Hierarchical Bayesian models of transcriptional and translational regulation processes with delays

Abstract

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.

Systematic inference identifies a major source of heterogeneity in cell signaling dynamics: The rate-limiting step number

“…However, for complex systems, estimation using low-order moments can be biased as it exploits only partial information of the system. This issue could be resolved by tracking all individual data points in the inference ( 59 ), which would be interesting in future work. Another important piece of follow-up research is to generalize the gamma hyper prior assumption of the mixed-effects model used in our and previous studies ( 25 , 60 ), because this makes it challenging to capture the multimodality of population distributions ( 61 ).…”

Inferring delays in partially observed gene regulatory networks

“…If we measure the trajectories at discrete time points t = 0,…, T , and denote these measurements by z , then according to Choi et al . (2020); Cortez et al . (2022), an approximate likelihood function, , is given by where r = ( r 1 ,…, r v ) is a vector of reaction counts completing within each of the time intervals, i.e., r k = ( r k 1 ,…, r kT ) and r ki is the count of reaction of type k that completes within the time interval ( i – 1, i ].…”

“…Recently, inference methods have been proposed based on the assumption that the unobserved processes are sequential, and thus can be modeled by introducing a delay (Jiang et al ., 2021; Heron et al ., 2007; Calderazzo et al ., 2018; Choi et al ., 2020; Cortez et al ., 2022; Barrio et al ., 2013; Leier et al ., 2014; Gomez et al ., 2016; Kim et al ., 2022). The resulting models are non-Markovian, as system dynamics depends not only on the present, but also past states.…”

“…Currently available Bayesian Markov chain Monte Carlo (MCMC) methods for inference of non-Markovian systems are not well suited to address this question. While these methods are applicable even to systems with high intrinsic noise (i.e., low copy numbers of molecules) (Choi et al, 2020) and cell-to-cell heterogeneity (Cortez et al, 2022), they rely on the assumption that all proteins in a GRN are observed. When some protein counts are unobserved, extending such methods directly requires that we characterize reactions involving the unobserved proteins at each measurement time.…”

DelaySSAToolkit.jl: stochastic simulation of reaction systems with time delays in Julia

“…In Figure 1, we summarize the workflow and showcase the software applied to an epidemic model, a birth-death stochastic model with heterogeneous delays (Cortez et al, 2021) and an auto-regulatory network of oscillatory gene expression with delays (Jiang et al, 2021).…”