Transcriptional Bursting in Gene Expression: Analytical Results for General Stochastic Models

Abstract

Gene expression in individual cells is highly variable and sporadic, often resulting in the synthesis of mRNAs and proteins in bursts. Such bursting has important consequences for cell-fate decisions in diverse processes ranging from HIV-1 viral infections to stem-cell differentiation. It is generally assumed that bursts are geometrically distributed and that they arrive according to a Poisson process. On the other hand, recent single-cell experiments provide evidence for complex burst arrival processes, highlighting the need for analysis of more general stochastic models. To address this issue, we invoke a mapping between general stochastic models of gene expression and systems studied in queueing theory to derive exact analytical expressions for the moments associated with mRNA/protein steady-state distributions. These results are then used to derive noise signatures, i.e. explicit conditions based entirely on experimentally measurable quantities, that determine if the burst distributions deviate from the geometric distribution or if burst arrival deviates from a Poisson process. For non-Poisson arrivals, we develop approaches for accurate estimation of burst parameters. The proposed approaches can lead to new insights into transcriptional bursting based on measurements of steady-state mRNA/protein distributions.

Fig 1. Kinetic scheme for the gene expression with general arrival time distributions.

Bursts of mRNAs arrive with a general arrival time distributions f(t). Each mRNA produces proteins with rate k _p and mRNAs and proteins decay with rates μ _m and μ _p, respectively.

Fig 2. Steady state moments for proteins.

(a) Kinetic scheme for the two-state random telegraph model. For this model, steady state variance (scaled by 10⁻⁵) and third central moment ν ₃ (scaled by 10⁻⁶) of proteins as a function of μ _m/μ _p are plotted in (b) and (c) respectively: lines represent analytic estimates and points correspond to the simulation results. Parameters are: α = 0.5, β = 0.25, k _m = 2, ⟨m _b⟩ = 5, k _p = 0.5.

Fig 3. Schematic representation of the general kinetic scheme with promoter switching.

Thick line from inactive state D ₀ to active state D _a represents a general kinetic scheme with g(t) as the waiting-time distribution for the promoter to switch to the ON state.

Fig 4. Estimation of mean burst size from sequence size function ϕ(τ).

For the transcriptional scheme shown in (a), the variations of ϕ″(τ) and ϕ(τ) as a function of time τ (scaled by 10³) are shown in (b) and (c) respectively. The three lines correspond to three different values of β, 50 (dashed line), 100 (dotted line) and 200 (dashed-dotted line), while keeping k _m = 500: Exact burst size for these three cases are 11, 6 and 3.5, respectively. Estimated mean burst size has been indicated by filled symbols and the inflexion points in the sequence size function are shown by empty symbols. Other parameters: α ₁ = 1,α ₂ = 0.5,α ₃ = 0.25,α ₄ = 0.75,β ₁ = 0.1,β ₂ = 0.2,β ₃ = 0.5.

Fig 5. Effects of extrinsic noise on burst estimation.

For the transcriptional scheme shown in the inset, the relative error Δ_σ(⟨m _b⟩) = (⟨m _b⟩₀−⟨m _b⟩_σ)/⟨m _b⟩₀ is plotted. Parameters as α ₁ = 1, α ₂ = 0.5, β = 50, ⟨k _m⟩ = 500 and μ _m = 1.

Fig 6. Signatures for non-Poisson arrival.

The quantities ${\mathcal{D}}_m$, ${\mathcal{D}}_p$ and ${\mathcal{D}}_{mp}$ are plotted for the model shown in Fig 2a as a function of off rate β. Analytic estimates are shown by lines whereas points correspond to the simulation results with parameters: α = 0.25, k _m = 2, ⟨m _b⟩ = 5, k _p = 0.5, μ _m = 1, μ _p = 0.01.