# Control Analysis of Cooperativity and Complementarity in Metabolic Regulations: The Case of NADPH Homeostasis

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

## 3. Results

#### 3.1. Distribution of Control Coefficients in Absence of Feedback Regulation

^{+}binding to G6PD provides a primary source of oxPPP flux increase in response to NADPH depletion where $0<{C}_{gr}^{J,ppp}<1$ (and $-1<{C}_{gr}^{S,nh}<0$) without the need of regulations. In addition, the main flux-controlling steps are G6PD, but also PFK1 and PGI, which is consistent with the notion that reduced enzyme activity in upper glycolysis leads to flux rerouting into oxPPP. However, some other features of the control pattern do not seem to match with expectations or experimental evidences. For instance, ${C}_{gr}^{J,ppp}$ and ${C}_{gr}^{S,nh}$ significantly decrease for flux state domains characterized with ${J}_{pgi}<0$, indicating an unlikely context-dependency of NADPH homeostasis. As well, 6PG and G6P concentration control coefficients ${C}_{gr}^{S,6pg/g6p}$ are strongly negative contradicting the numerous experimental evidences reporting a few-fold increase in 6PG and a moderate increase in G6P in response to oxidative stress [17,27,31]. These features suggest the involvement of regulation to enable ${C}_{gr}^{S,6pg/g6p}>0$ and to increase ${C}^{J,ppp}$ or ${C}^{S,nh}$ for a broader range of flux states.

#### 3.2. Feedback Inhibitions of PPP and Upper Glycolysis Synergistically Cooperate for Efficient PPP Flux Rerouting

^{+}as a cofactor of G6PD enzyme, which provides a maximum flux control of $max({C}_{gr}^{J,ppp})={S}_{nh}$ (maximum for ${J}_{ppp}=0$) (Equation (A11a) and Figure 3A). The regulation ${r}_{1}$ (NADPH-dependent inhibition of G6PD) can efficiently promote such PPP flux control to a maximum extent of ${C}_{gr}^{J,ppp}/{S}_{nh}=1+{r}_{1}$ for small enough ${S}_{nh}$ and ${J}_{ppp}$ (Equation (A11b) and Figure 3B). The regulation ${r}_{3}$ alone (6PG-dependent inhibition of PGI) does not promote PPP flux control (Equation (A11d) and Figure 3C). In sharp contrast, such allosteric regulation strongly enhances PPP flux control in presence of ${r}_{1}$ (Equations (A11e) and (A11f) and Figure 3D). This synergistic effect coincides with a positive control of G6P, which itself requires a strong positive control of 6PG mediated by ${r}_{1}$ (low panels of Figure 3B,D). The importance of a positive concentration control of 6PG and G6P is confirmed by the loss of synergistic effect for ${r}_{2}\ge {r}_{1}$ related to a loss of positive concentration control for G6P and 6PG (Figure 3E).

#### 3.3. Ros-Dependent Inhibition of Glycolytic Enzymes Expands NADPH Homeostatic Abilities

_{2}O

_{2}, a major source of ROS, directly interacts with and inhibits several glycolytic enzymes, notably GAPD and PFKFB3, through S-gluthationylation modifications. We therefore apply now metabolic control analysis to a context where the increase in GR-dependent oxidation of NADPH into NADP

^{+}is mediated by an increased production of H

_{2}O

_{2}, shifting the nature of parametric perturbation from k

_{gr}to k

_{ox}for which control coefficients are computed. The model incorporates now the H

_{2}O

_{2}-dependent oxidative inhibitions of GAPD and PFK1. In this scenario, control manifold equations can be derived (see Appendix C.2) to obtain a simple general expression for the NADPH control coefficient:

_{2}O

_{2}promotes significant NADPH homeostasis, but only for negative enough level of ${J}_{pgi}$ (compare Figure 4A–D). Enhanced NADPH homeostasis correlates with positive values of ${C}_{ox}^{S,f6p}$ and high values of ${J}_{pgi}^{-}$ consistently with the term ${J}_{pgi}^{-}{C}_{ox}^{S,f6p}$ of Equation (6). It is to note that inhibitions of PFK1 and GAPD exhibit qualitatively the same control effect. The interplay between the feedback control of G6PD and the inhibition of lower glycolysis is depicted in Figure 4E, where the upper bound for ${C}_{ox}^{S,nh}$ increases with both ${r}_{1}$ and ${C}_{gr}^{S,f6p}$ in independent manner. This interplay nevertheless requires a large bidirectional flux in PGI reaction as compared to the net flux (${J}_{pgi}^{+}+{J}_{pgi}^{-}\gg {J}_{pgi}$) as shown by plotting control coefficients onto the control manifold associated with Equation (6) (Figure 4F).

## 4. Discussion

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

MCA | Metabolic control analysis |

PPP | Pentose phosphate pathway |

ROS | Reactive oxygen species |

G6P | Glucose-6-phosphate |

F6P | Fructose-6-phosphate |

FBP | Fructose-1,6-bisphosphate |

GAP | Glyceraldehyde-3-phosphate |

6PG | 6-phosphogluconate |

R5P | Ribose 5-phosphate |

PKM2 | Pyruvate kinase muscle isozyme M2 |

PFKFB3 | Phosphofructo-2-kinase fructose-2,6-bisphosphatase-3 |

NH,NADPH | Nicotinamide adenine dinucleotide phosphate hydrogen |

N,NADP | Nicotinamide adenine dinucleotide phosphate |

OX | Oxidative stress |

GR | Glutathione reductase |

HK | Hexokinase |

G6PD | G6P dehydrogenase |

6PGD | 6PG dehydrogenase |

PRP | Phosphoribosyl pyrophosphate |

PGI | Phosphoglucose isomerase |

PFK | Phosphofructokinase (type 1) |

FBPase | Fructose-1,6-bisphosphatase |

ALD | Fructose 1,6 bisphosphate aldolase |

GAPD | GAP dehydrogenase |

TKT | Transketolase |

## Appendix A. Kinetic Model

- RPI, RPE, TKT1, TKT2, and TAL are pooled to form a single reaction for the nonoxidative branch of the PPP (S7P, E4P metabolites are not included).
- ALD and TPI are pooled (DHAP metabolite is not included).
- GP and GRX are pooled (GSSG, GSH are not included).
- Catalase reaction degrading H
_{2}O_{2}is neglected.

## Appendix B. Matrix Equation for Control Coefficients

## Appendix C. Computation of Control Manifolds

#### Appendix C.1. Control Analysis of Regulatory Crosstalk r_{1,2,3}

- All those asymptotic relations are proportional to ${S}_{nh}$, justifying the use of the normalized control coefficient ${C}_{gr}^{J,ppp}/{S}_{nh}$ (Figure 3).
- Equation (A11a) shows that a PPP flux control driven by NADPH
_{+}cofactor binding to G6PD (no regulation) has an upper bound of ${S}_{nh}$. - Equation (A11b) expresses that ${r}_{1}$ promotes PPP flux control (i) independently on ${r}_{2}$, (ii) especially for small ${S}_{nh}$.
- Equation (A11c) defines the complex nonlinear interplay between contributions of ${r}_{1}$, ${r}_{2}$, and ${r}_{3}$.
- Equation (A11d) shows indeed that ${r}_{3}$ alone, even very large, cannot increase PPP flux control.

#### Appendix C.2. Control Analysis of Regulatory Crosstalk r 1,4,5

_{2}O

_{2}or ROS (i.e., ${k}_{ox}$) where ${\pi}_{ox}={[0,0,0,0,0,0,0,0,0,0,1]}^{T}$ and ${a}_{ox}=1$ in Equation (A4). For such perturbation scheme, the relevant set of control equations becomes:

- Positive values of ${C}_{ox}^{S,f6p}$ which is increased by regulation ${r}_{4}$ following Equation (A12f), but also regulation ${r}_{5}$.
- The effect of those regulations is enhanced by large directional PGI flux ${J}_{pgi}^{-}$ from F6P to G6P.
- The effect of ${r}_{1}$ is amplified by large directional PGI flux ${J}_{pgi}^{+}$ from G6P to F6P.
- The two items above indicate that efficient regulation of NADPH homeostasis by ${r}_{1,4,5}$ requires high values of both ${J}_{pgi}^{+,-}$ with an upper bound given by:$${C}_{ox}^{s,nh}=\frac{{C}_{ox}^{S,f6p}-1}{-{\u03f5}_{1}}.$$

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**Figure 1.**Control analysis of metabolic regulation involved in NADPH homeostasis. (

**A**) A simplified metabolic network comprising the upper glycolysis and pentose phosphate pathways which includes a selected set of feedback regulation ${r}_{i}$ contributing to NADPH homeostasis. Legend is shown in inset. (

**B**) Framework combining metabolic control analysis (Equations (3) and (4)) and sampling analysis of regulatory crosstalk.

**Figure 2.**Control pattern without feedback regulation. (

**A**) Partitioning of the parameter sampling procedure in different domains of the two-dimensional polytope of possible flux configurations. (

**B**) Whisker-plot distribution of flux control ${C}_{i}^{J,ppp}$ (

**B**) and concentration control ${C}_{gr}^{S,j}$ (

**C**), obtained from parameter sampling of the 6 subdomains of the flux space (color code in (

**A**)).

**Figure 3.**Synergistic feedback regulations for PPP flux rerouting. Comparative analysis of 5 regulatory architectures schematically represented on top: (

**A**) ${r}_{1,2,3}=0$, (

**B**) ${r}_{1}=2$, (

**C**) ${r}_{3}=2$, (

**D**) ${r}_{1,3}=2$, and (

**E**) ${r}_{1,2,3}=2$. Middle and bottom panels represents the ${C}_{gr}^{J,ppp}/{S}_{nh}$ and ${C}_{gr}^{S,g6p}$ as function of ${C}_{gr}^{S,6pg}$, obtained from random sampling of kinetic parameters. Color code indicates different classes of behavior (Red: ${C}_{gr}^{S,6pg}>0$ and ${C}_{gr}^{S,g6p}<0$; Orange: ${C}_{gr}^{S,g6p}>0$, blue otherwise.) (

**F**) For the architecture of panel (

**D**), ${C}_{gr}^{J,ppp}/{S}_{nh}$ as function of ${J}_{ppp}$. (

**G**) For the architecture of panel (

**D**), ${C}_{gr}^{J,ppp}$ as function of ${S}_{nh}$ and ${C}_{gr}^{S,g6p}>0$ mapped onto the manifold related to Equation (A9). (

**H**) Scheme based on Equations (5) and (A8) (colored arrows) recapitulating the interplay of ${r}_{1}$, ${r}_{2}$, and ${r}_{3}$ and key steady-state variables on the control associated with NADPH homeostasis.

**Figure 4.**Complementary feedback regulation for NADPH homeostasis. Sampling space is restricted to the criteria ${S}_{nh}={S}_{n}$. (

**A**–

**D**) ${C}_{ox}^{S,nh}$ as function of ${J}_{pgi}$ for a random sampling of kinetic models where ${S}_{n}={S}_{nh}$. for 4 regulatory architecture ((

**A**): ${r}_{i}=0$, (

**B**): ${r}_{1}=5$, (

**C**): ${r}_{1}=5$, ${r}_{4}=2$, (

**D**): ${r}_{1}=5$, ${r}_{5}=2$). Orange colors are associated with sampled model satisfying ${J}_{pgi}^{-}>1$ and ${C}^{S,f6p}>0$. (

**E**) ${C}_{ox}^{S,nh}$ as function ${C}_{ox}^{S,f6p}$ for random model sampling, highlighting the maximal bound for NADPH homeostasis. (

**F**) ${C}_{ox}^{S,nh}$ as function ${C}_{ox}^{S,f6p}$ and ${J}_{pgi}^{-/+}$ where random model sampling are shown on the manifold obtained from Equation (6), intersected with the condition ${J}_{pgi}=0$. (

**G**) Scheme based on Equations (6) and (A12) (colored arrows) recapitulating the interplay of ${r}_{1}$, ${r}_{4}$ and ${r}_{5}$ and key steady-state variables on the NADPH homeostasis control coefficient.

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**MDPI and ACS Style**

Pfeuty, B.; Hurbain, J.; Thommen, Q.
Control Analysis of Cooperativity and Complementarity in Metabolic Regulations: The Case of NADPH Homeostasis. *Metabolites* **2023**, *13*, 485.
https://doi.org/10.3390/metabo13040485

**AMA Style**

Pfeuty B, Hurbain J, Thommen Q.
Control Analysis of Cooperativity and Complementarity in Metabolic Regulations: The Case of NADPH Homeostasis. *Metabolites*. 2023; 13(4):485.
https://doi.org/10.3390/metabo13040485

**Chicago/Turabian Style**

Pfeuty, Benjamin, Julien Hurbain, and Quentin Thommen.
2023. "Control Analysis of Cooperativity and Complementarity in Metabolic Regulations: The Case of NADPH Homeostasis" *Metabolites* 13, no. 4: 485.
https://doi.org/10.3390/metabo13040485