Transcriptional regulation by the numbers: models

Abstract

The expression of genes is regularly characterized with respect to how much, how fast, when and where. Such quantitative data demands quantitative models. Thermodynamic models are based on the assumption that the level of gene expression is proportional to the equilibrium probability that RNA polymerase (RNAP) is bound to the promoter of interest. Statistical mechanics provides a framework for computing these probabilities. Within this framework, interactions of activators, repressors, helper molecules and RNAP are described by a single function, the "regulation factor". This analysis culminates in an expression for the probability of RNA polymerase binding at the promoter of interest as a function of the number of regulatory proteins in the cell.

Figure 1

Probability of promoter occupancy (a) Schematic showing how, in the simple model, the DNA molecule serves as a reservoir for the RNAP molecules, almost all of which are bound to DNA. (b) Illustration of the states of the promoter – either with RNAP not bound or bound and the remaining polymerase molecules distributed among the non-specific sites. The statistical weights associated with these different states of promoter occupancy are also shown. (c) Probability of binding of RNAP to promoter as a function of the number of RNAP molecules for two different promoters. We assume the number of non-specific sites is N_NS = 5 × 10⁶, and calculate the binding energy difference using the simple relation $\mathrm{\Delta }\epsilon pd=k_{B}Tln(K_{pd}^S/K_{pd}^{NS})$, where the equilibrium dissociation constants for specific binding ( $K_{pd}^S$) and non-specific binding ( $K_{pd}^{NS}$) are taken from in vitro measurements. In particular, making the simplest assumption that the genomic background for RNAP is given only by the non-specific binding of RNAP with DNA, we take $K_{pd}^{NS}=10000\text{nM}$ [37], for the lac promoter $K_{pd}^S=550\text{nM}$ [38] and for the T7 promoter, $K_{pd}^S=3\text{nM}$ [39]. For the lac promoter, this results in Δε_pd = −2.9k_BT and for the T7 promoter, Δε_pd = −8.1k_BT.

Figure 2

Statistical mechanics of recruitment (a) Schematic showing the relationship between the various states of the promoter and its regulatory region, and their corresponding weights within the statistical mechanics framework. (b) Fold-change in promoter activity as a function of the number of activated (inducer-bound) CRP molecules, according to Equations 5 and 8, for different values of the adhesive interaction energy between activator and RNAP. As in Figure 1, $\mathrm{\Delta }{\epsilon }_{ad}=k_{B}Tln(K_{ad}^S/K_{ad}^{NS})$, with $K_{ad}^{NS}=10000\text{nM}$ [40] and $K_{ad}^S=0.02\text{nM}$ [41]. These in vitro numbers are chosen as a representative example to provide intuition for the action of activators. Applications to in vivo experiments are given in the accompanying paper [1^••]. Several different representative values of the adhesive interaction ε_ad that are consistent with measured activation are chosen to illustrate how activation depends upon this parameter.

Figure 3

DNA bending in transcription regulation. (a) DNA looping enables Lac repressor to bind to the main and the auxiliary operators simultaneously, thereby increasing the weight of the states in which the promoter is unoccupied. This leads to stronger repression than in the single operator case. (b) DNA bending by the activator leads to cooperative binding of the two activators because the free energy cost of bending is paid only once. This leads to a boost in activation above that provided by independent binding of the two activators [45].