Nascent RNA kinetics: Transient and steady state behavior of models of transcription

Regulation of transcription is a vital process in cells, but mechanistic details of this regulation still remain elusive. The dominant approach to unravel the dynamics of transcriptional regulation is to first develop mathematical models of transcription and then experimentally test the predictions these models make for the distribution of mRNA and protein molecules at the individual cell level. However, these measurements are affected by a multitude of downstream processes which make it difficult to interpret the measurements. Recent experimental advancements allow for counting the nascent mRNA number of a gene as a function of time at the single-cell level. These measurements closely reflect the dynamics of transcription. In this paper, we consider a general mechanism of transcription with stochastic initiation and deterministic elongation and probe its impact on the temporal behavior of nascent RNA levels. Using techniques from queueing theory, we derive exact analytical expressions for the mean and variance of the nascent RNA distribution as functions of time. We apply these analytical results to obtain the mean and variance of nascent RNA distribution for specific models of transcription. These models of initiation exhibit qualitatively distinct transient behaviors for both the mean and variance which further allows us to discriminate between them. Stochastic simulations confirm these results. Overall the analytical results presented here provide the necessary tools to connect mechanisms of transcription initiation to single-cell measurements of nascent RNA.

Figure 1

Model: An arbitrary promoter architecture with $N$ number of promoter states is considered. Here we show a model with five different states. The different promoter states are 1–5. The rate of transition from the $i\mathrm{th}$ to the $j\mathrm{th}$ state is given by $k_{ji}$. The rate of initiation from the first state is $r$. The rate of initiation from every other promoter states is zero. After initiation, each polymerase molecule traverses the gene at a speed $v_{\mathrm{EL}}$, where $L$ is the length of the gene. Time taken to traverse the gene is, therefore, $T=L/v_{\mathrm{EL}}$.

Figure 2

(a) Poisson initiation model. The promoter always remains active and initiates transcription at a rate $r$. Using our analytical results, we explore the mean and the Fano factor as functions of time, for this model. We also use Gillespie simulations to confirm the analytical results. To make these plots, we use the following parameters: $r=0.16/\min$, $v_{\mathrm{EL}}=0.8\mathrm{kb}/\min$, and $L=15\mathrm{kb}$, i.e., $T=18.75\min$. These parameters are characteristic of various genes (such as PDR5, MDN1, etc.) in yeast, as reported in [6]. (b) ON-OFF initiation model. The promoter switches between two states: an active and an inactive one. The rate of switching from the active state to the inactive state is $k_{\mathrm{OFF}}$, and from the inactive to the active state is $k_{\mathrm{ON}}$. From the active state, transcription initiation occurs at a rate $r$. Mean and Fano factor profiles: From the analytical expressions we have obtained, we explore the mean and Fano factor as functions of time. Simulations results are also shown. To illustrate this point, we use the following: $k_{\mathrm{ON}}=0.435/\min$, $k_{\mathrm{OFF}}=5$, $k_{\mathrm{INI}}=5/\min$, $L=4436\mathrm{bps}$, and $v_{\mathrm{EL}}=0.8\mathrm{kb}/\min$, which are characteristic of the PDR5 promoter in yeast, as reported in data published in [6]. (c) Two-step model of initiation. Initiation happens in two sequential steps: the formation of the preinitiation complex at the promoter occurs with rate $k_{\mathrm{LOAD}}$ followed by RNA polymerase escaping the promoter leading to an initiation event at rate $r$. Predictions for the mean and Fano factor profiles are shown as functions of time. Simulations results are shown. For the two-step model, we use $k_{\mathrm{LOAD}}=0.14/\min$, $r=0.14/\min$, and $v_{\mathrm{EL}}=0.8\mathrm{kb}/\min$, characteristic of the MDN1 promoter, which we find by analyzing the data reported in [25].