Team:NUDT CHINA/Modeling

Fig. 1 Cascade Regulatory Framework Fig. 2 Cascade Regulatory Pathway in DNA

According the cascade regulatory framework (Fig. 1) to solve the shortest path problem, we can build the cascade regulatory path in the plasmid of E.coli (Fig. 2). Now, we divide the whole cascade regulatory pathway into five units, which share same structures and similar properties (Fig. 3). Every unit can perform three common behaviours, i.e. promotion, transcription and translation. Usually, we can combine the process of promotion and translation when building and calculating the mathematic model of cascade regulatory. After combination, it is reasonable to assume that the transcriptional rate is in direct proportion to the extent of promotion. So we now get five easier units which can achieve two separated functions: promotion & transcription and translation.

The logic of the cascade regulation is:

• the translation of this cascade is regulated by the product (protein) of the upstream cascade;
• and identically, the product (protein) of this cascade regulates the translation of the downstream.

In addition, we need to take the temporal degradation of mRNA and protein into account.

I. Analyses of Cascade Regulatory Pathway

• the concentration of promoter binding sites in unit i is $D\_i$;
• the concentration of upstream promoter protein in unit i is $P\_i(t)$;
• the concentration of mrna in unit i is $R\_i(t)$;
• the concentration of product protein in unit i is $X\_i(t)$;
• the hill coefficient of promotion in unit i is $n\_i$;
• the dissociation constant in unit i is K_(P_i)_(D_i)^(n_i).

where i=1,2,3,4,5. (See Fig. 3)

Fig. 3 Five Units of the Cascade Regulatory Pathway and the Statement of Symbols

III. Mathematic Model