The proposed mathematical model can quantitatively describe the effects of varying quantities of siRNA onto the degradation of the target mRNA. By using a system of ordinary differential equations, the model describes mRNA transcription and the siRNA degradation. For the mRNA transcription a km value range was used, this represents the rate from the promoter that transcribes the mRNA to be targeted. This model also considers degradation mRNA due to RNAi with the following equation δ (Xm, X ). This function, δ (Xm, Xs ), depends on both the mRNA (Xm)and siRNA levels (X). The main equation was extracted from “Modeling RNA interference in mammalian cells” [1].
We based the siRNA degradation behavior on the model described in the paper “Computational Modeling of Post-Transcriptional Gene Regulation by MicroRNAs” . The siRNA degradation uses a standard Hill-kinetic model to describe the post-transcriptional effects of microRNAs on the gene expression (Khanin, 2008). As the most important difference between siRNA and miRNA is that siRNA have highly specific targets and miRNA have multiple targets, we decided to take this model (Lam et al, 2015). This model is a Hill-type enzymatic it has a hill coefficient bigger than 1, the model can be used for siRNA for multiple binding sites on the same mRNA. Other models have been attempted before but all rely on chemical or biochemical reactions using stoichiometry. This model considers two kinetic parameters: d and θ. These parameters depend on the siRNA efficiency to bind to its mRNA target. The maximal degradation rate of the mRNA due to interference is shown as d and the siRNA concentration to achieve half of the maximal degradation rate is θ (Khanin, 2008).
The above equation implies that for X < θ, the increase in the RNAi mediated degradation is linear with X^h while it saturates at higher levels of X , reaching the maximal degradation rate d.
Khanin et al. used the following equations:
How did we upgrade the previous model?
It was stated at the beginning of the competition that we would attempt to have the first adult primary culture of Diaphorina Citri, after trying several times it wasn’t possible, some limitations where the lack of an specific medium and time. But the objective was set and the math modeling considers the use of a cytometer with that primary culture, one of our SiRNA has an alexa fluor 647 tag in order to use it in the equipment and with this determine the transfection efficiency. We know the importance that transfection plays when using RNAi technology, there are several points to consider transfection efficiency such as: cell health, cell viability, number of passages, the quality and quantity of the nucleic acid. The experiment to prove our math modeling considers an in vivo transfection of the whole Diaphorina Citri, this method can be view in protocols. Taking into account a transfection efficiency of 50%-75% we continued to solve the following equations. The transfection efficiency variable would be “L”, this variable multiplicates directly to the X, siRNA levels, giving us the real sirna amount within in the cell walls in the sample.
The following videos represent the behaviour of our math modeling considering L.
First transcription and hill affinity take place, then and d take part in the mRNA degradation by siRNA activities.

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