# Modeling

## Contents

### Overview

To synthesize amino acids using synthetic biology, our team simulated the synthesis of glutamic acid from ammonia as substrate, which were synthesized by nitrogenase activity using Matlab. For the expression of glutamic acid, we designed the system of using operon that include σ-dependent promoter that transcribes GluDH gene. σ factor gene is designed downstream of GluDH gene.

### The aim of the modeling

This simulation can theoretically prove the efficiency of glutamate dehydrogenation activity.

### Background

The purpose of our project is to synthesize ammonia from nitrogen with nitrogenase and then synthesize glutamic acid with glutamate dehydrogenase. It is necessary to prove that synthesis of glutamic acid by glutamate dehydrogenation in the bacterial cells is realistic. We carried out simulation by using Matlab.

### The flow of whole modeling project

We first simulated how much glutamate dehydrogenase is expressed by the effect of the designed DNA. The designed DNA is shown in the figure below. The σ factor acts as an activator and transcription proceeds by binding to a σ-dependent promoter (Psigma). GluDH is translated into glutamate dehydrogenase and σ is translated into σ factor.

The translated σ factor binds to upstream Psigma and that promotes more transcription. The translated glutamate dehydrogenase synthesizes glutamate using ammonia as a substrate.

Glutamate dehydrogenase catalyze reaction below.

The variables and constants used for modeling were defined as follows.

Variable | Character |
---|---|

σ factor concentration | σ |

mRNA concentration | m |

GluDH concentration | [GluDH] |

Ammonia concentration | [NH_{3}] |

2-Oxoglutamate concentration | [2-Oxoglu] |

NADH_{2} concentration | [NADH_{2}] |

L-Glutamate concentration | [Glu] |

NAD concentration | [NAD] |

σ factor decomposition rate | g_{σ} |

σ factor translation rate | k_{σ} |

mRNA degradation rate | g_{m} |

mRNA transcription rate | a |

GluDH degradation rate | g_{gluDH} |

GluDH translation rate | k_{gluDH} |

E. coli concentration | N |

Parameters | Character |
---|---|

σ factor decomposition rate constant | δ_{σ} |

mRNA decomposition rate constant | δ_{m} |

GluDH decomposition rate constant | δ_{gluDH} |

Glu decomposition rate constant | δ_{glu} |

σfactor/Psigma dissociation constant | K |

mRNA transcription rate constant | t_{c} |

σ factor translation rate constant | t_{e} |

GluDH reaction rate constant | k_{cat} |

GluDH reaction rate ([NH_{3}]>>[GluDH]) | V_{max} |

Substrate concentration at 1 / 2Vmax | K_{m} |

Environment capacity | K_{e} |

Internal natural growth rate | r |

Based on the above variables and constants, we construct the model and simulated the behavioral model of *Escherichia coli* cells when Psigma-gluDH-σ was introduced, the logistic equation by *Escherichia coli* proliferation, and the enzyme reaction model of glutamate dehydrogenase using Michaelis-Menten.
And we solved using the ode45 solver available in Matlab.

### System

#### Behavioral model of E. coli cells incorporating Psigma-gluDH-σ

The σ factor concentration changes and the mRNA concentration changes in *E.coli* cells which Psigma-gluDH-σ were introduced, are shown in the following three simultaneous equations.

σ：σ factor concentration

m：mRNA concentration

t_{e}：σ factor translation rate constant

δ_{σ}：σ factor decomposition rate constant

t_{c}：RNA transcription rate constant

δ_{m}：mRNA decomposition rate constant

K：σfactor/Psigma dissociation constant

[GluDH]：GluDH concentration

k_{gluDH}：GluDH translation rate

δ_{gluDH}：GluDH decomposition rate constant

### Equation 1

The concentration change of σ factor depends on translation rate of mRNA and degradation rate of σ factor. The translation rate of mRNA is proportional to the mRNA concentration, and the degradation rate is proportional to the concentration of the σ factor.

### Equation 2

The concentration change of mRNA depends on the transcription rate of DNA and the degradation rate of mRNA. The rate of mRNA degradation is proportional to the concentration of mRNA. Since the transcription rate of DNA is proportional to the rate of the total promoter that the sigma factor bound to, it can be shown using the Hill equation.

### Equation 3

The concentration change of GluDH depends on the translation rate of mRNA and degradation rate of GluDH. The translation rate of mRNA is proportional to the mRNA concentration, and the degradation rate is proportional to the concentration of the GluDH.

### Result

#### Logistic equation by E. coli multiplication

When Escherichia coli divides and proliferates, there is no such thing that they can grow without limit in a constant pace, and as the number of bacteria increase, the growth rate decreases due to quorum sensing and nutritional conditions, and the number of bacteria will stay at the some point. And this phenomenon can be shown by the equation below.

N：E. coli concentration

r：Internal natural growth rate

Ke：Environment capacity

### result

#### Enzymatic reaction model of glutamate dehydrogenase

An enzyme kinetic modeling of the glutamate dehydrogenase is done in this study. The work reported uses Michaelis-Menten assumptions to model the enzyme activity and rate of production of substrates. The Michaelis-Menten kinetics represents the enzymatic reaction as a two-step process. This may be represented schematically below.

k_{f},k_{r},k_{cat}：rate constant

E：Enzyme

S：Substrate,[S]：Substrate concentration

ES：enzyme-substrate binding

P：Product,[P]：Product concentration

A set of ordinary differential equations were used to represent change in concentration of different substrates in the pathway using the Michaelis-Menten equations.
The given differential equations are the rates of production of Ammonia, enzyme-substrate binding and Glutamate.