Team:BIOSINT Mexico/Modeling

Modeling

This year we designed a construct where a lot of different genes interact in several ways. Eight sets of genes were constructed, using five distinctive kinds of genetic expression. They can be classified under the following categories:
a) Genes constitutively expressed
b) Genes that form a protein complex
c) Auto regulated networks
d) Simply induced sequences
e) Complex induced sequences

Equations for Phytochrome-PIF6 complex
In order to produce proteins, in the cell, two main reactions happen. First, DNA is transcripted to mRNA and then it is translated to a peptide sequence, it could be represented in the reaction by:

All these reactions are mediated by two specific enzymes, which are RNA polymerase (transcription) and ribosome (translation). These reactions have been well described in literature and can be modeled to a Ordinary Differential Equation system.
The rate of production of the PhyB construct is given by the rate of transcription of the protein from the mRNA molecule and the degradation of the messenger. Therefore:

Where γ is the translation rate of the cell, and α is the degradation of the degradation rate of the protein.
Also, the concentration of the messenger molecule is given by the equation

Where a represents the activity of the promoter, that is related to the capacity of being activated or suppressed by a transcription factor; β is the maximal production rate of the CaMV promoter (which is attached upstream to PhyB) and is a linear function dependent on the time of the reaction; as in the last equation, α is the degradation rate.
Also, since PCaMV is a constitutive promoter, the value of its activity is equal to 1. So the equation is reduced to

As PIF6 construct is connected to the same promoter, the equations of its expression are deduced by the same way, so:

In our system, both expressed gene constructs are part of another reaction in presence of deep red light (660 nm lightwave). When photoreceptor protein (PhyB construct) detects deep red light, it fuses to its interaction factor (PIF6 construct) and form a protein
For the complex creation, follow the following reaction in presence of 660 nm light.

Where Ku is the constant of the reaction, and is on concentration units. And K-u is the constant for the inverse reaction (separation of the complex), and the active form of the complex can be abbreviated as CPPA. Then, we can infer the concentration of active complex by the equation.

Also, in presence of far red light, the complex passes to its inactive form, represented as CPPI, by the following reaction:

Where KD is the constant of the reaction and K-D equals zero, because deactivated complex can’t return to its active form.

Then, if we despise the degradation rate of the inactivation of the protein complex, we can simplify the calculi of concentration of active photo protein by replacing 6 in 5.

Equations for mer Operon production
For the mer operon expression we start from the reaction

Of all the genes of this operon we used two specific proteins. merE is a transport protein in charge of the active transport od methyl-mercury from the environment of the cell to its interior. Once in the inner cytoplasm, merB protein attacks the molecule and breaks the covalent bond releasing methane and Hg2+(s) to the cell. Methane is metabolized and the quicksilver is accumulated.
Then, for the merE production rate we know that

And, as mer operon is attached to a minimal constitutive promoter (PCMV), the production rate of the messenger molecule is

The same process is applied to merB from the operon. Therefore: