Mutations are changes to DNA that provide the raw material upon which evolution can act. Therefore, to understand evolution, we need to know the effects of mutations, and how those mutations interact with each other (a phenomenon referred to as epistasis). So far, few mathematical models allow scientists to predict the effects of mutations, and even fewer are able to predict epistasis.

Biological systems are complex and consist of many proteins and other molecules. Genes are the sections of DNA that provide the instructions needed to produce these molecules, and some genes encode proteins that can bind to DNA to control whether other genes are switched on or off. Lagator, Paixão et al. have now used mathematical models and experiments to understand how the environment inside the cells of a bacterium known as E. coli, specifically the amount of particular proteins, affects epistasis.

These mathematical models are able to predict interactions between mutations in the most abundant class of DNA-binding sites in proteins. This approach found that the nature of the interaction between mutations can be explained through biophysical laws, combined with the basic knowledge of the logic of how genes regulate each other’s activities. Furthermore, the models allow Lagator, Paixão et al. to predict interactions between mutations in several different environments, such as the presence of a new food source or a toxin, defined by the amounts of relevant DNA-binding proteins in cells.

By providing new ways of understanding how genes are regulated in bacteria, and how gene regulation is affected by mutations, these findings contribute to our understanding of how organisms evolve. In addition, this work may help us to build artificial networks of genes that interact with each other to produce a desired response, such as more efficient production of fuel from ethanol or the break down of hazardous chemicals.

The interaction between individual mutations – epistasis – determines how a genotype maps onto a phenotype (Wolf et al., 2000; Phillips, 2008; Breen et al., 2012). As such, it determines the structure of the fitness landscape (de Visser and Krug, 2014) and plays a crucial role in defining adaptive pathways and evolutionary outcomes of complex genetic systems (Sackton and Hartl, 2016). For example, epistasis influences the repeatability of evolution (Weinreich et al., 2006; Woods et al., 2011; Szendro et al., 2013), the benefits of sexual reproduction (Kondrashov, 1988), and species divergence (Orr and Turelli, 2001; Dettman et al., 2007). Studies of epistasis have been limited to empirical statistical descriptions, and mostly focused on interactions between individual mutations in structural proteins and enzymes (Phillips, 2008; Starr and Thornton, 2016). While identifying a wide range of possible interactions (Figure 1), these studies have not led to a consensus on whether there is a systematic bias on the sign of epistasis (Lalić and Elena, 2013; Kussell, 2013; Velenich and Gore, 2013; Kondrashov and Kondrashov, 2015), a critical feature determining the ruggedness of the fitness landscape (Poelwijk et al., 2011). Specifically, it is only when mutations are in sign epistasis that the fitness landscape can have multiple fitness peaks - a feature that determines the number of evolutionary paths that are accessible to Darwinian adaptation (de Visser and Krug, 2014). Furthermore, even a pattern of positive or negative epistasis has consequences for important evolutionary questions such as the maintenance of genetic diversity (Charlesworth et al., 1995) and the evolution of sex (Kondrashov, 1988; Otto and Lenormand, 2002). While the absence of such a bias does not reduce the effect of epistasis on the response to selection, it does demonstrate that predicting epistasis remains elusive.

Figure 1

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Scarcity of predictive models of epistasis comes as no surprise, given that most experimental studies focused on proteins. The inability to predict structure from sequence, due to the prohibitively large sequence space that would need to be experimentally explored in order to understand even just the effects of point mutations (Maerkl and Quake, 2009; Shultzaberger et al., 2012), let alone the interactions between them, prevents the development of a predictive theory of epistasis (Lehner, 2013; de Visser and Krug, 2014). In fact, the only predictive models of epistasis focus on tractable systems where it is possible to connect the effects of mutations to the underlying biophysical and molecular mechanisms of the molecular machinery (Dean and Thornton, 2007; Lehner, 2011); namely, RNA sequence-to-shape models (Schuster, 2006), and models of metabolic networks (Szathmáry, 1993). Even though these studies have provided accurate predictions of interactions between mutations, applying their findings to address broader evolutionary questions remains challenging. For RNA sequence-to-shape models, the function of a novel phenotype (new folding structure) is impossible to determine without experiments. In addition, this approach cannot account for the dependence of epistatic interactions on even simple variations in cellular environments, which are known to affect epistasis (Flynn et al., 2013; Caudle et al., 2014). On the other hand, metabolic network models are limited to examining the effects of large effect mutations, like deletions and knockouts, and lack an explicit reference to genotype.

In order to overcome the limitations of existing theoretical approaches to predicting epistasis, we focused on bacterial regulation of gene expression as one of the simplest model systems in which the molecular biology and biophysics of the interacting components are well understood. We analyze the effects of mutations in a prokaryotic cis-regulatory element (CRE) – the region upstream of a gene containing DNA-binding sites for RNA polymerase (RNAP) and transcription factors (TFs). As such, we study a molecular system where an interaction between multiple components, rather than a single protein, determines the phenotype. Promoters that are regulated by competitive exclusion of RNAP by a repressor are particularly good candidates for developing a systematic approach to understanding epistasis as, in contrast to coding regions as well as more complex CREs and activatable promoters (Garcia et al., 2012), the phenotypic effects of mutations in binding sites of RNAP and repressor are tractable due to their short length and the well-understood biophysical properties of protein-DNA interactions (Bintu et al., 2005b; Saiz and Vilar, 2008; Vilar, 2010). Understanding the effects of point mutations in the cis-element on the binding properties of RNAP and TFs allows for the construction of a realistic model of transcription initiation (Bintu et al., 2005a; Kinney et al., 2010), while providing a measurable and relevant phenotype - gene expression level - for the analysis of epistasis.

Here we studied epistasis between point mutations in the canonical lambda bacteriophage CRE (Ptashne, 2011) (Figure 2). We employ a fluorescent reporter protein that is under the control of the strong lambda promoter P_R (Figure 2a), which is fully repressed by an inducible TF, CI (Figure 2b). RNAP and CI have overlapping binding sites in this CRE, and hence compete for binding. We created a library of 141 random double mutants in the CRE, with all their corresponding single mutants (Supplementary file 1). This design allows us to calculate epistasis between the mutations in the cis-regulatory element in two environments: in the absence of CI, when only RNAP determines expression; and in the presence of CI when the two proteins compete for binding.

Figure 2

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Most double mutants change the sign of epistasis between the two environments

Throughout we assume a multiplicative model of epistasis, which defines epistasis as a deviation of the observed double mutant expression level (relative to the wildtype) from the product of the relative single mutant expression levels (Phillips, 2008). It should be noted that there is no a priori expectation for the sign of epistasis, even if most mutations are deleterious: epistasis denotes only deviations from the expected phenotype of the double mutant, and can be either positive or negative (Figure 1). First, we measured expression levels in the absence of CI (Figure 3—figure supplement 1a, Figure 3—figure supplement 2a). We observe that the majority of double mutants are in negative epistasis (Figure 3a) — the observed double mutant expression level is lower than the multiplicative expectation based on single mutant expression levels (Pearson’s χ²_1,112=43.82, p<0.0001). Specifically, we observe negative epistasis in 83% of 113 mutants that display statistically significant epistasis, while 28 double mutants do not display significant epistasis (Figure 3a, Figure 3—source data 1).

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Figure 3—source data 1

Fluorescence measurements of single and double mutants, and the calculated values for epistasis for the random mutant library.

Multiplicative epistasis, both in the absence and in the presence of the repressor CI, for each double mutant from the random mutant library is provided along with the standard deviation for the measurement, the t-test value (5 degrees of freedom), and the FDR-corrected p value. Double mutants that do not exhibit a significant epistatic interaction are marked in green. Wildtype normalized fluorescence measurement of each single and double mutant, from which the epistasis values were calculated, is also provided for both environments.

https://doi.org/10.7554/eLife.25192.006

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Next, we estimated epistasis at high CI concentration, when gene expression depends on the competitive binding between RNAP and CI (Figure 3b, Figure 3—figure supplement 1b, Figure 3—figure supplement 2b, Figure 3—source data 1). In a repressible promoter, the effects of mutations on the binding of the two proteins have opposite effects on gene expression — a reduction in RNAP binding leads to a decrease in gene expression, while a reduction in CI binding leads to higher expression levels. By comparing epistasis between two environments – absence of CI and high CI concentration – we find that the 141 tested random double mutants show a strong dependence on the environment (ANOVA testing for a GxGxE interaction: F_1,280 = 21.77; p<0.0001), in line with previous observations in another bacterial regulatory system (Lagator et al., 2016). Interestingly, 58% of double mutants display a change in the sign of epistasis between the two environments (Figure 4). Especially prevalent is a switch from negative epistasis in the absence of CI, to positive epistasis in its presence (Figure 4). Strikingly, the proportion of double mutants exhibiting reciprocal sign epistasis (when the sign of the effect of each mutation changes in the presence of the other mutation) is greater in the presence (66%) than in the absence (8%) of CI (Supplementary file 2). This difference likely arises from the molecular architecture of a repressible strong promoter. Mutations affect the binding of both DNA-binding proteins, but in the presence of CI the effect on the binding of RNAP is only unmasked when CI does not fully bind, a scenario that is more likely in the presence of two mutations.

Figure 4

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Generic model of a simple CRE

In order to understand these observations, we created a model of gene regulation that relies on statistical thermodynamical assumptions to model the initiation of transcription, originally developed to describe gene regulation by the lambda bacteriophage repressor CI (Ackers et al., 1982). Importantly, our model is generic, as it does not consider the details of any specific transcription factors involved in regulation. Instead, we model competitive binding between two generic transcription factors that share a single binding site (Figure 5a). The binding of one of these TFs leads to an increase in the gene expression level, in a manner similar to the function of a typical RNAP or an activator. The other is a repressor molecule, the binding of which has a negative effect on gene expression level, achieved by blocking access of the activator to its cognate binding site. In order to draw a parallel to our experimental system, we refer to these two TFs in the generic model as ‘RNAP’ and ‘repressor’, without actually relying on any specific properties of the two molecules, such as CI dimerization, or cooperative binding of CI dimers to multiple operator sites.

Figure 5

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In the thermodynamic model of transcription, each DNA-binding protein is assigned a binding energy (E_i) to an arbitrary stretch of DNA. In our formulation, we assume that each position along the single DNA-binding site under consideration contributes additively to the global free binding energy – an assumption found to be accurate at least for a few mutations away from a reference sequence (Vilar, 2010). These energy contributions can be determined experimentally (Kinney et al., 2010) and are typically represented in the form of an energy matrix. Given a set of DNA-binding proteins (specifically, their energy matrices) and a promoter sequence, a Boltzmann weight can be assigned to any configuration of these proteins on the promoter. The Boltzmann weight is proportional to the probability of finding the system in each of the possible configurations. By assigning a Boltzmann weight to all configurations, one can calculate the probability of finding the system in a particular state (a set of configurations sharing a common property). Specifically, one can calculate the probability of finding the system in a configuration that leads to the initiation of transcription (Figure 5a).

In our generic model, we consider only a single binding site to which ‘repressor’ and ‘RNAP’ compete for binding. Note that the model does not make any assumptions about the identity of the TFs that are binding DNA and hence does not utilize any specific energy matrix. The model is, therefore, general in nature, relying only on the physical and mechanistic properties of protein-DNA binding. In such a system, three basic configurations are possible: no proteins bound to DNA, only ‘RNAP’ bound, or only ‘repressor’ bound (Figure 5a). Each of these states is assigned a Boltzmann weight (Z) based on its free binding energy E_i: 1; ${\displaystyle \left[P\right]e^{-\beta E_P}}$; and ${\displaystyle [R]e^{-\beta E_R}}$, respectively, where ${\displaystyle \beta }$ is 1/k_BT; subscript ${\displaystyle P}$ refers to ‘RNAP’, subscript R to the ‘repressor’; [P] and [R] to the exponential of the chemical potential for the ‘RNAP’ and the ‘repressor’ which for simplicity we equate to the concentrations of the two molecules; and ${\displaystyle E_i}$ corresponds to the change in Gibbs free energy of the reaction of the binding between protein and DNA. Assuming that the system is in thermodynamic equilibrium, we can calculate the probability of finding the system in a configuration leading to transcription (p_ON) – when RNAP is bound:

${\displaystyle p_{ON}=\frac{\left[P\right]e^{-\beta E_P}}{1+\left[P\right]e^{-\beta E_P}+\left[R\right]e^{-\beta E_R}}}$

The phenotype of a mutant is obtained by calculating ${\displaystyle p_{ON}}$ for a free energy ${\displaystyle E_i^{\prime }=E_i+\mathrm{\Delta }}$, where ${\displaystyle \mathrm{\Delta }}$ represents the effect of the mutation on the binding of the protein to the sequence. The energies of single mutants and double mutants are ${\displaystyle E_P^{m_1}=E_P+p_1}$ and ${\displaystyle E_R^{m_1}=E_R+p_1}$; and ${\displaystyle E_P^{m_2}=E_P+p_2}$ and ${\displaystyle E_R^{m_2}=E_R+p_2}$; and ${\displaystyle E_P^{m_{12}}=E_P+p_1+p_2}$ and ${\displaystyle E_R^{m_{12}}=E_R+p_1+p_2}$, respectively, where p_i stands for the effect of mutation ${\displaystyle i}$ on the binding of ‘RNAP’ and ${\displaystyle r_i}$ for the effect on ‘repressor’ binding. From these measures of the mutational effects, we calculated epistasis against a multiplicative model, in the same manner as done for the experimental measurements:

${\displaystyle p_{ON}^{m_{12}}=ϵp_{ON}^{WT}\frac{p_{ON}^{m_1}}{p_{ON}^{WT}}\frac{p_{ON}^{m_2}}{p_{ON}^{WT}}}$

With the generic model, we ask only about the sign of epistasis and say that it is positive when ε >1 and negative when ε <1. The generic model cannot predict the magnitude of epistasis in any particular biological system without accounting for the underlying energy matrices and intracellular concentrations of relevant TFs. As the model does not account for the details of any specific regulatory system, it considers only the direct, primary effects of a mutation on binding affinity (Bintu et al., 2005a), and does not consider any potential interactions arising from secondary effects, namely the effects of a mutation on the structure of DNA (Rajkumar et al., 2013), accessibility to the binding sites (Levo and Segal, 2014), protein cooperativity (Todeschini et al., 2014), looping (Levine et al., 2014), or any other potential regulatory structures.

Independent validation of the generic model predictions

The experimental data from the random mutant library (Figures 3 and 4) shows that the patterns of epistasis between two environments follow the generic model predictions, specifically that epistasis switches sign between environments in many mutants. However, our experimental design, where we only measure gene expression levels, does not allow us to identify the effects of a mutation on CI binding alone. For example, if a mutation decreases gene expression level in the presence of CI, we cannot know if it decreases RNAP binding, increases CI binding, or both. This prevents a more thorough verification of the generic model. In order to independently experimentally validate the generic model predictions (Table 1), it is necessary to know the effects of CRE mutations on RNAP and CI. To obtain this information, we used the experimentally determined energy matrices for RNAP (Kinney et al., 2010) and CI (Sarai and Takeda, 1989), and utilized it to create five random double mutants for each possible combination of single mutation effects shown in Table 1. Due to the high specificity of binding of both RNAP and CI, we could not identify point mutations that simultaneously improved the binding of both (Supplementary file 3). Therefore, we validate the model by measuring epistasis in 30 double mutants (five for each of the six possible combinations of single mutant effects) in the two environments. We find no difference between the predicted and experimental estimates of the sign of epistasis and its dependence on the two experimental environments (Pearson’s χ²_2,30=0.68; p=0.72) (Figure 6). As such, the predictions about the sign of epistasis that arise from the generic model (Table 1) hold true in our experimental system.

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Figure 6—source data 1

Fluorescence measurements of single and double mutants, and the calculated values for epistasis for the validation mutant library.

Multiplicative epistasis, both in the absence and in the presence of the repressor CI, for each double mutant from the 30-mutant validation library, is provided along with the standard deviation for the measurement, the t-test value (5 degrees of freedom), and the FDR-corrected p value. Double mutants that do not exhibit a significant epistatic interaction are marked in green. Wildtype normalized fluorescence measurement of each single and double mutant, from which the epistasis values were calculated, is also provided for both environments.

https://doi.org/10.7554/eLife.25192.014

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Furthermore, we tested if a simple thermodynamic model that incorporates the two energy matrices (Sarai and Takeda, 1989; Kinney et al., 2010) can predict not only the sign, but also the magnitude of epistasis in the two environments. Because such a model depends on the concentrations of RNAP and CI, we estimated the values for these parameters so as to maximize the correlation between model predictions and empirical values of epistasis. When we excluded those double mutants which did not empirically exhibit significant epistasis, we found a significant fit between experimental measurements and model predictions of the magnitude of epistasis in the absence (F_1,15 = 9.86; p<0.01) and in the presence of CI (F_1,15 = 4.59; p<0.05) (Figure 6—figure supplement 1). As such, the model predicts not only the general patterns of epistasis (sign), but is also reasonably accurate at predicting its magnitude, which is remarkable since the model does not consider detailed molecular aspects of the experimental system, such as CI dimerization or cooperativity.

The theory we present here, which is based on mechanistic properties of protein-DNA binding without accounting for any details of the molecular system studied, provides an accurate prediction of the sign of epistasis and its environmental dependence for a repressible promoter system - the most common form of gene regulation in E. coli (~40% of all regulated genes [Salgado et al., 2013]). Furthermore, the fact that we use a generic model with no reference to any particular empirical measures means that our results are derived from first principles. As such, the presented results should hold as long as the effects of mutations on gene expression are mainly driven by their direct impact on TF-DNA binding, as represented by the energy matrix for a given TF. Under such conditions, the thermodynamic model, rather than the multiplicative (or additive) expectation, provides a meaningful null model for the sign of epistasis in CREs.

The sign of the deviations from a multiplicative expectation can have important evolutionary consequences, such as for the evolution of sex (Otto and Lenormand, 2002) or the maintenance of genetic variation (Charlesworth et al., 1995). A particularly important pattern of epistasis is sign epistasis, where the sign of the effect of a particular substitution depends on the genetic background. Sign epistasis can lead to the existence of multiple optima (local peaks). In the system we analyze here, sign epistasis cannot exist in the absence of a repressor, since there is an optimum binding site sequence and the effects of mutations have a definite sign toward this optimal sequence. In the presence of a repressor, however, sign epistasis is possible (Poelwijk et al., 2011). Furthermore, we show that the sign of epistasis very often reverses between environments. This phenomenon, previously observed in a different system (de Vos et al., 2013; Lagator et al., 2016), could alleviate constraints coming from the existence of multiple peaks in a particular environment. The thermodynamic model provides a mechanistic basis for this observation: RNAP and repressor have opposite effects on gene expression and this, when combined with the specific shape of response induced by the thermodynamic model, can lead to the environmental dependence of the sign of epistasis.

Our results concern the combined effect of mutations (epistasis) on phenotype, as opposed to fitness. Phenotypes logically precede fitness and even though it could be argued that fitness is ‘what matters’ for evolution, since mutations spread in part based on their fitness effects, determining the fitness effects of mutations depends on the environment which may or may not be representative of ‘natural’ conditions. Moreover, knowledge about one environment is hardly informative about the fitness patterns in a novel environment. Our results allow for the prediction of patterns of phenotypic epistasis across different environmental conditions, independent of the selection pressures applied to this phenotype. The evolutionary consequences of these patterns of epistasis can then be inferred from the knowledge (or assumptions) of how selection is acting on this phenotype, or in other words, how the phenotype maps onto fitness.

In order to predict the sign of epistasis in a regulatory system, the thermodynamic model accounts for the underlying physical mechanisms that impose constraints on the genotype-phenotype map under consideration. Incorporating details of physical and molecular mechanisms into models of more complex regulatory elements, as well as coding sequences (Dean and Thornton, 2007; Li et al., 2016), can elucidate how epistasis impacts genotype-phenotype maps and their dynamic properties across environments, helping us to understand the environmental dependence of fitness landscapes.

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Author details

1. Mato Lagator

   Institute of Science and Technology Austria, Klosterneuburg, Austria

   Contribution

   ML, Conceptualization, Data curation, Formal analysis, Investigation, Visualization, Methodology, Writing—original draft

   Contributed equally with

   Tiago Paixão

   Competing interests

   The authors declare that no competing interests exist.

   "This ORCID iD identifies the author of this article:" 0000-0001-7847-3594

2. Tiago Paixão

   Institute of Science and Technology Austria, Klosterneuburg, Austria

   Contribution

   TP, Conceptualization, Data curation, Software, Formal analysis, Funding acquisition, Investigation, Visualization, Methodology, Writing—original draft

   Contributed equally with

   Mato Lagator

   Competing interests

   The authors declare that no competing interests exist.

   "This ORCID iD identifies the author of this article:" 0000-0003-2361-3953

3. Nicholas H Barton

   Institute of Science and Technology Austria, Klosterneuburg, Austria

   Contribution

   NHB, Conceptualization, Supervision, Funding acquisition, Project administration, Writing—review and editing

   Competing interests

   The authors declare that no competing interests exist.

4. Jonathan P Bollback

   1. Institute of Science and Technology Austria, Klosterneuburg, Austria
   2. Department of Integrative Biology, University of Liverpool, Liverpool, United Kingdom

   Contribution

   JPB, Conceptualization, Supervision, Funding acquisition, Project administration, Writing—review and editing

   Competing interests

   The authors declare that no competing interests exist.

5. Călin C Guet

   Institute of Science and Technology Austria, Klosterneuburg, Austria

   Contribution

   CCG, Conceptualization, Supervision, Funding acquisition, Methodology, Project administration, Writing—review and editing

   For correspondence

   calin@ist.ac.at

   Competing interests

   The authors declare that no competing interests exist.

   "This ORCID iD identifies the author of this article:" 0000-0001-6220-2052

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