A spatio-temporal model to reveal oscillator phenotypes in molecular clocks: Parameter estimation elucidates circadian gene transcription dynamics in single-cells

We propose a stochastic distributed delay model together with a Markov random field prior and a measurement model for bioluminescence-reporting to analyse spatio-temporal gene expression in intact networks of cells. The model describes the oscillating time evolution of molecular mRNA counts through a negative transcriptional-translational feedback loop encoded in a chemical Langevin equation with a probabilistic delay distribution. The model is extended spatially by means of a multiplicative random effects model with a first order Markov random field prior distribution. Our methodology effectively separates intrinsic molecular noise, measurement noise, and extrinsic noise and phenotypic variation driving cell heterogeneity, while being amenable to parameter identification and inference. Based on the single-cell model we propose a novel computational stability analysis that allows us to infer two key characteristics, namely the robustness of the oscillations, i.e. whether the reaction network exhibits sustained or damped oscillations, and the profile of the regulation, i.e. whether the inhibition occurs over time in a more distributed versus a more direct manner, which affects the cells’ ability to phase-shift to new schedules. We show how insight into the spatio-temporal characteristics of the circadian feedback loop in the suprachiasmatic nucleus (SCN) can be gained by applying the methodology to bioluminescence-reported expression of the circadian core clock gene Cry1 across mouse SCN tissue. We find that while (almost) all SCN neurons exhibit robust cell-autonomous oscillations, the parameters that are associated with the regulatory transcription profile give rise to a spatial division of the tissue between the central region whose oscillations are resilient to perturbation in the sense that they maintain a high degree of synchronicity, and the dorsal region which appears to phase shift in a more diversified way as a response to large perturbations and thus could be more amenable to entrainment.Author summary: We develop a method for estimating and modelling transcriptional processes in living tissue samples using a model which approximates the end product of a series of complex chemical interactions using a density function—and thus is amenable to parameter estimation—but can realistically account for the intrinsic noise and rhythm generation inherent in the single-cell. The model incorporates a form of dependence between nearby cells using a spatial prior distribution over the parameters. The model thus describes the cyclical behaviour of the production of the population of some molecular species within cells, along with the spatial variation of the process across a network of cells. This approach is suitable for modelling circadian gene expression in the suprachiasmatic nucleus (SCN), the region of the brain which is responsible for the ‘circadian master clock’ which coordinates the bodies daily rhythms. This model is applied to three sample tissues from mice SCN. Based on the inferred dynamic behaviour of the cells, we are able to divide the organ into two regions: a central core in which the rhythm is intrinsic and resilient and the more entrainable outer region which is much more heavily influenced by external stimuli. The findings highlight a trade-off between resilient cyclic behaviour and ability to adapt to environmental cues.