Abstract
Noise in gene expression can be substantively affected by the presence of production delay. Here we consider a mathematical model with bursty production of protein, a one-step production delay (the passage of which activates the protein), and feedback in the frequency of bursts. We specifically focus on examining the steady-state behaviour of the model in the slow-activation (i.e. large-delay) regime. Using a formal asymptotic approach, we derive an autonomous ordinary differential equation for the inactive protein that applies in the slow-activation regime. If the differential equation is monostable, the steady-state distribution of the inactive (active) protein is approximated by a single Gaussian (Poisson) mode located at the globally stable fixed point of the differential equation. If the differential equation is bistable (due to cooperative positive feedback), the steady-state distribution of the inactive (active) protein is approximated by a mixture of Gaussian (Poisson) modes located at the stable fixed points; the weights of the modes are determined from a WKB approximation to the stationary distribution. The asymptotic results are compared to numerical solutions of the chemical master equation.
Highlights
Gene expression in individual cells involves the interaction of molecules which are present at low copy numbers (Eldar and Elowitz 2010; Munsky et al 2012)
The intrinsic noise generated by the low-copy-number reactions is passed down to the end product of gene expression, the protein, and results in temporal fluctuations and cell-to-cell heterogeneity of the protein copy number (Taniguchi et al 2010; Suter et al 2011)
Combining (8) and (41), we express the WKB approximation to the joint distribution p(x, s; ε) of the inactive protein concentration x and the active protein copy number s in the form of p(x, s; ε)
Summary
Gene expression in individual cells involves the interaction of molecules which are present at low copy numbers (Eldar and Elowitz 2010; Munsky et al 2012). Previous results indicate that large one-step (exponential) and multi-step (Erlang/phase-type) delays reduce the super-Poissonian noise in a bursty protein down to Poissonian levels (Singh and Bokes 2012; Stoeger et al 2016; Smith and Singh 2019). This effect is seen experimentally with buffered noise in cytoplasmic mRNA levels compared to nuclear mRNA levels due to transport delays (Battich et al 2015). We will use the WKB approximation to obtain reliable estimates of the stationary distribution of the active (and the inactive) protein in the slow activation (large-delay) regime.
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