Urn models for stochastic gene expression yield intuitive insights into the probability distributions of single-cell mRNA and protein counts

Abstract

Fitting the probability mass functions from analytical solutions of stochastic models of gene expression to the single-cell count distributions of mRNA and protein molecules can yield valuable insights into mechanisms underlying gene expression. Solutions of chemical master equations are available for various kinetic schemes but, even for the basic ON-OFF genetic switch, they take complex forms with generating functions given as hypergeometric functions. Interpretation of gene expression dynamics in terms of bursts is not consistent with the complete range of parameters for these functions. Physical insights into the probability mass functions are essential to ensure proper interpretations but are lacking for models considering genetic switches. To fill this gap, we develop urn models for stochastic gene expression. We sample RNA polymerases or ribosomes from a master urn, which represents the cytosol, and assign them to recipient urns of two or more colors, which represent time intervals in which no switching occurs. Colors of the recipient urns represent sub-systems of the promoter states, and the assignments to urns of a specific color represent gene expression. We use elementary principles of discrete probability theory to solve a range of kinetic models without feedback, including the Peccoud-Ycart model, the Shahrezaei-Swain model, and models with an arbitrary number of promoter states. In the last case, we obtain a novel result for the protein distribution. For activated genes, we show that transcriptional lapses, which are events of gene inactivation for short time intervals separated by long active intervals, quantify the transcriptional dynamics better than bursts. We show that the intuition gained from our urn models may also be useful in understanding existing solutions for models with feedback. We contrast our models with urn models for related distributions, discuss a generalization of the Delaporte distribution for single-cell data analysis, and highlight the limitations of our models.

Coordination of gene expression noise with cell size: analytical results for agent-based models of growing cell populations

“…Similarly, bursty protein expression from a two-state promoter arising from a three-stage gene expression model [54,56] (figure 2c),…”

“…1 This follows from the fact that F(x) = 2 F 1 (a, λ; γ + λ; b x) is the moment-generating function of κ ∼ Gamma(a, (1 + b r) −1 ) and r ∼ Beta(λ, γ). Letting E P [z p js] ¼ F(s(z À 1)), a = a − /α, λ = a − α and λ + γ = (k on + k off )/α then yields the factorial-moment generating function solutions in [54,56].…”

Coordination of gene expression noise with cell size: analytical results for agent-based models of growing cell populations

“…Similarly, bursty protein expression from a two-state promoter arising from a three-stage gene expression model 54,56 (Fig. 2c),…”

Stationarity and inference in multistate promoter models of stochastic gene expression via stick-breaking measures

“…Such an ansatz corresponds to the idea that each individual mRNA independently produces protein at rate α > 0 and each protein degrades at rate γ > 0. A more involved model, which we leave to future consideration, would involve 'feedback' between the protein levels and the promoter-mRNA dynamics A version of this model was indicated in [19], and stationary distributions in the 'refractory' case, when only one β i is positive, were found in [6]. In this context, our goal will be to derive the stationary distribution in the general (β, δ, α, γ, G) model through the stick-breaking apparatus.…”

“…We mention, the identification of the stationary distribution in such a model, even in this 'non-feedback' network, was posed as an open problem in [19]; we leave to future work to consider ramifications in models with 'feedback'. Recently, in [6], protein interactions have been considered in the 'refractory' case where β i > 0 for exactly one state E = i in terms of Pólya urn models. We mention also previous work on on-off promoter models [33].…”