Transcription stochasticity of complex gene regulation models

Abstract

Transcription is regulated by a multitude of factors that concertedly induce genes to switch between activity states. Eukaryotic transcription involves a multitude of complexes that sequentially assemble on chromatin under the influence of transcription factors and the dynamic state of chromatin. Prokaryotic transcription depends on transcription factors, sigma-factors, and, in some cases, on DNA looping. We present a stochastic model of transcription that considers these complex regulatory mechanisms. We coarse-grain the molecular details in such a way that the model can describe a broad class of gene-regulation mechanisms. We solve this model analytically for various measures of stochastic transcription and compare alternative gene-regulation designs. We find that genes with complex multiprotein regulation can have peaked burst-size distributions in contrast to the geometric distributions found for simple models of transcription regulation. Burst-size distributions are, in addition, shaped by mRNA degradation during transcription bursts. We derive the stochastic properties of genes in the limit of deterministic switch times. These genes typically have reduced transcription noise. Severe timescale separation between gene regulation and transcription initiation enhances noise and leads to bimodal mRNA copy number distributions. In general, complex mechanisms for gene regulation lead to nonexponential waiting-time distributions for gene switching and transcription initiation, which typically reduce noise in mRNA copy numbers and burst size. Finally, we discuss that qualitatively different gene regulation models can often fit the same experimental data on single-cell mRNA abundance even though they have qualitatively different burst-size statistics and regulatory parameters.

Figure 1

Molecular-ratchet model of the basal design of eukaryotic transcription. (a) A eukaryotic gene is displayed that switches between on- and off-states via a sequence of transitions that involve reversible protein complex formation on chromatin followed by covalent-histone modifications that mark progress and sensitize chromatin for the next protein complex assembly. (b) Coarse-grained view of the molecular ratchet where the duration distributions of the on- and off-states and transcription initiation (involving PIC formation, open complex formation, and promotor escape) are given by general first-passage time distributions f(t), g(t), and h(t). The first-passage time distributions can be obtained from a molecular mechanism for ratchet transitions.

Figure 2

First-passage time (duration) distribution for a single-transition mechanism. A single-ratchet transition composed out of reversible protein complex assembly followed by irreversible covalent histone modification has a peaked waiting time distribution (solid line) that can be approximated by a gamma-distribution with the same mean and variance (dotted line).

Figure 3

Nonexponential on-state time distributions make the burst-size distribution peaked. The on-to-off transition and the initiation mechanism were modeled with Erlang (solid, light-shaded) or exponential (dark-shaded) distributions. The average duration of the on-state was kept constant while the initiation rate was adjusted to achieve a mean burst-size of 25 initiations per on-phase. The initiation time is exponentially distributed (solid line) and Erlang-distributed (light-shaded line). A nonexponentially distributed initiation time has a small effect on the burst-size dispersion.

Figure 4

High mRNA turnover or nonexponentially distributed on-lifetime can make the effective burst-size distribution peaked. All models have an exponential distribution for the initiation mechanism. The on-to-off transition was modeled with an Erlang distribution (N = 10, dotted black line) and with exponential distributions otherwise with a fixed average duration. For all models the initiation rate constant was adjusted to fix the mean effective burst-size to 25. The average mRNA lifetimes are 10τ_on (solid black, dotted black), 1τ_on (dark-shaded), and 0.1τ_on (light-shaded). At high turnover (light-shaded), mRNA can almost attain its steady-state level given transcription and degradation kinetics and the mRNA burst-size becomes peaked. At low turnover of mRNA (black lines), the burst-size can become peaked for nonexponentially distributed on-to-off transition durations (dotted black).

Figure 5

Stationary mRNA copy number distributions for the deterministic gene switch as function of on- and off-durations. A gene controlled by a deterministic switch with transcription rate 0.5 mRNA min⁻¹ and a mRNA lifetime of 40 min. If the gene is always in the on-state, the mRNA is Poisson-distributed with mean 20 molecules per cell (black). With 40 min in the on- and the off-state, the mean mRNA level goes down (dark-shaded). With infrequent switching (100 min in on- and off-state), the mRNA distribution becomes bimodal (light-shaded). When the gene is 1 min in the on-state and 60 min in the off-state, the distribution becomes nearly exponential with a mean of ≈0.3 molecules per cell (dashed black).

Figure 6

Stationary single-cell mRNA data contain only very limited information about the transcription mechanism. (a) Lifetime distributions of on- and off-states that were determined from choosing a combination of shape parameters for both states and then determining the rate parameters for both lifetime distributions as well as the rate constant for h(t) by matching the first three moments of the experimental data: (i.) (Red) N_on = 1, N_off = 1. (ii.) (Blue) N_on = 1, N_off = 10. (iii.) (Green) N_on = 5, N_off = 5; (magenta) N_on = 10, N_off = 1; and (orange) N_on = 10, N_off = 10. (b) Experimental data for PDR5 from Chubb et al. (8) (shaded bars) and the mRNA distributions of the fitted models. (c) Burst size distributions vary greatly between the fitted models. (d) Time-traces for the models described in panel a. (Colored lines) On- and off-states. (Black lines) Time-traces of simulations (50). (Numbers on top) Burst sizes.

Figure 7

Parameters estimated with the two-state model with exponentially distributed waiting times can be misleading. (a) Histogram shows the copy number distribution obtained by simulating a model with nonexponentially distributed lifetimes for on- and off-state. Both distributions were modeled as Erlang distributions with four steps and average times of 〈t_on〉 = 1 and 〈t_off〉 = 3. (Solid line) Best fit to the first three moments of the simulated data using the exponential two-state model. (Dashed line) Distribution for an exponential two-state model that has the same average lifetimes for on- and off-state as the nonexponential model used to simulate the data. (b–d) Contours show 99% and 90% confidence levels for a parameter fit to the exponential two-state model. (Solid dot) Best fit. (Asterisk) Parameters of the nonexponential model used to generate the data.