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Modeling
Introduction
Expected pathway
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Figure 1: Simplified representation of miRNA translational repression. Out of the many repression pathways known to be used, the figure displays increased degradation of mRNA directed by miRNA.
Ordinary Differential Equations
To mirror how miRNA interacts with mRNA in the process of translation, we used mechanistic equations and relative kinetic constants taken from Nadya Morozova’s “Kinetic signatures of microRNA modes of action”. (2012) [1]
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Figure 2: Collection of ODEs used to model miRNA translational repression. M0 represents new mRNA created by transcription. F0, M, and F represent mRNA ribosomal complexes in various stages of initiation, with R being the finished mRNA-ribosomal complex ready for translation. M’0, F’0, M’, F’, and R’ represent their respective RNA constructs with miRNA bound. P is the protein output. B is mRNA sequestered in P bodies and is ignored for purpose of simplicity.
It’s important to note the assumptions this system operates under. It assumes the environment to be aqueous for purpose of kinetics. Rate of translation for target mRNA is set at a constant. Also, rate of miRNA binding is a single order function that only takes into account concentration of target mRNA, with miRNA assumed to be free floating and at sufficiently high concentration such that binding with mRNA doesn’t significantly affect miRNA concentration. The same applies for recruitment of ribosomal subunits for translation. Moreover, miRNA mediated degradation is only represented by an improved degradation constant, without too much modeling of directed cleavage kinetics.
Analysis
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Figure 3: ODEs with miRNA binding improving degradation rate ten times above unbound. All other repression mechanisms were turned off. Note that time axis is in increments of the inverse of the degradation rate. The quantification for mRNA and protein output have no units and are relative amounts.
The graph illustrates several assumptions and simplifications that can be used for further analysis. The most apparent behavior is that equilibrium protein output is directly proportional to mRNA amount.
Assuming that equilibrium protein level is directly proportional to mRNA levels, we can compare mRNA levels before and after introduction of miRNAs to get an idea of decrease in protein expression.
Observable Differences
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Figure 4: Simplified representation of the miRNA repression system. kt is the rate of transcription at which mRNA is being constructed, kd represents the rate at which unbound mRNA is lost, including degradation and post-translational cleavage, kb is the rate of binding of mRNA by miRNA, and k’d represents the rate at which bound mRNA is lost.
We can simplify the system by removing protein and using the ratio of mRNA levels with and without miRNA to determine how much protein expression decreases. The ordinary differential equations for the system can be used to solve for equilibrium conditions.
Without miRNA
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At equilibrium
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