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

Kumar N, Singh A, Kulkarni RV - PLoS Comput. Biol. (2015)

Bottom Line: 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.The proposed approaches can lead to new insights into transcriptional bursting based on measurements of steady-state mRNA/protein distributions.

View Article: PubMed Central - PubMed

Affiliation: Department of Physics, University of Massachusetts Boston, Boston, Massachusetts, United States of America.

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.

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Signatures for non-Poisson arrival.The quantities ,  and  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, km = 2, ⟨mb⟩ = 5, kp = 0.5, μm = 1, μp = 0.01.
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pcbi.1004292.g006: Signatures for non-Poisson arrival.The quantities , and 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, km = 2, ⟨mb⟩ = 5, kp = 0.5, μm = 1, μp = 0.01.

Mentions: To illustrate the prescription derived for determining non-Poisson arrival processes, we consider a specific kinetic scheme, Fig 2a. For this kinetic scheme, the mRNA arrival time distribution in the Laplace domain is given by (Eq (S3–9) Supplementary S3 Text)fL(s)=km(α+s)km(α+s)+s(α+β+s).(23)Using this in Eq (5) we find the gestation factor, Kg, and hence the mean, Fano factor and skewness for both mRNAs and proteins. Finally, we derive exact analytic expressions for , and from Eqs (20), (21) and (22) respectively. The expression for readsDm=2kmβ(1-⟨mb⟩)θ(1+(1+α)(α+β)⟨mb⟩kmθ(⟨mb⟩-1))(2+α+β)(θ+βkm)((2⟨mb⟩-1)θ+2km⟨mb⟩β),(24)whereθ=(α+β)(α+β+1),(25)and we have set μm = 1 for simplicity. As expected, we note that vanishes for the Poisson arrival processes, i.e., either when β is zero, or when the switching rates α and β are very large compared to the rate of transcription, km. The general expression for is complicated. However, to gain insight about the arrival process, we can write down a simpler expression for in the burst limit, μm = 1 ≫ μp:Dp=-2⟨mb⟩2km2kp2αβ(α+β)4+3⟨mb⟩kp(α+β)2ψ+2⟨mb⟩2kp2ψ2,(26)whereψ=kmβ+(α+β)2.(27)Again, for Poisson arrival processes vanishes. Finally, we obtain an analytic expression for , which is given byDmp=β(μp-μm)α(α+β+μm)(α+β+μp),(28)and as expected, we note that vanishes for Poisson arrivals and is negative for μp < μm. In Fig 6, we have plotted the three analytic expressions together with simulation results as a function of β.


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

Kumar N, Singh A, Kulkarni RV - PLoS Comput. Biol. (2015)

Signatures for non-Poisson arrival.The quantities ,  and  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, km = 2, ⟨mb⟩ = 5, kp = 0.5, μm = 1, μp = 0.01.
© Copyright Policy
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC4608583&req=5

pcbi.1004292.g006: Signatures for non-Poisson arrival.The quantities , and 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, km = 2, ⟨mb⟩ = 5, kp = 0.5, μm = 1, μp = 0.01.
Mentions: To illustrate the prescription derived for determining non-Poisson arrival processes, we consider a specific kinetic scheme, Fig 2a. For this kinetic scheme, the mRNA arrival time distribution in the Laplace domain is given by (Eq (S3–9) Supplementary S3 Text)fL(s)=km(α+s)km(α+s)+s(α+β+s).(23)Using this in Eq (5) we find the gestation factor, Kg, and hence the mean, Fano factor and skewness for both mRNAs and proteins. Finally, we derive exact analytic expressions for , and from Eqs (20), (21) and (22) respectively. The expression for readsDm=2kmβ(1-⟨mb⟩)θ(1+(1+α)(α+β)⟨mb⟩kmθ(⟨mb⟩-1))(2+α+β)(θ+βkm)((2⟨mb⟩-1)θ+2km⟨mb⟩β),(24)whereθ=(α+β)(α+β+1),(25)and we have set μm = 1 for simplicity. As expected, we note that vanishes for the Poisson arrival processes, i.e., either when β is zero, or when the switching rates α and β are very large compared to the rate of transcription, km. The general expression for is complicated. However, to gain insight about the arrival process, we can write down a simpler expression for in the burst limit, μm = 1 ≫ μp:Dp=-2⟨mb⟩2km2kp2αβ(α+β)4+3⟨mb⟩kp(α+β)2ψ+2⟨mb⟩2kp2ψ2,(26)whereψ=kmβ+(α+β)2.(27)Again, for Poisson arrival processes vanishes. Finally, we obtain an analytic expression for , which is given byDmp=β(μp-μm)α(α+β+μm)(α+β+μp),(28)and as expected, we note that vanishes for Poisson arrivals and is negative for μp < μm. In Fig 6, we have plotted the three analytic expressions together with simulation results as a function of β.

Bottom Line: 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.The proposed approaches can lead to new insights into transcriptional bursting based on measurements of steady-state mRNA/protein distributions.

View Article: PubMed Central - PubMed

Affiliation: Department of Physics, University of Massachusetts Boston, Boston, Massachusetts, United States of America.

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.

No MeSH data available.


Related in: MedlinePlus