An information transmission model for transcription factor binding at regulatory DNA sites

Background Computational identification of transcription factor binding sites (TFBSs) is a rapid, cost-efficient way to locate unknown regulatory elements. With increased potential for high-throughput genome sequencing, the availability of accurate computational methods for TFBS prediction has never been as important as it currently is. To date, identifying TFBSs with high sensitivity and specificity is still an open challenge, necessitating the development of novel models for predicting transcription factor-binding regulatory DNA elements. Results Based on the information theory, we propose a model for transcription factor binding of regulatory DNA sites. Our model incorporates position interdependencies in effective ways. The model computes the information transferred (TI) between the transcription factor and the TFBS during the binding process and uses TI as the criterion to determine whether the sequence motif is a possible TFBS. Based on this model, we developed a computational method to identify TFBSs. By theoretically proving and testing our model using both real and artificial data, we found that our model provides highly accurate predictive results. Conclusions In this study, we present a novel model for transcription factor binding regulatory DNA sites. The model can provide an increased ability to detect TFBSs.

Results: Based on the information theory, we propose a model for transcription factor binding of regulatory DNA sites. Our model incorporates position interdependencies in effective ways. The model computes the information transferred (TI) between the transcription factor and the TFBS during the binding process and uses TI as the criterion to determine whether the sequence motif is a possible TFBS. Based on this model, we developed a computational method to identify TFBSs. By theoretically proving and testing our model using both real and artificial data, we found that our model provides highly accurate predictive results.
Conclusions: In this study, we present a novel model for transcription factor binding regulatory DNA sites. The model can provide an increased ability to detect TFBSs.

Background
The transcription of genes is controlled by transcription factors (TFs), which bind to short DNA motifs that are known as transcription factor binding sites (TFBSs). Identification of TFBSs lies not only at the very heart of expanding our knowledge of regulatory elements in the genome by helping to decode genomic data, discover regulatory patterns in gene expression, and establish transcription regulatory networks, but also of explaining the origins of organismal complexity and development [1]. Computational identification of TFBSs is a rapid, cost-efficient way to locate unknown TFBSs. With increased potential for highthroughput genome sequencing, the availability of accurate computational methods for TFBS prediction has never been as important as it currently is. However, DNA regulatory elements are frequently short and variable, making the computational identification of them a challenging problem because the real TFBSs might be easily lost in random DNA sequences, i.e., the "background noise".
To date, many models have been developed for transcription factor binding of regulatory DNA sites, and based on those models, numerous computational algorithms have been established to identify TFBSs. Several studies have utilised the structural information of DNA and protein to build predictive models for DNA binding sites [2][3][4][5]. These algorithms are able to identify previously uncharacterised binding sites for TFs and have improved performance over simple sequence profile models [6]. However, these algorithms have not been generally used because their parameters depend on the knowledge of the solved protein-DNA complex structures, which is a limited data set.
Several methods use pattern recognition algorithms derived from computer science or other research areas. These methods include support vector machines (SVMs) [7], self-organising maps (SOMs) [8], and Bayesian networks [9]. These algorithms can automatically provide objective and non-user-defined thresholds by training the programme with known data. Nevertheless, the biggest limitation of these methods might be the lack of explicitly biochemical or biophysical explanations.
Currently, position weight matrix (PWM) is the most common model for TFBS recognition. Many methods or programmes are based on the PWM model or its expansion, such as Match [10], the expectation-maximisation (EM) algorithm [11], and the stochastic variant of EM, the Gibbs sampling method [12,13]. In PWM, an L-long sequence motif is represented by a 4*L matrix, with weights giving the frequency of the four DNA bases (or the logarithm) in each of the L positions [6,14,15]. The basic PWM model is based on the biophysical considerations of protein-DNA interactions and uses the relative entropy, which is also known as the information content, as the criterion to determine whether an input sequence is a TFBS. According to this theory, the affinity between the factor and its TFBS is related to the free energy, which correlates with the relative entropy [6,14,15]. Therefore, in order for a sequence to be a TFBS, it must have higher relative entropy. Consequently, the relative entropy can be used as the criterion to detect a TFBS.
The PWM approach assumes that the contribution of each nucleotide position within a TFBS to the free energy is independent and that the effect on the binding strength is cumulative. We call this hypothesis the "independent hypothesis" because it supposes that each base of the motif is independent of the others. Methods based on the independent hypothesis are simple and have small numbers of parameters, making them easy to implement. These methods are widely used and often considered acceptable models for binding-site predictions.
The PWM model can suffer from high false-positive (FP) rates if motifs are degenerate. In addition, in some real cases, the affinity between factors and their TFBSs is weak, causing a high false-negative (FN) rate while using these methods. More importantly, the independent hypothesis can lead to deviations in the scoring mechanism and produce inaccurate results. Experimental evidence [16][17][18][19][20] suggests that there is interdependence among positions in the binding sites, which has prompted the development of models that incorporate position dependencies. The related methods include Bayesian networks [21], permuted Markov models [22], Markov chain optimisation [23], hidden Markov models [24], non-parametric models [25], and generalised weight matrix models [26]. Methods based on positiondependency models usually have better binding site prediction accuracy with lower FP rates. However, these methods require more complicated mathematical tools with more parameters to estimate and more experimental data than are typically available [27].
Orthogonal information from comparative genomics and information on co-regulation at the transcriptional level have also been integrated into these methods to identify cisregulatory sites [28][29][30][31]. Methods have also been proposed to discover the composite regulatory module (CMA) [32,33]. Because most of these methods rely on the basic algorithms proposed previously, their performances are mainly determined by these basic algorithms.
Therefore, although significant progress has been made, the accuracy of the computational identification of TFBSs can still be improved. To tackle the general problem of binding site identification in the absence of high-throughput experimental data, theoretical models of binding sites are still required.
One aim of this work is to develop a new model that incorporates position interdependencies in effective ways to improve the computational prediction of TFBSs. Based on information theory [34,35], in this study, we propose a novel computational model. By theoretically proving and testing our model using both real and artificial data, we find that our model gives highly accurate predictive results.

An information transfer model for TFBS binding
In this paper, we treat the complex between a transcription factor and its binding site as a stand-alone system. During the binding process, energy exchanges occur between the TF and TFBS, and the spatial structure and physical state of the system change. We assume that the total amount of information in the system remains unchanged in this process and that information is only transferred between the factor and the site: In this equation, INFO is the total information contained by the system, IF1 and IF2 are the information carried by the transcription factor in the unbound and bound states, respectively, and IS1 and IS2 are the information possessed by the DNA site at the unbound and bound states, respectively. During the binding process, the information flows from the TFBS to the factor ( Figure 1). The transferred information is Taking an L-bp sequence (seq) as the input sequence to be scanned, the jth base of seq is seq(j). The background probability of A, T, C, and G is q(i). In this formula, i represents the base A, T, C, or G, and the background probability can be obtained by Figure 1 Information transferred from TFBS to TF during binding. In this figure, we assume the transcription factor and its binding site are a stand-alone system. scanning the chromosome sequences of the species. Before binding, the occurrence probability of base seq(j) is q (seq(j)). According to information theory [25,26], the information carried by the jth base is -log 2 (q(seq(j))). With the independent hypothesis, the total information carried by the input sequence can be simply calculated by summing all of the information carried by each base of seq: Suppose that a transcription factor and its known TFBSs are aligned by an appropriate algorithm. In this study, we use L to represent the length of the aligned motif, j to represent the base position and p j (i) to represent the occurrence probability that the base i (A, T, C or G) appears at the position j according to the motif.
After the TF binding to its site, the state of the DNA sequence changes. The occurrence probability of base seq(j) changes to p j (seq(j)); therefore, the information carried by the jth base becomes -log 2 p j (seq(j)), and with the independent hypothesis, the total information IS2 is as follows: The TI can be described as We hypothesise that a factor binds to a TFBS only if enough information is transferred from the site to the factor. We can use a basic criterion to determine whether the factor can bind to the sequence: the TI of the sequence must be larger than a threshold value. This value can be defined as the minimum transferred information (MTI), which is the natural and objective threshold used to determine whether the binding can occur. That is, Once the TI of an input sequence is larger than MTI, then it is accepted as a possible TFBS.

Enhancement of the model to be universal
The independent model might lead to inaccurate predictive results. In this section, we discuss in detail how this can happen by example and in theory and how we enhanced our model to be independent of this hypothesis.
An example of the correlation among different bases is shown in Figure 2. The same example is used by GuhaThakurta [1] to show the basic concept of the PWM and relative entropy methods. We can see that the 1 st and 11 th bases are correlated: when the 1 st base is C, the 11 th base is strictly T. We can find that P 1,11 (C,T) = 0.5, and P 1 (C) P 11 (T) = 0.5*0.75 = 0.375. As P 1,11 (C,T)>P 1 (C)P 11 (T); therefore, we conclude that these bases are positively correlated. When position 1 is C, there is a high probability that position 11 is T. For these two positions, P 1,11 (T,T) = 0.25, P 1 (T)P 11 (T) = 0.375. So P 1,11 (T,T)<P 1 (T)P 11 (T); therefore, we conclude that the positions are negatively correlated, which means that when position 1 is T, there is a high probability that position 11 is not T. Such correlations are not rare, as they can be found in most of the real TFBS data set.
Based on this observation, we propose a formal definition of positive and negative correlations of the bases in a motif: if P seq i 1 negatively correlated. In this formula, seq(i) is the ith base of the sequence seq. For example, the 4 th and 6 th bases are positively correlated and contain no more or less information than only the individual base. Therefore, the independent hypothesis leads to an inaccurate estimation of the TI, thereby making an erroneous prediction of the TFBS. Similarly, the use of other methods that are based on the independent hypothesis also results in incorrect scores and leads to inaccurate predictive results. To avoid this inaccuracy, the model was enhanced to address the correlations such that it is capable of determining the correct TI despite the inaccuracy of the independent hypothesis. First, we know that after binding of the TF to the TFBS, the information encoded by the TFBS seq, is IS2 ¼ TFBSs of the TF. Due to unknown TFBSs and lack of statistical data of the known TFBSs, we cannot determine p seq or I L directly, but these terms can be estimated from the known TFBSs. We use the information of r-base sub-sequences  This probability can reveal the correlation among these bases. For example, if base i 1 is fully and positively correlated with base i k , as the 4 th and 6 th positions in Figure 2 are, then hence, the information of these two bases is , it is only half of the information of the independent situation). If base i 1 is independent from base i k , then p seq i 1 As there are C r L such r-base sub-sequences in total, the average information for r-base sub-sequences is À1 C r L P i 1 >i 2 >⋯>i r log 2 P seq i 1 ð Þ;⋯;seq i r ð Þ . Because the length of these subsequences is r, then the average information carried by one base is À1 rC r L P i 1 >i 2 >⋯>i r log 2 P seq i 1 ð Þ;⋯;seq i r ð Þ . The information of the whole sequence can be estimated by simply multiplying by the length L: Similar to the example in Figure 2, if there is a strong tendency for the bases to be positively correlated, then it can be assumed that Under this assumption, we can prove an important relationship, as follows: From (8) we know that Next, we obtain According to (11), we can infer that If we assume that I rþ1 ≥I r , then we immediately obtain I rþ1 ≥I 1 , which contradicts (11). Therefore, it must be the case that I rþ1 < I r . Hence, (10) is proved.
Immediately, we know that when the correlation is positive Similarly, if the correlation tends to be negative, then the following must be true: Therefore, If the TFBS seq conforms to the independent hypothesis, according to (5) and (8), its information is Therefore, conversely, the tendency of I r can be used to judge if the correlation is positive, negative, or independent. We now know that I independent would overestimate (when the correlation is positive) or underestimate (when the correlation is negative) the I L if the independent hypothesis is not true. Again, this finding can explain why using the independent hypothesis can lead to inaccurately predicted results. More importantly, from (13) and (15), we know that I r is more accurate than I independent when r ≥ 2. So, we can use I r (r ≥ 2) to estimate the information and obtain the predictive results with more accuracy.
The method for calculating the background probabilities must be revised accordingly to adapt to the enhanced model. Instead of counting each single base by scanning the chromosome sequences to obtain the background probability under the independent hypothesis, a window of length L slides through the chromosomes, and all of the r-base sub-sequences in this window are counted. After the scanning, 4 r probabilities are calculated for all of the 4 r possible r-base sub-sequences. These values are used to estimate the information carried by the TFBS before the binding event: In this equation, q seq i 1 ð Þ;⋯;seq i r ð Þ is the background correlation probability, calculated as described previously.
In addition, the formula for estimating the transferred information is changed as follows: Once TI r ≥ MTI r (factor), then seq is accepted as a possible TFBS.

Performance in Saccharomyces cerevisiae promoter regions
We tested our model by calculating the TI for all of the known TFBSs of 10 wellcharacterised transcription factors in the yeast S. cerevisiae promoter database (SCPD) [36].
We found that most of the TFBSs have a TI larger than 0. This evidence strongly supports our TI hypothesis that the information is transferred from the TFBS to the factor, and binding of the TF to the TFBS only happens if enough information is transferred. First, we use 100% of the known TFBSs as the training set to work out the MTI for each TF and test our method with r = 1, 2, 3, and 4 on this data set. We observed that the number of predicted TFBS decreases more than twofold when r changes from 1 to 2 (Figure 3), which guarantees an increase in accuracy as Figure 4 shows. From Figure 3 and Figure 4, we observe that when r ≥ 2, the performance increases with increasing r-values but not as significantly as when r changes from 1 to 2. Because the computational complexity of our method rapidly increases as r increases, r = 3 is a proper value to obtain good performance and maintain a low level of computational complexity. Therefore, the results of r = 3 were used to compare this method with the others.
Next, we examined how the average performance changes as the proportion of the training set increased from 25% to 100% with r = 3 ( Figure 5). We found that as the proportion increased, 1-FN increased linearly; hence, more of the real TFBSs were identified. Moreover, this curve indicated that the method is powerful when little is known about the TFBS. For example, 49% of TFBSs were identified when the model was trained by 25% known TFBSs. Additionally, the FP rate increased little when the proportion of the training set increased, and it was always below 0.3.
In this study, we illustrate several snapshots of TI by scanning several sequences of S. cerevisiae. These sequences cover the coding regions, the regulatory regions and the "flank" regions.
In Figure 6, we illustrate a snapshot of TI by scanning a promoter region of S. cerevisiae. With 75% of the real TFBSs as the training set, we obtained the MTI for the factor. The highest peaks are precisely the real TFBSs, and there are also peaks on the opposite strand that do not reach the threshold.

A comparison of the TI model with other methods
To illustrate the performance of the information transmission model, we implemented this novel model with a programme named tfbsInfoScanner and compared it with Figure 3 The number of predicted TFBS. With 100% of the known TFBSs as the training set, the number of predicted TFBS decreases more than two times when r changes from 1 to 2, which guarantees an increase in accuracy.
commonly used motif identification programmes, such as SOMBRERO, MEME and AlignACE. Mahony et al. [17] proposed the TFBS prediction method SOMBRERO and compared the results derived from SOMBRERO with those from two popular motif finding programmes, MEME [37] and AlignACE [11]. These researchers used the same real data set that we used. To efficiently analyse the performance of our method and to avoid repetitive and time-consuming computation, we used the same real sequence data set and compared results derived from our method to those obtained from SOM-BRERO, MEME and AlignACE. Table 1 shows a performance comparison of our method and three other programmes. The results indicate that when the proportion of the training set is larger than or equal to 50%, our method achieves the best performance in most cases.

Performance on artificial sequences
To examine the performance of our method in discovering "unknown" TFBSs, we subsequently trained our method with all of the known TFBSs and embedded the artificial sequences with pseudo-motifs. Similar to Mahony et al. [17], we also generated three artificial test set, although using our own method. In the artificial test set used by Mahony et al., each set comprises 10 data sets, each of which comprises 10 sequences; each sequence harbours a random number of occurrences (0~3) for each of the Figure 4 Variety of performance as r changes from 1 to 4. In this figure, the proportion of the training set = 100%; however, the figures for proportion of training set = 25%, 75% and 100% are similar. In this figure, PERF=(k P)/(K P), where K is the set of known motif sites and P is the set of predicted motif sites. RT/ PT is defined as the ratio of the real TFBSs to the predicted TFBSs. binding motifs for gcn4, gal4 and mat1 (generated from PWMs). The total lengths of these three sets of 100 sequences are 4500, 8000 and 12500 bp, respectively. The average length of one sequence is therefore 45 bp, 80 bp or 125 bp, but each sequence harbours at most 9 occurrences of the motifs. We believe this number of occurrences may be too dense, and perhaps a high occurrence of pseudo-TFBSs may be encoded by these sequences.
In our modified method, we also generated three artificial test sets with different sequence lengths (450, 800 and 1250 bp), and each test set consists of 10 sequences that were randomly generated according to the GC content of S. cerevisiae. Each sequence harbours a random number of occurrences (0~3) for each of the binding motifs for gcn4, gal4 and mcb (randomly generated from PWMs). Mahony et al. [17] used mat1 as a test object, but in the new version of SCPD, the TFBS of mat1 is split into  Table 1 Performance comparison between our TI method (r = 3) and three other programmes: SOMBRERO, MEME and AlignACE The best performances of the other three programmes are underlined. The performances of our method that are better than the best of the other three programmes are in bold and underlined. The performances that were close to the best of the other three programmes are in bold. The results show that when the proportion of the training set is larger than or equal to 50%, our method achieves the best performance in most cases.
mat1_alpha and mat1_beta; therefore, we arbitrarily chose mcb as a substitute for mat1. This test set is more rigorous because these artificial sequences are 10 times longer, leading to an increase in the number of random sequences, which may result in a higher FP rate. As our method is still under development, in this test, the pseudo-TFBSs are also generated from the PWMs. Because the PWM method assumes that the independent hypothesis is true, these pseudo-TFBSs cannot correctly indicate correlation among the bases. This deficiency might lead to a lower TI, and, therefore, some pseudo-TFBSs may not be identified by our method. However, we can investigate what happens when scanning these artificial sequences.
The average performance in each test set using r = 3 is summarised in Table 2. As demonstrated, most pseudo-TFBSs of gcn4 and mcb were recognised. As for gal4, almost none of these sites was identified. Almost none of the unreal result was predicted at the same time. This result was observed mainly because the correlations of actual TFBSs are strong, while the pseudo-motifs do not have such correlations, and therefore their TI is far below the MTI. However, this finding does not mean that our method is ineffective in identifying pseudo-TFBSs. A typical snapshot of the artificial regulatory region that harbours the unrecognised pseudo-TFBSs is shown in Figure 7. Although these pseudo-TFBSs have no correlation in their sites, our method can still identify a strong TI.
The average performance for r = 1 is also summarised in Table 3. When r = 1, our method is equivalent to the assumption that the independent hypothesis is true; therefore, all of the pseudo-TFBSs were identified. Not surprisingly, the FP rate was high. According to both sets of results, the FN rate was low. The FP rate was maintained at a moderate level for r = 3, even though these artificial sequences were 10 times longer than those described previously.

Discussion
During evolution, regulatory instructions or information were encoded in the DNA sequence. Redundant coding (or correlated coding) is utilised to ensure that the important regulatory information will be inherited and transferred correctly. During the binding Table 2 Average performance of the artificial sequence data set (r = 3), perf = (kz∩P)/(K∪P), where K is the set of known motif sites and P is the set of predicted motif sites [30]  process, the transcription factor reads the regulatory instructions from the TFBS and subsequently guides transcription according to the regulatory information. In other words, the factor reading the special regulatory instruction from the TFBS then instructs the transcription according to the regulatory information obtained from the TFBS. Nucleic acids needed to be coded in a redundant manner to ensure that the regulatory information can be transferred correctly, and therefore these sites are not independent of the others.

Figure 7
A typical TI spectrum of the artificial regulatory region harbours unidentified pseudo-TFBSs. The TI spectrum is on the bottom, and the TI of the complementary side is on the right. In this figure, the horizontal axis indicates the centre of the input sub-sequence with length that equals the aligned TFBSs of the specific factor. The data shows two embedded pseudo-gal4 TFBSs in the sequence. We can see that there are two TI peaks that are distinct but do not reach the threshold. With our model, for the sequences encoding motifs, such as TFBSs, the input sequences can be scanned, and the sub-sequences for which the TI is greater than the MTI of the motif can be taken as the predictive hits.
In our observations, most of the real TFBSs had a positive correlations because with the positively correlated coding, the information that they contained decreased accordingly, but the information was transferred correctly.
Interestingly, we find that if there is a real TFBS encoded by one strand, then there often are peaks on both strands, but the peaks on the opposite strand are usually lower. We think this phenomenon happens for two reasons: first, certain factors bind to their TFBS by inserting a domain into the DNA grooves. In this case, both strands of DNA could have physical contact with the transcription factor; hence, both sides could transfer the regulatory information to the factor, which is detected by our method. Second, it is not known from which strand the background noise comes. Therefore, for example, for r = 2, the occurrence probability of AG equals TC. Therefore, the complementary strand of a real TFBS can have a high TI.
Furthermore, this information transmission model has the potential to be useful in other research areas, for example, in the computational identification of other motifs.

Concluding remarks
In this work, we present a novel model for transcription factor binding regulatory DNA sites. This information transmission model is based on information theory and effectively incorporates position interdependencies. By testing the model on both real and artificial data sets, we have illustrated that our method is efficient at predicting unknown TFBSs.

Data set preparation
The TFBSs of the 11 TFs and regulatory region sequences were obtained from the yeast S. cerevisiae Promoter Database (SCPD, http://rulai.cshl.edu/SCPD) [28]. This data set includes 68 regulatory regions with a total length of 30299 bp. These sequences harbour 309 experimentally mapped TFBS, including 141 real TFBSs of the 11 TFs. The chromosome sequences of S. cerevisiae were obtained from the National Center for Biotechnology Information (NCBI) reference sequence database.
The artificial sequences used in the test were randomly generated, taking into account the GC content of the S. cerevisiae genome. The pseudo-TFBSs of gcn4, gal4 and mcb were randomly generated from PWMs. We did not generate the correlated TFBSs directly because it is difficult to make the pseudo-TFBSs conform to the correlation relationships, as real TFBSs do.

Background probabilities calculation
Background probabilities are used to estimate the information carried by the TFBS before the binding event. An L-base window slides through the chromosomes, and all of the r-base sub-sequences in this window are counted. After the scanning, 4 r probabilities are calculated for all the 4 r possible r-base sub-sequences. This computation is time-consuming, but once the background probabilities are worked out, they can be reused in all of the TFBS predictions of this species without being recalculated.