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.

中文翻译：

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具有端粒长度维持的细胞中端粒长度的平稳分布及其参数推断

端粒是位于每个真核染色体末端的核苷酸帽。在正常生理条件下以及在培养中，它们在每个 DNA 复制轮中都会缩短。短端粒启动细胞增殖停滞，称为“复制性衰老”。然而，具有无限复制潜力的癌细胞可以通过端粒维持机制避免衰老，从而抵消端粒损失。因此，癌细胞具有足够长的端粒，即使它们的长度明显短于它们的正常对应物。这意味着磨损率和伸长率在决定细胞最终是否以及何时致癌方面起着至关重要的作用。在这项研究中，我们提出了一个简洁的数学模型，该模型显示了每次细胞分裂时的最短端粒长度，并证明了与磨损和伸长率相关的数学条件，这是端粒长度平稳分布存在的必要和充分条件。此外，我们基于频率论和贝叶斯方法估计端粒长度维持过程的参数。这项研究扩展了我们对癌细胞中端粒磨损和伸长率之间数学关系的认识，这很重要，因为端粒长度动态是癌症诊断和预后的有用生物标志物。我们基于频率论和贝叶斯方法估计端粒长度维持过程的参数。这项研究扩展了我们对癌细胞中端粒磨损和伸长率之间数学关系的认识，这很重要，因为端粒长度动态是癌症诊断和预后的有用生物标志物。我们基于频率论和贝叶斯方法估计端粒长度维持过程的参数。这项研究扩展了我们对癌细胞中端粒磨损和伸长率之间数学关系的认识，这很重要，因为端粒长度动态是癌症诊断和预后的有用生物标志物。