Reconciling kinetic and thermodynamic models of bacterial transcription

Abstract

The study of transcription remains one of the centerpieces of modern biology with implications in settings from development to metabolism to evolution to disease. Precision measurements using a host of different techniques including fluorescence and sequencing readouts have raised the bar for what it means to quantitatively understand transcriptional regulation. In particular our understanding of the simplest genetic circuit is sufficiently refined both experimentally and theoretically that it has become possible to carefully discriminate between different conceptual pictures of how this regulatory system works. This regulatory motif, originally posited by Jacob and Monod in the 1960s, consists of a single transcriptional repressor binding to a promoter site and inhibiting transcription. In this paper, we show how seven distinct models of this so-called simple-repression motif, based both on thermodynamic and kinetic thinking, can be used to derive the predicted levels of gene expression and shed light on the often surprising past success of the thermodynamic models. These different models are then invoked to confront a variety of different data on mean, variance and full gene expression distributions, illustrating the extent to which such models can and cannot be distinguished, and suggesting a two-state model with a distribution of burst sizes as the most potent of the seven for describing the simple-repression motif.

Fig 1. An overview of the simple repression motif at the level of means.

(A) Schematic of the qualitative biological picture of the simple repression genetic architecture. (B) and (C) A variety of possible mathematicized cartoons of simple repression, along with the effective parameter ρ which subsumes all regulatory details of the architecture that do not directly involve the repressor. (B) Simple repression models from a thermodynamic perspective. (C) Equivalent models cast in chemical kinetics language. (D) The “master curve” to which all cartoons in (B) and (C) collapse.

Fig 2. Comparison of different models for noise in the constitutive promoter.

(A) The left column depicts various plausible models for the dynamics of constitutive promoters. In model (1), transcripts are produced in a Poisson process [32, 33]. Model (2) features explicit treatment of RNAP binding/unbinding kinetics [45]. Model (3) is a more detailed generalization of model (2), treating transcription initiation as a multi-step process proceeding through closed and open complexes [46]. Model (4) is somewhat analogous to (2) except with the precise nature of active and inactive states left ambiguous [17, 21, 47]. Finally, model (5) can be viewed as a certain limit of model (4) in which transcripts are produced in bursts, and initiation of bursts is a Poisson process. The right column shows the Fano factor ν (variance/mean) for each model. Note especially the crucial diagnostic: (2) and (3) have ν strictly below 1, while only for (4) and (5) can ν exceed 1. Models with Fano factors ≤ 1 cannot produce the single-cell data observed in part (B) without introducing additional assumptions and model complexity. (B) Data from [33]. Mean mRNA count vs. Fano factor (variance/mean) for different promoters as determined with single-molecule mRNA Fluorescence in situ Hybridization. The colorbar indicates the predicted binding affinity of RNAP to the promoter sequence as determined in [48]. Numbers serve for cross comparison with data presented in Fig 3.

Fig 3. Constitutive promoter posterior inference and model comparison.

(A) The joint posterior density of model 5, the bursty promoter with negative binomially-distributed steady state, is plotted with MCMC samples. 1D marginal probability densities are plotted as flanking histograms. The model was fit on lacUV5 data from [33]. (B) The empirical cumulative distribution function (ECDF) of the observed population distribution of mRNA transcripts under the control of a constitutive lacUV5 promoter is shown in black. The median posterior predictive ECDFs for models (1), Poisson, and (5), negative binomial, are plotted in dark green and dark blue, respectively. Lighter green and blue regions enclose 95% of all posterior predictive samples from their respective models. Model (1) is in obvious contradiction with the data while model (5) is not. Single-cell mRNA count data is again from [33]. (C) Joint posterior distributions for burst rate k_i and mean burst size b for 18 unregulated promoters from [33]. Each contour indicates the 95% highest posterior probability density region for a particular promoter. Note that the vertical axis is shared with (D). (D) Plots of the burst rate k_i vs. the binding energy for each promoter as predicted in [48]. The dotted line shows the predicted slope according to Eq 22, described in text. Each individual promoter is labeled with a unique number in both (C) and (D) for cross comparison and for comparison with Fig 2B.

Fig 4. Simple repression parameter inference and comparison.

(A) Contours which enclose 50% and 95% of the posterior probability mass are shown for each of several 2D slices of the 9D posterior distribution. The model assumes one unbinding rate for each operator (Oid, O1, O2) and one binding rate for each aTc induction concentration (corresponding to an unknown mean repressor copy number). (B, upper) Ratios of our inferred unbinding rates are compared with operator binding energy differences measured by Garcia and Phillips [41] (triangles) and Razo-Mejia et. al. [43] (circles). Blue glyphs compare O2-O1, while orange compare O1-Oid. Points with perfect agreement would lie on the dotted line. (B, lower) Unbinding rates for O1 (green) and Oid (red) inferred in this work are compared with single-molecule measurements from Hammar et. al. [55]. We plot the comparison assuming illustrative mRNA lifetimes of γ⁻¹ = 3 min (triangles) or γ⁻¹ = 5 min (squares). Dotted line is as in upper panel. (C) Theory-experiment comparison are shown for each of the datasets used in the inference of the model in (A). Observed single-molecule mRNA counts data from [33] are plotted as black lines. The median of the randomly generated samples for each condition is plotted as a dark colored line. Lighter colored bands enclose 95% of all samples for a given operator/repressor copy number pair. The unregulated promoter, lacUV5, is shown with each as a reference.