We obtained chemically synthesized nucleotide sequences of random nucleotides 200 bp in length (Purimex, Germany). Each sequence had defined 5′ and 3′ ends to allow PCR amplification. Within these constant regions, restriction sites for BamHI and XhoI were present. The intervening sequence was made up of 157 bp of random nucleotides (5′-CCTTTCGTCTTCACCTCGAG-(N157)-GGGATCCTCTGGATGTAAGAAGG-3′). However, as coupling of base pairs during oligonucleotide synthesis is not always successful and strand breaks can frequently occur in long oligonucleotides, many oligonucleotides were shorter than 200 bp in length. We used PCR to generate double-stranded DNA from the single-stranded oligonucleotides using forward and reverse primers matching the defined 5′ and 3′ ends. We gel-purified the double-stranded PCR product and double-digested it using BamHI and XhoI. After column purification, sequences were ligated into a version of the low-copy plasmid pUA66, which contains a gfpmut2 open reading frame downstream of a strong ribosomal binding site (Zaslaver et al., 2006). The vector was modified to remove a weak σ70 binding site present 24 bp upstream of the GFP open reading frame (two point mutations, A → G and T → G, were introduced, changing the putative σ70 binding site from TAGATT to TGGATG, with the consensus σ70 binding site being TATAAT). The ligation was performed using T4 DNA ligase (NEB) at 16°C for 24 hr. The ligation product was then column purified and electroporated into E. coli DH10B cells. This protocol resulted in extremely high transformation yields (approximately 10⁶ individual clones per transformation).

Cultures of transformed cells were regenerated for 1 hr in 1 ml SOC medium (Super Optimal Broth supplemented with 20 mM glucose) and afterwards 1 ml SOC containing 50 μg/ml kanamycin was added for overnight growth, ensuring that only cells containing the plasmid could grow. These cultures were then diluted 500-fold (approximately 5 × 10⁶ cells in total) into M9 minimal media supplemented with 0.2% glucose and grown for 2.5 hr with shaking at 200 rpm. The distribution of GFP fluorescence levels was measured for each culture using FACS in a FACSAria IIIu (BD Biosciences), with excitation at 488 nm and a 513/17 nm bandpass filter used for emission.

We used this distribution of fluorescence values to designate a selection gate. The position of the gate was determined by measuring the mean fluorescence of two reference promoters (Zaslaver et al., 2006): gyrB which exhibits a mean expression level that is at the 50th percentile all E. coli promoters; and rpmB, which exhibits a mean expression level that is at the 97.5th percentile of all E. coli promoters (Silander et al., 2012). For each of these reference genes, the mean fluorescence level was measured and a selection gate was constructed, centered on this mean expression level, such that 5% of all clones in the population fell within the gate. For each round of selection, we sorted 200,000 cells contained within this gate. Sorted cells were then transferred to 4 ml Luria Broth (LB) media (containing 50 μg/ml kanamycin) and grown overnight. These cultures were stored supplemented with 7.5% glycerol at −80°C for subsequent analysis.

For each expression level (i.e., reference gene) we evolved three replicate populations. We refer to these as the medium expressers (those promoters selected based on the gyrB reference gate) and high expressers (those promoters selected based on the rpmB reference gate).

Following FACS-based selection on fluorescence, we introduced novel genetic variation into the populations using PCR mutagenesis. We first re-grew the cells overnight and used this culture to prepare plasmid DNA. We amplified the promoter sequences from these plasmids using the GeneMorph II Random Mutagenesis Kit (Stratagene) with the primers referred to previously that matched the defined regions of the promoters. We used 0.01 ng of DNA as starting material and 35 cycles for amplification. This resulted in a mutation rate of around 0.01 per bp (such that we expect that, in 200 bp, 95% of the promoters will contain between zero and four mutations). These PCR products were then digested with XhoI and BamHI, ligated back into the vector, and again transformed into DH10B cells. After an initial round of selection on the initial library, this entire process (PCR mutagenesis, transformation, and selection) was repeated four times in total. At this point, the plasmid libraries of synthetic promoters were isolated and transformed into E. coli K12 MG1655 for comparison with a library of native E. coli promoters (see below).

To quantify fluorescence on a single-cell level, we used flow cytometry with a FACSCanto II (BD Biosciences), with excitation at 488 nm and a 513/17 nm band-pass filter used for emission. We collected data for at least 50,000 events. We then gated this data as outlined in Silander et al. (2012), identifying approximately 5000 cells most similar in forward scatter (FSC) and side scatter (SSC). We then calculated the mean and variance in log-fluorescence using these cells, using a Bayesian procedure that accounts for outliers (Appendix 1). We randomly selected 479 promoters from the evolved set (72 medium expressers and 72 high expressers after three rounds of selection; 168 medium expressers and 167 high expressers after five rounds of selection) and quantified mean and variance in fluorescence. We used the same measurement procedures to calculate mean and variance for all promoters contained in a library of E. coli promoters also placed upstream of the gfpmut2 open reading frame on the pUA66 plasmid (Zaslaver et al., 2006). We refer to the promoters from this library as native E. coli promoters. For 288 promoters, we quantified fluorescence in three independent cultures and found that both mean and variance in expression were reproducible across replicate biological experiments (Figure 1—figure supplement 4). Additionally, we sequenced 378 sequences from our set of 479 promoter sequences, which showed that even after five rounds of selection, the promoters were quite diverse (Figure 1—figure supplement 1). To confirm the sensitivity and accuracy of the FACS measurements, we selected 10 promoters and used fluorescence microscopy to measure their mean and variance in fluorescence. The cells were grown in the same conditions described above, placed on 1% agarose pad, and images were obtained using a CoolSNAP HQ CCD camera (Photometrics) connected to a DeltaVision Core microscope (Applied Precision) with a UPlanSApo 100×/1.40 oil objective (Olympus). Image-processing was done in soft-WoRx v3.3.6 (Applied Precision) and fluorescence values were extracted based on DIC image-mediated cell detection in MicrobeTracker Suite (Sliusarenko et al., 2011). For each cell, we calculated fluorescence per cell volume by summing all pixel values and dividing by the volume of the cell as estimated by MicrobeTracker. Cells undergo substantial phenotypic changes when they are put on agar, including changes in the distribution of cell sizes. Consequently, it is problematic to compare absolute variance measurements directly between FACS and microscope. We therefore compared the relative noise levels of different promoters. The 10 selected native promoters consist of five pairs with almost identical mean expression values (as measured by FACS) but with noise levels that vary by different amounts. For each of the five pairs we calculated the ratio of the noise levels of the higher and the lower noise promoter as measured by both FACS and the microscope. As shown in Figure 1—figure supplement 5, with the exception of one pair of promoters that showed almost equal noise levels in FACS but a 50% difference in noise in the microscope, all other pairs showed good correlation of the relative noise levels in FACS and in the microscope, confirming that relative noise levels are similar in FACS and microscope measurements.

To determine the correspondence between fluorescence intensities and absolute GFP numbers per cell, eight individual promoter clones were grown in three biological replicates using the same media conditions as in the experimental evolution. The cells were then re-suspended in SDS sample buffer, heated for 5 min at 95°C, and proteins were resolved by 12% SDS-PAGE. Quantification was done by loading a standard curve consisting of 10, 25, 50, 75, and 100 ng of GFP (#632373; Clonetech). Proteins were transferred to a Hybond ECL membrane (GE Healthcare, Life Sciences), which was then blocked in TNT (20 mM Tris pH 7.5, 150 mM NaCl, 0.05% Tween 20) with 1% BSA and 1% milk powder. Detection was performed with the ECL system after incubation with rabbit anti-GFP and polyclonal pig anti-rabbit. Western intensities for each sample were extracted using ImageJ (Figure 1—figure supplement 2). The number of cells loaded was estimated by calculating the relationship between OD600 and CFU counts. Details of the data analysis procedures are given in Appendix 1.

Native and evolved single-promoter populations were grown in three biological replicates by diluting overnight LB cultures 500-fold into M9 media supplemented with glucose. These cultures were grown for 2.5 hr, stabilized with an equal volume of RNA Later (Sigma–Aldrich) and RNA was extracted using the Total RNA Purification 96-Well Kit (Norgen Biotek Corp) with on-column DNAse I digestion. Reverse transcription was done using random hexamers and qPCR with TaqMan probes and performed by Eurofins Medigenomix GmbH (Germany). Three technical replicates were performed. The efficiency of the primers and probes used were validated in a dilution series. Relative RNA levels per cell were obtained by normalizing to the reference gene ihfB using a Bayesian procedure for integrating data from the replicates and accounting for failed measurements (Appendix 1). The primers and probes used were: GFP forward primer: 5′-CCTGTCCTTTTACCAGACAA-3′; GFP reverse primer: 5′-GTGGTCTCTCTTTTCGTTGGGAT-3′; GFP probe: 5′-TACCTGTCCACACAATCTGCCCTTTCG-3′, ihfB forward primer: 5′-GTTTCGGCAGTTTCTCTTTG-3′, ihfB reverse primer: 5′-ATCGCCAGTCTTCGGATTA-3′, ihfB probe: 5′-ACTACCGCGCACCACGTACCGGA-3′).

In a simple model of gene expression in which there are constant rates of transcription, translation, mRNA decay, and protein decay, the probability distribution for the number of proteins per cell is a negative binomial with variance proportional to the mean $\langle n\rangle$: $\text{var}\left(n\right)=\left(b+1\right)\langle n\rangle$, where the constant b is the ratio between the mRNA translation rate and the mRNA decay rate, which is often referred to as ‘burst size’ (Shahrezaei and Swain, 2008). However, in general there are also cell-to-cell fluctuations in the transcription, translation, and decay rates, which are proportional to these rates themselves. These fluctuations lead to an additional term in the variance var(n) which is proportional to the square of the mean: $\text{var}\left(n\right)=\beta \langle n\rangle +{\sigma }_{ab}^2{\langle n\rangle }^2$, where β is a renormalized burst size and ${\sigma }_{ab}^2$ is the relative variance of the product of transcription, translation, and decay rates across cells (Appendix 1).

The total fluorescence in a cell (measured in units equivalent to number of GFP proteins) n_meas can then generally be written as: $n_{\text{meas}}=n_{\text{bg}}+\langle n\rangle +ϵ\sqrt{\text{var}\left(n\right)}$, where n_bg is background fluorescence and ϵ is a fluctuating quantity with mean zero and variance one. Assuming that the fluctuations are small relative to the mean, we then find for the variance of the logarithm of n_meas:

We fit this functional form to the minimum variance var(log[n_meas]) as a function of the mean, with ${\sigma }_{ab}^2=0.025$ and β = 450. We defined the excess variance as the difference between the measured variance and this fitted minimal variance. A more detailed derivation is given in Appendix 1.

By comparing the distributions of the population's expression levels before and after rounds of selection (without intervening mutation of the promoters), we found that the probability that a cell with expression level x is selected by the FACS is well-approximated by $f\left(x|{\mu }_{\text{*}},\tau \right)=\text{exp}\left[-\frac{{\left(x-{\mu }_{\text{*}}\right)}^2}{2{\tau }^2}\right]$, with μ_* the desired expression level and τ the width of the selection window. For the last three rounds of selection for medium expression, the selection gates in the FACS were relatively constant, and we estimated τ ≈ 0.03 and μ_* fluctuated slightly around an average value of μ_* ≈ 8.1 for these selection rounds.

With this selection function, a promoter genotype that exhibits a distribution of expression values with mean μ and standard deviation σ has a fitness (fraction of cells selected in the FACS) of

This estimated fitness function indicated that the fitness of promoter genotypes strongly depends on their mean μ and is almost independent of their excess noise (Appendix 2—figure 3 and Appendix 2—figure 4). In addition, applying additional rounds of selection of varying strengths to the population of evolved promoters did not systematically alter their distribution of excess noise levels. Details of the analysis of the FACS selection are given in Appendix 2.

Although the model we present can be extended to include the evolution of gene regulation for multiple genes, for simplicity we focused on the evolution of a single gene and its promoter. We assume that the population experiences a sequence of different environments and that, in each environment, the fitness of each organism is a function of its gene expression level. We characterized the fitness function in each environment by two parameters: the desired level μ_e that maximizes the fitness and a parameter τ that quantifies how quickly fitness falls away from this optimum. For simplicity and analytical tractability, we assumed a Gaussian form: $f\left(x|{\mu }_e,\tau \right)=\text{exp}\left[-\frac{{\left(x-{\mu }_e\right)}^2}{2{\tau }^2}\right]$. Similarly, although it is straightforward to allow the variance τ² to vary across conditions, the results are more transparent when we assume τ² is the same in all environments. Note that this fitness function has the same form as the FACS selection function. Consequently, the fitness $f\left(\mu ,\sigma |{\mu }_e,\tau \right)$ of a promoter with mean μ and variance σ² is given by Equation 2 as well, with μ_e replacing μ_*.

The total number of offspring that a promoter will leave behind after experiencing all environments is given by the product of its fitness in each of the environments. Equivalently, the log-fitness of a promoter is proportional to its average log-fitness across all environments. For an unregulated promoter with fixed mean μ and variance σ² in expression, we then find for the log-fitness:

where $\langle {\mu }_e\rangle$ is the average of the desired expression levels across environments and var(μ_e) is the variance in the desired expression levels across environments. If we do not consider gene regulation but simply optimize the promoter's mean expression and noise level, then we find optimal log-fitness occurs when $\mu =\langle {\mu }_e\rangle$ and σ² = 0 (when var(μ_e) < τ²) or σ² = var(μ_e) − τ² otherwise. That is, when the desired expression level varies more than the width of the selection window, fitness is optimized by increasing noise so as to ensure the distribution overlaps the desired levels across all conditions. This result is equivalent to previous results on the evolution of phenotypic diversity in fluctuating environments (Bull, 1987).

To increase fitness, a promoter can evolve to become regulated by one of the regulators existing in the genome. Instead of having a constant mean expression μ, the promoter's mean expression will then become a function of the environment e: μ(e) = μ + cr_e, where r_e is the mean expression (or more generally regulatory activity) of the regulator in environment e, and c is the coupling strength. Note that, for simplicity, we thus assume a linear coupling between the means of regulator and target. Since any gene will have some variability in its expression, we assumed that the actual expression/activity of the regulator in each environment e is Gaussian distributed with a variance ${\sigma }_r^2$. As for the width of the fitness function τ, it would again be straightforward to allow ${\sigma }_r^2$ to vary across conditions (as it likely does in reality). However, the results are analytically more transparent and bring out the main features of the model better if we assume the regulator's noise ${\sigma }_r^2$ is the same in all conditions.

When coupled to the regulator, the promoter's total expression variance will become ${\sigma }_{\text{tot}}^2={\sigma }^2+c^2{\sigma }_r^2$ and the log-fitness of the promoter becomes:

Assuming that the basal expression level μ is optimized to maximize log-fitness, that is, $\mu =\langle {\mu }_e\rangle -c\langle r_e\rangle$, this log-fitness can be rewritten as:

where X measures the coupling strength $\left(X^2=\frac{c^2{\sigma }_r^2}{{\tau }^2+{\sigma }^2}\right)$, Y is the expression mismatch that measures how much the desired expression level varies across environments $\left(Y^2=\frac{\text{var}\left({\mu }_e\right)}{{\tau }^2+{\sigma }^2}\right)$, S is the signal-to-noise of the regulator $\left(S^2=\frac{\text{var}\left(r_e\right)}{{\sigma }_r^2}\right)$, and R is the Pearson correlation between the desired expression levels μ_e and the activity levels r_e of the regulator. The change in log-fitness between the situation before and after adding of the regulatory interaction is obtained by subtracting log[f (0, Y, S, R)] from log[f (X, Y, S, R)], yielding

Note that this basic argument can be iterated. After the promoter has been coupled to a regulator, the residual deviations between the desired and actual expression levels are given by ${\tilde{\mu }}_e={\mu }_e-c{r}_e$ and the new noise level of the promoter is given by ${\tilde{\sigma }}^2={\sigma }^2+c^2{\sigma }_r^2$. If we define a new expression mismatch ${\tilde{Y}}^2=\text{var}\left({\tilde{\mu }}_e\right)/\left({\tilde{\sigma }}^2+{\tau }^2\right)$, then we can calculate the log-fitness changes associated with adding another regulatory interaction using exactly the same expressions as above, replacing Y by $\tilde{Y}$.

In addition, because the coupling between the activity of the regulator and the expression of the promoter is linear, coupling the promoter to an arbitrary linear combination of different regulators can be modeled as coupling the promoter to a single ‘effective’ regulator; that is, if a promoter is coupling to different regulators rⁱ with coupling constants c_i, then in environment e we have $\mu \left(e\right)=\mu +{{\displaystyle \sum }}_i{c}_i{r}_e^i$, which is equivalent to coupling with constant c to a regulator with mean $r_e={{\displaystyle \sum }}_i{c}_i{r}_e^i/c$. If ${\sigma }_i^2$ is the noise level of regulator i and R_ij is the Pearson correlation in the fluctuations of regulators i and j, then this composite regulator has a total variance ${\sigma }_r^2={{\displaystyle \sum }}_i{c}_i^2{\sigma }_i^2+{{\displaystyle \sum }}_{i\ne j}{R}_{ij}{\sigma }_i{\sigma }_j$.

As can be easily seen from Equation 1 in the main text, if the best linear combination of regulators provides a correlation R with the promoter's desired levels, the optimal value S_* of the signal-to-noise of this composite regulator is given by S_* = RY/X. Substituting this back into Equation 1, we find for the optimal coupling strength $X_{\text{*}}^2=\text{max}\left[0,\left(1-R^2\right)Y^2-1\right]$. This function is plotted in Figure 3—figure supplement 1, together with the values of S_* as a function of Y and R. Note that (1 − R²)Y² is the part of the expression mismatch that is not accounted for by the condition-response effect of the regulators. Whenever this remaining expression mismatch is less than 1, noise-propagation is a detrimental side-effect of regulation and regulators will be selected to be as accurate as possible. However, when (1 − R²)Y² > 1, noise-propagation will be selected for, and the increase in the total noise is equal to the amount of expression mismatch not accounted for by the condition-response.

Additional details on the derivation of our model and analysis of the behavior of the fitness function as a function of its parameters are given in Appendix 3.

We re-annotated the promoter fragments of Zaslaver et al., 2006 by mapping the published primer pairs to the E. coli K12 MG1655 genome. Of the 1816 promoter fragments, 1718 could be unambiguously associated with a gene that was immediately downstream, and the 1718 promoter fragments were associated with 1137 different downstream genes (for some genes there were multiple or repeated upstream promoter fragments). We used the operon annotations of RegulonDB (Salgado et al., 2013) to extract, for each promoter, the set of additional downstream genes that are part of the same operon as the first downstream gene. We obtained known regulatory interactions between TFs and genes from RegulonDB and counted, for each E. coli gene, the number of TFs known to regulate the gene. We defined the number of regulatory inputs of a promoter to equal the average of the number of inputs for all genes in the operon downstream of the promoter. We sorted promoters by their excess noise and, as a function of a cut-off on excess noise level, calculated the mean and standard error of the number of regulatory inputs for all promoters with excess noise level above the cut-off. We obtained genome-wide gene expression measurements from the Gene Expression Database (http://genexpdb.ou.edu/). For each E. coli gene, we obtained 240 log fold-change values x corresponding to the logarithm of the expression ratio of the gene in a perturbed and a reference condition. We defined the variance in expression of a gene as the average of x² across the 240 experiments. We again sorted promoters by their excess noise and, as a function of a cut-off on excess noise level, calculated the mean and standard error of gene expression variances for all promoters with excess noise above the cut-off.

Using the RegulonDB database (Salgado et al., 2013), we constructed a binary matrix R of regulatory interactions, where the components R_pr = 1 when regulator r is known to target promoter p, and R_pr = 0 otherwise. Following previous work from our group in which we modeled gene expression patterns in mammals in terms of regulatory sites (FANTOM Consortium et al., 2009; Balwierz et al., 2014), we use a simple linear model to relate the excess noise E_p of each promoter p to the (unknown) noise-propagation strengths V_r of each regulator r:

We assume the noise is Gaussian distributed with unknown variance, and we use a Gaussian prior $P\left(V_r\right)\propto e^{-\lambda V_r^2/2}$ on the noise-propagation strengths V_r to avoid over-fitting. The hyper-parameter λ is chosen using a cross-validation, fitting the V_r on a random fraction of 80% of the promoters, and maximizing the quality of the predictions on the remaining 20% of the promoters. The quality of fit is quantified by the fraction of the variance in noise levels E_p that is explained by the fit. For our dataset, 17.1% of the variance of the overall dataset was explained by the fit.

As explained in the ‘Materials and methods’, for both the medium and high expression evolutionary runs, the desired expression level μ_* is taken from the expression level of a reference promoter from the library of E. coli promoters. At each selection round we measure the expression μ_* of the reference promoter, and set the center of the FACS selection window to μ_*. We then set the width of the selection window such that 5% of the cells have expression levels within the selection window.

Although, in principle, the FACS selection should work such that a cell with expression level anywhere within the selection window has 100% probability to be selected, and 0% probability to be selected if the cell's expression is anywhere outside the selection window, it is unrealistic to assume that the boundaries of the selection window are so precisely defined in practice. As illustrated below, comparison of the population's expression levels before and after selection shows that the probability for a cell with log-fluorescence x to be selected can be well-approximated by

where μ_* is the desired expression level and τ corresponds to the width of the selection window.

Note that, for a promoter with mean expression μ and variance σ², the fraction $P\left(x|\mu ,\sigma \right)$ of its cells that have expression level x is given by

Consequently, the ‘fitness’ of this promoter, that is, the fraction of its cells that are selected in the FACS, is given by

To infer the values of μ_* and τ that apply to our evolutionary runs, we performed a number of experiments in which we:

1. Took a population from one of the rounds of our evolutionary runs.

2. Measured its distribution of log-fluorescence levels.

3. Set the selection window [μ_* − δ, μ_* + δ] such that a percentage p of the population has log-fluorescence levels within this selection window.

4. Performed selection and re-measured the log-fluorescence levels of the selected population.

As shown in Figure 1B of the main paper, in the evolutionary runs in which we are selecting for high expression, the selection window is changing at every round of the evolutionary run. In contrast, in the evolutionary runs selecting for medium expression, the selection window is essentially constant from round three through round five. We thus decided to focus on inferring the precise fitness function that acted during these three rounds of selection.

We took the evolved populations from the third and fifth rounds of the evolutionary runs selecting for medium expression and performed another round of selection on them, selecting 5% of the population closest to the desired log-fluorescence μ_*. In addition, we also performed a round of less stringent selection on these populations, selecting 25% closest to the desired level, and a round of more stringent selection, selecting only the 1% of the population closest to the desired level. Besides measuring the log-fluorescence levels of the population both before and after the round of selection, we also selected dozens of clones from the populations before and after the selection and measured the entire distribution of log-fluorescence levels for these clones. Appendix 2—figure 1 shows the means and variances of the log-fluorescence distributions of these clones.

Appendix 2—figure 1

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Intuitively, one might think that the relative fitness f(x) of each log-fluorescence level x could be easily estimated by simply measuring the ratio of the fraction of the population p′(x) with log-fluorescence level x after selection and the fraction p(x) with log-fluorescence x before selection. However, the single cells that were selected in the FACS each grow into an entire population of cells before the ‘after selection’ population is measured again. Thus, a selected cell containing a promoter with a given mean μ and variance σ² will contribute an entire population of cells with this distribution, even though the individual cell may itself have had a log-fluorescence that was in one of the tails of this distribution. Thus, in general the distribution of log-fluorescence levels in the population after selection may be much wider than the actual selection window itself.

Before selection, the population consisted of a mixture containing (unknown) fractions ρ(μ, σ) of cells containing promoters with mean μ and variance σ². This gives rise to an overall distribution p(x) of expression levels given by

Unfortunately we cannot uniquely infer ρ(μ, σ) from knowing only the distribution p(x). However, as shown in Appendix 2—figure 1, for the clones in these populations, the large majority of promoters have variances σ² lying in a narrow band as a function of μ. We thus chose to make the approximation that all promoters in the population have variances σ² that are uniquely determined by their mean expression μ, and we used the fit σ²(μ) shown as the blue curve in Appendix 2—figure 1. This simplifies the problem from inferring a two-dimensional distribution ρ(μ, σ) to inferring a one-dimensional distribution ρ(μ), that is,

Applying selection with target log-fluorescence μ_* and width τ to this distribution, we obtain a new distribution

where C is a normalization constant that ensures ${\displaystyle \int }d\mu \rho \prime \left(\mu \right)=1$. The population distribution of log-fluorescence levels after selection is then

To infer the parameters of the fitness function for each selection that we performed, we fit the distribution ρ(μ) and parameters μ_* and τ that lead to an optimal fit to the observed distributions p(x) and p′(x). Appendix 2—figure 2 shows the inferred and observed distributions, as well as the inferred fitness function, for each of the six selection experiments that we performed.

Appendix 2—figure 2

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The figure shows that the distributions p(x) and p′(x) can be well fit by this model, illustrating that the form of the selection window (Equation 69) can well describe the effects of selection in the FACS machine. Moreover, we see that the distributions p(x) and p′(x) are typically significantly wider than the selection window $f\left(x|{\mu }_{\text{*}},\tau \right)$. Moreover, the fitted values of τ are almost perfectly proportional to the fraction of the population that was selected, with a value of τ ≈ 0.03 corresponding to the selection of 5% of the population that was used during the evolutionary runs. The fits also show that, although we in each experiment determine μ_* from the expression of the same reference promoter, there is some variability in μ_* from one experiment to the next. From these six experiments, we find that on average $\langle {\mu }_{\text{*}}\rangle =8.115$ and σ(μ_*) = 0.133.

Thus, in each selection round the fitness of a promoter with mean μ and variance σ² is given by Equation 71, where τ = 0.03 and μ_* fluctuates around $\langle {\mu }_{\text{*}}\rangle =8.115$. The effective fitness experienced by this promoter is thus given by the geometric average of Equation 71 with fluctuating values of μ_* and this is given by

Appendix 2—figure 3 shows a contour plot of this fitness function with the inferred parameters of $\langle {\mu }_{\text{*}}\rangle$, σ(μ_*) and τ as a function of the mean fitness μ, and the excess noise level η = σ² − σ_min²(μ), where ${\sigma }_{\text{min}}^2\left(\mu \right)$ is the minimal variance as a function of mean expression level μ (Equation 67). Note that, for these measurements, the plasmids were transformed into a different strain than those used to compare with the native E. coli promoters. We noticed that the minimal noise level ${\sigma }_{\text{min}}^2\left(\mu \right)$ as a function of mean μ is slightly different. Although the background fluorescence and burst size parameter β = 450 are the same, the parameter ${\sigma }_{ab}^2$ is smaller, that is, ${\sigma }_{ab}^2=0.006$ instead of ${\sigma }_{ab}^2=0..025$.

Appendix 2—figure 3

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Appendix 2—figure 3 clearly shows that fitness drops far more dramatically as a function of mean μ than as a function of excess noise η, that is, except for right at the optimal mean μ_*, the contours are running almost vertically in the plot. Second, we do not observe promoters with high excess noise, even though their fitness would easily allow it. For example, a promoter with mean expression near the optimum 8.11 but excess noise as high as 0.25 (i.e., significantly higher than observed for any of the clones) would have higher fitness than any promoter with mean less than μ = 7.76 or larger than μ = 8.47 (independent of their noise), and we do observe many promoters with means that deviate this far from the optimum. The fact that we do not observe high excess noise promoters, even though they would not be selected against, strongly suggests that such high noise promoters are uncommon a priori, that is, among all random sequences that drive expression at a medium level, the large majority have low excess noise levels and high noise promoters are rare. Moreover, note also that, for promoters that are not near the optimum, the optimal excess noise level is typically larger than those of the observed clones, for example, the optimal excess noise for a promoter with mean μ = 7.5 is η = 0.22. These observations all indicate that promoters have not experienced significant selection on their noise levels.

To further support this conclusion, Appendix 2—figure 4 shows the inferred fitness values for the observed clones both as a function of their mean expression (left panel) and as a function of their excess noise (right panel). The figure shows that, whereas the fitness of a clone can be accurately predicted from its mean μ, fitness is almost entirely uncorrelated to a promoter's excess noise η.

Appendix 2—figure 4

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Finally, if there was significant selection on noise levels, then we expect noise levels to systematically shift under selection. Appendix 2—figure 5 shows cumulative distributions of excess noise levels for clones obtained from different populations of cells.

Appendix 2—figure 5

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The figure shows that, surprisingly, the excess noise levels seem to increase from the third to the fifth round in the evolutionary run. However, given the limited number of clones involved, the change in excess noise levels is only marginally significant (p = 0.004 in a t-test). Similarly, the effect of selection on excess noise levels seems to be opposite on the populations of round 3 and round 5 (center and right panels in Appendix 2—figure 5). We suspect that there are some systematic experimental fluctuations that make measured excess noise levels vary across days, and that the observed distributions of excess noise levels are more a reflection of experimental variability than of true shifts in the distribution. Importantly, excess noise levels larger than 0.1, which are observed for a substantial fraction of native promoters, are very rare for all these populations.

In summary, our in-depth analysis of the FACS fitness function and the effects of selection show that the noise properties of the synthetic promoters have not been significantly shaped by selection. Already the synthetic promoters at the third round of selection are tightly concentrated in a low noise band, even though selection does not select against noise, and promoters with mean away from the optimum would benefit from having higher noise. Two additional rounds of mutation and selection on the promoters from the third round do not substantially change the distribution of noise levels, confirming that there are no substantial fitness differences among promoters with different noise. Similarly, performing additional rounds of selection (be it very stringent, normal, or lenient) also does not substantially change the observed noise levels of the selected promoters. Thus, our results show that promoters selected from a large collection of random sequences naturally display low noise levels. Importantly, this implies that the native promoters with substantially higher noise levels must have experienced some selective pressures that caused them to increase their noise.

Given a particular environment, the fitness, for example, growth rate or survival probability of a cell, depends on the expression level of its genes. Note that the fact that gene regulatory mechanisms have evolved already demonstrates that different environments require different gene expression patterns, that is, expressing a gene at the ‘wrong’ level for a given environment has negative effects on fitness/growth rate. Although our model can be developed for an arbitrary number of genes, for simplicity we will focus on a single gene. We assume that, in each given environment, there is an optimal expression level μ_*. Given that, as we have seen, expression levels are roughly log-normally distributed, we will express expression levels in log-space, that is, the logarithm of the number of proteins per cell. We define the fitness at the optimal expression level μ_* as f_o. Fitness will fall as the expression level x moves away from this optimum. In this simple conceptual model, we will assume that, like in our FACS selection, the fitness f(x) falls approximately Gaussian away from the optimum, that is,

where τ is again a parameter that determines how fast the fitness falls when the expression x moves away from the optimum μ_*.

To justify the Gaussian form of the fitness function, assume that expression levels are maintained for some characteristic time t. Over this time, cells that grow at rate ρ will expand by a factor of f = e^ρt. The growth rate ρ is optimal at x = μ_*, and to second order in the difference between x and μ_*, we can write

Defining f_o = e^ρt and $1/{\tau }^2=t{\Vert \frac{d^2\rho }{d{x}^2}\Vert }_{x={\mu }_{\text{*}}}$, we obtain the fitness function defined above. Note that, in this interpretation, the width of the selection function is inversely proportional to the time for which fitness fluctuations are maintained.

In complete analogy with the FACS selection case, the fitness $f\left(\mu ,\sigma |{\mu }_{\text{*}},\tau \right)$ of a promoter with mean μ and variance σ² in an environment with optimal expression level μ_* and width τ is given by the integral of the product of the fitness function (Equation 77) and the Gaussian distribution of expression levels, giving

Note that this functional form is a good approximation to the fitness function as long as expression levels are roughly log-normally distributed and as long as the integral of expression levels and fitness function can be approximated using the standard Laplace approximation, that is, expanding the logarithm of fitness to second order around its maximum.

We now extend this simple situation in two respects. First, instead of assuming that the optimal level μ_* is fixed, we imagine that the population of cells has gone through several different ‘environments’ where, in each environment e, there was an optimal expression level μ_e. Although we could develop our theory assuming the width τ varies across environments, the analytical results are much more transparent if we assume τ is the same in each environment, and we will make this assumption for clarity of presentation.