Zinc finger proteins are the largest family of transcription factors in the human genome. The EGR sub-family of C2H2 zinc finger proteins has been extensively studied to determine the basis of DNA-protein binding specificity. The structure of the DNA-protein complex has been determined for the wild-type EGR1 (zif268) protein bound to its consensus site 12 and for several other variants of the interaction 345. From the structure, the interaction appears very modular with each protein containing several zinc finger domains and each finger interacting with adjacent 3 base-pair (or overlapping 4 base-pair) segments of the binding site. Analysis of binding sites for this family of proteins suggested there were simple rules that relate the sequence of the zinc finger protein to its preferred binding site sequence 6, and that those rules could be used to design proteins with desired specificities 78. Soon after, experimental techniques of in vitro randomization and selection were employed to greatly expand the collection of protein-DNA high affinity interactions 9101112. Several reviews 4131415161718 have analyzed the protein-DNA crystal structures, summarized the results of the in vitro selection experiments, described rules for predicting high affinity protein-DNA interacting pairs and assessed the success of those rules for designing proteins to recognize particular sequences. Most of the recognition rules that have been developed are qualitative, specifying the amino acid and base-pair combinations that are preferred at each position in the binding sites 18. Such rules can be effectively used to design proteins with preferred binding sites that are desired 19.

Despite the success of the qualitative recognition codes for designing proteins with desired preferred binding sites, the utility of such codes is still quite limited. If one compares the collection of known protein-DNA interacting pairs obtained in in vitro selection experiments, more than half of the fingers contain at least one amino acid/base-pair interaction that is not included in the code 20. Furthermore, the code only predicts the preferred binding site for each protein sequence, or preferred protein for each DNA binding site. But it does not, by its qualitative nature, attempt to predict differences in affinities to similar sequences. Because all of these proteins bind with limited specificity, sites that are very similar to the preferred binding site can often bind with only slightly reduced affinity. Therefore predicting the quantitative binding specificities is important for a comprehensive view of their functions.

Several quantitative binding models have been developed, either specifically for the zinc finger proteins or for general protein-DNA interactions 20212223242526. In many cases such codes can accurately predict the preferred binding sites as well as the qualitative codes, but the overall accuracy of the quantitative predictions is limited, undoubtedly for a combination of reasons. One reason is that there are limited data upon which to infer the model parameters using statistical approaches. Another reason is that many of the models are overly simplified, for instance assuming that each amino acid/base-pair contact is independent of any of the surrounding structure. We know, for instance, that the interactions of the protein and DNA are not completely additive 2728, and it is also known that both intermolecular and intramolecular interactions contribute to protein-DNA recognition (24). But it has also been shown that models which are additive over the DNA positions can be a reasonably good approximations, at least for some proteins 2930. Most studies of additivity have focused on the DNA binding site, testing whether independent models for each base-pair fit the binding data well 293132. But equally important to the recognition codes is whether additivity holds within the protein. In one example from the EGR family, additivity within the protein was shown to be approximately additive (within 0.5 kcal) for one pair of mutated amino acids 33. But very few studies have addressed the issue. Even though many variants of EGR family proteins have been used in SELEX and phage-display selection studies (see 20 for a summary), very few of the affinities have been quantified. Bulyk et al 28 did measure the affinity to each of 64 different binding sites for five different proteins, but the proteins were different at too many positions to be useful for determining additivity. One needs to have a set of single mutations and their double mutant combinations in order to determine whether the contributions to binding are independent or not. Several structural studies have highlighted the substantial rearrangements that can occur at the protein-DNA interface and can cause single amino acid or base-pair substitutions to influence the interactions at neighboring positions 3153435. Such context effects may limit the predictive accuracy of simple recognition codes, although it is also possible that additivity can hold approximately even in the presence of such rearrangements. In the Mnt protein, a single amino acid change can alter the preferred binding site primarily at two adjacent positions, and more weakly over a longer distance 3637. Nevertheless, a complete quantitative analysis of the adjacent positions that were primarily affected showed that the interaction was largely additive for a wide variety of amino acid substitutions 30.

In this study we analyze the additivity of the interaction in both the DNA binding sites and in the interacting positions of the protein. We measure binding affinities for each of six different proteins, with single and double mutations compared to the wild-type protein, to each of six different DNA sites, also with single and double mutations from the wild-type binding site. We show that for any specific protein or DNA an additive model fits the data quite well. However, there are clear context effects such that no single interaction model fits all of the protein-DNA combinations. But only a small modification to the additive model, with just three additional parameters, improves the fit significantly.

Figure 1 diagrams the direct interactions between the amino acids of finger 1 of the zif268 protein with the bases of the consensus binding site as determined by X-ray crystallography 12. In order to study the additivity of the interaction on the side of protein, we constructed wild-type zif268 and five mutants where mutations occur in finger one. These five mutants include two single mutants of zif268 at position -1 in which arginine (R18) (referred to as RE) was replaced by glutamine (Q) (referred to as QE) and aspartic acid (D) (referred to as DE), separately, one single mutant at position +3 where glutamic acid (E21) was mutated to asparagine (N) (referred to as RN), and two corresponding double mutants (referred as to QN and DN, respectively). The six DNA sites used for this study were chosen primarily based on the qualitative code that represents the correlations between amino acids located at different positions and the DNA bases that they specify 41534. Specifically, the anticipated base specificity for amino acids arginine, glutamine and aspartic acid at position -1 are G, A and C at position 9 in the DNA sequence, respectively. The favorable bases for amino acids glutamic acid and asparagine at position +3 are C and A at position 8. The oligos used to generate the six DNA sites are shown in Table 1. They share common sequences except for the DNA bases that are recognized by the amino acids at the position of +3 and -1 of finger 1, referred as CG, CA, CC, AG, AA, and AC, respectively. We measured the affinity of each of six proteins to each of six DNA sites, and we use these data to analyze the additivity in both the protein and the DNA binding sites.

For each protein we determined the relative affinity of each different binding site compared to the wild type site (CG) using the QuMFRA assay (Table 2). For the wild-type protein, the relative affinities of CA, CC, and AG to the reference site CG in this study are 0.27, 0.082 and 0.15, respectively. These data are in good agreement with the relative affinities previously determined by Miller and Pabo (0.21, 0.ll and 0.20, respectively 34). Table 2 shows only the wild-type protein (RE) binds preferentially to the wild-type binding site (CG), all of the other proteins preferring a different binding site sequence. The range of affinities varies considerably between the different proteins. RE has about a 25-fold difference between the highest and lowest sites, while QE only varies by about 2-fold between the highest and lowest. We also measured the absolute binding affinity of each protein to one of the DNA binding sites with a Scatchard analysis (Table 3). The K_d for wildtype zif268 binding to the DNA site CC is 3.0 × 10^-8 M, which converts to a K_d for wildtype binding site CG of 2.5 × 10^-9 M. This value is almost the same as that determined by Hamilton et al (2.2 × 10^-9 M) 41 (previously reported values for this K_d range from 0.04 to 6.5 nM, depending on the binding condition used 33). No similar data exist for the other proteins in our collection. Combining the data from Tables 2 and 3, we derive the association constant of each protein for each different DNA sequence, which differ by over 300-fold between the highest and lowest affinities (Table 4).

From the binding data we can assess the additivity of the interaction for both the protein and the DNA. In a perfectly additive interaction the binding energy for each sequence would be the sum of the independent contributions at each position. For example, for any protein j, the binding energy to any DNA sequence XY, would be the sum of the interactions with base X and base Y:

ΔG_j(X₈Y₉) = ΔG_j(X₈) + ΔG_j(Y₉).     (1)

The important assumption of the additive model is that the interaction energy at position 8, for example, doesn't depend on which base occurs at position 9. We do not expect additivity to hold precisely 302728, but it can be a very good approximation, at least for some proteins 2729. Previously, studies of additivity have focused on whether the positions in the DNA binding site contribute independently to the binding of a particular protein. Using the data of Table 4 we can also determine whether the positions in the protein contribute additively to the binding of a particular DNA site. That is, we can reverse the symbols of equation 1 to refer to the binding of a particular DNA sequence, i, to a protein sequence UV:

ΔG_i(U_-1V₃) = ΔG_i(U_-1) + ΔG_i(V₃).     (2)

Of course, we have not measured affinities to all possible DNA sequences or for all possible protein sequences, but because we have both single and double mutants in both the protein and the DNA, and have measured the binding affinities of all combinations, we can determine how well additivity holds on both sides, the DNA and the protein, at least for this limited set of variants.

We cannot actually measure the binding affinities to single positions because they always occur in some context. But we can find the "best fit" values for the independent interactions, and then determine how well the total data fits the additive model using those values. One method to obtain the best fit independent parameters is to apply multiple linear regression to the total data 3132. However, we have argued previously 29 that a better criterion is to minimize the difference in total free energy between the observed data and the model.

The and values are those obtained as the best fit parameters (those which minimize M) for each position assuming independence. The ω refers to either the protein or the DNA, and α,β refer to the residues at the two interacting positions. The first term inside the sum represents the probability that each particular residue sequence will be bound, and so weights the energy differences by their contribution to the total free energy of the system. As can be seen in the last form of the equation, M is the "mutual information" between the positions, the amount of total information content in the data that cannot be explained by the best independent model. We use log₂ so that the mutual information is measured in bits.

Given the best fit independent parameters we can calculate the specificity information, I_spec, of each position independently 42. For example the specificity information for the protein or DNA ω at the first interacting position is

I_spec measures the amount of specificity in the interaction in bits; any non-specific protein or DNA would have I_spec = 0. Figure 2 shows sequence logos 43 for each of the six proteins and the six DNA sequences for which we have measured the affinity. We have added the symbol "M" to each one which shows the amount of mutual information in each interaction 442730. That is the amount of total free energy, or specificity information, which is not captured by the best fit additive model. Half of the total mutual information is displayed above each position.

Several interesting results are evident in Figure 2. As stated above, the proteins vary considerably in their specificity, with RE (shown as "ER" in the figure) showing large discrimination between the different DNA sites, whereas QE and DE are fairly non-specific. The same holds for the different DNA sites, where CG is much more specific than CC or AC. It is interesting that every DNA site prefers R at position -1 of the protein, showing that it contributes to the total affinity of each protein as well as to the specificity of some proteins. The small degree of mutual information, the "M" in each logo, means that every interaction fits well with an additive model. Not only do the DNA positions contribute very additively, as has been shown previously for this family of proteins 29, but the contributions of the amino acids in the protein are also largely additive. The conclusion that additive models are good approximations to the true data holds for every DNA site and every protein included in the analysis. However, it is also true that there is not a single set of additive parameters that fit well for every case. This is consistent with the context effects previously noted for this family 1534. For example, R prefers to bind to G over A or C, but the magnitude of that preference is much larger if position +3 is an E instead of N. And an N at position +3 always prefers an A over C in the binding site, but that preference is much weaker with an R at position -1 than with a Q or D. Similarly, E at position +3 prefers a C very strongly in the context of an R, but is quite non-specific with either a Q or D at position -1. Similar effects, but of smaller magnitude, can be seen in the context effects of the DNA sites. These results show that additive models can be good approximations not only for the DNA sites in binding to any particular protein as has been seen before 29, but also for the proteins in binding to any particular DNA site. But the results also show that additivity for specific proteins and DNA sites is not sufficient to generate a general recognition code because context effects can still be important when both the DNA and protein can be variable. The small amounts of mutual information observed for any specific protein or DNA can be reinforced to give much larger amounts when measured over combinations of both components.

To get a more detailed view of the dependencies in the data, it is useful to reformat it as in Figure 3A. Those data are the same as in Table 4 except that it has been normalized to a sum of 1000. In an experiment where every protein and DNA was equally available for binding, those elements in the table are 1000-times the probability of picking that particular combination from all of those in the bound state. The data are arranged in a four-dimensional (4D) table, with one dimension for each of the two positions in the protein and the two positions in the DNA. For example, the 335 at the RE-CG element of the table corresponds to the wild-type association constant of 406 from Table 4 after normalization. From the data in Figure 3A it is easy to obtain different lower dimensional views by summing over the other dimensions. For example, Figure 3B shows the 2D view of the interaction of the amino acid at position -1 with the base-pair at position 9 obtained by summing over all of the combinations of E,N at protein position +3 and C, A at binding site position 8 (inside the bold lines of Figure 3A). Similarly, Figure 3C shows a 2D view of the interaction between the amino acid at position +3 and the binding site position 8. Those two 2D views are orthogonal and together cover the 4D space of Figure 3A. We also show the remaining 2D views in Figures 3D–G. The pairs in Figure 3D,E and 3F,G are also orthogonal and together cover the 4D space of the data. If the binding interaction was completely additive, the true data of 3A could be calculated as the (renormalized) outer product of any pair of orthogonal matrices. Such predictions are not too bad, but demonstrate limitations of the additive model (see below).

Because the data in Figure 3 are in probabilities (if divided by 1000), the information specificity can be calculated more easily than in equation (4):

I_spec(α) = log₂N_α - H_α     (5)

where α is any of the positions or combination of positions, H_α is the Shannon entropy of the data at those positions and N_α is the number of entries in the data. For example, position -1 of the protein has three entries, R, Q and D, with overall probabilities of 0.852, 0.093 and 0.054, respectively, which gives I_spec(- 1) = 0.84 bits. The upper half of Table 5 shows the specificity information for each of the positions (along the diagonal) as well as the specificity information for each of the pairs of positions (from the data shown in Figure 3). If the two positions contribute independently to the total specificity then the information for the paired positions is just the sum of the information at the each position. In this case the mutual information between the positions is the amount of information in the pair that exceeds the sum of the individual positions:

M(α,β) = I_spec(α,β) - (I_spec(α) + I_spec(β))     (6)

Those values are shown in the lower half of Table 5. From the standard model of interaction between the DNA and protein we would expect there to be very little mutual information for any of the 2D datasets of Figure 3D–G, and that expectation is met. But we do expect high mutual information for the datasets in Figure 3B and 3C because those are the interacting positions. Just as we get high mutual information for positions that interact in RNA structures 44, we expect to see compensating changes between the amino acids and base-pairs that interact. That expectation is met for the combination of protein position +3 and base-pair position 8 (Figure 3C) where there is a clear preference for E binding to C and for N binding to A. In that case the mutual information is 0.19 bits, which is the main contribution to the total information of that pair, 0.24 bits. However, protein position -1 and base-pair position 9 also interact but show little mutual information because R is the preferred amino acid for each different DNA sequence and G is the preferred base-pair for each different protein. That pair has high specificity information, 1.09 bits, but it is very additive with only 0.02 bits of mutual information.

The total specificity information in the complete data of Figure 3A is 1.46 bits. The sum of the information for the interacting pairs, -1,9 and 3,8, is 1.33 bits, which shows that the complete specificity is reasonably well fit by assuming independent contributions from those interacting positions, as in most recognition code models 18. If one predicts the complete data of Figure 3A as the outer-product of the matrices of Figure 3B and 3C (not shown), the correlation coefficient between the observed and predicted binding energies is 0.87 (Model 1 of Figure 5), similar to what had been observed previously for data in which only the DNA site had been varied 29. While that result is reasonably good overall, examination of the complete data in Figure 3A identifies one clear source of context dependence between the interacting positions. When protein position -1 is R and the base-pair at position 9 is either G or A, there is a clear preference for the specific combination of E with C and a weak preference for N with A. But for all other combinations of positions -1 and 9, there is a strong preference for N with A, but very little preference for E. That is, the preference of E for C depends on the R with G or A combination being adjacent. In the structure of zif268 with the wild-type DNA there is no hydrogen bound between the position +3 E and the C base-pair, but rather it interacts with the backbone and with the neighboring R amino acid 21. Various qualitative codes for the interactions of this protein family do not include E as an acceptable amino acid at position +3 415. But in the compilation of SELEX and phage-display results used by Benos et al 20, the combination of RE-CG was much more frequent than expected from the individual or pair occurrences (p-value less than 0.001). That is consistent with our result that in general E contributes little to the specificity of the binding site at position 8 except in the case where the adjacent interaction is R with G or A. Such context dependencies are not included in the simple recognition code models, but we can easily add that to the basic model. In Figure 4 we show two different specificity tables for the interaction of positions +3 and 8. Figure 4A represents the general case, and Figure 4B is for the special case of R with G or A at positions -1 and 9. If we now predict the complete data using these models, combined with the general model for positions -1 and 9 in Figure 3B, we obtain the values shown in Figure 4C. The specificity information of this data is 1.44 bits, showing that it models quite accurately the complete data. The correlation coefficient for those predicted binding energies with the measured energies is 0.96, a significant improvement over the model without the context dependent parameters (Model 2 of Figure 5). This improvement is at the cost of only three additional parameters due to the separation into two distinct classes depending on whether or not position -1 is an R that interacts with G or A. The completely additive model has 8 free parameters for the interaction of positions -1 and 9 (the 9 values in Figure 3B minus 1 for the total fixed sum) and 3 free parameters for the interaction of positions +3 and 8 (from the 4 values in Figure 3C). By separating the matrix of Figure 3C into two separate cases, shown in Figure 4A,B, we need 3 additional parameters in the model, for a total of 14. The model is used to predict data with 35 free values (the 36 elements of Figure 3A minus 1 for the fixed sum), so the additional parameters are only a small reduction in the degrees of freedom remaining to assess the fitness of the model.

The EGR family of proteins is an ideal case to study the effectiveness of a recognition code for protein-DNA interactions. The collection of crystal structures along with a large number of examples from selection experiments provides a wealth of information for determining the relationship between the protein sequence and the affinity for different DNA sequences. Simple qualitative models that predict the preferred interactions can be very effective and useful for designing new TFs 1419. Quantitative models, that predict relative binding affinities to multiple DNA sites, are more challenging but some success has been achieved by statistical approaches as well as by structure based approaches 20212223242526. Most current models of this type assume independence of the contributions to binding between the positions in the interactions. In this work we show that additive models can be a good approximation for any particular EGR protein and also for binding to any particular DNA site; additivity holds well for both the DNA and protein side of the interaction. But we also show that there is not a universal set of parameters that work for all proteins or all DNA sites, rather there is context dependence in the interactions. However, at least in the cases studied here, a simple addition to the independent model that divides sites into two classes provides a much better fit. This holds promise that, even though additivity does not hold precisely, it may still be possible to determine an additive recognition code by identifying a small set of classes that cover the entire set of interactions. How many classes will be needed is unknown at this time. The 36 combinations in our study required only two classes to give a very good fit but this is still far from a comprehensive analysis. The total number of adjacent amino acid pairs is 400 and the number of di-nucleotide combinations is 16, so there are 6400 possible combinations of the two. Quantitative analyses that cover all possible combinations of even a single zinc finger are impossible at this time. But more thorough sampling of the space of high affinity interactions, followed by quantitative binding assays, will provide much valuable information regarding the nature of recognition codes. While a completely additive model for the interaction of the protein and DNA is not correct, it may be that only relatively minor modifications are needed to make significantly better predictions.

By determining the binding affinities of single and double mutants in both the DNA binding site and in the protein we were able to assess the degree of additivity in both halves of the interaction. Although only a limited number of combinations were tested, we find that for every DNA sequence and for every protein sequence an additive model is a good approximation to the real binding data. However, when all of the data are considered together there are clear context effects that are not well fit by a single additive model. A slightly more complex model does provide a good fit to the observed data, suggesting that quite simple may still be employed to predict quantitative binding interactions of proteins with DNA. Further data are needed to determine how well these findings generalize to more variations and to other protein families.

JL performed all of the experiments, which GS helped to design. Both authors contributed to the analysis of the data and the writing of the paper.