Self-consistent theory of transcriptional control in complex regulatory architectures

Abstract

Individual regulatory proteins are typically charged with the simultaneous regulation of a battery of different genes. As a result, when one of these proteins is limiting, competitive effects have a significant impact on the transcriptional response of the regulated genes. Here we present a general framework for the analysis of any generic regulatory architecture that accounts for the competitive effects of the regulatory environment by isolating these effects into an effective concentration parameter. These predictions are formulated using the grand-canonical ensemble of statistical mechanics and the fold-change in gene expression is predicted as a function of the number of transcription factors, the strength of interactions between the transcription factors and their DNA binding sites, and the effective concentration of the transcription factor. The effective concentration is set by the transcription factor interactions with competing binding sites within the cell and is determined self-consistently. Using this approach, we analyze regulatory architectures in the grand-canonical ensemble ranging from simple repression and simple activation to scenarios that include repression mediated by DNA looping of distal regulatory sites. It is demonstrated that all the canonical expressions previously derived in the case of an isolated, non-competing gene, can be generalised by a simple substitution to their grand canonical counterpart, which allows for simple intuitive incorporation of the influence of multiple competing transcription factor binding sites. As an example of the strength of this approach, we build on these results to present an analytical description of transcriptional regulation of the lac operon.

Fig 1. States and their weights in the simple repression architecture.

All allowed states of the simple repression architecture are shown with their associated energies and statistical weights. ϵ_P is the binding energy of the RNAP onto the promoter site, ϵ_R the binding energy of a repressor molecule onto the operator site. The third column shows the grand canonical weights, where the λ_i is the fugacity of the RNAPs (i = P) or repressors (i = R), and $x_i=e^{-{ϵ}_i/k_{\text{B}}T}$. The right column lists the weights in the canonical ensemble where P is the number of RNA-polymerase molecules, R the number of repressors, and N_ns the number of non-specific binding sites of the genome.

Fig 2. Grand canonical states and weights in the looping architecture.

(A) Looping architecture where a repressor bound to the main operator and RNAP binding are mutually exclusive. (B) Additional states and weights for the exclusive looping scenario. In this scenario, repression is only effective in the looped state.

Fig 3. Fold-change and occupation for the looping scenarios.

(A) Fold-change as a function of the fugacity λ_R for the looping scenario (blue curve, Eq (17)) and the exclusive looping scenario (green curve, Eq (20)). The pink curve is the simple repression scenario. (B) Average occupation of repressors to a single gene 〈R_ads〉/N in Eq (22) in the looping architecture. (C) Fold-change as a function of the total number of repressor molecules R for the looping scenario (blue curve) and exclusive looping (green curve) scenario. (D) the repressor fugacity as function of the total number of repressor molecules R for both the looping and exclusive looping scenario. The value of ${ϵ}_{\text{R}}^m={ϵ}_{\text{R}}^a=-17.3k_{\text{B}}T$, and F_L = +10k_BT as in [34]. Furthermore, we took the number of promoters to be N = 10 and the number of non-specific sites to be N_ns = 5 × 10⁶.

Fig 4. Transcription activity data of simple repression and looping regulated genes.

Transcription activity data for the simple repression architecture from [17, 37], as previously shown in [38], as well as data for the looping scenario from [55, 56], rescaled to the scaling factor z appropriate to its architecture. For simple repression scenarios, z = λ_R exp(−βϵ_R). For the looping scenario, z_L is calculated using Eqs (18) and (26). The solid blue line signifies the scaling function (1 + z)⁻¹. The repressor binding energies are taken from [37] as ${ϵ}_{\text{R}}^{\text{Oid}}=-17k_{\text{B}}T$, ${ϵ}_{\text{R}}^{\text{O}1}=-15.3k_{\text{B}}T$,${ϵ}_{\text{R}}^{\text{O}2}=-13.9k_{\text{B}}T$ and ${ϵ}_{\text{R}}^{\text{O}3}=-9.7k_{\text{B}}T$. Values for promoter copy numbers N and competitor sites N_c are taken from [17] (simple repression) and [55] (looping). The value for the looping free energy, F_L = +9.1k_BT, was taken from Fig 3b in [55] as the average looping free energy for a loop that has a length in between 76 and 84 base pairs. For each data set, λ_R is calculated by solving the mass balance appropriate for the architecture, Eq (7) (simple repression) or Eq (24) (looping).

Fig 5. States and weights for the simple activation scenario.

An activator and the RNA polymerase can bind to the activator binding site and to the promoter site with energies ϵ_A and ϵ_P, respectively. The state where both molecules are bound simultaneously includes an additional energy ϵ_AP, which reflects the adhesive interaction between activator and RNA polymerase.

Fig 6. Fold-change in the grand canonical and the canonical ensemble for a variety of regulatory architectures.

The promoter is indicated by a red patch on the DNA, with the transcription start site denoted by the straight arrow. Interactions between transcription factors bound to a site are specified by a solid curve ending in an arrow tip (activation), in a bar (repression) or unadorned (unspecified interaction). Dashed curved lines signify looping between two sites.

Fig 7. List of all allowed states of the lac operon, and their grand canonical weights.

The lac operon has three binding sites (O1, O2, O3) for the lac repressor (LacI) and one binding (A) site for a CRP activator. LacI has two binding heads and can bind to two sites simultaneously. In those cases the DNA in between the binding sites forms a loop. States where RNAP is bound to the promoter (p) and LacI is bound to the O1 operator sites are not allowed, as well as looped states where RNAP is bound to the promoter.

Fig 8. Fugacities of the transcription factors for the lac operon.

(A) Fugacity of activators as a function of the number of activator molecules in the cell, in the absence of repressor (blue curve) and in the presence of a high concentration of repressor (green dotted curve). (B) Fugacity of repressors as a function of the number of repressors in the cell, in the absence of activator (blue curve) and in the presence of a high concentration of activator (green dotted curve). In both cases the copy number of the gene is N = 1. Note that the presence of repressor causes a slight shift in the activator fugacity. The parameters used are listed in Table 2.

Fig 9. Fold-change of lac operon constructs from the literature.

Theoretical fold-change according to Eq (39) compared to the experimental fold-change as determined in [56, 62]. The dashed line is the x = y line. The blue squares correspond to the 1994 paper, the green diamonds to the older 1990 paper. For some strongly repressive constructs, Oehler et al. were only able to measure a lower bound to the level of repression. These points were marked with a cross.

Fig 10. Fold-change of the lac operon.

Fold-change as a function of activator and repressor concentrations for N = 1 (yellow surface) and N = 10 (translucent blue surface). When only a single copy of the lac operon is present in the cell, the action of LacI is significant: the introduction of as little as 2 or 3 copies of LacI cause a 100-fold drop in the transcription rate. In vivo, E. coli cells typically contain 10¹ instances of LacI, keeping the activity of the lac operon low. When there are multiple copies of the lac operon present, all copies have to compete for the availability of LacI and significant repression only occurs when the number of LacI exceeds the operon copy number. Due to this titration effect, the transcription rate becomes sensitive to fluctuations in wildtype LacI availability. A similar titration effect occurs for the availability of CRP, but since CRP is already strongly competed for, the addition of multiple gene copies has no significant additional effect.

Fig 11. Activators increase the dynamic range of repression of the lac operon.

(A) Gene expression normalized to the gene expression at R = 0 (fold-change_R) as a function of number of repressors, in the absence (blue curve) and presence (green) of activators. (B) Gene expression normalized to the gene expression at A = 0 (fold-change_A) as a function of the number of activators, in the absence (blue curve) and presence (green curve) of repressors. Bound activator causes a sharp bend in the DNA that facilitates the loop between O1 and O3. This causes an additional, cooperative repression effect on top of the (uncooperative) activation behaviour of the activators.

Fig 12. Effect of the competitive environment on activation.

(A) Gene expression normalized to the gene expression at A = 0 (fold-change_A) and (B) fugacity of activators, as a function of the number of activators in the isolated gene case (blue curves), and in the case where the activators are competed for by 350 additional competitor sites in the cell (green curves). In the interacting gene model the effective concentration of CRP is lower due to binding to competitor sites. Consequently, the transcription rate is significantly lower.