Transcriptional bursting in gene expression: analytical results for general stochastic models

Niraj Kumar; Abhyudai Singh; Rahul V. Kulkarni

Source-linked AI summary

Transcriptional bursting in gene expression: analytical results for general stochastic models

Niraj Kumar, Abhyudai Singh, Rahul V. Kulkarni

arXiv:1412.8634v1Dec 30, 2014q-bio.MNcond-mat.stat-mechphysics.bio-ph

TL;DR

The paper asks how steady-state data can identify departures from geometric burst sizes and Poisson burst arrivals in stochastic gene expression. It maps general gene-expression models to queueing systems to derive exact steady-state moments, then uses them to formulate experimentally measurable detection conditions and estimate burst parameters. The results apply across general kinetic schemes and can use mRNA, protein, or combined measurements.

Problem
Steady-state means and variances do not suffice to estimate burst parameters in general gene-expression models with complex promoter regulation, while burst arrivals and sizes may deviate from standard assumptions.
Method
The paper maps general gene-expression models with renewal-type mRNA arrivals to queueing systems and derives exact analytical expressions for mRNA and protein steady-state moments.
Results
The derived moments yield experimentally measurable conditions for detecting non-Poisson arrivals and non-geometric bursts, with analytical estimates agreeing well with simulations.
Takeaways & Limitations
The framework supports inference of arrival-process statistics and iterative estimation of burst parameters from mRNA steady-state data, protein steady-state data, or both.

Abstract

from arXiv · show

Gene expression in individual cells is highly variable and sporadic, often resulting in the synthesis of mRNAs and proteins in bursts. Bursting in gene expression is known to impact cell-fate in diverse systems ranging from latency in HIV-1 viral infections to cellular 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 more general stochastic models. To address this issue, we invoke a mapping between general 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 moments are then used to derive 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.

I. INTRODUCTION

The paper addresses whether steady-state measurements can reveal non-geometric burst sizes and non-Poisson burst arrivals in stochastic gene-expression models. It maps general expression models to queueing systems to derive exact moments and experimentally testable conditions.

Motivation: Single-cell experiments show sporadic transcriptional bursts that generate substantial variability and phenotypic diversity among genetically identical cells.These effects are relevant to cellular responses and bet-hedging across processes including viral infection and bacterial competence.
Motivation: Steady-state means and variances alone cannot estimate burst parameters for general gene-expression models, motivating higher-moment analysis.Complex promoter regulation can produce non-Poisson burst arrivals, while analytical higher-moment results had been unavailable.
Model scope: The paper considers general mRNA burst arrival processes and arbitrary burst distributions rather than assuming exponential waiting times or geometric burst sizes.The model allows mRNAs to produce proteins at rate k_p and permits arbitrary protein burst distributions under independent production.
Main outcomes: The derived moments provide experimentally measurable conditions for detecting non-geometric bursts or non-Poisson arrivals and support burst-parameter estimation.The conditions can use mRNA steady-state data, protein steady-state data, or both, while the expressions also describe noise contributions from arrival and burst-distribution deviations.
Analytical approach: Mapping gene-expression dynamics to queueing theory yields exact analytical expressions for steady-state mRNA and protein moments.mRNAs and proteins correspond to queued customers, burst production to batch arrivals, and degradation to service completion; exact GIX/M/∞ results provide the moment expressions.
Validation: The analytical estimates for the two-state model agree well with simulation results.The comparison is shown for the protein steady-state variance and third central moment.

IV. SIGNATURES FOR NON-GEOMETRIC BURSTS

The paper derives moment-based signatures for detecting deviations from conditional geometric mRNA bursts, including under arbitrary burst-arrival processes.

The conditional geometric framework unifies single-mRNA arrivals with geometrically distributed bursts through the mean burst-size parameter.
The signature is derived assuming only a conditional geometric mRNA burst distribution and uses the first three steady-state moments.
A deviation of Gm from 1 identifies non-geometric mRNA bursts using experimentally measurable steady-state moments.The criterion requires measurements at mRNA degradation rates µm and 2µm.
For negative binomial bursts, r = 1 recovers the geometric distribution and yields Gm = 1, whereas non-geometric cases produce deviations.
An analogous condition can be derived from protein steady-state moments to assess geometric burst distributions.

V. SIGNATURES FOR NON-POISSON ARRIVALS

The paper examines whether burst arrivals follow a Poisson process while retaining conditional geometric mRNA bursts and geometric protein bursts.

The analysis tests Poisson-arrival assumptions using mRNA data, protein data, or both types of steady-state distribution measurements.
The conditional geometric mRNA model accommodates both single mRNA arrivals and geometric mRNA bursts within the arrival-process analysis.
The protein burst distribution is taken to be geometric, consistent with multiple experimental observations.

A. Using moments of mRNA steady-state distributions

mRNA steady-state moments provide experimentally measurable signatures for rejecting Poisson burst arrivals.

Under Poisson arrivals, the gestation factors at µm and 2µm equal 1, linking the mRNA Fano factor to the mean burst size.
A nonzero Dm is a signature of non-Poisson mRNA burst arrivals.
The mRNA criterion uses measurable quantities including the mean, variance, skewness, and degradation rate.

B. Using moments of protein steady-state distributions

Protein moments and combined mRNA–protein measurements provide alternative signatures for non-Poisson burst arrivals, including one that does not require geometric mRNA bursts.

A nonzero Dp identifies non-Poisson arrivals using protein steady-state moments.
The combined quantity Dmp detects non-Poisson arrivals without assuming geometric mRNA bursts or measuring third moments.
Signatures for a simple kinetic scheme: For the illustrated kinetic scheme, Dm vanishes in Poisson limits including β = 0 or switching rates much faster than transcription.
Signatures for a simple kinetic scheme: For Poisson arrivals, Dp and Dmp vanish; in the burst limit, Dmp is negative when µp < µm.

VI. ESTIMATION OF BURST PARAMETERS

The paper develops procedures for estimating burst parameters when mRNA burst arrivals and sizes may follow general stochastic processes. These procedures use sequence-size functions, steady-state moments, and iterative waiting-time representations validated against simulations.

A. Estimation of mean burst size from f(t): The sequence-size function categorizes intervals longer than τ as separating bursts, and its value at τx estimates the actual mean burst size.This relies on a separation between within-burst and between-burst timescales.
A. Estimation of mean burst size from f(t): Mean burst size is estimated as twice the sequence-size function at its inflection point, τx, when f(t) is available.The sequence-size function is obtained from the waiting-time distribution for single-mRNA arrivals.
B. Estimation of f(t) from steady-state moments: The estimation approach was validated by stochastic simulations for promoter models with complex waiting-time distributions between bursts.Figure 4 illustrates estimation of mean burst size for two transcriptional schemes.
B. Estimation of f(t) from steady-state moments: For experimentally inaccessible f(t), the paper estimates it from steady-state mRNA or protein moments using a rational Laplace-transform representation of g(t).The representation is consistent with phase-type waiting-time distributions and requires m + n + 2 measurements for its parameters.
B. Estimation of f(t) from steady-state moments: An iterative procedure starts with the simplest waiting-time representation, tests higher-moment predictions, and adds kinetic complexity when observations are inconsistent.A two-step representation can estimate burst size well even when it is only an approximate reduction of a more complex promoter process.

VII. DISCUSSION

The paper maps general renewal-type mRNA arrival models to queueing systems to derive experimentally testable steady-state moment conditions. These conditions identify non-Poisson arrivals and support iterative estimation of burst parameters from mRNA, protein, or combined measurements.

VII. DISCUSSION: The queueing-theory mapping yields analytical mRNA and protein steady-state moments for general renewal-type mRNA arrival processes.The results include exact expressions for the moments of both distributions.
VII. DISCUSSION: Analytic conditions based on experimentally measurable quantities can identify whether mRNA burst arrivals are non-Poisson.Measurements of mRNA distributions, protein distributions, or both can be used.
VII. DISCUSSION: When arrivals are non-Poisson, the derived steady-state results support accurate burst-parameter estimation through an iterative approach.The approach is intended for use with available experimental measurements.

Appendix A: Derivation of steady-state moments for mRNAs and proteins

The appendix derives steady-state mRNA and protein moments by mapping gene expression to a GIX/M/∞ queue. It provides exact mRNA expressions, burst-limit protein expressions, and simulation-supported higher-moment approximations.

Derivation framework: The GIX/M/∞ mapping provides exact steady-state moments for gene-expression models by translating queueing-theory results into mRNA and protein distributions.The appendix explicitly provides the first four steady-state moments through this mapping.
mRNA moments: The first three mRNA copy-number moments derived from burst statistics are exact across all parameter ranges.The burst-size parameters are expressed using moments of the mRNA burst size.
Protein moments: Protein variance and skewness expressions are exact in the burst limit µm ≫ µp and approximate beyond that limit.The protein burst parameters account for the random number of proteins produced by each mRNA.
Higher moments: The appendix extends the derivation to fourth central moments, with exact mRNA results and burst-limit protein results.Beyond the burst limit, protein fourth moments are approximated by matching burst-limit and two-stage-model results.
Higher moments: Analytical fourth-moment estimates show good agreement with simulation results for the illustrated protein model.Figure 5 compares analytic lines with simulation points using α = 0.5, β = 0.25, km = 2, ⟨mb⟩ = 5, and kp = 0.5.

Appendix B: Illustrative examples for condition identifying non-geometric bursts

This appendix section introduces illustrative examples for the condition identifying non-geometric mRNA burst distributions.

Appendix B: Illustrative examples for condition identifying non-geometric bursts: The section considers illustrative examples for the condition relating to the assumption of geometric mRNA burst distributions.

a. Poisson arrival of negative binomial bursts

For Poisson arrival with negative-binomial bursts, the analytical moment condition gives Gm = 1 for geometric bursts and detects deviations otherwise; the two-state telegraph model satisfies the geometric-burst condition.

a. Poisson arrival of negative binomial bursts: Gm = 1 for geometric bursts, whereas non-geometric bursts produce deviations of Gm from 1.The geometric case corresponds to r = 1; deviations are illustrated in Fig. 6.
b. Two-state random telegraph model: The two-state random telegraph model switches between OFF and ON states, produces mRNA in the ON state, and includes mRNA degradation.The switching rates are α and β, transcription occurs at rate km, and degradation occurs at rate µm.
b. Two-state random telegraph model: The waiting-time distribution for mRNA-burst arrival is obtained from the first-passage distribution of production after the gene starts active.The calculation uses the OFF and ON probabilities and their master equation in the Laplace domain.
b. Two-state random telegraph model: Using the resulting waiting-time transform, the first three mRNA moments yield Gm = 1 for the two-state random telegraph model.This agrees with the model's geometric burst distribution.

c. Transcription from two promoter states

When mRNAs are produced from two promoter states, combining both transcriptional routes produces deviations from the geometric-burst condition, quantified through Gm.

c. Transcription from two promoter states: Using both promoter transcriptional routes produces deviations from Gm = 1, unlike either route alone.The model has promoter states D0, D1, and D2, with mRNA production from D1 and D2 at rates km1 and km2.
c. Transcription from two promoter states: The three-state promoter model derives steady-state mRNA moments by summing its probability evolution equations over m.The first three moments are obtained from the corresponding moment equations.
c. Transcription from two promoter states: Gm approaches 1 when km2 = 0, but significant deviations appear as transcription from the second promoter state increases.For fixed remaining parameters, Fig. 8 shows Gm as a function of km2 for two α values.

Appendix C: Condition for non-geometric bursts using protein steady-state moments

In the protein burst limit, steady-state protein moments provide a condition for geometric bursts: Gp ≠ 1 indicates deviations in both protein and mRNA burst distributions, while the telegraph model gives Gp = 1.

Appendix C: Condition for non-geometric bursts using protein steady-state moments: In the burst limit, protein steady-state moments yield an experimentally measurable condition for geometric protein bursts.The condition applies when mRNA degradation is much faster than protein degradation.
Appendix C: Condition for non-geometric bursts using protein steady-state moments: Gp ≠ 1 indicates that both protein and mRNA burst distributions differ from the geometric distribution.A geometric protein burst distribution implies a conditional geometric underlying mRNA burst distribution.
Appendix C: Condition for non-geometric bursts using protein steady-state moments: For the two-state random telegraph model, substituting the protein moments into the condition gives Gp = 1.This agrees with the model's geometric protein-burst distribution.
Appendix C: Condition for non-geometric bursts using protein steady-state moments: The appendix tests the two-state telegraph representation using first three mRNA moments to estimate parameters and a fourth-moment relation.The resulting diagnostic is expressed as an equation involving the fourth central moment and the mRNA Fano factor.