Population density approach for discrete mRNA distributions in generalized switching models for stochastic gene expression

Adam R. Stinchcombe, Charles S. Peskin, and Daniel Tranchina

Phys. Rev. E 85, 061919 – Published 22 June 2012

Abstract

We present a generalization of a population density approach for modeling and analysis of stochastic gene expression. In the model, the gene of interest fluctuates stochastically between an inactive state, in which transcription cannot occur, and an active state, in which discrete transcription events occur; and the individual mRNA molecules are degraded stochastically in an independent manner. This sort of model in simplest form with exponential dwell times has been used to explain experimental estimates of the discrete distribution of random mRNA copy number. In our generalization, the random dwell times in the inactive and active states, $T_{0}$ and $T_{1}$ , respectively, are independent random variables drawn from any specified distributions. Consequently, the probability per unit time of switching out of a state depends on the time since entering that state. Our method exploits a connection between the fully discrete random process and a related continuous process. We present numerical methods for computing steady-state mRNA distributions and an analytical derivation of the mRNA autocovariance function. We find that empirical estimates of the steady-state mRNA probability mass function from Monte Carlo simulations of laboratory data do not allow one to distinguish between underlying models with exponential and nonexponential dwell times in some relevant parameter regimes. However, in these parameter regimes and where the autocovariance function has negative lobes, the autocovariance function disambiguates the two types of models. Our results strongly suggest that temporal data beyond the autocovariance function is required in general to characterize gene switching.

Received 19 April 2012

DOI:https://doi.org/10.1103/PhysRevE.85.061919

Authors & Affiliations

Adam R. Stinchcombe^*, Charles S. Peskin^†, and Daniel Tranchina^‡

Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, New York 10012, USA

^*stinch@courant.nyu.edu
^†peskin@courant.nyu.edu
^‡tranchina@courant.nyu.edu; Department of Biology, New York University, New York, New York 10003, USA.

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Vol. 85, Iss. 6 — June 2012

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Images

Figure 1
Steady-state mRNA distributions for four sets of parameters of the exponential dwell times model and the gamma dwell times model: $(δ / λ, δ / γ)$ and $(k_{0}, θ_{0} δ, k_{1}, θ_{1} δ)$ with $ν / δ = 10$ for both models. The probability mass function for the random mRNA copy number in the fully discrete problem for the exponential dwell times model (black bars) and the gamma dwell times model (white bars) are shown along with the parent probability density function from the continuous problem for the exponential dwell times model (solid) and the gamma dwell time model (dashed). The left column corresponds to fast activation (FA) and the right to slow activation (SA), while the first row corresponds to fast inactivation (FI) and the second to slow inactivation (SI).Reuse & Permissions
Figure 2
Details of the parameter estimation process for the parameters: $(k_{0}, θ_{0} δ, k_{1}, θ_{1} δ, δ / ν) = (2.0, 2.5, 2.0, 2.5, 0.10)$ , $({(δ / λ)}^{M L}, {(δ / γ)}^{M L}, {(δ / ν)}^{M L}) = (4.0, 4.0, 0.11)$ . (a) The white bars are the underlying distribution, the gray bars are a histogram of the simulated data from 1000 cells, and the black bars are the probability mass function for the maximum likelihood parameters of the exponential dwell times model. (b) Histogram of the -log-likelihood values of 1000 data sets, each consisting of simulated measurements of mRNA copy numbers in 1000 cells. The bin containing the likelihood of the data above (2475) is identified. (c) Contours of log-likelihood as a function of model parameters. Slices of the 3D function occur at $λ / δ = {(λ / δ)}^{M L}$ , $γ / δ = {(γ / δ)}^{M L}$ , and $ν / δ = {(ν / δ)}^{M L}$ . The largest contour value is $- 2550$ , and the contour values have a uniform spacing of 130.Reuse & Permissions
Figure 3
mRNA autocovariance functions for the exponential dwell times model (solid) and the gamma dwell times model (dashed) for the four parameter sets in Fig. 1.Reuse & Permissions
Figure 4
Steady-state autocovariance functions. (a) $Φ_{Z Z} (τ)$ for maximum likelihood parameters of the exponential dwell times model (solid) with $Φ_{Z Z} (τ)$ of the model underlying the simulated data (dashed). The open circles are an estimate of the autocovariance function from a Monte Carlo simulation. (b) Same for $Φ_{M M} (τ)$ .Reuse & Permissions
Figure 5
Contour plot of the extreme negative value of the autocovariance function, $Φ_{M M}$ , as a fraction of its value at zero in the gamma dwell times model with $k_{0} = 5$ and $k_{1} = 2$ over a range of mean dwell times in active and inactive gene states. The contours range from 0 $%$ (black) to 16 $%$ (white) in increments of 0.3 $%$ . Shown below are the autocovariance functions with parameters given by the corresponding dot above. The cross corresponds to the autocovariance function from Fig. 3b.Reuse & Permissions

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