|Home | About | Journals | Submit | Contact Us | Français|
The recent back-to-back articles by Hay et al.1 and Shin et al.2 both addressed the important question of how the constituent enhancers of a so-called “super-enhancer” combine to activate the expression of a target gene. Super-enhancers are collections of closely spaced genomic regions that exhibit hallmarks of enhancers, such as binding by the Mediator complex and acetylation of histone H3 at lysine 27 (H3K27ac)3–5. As these authors noted, there is continuing controversy over whether super-enhancers genuinely represent a new paradigm in transcriptional regulation or whether they may essentially just be clusters of conventional enhancers that together produce a strong transcriptional response6.
At the heart of this question is whether the activity of a super-enhancer is simply given by the sum of its constituent enhancers—that is, whether it is additive—or whether these components instead exhibit some kind of synergy. Indeed, this question of additivity is of general interest, whether or not super-enhancers are qualitatively distinct from other loci. Hay et al.1 and Shin et al.2 addressed this question by carefully dissecting the highly expressed α-globin and Wap loci, respectively, and measuring the reductions in gene expression resulting from several individual and combined knockouts of constituent enhancers. Both articles described highly variable effects on gene expression from different individual knockout experiments, and both reported that it was necessary to disable multiple enhancers to abolish, or nearly abolish, expression. On the question of additivity, however, the two articles reached strikingly different conclusions: Hay et al. reported that the constituent enhancers at the α-globin locus acted “independently and in an additive fashion,” whereas Shin et al. reported that their observations of the Wap super-enhancer supported a “temporal and functional hierarchy” of constituent enhancers that is presumably non-additive.
It was notable that neither of these articles offered a precise definition for “additivity” or “hierarchy”. Moreover, neither article explicitly compared a null hypothesis of additivity against an alternative hypothesis. In reviewing these two works, we became interested in the various ways in which a super-enhancer’s activity could plausibly be modeled using a linear function of the activity of its constituent enhancers, possibly together with a simple nonlinear “link” function7, and in whether the data would allow a null hypothesis of such generalized linearity to be formally rejected. Here we show, by reanalyzing these two data sets, that they are both consistent with a generalized linear model that has a simple biophysical interpretation and does not require any hierarchy or synergy among constituent enhancers. Thus, we argue that it still remains to be demonstrated that a super-enhancer is greater than the sum of its parts.
Perhaps the simplest linear model would assume each constituent enhancer makes an additive contribution directly to the expression level of the target gene, such as might be the case if the constituent enhancers separately contribute to transcription. (This appears to be the model that Hay et al.1 had in mind.) Specifically, let us define the “activity” of the super-enhancer by the affine (linear plus constant) function,
where is a vector of binary variables indicating whether each constituent enhancer xi is present (xi=1) or absent (xi=0) and is a corresponding vector of real-valued coefficients, with β0 as an intercept term. Then, we can model measurements of the expression of the target gene, R(x), as a combination of this activity function and a variable ε representing some combination of biological and experimental noise. Thus, we write,
In practice, we consider alternative noise models and find that a log-normal model fits the data best for all of the models that we consider (see Supplementary Note for details).
Another plausible scenario is that the constituent enhancers combine multiplicatively, rather than additively, in determining R(x). This multiplicative relationship might be expected, say, if the constituent enhancers act to promote transcription in a sequential manner, with each step having the opportunity to amplify or dampen the outputs of previous steps. This relationship can be captured simply by making R(x) an exponential, rather than an additive, function of the activity A(x). Because the scale of A(x) is determined by free parameters, the base associated with the exponent is unimportant. By convention, we use base e and write,
Equation 3 can be considered a generalized linear model with inverse link function ex (in the language of GLMs7). Importantly, it is fully determined by a linear activity function, with no explicit consideration of interactions between the constituent enhancers.
Notably, this model can be given an alternative biophysical interpretation. Let us assume a physical system with two broadly defined “states,” a low-energy state associated with active transcription and a higher-energy baseline state. (The model is abstract: in reality, these “states” may each correspond to large ensembles of particular configurations of molecules.) Furthermore, let us interpret A(x) as a measure of the reduction in energy of the transcription-associated state relative to the baseline state. Statistical mechanics tells us that the occupancy of the low-energy state should be given by a Boltzmann distribution and be proportional to A(x)/Z, where Z is the partition function. If we further assume that the system is far from its optimum, then the occupancy of the low-energy state will be approximately proportional to eA(x). Equation 3 can therefore be interpreted as the model that results from assuming transcription is proportional to occupancy of the low-energy state in this suboptimal regime.
This physical interpretation, with a two-state system and a linear energy function, leads naturally to a third generalized linear model. In this case, we abandon the “suboptimal” approximation and consider the full Boltzmann distribution for the system8,9. In the two-state model, we can explicitly calculate the partition function Z and write, , which is known as a logistic function of A(x). Thus, we can fully describe the fraction of time the low-energy state is occupied using a generalized linear model with the logistic function as the inverse link function. Assuming again that gene expression is proportional to the occupancy of the low-energy state, we write,
where γ defines the maximum expected level of gene expression (for a similar model applied to enhancers, see reference ). Equation 4 will behave similarly to equation 3 when A(x) is far from its optimum but it will capture the phenomenon of diminishing returns in transcriptional output as the energetics of productive transcriptional elongation approach an optimum and gene expression is limited by other features of the system (saturation).
We fitted these three models (equations 2–4) to the raw data from Hay et al.1 and Shin et al.2 by maximum likelihood using a numerical algorithm for optimization. The data consisted of all replicates for each tested configuration (wild type and knockout) of the three constituent enhancers of the Wap super-enhancer and the five constituent enhancers of the α-globin superenhancer (see Supplementary Note for complete details). We compared the goodness-of-fit of the models using the Bayesian Information Criterion (BIC), which penalizes more complex models for their additional parameters. (Here, the linear-logistic model has one additional parameter, γ.)
For the α-globin data set1, for which the authors claimed additivity, we found that the additive model did indeed fit the data fairly well (Figure 1A). Nevertheless, the linear-logistic model was preferred over the additive model according to the BIC, despite its additional parameter. For the Wap data set2, the linear-logistic model is the best-fitting model by a substantial margin. Thus, for both of these data sets, the linear-logistic model explains the observed data better than any other generalized linear model (Figure 1B&C), and therefore is a better null model than the additive model. Notably, the linear-logistic model explains both data sets well despite several important differences between the two loci (e.g., the Wap component enhancers are substantially more tightly clustered and closer to the TSS than those for α-globin) and between the knock-out strategies used (Shin et al. deleted STAT5-binding sites whereas Hay et al. deleted larger DNase-I hypersensitive regions), which underscores the flexibility and generality of this simple model.
But do the data of Shin et al.2 for the Wap super-enhancer truly support something more complex than a generalized linear relationship, as the authors seem to claim? We attempted to address this question quantitatively in our framework by introducing interaction terms for the two pairs of constituent enhancers that were simultaneously knocked out in that study (ΔE1a/ΔE2 & ΔE2/ΔE3). We found that models allowing for interactions between constituent enhancers do have slightly higher likelihoods than the simple linear-logistic model, as they must, but, according to the BIC, these improvements are not sufficient to justify the use of an additional parameter (Figure 1D; see Supplementary Note for details). Thus, we find not only that the linear-logistic model fits both the α-globin and Wap data sets reasonably well, but also that this model cannot confidently be rejected in favor of one that allows for interactions between constituent enhancers.
It is possible, of course, that interactions between component enhancers do occur in reality, but the data collected so far are insufficiently abundant or precise to reject a generalized linear null model. In addition, our models are limited in that they address only the knockout data from these studies. In particular, our models do not address Shin et al.’s observation that the E1 enhancer is occupied by key transcription factors first during pregnancy, suggesting possible non-additivity in temporal establishment of the Wap super-enhancer, if not in its subsequent regulatory behavior. Finally, it is worth emphasizing that our abstract modeling approach provides no direct mechanistic insights into transcriptional regulation at either of these loci. Nevertheless, we have shown that the observed knockout data for both of these super-enhancers can be explained fairly well by a very simple generalized linear model, and this observation can at least constrain the family of possible mechanistic models. More broadly, we argue that the transcription field would benefit from clearer definitions of null models and more rigorous criteria for rejecting them before concluding that complex behaviors occur.
We thank the authors of references  and  for providing their raw data. We thank Barak Cohen, Charles Danko, and John Lis for comments on the manuscript, and Justin Kinney for useful discussions about biophysical models. This research was supported in part by US National Institutes of Health grants GM102192 and HG007070. The content is solely the responsibility of the authors and does not necessarily represent the official views of the US National Institutes of Health.
Note: The computer code developed for this analysis is available on GitHub (https://github.com/CshlSiepelLab/super-enhancer-code).
Author Contributions. N.D., Y.F.H, and B.G. contributed equally to this work. N.D., Y.F.H., and B.G. designed the experiments and analyzed the data. N.D., Y.F.H, B.G., and A.S. wrote the manuscript.
Competing Financial Interests. The authors declare no competing financial interests.