# Team:Nanjing-China/Model

Nanjing-China2018

This year our team created a mathematical model to optimize the arrangement of the nif gene cluster. This model helped we refined our design and provided some new perspectives of our nitrogen-fixation system attranscriptional level.

We developed this model with two goals in mind:
1. We want to achieve the putative best stoichiometric proportion of each nif gene, which is nifB: nifH: nifD: nifK: nifE: nifN: nifX: nifV=1: 3: 4: 4: 1: 1: 1: 1.
2. We want our system as simple as possible, that means minimizing numbers of promoters and each nif gene.

1. There are two kinds of promoters, both of which can successfully launch the expression of every nitrogen fixation gene involved in our system.
2. One promoter is stronger(called H) while the other is relatively weak(called L). Under promoter H, each gene’s transcription level is double that of under promoter L.
3. The order of genes has little influence on their transcription level.

We conducted Real-time Quantitative PCR to detect the transcription level of nif gene cluster and the experimental data we received became an important reference for our modeling.

 gene Average value of Cq Relative expression level 16S DNA 6.33 nifB 19.97 7.80E-05 nifH 17.37 4.74E-04 nifD 18.34 2.42E-04 nifK 20.77 4.48E-05 nifE 22.20 1.66E-05 nifN 22.24 1.62E-05 nifX 22.92 1.01E-05 nifV 21.25 3.22E-05

Table1 The result of qPCR

### Method:

To start with, we put all genes into two groups. One group is under the strong promoter while the other is under the weak one. We constructed two arrays, weak[i] and expected[i].

 Parameters(i=1,2,3,4,5,6,7,8) Meanings weak[i] the relative expression level of each nif gene under the weak promoter weak[i]* the relative expression level of each nif gene under the weak promoter after normalization expected[i] the ideal stoichiometric proportion expected[i]* the ideal stoichiometric proportion after normalization strong[i] the relative expression level of each nif gene under the strong promoter after normalization ei the ideal stoichiometric proportion of the ith gene after all preprocessings ai the relative expression level of the ith gene under the weak promoter after all preprocessings mi the number of the ith gene under the strong promoter ni the number of the ith gene under the weak promoter

Table 2 The table of parameters in our model

Then we did some necessary preprocessings. Firstly, we found the smallest data in weak[i] and called it “min”. We normalized all the other data accordingly by doing:

We constructed strong[i]:

strong[i]=2×weak[i]*

Secondly, to guarantee the existence of a solution, we adjusted expected[i]* by examining whether it is no less than the corresponding weak[i]*, if not, we did:

expected[i]*=weak[i]*

After that, we began the organization. In order to minimize the total numbers of genes, we arranged the strong promoter group first, and considered the weak group later. Because each gene can be considered separately, here we only describe the organization of the ith gene as an example.
For the ith gene, we tried adding one copy of it under the strong promoter. If

|ei-2×ai|<ei,

we actually added it. Until we have added (mi+1) ith genes, and got

|ei-2(mi+1)×ai|>=|ei-2mii×ai|

Then we stopped adding it and recorded that we have added mi ith genes under the strong promoter.

For the weak promoter group, we applied a similar method. For the ith gene, we tried adding one copy of it under the weak promoter. If

|ei-2×mi×ai-ai|<|ei-2×mi×ai|,

we actually added it. Until we have added (ni+1) ith genes, and got

|ei-2×mi×ai-(ni+1)×ai|>=|ei-2×mii×ai-ni×ai|

Then we stopped adding it and recorded that we have added ni ith genes under the weak promoter.

In that way, we were able to determine numbers of the ith gene under the two promoters with which the deviation was the smallest.

Fig 1. A flow diagram describing the idea of our modeling process

According to this flow diagram, we programmed with Python and got the following results:

Fig 2. The best arrangement of nif genes according to our calculation

With this arrangement, the proportion of nifB: nifH: nifD: nifK: nifE: nifN: nifX: nifV= 15.44: 46.93: 71.88: 62.10: 16.44: 16.04: 16.0: 15.94, which is close enough to the ideal proportion among all the solutions.

## Refinement of our model:

We modified the putative best expression level of nifB:nifH:nifD:nifK:nifE:nifN:nifX:nifV to 5:3:4:4:1:1:1:1. We believed in this way, we could better simulate the expression of nitrogenase in our engineered E.coli strains. We made this change because of three reasons.

Firstly, nifB is indispensable for nitrogenase assembly no matter in diazotrophs or engineered E. coli strains. Apart from the minimal nitrogen fixation gene cluster, the genomic DNA of wide type Paenibacillus polymyxa includes analogues of nifM, nifU, nifS and other genes which exist in other nitrogen-fixing microorganisms and are essential for the correct folding of nitrogenase iron protein. However, the E. coli genome doesn’t have such analogues. Nevertheless, it has been reported that the excessive expression of nifB can compensate for the absence of nifU and nifS. That is, if nifB is overexpressed in E. coli, these auxiliaries are not necessary. Therefore, the expression level of nifB should be the highest 5.

Secondly, compared with nitrogen-fixing microorganisms, E. coli also lacks some genes that provide electron transfer function, such as nifF and nifJ. So the intracellular reductive power of E. coli is insufficient to accomplish nitrogen fixation. Thus it is necessary to overexpress nifH(nitrogenase reductase) and the value is set to 3 instead of 5 because our semiconductor, the CdS part, can provide additional electrons.

Thirdly, we set the expression level of nifD and nifK to be 4 because molybdenum iron protein is an ɑ2β2 allotetramer and is the core of nitrogenase.

Based on the new ideal stoichiometric proportion, we adjusted the code and received a more accurate result.

Fig 3 The best arrangement of nif genes version 2.0.

The achieved stoichiometric proportion of nifB: nifH: nifD: nifK: nifE: nifN: nifX: nifV=77.23: 46.93: 71.88: 62.10: 16.44: 16.04: 16.0: 15.94, which is close enough to the ideal 5:3:4:4:1:1:1:1.

This model provided a potential strategy for the improvement of biological activity of nitrogenase expressed in our engineered E. coli strain and offered a great help to our further experiments.