Ab initio thermodynamic modeling of distal multisite transcription regulation

Abstract

Transcription regulation typically involves the binding of proteins over long distances on multiple DNA sites that are brought close to each other by the formation of DNA loops. The inherent complexity of assembling regulatory complexes on looped DNA challenges the understanding of even the simplest genetic systems, including the prototypical lac operon. Here we implement a scalable approach based on thermodynamic molecular properties to model ab initio systems regulated through multiple DNA sites with looping. We show that this approach applied to the lac operon accurately predicts the system behavior for a wide range of cellular conditions, which include the transcription rate over five orders of magnitude as a function of the repressor concentration for wild type and all seven combinations of deletions of three operators, as well as the observed induction curves for cells with and without active catabolite activator protein. Our results provide new insights into the detailed functioning of the lac operon and reveal an efficient avenue to incorporate the required underlying molecular complexity into fully predictive models of gene regulation.

Figure 1.

Operators controlling expression of the lacZ, lacY, and lacA genes in the lac operon. (A) Location of the main (O₁) and the two auxiliary (O₂ and O₃) operators, shown as orange rectangles on the thick black segment representing DNA. Binding of the lac repressor to O₁ prevents transcription of the three lacZYA genes. (B, C) The bidentate repressor can bind to any of the three operators and simultaneously to any two of them by looping the intervening DNA, resulting in different protein–DNA complexes. Two of the three possible loops are shown: (B) one lac repressor (shown in red) loops DNA by binding simultaneously to O₁ and O₃ (loop L13) whereas another repressor binds to O₂; (C) only one repressor is bound to DNA, forming a loop between O₁ and O₂ (loop L12). In both cases, the lacZYA genes are repressed. The different contributions to the free energy of the lac repressor–DNA complexes, which include positional (p), interaction (e₁, e₂, and e₃), and conformational (c_L12, c_L13, and c_L23) contributions, are explicitly indicated in these two cases.

Figure 2.

Repression level as a function of the repressor concentration for WT and all seven combinations of deletions of three operators. The repression levels in the presence of active CAP were obtained for WT and seven mutants accounting for all the combinations of deletions of the three operators. For each of the eight cases, the results of the model (red curves) as a function of the repressor concentration are compared with the experimental data (shaded blue squares) (22) available for three concentrations corresponding to 10 (WT cells), 50, and 900 repressors per cell. The particular set of WT or deleted (X) operators is indicated for each curve; for instance, O₃-O₁-O₂ corresponds to WT lac operon and X-X-X, to the mutant with all three operators deleted. The excellent agreement indicates that the model not only captures the repression values quantitatively but also the shapes of the curves, which are very different depending on the mutant. The values of the parameters used are: e₁=−27.8 kcal/mol, e₂=−26.3 kcal/mol, e₃=−24.1 kcal/mol, c_L12 = 23.35 kcal/mol, c_L13 = 22.05 kcal/mol, c_L23 = 23.50 kcal/mol, p°= 15 kcal/mol, and χ = 0.03. A deleted operator is modeled by increasing its free energy by 5 kcal/mol. The two repression levels represented by broken squares in the graphs for O3-O1-O2 and O3-O1-X were not accurately measured in the experiments, as described by the authors in a subsequent publication (16).

Figure 3.

Inducer binding to the lac repressor. An inducer molecule, such as IPTG, can bind independently to each of the four identical monomeric units that form the lac repressor and impair its function. Without IPTG bound, each of the two dimers of the tetrameric repressor (in red) has a functional DNA-binding domain and can bind strongly to the operators (orange rectangle) and loop the intervening DNA (A). With one or two IPTG molecules (small yellow circles) bound to the same domain, the repressor can bind strongly to one operator but cannot loop DNA (B,C). With two or more IPTG molecules bound to different domains of the repressor, strong operator binding is abolished (D,E,F,G).

Figure 4.

Induction curves of the lac operon for cellular conditions with and without active CAP. (A) The repression level in the presence of active CAP (lower curves labeled ‘CAP’) was obtained as a function of the concentration of the inducer IPTG for the model given by ΔG(s), τ(s), and (dashed red lines) and compared with the experimental data from Ref. (17) (blue squares). The values of the parameters are the same as in Figure 2, with additional parameters K_I = 6.9μM and n_T = 15 nM. In the absence of active CAP (upper curves, labeled ‘No CAP’), the model (dashed red lines) does not accurately reproduce the experimental data from Ref. (17) (blue diamonds). We consider that without active CAP, in addition to a reduced transcription , the formation of the O₁-O₃ loop is 0.9 kcal/mol more costly (c'_L13 = 22.95 kcal/mol) than with CAP. (B) The excellent agreement with induction experiments in the absence of active CAP is recovered when the model is generalized to include an arbitrary number of repressor molecules per cell through the expressions ΔG_F(s), τ_F(s), n, n_d, and n_i, with same values of the parameters as in panel A, with the additional parameter r = 2.8 kcal/mol and 200 repressors per cell (n_T = 300 nM) (upper continuous red line). In the presence of active CAP, the generalized model (lower continuous red line) recovers the results in panel A for low number of repressors (n_T = 15 nM).