- Open Access
Explicit equilibrium modeling of transcription-factor binding and gene regulation
© Granek and Clarke; licensee BioMed Central Ltd. 2005
Received: 3 May 2005
Accepted: 30 August 2005
Published: 30 September 2005
We have developed a computational model that predicts the probability of transcription factor binding to any site in the genome. GOMER (generalizable occupancy model of expression regulation) calculates binding probabilities on the basis of position weight matrices, and incorporates the effects of cooperativity and competition by explicit calculation of coupled binding equilibria. GOMER can be used to test hypotheses regarding gene regulation that build upon this physically principled prediction of protein-DNA binding.
Transcription is regulated by the binding of proteins to specific DNA sequences. Until recently, binding and regulation could only be studied at the level of individual genes, but they can now be studied as a complex system due to the availability of genome-wide data on expression and transcription factor binding. Computational models are needed, however, to evaluate co-regulated genes and the sequence motifs associated with them.
A general strategy for testing the relevance of a DNA binding motif to gene regulation is to quantify the association of the motif with co-regulated genes. This can be done by comparing the regulatory sequences of co-regulated genes with the regulatory sequences of all other genes [1–4]. One simple test is to score for the occurrence of a consensus site within a prescribed distance 5' to the start of transcription. If the fraction of regulated genes with a consensus site is significantly larger than the fraction of unregulated genes, as it often is, then the test has some predictive power [1, 5–7]. As with all statistical tests, there is a model implicit in this test: in this case, the implicit model is that gene regulation is mediated by a single consensus binding site.
There are problems with such a simple model. First, the use of consensus binding sites, even if degenerate, underestimates the importance of motifs that resemble the consensus but do not match it . At the same time, degenerate consensus sites fail to distinguish among motifs that match the consensus even if the motifs that match differ in affinity. Second, regulated genes often contain more than one binding site for a given factor, so scoring based on a single site (or any other threshold number of sites) is arbitrary. Third, the binding of a factor is typically affected by cooperative and competitive interactions with other proteins, so binding sites for those other proteins may need to be considered. Fourth, gene expression can be affected by the location, orientation and spacing of bound transcription factors. Therefore, to be realistic, a model for gene regulation should use to full advantage an accurate representation of binding specificity, integrate over multiple binding sites of different strength, account for cooperative and competitive interactions, and be flexible enough to model the variable effects that binding can have on gene expression.
We previously described an algorithm for predicting the probability that a transcription factor binds within a promoter region . The algorithm predicts the relative affinity of binding sites using a position weight matrix (PWM) in which the elements of the PWM represent contributions to the free energy of binding for all possible bases at each position in a binding site . The algorithm then integrates over the affinities of all possible binding sites within a region of interest by calculating the probability that at least one site is bound at a given assumed protein concentration. Using a PWM defined by extensive binding equilibrium measurements of yeast Leu3p, we showed that this method was able to predict the set of known target genes for Leu3p better than could be achieved by simple enumeration of discrete binding sites .
Building on those results, we report here a very general physically principled model for transcription factor localization based on protein-DNA and protein-protein binding equilibria. The model, which we have named GOMER (generalizable occupancy modeling of expression regulation), uses PWMs to predict explicitly the relative affinity of binding sites, taking into account the effect of cooperative and competitive interactions. Based on the binding predictions, GOMER predicts gene regulation by weighting binding sites according to their location and orientation. The weights are calculated from functions specified or defined by the user. These functions and their parameters allow the user to test alternative hypotheses concerning the control of co-regulated genes.
Here we describe GOMER and give examples of its application. We use the program to analyze the effect of cooperativity between forkhead proteins and the transcription factor Mcm1p in controlling the expression of a set of cell-cycle regulated genes in yeast [7, 10]. Although in vitro experiments show that direct interactions between these factors occur over very short distances , we find evidence that cooperative interactions can extend over a distance of 100 base pairs (bp) or more. We also use the model to investigate the role of competition between two transcription factors, Ndt80p and Sum1p, in distinguishing between mitotic and meiotic programs of gene expression . Competition between these proteins better explains a set of genes that is regulated by both transcription factors than does simple non-competitive binding. Finally, we evaluate the correlation between predicted and observed binding of Rap1p in a chromatin immunoprecipitation microarray (ChIP-array) experiment . We show that the correlation between predicted and observed binding can be dramatically improved by a model that accounts for hybridization to a spot on the array (an array feature) that is due to binding to sites outside the sequence of the array feature itself. The GOMER program is freely available.
Realistic modeling of promoter regions using binding site weight functions
In GOMER, regulatory regions are defined by user specified functions that assign a weight to each binding site based on its location. For example, it is common practice to assume that yeast regulatory regions consist of the 600 bp 5' to the start of translation [1, 5, 16]. To model this regulatory region in GOMER we used a function that simply assigns a weight of 1 to all sites that lie within the region and 0 to all sites outside. The region itself is defined by parameters to the function that specify the endpoints of the region with respect to the 5' end of an open reading frame (ORF). Figures 1a and 1c show the effect of varying the parameters for this simple model (the beginning and end points of the regulatory region).
While the conventional 600 bp definition of the regulatory region works well (ROC AUC = 0.75), alternative parameters explain the CLB2 cluster genes somewhat better. The choice of parameters that works best defines a regulatory region extending from 650 bp 5' to the ORF to 150 bp inside the ORF (ROC AUC = 0.78). Exclusion of the 150 bp inside the ORF makes the model perform somewhat less well (ROC AUC = 0.75), which means that sites within the first 150 bp contribute to our ability to distinguish true forkhead regulated genes from other genes that happen to have forkhead binding sites. Thus, there may be weak but biologically relevant binding sites within the coding region of some forkhead-regulated genes.
Real regulatory regions rarely have strict boundaries like the 600 bp definition used by convention in yeast, and sites within these regions can differ substantially in their functional importance. One advantage of the GOMER approach is that it allows users to evaluate more realistic models of gene regulation by defining their own regulatory-region weight functions. Figure 1b illustrates, as an example, a Gaussian weight function, and Figure 1d shows the results of using this function with various parameters. The Gaussian weight function models a regulatory mechanism in which there is an optimal position for a bound protein to affect gene expression. The effect of a bound protein decreases with distance from this optimal position. Unlike the uniform weight function, there is no sudden and substantial drop in weights (though weights below a user-specified threshold are rounded down to zero in the interests of computational efficiency).
Figures 1c and 1d compare the effectiveness of the uniform and Gaussian functions over an equivalent range of parameters. The two functions achieve similar ROC AUC values using their optimal parameters, but the uniform weight function is much more sensitive to the choice of parameter values than is the Gaussian function. This is evident from the irregular contours in Figure 1c, which are a consequence of the hard cutoffs imposed by the uniform weight function. Thus, GOMER's flexible definition of gene regulatory regions allows for regulatory models that are both more realistic and more robust.
Homotypic and heterotypic cooperativity in the regulation of cell-cycle genes by forkhead transcription factors
The forkhead PWM is able to distinguish CLB2 cluster genes reasonably well using either the uniform-weight definition of the regulatory region or the Gaussian-weight definition. However, neither model is exceptionally good: both have ROC AUC values of approximately 0.78. In vitro experiments suggest that Fkh2p binds cooperatively to DNA with itself and with the transcription factor Mcm1p [11, 17]. The lack of cooperative interactions in the models might explain their suboptimal performance.
The model for cooperativity used in this analysis is extremely simple: all sites within a given distance are considered to be equally capable of interacting cooperatively. However, GOMER's 'plug-in' weight functions make it easy to explore more elaborate models for cooperativity (see Materials and methods).
Competitive interactions between Sum1 and Ndt80
Competition among transcription factors is a potentially important mechanism for controlling complex biological responses. We have incorporated a realistic model for competitive interactions into GOMER (see Materials and methods) and have used this model to study the interaction of yeast transcription factors Ndt80p and Sum1p. Ndt80p is an activator of genes expressed during the middle stage of sporulation [5, 19]. Sum1p represses genes during mitotic growth and the early stage of sporulation [12, 20]. A number of the genes induced by Ndt80p during middle sporulation are targets of repression by Sum1p. Ndt80p and Sum1p have overlapping binding specificities, which suggests that competition between these transcription factors may be important for regulation. Competition for binding has been demonstrated in vitro by gel-shift assays and in vivo using reporter constructs .
If competition between Sum1p and Ndt80p were relevant to the regulation of a particular gene, we would expect the regulatory sequence for that gene to be sensitive to the concentrations of the two transcription factors. To test this, we fixed the concentration of Ndt80p in the model and explored the effect of increasing concentrations of competing Sum1p. Importantly, the genes that are regulated by both proteins, and therefore are the best candidates for being affected by competition between the proteins, show the greatest sensitivity to competition by Sum1p (Figure 3). At higher Sum1p concentrations there is substantially less specific binding by Ndt80p to these genes, as reflected in lower ROC AUC values for this set of genes when scored for Ndt80p occupancy. Substantially smaller effects of Sum1p concentration are seen for the genes that are regulated by Ndt80p alone, consistent with the observation that Sum1p does not regulate these genes. A similar conclusion was recently reported independently by Wang et al. . These results suggest that binding site variants that are found in genes regulated by both Ndt80p and Sum1p have been tuned by evolution to be sensitive to the relative concentration of the two proteins. Binding sites in genes regulated by only one of the transcription factors tend to more closely match the specificity of that particular transcription factor and are, therefore, less sensitive to the effects of the competing factor.
Improved correlation between predicted and observed binding in ChIP-array experiments
This difference is largely due to the naïve model we used initially to score sequence features on the array. This model considers only the sequence of the array feature itself (Figure 4a). Because bound DNA is sheared to a size of several hundred base pairs in the ChIP procedure, some of the molecules that are immunoprecipitated due to binding to a site within one array feature overlap the sequence of a neighboring feature, as previously pointed out by Lieb et al. . We can model this effect in GOMER using a weight function that allows sites outside the feature to contribute in proportion to the fraction of immunoprecipitated molecules we expect to hybridize. Doing so dramatically improves our ability to explain the Rap1p ChIP data, especially for ORF features (Figure 4b, right). This suggests that much of the experimental enrichment of ORF features is actually due to binding sites that are in sequences flanking the ORFs (that is, in intergenic regions).
The GOMER scoring function uses PWM scores to estimate relative free energies of binding to potential sites. How well this works depends on how well the PWM represents the contributions of each base to the free energy of binding. These free energy contributions can be directly estimated from thermodynamic binding data, but more commonly they are inferred from the base frequencies in known binding sites. This is possible because there is a connection between the information content in a set of sequences and the thermodynamics and specificity of binding (see Materials and methods for a fuller discussion) . A variation on this idea treats the PWM as a predictor of affinity and then uses protein concentration as a variable to maximize the likelihood of observing the set of known sequences . Regardless of how the PWM is defined, GOMER itself can be used to compare the predictive value of different PWMs by assessing their ability to explain experimental binding or expression data. For example, we have shown that PWMs defined either by direct measurement of binding affinities or by computational motif discovery are equally good at explaining an independent ChIP experiment. 
A key attribute of GOMER is its flexibility. GOMER uses weight functions, specified by the user, to create position-dependent models that define the size and shape of regulatory regions and describe the nature of cooperative and competitive interactions (see Materials and methods). These functions can be as complex as the user desires, although care should be taken not to use more parameters than is justified by the data. The power of this approach for modeling gene regulation will become more valuable as more data become available.
One parameter used by GOMER is the free concentration of transcription factors, which is needed for calculating binding site occupancies based on predicted affinities (see Materials and methods). When a single, non-cooperative factor is analyzed, concentration has only a marginal effect on the ROC curve. This is because only the ranks of the genes are relevant to the curve, not the absolute occupancy of the gene by the transcription factor. (There can be a modest effect of concentration in this case because the occupancy score for a gene with a single high-affinity site changes with concentration somewhat differently than does the occupancy score of a gene with several lower-affinity sites .) Varying the concentration can, however, have a much more substantial effect when cooperative and competitive interactions are included in the model (Figures 2 and 3). Because cooperative and competitive interactions are common in gene regulation, the explicit consideration of concentration is likely to be necessary for a complete understanding of gene regulation.
GOMER is a physically principled method because of the way it uses PWMs to estimate binding affinities but also because its weight functions and parameters can be understood in terms of specific physical and biological models. This distinguishes GOMER from machine learning methods that search for rules describing gene regulation without the assumption of an underlying physical model . GOMER also differs in philosophy from purely empirical algorithms. For example, rules for defining clusters of binding sites have been developed that help distinguish regulated genes from other genes that have a comparable number of binding sites [24–26]. GOMER, on the other hand, can distinguish genes with clustered binding sites from genes whose sites are dispersed by modeling cooperative binding interactions. These cooperative interactions are likely to be the reason why sites are clustered in the first place.
We showed previously that gene scoring functions that are based on enumeration of binding sites are typically poorer predictors of gene regulation than is the simple GOMER occupancy score, which integrates over binding sites of differing predicted affinities . We expect, of course, that any motif searching algorithm that uses PWMs in a related way, and which ranks genes based on the scores for all sites, would perform similarly. To verify this, we ran the motif searching program PATSER using the FKH1 and NDT80 PWM, and obtained scores for the top five sites upstream of every gene . PATSER does not provide an integrated binding site score for each gene, so we ranked genes according to their highest scoring site. In the event of ties, the second highest scoring site was used as a tie breaker, then the third highest scoring site, and so on. For the largest gene set analyzed, the genes regulated by NDT80 only, the ROC AUC values for the simple GOMER function and the PATSER-based ranking algorithm are nearly identical (between 0.70 and 0.71). For two smaller gene sets, the simple GOMER function performed better than the PATSER-based algorithm in one case (0.78 versus 0.72 for the CLB2 cluster genes) and less well in the other (0.80 versus 0.89 for the genes regulated by both NDT80 and SUM1).
The purpose of this paper is to demonstrate that a substantial improvement in these scores can be obtained using GOMER's cooperative and competitive modeling functions. GOMER is unique thus far in its ability to model cooperative and competitive interactions, so we are not able to compare these important features of GOMER to other algorithms. We hope the availability of GOMER and the data sets used in this paper will permit others in the field to test GOMER against new algorithms as these new algorithms are developed.
Computational models of gene regulation are far from perfect because gene regulation is a complex phenomenon. It is because of this complexity, though, that it is important to develop realistic, quantitative models like GOMER. By assessing how well (or poorly) we can predict the effect of mutations or environmental signals, we can better identify deficiencies in our understanding of gene regulation and allow the development of new additions to the model. GOMER can be applied to other organisms besides yeast, and indeed we have begun using it to study developmentally important transcription factors in Caenorhabditis elegans. GOMER can also be used to study sequence signals that regulate transcription termination , and it could be adapted to study any regulatory mechanism that involves sequence specific binding, not just transcription. In the future, we anticipate incorporating experimental data on the distribution of nucleosomes and nucleosome modifications, and we will begin to address differences in the kinetics of binding in addition to differences in affinity.
Materials and methods
Representation of binding specificity and the prediction of binding affinity
The elements of a PWM are base-specific scores for each position of a potential binding site. The values of the PWM can be defined by direct measurement of binding affinities [3, 12], but more often they are estimated from the frequency of occurrence of each base at each position in a set of presumptive binding sites. These sites are determined experimentally, for example by binding site selection  or by computational analysis of co-regulated genes [2, 30, 31], or by a combination of selection and computational analysis . Typically, a PWM element [b,j], is derived from the ratio fb,j/pb where fb,j is the observed frequency of base b at position j, and pb is the prior probability of base b (usually the frequency of b in the genome). The ratio fb,j/pb can be thought of as an equilibrium constant between the protein binding to sites that contain base b at position j and the protein binding to a mixture of sites containing each of the four bases at position j, with the frequency of the bases the same as that found in the genome . It follows that if PWM elements are calculated as RTln(fb,j/pb) (where R is the gas constant and T is the temperature), the value of element [b,j] can be interpreted as the contribution to the relative free energy of binding to a base b at position j in a particular sequence. In practice, GOMER generates the PWM internally from data supplied by the user: a probability matrix (PM) file (which contains the position-specific base frequencies), and the expected base frequencies (calculated from the sequences). A temperature of 300 K is used in calculating the PWM from the PM.
Sequence windows (potential binding sites) are scored by summing the appropriate base-specific values for each position in the window, as defined by the PWM. The score for a site is computed as the sum of position scores, based on the assumption that each base makes an independent contribution to the free energy of binding to the site. This assumption is a good approximation in most cases . The PWM score for a window can be interpreted, therefore, as a relative free energy of binding and from that value an equilibrium binding constant (Kd = e-ΔG/RT) can be calculated. A default temperature of 300 K is used to calculate the equilibrium constant from the PWM; however, the temperature parameter can be varied, changing the relative affinity for favored bases over disfavored.
Probability of protein occupancy for regulatory sequences
Once an equilibrium constant has been calculated for a sequence window, i, the probability of binding to that site, Pi, can be calculated from the standard equation for a simple binding isotherm:
where Kd,X,i is the predicted equilibrium dissociation constant for X binding to window i and [X] is the free concentration of X. Although [X] represents a real physical quantity, it is exceedingly difficult to determine its in vivo value experimentally , so for most purposes [X] is an adjustable parameter. By default, [X] is set equal to the Kd,X for the optimal binding site, resulting in an occupancy score of 0.5 for optimal sites.
The probability of binding is calculated for all sequence windows within a regulatory sequence. GOMER then integrates over all sequence windows by calculating the probability, Pocc, that the protein is bound to at least one site within the regulatory sequence based on the probability of binding to each site, Pi.
The probability of not being bound at site i, 1 - Pi, is
where Ka,X,i, is an equilibrium association constant and is the reciprocal of Kd,X,i.
(We used Kd, the equilibrium dissociation constant, at the beginning of the derivation because its use in the standard binding isotherm equation is familiar to biochemists, but we switch here to Ka, the equilibrium association constant, because the final form of the GOMER scoring function is visually less complicated using this substitution).
Regulatory regions are defined in GOMER by user-specified weight functions
Generally, we want to use GOMER to predict the probability of a gene being regulated rather than just the probability that a transcription factor binds in its vicinity. To determine this functional probability, we need to weight binding sites by their expected relevance to regulation. In GOMER, equilibrium constants are modified by weights calculated from user-specified functions. These functions weight sites based on their location and/or orientation with respect to genome features (for example, the start of transcription). Thus, we define a GOMER score, S, which is similar to Pocc but which incorporates functional weights.
where Ka,eff,X,i = κiKa,X,i and κi is the weight for site i based on the user-specified function.
The cooperative binding of proteins X and Y to DNA can be separated thermodynamically into the formation of an XY dimer and the binding of that dimer to DNA; this is thermodynamically equivalent to protein X binding to its site with higher affinity in the presence of pre-bound Y. This leads to a conceptually simple means for incorporating cooperative interactions into the GOMER model: the probability that a given site i is occupied by X depends not only on the probability that it is occupied by monomeric X but also on the probability that it is occupied by XY. Calculating the probability of occupancy by the XY dimer requires us to take into account all possible pairs of binding sites that consist of a site i to which X binds and a second site, j, to which Y binds. These site pairs need not be contiguous. Extending the expression derived above for monomer binding, the expression for calculating the GOMER score, accounting for cooperative interactions, is:
Ka,eff,XY,i,j is the equilibrium constant for an XY dimer binding to a site that consists of windows i and j; it is analogous to Ka,eff,X,i, the equilibrium constant for monomeric X binding to site i in that it is the product of an intrinsic binding affinity, Ka,XY,i,j, and a weight, κC,i,j, which is assigned by a user-specified weight function κC. GOMER assumes that the intrinsic binding affinity of the dimer, Ka,XY,i,j, is the product of the binding constants of the two proteins, X and Y, for their respective sites, i and j. Thus:
Ka,eff,XY,i,j = κC,i,j·Ka,X,Y,i,j = κC,i,j·(Ka,eff,X,i·Ka,Y,j)
where the affinity of protein Y for its site j (Ka,Y,j) is calculated from a PM in the same way as we have described for protein X. There is no need to apply a functional weight to Y binding because the only role for Y in the model is modification of X binding, rather than a direct role in modulating expression.
The weight function, κC, will typically define weights depending on the spacing and orientation of site j with respect to site i. For example, if two sites must be adjacent for cooperative binding to occur, then a simple weight function can be used that assigns a weight of 1 for adjacent sites and a weight of 0 to all other sites. The concentration of the dimer, [XY], is the product of [X], [Y], and the dimerization constant, Ka,dimer ([XY] = [X][Y]Ka,dimer). By default, [X] and [Y] are set equal to the Kd for their respective optimal sites, and Ka,dimer is equal to the Ka for binding of monomeric Y to its optimal site. All these values are parameters in the model. The strength of the cooperative interaction can be adjusted by varying the affinity between X and Y (Ka,dimer).
There is no limit to how many transcription factors can bind cooperatively with protein X. The product is therefore taken over all cooperative factors, Y. Homotypic cooperativity is simply a special case, where the same transcription factor matrix is supplied for both X and Y:
For a single competitor protein, Q, binding in direct competition to the same sites as protein X, the higher the concentration of Q or the stronger its affinity for window i, the lower the probability that X will be bound to that window. Formally:
where Ka,Q,i is the predicted binding constant for Q at site i based on the PM for protein Q. More generally:
The competition term, Ci, incorporates all potential competitors binding at any window, k, that affects binding of protein X to site i. κQ,i,k is a weight defined by a user-specified function that determines the effect of protein Q binding at site k on the binding of protein X at site i. For a simple competition weight function, the weight might be a binary function of the distance between sites i and k, such that for sites closer than a distance threshold, the weight is 1 (binding of Q completely occludes binding of X) and for sites further than the threshold distance the weight is 0 (no competition). This function models simple steric exclusion, but more complex functions of the distance and orientation between sites can be used to model more complex interactions.
The complete GOMER model
Adding the effect of competition to the scoring function derived above for cooperative interactions, we obtain the complete model for in vivo binding and gene regulation, as implemented by the GOMER program.
GOMER reports the GOMER score, S, for all regulatory sequences that are of interest. These can be specified in several ways: by reference to a gene annotation file (for example, the 1,000 bases 5' to the start of an ORF or a snRNA gene); using a list of genome sequence coordinate pairs (see the analysis of ChIP-array data below); or providing FASTA-formatted sequence files. In addition to the scores for each sequence of interest, GOMER also reports statistical measures that quantify the ability of a model to distinguish sequences that have been classified as regulated from those that are not. Here, we restrict our discussion to the ROC AUC . A fuller discussion of evaluation metrics is available elsewhere .
Genome sequence, regulated gene sets and probability matrices
All analyses were performed using Saccharomyces cerevisiae genome sequence and genome annotation files obtained from the Saccharomyces Genome Database  on January 29, 2004. PMs used in this work and lists of regulated genes are available as Additional data files 2, 3, 4, 5, 6. For analysis of ChIP-array data, the genome sequence coordinates that define each microarray spot were determined from the sequences of the PCR primers used to make the array (see supplementary methods in Additional data file 1 for details).
The GOMER program was written in Python . Weight functions are Python modules with a defined programming interface so users can create novel functions to fit their regulatory system of interest without needing to know the internal design of GOMER. The software and a manual for its use are available from the GOMER web site .
Additional data files
The following additional data are available with the online version of this paper. Additional data file 1 is a PDF file providing supplementary methods. Additional data file 2 is a table of the Fkh1p binding probability matrix. Additional data file 3 is a table of the Mcm1p binding probability matrix. Additional data file 4 is a table of the Sum1p binding probability matrix. Additional data file 5 is a table of the Ndt80p binding probability matrix. Additional data file 6 is a table of the Rap1p binding probability matrix. Additional data file 7 is a table listing the CLB2 cluster (Fkh/Mcm1 regulated genes). Additional data file 8 is a table listing open reading frames regulated by Sum1p (derepressed in a Sum1 knockout). Additional data file 9 is a table listing listing open reading frames regulated by Ndt80p (induced by Ndt80p overexpression). Additional data file 10 is a table listing open reading frames regulated by both Sum1p and Ndt80p (intersection of Sum1p regulated ORFs and Ndt80p regulated ORFs). Additional data file 11 is a table listing open reading frames that are chromatin immunoprecipitated by Rap1p. Additional data file 12 is a table listing intergenic regions that are chromatin immunoprecipitated by Rap1p.
We thank David Noll for thoughtful contributions and suggestions throughout this work. We also thank Jason Lieb for his careful reading of an earlier version of the manuscript and his helpful comments. This work was supported by a grant from the National Institute of General Medical Sciences Health to N.D.C. and a National Science Foundation Graduate Research Fellowship to J.A.G.
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