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## Results: 6

1.

(A) Simple two-state bacterial promoter undergoing stochastic activation by a transcriptional activator binding to a single operator site. The rates of activator association and dissociation are given by and , respectively and the rates of mRNA production for the basal and active states are and respectively. The mRNA degradation rate is assumed to be constant for each molecule, and is given by the parameter . (B) List of all possible stochastic transitions affecting either the copy number of mRNA (m) or the state of the promoter (s) and their respective statistical weight. State 1 has the operator free. State 2 is the activator bound state. The weights represent the probability that each change of state will occur during a time increment . The master equation is constructed based on these rules.

2.

(A) Kinetic mechanism of repression for a dual-repression architecture. The parameters and are the rates of repressor dissociation and association to the operators, and is a parameter reflecting the effect of cooperative binding on the dissociation rate. For independent binding, and for cooperative binding (see Table 1). (B) Fold-change in the mRNA noise caused by gene regulation for independent (red) and cooperative (black) repression as a function of the mean mRNA copy number. Inset: Prediction for a variant of the λ P

_{R}promoter where the upstream operators O_{L1}, O_{L2}and O_{L3}are deleted. The promoter mRNA noise is plotted as a function of the mean mRNA number for both wild-type cI repressor (blue line) and a repressor mutant (Y210H) that abolishes cooperativity (red line). Parameters taken from [43], [97]. The lifetime of the O_{R1}-cI complex is 4 min. Lifetime of O_{R2}-cI complex is 9.5s. (C) mRNA distribution for the same parameters used in (B).3.

(A) Kinetic mechanism of dual activation. The parameters and are the rates of activator dissociation and association to the operators, and is a parameter reflecting the effect of cooperative binding on the dissociation rate. (B) Fano factor as a function of the mean mRNA for independent (, black), cooperative (, red), and for simple activation (blue). The parameters are taken from Table 1 and , , , and ;

*f*is the enhancement factor. (C) A stochastic simulation shows the effect of independent and cooperative binding in creating a sustained state of high promoter activity, resulting in high levels of mRNA in the active state and large cell-to-cell variability. (D) Prediction for the r1-P_{RM}promoter (a P_{RM}promoter variant that does not exhibit O_{R3}mediated repression [51]). This promoter is activated by cI, which binds cooperatively to O_{R1}and O_{R2}. The prediction is shown for wild-type cI () and for a cooperativity deficient mutant (Y210H, ). Parameters are taken from [33], [43], [58], [97]. The lifetime of O_{R1}-cI complex is 4 min. Lifetime of O_{R2}-cI complex is 9.5 s.4.

(A) The Fano factor is plotted as a function of the fold-change gene expression (blue line). In red, we show the effect of reducing operator strength (i.e., reducing the lifetime of the operator-activator complex) by a factor of 10. Just as we observed with single repression, weak activator binding operators generate less promoter noise than strong activating operators. The parameters used are shown in Table 1 with the exception of , where

*f*is the enhancement factor. Inset: Prediction for the activation of the P_{lac}promoter. The fold-change in noise is plotted as a function of the fold-change in mean mRNA expression for both the wild-type P_{lac}(CRP dissociation time = 8 min), represented by a blue line, and a P_{lac}promoter variant where the*lac*CRP binding site has been replaced by the weaker*gal*CRP binding site (dissociation time = 1 min). The enhancement factor was set to [33]. These parameters are taken from [67] and [33]. The remaining parameters are taken from Table 1. (B) Fano factor as a function of for a repressor (black) and an activator (red) with the same transcription factor affinity. The transcription rate in the absence of activator is assumed to be zero. The transcription rate in the fully activated case is equal to the transcription rate of the repression construct in the absence of repressor and is as specified by Table 1. For low expression levels simple activation is considerably noisier than simple repression. (C) The results of a stochastic simulation for the simple activation and simple repression architectures. We assume identical dissociation rates for the activator and repressor, and identical rates of transcription in their respective active states. As shown in (B), low concentrations of an activator result in few, but very productive transcription events, whereas high concentrations of a repressor lead to the frequent but short lived excursions into the active state.5.

(A) Time traces for promoter activity, mRNA and protein copy number are shown for both the weak operator and the strong operator. The mRNA histograms are also shown. The weaker operator with a faster repressor dissociation rate leads to small promoter noise, and an mRNA probability distribution resembling a Poisson distribution (shown by the blue-bar histogram), in which most cells express mRNA near the population average. In contrast, the stronger operator with a slower repressor dissociation rate, leads to larger promoter noise and strongly non-Poissonian mRNA statistics. (B) Kinetic mechanism of repression for an architecture involving a single repressor binding site. The repressor turns off the gene when it binds to the promoter (with rate ), and transcription occurs at a constant rate

*r*when the repressor falls off (with rate ). (C) Normalized variance as a function of the fold-change in mean mRNA copy number. The parameters used are drawn from Table 1. The value of from Table 1 corresponds to the*in vitro*dissociation constant of the Lac repressor from the Oid operator (black). The results for an off-rate 10-times higher are also plotted (red). As a reference for the size of the fluctuations, we show the normalized variance for a Poisson promoter. (D) Fano factor for two promoters bearing the same off-rates as in (B). Inset. Prediction for the Fano factor for the Δ_{O3}Δ_{O2}P_{lacUV5}promoter, a variant of the P_{lacUV5}promoter for which the two auxiliary operators have been deleted. The fold-change in mRNA noise is plotted as a function of the fold-change in mean mRNA copy number for mutants of the promoter that replace O1 for Oid, O2 or O3. The parameters are taken from Table 1 and [33]. Lifetimes of the operator-repressor complex are 7 min for Oid, 2.4 min for O1, 11s for O2 and 0.47 s for O3. (E) Fold-change in protein noise as a function of the fold-change in mean expression. As expected, the effect of operator strength is the same as observed for mRNA noise.6.

(A) Kinetic mechanism of repression. and are the rates of repressor dissociation and association. The rate of loop formation is , where can be thought of as the local concentration of repressor in the vicinity of one operator when it is bound to the other operator. The rate of dissociation of the operator-repressor complex in the looped conformation is given by . The parameter

*c*captures the rate of repressor dissociation in the looped state relative to the rate of dissociation in a non-looped state. (B) Effect of DNA looping on cell-to-cell variability. The Fano factor is plotted as a function of the fold-change in the mean expression level, in the absence (blue) and presence (black) of the auxiliary operator, and assuming that dissociation of the operator from Om is the same in the looped and the unlooped state (*c*= 1). The presence of the auxiliary operator, which enables repression by DNA looping, increases the cell-to-cell variability. The regions over which the state with two repressors bound, the state with one repressor bound, or the looped DNA state are dominant are indicated by the shading in the background. The noise is larger at intermediate repression levels, where only one repressor is found bound to the promoter region, simultaneously occupying the auxiliary and main operators through DNA looping. The rate of DNA loop formation is [33]. We also show the effect of DNA looping in the case where the kinetics of dissociation from the looped state are 100 times faster than the kinetics of dissociation from the unlooped state: (red). In this limit, the presence of the auxiliary operator leads to less gene expression noise. (C) Prediction for a library of P_{lacUV5}promoter variants, harboring an O2 deletion, and with the position of O3 moved upstream by multiples of 11 bp while keeping its identity (red), or replaced by the operator by Oid (black). Parameters are taken from the analysis in [33] of the data in [98]. We assume a concentration of 50 Lac repressor tetramers per cell. The association rate of the tetrameric repressor to the operators is taken from Table 1. The lifetimes of the operator-repressor complex are given in the caption to Figure 2. The dependence of the rate of DNA looping on the inter-operator distance is taken from [33], and equal to: , where , , , . Note that the Fano factor is not plotted as a function of the mean, but as a function of the inter-operator distance*D*. In this case, as we change*D*, we vary both the mean and the Fano factor.