# Model Reduction for Kinetic Models of Biological Systems

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## Abstract

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## 1. Introduction

## 2. Proper Orthogonal Decomposition for Differential Equations

## 3. Application of the POD-DEIM Approach to Kinetic Model Examples

#### 3.1. Kinetic Model of the Metabolic-Genetic Network

#### Application of POD-DEIM to the Diauxic-Switch Scenario

`ode23s`with tolerances $RTOL=ATOL={10}^{-6}$ to compute a numerical solution of the ODE system (A2) with parameters as given in Table A2 in the time interval [0, 5] hr using a non-equidistant output time-grid. The initial concentrations of internal metabolites are assumed to be $(A,B,C,D,E,F,G,H,ATP,NADH,{O}_{2})=(0,1,0,0,0,0.2,0,0.03,6,5,0)$. The initial concentrations of external metabolites are taken from [26] for the different scenarios. This numerical solution yields the snapshot matrix $\mathbb{X}$ and the nonlinear snapshot matrix $\mathbb{F}$. We used the MATLAB function

`svd`to calculate the singular value decomposition of $\mathbb{X}$ and $\mathbb{F}$. The singular values are depicted in Figure 2.

`timeit`.

#### 3.2. Kinetic Model of the Yeast Metabolic Network

`ode23tb`with the default setting and initial values taken from the database model. The behavior of the singular values of the snapshot matrix is depicted in Figure 7. We could observe a fast decay in the singular values with a large number of values of magnitude ${10}^{-12}$ indicating that neglecting these values would not result in any considerable loss of information in the reduced order model.

#### 3.3. Kinetic Model of the E. coli Metabolic Network

`ode23tb`setting $ATOL={10}^{-3}$ and the initial values taken from the database model. The behavior of the singular values of the snapshot matrix is depicted in Figure 10.

## 4. POD for Kinetic Models With Different Initial Conditions

#### Application to the Kinetic Model of a Metabolic-Genetic Network

`ode23s`with tolerances $RTOL=ATOL={10}^{-6}$ and initial conditions

`ode23s`with tolerances $RTOL=ATOL={10}^{-6}$ and initial conditions

`svd`to calculate the singular value decompositions of $\mathbb{Z}=\left[\mathbb{X}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\mathbb{Y}\right]$. The singular values are depicted in Figure 13.

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

#### Appendix A.1. Derivation of Kinetic Model Equations

**Table A1.**Regulatory rules for the metabolic network Figure 1.

Regulatory Proteins | Transcriptional Regulation | ||
---|---|---|---|

$RP{c}_{1}$ | IF (${C}_{1}$) | $tT{c}_{2}$ | IF NOT ( $RP{c}_{1}$) |

$RPh$ | IF (${\nu}_{Th}>0$) | $tR8a$ | IF NOT ( $RPh$ ) |

$RPb$ | IF (${\nu}_{R2b}>0$) | $tR2a,tR7$ | IF NOT ( $RPb$ ) |

$RP{O}_{2}$ | IF Not (${O}_{ext}$) | $tRres,tR5a$ | IF NOT ($RP{O}_{2}$) |

#### Appendix A.2. Model Fitting

**Table A2.**The set of all parameters of the model that are estimated of two datasets of the diauxic switch and aerobic/anaerobic-diauxie scenarios using the D2D toolbox.

Constant Rates | Value | Unit | |
---|---|---|---|

${K}_{mc1}$ | $0.38$ | mM | estimated |

${K}_{mc2}$ | $0.38$ | mM | estimated |

${K}_{mf}$ | $6.2$ | mM | estimated |

${K}_{m{o}_{2}}$ | $5.6\xb7{10}^{-5}$ | mM | estimated |

${K}_{md}$ | $1.1\xb7{10}^{-4}$ | mmol/gDW | estimated |

${K}_{me}$ | ${10}^{-5}$ | mmol/gDW | estimated |

${K}_{mh}$ | 41 | mM | estimated |

${\tilde{k}}_{1}$ | 980 | mmol^{3}/gDW^{3} · h^{−1} | estimated |

${\tilde{k}}_{2}$ | 1000 | mmol^{2}/gDW^{2} · h^{−1} | estimated |

${\tilde{k}}_{3}$ | 300 | mmol/gDW · h^{−1} | estimated |

${\tilde{k}}_{4}$ | 140 | mmol^{3}/gDW^{3} · h^{−1} | estimated |

${\tilde{k}}_{5}$ | 23 | mmol/gDW · h^{−1} | estimated |

${\tilde{k}}_{6}$ | 25 | mmol/gDW · h^{−1} | estimated |

${\tilde{k}}_{7}$ | $1.6$ | mmol/gDW· h^{−1} | estimated |

${\tilde{k}}_{8}$ | 1000 | mmol/gDW· h^{−1} | estimated |

${\tilde{k}}_{9}$ | 13 | mmol/gDW· h^{−1} | estimated |

${\tilde{k}}_{10}$ | $2.9$ | mmol^{2}/gDW^{2}· h^{−1} | estimated |

${\tilde{k}}_{11}$ | 150 | mmol^{3}/gDW^{3} · h^{−1} | estimated |

${\tilde{k}}_{12}$ | $7.9\xb7{10}^{-5}$ | mmol/gDW· h^{−1} | estimated |

${\tilde{k}}_{13}$ | 170 | mmol^{2}/gDW^{2} · h^{−1} | estimated |

$\zeta $ | ${10}^{-5}$ | mM | estimated |

$\gamma $ | $0.024$ | mmol/gDW | estimated |

$\beta $ | $0.11$ | mM | estimated |

$\alpha $ | 290 | mmol/gDW | estimated |

$\omega $ | 1 | gDW/mmol | assumed |

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**Figure 1.**A network of chemical reactions of the core carbon metabolic network [26].

**Figure 2.**Singular values of snapshot matrix $\mathbb{X}$ (left) and nonlinear snapshot matrix $\mathbb{F}$ (right) for the simple metabolic-genetic network example of [26].

**Figure 3.**Comparison of some of the model components in the original model and the reduced order models (proper orthogonal decomposition (POD) and POD-discrete empirical interpolation method (DEIM)) of dimension $k=16$, $\ell =16$.

**Figure 4.**Comparison of some of the model components in the original model and the reduced order models (POD and POD-DEIM) of dimension $k=15$, $\ell =16$.

**Figure 5.**Comparison of some of the model components in the original model and the reduced order models (POD and POD-DEIM) of dimension $k=14$, $\ell =16$.

**Figure 6.**Comparison of some of the model components in the original model and the reduced order models (POD and POD-DEIM) of dimension $k=14$, $\ell =13$.

**Figure 8.**Comparison of the behavior of some metabolites in the yeast model for the original and the POD reduced model (Scenario 1).

**Figure 9.**Comparison of the behavior of some metabolites in the yeast model for the original and the POD reduced model (Scenario 2).

**Figure 10.**Singular values of snapshot matrix $\mathbb{X}$ for the kinetic model of the E. coli metabolic network of [31].

**Figure 11.**Comparison of the behavior of some metabolites in the yeast model for the original and the POD reduced model.

**Figure 12.**Comparison of the behavior of some metabolites in the yeast model for the original and the POD reduced model.

**Figure 13.**Singular values of snapshot matrix $\mathbb{Z}$ for the simple metabolic-genetic network example of [26].

**Figure 14.**Comparison of the behavior of some metabolites in the diauxic switch scenario for the original and the POD reduced model.

**Figure 15.**Comparison of the behavior of some metabolites in the aerobic/anaerobic-diauxie scenario for the original and the POD reduced model.

Original Model | POD ROM | POD-DEIM ROM | |
---|---|---|---|

Scenario 1 ($k=16$, $\ell =16$) | 0.3 s | 1.4 s | 0.7 s |

Scenario 2 ($k=15$, $\ell =16$) | 0.3 s | 1.4 s | 0.6 s |

Scenario 3 ($k=14$, $\ell =16$) | 0.3 s | 1.9 s | 0.9 s |

Scenario 4 ($k=14$, $\ell =13$) | 0.3 s | 1.9 s | 0.9 s |

Original Model | POD ROM | |
---|---|---|

Scenario 1 ($k=45$) | 0.28 s | 0.20 s |

Scenario 2 ($k=40$) | 0.28 s | 0.46 s |

Original Model | POD ROM | |
---|---|---|

Scenario ($k=45$) | 0.10 s | 0.014 s |

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Ali Eshtewy, N.; Scholz, L.
Model Reduction for Kinetic Models of Biological Systems. *Symmetry* **2020**, *12*, 863.
https://doi.org/10.3390/sym12050863

**AMA Style**

Ali Eshtewy N, Scholz L.
Model Reduction for Kinetic Models of Biological Systems. *Symmetry*. 2020; 12(5):863.
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**Chicago/Turabian Style**

Ali Eshtewy, Neveen, and Lena Scholz.
2020. "Model Reduction for Kinetic Models of Biological Systems" *Symmetry* 12, no. 5: 863.
https://doi.org/10.3390/sym12050863