# Construction of a Genome-Scale Kinetic Model of Mycobacterium Tuberculosis Using Generic Rate Equations

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methods

#### 2.1. Enzyme Kinetics and Rate Equations

_{1}, a

_{2}, ...) and the concentrations of products are represented by a vector b = (b

_{1}, b

_{2}, ...). The flux v(a, b) is defined as:

_{Ai}represents the parameter (substrate constants) of the i

^{th}substrate and K

_{Bj}that of the j

^{th}product of the reaction, e

_{0}is the concentration of enzyme, V

^{+}is the substrate turnover rate and V

^{-}is the product turnover rate.

#### 2.2. Parameter Estimation

#### 2.3. Parameter Variability Analysis (PVA)

**Figure 1**. After each run of estimation, the objective function, which is a measure of the fit of the estimation to the original data, and kinetic parameter values are stored in a data file in a tabbed-delimited format. The results of PVA can then be exported to spreadsheet software and statistically analysed. The PVA function has now been fully integrated into the GRaPe software.

## 3. Results

#### 3.1. The Genome-Scale Kinetic Model of Mycobacterium Tuberculosis

#### 3.2. Parameter Estimation

#### 3.3. Model Validation

**Figure 2.**Main response of Mycobacterium tuberculosis to glycerol uptake rates at 0, 0.5 and 1.0 mmol/gDW/h. The network shows a selected set of reactions in the central metabolic pathways of M. tuberculosis. Reactions are represented using arrows and the positive direction of flux is indicated by the direction of the arrow (reaction reversibility is not represented). Numbers next to the arrows indicate flux values with black, blue and red colours corresponding to a glycerol uptake rate of 0, 0.5 and 1.0 mmol/gDW/h respectively; black numbers starting with R represent reaction numbers.

#### 3.4. Parameter Variability Analysis

_{f}, the velocity of the forward reaction, is the most constrained parameter having the smallest standard deviation (Table 1). Since V

_{f}is directly related to the amount of enzyme and the expression level of the corresponding gene(s), it is expected to be more tightly linked to a particular condition, thus more constrained by a given flux distribution.

**Figure 3.**Relative changes in flux with changes in glycerol consumption rate. The response of selected reactions when the glycerol uptake rate is at 0 and 0.5 mmol/gDW/h is compared in Experiment A, between 0 and 1 mmol/gDW/h in Experiment B, and between 0.5 and 1 mmol/gDW/h in Experiment C. Only reactions with an absolute change greater than 10 % (0.1) and an absolute flux greater than 0.01 are shown. Green indicates a decrease and red indicates an increase in flux. Reactions identifiers are the same as in the Beste model.

**Figure 4.**Average parameter values and standard deviations of estimated kinetic parameters after repeating the genetic algorithm 100 times. The parameters were classified into five reaction types: uni-uni (black), uni-bi (red), bi-uni (blue), bi-bi (purple) and convenience kinetics (green). Both axes of the graph are in logarithmic scale. Vf and Vr are the forward and backward reaction velocity, respectively; KmA, KmB, KmP and KmQ are Michaelis constants; KiA, KiB, KiP and KiQ are dissociation constants; KA, KB, KP and KQ are parameters of the convenience kinetics as defined in Equation (1).

**Table 1.**Average parameter values and standard deviation (Stdev) for the most constrained parameters in logarithmic scale over 100 iterations of parameter variability analysis (PVA). Reactions are of type uni-uni, uni-bi, bi-uni, bi-bi, or convenience kinetics (CK).

Most constrained parameters | |||
---|---|---|---|

Parameter | Reaction type | Average | Stdev |

V_{f} | CK | -0.78 | 2.35 |

V_{f} | uni-bi | -1.39 | 1.41 |

V_{f} | bi-uni | -1.71 | 1.31 |

V_{f} | bi-bi | -1.31 | 1.75 |

^{-4}and limiting the data points to three decimal places in the input dataset. The objective function is the summed squared mean distance measured between the simulated data and input data. Reducing the objective function increased computing time but improves the quality of the parameter fit to input data. We performed an experiment to determine the relationship between the value of the objective function and the time taken to compute PVA for one reaction with two substrates, two products, one enzyme and six kinetic parameters (Figure 5). The results of this experiment indicate that the computing time for parameter estimation increases significantly when the objective function is reduced to 10

^{-10}and beyond. The relationship that is observed between the objective function and computing time appears to be linear (PVA was computed on a desktop computer with a quad CPU having 3.00 GHz, 2.99 GHz processor speed and 4 GB of RAM).

**Figure 5.**Computing times of parameter variability analysis (PVA) against changes in objective function. PVA was performed for a reaction with two substrates, two products, one enzyme and six kinetic parameters. For each PVA run, the summed squared mean distance measured between the simulated data and input data, known as the objective function, was set and the time taken to compute PVA results (running the genetic algorithm 100 times) was recorded. The results indicate a linear relation between the objective function and the computing time until the limits of computational precision are reached. Both axes of the graph are in logarithmic scale.

**Figure 6.**Relationship between number of input data points and computing time. PVA was performed for a single reaction of two substrates, two products, one enzyme and six kinetic parameters. PVA was repeated six times and for each iteration the number of data points in the input dataset for parameter estimation was increased from 3 to 30. The results show a rising curve in a non-linear shape.

#### 3.5. Validation on Model Integrity

^{-4}. The range of objective functions observed for individual reactions was between 10

^{-8}and 10

^{‑20}. After parameter estimation, three steady-state analyses were performed with glycerol uptake at 0, 0.5 and 1 mmol/gDW/h using COPASI.

_{f}and V

_{r}), which can vary with different expression of the corresponding enzymes, should be allowed to vary in different conditions, whereas other parameters should remain the same. It is not currently possible to specify different levels of parameter constraints for different conditions in GRaPe, but this possibility may be added in the future.

## 4. Discussion

_{f}, the velocity of the forward reaction, is the most constrained parameter. The rest of the parameters in our model exhibit a high degree of redundancy. Banga [13] suggests that global optimisation methods are needed in an attempt to avoid finding local solutions. Additionally, there are suggestions indicating that due to the stochastic nature of biological systems, parameter estimation must account for this degree of stochasticity [33]. Reducing the value of the objective function in parameter estimation improves the quality of the kinetic parameters. However, we observed a significant increase in computing time when the objective function was reduced beyond 10

^{-8}. The compromise between computing time and more precise parameter values must always be considered when performing parameter estimation. Furthermore, our results also show that computing time increases non-linearly with the number of data points in the parameter estimation training data. When parameter estimation is being carried out for a system in steady-state, the number of data points can be reduced to lower the computing time.

## 4. Conclusions

## Supplementary files

## Acknowledgments

## Conflict of Interest

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## Supplementary Files

**Supplementary File 1:**

**Supplementary File 2:**

**Supplementary File 3:**

**Supplementary File 4:**

**Supplementary File 5:**

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**MDPI and ACS Style**

Adiamah, D.A.; Schwartz, J.-M.
Construction of a Genome-Scale Kinetic Model of Mycobacterium Tuberculosis Using Generic Rate Equations. *Metabolites* **2012**, *2*, 382-397.
https://doi.org/10.3390/metabo2030382

**AMA Style**

Adiamah DA, Schwartz J-M.
Construction of a Genome-Scale Kinetic Model of Mycobacterium Tuberculosis Using Generic Rate Equations. *Metabolites*. 2012; 2(3):382-397.
https://doi.org/10.3390/metabo2030382

**Chicago/Turabian Style**

Adiamah, Delali A., and Jean-Marc Schwartz.
2012. "Construction of a Genome-Scale Kinetic Model of Mycobacterium Tuberculosis Using Generic Rate Equations" *Metabolites* 2, no. 3: 382-397.
https://doi.org/10.3390/metabo2030382