# An Automated Model Reduction Method for Biochemical Reaction Networks

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## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

#### 2.1. Mathematical Models of Reaction Networks

#### 2.2. Kron Reduction of Mathematical Models Corresponding to Reaction Networks

**Definition**

**1.**

## 3. Equivalent Mathematical Models

**Definition**

**2.**

**Definition**

**3.**

**Lemma**

**1.**

**Proof.**

**Remark**

**1.**

**Proposition**

**1.**

**Proof.**

**Remark**

**2.**

- M1.
- Elimination of certain species from the network (rewriting the corresponding concentrations in terms of the concentrations of the other species), after which certain complexes are composed of the same species.
- M2.
- Application of Lemma 1 in order to make such complexes identical to each other.
- M3.
- Joining the reactions with identical complexes into a single linkage class.

## 4. Automatic Reduction Procedure

- The complex composition matrix $Z\in {\mathbb{R}}^{m\times c}$, where m and c are the number of the species and complexes of the network, respectively.
- The incidence matrix $B\in {\mathbb{R}}^{c\times r}$ of the network, where r is the number of reactions of the network.
- The vector $k\in {\mathbb{R}}_{+}^{r}$ of rate constants of the reactions.
- The vector $d\left(x\right)\in {\mathbb{R}}_{+}^{r}$ of rational terms in the expressions of reaction rates.
- The vector ${x}_{0}\in {\mathbb{R}}_{+}^{m}$ of initial concentrations.
- The threshold value of the error integral (9), i.e., the maximum admissible value of E.

- The mathematical model and the complex graph corresponding to the original network.
- The mathematical model and the complex graph corresponding to the reduced network.
- The final value of the error integral.

- Step 1: Mathematical model of the network.

- Step 2: Settling time of the network.

- Step 3: Selecting species to be eliminated from the network.

- Step 4: Mathematical model of the equivalent network.

- M1.
- As the eliminated species are no longer participating in the equivalent network, we delete the corresponding rows from Z. Subsequently, the vector $\tilde{x}\in {\mathbb{R}}^{\tilde{m}}$ of species’ concentrations is defined by deleting the $\beta $th, $\beta \in I$, element of x. Similarly, we obtain the vector ${\tilde{x}}_{0}\in {\mathbb{R}}^{\tilde{m}}$ of initial concentrations of the equivalent network. The vector $\tilde{d}\left(\tilde{x}\right)$ of the rational functions can be derived from $d\left(x\right)$ in the following way. For every $\beta \in I$, if the species ${X}_{\beta}$ is participating in the substrate complex of jth reaction, then ${\tilde{d}}_{j}\left(\tilde{x}\right)={x}_{\beta}^{-{S}_{\beta j}}{d}_{j}\left(x\right)$, which ensures that the reaction fluxes of the network obtained at this step still obey the Equation (6). For the network (12), it is clear that the species ${X}_{3}$ is participating in the substrate complex of the second reaction. After rewriting the concentration function ${x}_{3}$ in terms of concentrations of the other species, we multiply ${d}_{2}$ by ${x}_{3}$. Thus, ${\tilde{d}}_{2}\left(\tilde{x}\right)={x}_{3}{d}_{2}\left(x\right)$. Similarly, we have ${\tilde{d}}_{3}\left(\tilde{x}\right)={x}_{5}{d}_{3}\left(x\right)$, ${\tilde{d}}_{4}\left(\tilde{x}\right)={x}_{6}{d}_{4}\left(x\right)$, and ${\tilde{d}}_{5}\left(\tilde{x}\right)={x}_{8}{d}_{5}\left(x\right)$. We therefore write the reactions of the network (12) in the following form.$$\begin{array}{ccc}\hfill {X}_{1}+{X}_{2}& \u2942& 2{X}_{4}\hfill \\ \hfill 3{X}_{4}& \u2942& {X}_{7}\hfill \\ \hfill 5{X}_{7}& \u2942& {X}_{9}+{X}_{10}\hfill \end{array}$$
- M2.
- After the elimination of species, the columns of Z corresponding to the complexes that share species become multiples of each other. Consequently, in order to make these columns identical to each other, we multiply each of them by the corresponding constant from (11). Likewise, we divide each rate constant ${k}_{j}$ by the corresponding constant given in (11). In the case of the network (12), the vector of rate constants of its corresponding equivalent network is $\tilde{k}=\left[\frac{{k}_{1}}{15},\frac{{k}_{-1}}{30},\frac{{k}_{2}}{30},\frac{{k}_{-2}}{10},\frac{{k}_{3}}{10},\frac{{k}_{-3}}{2}\right]$. We therefore rewrite the reactions (14) in the following form.$$\begin{array}{ccc}\hfill 15{X}_{1}+15{X}_{2}& \u2942& 30{X}_{4}\hfill \\ \hfill 30{X}_{4}& \u2942& 10{X}_{7}\hfill \\ \hfill 5{X}_{7}& \u2942& 2{X}_{9}+2{X}_{10}\hfill \end{array}$$
- M3.
- We now delete the duplicate columns of Z and keep only one of them in order to find the complex composition matrix $\tilde{Z}$ of the equivalent network. Suppose that $\tilde{Z}\in {\mathbb{R}}^{\tilde{m}\times \tilde{c}}$, where $\tilde{m}$ and $\tilde{c}$ are the number of species and complexes in the equivalent model, respectively. It is clear that, if $n\left(I\right)={n}_{1}$ and the number of duplicate columns is ${n}_{2}$, then $\tilde{m}=m-{n}_{1}$ and $\tilde{c}=c-{n}_{2}+1$.Let the pth and qth complexes be a pair of identical complexes. We first replace the pthrow of B with$$\begin{array}{c}\hfill {\tilde{B}}_{pj}=\left\{\begin{array}{cc}{B}_{qj},\hfill & \mathrm{if}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}{B}_{pj}=0\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\mathrm{and}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}{B}_{qj}\ne 0,\hfill \\ {B}_{pj},\hfill & \mathrm{otherwise},\hfill \end{array}\right.\end{array}$$The complex composition matrix $\tilde{Z}\in {\mathbb{R}}^{6\times 4}$ and the incidence matrix $\tilde{B}\in {\mathbb{R}}^{4\times 6}$ of the equivalent network corresponding to the network (12) are$$\begin{array}{cc}\hfill \tilde{Z}=\left[\begin{array}{cccc}15& 0& 0& 0\\ 15& 0& 0& 0\\ 0& 30& 0& 0\\ 0& 0& 10& 0\\ 0& 0& 0& 2\\ 0& 0& 0& 2\end{array}\right],& \tilde{B}=\left[\begin{array}{cccccc}-1& 1& 0& 0& 0& 0\\ 1& -1& -1& 1& 0& 0\\ 0& 0& 1& -1& -1& 1\\ 0& 0& 0& 0& 1& -1\end{array}\right].\hfill \end{array}$$

- Step 5: Independent subnetworks.

- Step 6: Selecting complexes for deletion.

- Step 7: Mathematical model of the reduced network.

## 5. Application to Real-Life Reaction Networks

#### 5.1. Neural Stem Cell Regulation

#### 5.2. Hedgehog Signaling Pathway

## 6. Conclusions and Discussion

## Supplementary Materials

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

CRN | biochemical reaction networks |

HSP | hedgehog signaling pathway |

NSCR | neural stem cell regulation |

ODE | ordinary differential equation |

QSSA | quasi-steady-state assumption |

ADP | adenosine diphosphate |

ATP | adenosine triphosphate |

## Appendix A. Detailed Explanation of the Reduction Procedure

#### Appendix A.1. Reduction of Neural Stem Cell Regulation

**Table A1.**Quantitative comparison of the original model and the reduced model of neural stem cell regulation.

Species | Reactions | |
---|---|---|

Original Model | 21 | 10 |

Reduced Model | 14 | 7 |

#### Appendix A.2. Reduction of Hedgehog Signaling Pathway

**Table A2.**Quantitative comparison of the original model and the reduced model of hedgehog signaling pathway.

Species | Reactions | |
---|---|---|

Original Model | 21 | 11 |

Reduced Model | 14 | 7 |

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**Figure 1.**The left-hand panel is the complex graph corresponding to the original model of neural stem cell regulation. The complex graph corresponding to the equivalent model obtained after eliminating the species ${X}_{5}$, ${X}_{6}$, and ${X}_{20}$ from the original model is given in the middle panel. Deletion of the intermediate complexes ${X}_{4}$, ${X}_{8}$, ${X}_{13}$, and ${X}_{21}$ leads to a reduced network with the corresponding complex graph represented in the right-hand panel. The difference between the original model and the reduced model, as measured by the error integral, is 4.85%.

**Figure 2.**Concentrations of the important species of neural stem cell regulation in the original model and in the reduced model. The difference between these models, as measured by the error integral, is 4.85%.

**Figure 3.**The left-hand panel is the complex graph of the original model of hedgehog signaling pathway. The complex graph corresponding to the equivalent model obtained after eliminating the species ${X}_{1}$, ${X}_{11}$, and ${X}_{12}$ from the original model is given in the middle panel. Deletion of the intermediate complexes ${X}_{4}$, ${X}_{8}$, ${X}_{13}$, and ${X}_{18}$ leads to a reduced model with the corresponding complex graph shown in the right-hand panel. The difference between the original model and the reduced model, as measured by the error integral, is 6.59%.

**Figure 4.**Concentrations of the important species of hedgehog signaling pathway in the original model and in the reduced model. The difference between these models, as measured by the error integral, is 6.59%.

**Table 1.**The amount of deleted species and the value of the error integral (in%) corresponding to each example after its model reduction using our method.

Model | Deleted Species (%) | Error Integral (%) |
---|---|---|

NSCR | 33.33 | 4.85 |

HSP | 33.33 | 6.59 |

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Gasparyan, M.; Van Messem, A.; Rao, S.
An Automated Model Reduction Method for Biochemical Reaction Networks. *Symmetry* **2020**, *12*, 1321.
https://doi.org/10.3390/sym12081321

**AMA Style**

Gasparyan M, Van Messem A, Rao S.
An Automated Model Reduction Method for Biochemical Reaction Networks. *Symmetry*. 2020; 12(8):1321.
https://doi.org/10.3390/sym12081321

**Chicago/Turabian Style**

Gasparyan, Manvel, Arnout Van Messem, and Shodhan Rao.
2020. "An Automated Model Reduction Method for Biochemical Reaction Networks" *Symmetry* 12, no. 8: 1321.
https://doi.org/10.3390/sym12081321