Bayesian inference of distributed time delay in transcriptional and translational regulation

Abstract

Motivation: Advances in experimental and imaging techniques have allowed for unprecedented insights into the dynamical processes within individual cells. However, many facets of intracellular dynamics remain hidden, or can be measured only indirectly. This makes it challenging to reconstruct the regulatory networks that govern the biochemical processes underlying various cell functions. Current estimation techniques for inferring reaction rates frequently rely on marginalization over unobserved processes and states. Even in simple systems this approach can be computationally challenging, and can lead to large uncertainties and lack of robustness in parameter estimates. Therefore we will require alternative approaches to efficiently uncover the interactions in complex biochemical networks.Results: We propose a Bayesian inference framework based on replacing uninteresting or unobserved reactions with time delays. Although the resulting models are non-Markovian, recent results on stochastic systems with random delays allow us to rigorously obtain expressions for the likelihoods of model parameters. In turn, this allows us to extend MCMC methods to efficiently estimate reaction rates, and delay distribution parameters, from single-cell assays. We illustrate the advantages, and potential pitfalls, of the approach using a birth-death model with both synthetic and experimental data, and show that we can robustly infer model parameters using a relatively small number of measurements. We demonstrate how to do so even when only the relative molecule count within the cell is measured, as in the case of fluorescence microscopy. Introduction 1The dynamics of intracellular processes are determined by the structure and rates of interactions between 2 different molecular species. However, stochasticity and limitations of experimental methods make it difficult 3 to infer the characteristics of these interactions from data. On the single-cell level, different molecular 4 species can occur in small number, correlate with phenotype, and localize within different parts of the cell. 5The resulting dynamics can thus be highly variable over time, and across the population. Averaging over 6 such fluctuations can lead to inaccurate representations of the underlying biology [1], and inference methods 7 therefore need to account for stochasticity within individual cells, and variability across the population [2-5].8 Different statistical approaches have been developed to fit stochastic models to data from single-cell 9 assays, offering a window into the dynamical processes within individual cells [6][7][8][9][10][11][12][13]. Among these, Bayesian 10 methods have been particularly promising. To apply Bayesian techniques, one typically assumes a model 11 for the network of interactions, postulates a prior over model parameters, and uses experimental data to 12 determine a posterior and estimates of unknown parameters [6,[13][14][15]. However, Bayesian approaches can 13 suffer from the curse of dimensionality [16], and are thus difficult to i...

Hierarchical Bayesian models of transcriptional and translational regulation processes with delays

“…Unexpectedly, we find that delay is not strongly correlated with production rate or growth rate. Our mean delay estimates are higher than those reported previously [Choi et al ., 2020]. Using synthetic data we show that this difference can be explained by a bias in mean delay estimates when assuming all cells are identical, and illustrate that the parameters inferred using a hierarchical approach reproduce the experimentally observed YFP dynamics in individual cells.…”

“…S1 e). Since dilution rates can be measured by tracking cell growth, death rates can be estimated from experimental data [Choi et al ., 2020]. Hence, inferring delays and birth rates from realistic amounts of data (e.g., 40 min in this case) is possible [Cheng et al ., 2017] if the B n can be directly measured.…”

“…Integrating with time, we have , and , where y denotes the trajectory of all chemical species from time 0 to time T , and θ and Δ denote the vector of all the rate and delay parameters, respectively. Building on this result, Choi et al [2020] derived likelihood functions for reaction rate and delay parameters for an individual stochastic process described by Eq. (1).…”

“…(6), we use Gibbs sampling: For every individual cell, n , we obtain samples for θ n and Δ n from their conditional posterior distributions by using the Metropolis-Hastings algorithm [Hastings, 1970; Metropolis et al ., 1953]. As described by Choi et al [2020], knowledge about the number of completed reactions, r nki , is needed in the sampling of these individual parameters. Since the discrete-time measurements are not sufficient to uniquely determine the number of reactions, we sample r nki in each Gibbs step.…”

“…These approaches, however, are effective only when molecule counts are high and stochastic differential equations accurately capture system dynamics [Gupta et al ., 2014]. Choi et al [2020] developed an alternative Bayesian approach using non-Markovian models to develop inference algorithms for rate and delay parameters in common biochemical reactions. This approach works well with synthetic and experimental data, and is effective even when molecule counts are low because the model is based on the chemical master equation.…”