The population genetics of cooperative gene regulation
Changes in gene regulatory networks drive the evolution of phenotypic diversity both within and between species. Rewiring of transcriptional networks is achieved either by changes to transcription factor binding sites or by changes to the physical interactions among transcription factor proteins. It has been suggested that the evolution of cooperative binding among factors can facilitate the adaptive rewiring of a regulatory network.
We use a population-genetic model to explore when cooperative binding of transcription factors is favored by evolution, and what effects cooperativity then has on the adaptive re-writing of regulatory networks. We consider a pair of transcription factors that regulate multiple targets and overlap in the sets of target genes they regulate. We show that, under stabilising selection, cooperative binding between the transcription factors is favoured provided the amount of overlap between their target genes exceeds a threshold. The value of this threshold depends on several population-genetic factors: strength of selection on binding sites, cost of pleiotropy associated with protein-protein interactions, rates of mutation and population size. Once it is established, we find that cooperative binding of transcription factors significantly accelerates the adaptive rewiring of transcriptional networks under positive selection. We compare our qualitative predictions to systematic data on Saccharomyces cerevisiae transcription factors, their binding sites, and their protein-protein interactions.
Our study reveals a rich set of evolutionary dynamics driven by a tradeoff between the beneficial effects of cooperative binding at targets shared by a pair of factors, and the detrimental effects of cooperative binding for non-shared targets. We find that cooperative regulation will evolve when transcription factors share a sufficient proportion of their target genes. These findings help to explain empirical pattens in datasets of transcription factors in Saccharomyces cerevisiae and, they suggest that changes to physical interactions between transcription factors can play a critical role in the evolution of gene regulatory networks.
KeywordsTranscription Factor Binding Site Deleterious Mutation Fitness Landscape Selective Coefficient Cooperative Regulation
It is often difficult for a population to acquire an adaptive phenotype that requires simultaneous changes in the co-expression of multiple genes [1, 2, 3, 4, 5, 6, 7, 8, 9, 10]. If selection favours a change in the way a group of genes are regulated, then each of the target genes must independently gain novel binding sites and/or lose existing ones [8, 9, 11, 12]. This has led to the proposal that adaptive rewiring of a regulatory network can be accelerated if pairs of transcription factors bind their targets cooperatively, through a physical interaction between the transcription factor proteins themselves [8, 9, 11].
A number of previous studies have explored the mechanistic details of cooperative transcription factor binding at a given target gene [13, 14, 15]. Such biophysical studies focus on transcription factor binding at a single target gene and are able, with remarkable accuracy, to account for a number of the physical properties of binding sites [14, 16, 17]. However, the evolution of cooperative binding occurs through mutations at transcription factor proteins, and such mutations can alter transcription factor binding at every binding site across the genome. To understand the fitness effects of such a mutation therefore requires that we understand the evolution of the whole ensemble of binding sites for a transcription factor. The population genetics of such an ensemble cannot be understood in a simple way just by focusing on the details of a single member of the ensemble. They depend critically on the population-genetic parameters of the ensemble, such as number of target genes, overall mutation rates and selection coefficients, and population size. Therefore in this paper, we do not focus on the details of a cooperative binding at a single target gene. Instead our analysis is in terms of these population-genetic parameters, and whilst we estimate selective coefficients from biophysical studies, we do not specify the mechanistic details of protein-DNA interactions that give rise to them.
We use a mathematical model to study the conditions under which cooperative binding between pairs of transcription factors is favoured. We first determine the evolutionary conditions that favour cooperative binding under stabilising selection, in terms of the basic evolutionary parameters of the population: the strength of selection on binding sites, the rate of mutation, and the population size. We then study the influence of cooperative regulation on the capacity for a transcriptional circuit to adapt under positive selection. We calculate the time required for a target gene to gain a new, adaptive transcription factor binding site, in the presence or absence of cooperative interactions among its regulators. We confirm our analytical results on the evolution of cooperative regulation by comparison to Monte-Carlo simulations of the Wright-Fisher process associated with our system, and we compare our qualitative conclusions to systematic empirical data.
Our population-genetic model describes a pair of transcription factors, each with its own set of target genes, with some degree of overlap between these sets (Figure 1). According to our model, which is specified in detail below, a target gene that is regulated by both factors has two corresponding binding sites, while a target gene that is regulated by only one of the factors has a single binding site. We assume that mutations that result in loss of function can occur at any binding site, and that non-functional binding sites can also undergo gain of function mutations. When there is no cooperative regulation between the two transcription factors, binding to each of their targets is determined solely by their binding sites. If a binding site is not functional, this results in reduced fitness. When cooperative binding is present, two conflicting effects occur: On the one hand, cooperative binding partially compensates for the deleterious effects of loss of function mutations to the binding sites at shared targets. On the other hand, cooperative binding results in some degree of mis-regulation at each of the targets that are not shared, and this has a deleterious impact on fitness. By constructing our model in terms of these fitness benefits and costs we are able to study the evolutionary dynamics of the system, and determine the effects of varying different population-genetic parameters on the evolution of cooperative gene regulation. This approach therefore complements the detailed mechanistic models of gene regulation that have been studied elsewhere [13, 14, 15].
Results and discussion
Stabilising selection without cooperative binding
We consider a pair of transcription factors, labelled 1 and 2, that have K1 and K2 targets, respectively. A fraction β of the binding sites are at shared target genes, so that the number of binding sites at genes that are co-regulated by the pair is β(K1 + K2), as illustrated in Figure 1. Loss of function mutations occur at binding sites at a rate u l , and back mutations, which result in a functional binding site being gained at a target, occur at rate u g . An individual incurs a fitness penalty s, where 0≤s < 1, for each non-functional binding site, and fitness is assumed to be multiplicative across loci. Therefore the fitness of an individual that lacks i≤K1 + K2 of its required binding sites is w i =(1−s) i . The fitness landscape associated with our model thus has a single peak at i=0; and for each transcription factor binding site that is lost, fitness is reduced by an additional factor (1−s). Empirical estimates of the strength of selection on transcription factor binding sites suggest that typically Ns∼10 , suggesting that s is small. We assume that s is the same for all binding sites, an assumption which we relax in the Methods section.
where , and P ij is the probability a genotype lacking j functional binding sites mutates to a genotype lacking i functional binding sites (see Methods). Previous work [19, 20, 21] has shown that at equilibrium, when rates of forward and back mutations are identical (u l =u g ), the solution to Equation 1 is a binomial distribution. In the more general case of a finite population, with u l ≠u g , we find that the equilibrium continues to be well approximated by a binomial distribution, with mean (K1 + K2)a s . The term a s is the probability that a binding site will be non-functional in a randomly chosen individual at equilibrium. The probability a s depends on the strength of selection against non-functional binding sites, s, population size, N, and the rates of forward and back mutation, u l and u g (see Methods and [20, 21]).
(see Methods). These equations have an intuitive interpretation: When 2Ns > 1 the first term describes the effect of genetic drift which tends to push the system towards its neutral equilibrium, a0=u l /(u l + u g ), and the second term describes the effect of selection. In the limit N→∞, a s equals u l /s, which is the standard result for the frequency of a deleterious allele in an infinite population under mutation-selection balance. When 2Ns < 1, evolution is nearly neutral and drift dominates, so the system is close to the neutral equilibrium a0.
Stabilising selection with cooperative binding
Here we modify our model to account for cooperative regulation by a pair of factors. This allows us to ask when cooperative regulation is favored by evolution. A mutation that results in cooperative binding between a pair of transcription factors has two effects on the fitness of a transcriptional circuit. For a target that is regulated by both transcription factors, we assume that cooperative binding mitigates the effects of deleterious mutations at transcription factor binding sites [7, 8, 9]. This results in a reduced fitness penalty for a mutation at the β(K1 + K2) shared targets, so that (1−s) is replaced by (1−hs) for some constant 0≤h≤1. Nonetheless, there are also (1−β)(K1 + K2) targets that are regulated by only one or the other of the transcription factors. We assume that the cooperative binding of the transcription factors causes pleiotropic mis-regulation at these targets (since the other transcription factor, which does not have a binding site at such sites, now binds to the first transcription factor through a physical interaction). This results in a fitness penalty t at each of the (1−β)(K1 + K2) targets that are not co-regulated. Fitness is again assumed to be multiplicative, so that the cost of pleiotropy associated with cooperative binding is .
Provided u l ,u g ≪ 1, genes that are co-regulated and genes that are not co-regulated have equilibrium distributions described by independent binomial distributions with means a hs and a s respectively, which are approximated by Equation 2 (substituting hs for s appropriately, see Methods). We can now specify the conditions for the invasion of cooperative gene regulation. A mutation resulting in cooperative binding between a pair of factors will be favoured if the expected fitness of the mutant is greater than the equilibrium mean fitness. Using the expressions for mean fitness given above, this occurs when . Assuming t,s ≪ 1, this expression can be simplified to give . This means that, when the fraction of binding sites at shared targets, β, is greater than a threshold depending on s, h, t and a s , a mutation that results in cooperative binding can invade a population at equilibrium.
Similarly, a mutation that results in the loss of cooperative binding in a population where it is present will be favoured when . Again assuming t,s ≪ 1, this expression can be simplified to give so that, when the fraction of binding sites at shared targets, β, is less than a threshold depending on s, h, t and a hs , a mutation that results in loss of cooperative binding can invade a population at equilibrium.
Since the first expression in Equation 2 is monotonically decreasing in s, and the second expression is independent of s, it is always true that a hs ≤a s , i.e populations that have cooperative binding accumulate more deleterious mutations, that result in weaker transcription factor binding sites, than populations that lack it. As a result there is a range of β for which both a population that lacks cooperative binding, and a population that has cooperative binding are not invadable by mutations that gain or remove cooperative binding respectively. In this range, the evolutionary dynamics of the system are bi-stable. In this range, we expect to find some genes that are regulated by pairs of transcription factors that act cooperatively and some that don’t.
Similarly, from Equation 4 for a population with cooperative binding, we see that when N and/or hs is large, so that 2Nhs > 1, the threshold number of shared targets βbelow which cooperative binding becomes disadvantageous is independent of the strength of selection s (Figure 2a). As before, the threshold decreases as the mutation buffering effect of cooperative binding increases (i.e. as h decreases, Figure 2b) and the threshold increases with population size N (Figure 2c), and the cost of pleiotropy t (Figure 2d). In contrast, when N and/or hs is small, so that 2Nhs < 1, drift dominates and the threshold number of shared targets βis independent of population size N (Figure 2c), but decreases with the strength of selection s (Figure 2a). The size of the bistable region is largest when s is large and h is small, and for intermediate values of N and t, as shown in Figure 2. As this analysis demonstrates, there is a broad range of possible evolutionary outcomes and, crucially, cooperative binding can evolve under a wide range of circumstances despite the deleterious pleiotropic effects associated with physical interactions among transcription factors.
Adaptation of transcriptional circuits under positive selection
We study adaptive change that involves replacement of an existing transcription factor by a new one that confers higher fitness. We assume that the target gene must first suffer an initially deleterious mutation at its existing binding site before a newly adaptive binding site can be acquired (Figure 3) [8, 9, 11]. The newly adaptive binding site is produced from binding sites that have already mutated at a rate u r . The expected waiting time for such a gene to produce a newly adaptive binding site therefore depends on the number of binding sites in the population that harbor a deleterious mutation, which is proportional to a s when cooperativity is absent and a hs when it is present. Since a hs > a s , this number is greater when cooperative binding is present than when it is absent.
Cooperative binding and the fraction of shared targets in yeast
Cooperative binding in the yeast sex determination network
The ability of cooperative transcription factors to facilitate adaptation also has empirical support, from observations in the sex determination networks of different yeast species [7, 8, 9]. The acquisition of a protein-protein interaction between the mating factor MATα2 and Mcm1 was able to buffer the deleterious effects of mutations that strengthened Mcm1 binding sites . Prior to the emergence of a protein-protein interaction, sex determining genes were activated only in the presence of Mcm1 and MATa2 together . The buffering effects of the protein-protein interaction allowed Mcm1 binding sites to acquire strengthening mutations such that sex determining genes became activated by Mcm1 alone. As a result, MATa2 became redundant and was lost . The result was a significant upstream reorganization of the yeast sex determination network without the need for any parallel changes to the downstream output of the network. Similar patterns, in which acquisition of cooperative binding between transcription factors is followed by changes to the regulation of their shared targets, are observed across the yeast transcriptome , and support the prediction of our analysis of positive selection on transcriptional networks.
We have shown that cooperative binding between a pair of transcription factors is favoured under stabilising selection, provided the overlap between their targets is sufficiently large. The threshold fraction of shared targets depends upon the strength of selection on binding sites, the cost of pleiotropy associated with protein-protein interactions, and the rates of mutations. It also depends on the population size. Just as in models that consider the evolution of redundancy [20, 29], we find that greater redundancy (i.e. cooperative regulation) is more strongly favoured in smaller populations; and that for intermediate population sizes the evolutionary dynamics are bistable, such that cooperative binding is maintained if it is already present, but cannot evolve if it is absent. Finally, we found that cooperative binding facilitates the rewiring of transcriptional circuits under positive selection.
This study shows that, even when the deleterious effects of pleiotropy are taken into account, mutations that change transcription factor function can play an important role in the evolution of gene expression. Taking account of mutations both to regulatory binding sites and to the transcription factors themselves reveals a rich set of evolutionary dynamics that helps explain how complex transcriptional networks can rapidly rewire large sets of genes in order to adapt to new environments.
To find the equilibrium relative abundances of the hamming classes x i that give the solution to Equation 1, we follow [20, 21] and look for a solution of the form . Given this assumed form, the mean fitness of the population at equilibrium is . Since ∑ i x i =1 it is easy to show that . Taking p=a(1−s)/(1−a), this gives a mean fitness of , which is the form given in the main text.
and at equilibrium . Because we are assuming that w ij is just the product of the two independent fitness landscapes associated with the different selective coefficients, i.e w ij =(1−hs) i (1−s) j using our assumed form of x ij results in values of a s and a hs as given by Equation 5 for the independent distributions with the appropriate selective coefficients.
Assuming u l ,u g ≪ s, we obtain the Taylor expansion of a s to first order, in terms of 1/(2Ns) (which is relevant when 2Ns > 1) and in terms of 2Ns (relevant when 2Ns < 1) to obtain Equation 2.
According to this inequality, cooperative binding is advantageous only when the fraction of targets shared by the pair of transcription factors is greater than a threshold. Since by definition β≤1, Equation 7 says that cooperative binding can only increase population mean fitness when 2Nhs < 1 and 2Ns > 1, i.e if the benefit of cooperativity, h, is sufficient to make mutations at transcription factor binding sites that are deleterious in the absence of cooperativity nearly neutral when it is present.
Therefore the ratio of expected waiting times with and without cooperative binding is independent of the number of genes to be rewired, and depends only on the ratio a hs /a s .
Variation in selection strength across sites
Up to this point we have assumed that the selective coefficients, s and h, are constant across binding sites. However it is obviously possible that these parameters may vary between binding sites. Such a generalization of our model represents a significant complication, and a full treatment is outside the scope of this paper. However, it can be analyzed in the simple case that the coefficients associated with each binding site i satisfy s i ≪ 1, such that the fitness landscape is approximately additive.
We assume that there are a finite set of selective coefficients, s α and h γ , where the super-scripts αand γindex the different sets, and that the number of binding sites with a given coefficient s α or h γ are distributed according to some function F(s) and G(h). We also assume that the coefficients s and h are distributed independently of one another. Each binding site i has a value s i associated with it, drawn according to F(s). In the quasi-species regime the probability that binding site i has a mutation is simply given by , as given by Equation 5. In this case the distribution of hamming classes, rather than being binomial, is poisson binomial, paramaterized by . Similarly, when cooperative binding is present, the distribution is poisson binomial with the modification that shared targets have a mutation with probability where h i is drawn independently from the distribution G(h). The system is easiest to analyse if we separate binding sites into sub-classes, α, of binding sites with the same selective coefficients, where the size of each sub-class, n α is given by n α =F(s α )(K1 + K2). The number of mutations in each subclass is then given by a binomial distribution.
When the fitness landscape is close to additive, the method of  can be applied independently to each sub-class to determine the expected number of mutations in the sub-class. This is only true in an additive fitness landscape, or in a multiplicative fitness landscape in which cross terms between subclasses are sufficiently small that they can be neglected. When this condition holds, the value of associated with each sub-class is given by Equation 6.
From these the invasion probabilities and threshold values of βcan be calculated in the same way as in the case of constant s and h above. The only difference in the case with variable selective coefficients is that the invisibility criteria for a mutation resulting in gain or loss of cooperative biding dependants on the average of a s s across the distribution F(s), and on the average a hs hsacross the joint distribution F(s)G(h). Investigating different forms of the functions F(s) and G(h) represents an interesting avenue for further work.
JBP and AJS acknowledge funding from the Burroughs Wellcome Fund, the David and Lucile Packard Foundation, the James S. McDonnell Foundation, the Alfred P. Sloan Foundation, grant #D12AP00025 from the U.S. Department of the Interior and Defense Advanced Research Projects Agency, and grant RFP-12-16 from the Foundational Questions in Evolutionary Biology Fund. AP acknowledges grants from the Natural Environment Research Council (NE/G00563X/1) and the Engineering and Physical Sciences Research Council (EP/F500351/1, EP/I017909/1). AJS also acknowledges an EPSRC PhD Plus fellowship.
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