0.166667em}{0ex}}\mathbf{AA}\mathbf{1}}=1$ | ${K}_{\mathit{KREBS},\phantom{\rule{0.166667em}{0ex}}\mathbf{BOX}}=1$ | ${K}_{\mathit{NADH},\phantom{\rule{0.166667em}{0ex}}\mathbf{GLYC}\phantom{\rule{0.166667em}{0ex}}\mathbf{KREBS}\phantom{\rule{0.166667em}{0ex}}\mathbf{AAS}\phantom{\rule{0.166667em}{0ex}}\mathbf{PHOX}}=1$ |

${K}_{\mathit{NCD},\phantom{\rule{0.166667em}{0ex}}\mathbf{AA}\mathbf{1}\phantom{\rule{0.166667em}{0ex}}\mathbf{KREBS}}=1$ | ||

${K}_{\mathit{NCD},\phantom{\rule{0.166667em}{0ex}}\mathbf{AA}\mathbf{1}\phantom{\rule{0.166667em}{0ex}}\mathbf{AA}\mathbf{2}}=2$ | # Parameters for PHOX | # Parameters for FERM |

${K}_{\mathit{NCD},\phantom{\rule{0.166667em}{0ex}}\mathbf{AA}\mathbf{1}\phantom{\rule{0.166667em}{0ex}}\mathbf{AA}\mathbf{2}\phantom{\rule{0.166667em}{0ex}}\mathbf{KREBS}}=2$ | ${K}_{\mathit{PHOX}}=0$ | ${K}_{\mathit{FERM}}=0$ |

${K}_{\mathit{PHOX},\phantom{\rule{0.166667em}{0ex}}\mathbf{PC}}=1$ | ${K}_{\mathit{FERM},\phantom{\rule{0.166667em}{0ex}}\mathbf{EP}}=1$ |

**Table 2.**Validation matrix for the cell metabolism regulation model. Each row (resp. column) represents an experimental condition (resp. an observable systemic variable). Thus, each box of the table formalizes the known behavior of that observable variable in that experimental condition. Grey boxes refer to missing knowledge depending on the cell type or more generally knowledge which we do not consider as consensus. Lines 16, 17, 18, 34, 35 and 36 formalize the Warburg/Crabtree effect.

Biological Context | FA | exO2 | GLC | AA | ATP (0–2) | O2 (0–1) | GLYC (0–2) | nLBP (0–1) | LBP (0–1) | FERM (0–1) | KREBS (0–2) | PHOX (0–1) | NADH (0–1) |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Neither lipids nor oxygen supply | |||||||||||||

1 | 0 | 0 | 0 | 0 | td(0) | td(0) | td(0) | td(0) | td(0) | td(0) | td(0) | td(0) | td(0) |

2 & 3 | 0 | 0 | 0 | 1 & 2 | td(0) | ||||||||

4 | 0 | 0 | 1 | 0 | !td(0) | td(0) | osc(0-1) | !td(0) | td(0) | osc | |||

5 | 0 | 0 | 1 | 1 | osc | td(0) | osc(0-1) | !td(0) | !td(0) | td(0) | osc | ||

6 | 0 | 0 | 1 | 2 | osc | td(0) | osc | !td(0) | !td(0) | !td(0) | td(0) | osc | |

7 | 0 | 0 | 2 | 0 | !td(0) | td(0) | osc | !td(0) | td(0) | osc | |||

8 | 0 | 0 | 2 | 1 | osc | td(0) | osc | !td(0) | !td(0) | td(0) | osc | ||

9 | 0 | 0 | 2 | 2 | osc | td(0) | osc | !td(0) | !td(0) | !td(0) | td(0) | osc | |

No lipids but oxygen supply | |||||||||||||

10 | 0 | 1 | 0 | 0 | td(0) | !td(0) | td(0) | td(0) | td(0) | td(0) | td(0) | td(0) | td(0) |

11 & 12 | 0 | 1 | 0 | 1 & 2 | !td(0) | ||||||||

13 | 0 | 1 | 1 | 0 | !td(0) | osc | osc | td(0) | osc | osc | osc | ||

14 | 0 | 1 | 1 | 1 | osc | osc | osc | !td(0) | td(0) | osc | osc | osc | |

15 | 0 | 1 | 1 | 2 | osc | osc | osc | !td(0) | !td(0) | td(0) | osc | osc | osc |

16 | 0 | 1 | 2 | 0 | !td(0) | !td(0) | osc | !td(0) | !td(1) | osc | |||

17 & 18 | 0 | 1 | 2 | 1&2 | osc | !td(0) | osc | !td(0) | !td(0) | !td(0) | !td(1) | osc | |

Lipids but no oxygen supply | |||||||||||||

19 | 1 | 0 | 0 | 0 | td(0) | td(0) | td(0) | td(0) | td(0) | td(0) | td(0) | td(0) | td(0) |

20 & 21 | 1 | 0 | 0 | 1 & 2 | td(0) | ||||||||

22 | 1 | 0 | 1 | 0 | osc | td(0) | osc(0-1) | !td(0) | !td(0) | td(0) | osc | ||

23 | 1 | 0 | 1 | 1 | osc | td(0) | osc(0-1) | !td(0) | !td(0) | !td(0) | td(0) | osc | |

24 | 1 | 0 | 1 | 2 | osc | td(0) | osc | !td(0) | !td(0) | !td(0) | td(0) | osc | |

25 | 1 | 0 | 2 | 0 | osc | td(0) | osc | !td(0) | !td(0) | td(0) | osc | ||

26 & 27 | 1 | 0 | 2 | 1 & 2 | osc | td(0) | osc | !td(0) | !td(0) | !td(0) | td(0) | osc | |

Lipids and oxygen supply | |||||||||||||

28 | 1 | 1 | 0 | 0 | td(0) | !td(0) | td(0) | td(0) | td(0) | td(0) | td(0) | td(0) | td(0) |

29 & 30 | 1 | 1 | 0 | 1 & 2 | !td(0) | ||||||||

31 | 1 | 1 | 1 | 0 | osc | osc | osc | !td(0) | td(0) | osc | osc | osc | |

32 | 1 | 1 | 1 | 1 | osc | osc | osc | !td(0) | !td(0) | td(0) | osc | osc | osc |

33 | 1 | 1 | 1 | 2 | osc | osc | osc | !td(0) | !td(0) | osc | osc | osc | |

34 | 1 | 1 | 2 | 0 | osc | !td(0) | osc | !td(0) | !td(0) | !td(1) | osc | ||

35 & 36 | 1 | 1 | 2 | 1 & 2 | osc | !td(0) | osc | !td(0) | !td(0) | !td(0) | !td(1) | osc |

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Gibart, L.; Khoodeeram, R.; Bernot, G.; Comet, J.-P.; Trosset, J.-Y.
Regulation of Eukaryote Metabolism: An Abstract Model Explaining the Warburg/Crabtree Effect. *Processes* **2021**, *9*, 1496.
https://doi.org/10.3390/pr9091496

**AMA Style**

Gibart L, Khoodeeram R, Bernot G, Comet J-P, Trosset J-Y.
Regulation of Eukaryote Metabolism: An Abstract Model Explaining the Warburg/Crabtree Effect. *Processes*. 2021; 9(9):1496.
https://doi.org/10.3390/pr9091496

**Chicago/Turabian Style**

Gibart, Laetitia, Rajeev Khoodeeram, Gilles Bernot, Jean-Paul Comet, and Jean-Yves Trosset.
2021. "Regulation of Eukaryote Metabolism: An Abstract Model Explaining the Warburg/Crabtree Effect" *Processes* 9, no. 9: 1496.
https://doi.org/10.3390/pr9091496