It is well known that the expression noise is lessened by natural selection for genes that are important for cell growth or are sensitive to dosage. In theory, expression noise can also be elevated by natural selection when noisy gene expression is advantageous. Here we analyze yeast genome‐wide gene expression noise data and show that plasma‐membrane transporters show significantly elevated expression noise after controlling all confounding factors. We propose a model that explains why and under what conditions elevated expression noise may be beneficial and subject to positive selection. Our model predicts and the simulation confirms that, under certain conditions, expression noise also increases the evolvability of gene expression by promoting the fixation of favorable expression level‐altering mutations. Indeed, yeast genes with higher noise show greater between‐strain and between‐species divergences in expression, even when all confounding factors are excluded. Together, our theoretical model and empirical results suggest that, for yeast genes such as plasma‐membrane transporters, elevated expression noise is advantageous, is subject to positive selection, and is a facilitator of adaptive gene expression evolution.
Individual cells with the same genotype often have different levels of gene expression even under the same condition, a phenomenon known as gene expression noise. Expression noise arises from stochastic events in gene expression, including those in transcription initiation, mRNA degradation, translation initiation, and protein degradation, and differences between cells, either in local environment or in the concentration or activity of any factor influencing gene expression (Raser and O'Shea, 2005). Gene expression noise can be harmful because it ruins cellular homeostasis in metabolism and developmental programs, affects precise controls of biochemical processes in cells, and breaks the stoichiometric balances among members of protein complexes. Indeed, yeast genes important for cell growth have lower expression noise than less important ones (Lehner, 2008). However, some genes are known to have high expression noise (Newman et al, 2006), raising the intriguing possibility that high noise is beneficial to the cell under certain circumstances. However, it is unclear whether the observed high noise of some genes is due to positive selection for elevated noise or ‘a mere result of lack of constraint on the variability in expression of such genes’ (Bar‐Even et al, 2006).
In this study, we address the above question in yeast, because expression noise at the protein level has been measured for thousands of yeast genes (Newman et al, 2006). We show that plasma‐membrane transporters have significantly greater‐than‐expected noise, given their levels of gene importance (Figure 1). This result holds even after we control for all known factors that are potentially associated with the relaxation of purifying selection on noise. Thus, the greater‐than‐expected expression noise of plasma‐membrane transporters is most probably caused by positive selection for elevated noise. We separated all genes in the genome into 21 bins according to their relative importance to cell growth. As shown in Figure 1B, the average noise level is only slightly higher in the last bin (most unimportant) than that in the first bin (most important). By contrast, the noise levels of the plasma‐membrane transporters in the last bin are much higher than an average gene in the last bin, suggesting that the effect of positive selection in elevating expression noise can be substantial. Twenty noisiest plasma‐membrane transporters in yeast are shown in Figure 1C.
Why would high noise of plasma‐membrane transporters be advantageous? We suggest that, because plasma‐membrane transporters interact with the environment directly, the mean expression level of a plasma‐membrane transporter is often suboptimal, due to frequent, sudden, and unpredictable environmental changes. We prove mathematically that when the fitness landscape (i.e. fitness of a cell as a function of the level of gene expression) is convex, the noisier genotype is fitter than the quieter genotype when the two genotypes have the same mean expression level (Figure 2). This is due to the likely non‐linear asymmetry of the fitness landscape at any suboptimal expression level (Figure 2A–D). By contrast, if the fitness landscape is concave, the quieter genotype is fitter (Figure 2C and D). If the fitness landscape is linear, the two genotypes are equally fit (Figure 2E and F). As the fitness landscape is likely to be bell shaped, the landscape is locally convex if the mean expression level is far from the optimal. Consequently, high noise is favored over low noise. This scenario may explain why high expression noise of plasma‐membrane transporters has been selected for.
On the basis of our theoretical model, we further prove mathematically that, under certain conditions, positive selection for advantageous mutations that alter the mean expression level is stronger for noisy genotypes than for quiet genotypes. Our computer simulation confirms and illustrates this theoretical result. To empirically test our theoretical prediction, we measured gene expression divergence among strains of the baker's yeast and among closely related yeast species. We found a significant positive correlation between the noise level of a gene and its expression divergence between strains and between species. This result holds after we control for mutational effect, gene importance, and all other known confounding factors. Thus, theoretical analysis, computer simulation, and empirical data all show that expression noise can increase the evolvability of gene expression by promoting the fixation of favorable expression level‐altering mutations.
Together, our theoretical model and empirical results show that, to yeast genes such as plasma‐membrane transporters, elevated expression noise is advantageous, is subject to positive selection, and is a facilitator of adaptive gene expression evolution.
Positive selection drives the elevation of expression noise of yeast plasma‐membrane transporters.
Theoretical modeling explains why and under what conditions high‐noise genotypes are selectively favored over low‐noise genotypes.
Theoretical analysis, computer simulation, and empirical data all show that expression noise can increase the evolvability of gene expression by promoting the fixation of favorable expression‐level‐altering mutations.
Gene expression, as other biological processes, is subject to noise (Schrodinger, 1944), which is defined as the stochastic variation in the expression level of a gene among isogenic cells under the same condition. Here and elsewhere in the paper, expression level refers to the level of the protein product of the gene, as expression noise is usually measured at the level of protein. Gene expression noise has been measured in prokaryotes (Elowitz et al, 2002; Ozbudak et al, 2002; Rosenfeld et al, 2005), unicellular eukaryotes (Blake et al, 2003; Raser and O'Shea, 2004), and mammalian cells (Ramsey et al, 2006). These and other studies showed that the level of expression noise varies substantially among genes, is determined genetically, and is selectable (Blake et al, 2006; Newman et al, 2006; Maheshri and O'Shea, 2007; Ansel et al, 2008). Expression noise has both intrinsic and extrinsic sources (Orphanides and Reinberg, 2002; Rao et al, 2002; Blake et al, 2003; Kaern et al, 2005; Raser and O'Shea, 2004, 2005; Bar‐Even et al, 2006; Newman et al, 2006; Volfson et al, 2006). Stochastic events in gene expression, including those in transcription initiation, mRNA degradation, translation initiation, and protein degradation, generate intrinsic noise (Raser and O'Shea, 2005). Differences between cells, either in local environment or in the concentration or activity of any factor influencing gene expression, generate extrinsic noise (Raser and O'Shea, 2005). We focus on intrinsic noise in this study because only intrinsic noise is an intrinsic property of a gene.
Gene expression noise is often considered a two‐edged sword. On one hand, the noise could be deleterious because it ruins cellular homeostasis in metabolism and developmental programs, affects precise controls of biochemical processes in cells, and breaks the stoichiometric balances among members of protein complexes (Fraser et al, 2004; Batada et al, 2006; Lehner, 2008). Increased gene expression noise has been reported to result in disease (Cook et al, 1998; Kemkemer et al, 2002; Bahar et al, 2006). Several studies showed direct and indirect evidence for lessened expression noise of genes that are important to cell growth or sensitive to dosage (Fraser et al, 2004; Newman et al, 2006; Batada and Hurst, 2007; Lehner, 2008). Furthermore, various molecular mechanisms and regulatory network structures (e.g. negative feedbacks) are found to attenuate expression noise (Becskei and Serrano, 2000; Pedraza and van Oudenaarden, 2005). On the other hand, several benefits of expression noise have been suggested. In particular, it has been argued that stochastic noise is essential in cell‐fate determination (Colman‐Lerner et al, 2005; Kaern et al, 2005; Losick and Desplan, 2008) and thus is important in the development of multicellular organisms. In unicellular organisms, it has been shown both theoretically and experimentally that stochastic switching of expression level or high expression noise could be beneficial in the face of fluctuating environments or under acute environmental stresses (Thattai and van Oudenaarden, 2004; Blake et al, 2006; Acar et al, 2008). It is thus plausible that a certain fraction of genes in a genome have elevated expression noise driven by positive selection. Indeed, a study of 43 yeast genes showed that stress‐related genes are noisier than the rest of the genes (Bar‐Even et al, 2006). A subsequent characterization of expression noise of thousands of yeast genes identified several Gene Ontology (GO) categories with significantly elevated noise, compared with the genomic average (Newman et al, 2006). These GO categories include amino‐acid biosynthesis, oxidative phosphorylation, heat shock, and stress response. Although it is tempting to suggest that the higher‐than‐average noise of these genes is a result of positive selection (Newman et al, 2006; Lopez‐Maury et al, 2008; Raj and van Oudenaarden, 2008), one cannot exclude the possibility that ‘it is a mere result of lack of constraint on the variability in expression of such genes’, as has been previously argued (Bar‐Even et al, 2006). The genome‐wide analysis (Newman et al, 2006) also suffered from a lack of statistical correction for multiple testing when many GOs were evaluated. Hence, it is not clear whether there are genuinely noisier‐than‐average GOs.
In this study, we test the hypothesis of positive selection for elevated gene expression noise by controlling multiple factors potentially associated with the relaxation of purifying selection. We identify plasma‐membrane transporters as the only group in yeast that shows significantly greater noise than the neutral expectation. We propose a model explaining why and under what conditions high noise may be beneficial. We further show theoretically and empirically that high noise facilitates adaptive gene expression evolution.
Plasma‐membrane transporters are significantly noisier than the neutral expectation
Newman et al (2006) measured the expression noise for over 2000 genes of the budding yeast Saccharomyces cerevisiae in rich (YPD) medium. As they controlled for several extrinsic factors, their noise estimates can be approximately regarded as intrinsic noise (Newman et al, 2006). The noise level is commonly measured by the coefficient of variation (CV), which is the s.d. of the expression level divided by the mean. Newman et al (2006) found a genome‐wide pattern of lower CV for genes with higher mean expression (also see Bar‐Even et al (2006)). To control the influence of mean expression level on noise and allow among‐gene comparison of noise levels, they used a new measure of noise named DM. For a given gene, DM is the difference of its CV from the median CV of those genes that have a similar mean expression as the focal gene (Newman et al, 2006).
As there is good evidence that the expression noise is lessened by natural selection for genes important for cell growth (Fraser et al, 2004; Batada and Hurst, 2007; Lehner, 2008), we need to control for the ‘importance’ of a gene when evaluating whether it is noisier than the expectation. The importance of a gene in yeast cell growth can be measured by the reduction in growth rate (i.e. fitness) in YPD upon deletion of the gene from the genome. Fortunately, such data exist for virtually every yeast gene (Giaever et al, 2002; Steinmetz et al, 2002). We separate all genes with expression noise data into 21 bins of different importance levels, with the fitness of the deletion strains being in the ranges of <0.05, 0.05–0.10, 0.10–0.15, …, 0.95–1.00, and >1.00, respectively. The last bin is not empty because the fitness value of a gene‐deletion strain was originally measured relative to the mean of all viable gene‐deletion strains, rather than to the wild‐type strain (Steinmetz et al, 2002). To test whether the noise level of genes belonging to a given GO category exceeds the expectation, we randomly draw genes (with replacement) from the genome‐wide expression noise data to form a gene set that has the same number of genes in each of the 21 bins as the focal GO has. We repeat this process 20 000 times and calculate the proportion of times when the mean noise level of the GO is lower than that of the randomly constructed gene set. If this probability (P‐value in Table I) is lower than 5%, we regard the GO to be significantly noisier than expected. As we examine numerous GO categories, we further control for multiple testing using a 5% false discovery rate (Storey and Tibshirani, 2003). That is, only GOs with a Q‐value <0.05 are considered as truly significant. To ensure that there is sufficient statistical power to detect elevated noise of a GO, only those GOs with at least 30 genes were examined.
GO categories are organized into three groups: biological process, cellular component, and molecular function (Ashburner et al, 2000). The three groups characterize different aspects of a gene's function and are thus examined separately in our analysis. We found that in terms of biological process, 18 GOs related to metabolism and transport show significantly higher‐than‐expected noise (Table I). In terms of cellular component, five GOs related to organelles (particularly mitochondrion) have high noise (Table I). In terms of molecular function, four GOs related to catalytic activity and transporter activity have high noise (Table I). The high expression noise of proteins localized to the mitochondrion (and other low copy‐number organelles) was noted before and was thought to be caused by unequal partitioning of mitochondria (and other organelles) during mitosis (Newman et al, 2006). Further evidence for this explanation came from the experiment showing that the same protein expressed from the same promoter and locus is noisier when targeted to low copy‐number organelles than when localized to the cytosol (Newman et al, 2006). Thus, the high noise of mitochondrial proteins is unlikely the result of positive selection for elevated noise. Further, the high noise of enzymes is probably due to their special insensitivity to dosage, rather than positive selection for high noise, because it is well known that, in a metabolic pathway, even a considerable change in the concentration of an enzyme has a minimal effect on the flux of the pathway (Kacser and Burns, 1981). This phenomenon arises from the kinetic connection through the shared substrates/products of adjacent biochemical reactions such that the effect of changing the catalytic activity in one reaction tends to be buffered by the response to this of the other reactions (Kacser and Burns, 1981). Thus, to be conservative, we removed all mitochondrial proteins and all enzymes, and re‐tested each GO. This time, we identified plasma membrane as the only cellular component GO category and transporter activity as the only molecular function GO category that show significantly higher‐than‐expected noise (Table I). No biological process GO category is significantly noisier than expected. Our results are robust to the variation of the number of bins used (11–26) in controlling the effect of gene importance on noise (Supplementary Tables S1 and S2).
Haploinsufficient genes are sensitive to expression noise and should have reduced noise, as has been shown (Cook et al, 1998; Batada and Hurst, 2007; Lehner, 2008). To test whether the high noise of plasma‐membrane proteins and transporters is simply because they are less likely to be haploinsufficient than other genes in the genome, we further removed all haploinsufficient genes (Deutschbauer et al, 2005) from the genome and re‐tested every GO. We found that both ‘plasma membrane’ and ‘transporter activity’ GOs remain significantly noisier than expected (Q=0.0038 and 0.0056, respectively) and that the Q‐values are similar to those obtained without the control for haploinsufficient genes (Table I). This is probably due to the paucity of haploinsufficient genes in the yeast genome (Deutschbauer et al, 2005). For this reason, haploinsufficient genes are no longer controlled for in subsequent analysis unless otherwise noted. Previous direct and indirect evidence suggested that components of stable protein complexes are also sensitive to dosage and thus have reduced noise (Fraser et al, 2004; Lehner, 2008). We found that after the control for gene importance, protein complex members no longer have lower noise than other proteins (P=0.76, Mann–Whitney U test). Thus, there is no need to further control for protein complex membership in our analysis. Overexpressions of certain genes are detrimental; these genes could have reduced expression noise as well (Lehner, 2008). However, we found no significant correlation between gene expression noise (DM) and the fitness of gene overexpression strains (Sopko et al, 2006) (Spearman's rank correlation ρ=0.03, P=0.15). Thus, there is no need to consider the potential selection on noise due to gene overexpression. Taken together, the high expression noise of plasma‐membrane genes and transporter genes cannot be explained by relaxation of purifying selection because all known factors that could potentially lead to the relaxation of purifying selection on noise have been excluded; positive selection for elevated noise remains the most plausible explanation of their higher‐than‐expected noise.
We suspect that the significant results from ‘plasma membrane’ and ‘transporter activity’ GOs are because of the high noise of plasma‐membrane transporters. Indeed, plasma‐membrane transporters are significantly noisier than expected after the control for gene importance and the removal of enzymes and mitochondrial proteins (P=3.3 × 10−6; two‐tail Z‐test), whereas plasma‐membrane proteins that are non‐transporters (P=0.77) and transporters that are not localized to the plasma membrane (P=0.21) are not significantly different from the expectation (Figure 1A). A careful examination shows that the majority of plasma‐membrane transporters (79%) belong to the last bin of gene importance (i.e. fitness of the gene‐deletion strain >1.00) (Figure 1B). For this bin, the genomic average noise level is DM=0.87±0.16, only slightly, although significantly, greater than the mean noise (−0.10±0.18) of the first bin (i.e. fitness <0.05), suggesting that the effect of negative selection in reducing the expression noise of important genes is overall relatively small (Figure 1B). By contrast, the mean noise of the plasma‐membrane transporters in the last bin is DM=5.62±1.00, suggesting that the effect of positive selection in elevating expression noise can be substantial (Figure 1B). Again, the above comparison is based on the dataset after the removal of enzymes and mitochondrial proteins. Figure 1C lists the 20 noisiest plasma‐membrane transporters. These proteins transport a diverse array of chemicals, such as amino acids, glucose, ions, thiamine, polyamine, oligopeptides, and nucleotides, across the cell membrane. They are involved in the uptake of nutrients and ions, excretion of end products of metabolism and deleterious substances, and communication between cells and the environment. We also examined the yeast expression noise data obtained under the minimal (SD) medium (Newman et al, 2006) and confirmed that plasma‐membrane transporters is the only group with significantly greater noise than expected after all the controls (i.e. gene importance, enzymes, and mitochondrial proteins) (Supplementary Table S3). We also confirmed that this result is robust to the variation of the number of bins used (11–26) in controlling the effect of gene importance on noise (Supplementary Tables S3–S5).
High noise can be beneficial when the mean expression level is suboptimal
Why would high noise be beneficial to plasma‐membrane transporters? It is likely that the optimal expression level of each transporter depends on environmental factors such as the nutrients available to the cell. The underexpression of a transporter may limit the nutrient uptake rate and hence limit the cell's Darwinian fitness. However, overexpression of a transporter could also be disadvantageous for two reasons. First, overexpression has a fitness cost due to the waste of energy in transcription and translation (Wagner, 2005; Stoebel et al, 2008). Second and more importantly, presence of unwanted transporters could reduce the metabolic efficiency and hence the fitness. For example, imagine that two carbon sources C1 (e.g. maltose) and C2 (e.g. lactose) are both present in the medium, but C1 is energetically more efficient than C2 for the cell to use. If the total number of carbon source molecules that the cell can catabolize per unit time is limited, it would be better for the cell to use C1 rather than C2. Thus, an overexpression of the transporter for C2 will reduce the number of carbon source molecules catabolized by the cell per unit time and thus will be deleterious. Certainly, many transporter genes are under transcriptional regulation such that the transporter concentrations differ under different environments. However, changes of expression by gene regulation take time and are energetically costly (Perez‐Ortin et al, 2007). More importantly, the cell does not have regulatory responses to all possible environmental changes. Thus, high expression noise of transporters allows, at least, some cells to have high fitness in an unpredictable environment. Below we show mathematically that, under certain conditions, genotypes with high expression noise can have greater Darwinian fitness than those with low noise.
Let us consider two genotypes A and B. The only difference between them is that A has a higher level of expression noise than B for gene X. The mean expression level (m) of X is identical between the two genotypes. The distribution of the expression noise (e) for gene X is described by probability density functions gA(e) and gB(e) for the two genotypes, respectively. Genome‐wide expression noise data showed that e generally follows a normal distribution (Bar‐Even et al, 2006; Newman et al, 2006). Let us assume that a population, having A and B cells, experiences an environmental change such that the mean expression level of X becomes suboptimal. Let f(x)=f(m+e) be the fitness of the cell that has an expression level of X equal to x. So, the fitness of genotype A, or the mean fitness of A cells, equals . Similarly, the fitness of genotype B equals . It can be shown that (i) when f(x) is a convex function (i.e. the second derivative of f(x) is positive), FA>FB; (ii) when f(x) is a concave function, FA<FB; and (iii) when f(x) is linear, FA=FB (Figure 2; Supplementary Figure S1; Supplementary information 1). As f(x) may not be concave or convex for all possible values of x, what matters is whether f(x) is concave or convex for the range of x realized in the majority (e.g. 95% or 99%) of A and B cells. Note that in our model, the optimal expression level can be either higher or lower than m (Figure 2). Although the shape of f(x) is generally unknown, it is reasonable to assume that, at least, for many genes if not most genes, it is bell shaped with the optimal expression level in the center (Kacser and Burns, 1981; Hartl et al, 1985; Bedford and Hartl, 2009). In such cases, f(x) is concave when x is close to the optimal expression level, but convex when x is far from the optimal. Thus, big environmental changes tend to generate conditions under which high noise is beneficial. Note that although we compared mean fitness values of cells with two different genotypes, there is no involvement of group selection in our model. When f(x) is convex, in a population fixed with the wild type, a mutant with a higher level of noise is expected to increase its frequency in the population because its fitness is greater than that of the wild type.
To see how large FA–FB is when realistic parameters are used in our model, we examined a few numerical examples. As the effective population size of yeast is of the order of 107 (Wagner, 2005), a fitness differential greater than 10−7 can be detected by natural selection. We found that FA‐FB is easily greater than 10−7. For example, in Figure 2A, we assumed . That is, f(x) is scaled by from the probability density function of normal distribution N(μ, σ), where μ and σ are the mean and s.d. of the normal distribution, respectively. We used μ=6.2 and σ=1. We further assumed that the expression noise in genotypes A and B follows N(0, 1.2) and N(0, 0.7), respectively, and that the mean expression levels of the two genotypes are both m=3. Given these parameters, we found that FA–FB=0.0728–0.0264=0.0464, five orders of magnitude greater than 10−7. Further analysis indicates that a large parameter space allows FA–FB to be substantially greater than 10−7, and this is true even for genes with a tiny fitness effect (e.g. <1%) upon deletion (Supplementary Figure S2). A previous site‐directed mutagenesis study showed that a single point mutation in the GAL1 promoter of yeast can more than triple the level of expression noise (measured by the s.d. of the expression level) (Blake et al, 2006). So, the assumed noise difference between genotypes A and B here can arise simply by a point mutation. This and other numerical examples, we tried, suggest that conditions under which the benefit of high noise is detectable by natural selection arise easily.
Our model predicts faster adaptive expression evolution of noisier genes
Under our model described above, it can be shown that, when the mean expression level of a genotype is lower than the optimal level and the third derivative of f(x) is positive, or when the mean expression level of a genotype is higher than the optimal level and the third derivative of f(x) is negative, a given amount of change in mean expression level towards the optimal level will result in a greater fitness increase for the genotype with a higher level of noise (Figure 3; Supplementary Figure S3; Supplementary information 1). This is because, under the above conditions, the same advantageous mutation increases the mean fitness of the noisier genotype more than that of the quieter genotype (Figure 3). Consequently, the strength of positive selection for the same advantageous mutation that changes the same amount of mean expression level is stronger in a noisier genotype than in a quieter genotype. For instance, in the numerical example depicted in Figure 3, we used f(x) = e−(x−μ)2/(2σ2), where μ=11 and σ=2.5. The expression noise in genotypes A and B follows N(0, 1.2) and N(0, 0.7), respectively, and the mean expression levels of the two genotypes are both m=3.0. The advantageous mutation shifts the mean expression of both genotypes to n=7.1. Under such conditions, the fitness of genotype A increases from 0.0178 to 0.4195 because of the mutation, whereas the fitness of genotype B increases from 0.0107 to 0.3978 because of the same mutation. Thus, the fitness gain for genotype A (0.4017) is greater than that (0.3871) for genotype B. We observed this trend in a large parameter space examined (Supplementary Figure S4). Here we assumed that the mutation size (n−m=4.1) is ∼3.5 times the noise level of genotype A and ∼6 times the noise level of genotype B. This assumption is realistic, because a previous study on the yeast GAL1 promoter showed that a single point mutation can change the mean expression level by more than 10 times the noise level (Blake et al, 2006).
As the same advantageous mutation can enhance the fitness of the noisier genotype more than the quieter genotype, we predict faster adaptive evolutionary changes in mean expression level for noisier genes than for quieter genes. To see to what extent the noise level impacts the rate of adaptation, we conducted a computer simulation. Let us consider a population of yeast cells all with genotype A and another population all with genotype B. The two genotypes have the same mean expression level that is suboptimal. Genotype A has a higher expression noise level than genotype B. The two populations have the same population size, mutation rate, and mutation spectrum. Mutations are randomly generated with a size that follows a normal distribution. Here, mutation size refers to the difference between the mean expression level of the mutant and that of the wild type. We assume that the level of expression noise does not change. As shown in Figure 4A, under the parameters detailed in Methods section, genotype A adapts its expression level to the optimal level significantly faster than genotype B (P<10−48, t‐test), and the difference in speed is on average 2.56‐fold. Figure 4 also shows the adaptation process from one simulation replication, in which the noisier genotype (Figure 4B) adapts to the optimal expression level in about one‐fifth the time required for the quieter genotype (Figure 4C). Thus, at least under some conditions, high expression noise leads to a substantially enhanced rate of adaptation of gene expression level because noise can facilitate positive selection for advantageous mutations. Note that although the number of generations required for adaptation seems very large in Figure 4, the actual time required can be much shorter if the mutations are larger or the mutation rate is higher. We found that our simulation result holds in a broad parameter space when we vary the mutation rate and the noise ratio of the high‐noise and low‐noise genotypes (Supplementary Figure S5).
Above, we considered beneficial mutations. In the case of deleterious mutations, it can be similarly shown that, under our model, when the mean expression level of a genotype is lower than the optimal level and the third derivative of f(x) is positive, or when the mean expression level of a genotype is higher than the optimal level and the third derivative of f(x) is negative, a deleterious mutation that renders the mean expression level further away from the optimal level will result in a larger fitness loss for the genotype with a higher level of noise (Supplementary Figure S3; Supplementary information 1). In other words, under such conditions, negative selection against deleterious mutations that affect the mean expression level will be stronger for noisier genotypes.
Empirical data show greater expression divergence of noisier genes
It will be interesting to empirically verify our prediction of higher rates of adaptive expression evolution in noisy genes than in quiet genes. As the available expression noise data are from one strain of Saccharomyces cerevisiae (Newman et al, 2006), it is better to estimate the evolutionary rate of gene expression using closely related species or even intraspecific strains such that the noise level may be considered constant in the evolution of gene expression level. We first compared two strains of S. cerevisiae, a laboratory strain BY4716 (derived from s288c) and a wild isolated strain RM11‐1a, using whole genome microarray gene expression data generated under the same condition (synthetic complete medium at 30°C) (Brem and Kruglyak, 2005). We observed a positive correlation between gene expression noise and gene expression divergence between the two strains (Spearman's rank correlation coefficient ρ=0.241, P<10−26) (Figure 5A; Table II). Using microarray gene expression data from five different conditions, we next measured the divergence of mean expression level among four closely related species of the Saccharomyces sensu stricto complex (Tirosh et al, 2006), and found that the divergence is also positively correlated with expression noise (ρ=0.291, P<10−32; Figure 5B; Table II), as was previously observed (Lehner, 2008). However, it is not trivial to show that the correlation reflects the prediction of our model rather than some other mechanism. Below we examine possible alternative mechanisms by calculating partial correlation (Fisher, 1924) and show that although some of them do have a role, they cannot fully explain the observed correlations.
First, it is possible that the number of mutation sites or the effect of each mutation (in terms of changing the mean expression level) varies among genes of different levels of noise. Indeed, as was previously noted (Landry et al, 2007), expression evolution, measured by expression variance among mutation‐accumulation lines of yeast, is positively correlated with expression noise (ρ=0.24, P<10−27). As the effective population size was controlled to be ∼10 cells in the mutation‐accumulation experiment (Landry et al, 2007), the majority of non‐lethal mutations behave neutrally and thus the rate of expression evolution of these lines reflect the rate and size of expression mutation, hereby collectively referred to as mutational effect. If the difference in mutational effect is the sole reason for the correlation between gene expression noise and expression divergence shown in Figure 5, we should not observe this correlation after the control for the expression variance in mutation‐accumulation lines. However, in fact, the partial correlation between gene expression noise and expression divergence remains significantly positive in both between‐strain (ρ=0.203, P<10−18; Table II) and between‐species (ρ=0.247, P<10−23; Table II) comparisons.
Second, negative selection against expression noise can also generate a positive correlation between the noise level and the rate of expression evolution, because important genes tend to have both low noise (Cook et al, 1998; Batada and Hurst, 2007; Lehner, 2008) and low rate of expression evolution (Tirosh and Barkai, 2008). However, after further controlling gene importance, we found the positive correlation between the noise level and the rate of expression evolution to remain significant in both between‐strain (ρ=0.191, P<10−10; Table II) and between‐species (ρ=0.223, P<10−11; Table II) comparisons.
Third, the above control of gene importance does not fully eliminate the among‐gene variation in the level of negative selection against noise. Thus, we further removed mitochondrial proteins, enzymes, and haploinsufficient proteins from our dataset. We found the positive correlation between the noise level and expression divergence to remain significant in both between‐strain (ρ=0.131, P<10−4; Table II) and between‐species (ρ=0.216, P<10−8; Table II) comparisons. As shown earlier, after the control for gene importance, membership in stable protein complexes no long correlates with expression noise. We therefore did not further control for complex membership here. Together, the above results provide empirical support to the prediction of our model that high noise can facilitate adaptive gene expression evolution.
By analyzing the yeast genome‐wide gene expression noise data, we identified plasma‐membrane transporters as the only group that shows significantly greater‐than‐expected noise after the exclusion of multiple factors related to the relaxation of negative selection against noise. Although this result suggests that the elevation of the expression noise in plasma‐membrane transporters is driven by positive selection, an alternative hypothesis is that the high noise is a by‐product of selection for something else rather than the direct target of selection. One particularly relevant subject here is the differential use of TATA boxes in the promoters of different groups of genes (Basehoar et al, 2004). For example, TATA‐containing genes are associated with responses to stress, are highly regulated, and preferentially utilize SAGA rather than TFIID when compared with TATA‐less genes (Basehoar et al, 2004). Interestingly, TATA‐containing genes have significantly larger expression noise than TATA‐less genes (Newman et al, 2006). Hence, the high noise of plasma‐membrane transporters could potentially be a by‐product of the use of TATA‐containing promoters, if plasma‐membrane transporter genes require TATA‐containing promoters for gene regulation. After removing enzymes and mitochondrial proteins, our dataset contains 1088 genes that have the information about gene importance, expression noise, and the presence/absence of a TATA box. Although only 13.4% of these 1088 genes contain a TATA box, the corresponding number is 54.5% among plasma‐membrane transporters (P<10−7, χ2 test). Nevertheless, even among TATA‐containing genes, the expression noise is significantly higher for plasma‐membrane transporters than for the other genes after the control for gene importance, enzymes, and mitochondrial proteins (P<0.001; two‐tail Mann–Whitney U test). The same is true among TATA‐less genes (P<0.02). Thus, the high noise of plasma‐membrane transporters is not fully attributable to the preferential use of TATA‐containing promoters, supporting direct positive selection for elevated expression noise of these genes. The result further suggests that multiple molecular mechanisms are used to achieve the high expression noise of plasma‐membrane transporters. We note that even if the high noise of plasma‐membrane transporters were fully attributable to the preferential use of TATA‐containing promoters, the hypothesis of direct selection for high noise could not be rejected, because of the possibility that the preferential use of TATA‐containing promoters is a by‐product of the selection for high noise; it would then become necessary to differentiate which is the direct target of selection and which is the by‐product.
When controlling for gene importance, we used the data of fitness reduction by gene deletion measured in laboratory rich media, which may not resemble closely the natural environments of yeast. However, the fact that this gene importance index significantly inversely correlates with the expression noise level (Lehner, 2008), also measured under the rich media, suggests that using this gene importance index in analyzing the rich media noise data is meaningful. Our subsequent analysis of the noise data from rich and minimal media showed that despite the large nutritional difference between the two media, the noise levels under the two media are highly correlated (ρ=0.53, P<10−15), suggesting that the noise levels measured in lab conditions may be good proxies for the true values in nature. Moreover, expression noise data from both the rich and minimal media identified plasma‐membrane transporters as the only group with higher‐than‐expected noise. Thus, it is unlikely that our result is an artifact due to the use of various data generated from lab conditions that are different from the natural environments of yeast.
One could argue that plasma‐membrane transporters may be regarded as special enzymes and thus their high noise may be related to the dosage‐buffering effect that enzymes generally suffer from (Kacser and Burns, 1981). We compared the noise level of enzymes and plasma‐membrane transporters by separating the enzyme genes into 21 bins based on their fitness effects upon deletion and drawing enzyme genes randomly according to the importance levels of the plasma‐membrane transporters. Repeating this process 10 000 times, we found that plasma‐membrane transporters are on average 2.92 times nosier than enzymes after the control for gene importance (P=0.001, Mann–Whitney U test). Thus, the high noise of plasma‐membrane transporters cannot be explained by the buffering effect even if the transporters behave as enzymes, further supporting positive selection as the evolutionary force behind their high noise.
We proposed a simple mathematical model and showed that a high‐noise genotype will have a greater fitness than a low‐noise genotype with the same mean expression, as long as the fitness function is convex. The key question is whether the cellular fitness, as a function of the expression level of a plasma‐membrane transporter, has a convex region. To our knowledge, there has been only one study that empirically determined the fitness function (Dekel and Alon, 2005). This study reported the relationship between the fitness of Escherichia coli cells and the expression level of Lac proteins; the fitness function seems to be concave in the ranges examined. However, this result does not preclude the existence of a convex region in unexamined expression ranges of Lac proteins, nor does it tell us the fitness functions of other genes. It is likely that the shape of the fitness function varies depending on the specific cellular role each gene has. Owing to the lack of sufficient empirical data on the fitness function, we decide to examine the theoretical possibilities, especially in the context of plasma‐membrane transporters. A simple theoretical model shows the existence of convex regions in the fitness function (Supplementary information 2 and Supplementary Figure S6). Although the jury is still out as whether the fitness function indeed contains a convex region, our theoretical modeling supports this possibility. As the natural environment of yeast may change abruptly and frequently and because plasma‐membrane transporters are directly involved in the interactions between a cell and its biotic and abiotic environment, conditions under which high expression noise of plasma‐membrane transporters is favored over low noise may arise relatively easily. By contrast, non‐plasma‐membrane transporters (e.g. those localized to the nuclear envelop) and plasma‐membrane proteins that are not transporters (e.g. those attaching the cell wall to the plasma‐membrane) generally do not face relevant environmental changes that are unpredictable, abrupt, and frequent. It should be noted that because mitochondrial proteins and enzymes were removed in our GO analysis, adaptive elevation of noise could not be tested for genes belonging to these two categories. A previous study identified genes related to stress to be noisier than the genomic average (Newman et al, 2006). In our analysis, the biological process GO category of ‘response to stress’ (GO0006950) was significantly noisier than the genomic average before the control for multiple testing (P=0.012), but not after the control (Q=0.082) (Supplementary Table S6). Regardless, although our analysis does not preclude the possibility that even some individual non‐plasma‐membrane‐transporter genes have elevated noise driven by positive selection, plasma‐membrane transporters are apparently among those that most frequently face large and unpredictable environment fluctuations. Hence, our results are biologically sensible. The strength of positive selection for high noise depends on how frequently favorable conditions occur and on the fitness functions of a gene under such conditions. It should be noted that high noise is advantageous only under unpredictable environmental changes. Repeated switching among a fixed set of environments may lead to the evolutionary emergence of gene regulation, with which low noise could be beneficial.
Our model of why high noise can be favored over low noise differs significantly from a previous model that is based on the bimodal distribution of gene expression (Thattai and van Oudenaarden, 2004). In the model, the expression level of a gene in a cell switches between two states. Given that empirical data show a normal distribution of noise (Bar‐Even et al, 2006; Newman et al, 2006), our model is more realistic and general. Our model is also more general than another earlier model in which the fitness function f(x) is assumed to be either 0 or 1 (Blake et al, 2006).
Our mathematical model further predicts faster adaptive evolution of gene expression toward the optimum for noisier genotypes than for quieter genotypes under certain conditions. Our model, again, is significantly different from previous models that are based on multiple expression attractors (Kaneko and Furusawa, 2008). Our prediction is supported by our observation of higher expression divergence between yeast strains and between yeast species for genes of higher noise even when all confounding factors are controlled for. We note that our result does not rely on the assumption that all or most gene expression divergence between strains (or species) is adaptive. The fact that, after all controls, expression noise explains only several percent of the among‐gene variation in expression divergence, is not inconsistent with the hypothesis that the majority of expression divergence is neutral (Khaitovich et al, 2006).
Taken together, high expression noise is not only selected for in certain yeast genes under unpredictable environmental changes, it also facilitates adaptive expression evolution when a directional environmental change occurs. We expect that all unicellular organisms that face unpredictable and frequent environmental changes would show a similar pattern of elevated expression noise in those genes whose expression levels are often suboptimal, and it will be interesting to test this prediction in the future when genome‐wide expression noise data become available for additional species. The power and versatility of natural selection in seizing and utilizing even seemingly harmful biological properties such as the stochasticity in gene expression to enhance organismal fitness is a wonderful tribute to the theory of evolution by means of natural selection.
Materials and methods
The yeast genome‐wide datasets of normalized gene expression noise level (DM) in rich and minimal media were from the study by Newman et al (2006). Gene expression data from BY4716 and RM11‐1a strains (Brem and Kruglyak, 2005) were retrieved using GEOquery in Bioconductor (Gentleman et al, 2004; Sean and Meltzer, 2007). Expression divergence was estimated by the log2 ratio of the relative intensity of hybridization signals in microarray experiments (Brem and Kruglyak, 2005). Gene expression divergence among four yeast species was similarly estimated and was directly taken from the study by Tirosh et al (2006). Gene expression divergence among mutation‐accumulation lines of yeast was measured by the variance of expression signal across lines and was supplied by Landry et al (2007). Gene importance was measured by the reduction in fitness upon gene deletion, and was acquired from earlier studies (Giaever et al, 2002; Steinmetz et al, 2002). Data on fitness effects of gene overexpression are obtained from Sopko et al (2006), in which the growth rates of gene overexpression strains are divided into five levels, from 1 (no growth) to 5 (normal growth). GoMiner (Zeeberg et al, 2003) was used to retrieve GO (Ashburner et al, 2000) information for yeast genes. The information about the presence and absence of TATA boxes in yeast genes was acquired from the study by Basehoar et al (2004).
Computer simulation was conducted to examine to what extent the noise level impacts the rate of adaptive expression evolution, in the face of mutation, drift, and selection. We considered a population of yeast cells all with genotype A and another population all with genotype B. The only difference between the two genotypes is that the expression noise for gene X is higher in A than in B. We assumed that the expression level of gene X in individual cells of the two populations follows the normal distribution N(μ, σ1) and N(μ, σ2), respectively. The two populations have the same population size L, mutation rate m, and mutation size distribution. Here, mutation size refers to the difference between the mean expression level of the mutant and that of the wild type. Mutations were randomly generated with a size that follows the normal distribution N(0, σ′). We assumed that the expression noise level does not change. The fitness function f(x) used was bell shaped (see Figure 4B and C): , where the normal probability density function , where a is a fitness scaling factor and b the optimal expression level. We used the following parameters in our simulation: L=1000, a=0.1, b=6, c=0.4, m =10−4, μ=4, σ1=0.6, σ2=0.3, σ′=0.1. We started the simulation by generating a population of L haploid cells. We then generated mutations in each cell. The relative frequency of each allele in the next generation was determined by its relative fitness in the population as well as genetic drift. When the mean expression level of the population reached within one s.d. from the optimal expression level, we considered the adaptive evolution to be successful and recorded the number of generations used. If after 104 generations the mean expression level of the population still had not reached the above cutoff, we stopped the simulation and recorded the time used as 104 generations. We conducted 200 simulation replications for each of the two populations.
We thank Christian Landry for providing the data of expression variance among mutation‐accumulation yeast lines and Meg Bakewell, Zhi Wang, and three anonymous reviewers for valuable comments. This study was supported by research grants from the National Institutes of Health and University of Michigan Center for Computational Medicine and Biology to JZ.
Conflict of Interest
The authors declare that they have no conflict of interest.
All supplementary files
Supplementary Notes 1‐2, Supplementary Tables S1‐S6, Supplementary Figures S1‐S6 [msb200958-sup-0001.pdf]
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