# 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 |

## References

- Gerdtzen, Z.P.; Daoutidis, P.; Hu, W.-S. Non-linear reduction for kinetic models of metabolic reaction networks. Metab. Eng.
**2004**, 6, 140–154. [Google Scholar] [CrossRef] - Briggs, G.E.; Sanderson Haldane, J.B. A note on the kinetics of enzyme action. Biochem. J.
**1925**, 19, 338. [Google Scholar] [CrossRef] [Green Version] - Michaelis, L.; Menten, M.L. Die Kinetik der Invertinwirkung. Biochem Z
**1913**, 49, 333–369. [Google Scholar] - Nelson, D.L.; Cox, M.M. Hormonal regulation of food metabolism. In Lehninger Principles of Biochemistry, 4th ed.; WH Freeman: New York, NY, USA, 2005; pp. 881–992. [Google Scholar]
- Sontag, E.D. Lecture Notes on Mathematical Systems Biology; Northeastern University: Boston, MA, USA, 2005. [Google Scholar]
- Snowden, T.J.; van der Graaf, P.H.; Tindall, M.J. Methods of model reduction for large-scale biological systems: A survey of current methods and trends. Bull. Math. Biol.
**2017**, 79, 1449–1486. [Google Scholar] [CrossRef] [Green Version] - Costa, R.S.; Rocha, I.; Ferreira, E.C. Model Reduction Based on Dynamic Sensitivity Analysis: A Systems Biology Case of Study. 2008. Available online: http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.619.3225 (accessed on 8 May 2020).
- Zi, Z. Sensitivity analysis approaches applied to systems biology models. IET Syst. Biol.
**2011**, 5, 336–346. [Google Scholar] [CrossRef] - Kuo, J.C.W.; Wei, J. Lumping analysis in monomolecular reaction systems. analysis of approximately lumpable system. Ind. Eng. Chem. Fund.
**1969**, 8, 124–133. [Google Scholar] [CrossRef] - Wei, J.; Kuo, J.C.W. Lumping analysis in monomolecular reaction systems. analysis of the exactly lumpable system. Ind. Eng. Chem. Fund.
**1969**, 8, 114–123. [Google Scholar] [CrossRef] - Okeke, B.E. Lumping Methods for Model Reduction. Ph.D. Thesis, University of Lethbridge, Lethbridge, AB, Canada, 2013. [Google Scholar]
- Pepiot-Desjardins, P.; Pitsch, H. An automatic chemical lumping method for the reduction of large chemical kinetic mechanisms. Combust. Theory Model.
**2008**, 12, 1089–1108. [Google Scholar] [CrossRef] - Okino, M.S.; Mavrovouniotis, M.L. Simplification of mathematical models of chemical reaction systems. Chem. Rev.
**1998**, 98, 391–408. [Google Scholar] [CrossRef] [PubMed] - Flach, E.; Schnell, S. Use and abuse of the quasi-steady-state approximation. IEE Proc.-Syst. Biol.
**2006**, 153, 187–191. [Google Scholar] [CrossRef] [Green Version] - Khalil, H.K.; Grizzle, J.W. Nonlinear Systems; Prentice Hall: Upper Saddle River, NJ, USA, 2002; p. 3. [Google Scholar]
- Tikhonov, A.N. Systems of differential equations containing small parameters in the derivatives. Matematicheskii Sbornik
**1952**, 73, 575–586. [Google Scholar] - Kuntz, J.; Oyarzún, D.; Stan, G.-B. Model reduction of genetic-metabolic networks via time scale separation. In A Systems Theoretic Approach to Systems and Synthetic Biology I: Models and System Characterizations; Springer: Berlin, Germany, 2014; pp. 181–210. [Google Scholar]
- Duan, Z.; Cruz Bournazou, M.N.; Kravaris, C. Dynamic model reduction for two-stage anaerobic digestion processes. Chem. Eng. J.
**2017**, 327, 1102–1116. [Google Scholar] [CrossRef] - Belgacem, I.; Casagranda, S.; Grac, E.; Ropers, D.; Gouzé, J.-L. Reduction and stability analysis of a transcription–translation model of RNA polymerase. Bull. Math. Biol.
**2018**, 80, 294–318. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Karhunen, K. Über Lineare Methoden in der Wahrscheinlichkeitsrechnung. Sana
**1947**, 37. [Google Scholar] - Loeve, M. Elementary probability theory. In Probability Theory I; Springer: Berlin, Germany, 1977; pp. 1–52. [Google Scholar]
- Brunton, S.L.; Kutz, J.N. Data-Driven Science and Engineering: Machine Learning, Dynamical Systems, and Control; Cambridge University Press: Cambridge, UK, 2019. [Google Scholar]
- Golub, G.H.; Reinsch, C. Singular value decomposition and least squares solutions. In Linear Algebra; Springer: Berlin, Germany, 1971; pp. 134–151. [Google Scholar]
- Strang, G. Introduction to Linear Algebra; Wellesley-Cambridge Press: Wellesley, MA, USA, 1993; Volume 3. [Google Scholar]
- Chaturantabut, S.; Sorensen, D. Nonlinear model reduction via discrete empirical interpolation. SIAM J. Sci. Comput.
**2010**, 32, 2737–2764. [Google Scholar] [CrossRef] - Covert, M.W.; Schilling, C.H.; Palsson, B. Regulation of gene expression in flux balance models of metabolism. J. Theor. Biol.
**2001**, 213, 73–88. [Google Scholar] [CrossRef] [Green Version] - Beattieand, C.; Gugercin, S.; Mehrmann, V. Model reduction for systems with inhomogeneous initial conditions. Syst. Control Lett.
**2017**, 99, 99–166. [Google Scholar] [CrossRef] [Green Version] - Chaturantabut, S.; Sorensen, D. A state space error estimate for POD-DEIM nonlinear model reduction. SIAM J. Numer. Anal.
**2012**, 50, 46–63. [Google Scholar] [CrossRef] - Afanasiev, K.; Hinze, M. Adaptive control of a wake flow using proper orthogonal decomposition1. Lect. Notes Pure Appl. Math.
**2001**, 1, 216. [Google Scholar] [CrossRef] - Stanford, N.J.; Lubitz, T.; Smallbone, K.; Klipp, E.; Mendes, P.; Liebermeister, W. Systematic Construction of Kinetic Models from Genome-Scale Metabolic Networks. PLoS ONE
**2013**, 8, e79195. [Google Scholar] [CrossRef] [Green Version] - Smallbone, K.; Mendes, P. Large-scale metabolic models: From reconstruction to differential equations. Ind. Biotechnol.
**2013**, 9, 179–184. [Google Scholar] [CrossRef] [Green Version] - Badreddine, A.-A.; Henrik, P. Simulation of switching phenomena in biological systems. In Biochemical Engineering for 2001; Springer: Berlin, Germany, 1992; pp. 701–704. [Google Scholar]
- Kremling, A.; Kremling, S.; Bettenbrock, K. Catabolite repression in escherichia coli—a comparison of modelling approaches. FEBS J.
**2009**, 276, 594–602. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Rosa, C.S.; Boris, L.; Rudy, E. Stability Preservation in Projection-based Model Order Reduction of Large Scale Systems. Eur. J. Control
**2012**, 18, 122–132. [Google Scholar] - Roland, P. Stability preservation in Galerkin-type projection-based model order reduction. arXiv
**2017**, arXiv:1711.02912. [Google Scholar] - Gesztelyi, R.; Zsuga, J.; Kemeny-Beke, A.; Varga, B.; Juhasz, B.; Tosaki, A. The Hill Equation and the Origin of Quantitative Pharmacology; Springer: Berlin, Germany, 2012; Volume 66, pp. 427–438. [Google Scholar]
- Hill, A.V. The possible effects of the aggregation of the molecules of hæmoglobin on its dissociation curves. J. Physiol.
**1910**, 40, i–vii. [Google Scholar] - Guldberg, C.M.; Waage, P. Etudes sur Les Affinités Chimiques; Brøgger & Christie: Oslo, Norway, 1867. [Google Scholar]
- Raue, A.; Steiert, B.; Schelker, M.; Kreutz, C.; Maiwald, T.; Hass, H.; Vanlier, J.; Tönsing, C.; Adlung, L.; Engesser, R.; et al. Data2Dynamics: A modeling environment tailored to parameter estimation in dynamical systems. Bioinformatics
**2015**, 31, 3558–3560. [Google Scholar] [CrossRef] [Green Version] - Raue, A.; Schilling, M.; Bachmann, J.; Matteson, A.; Schelke, M.; Kaschek, D.; Hug, S.; Kreutz, C.; Harms, B.D.; Theis, F.J.; et al. Lessons Learned from Quantitative Dynamical Modeling in Systems Biology. PLoS ONE
**2013**, 8, 1932–6203. [Google Scholar] [CrossRef]

**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