# Model

**Hypothesis**

Hypothesis 1：VHb has linear influence on available oxygen

Hypothesis 2：the gaining effect of available oxygen is orthogonal to the burden of producing VHb

**Formulas**

We assumed that VHb has an additional positive effect on the surrounding oxygen concentration. The adsorption coefficient of VHb which shows the relationship between incremental oxygen concentration and VHb protein concentration in the bacteria is designated as *K _{1}*, whose value is determined by the efficiency of VHb; And the conversion coefficient which convert the percentage of VHb into concentration of VHb is designated as

*K*. Therefore,

_{2}*K*will denote the incremental oxygen concentration surrounding the bacteria. The total oxygen available will be the original oxygen concentration plus the incremental oxygen adsorbed by VHb as shown in Formula 1.

_{1}K_{2}pVHbPrevious researches have reported that the growth rate of bacteria is affected by both the oxygen concentration and protein expression. Stolper et al. (2010) showed the positive relationship between oxygen concentration and growth rate; Scott et al. (2010) reported the negative effect of gene expression on growth rate since protein expression is an energy-consuming process. Given an unchanged level of total energy, additional protein expression will reduce the energy available for cell growth. Based on these two researches we got the relationship between growth rate and VHb concentration as shown in Formula 2.

Since the original growth rate *r* was measured under the original oxygen concentration, we made a correction using Formula 3 to remove the influence of the oxygen concentration, and the corrected *r _{c}* was applied to Formula 2.

symbol | description |

r | Original growth rate^{[2]} |

r_{c} |
Corrected r removing the effect of original oxygen concentration |

k | The slope constant^{[2]} |

K_{s} |
Half saturation constant with respect to growth |

pVHb | The percentage of VHb in total bacterial protein |

μ | Growth rate of the bacteria |

μ_{max} |
The maximum growth rate of the bacteria under the adjustion of oxygen concentration |

O_{2ori} |
The original oxygen concentration surrounding the bacteria |

O_{2useful} |
The total amount of oxygen available to the bacteria |

K_{2} |
The conversion coefficient of VHb concentration |

K_{1} |
The oxygen adsorption coefficient of VHb |

**Explaination of parameter**

Value | Reference | |

K_{s} |
121nM | R^{[1]} |

μ_{max} |
2h^{-1} |
Supposed from R^{[1]} |

r | 1.7h^{-1} |
R^{[2]} |

k | 0.034 | R^{[2]} |

K_{1} |
0.5 | hypothesis |

K_{2} |
100 | hypothesis |

O_{2ori} |
100nM | hypothesis |

**Result**