Team:BHSF/Model

Welcome To Our Model!

It's Nice To Meet You

Here we go

Model

 In this model, we are going to predict the theoretical production of carminic acid produced by gene-edited 
 E.coli from the Acetyl-CoA and Malonyl-CoA in it. Because of the difficulty of conducting 2-substrates 
 production, we had no choice but to assume we only got one substrate, which is Malonyl-CoA, because it 
 contributes most in the production(6 times more Malonyl-CoA will be consumed than Acetyl-CoA ). 

 Deduction of the function to calculate the concentration of the product:
 [S]->[X]+[P]
 This is the basic function of the reaction, in which [S] represents the concentration of the solute, 
 [X] represents the concentration of the cell, [P] represents the concentration of the product.


 Now, we are going to deduct the equation that tells us the rate of production t:
ΔMp=Mpin-Mpout+Mpmd-MpdsyΔMx=Mxin-Mxout+Mgth    ΔMs=Msin-Msout+Mpmd-Mrmd
                      

The Differential Equation Of Production

The change of Mp equals the input of product minus the output of the product plus the product made by the E.coli and minus the destroyed product.Then, the ΔMp can be represented by Vp⋅[P]. We define fout as the rate of output in volume, so we can now define the rate of output in mass by [P]⋅fout. We define Qp be the rate of product formation, which can be calculated by using the formula Qp=production/consumed (the production represents the grams of products made and the consumed means the grams of substrate used.) We now can represent the Mass of production by Qp⋅[X]⋅Vp. We define Qp be the rate of product formation, which can be calculated by using the formula Qp=gramsofproductmade/(gramsofcell⋅time) We now can represent the Mass of production by Qp⋅[X]⋅Vp. To make the result more simple and concern only the theoretical production, we can assume that the mass of production destroyed proportionate to a constant, Kd. We now can represent the Mass of destroyed by Kd⋅[P]⋅Vp. Also, because the initial concentration of product, Mpin, is zero, we can transform the equation above to this form: (dVp⋅[P])/dt=-[P]⋅fout+Qp⋅[X]⋅Vp-Kd⋅[P]⋅Vp The left hand is the representation of the rate of change of the product mass and the first term of the right hand represents the rate of output, the second represents the rate of change of mass of product is made and the third represents the rate of destroy. Then, we divide both sides Vp, the volume of liquid. We get: (d[P])/dt=-fout[P]/Vp+Qp[X] We define the fout/Vp as dilution factor, D and to make the result simpler, we ignored the destruction rate.

The Differential Equation Of Substrate

The change of Ms equals the input of product minus the 
output of the product plus the amount of growth of E.coli
(dVx⋅[X])/dt=[X]⋅fin-[X]⋅fout+u⋅[X]⋅V

The u represents a constant, which helps to calculate the rate of growth. 
To make the result simpler, we can ignore the [X]⋅fin. Then, we divide the 
both sides the volume. So, the final result is:
(d⋅[X])/dt=-fout[X]/V⋅[X]+u⋅[X]
                        

The Differential Equation Of Cell

The Mrmd in the equation represents the mass of removed of the cell, which can be represented by 
[S]->[X]+[P]. There are few cases of mass of removed:
① [S]->[X]
② [S]->[P]
③[S]->maintain the cell

We define Qp be the rate of product formation, which can be calculated by using the formula 
Qp=gramsofproductmade/(gramsofcell⋅time). We define the Ypls as the Yield coefficient, 
which is grams of product made divided by grams oof substrate used. 

We now can represent the Mass of removed from [P] by: (Qp⋅V⋅[X])/Ypls, the mass of removed from 
[X] by (u⋅V⋅[X])/Ypls, the removed mass from the maintaining in the cell by [X]⋅u⋅V

Thus, the differential equation is:
(dV⋅[S])/dt=[S]⋅fin-[S]⋅fout-([X]⋅Qp⋅V)/Ypls-([X]⋅u⋅V)/Ypls-u⋅[X]⋅V

We divide both sides V and we finally got: (We simply ignore the last term to make the result easier.)
(d[S])/dt=fin/V([S]in-[S]out)-([X]⋅Qp)/Ypls-([X]⋅u)/Ypls

Then, the ΔMp can be represented by Vp⋅[P]. We define fout as the rate of output in volume, 
so we can now define the rate of output in mass by [P]⋅fout.
                      

Final Results

Now, we only need to solve the differential equation to find the relationship between the concentration of 
the product, the concentration of cell and the concentration of substrate. The final function is:   

[P]=Qp/D⋅[X]

[X]=(D⋅([S]in-[S]out))/(Qp/Ypls+D/Ypls)

[S]=Dks/(Umax-D)
                    

Conclusions

There are 2 conclusions that can be concluded from our model:

1.Since the variant of the right side of the equation is [X], 
the production of carminic acid is determined by the concentration of E.Coli

2.Letting the E.coli secrete the product out of the cell can make the 
production more efficient and productive.
                      

Reflection

From the conclusion we had, we can increase the overall biomass because it will 
produce more our product. So we decide to use a reactor because it can potentially 
increase the biomass 10 times over traditional tools. 
                      

Reference

Dongsoo Yang, Production of Carminic Acid by Metabolically Engineered Escherichia coli 
(JOURNAL OF THE AMERICAN CHEMICAL SOCIETY, 2021)
                      

Special Thanks

Zeyu Tang’s lecture about biomodel 
Mingyang Xu’s lecture of mathematical deduction in biology