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Modeling
Introduction
Modeling is a useful method by which proposed mechanisms and experimental results can be compared. Through these comparisons, not only can reaction mechanisms be explored, the parameters for our circuits can also be tuned and perfected. This is especially relevant to our project given its intention to eventually be medically and the amount of precision tuning necessary for such systems. Modeling can help us analyze our data and figure out how to adjust our system parameters to match those expected to be found in vivo.
Expected pathway
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]
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
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
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
At equilibrium
With miRNA
At equilibrium
Protein drop-off can be estimated by the ratio of mRNA available for translation
Given that miRNA binding doesn’t directly block translation
If miRNA binding does directly block translation
In an in vitro experiment where we would transfect mammalian cells to observe the effect of miRNA repression, kt and kb will be functions of the transfection efficiency, and can be modeled as such. Thus, using the equilibrium ratios procured and randomly generating numbers for transfection efficiency and scatter, we can simulate expected behavior from flow-cytometry readouts.
Figure 5: Results from a simulated flow cytometry readout. Variable T was attached to the constants related to transfected parts and randomly distributed on a log scale with random variation. miRNA that increases degradation without preventing ribosomal attachment shows a step-down but does not cause a noticeable slope change. miRNA that interrupts ribosomal attachment would change slope.
The lack of a clear difference between no miRNA and miRNA that don’t inhibit ribosomal recruitment readouts can be explained by looking at its net mRNA ratio (1). The ratio has a net order of zero for transfection dependent terms, meaning the drop-off in protein expression is only a constant multiple. This will be shown as a vertical shift away down but will not cause a change in slope, making detection of miRNA activity in in vitro systems difficult.
On the other hand, miRNA blocking ribosomal attachment has a distinct pattern from unrepressed systems. This is because the drop-off rate increases with transfection efficiency, making the output visibly different on a loglog graph.
Conclusion
Given that output units are not precisely defined, it was originally planned to quantify L7Ae output from experimental results before modeling the high sensors. However, due to L7Ae’s overwhelming ability as a repressor, it was difficult to quantify and explain L7Ae dynamics.
The difficulty for miRNA repression systems that don’t block ribosomal binding may explain the ambiguousness of our low sensor data. Within in vivo systems where expression of circuit components are generally lower and more controlled, the threshold of the low sensor system may be sufficient to respond appropriately for diagnostic functions. However, that makes quantification and in vitro testing difficult at the moment without a way to clearly distinguish control groups from experimental groups. In the future, we plan to test the circuits in cell lines with native miRNA expression to gauge in vivo response.
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