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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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Related in: MedlinePlus

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 ν3 (scaled by 10−6) of proteins as a function of μm/μp 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, km = 2, ⟨mb⟩ = 5, kp = 0.5.
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pcbi.1004292.g002: 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 ν3 (scaled by 10−6) of proteins as a function of μm/μp 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, km = 2, ⟨mb⟩ = 5, kp = 0.5.

Mentions: The analytical results derived above for proteins are exact in the burst limit, which assumes that proteins are produced instantaneously from all the mRNAs in a burst. Going beyond the burst limit (i.e. not limited to μm ≫ μp), exact results for the higher moments of the protein steady-state distribution will, in general, depend on the details of the kinetic scheme for gene expression. However, we can derive approximate analytical expressions for general schemes by requiring that: a) the results reduce to the exact results in the burst limit and b) they match the exact results for the two-stage model of gene expression. For the two-stage model, exact results for the first four moments have been derived by Bokes et. al [55]. Comparing these exact results with our results derived in the burst limit, we observe that results of [55] can be reproduced by a suitable scaling of the burst-size parameters . For example, the exact expression for the noise is obtained by the following scaling [43]:(σps2⟨ps⟩2-1⟨ps⟩)→(σps2⟨ps⟩2-1⟨ps⟩)11+μpμm.(11)Similarly, for the expression for skewness, the parameters and are scaled as:A2p→A2p11+μpμmandA3p→A3p1(1+μpμm)(1+2μpμm).(12)As shown in Fig 2 (for the random telegraph model) the analytical expressions using this approach are in good agreement with results from simulations [56].


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

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

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 ν3 (scaled by 10−6) of proteins as a function of μm/μp 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, km = 2, ⟨mb⟩ = 5, kp = 0.5.
© Copyright Policy
Related In: Results  -  Collection

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getmorefigures.php?uid=PMC4608583&req=5

pcbi.1004292.g002: 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 ν3 (scaled by 10−6) of proteins as a function of μm/μp 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, km = 2, ⟨mb⟩ = 5, kp = 0.5.
Mentions: The analytical results derived above for proteins are exact in the burst limit, which assumes that proteins are produced instantaneously from all the mRNAs in a burst. Going beyond the burst limit (i.e. not limited to μm ≫ μp), exact results for the higher moments of the protein steady-state distribution will, in general, depend on the details of the kinetic scheme for gene expression. However, we can derive approximate analytical expressions for general schemes by requiring that: a) the results reduce to the exact results in the burst limit and b) they match the exact results for the two-stage model of gene expression. For the two-stage model, exact results for the first four moments have been derived by Bokes et. al [55]. Comparing these exact results with our results derived in the burst limit, we observe that results of [55] can be reproduced by a suitable scaling of the burst-size parameters . For example, the exact expression for the noise is obtained by the following scaling [43]:(σps2⟨ps⟩2-1⟨ps⟩)→(σps2⟨ps⟩2-1⟨ps⟩)11+μpμm.(11)Similarly, for the expression for skewness, the parameters and are scaled as:A2p→A2p11+μpμmandA3p→A3p1(1+μpμm)(1+2μpμm).(12)As shown in Fig 2 (for the random telegraph model) the analytical expressions using this approach are in good agreement with results from simulations [56].

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