Abstract
Regulatory molecules such as transcription factors are often present at relatively small copy numbers in living cells. The copy number of a particular molecule fluctuates in time due to the random occurrence of production and degradation reactions. Here we consider a stochastic model for a self-regulating transcription factor whose lifespan (or time till degradation) follows a general distribution modelled as per a multi-dimensional phase-type process. We show that at steady state the protein copy-number distribution is the same as in a one-dimensional model with exponentially distributed lifetimes. This invariance result holds only if molecules are produced one at a time: we provide explicit counterexamples in the bursty production regime. Additionally, we consider the case of a bistable genetic switch constituted by a positively autoregulating transcription factor. The switch alternately resides in states of up- and downregulation and generates bimodal protein distributions. In the context of our invariance result, we investigate how the choice of lifetime distribution affects the rates of metastable transitions between the two modes of the distribution. The phase-type model, being non-linear and multi-dimensional whilst possessing an explicit stationary distribution, provides a valuable test example for exploring dynamics in complex biological systems.
CÇ is supported by the Comenius University grant for doctoral students Nos. UK/201/2019 and UK/106/2020. PB is supported by the Slovak Research and Development Agency under the contract No. APVV-18-0308, by the VEGA grant 1/0347/18, and the EraCoSysMed project 4D-Healing. AS is supported by the National Science Foundation grant ECCS-1711548 and ARO W911NF-19-1-0243.
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Çelik, C., Bokes, P., Singh, A. (2020). Stationary Distributions and Metastable Behaviour for Self-regulating Proteins with General Lifetime Distributions. In: Abate, A., Petrov, T., Wolf, V. (eds) Computational Methods in Systems Biology. CMSB 2020. Lecture Notes in Computer Science(), vol 12314. Springer, Cham. https://doi.org/10.1007/978-3-030-60327-4_2
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