Motivation Simultaneous recordings of gene network dynamics across large populations have revealed that cell characteristics vary considerably even in clonal lines. Inferring the variability of parameters that determine gene dynamics is key to understanding cellular behavior. However, this is complicated by the fact that the outcomes and effects of many reactions are not observable directly. Unobserved reactions can be replaced with time delays to reduce model dimensionality and simplify inference. However, the resulting models are non-Markovian, and require the development of new inference techniques. Results We propose a non-Markovian, hierarchical Bayesian inference framework for quantifying the variability of cellular processes within and across cells in a population. We illustrate our approach using a delayed birth–death process. In general, a distributed delay model, rather than a popular fixed delay model, is needed for inference, even if only mean reaction delays are of interest. Using in silico and experimental data we show that the proposed hierarchical framework is robust and leads to improved estimates compared to its non-hierarchical counterpart. We apply our method to data obtained using time-lapse microscopy and infer the parameters that describe the dynamics of protein production at the single cell and population level. The mean delays in protein production are larger than previously reported, have a coefficient of variation of around 0.2 across the population, and are not strongly correlated with protein production or growth rates. Availability and implementation Accompanying code in Python is available at https://github.com/mvcortez/Bayesian-Inference. Contact kresimir.josic@gmail.com or jaekkim@kaist.ac.kr or cbskust@korea.ac.kr Supplementary information Supplementary data are available at Bioinformatics online.

中文翻译：

---

具有延迟的转录和平移调节过程的分层贝叶斯模型

动机 同时记录大量人群的基因网络动态表明，即使在克隆系中，细胞特征也有很大差异。推断决定基因动力学的参数的可变性是理解细胞行为的关键。然而，由于许多反应的结果和影响无法直接观察到，这使情况变得复杂。可以用时间延迟代替未观察到的反应，以降低模型维度并简化推理。然而，由此产生的模型是非马尔可夫模型，需要开发新的推理技术。结果我们提出了一个非马尔可夫、分层贝叶斯推理框架，用于量化群体中细胞内和细胞间细胞过程的变异性。我们使用延迟的出生-死亡过程来说明我们的方法。一般来说，推理需要分布式延迟模型，而不是流行的固定延迟模型，即使只对平均反应延迟感兴趣。使用计算机和实验数据，我们表明所提出的分层框架是稳健的，并且与其非分层对应物相比可以改进估计。我们将我们的方法应用于使用延时显微镜获得的数据，并推断出描述单细胞和群体水平的蛋白质生产动态的参数。蛋白质生产的平均延迟比之前报道的要大，整个人群的变异系数约为 0.2，并且与蛋白质生产或增长率没有很强的相关性。可用性和实施​​ Python 中的附带代码可在 https://github.com/mvcortez/Bayesian-Inference 获得。联系克雷西米尔。