# Post-transcriptional regulation in the nucleus and cytoplasm: study of mean time to threshold (MTT) and narrow escape problem

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## Abstract

Messenger RNAs (mRNAs) can be repressed and degraded by small non-coding RNA molecules. In this paper, we formulate a coarsegrained Markov-chain description of the post-transcriptional regulation of mRNAs by either small interfering RNAs (siRNAs) or microRNAs (miRNAs). We calculate the probability of an mRNA escaping from its domain before it is repressed by siRNAs/miRNAs via calculation of the mean time to threshold: when the number of bound siRNAs/miRNAs exceeds a certain threshold value, the mRNA is irreversibly repressed. In some cases, the analysis can be reduced to counting certain paths in a reduced Markov model. We obtain explicit expressions when the small RNA bind irreversibly to the mRNA and we also discuss the reversible binding case. We apply our models to the study of RNA interference in the nucleus, examining the probability of mRNAs escaping via small nuclear pores before being degraded by siRNAs. Using the same modelling framework, we further investigate the effect of small, decoy RNAs (decoys) on the process of post-transcriptional regulation, by studying regulation of the tumor suppressor gene, *PTEN*: decoys are able to block binding sites on *PTEN* mRNAs, thereby reducing the number of sites available to siRNAs/miRNAs and helping to protect it from repression. We calculate the probability of a cytoplasmic *PTEN* mRNA translocating to the endoplasmic reticulum before being repressed by miRNAs. We support our results with stochastic simulations.

## Keywords

Stochastic process Markov chain Gene expression mean first passage time Fokker Planck equation PTEN mRNA## Mathematics Subject Classification (2010)

92B05 60J28 60J70## References

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