# Team:Fudan Lux/Modeling

### From 2012.igem.org

# Mathematical Model

The model has three elements: the light sensor, which exists as a dimer, the Lux protein and the light intensity. The light sensor dimer not only be able to response to changing of light intensity, but also has the capability to bind to downstream “Tet on” promoter, and regulate the expression of Lux family. Once the dimer binds to the promoter, the transcription rate of the Lux is decreased more than 1000 times. The parameters of the system are listed in Table 1.

We built the corresponding Langevin mathematical model for this system as follows:

equation (1), the variables Ls, Ls1, Lux refer to the concentration of the light sensor, the active light sensor and Lux protein in the system, respectively. The variables Z , Z1 and Z2 denote the total light strength, intrinsic light strength and external light strength. The initial values of these variables are set to zero. N is the copy number of plasmid, K1 and K2 are expression capability of light sensor gene and Lux, respectively, K3 is the parameter describe the ability of give out light by Lux, K4 is attenuation rate of light propagation in colonial, K5 and K6 are parameters of logistic eqution, d is the degradation rate of light sensor and Lux protein, r means the dilution rate caused by cell volume grow. δ denotes the noise of gene expression. In order to calculate the total extern light strength, we used multiple integral to describe the contribution of whole colonial to this unit. Multiple logistic method was also be used to describe concentration of the active light sensor change by light sensor and total light strength.

# Simulation

Based on the above model, we stimulated the stochastic dynamics of this light communication system by *GeneCircuits tool.*

Paramemter | Value |

N | row 2, cell 1 |

K1 | 0.01 umol/min |

K2 | 0.01umol/min |

K3 | 8.4 |

K4 | 0.05 |

K5 | 20 |

K6 | 20 |

d2 | 0 |

d1 | 0 |

r | 0.02 |

# Spectrum analysis

Randomly, we chased more than 2000 sampling points, and extracted time serials data of each sampling points from our time lapse images. Use customized matlab code, we analyzed the spectrum of each sampling point, and then shown the distribution of frequency of all sampling points by histogram.