Stationary Distribution of Telomere Lengths in Cells with Telomere Length Maintenance and its Parametric Inference
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Stationary Distribution of Telomere Lengths in Cells with Telomere Length Maintenance and its Parametric Inference Kyung Hyun Lee1
· Marek Kimmel1
Received: 8 November 2019 / Accepted: 25 September 2020 / Published online: 20 November 2020 © Society for Mathematical Biology 2020
Abstract Telomeres are nucleotide caps located at the ends of each eukaryotic chromosome. Under normal physiological conditions as well as in culture, they shorten during each DNA replication round. Short telomeres initiate a proliferative arrest of cells termed ‘replicative senescence’. However, cancer cells possessing limitless replication potential can avoid senescence by the telomere maintenance mechanism, which offsets telomeric loss. Therefore, cancer cells have sufficiently long telomeres even though their lengths are significantly shorter than their normal counterparts. This implies that the attrition and elongation rates play crucial roles in deciding whether and when cells ultimately become carcinogenic. In this research, we propose a concise mathematical model that shows the shortest telomere length at each cell division and prove mathematical conditions related to the attrition and elongation rates, which are necessary and sufficient for the existence of stationary distribution of telomere lengths. Moreover, we estimate the parameters of the telomere length maintenance process based on frequentist and Bayesian approaches. This study expands our knowledge of the mathematical relationship between the telomere attrition and elongation rates in cancer cells, which is important because the telomere length dynamics is a useful biomarker of cancer diagnosis and prognosis. Keywords Markov chain · Stationary distribution · Probability generating function · Classification of states · Telomeres · Telomere maintenance mechanisms
This research was supported by National Institute of Health with the project code, 5R01HL128173-04. Electronic supplementary material The online version of this article (https://doi.org/10.1007/s11538020-00811-1) contains supplementary material, which is available to authorized users.
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Marek Kimmel [email protected] Kyung Hyun Lee [email protected]
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K. H. Lee, M. Kimmel
1 Introduction Telomeres are composed of repetitive nucleotide sequences at the end of chromosomes. In a normal cell, they are shortened during each cell division. Cells with critically short telomeres enter an irreversible arrest state of cell proliferation; therefore, telomeres behave as a mitotic clock that recognizes replicative senescence. It is widely hypothesized that replicative senescence due to telomere shortening is both a tumor suppressor and the implication of organismic aging (Campisi 1997). Modern theories of telomere biology are formed according to the following three principles (Bernadotte et al. 2016): First, cellular aging is programmed, and second, telomere shortening plays a role in signaling when the cell will enter the arrest state. Third, normal cells with
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