Maintaining Gene Expression Levels by Positive Feedback in Burst Size in the Presence of Infinitesimal Delay

Abstract

Synthesis of individual molecules in the expression of genes often occurs in bursts of multiple copies. Gene regulatory feedback can affect the frequency with which these bursts occur or their size. Whereas frequency regulation has traditionally received more attention, we focus specifically on the regulation of burst size. It turns out that there are (at least) two alternative formulations of feedback in burst size. In the first, newly produced molecules immediately partake in feedback, even within the same burst. In the second, there is no within-burst regulation due to what we call infinitesimal delay. We describe both alternatives using a minimalistic Markovian drift-jump framework combining discrete and continuous dynamics. We derive detailed analytic results and efficient simulation algorithms for positive noncooperative autoregulation (whether infinitesimally delayed or not). We show that at steady state both alternatives lead to a gamma distribution of protein level. The steady-state distribution becomes available only after a transcritical bifurcation point is passed. Interestingly, the onset of the bifurcation is postponed by the inclusion of infinitesimal delay.

Heavy-tailed distributions in a stochastic gene autoregulation model

“…Earlier studies argued that the subtleties that arise with feedback in burst size are an artefact of delay [32,33]. Indeed, the memoryless property of the exponential distribution of burst sizes implies a lack of control at the infinitesimal timescale of burst growth.…”

“…The explicit stationary solution to the drift-jump model has been extended to the case of feedback in protein stability [31]. However, in case of feedback in burst size, an explicit solution is unavailable, save for the special case of Michaelis-Menten-type response [32].…”

Exact and WKB-approximate distributions in a gene expression model with feedback in burst frequency, burst size, and protein stability

“…With (35), bursts arrive as per a Poisson process with rate ε −1 , and the sample paths are proportional to e −t on any time interval between two consecutive bursts.…”

Noise induced bimodality in genetic circuits with monostable positive feedback

“…Regulation of burst frequency (Fig 1, right top), and partly also burst size (Fig 1, right bottom), have previously been examined using the aforementioned modelling framework [26]- [29]. In the context of burst-size regulation, two alternative versions have been proposed depending on the inclusion or omission of a so-called infinitesimal delay [30], [31]. Here we shall focus specifically on the undelayed case in the sense of [30], [31].…”

“…In the context of burst-size regulation, two alternative versions have been proposed depending on the inclusion or omission of a so-called infinitesimal delay [30], [31]. Here we shall focus specifically on the undelayed case in the sense of [30], [31].…”