A simple model for Lutz and Bujard's controllable promoters and its application for analyzing a simple genetic oscillator

Abstract

We develop an exact and flexible mathematical model for Lutz and Bujard's controllable promoters. It can be used as a building block for modeling genetic systems based on them. Special attention is paid to deduce all the model parameters from reported (in vitro) experimental data. We validate our model by comparing the regulatory ranges measured in vivo by Lutz and Bujard against the ranges predicted by the model, and which are calculated as the reporter activity obtained under inducing conditions divided by the activity measured under maximal repression. In particular, we verify Bond et al. assertion that the cooperativity between two lac operators can be assumed to be negligible when their central base pairs are separated by 22 or 32 bp [Gene repression by minimal lac loops in vivo, Nucleic Acids Res, 38 (2010) 8072-8082]. Moreover, we also find that the probability that two repressors LacI bind to these operators at the same time can be assumed to be negligible as well. We finally use the model for the promoter P(LlacO-1) to analyze a synthetic genetic oscillator recently build by Stricker et al. [A fast, robust and tunable synthetic gene oscillator, Nature, 456 (2008) 516-519].

Keywords: Synthetic promoters; biochemical oscillations; lac and tet operators.

Fig.1

Topography and sequences of the central parts of the lac promoter P_lac and four of the Lutz and Bujard promoter/operators. The -10 and -33 hexamers are boxed and the transcriptional starting site is marked with an umlaut diacritic Ä. The corresponding sequences of the lac and tet operators (binding sites for the repressors LacI and TetR) are all underlined; they are between 18 and 21bp long; and their central base pairs are indicated with a tilde diacritic ≳. The promoters P_lac, P_A1lacO - 1, and P_LlacO - 1 include the lac operator; while P_LtetO - 1 and P_{N25tetO - 1} include the tet operator. The central base pairs of the operators in the promoters P_A1lacO - 1, P_LlacO - 1, and P_LtetO - 1 are respectively separated by 32, 22, and 24bp.

Fig.2

Graphic representation of the different states O ₀, $O_P^c$, O _R and O _2R assumed by the promoter P_LtetO - 1 in presence of the polymerase and the (dimeric) tetracycline repressor TetR. The transcriptional starting site is marked with the letter A, the -10 and -33 hexamers are indicated by small grey rectangles, the polymerase is represented by an orange kidney shaped figure, the tet operators are indicated by narrow red rectangles, and the repressors TetR are represented by big red ovals.

Fig.3

Average power spectra for the molecular counting of the active lactose repressor tetramer R, when the promoter is P_LlacO - 1 and the concentration of IPTG is: 0mbp, a; 50mbp, b; 100mbp, c; 150mbp, d; 200mbp, e; 250mbp, f. The vertical axis denotes normalized intensity in arbitrary units and the horizontal axis denotes frequency in miliHertz. Oscillations with an average period of 33.3 and 27.8 minutes can be observed when the concentration of IPTG is 100mpb and 150mpb, respectively.

Fig.4

Average power spectra for the molecular counting of the active lactose repressor tetramer R, when the promoter is P_lac and the concentration of IPTG is: 0mbp, a; 50mbp, b; 100mbp, c; 150mbp, d. The vertical axis denotes normalized intensity in arbitrary units and the horizontal axis denotes frequency in miliHertz.