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.


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


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Supplementary information

Supplementary data are available at Bioinformatics online.

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