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 ;
• the concentration of upstream promoter protein in unit i is ;
• the concentration of mrna in unit i is ;
• the concentration of product protein in unit i is ;
• the hill coefficient of promotion in unit i is ;
• the dissociation constant in unit i is .

where (See Fig. 3)

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

III. Mathematic Model

In the model of the cascade regulatory pathway, we can gets equations according to the mRNA & protein metabolism of each unit and the protein transmit among different units.

i. mRNA Metabolism of Each Unit

According to and the Hill equation, we suppose that mRNA degraded without enzyme. So the degradation rate is proportional to the concentration of the mRNA itself. Now, we can describe the rate of promotion & transcription by time:

      (3.1)

Where is the maximum formation rate of mRNA in unit i and is the degradation rate constant of mRNA in unit i.()

ii. Protein Metabolism of Each Unit

In the same way, we suppose that protein degraded without enzyme. So the degradation rate is proportional to the concentration of the mRNA itself. Now, we can descript the rate of translation by time:

      (3.2)

Where is the maximum formation rate of protein in unit i and is the degradation rate constant of protein in unit i. ()

iii. Protein Transmit Among Different Units

From the topological relationship of the cascade regulatory pathway (Fig. 3), we can get the equations by describing the protein transmitting among different units:

      (3.2)

Theoretically, we can get the function of , which tells the temporal concentration of green fluorescence protein (GFP), by simultaneous equations (3.1)(3.2)(3.3).

IV. Calculation and Results of the Mathematic Model

i. Calculation

Limited by the calculation scale and experimental data, we assume that parameters (except the input and output) of each units are identical.

      (4.1)
      (4.2)

In addition, we set the parameters above according more to calculation results than to experimental data so that the parameters have no units. In this way, we can still get useful semiquantitative result.

Accoring to the equations (3.1)(3.2)(3.3)(4.1)(4.2), we wrote calculation program in Matlab to conduct numerical simulation and get the curve of under the cascade regulation pathway “111”.

Similarly, by modifying the equation (3.1) we can get the the curve of under the cascade regulation pathway“110”, “001”, “101” and “011”, respectively.

The modification rule is show in the chart below.

110	001	101	011

where the rest equations are the same as “111”.

Now, we put the curve of “111”, “110”, “001”, “101” and “011”in one figure and label them in different colors. (Fig. 4) So we can find difference of the GFP formation rate among these cascade regulatory pathways.

ii. Result

Before giving the result, we list the cascade regulatory pathways of “111”, “110”, “001”, “101” and “011” by figures.

111	110	001
101	011

Obviously, the GFP formation is later in “110” than others because the absence of pathway ③. The result is a powerful support to our iGEM design to solve the shortest path problem.

Problems

There are still some problems that need more experiment and time to deal with during building and calculating the model.

i. During Building the Model

Including the neglected details of Hill model itself, our model do not take the enzymatic degradation of mRNA and protein which, in fact, is common in cells.

ii. During Calculation

We suppose the parameters of the five is identical and set the parameters based more on calculation results than on experimental data. But in fact, the difference between different regulatory units can be obvious.