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, X), depends on both the mRNA (Xm)and siRNA levels (X). The main equation was extracted from “Modeling RNA interference in mammalian cells” (Cuccato, 2011).

We based the siRNA degradation behavior on the model described in the article “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 behavior 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.

Then d and θ are added remember that these values are the 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.

Then by activating L, the transfection efficiency we can observe how to slope decreases demonstrating the degradation of the mRNA.

A different slope movement can be observed when d and are manipulated when transfection is in action.

Another elemented included is dm that describes the basal degradation of the mRNA, as you can see the initial concentration decreases.

Hill also affects the curve movement.

In conclusion

Our mathematical modeling describes the quantitative effects of varying SiRNA quantities on the targeted mRNA species, this model includes mRNA transcription, and siRNA degradation by using a standard Hill-kinetic model to describe the SiRNA binding to mRNA, all the above was included in a system of ordinary differential equations. Our upgrade in the described model is the inclusion of the cellular transfection efficiency. By adding this variable, we ensure that the siRNA activity will be carried out successfully by the cells.

References

Cuccato, G., Polynikis, A., Siciliano, V., Graziano, M., Bernardo, M. D., & Bernardo, D. D. (2011). Modeling RNA interference in mammalian cells. BMC Systems Biology, 5(1), 19. doi:10.1186/1752-0509-5-19
Khanin, R., & Vinciotti, V. (2008, 04). Computational Modeling of Post-Transcriptional Gene Regulation by MicroRNAs. Journal of Computational Biology, 15(3), 305-316. doi:10.1089/cmb.2007.0184
Lam, J. K., Chow, M. Y., Zhang, Y., & Leung, S. W. (2015, September). SiRNA Versus miRNA as Therapeutics for Gene Silencing. Retrieved from https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4877448/