Optimized mixed Markov models for motif identification
 Weichun Huang^{1, 2, 3}Email author,
 David M Umbach^{2},
 Uwe Ohler^{3} and
 Leping Li^{2}
DOI: 10.1186/147121057279
© Huang et al; licensee BioMed Central Ltd. 2006
Received: 08 May 2006
Accepted: 02 June 2006
Published: 02 June 2006
Abstract
Background
Identifying functional elements, such as transcriptional factor binding sites, is a fundamental step in reconstructing gene regulatory networks and remains a challenging issue, largely due to limited availability of training samples.
Results
We introduce a novel and flexible model, the O ptimized Mi xture Ma rkov model (OMiMa), and related methods to allow adjustment of model complexity for different motifs. In comparison with other leading methods, OMiMa can incorporate more than the NNSplice's pairwise dependencies; OMiMa avoids model overfitting better than the Permuted Variable Length Markov Model (PVLMM); and OMiMa requires smaller training samples than the Maximum Entropy Model (MEM). Testing on both simulated and actual data (regulatory ciselements and splice sites), we found OMiMa's performance superior to the other leading methods in terms of prediction accuracy, required size of training data or computational time. Our OMiMa system, to our knowledge, is the only motif finding tool that incorporates automatic selection of the best model. OMiMa is freely available at [1].
Conclusion
Our optimized mixture of Markov models represents an alternative to the existing methods for modeling dependent structures within a biological motif. Our model is conceptually simple and effective, and can improve prediction accuracy and/or computational speed over other leading methods.
Background
Biological sequences, including DNA, RNA and proteins, contain functionally important motifs, such as transcription factor binding sites (TFBS), RNA splice sites, and protein domains. With the increasingavailability of genome sequences, identification of such functional motifs not only plays important roles in gene finding and function prediction but also is a fundamental step in reconstructing gene regulatory networks and in revealing gene evolutionary mechanisms [2–6].
A commonly used model for motif identification is the Weight Matrix Model (WMM) proposed by Staden [7], also called the Position Weight Matrix (PWM) or Mononucleotide Weight Matrix (MWM). A PWM is usually generated from a set of aligned instances of known motif sequences, using the observed positionspecific base frequencies and/or prior information. Stormo and Fields [8] showed that the PWM score of a motif is proportional to the total binding energy contributed by individual bases. PWM has been used by many motif identification programs, e.g., Matlnspector [9] and Match [10], and performs reasonably well for motif identification. While a PWM can capture both nucleotide preferences at each position and different levels of position specificity, it does not account for functional dependencies between positions. Recent studies [11–15] indicate that there are often important interactions between positions, adjacent as well as nonadjacent, within a motif. The inability of the PWM to capture such dependencies is a limitation as the PWM model often produces a large number of false positives in a genomewide scan [16].
Many models have been developed to incorporate position dependencies. Motif models, such as the Dinucleotide Weight Matrix Model (DWMM) [17] and the Weight Array Model (WAM) [18], can incorporate dependencies between adjacent positions. To incorporate further dependencies of nonadjacent positions, Ponomarenko et al. [19] extended DWMM by introducing the Oligonucleotide Weight Matrix model, which includes a comprehensive set of oliogonucleotide matrices classified into 5 biological function categories. A WAM could also be extended to a high order WAM in principle, e.g., windowed 2^{nd} order WAM [2]. However, the exponentially increased number of parameters of these models makes them impractical due to insufficient training data. To address the weaknesses of WAM in incorporating longrange interactions, Burge and Karlin [2] proposed the Maximal Dependence Decomposition (MDD) model, which has a binary tree structure formed by a set of conditional WAMs. While the MDD model can capture nonadjacent dependencies through the conditional WAM models, it still requires a rather large number of training sequences, which are partitioned into smaller subsets to train all conditional WAMs. To alleviate the requirement of a large training set, Cai et al. [20] developed a Bayesian tree to model dependencies within RNA splice sites; Ellrott et al. [21] suggested a position order optimized Markov chain model, which reorders motif positions to bring distant but dependent positions into near neighbors. More recently, several other models have been developed, including Bayesian networks for modeling proteinDNA binding sites [22], Maximum Entropy Model (MEM) for splice site identification [23], Permuted Variable Length Markov Model (PVLMM) for finding transcription factor binding sites and splice sites [24]. For a biological motif with position dependencies, these models can show improvement in prediction accuracy over the models that assume independence. Incorporating position dependencies can also improve the accuracy of de novo motif discovery [25].
In this paper, we present a new and flexible motif model, the OMiMa, to incorporate position dependencies within a motif. OMiMa can not only adjust model complexity according to motif dependency structures but also minimize model complexity without compromising prediction accuracy. As an integrated part of OMiMa, we also introduce the Directed NeighborJoining (DNJ) method to optimally rearrange positions to minimize Markov order. We then describe and discuss the methods for selecting the best model. We implement our model into the OMiMa system that is freely available to the public.
Results
Mixed Markov models
Let X_{ i }be the discrete random variable associated with position i in a biological motif X of length w. For DNA sequences, X_{ i }takes values from set B = {A, C, G, T}; and for protein sequences, X_{ i }takes values from 20 different amino acids. X_{ i }follows a multinomial distribution. Let ${X}_{i}^{k}$ = X_{ik}...X_{i1}and ${x}_{i}^{k}$ = x_{ik}...x_{i1}, where k = 0,..., w  1; upper case X (X_{ i }) is a random variable and lower case x (x_{ i }) is a particular value. The ${x}_{i}^{0}$ denotes an empty sequence and Pr(${X}_{i}^{0}$ = ${x}_{i}^{0}$) = 1. Additionally, let X_{j}= X_{wj}, x_{j}= x_{wj}, where j = 0...w  1. If one uses the k^{ th } order Markov model (M_{ k }), the probability of observing a motif sequence x is just the product of conditional/transition probabilities. Let ${M}_{k}^{L}$ be a k^{ th } order Markov model of a linear chain, and ${M}_{k}^{C}$ be a k^{ th } order Markov model of a circular chain. The probability of a motif sequence is given by equation (1) for a linear chain and equation (2) for a circular chain, respectively.
Compared to a linear Markov chain, a circular Markov chain incorporates additional dependencies that may contain subtle signals that allow the model to distinguish true motifs from false ones, especially when false motifs are similar to true motifs.
Suppose a motif X can be divided into m independent submotifs, that is X = Y_{1},... Y_{ m }and each submotif is modeled as an independent Markov chain, that is ${M}_{X}={M}_{{Y}_{1}},\dots ,{M}_{{Y}_{m}}$, then the probability of the sequence (x) given the Markov models is:
Conceivably, the different parts of a motif could have distinct roles in the interaction with their partners. Motif positions involved in the same role can be highly dependent, whereas those involved unrelated roles are likely independent. A mixture of Markov models seems an ideal fit by modeling different signals with different submodels. A 0^{ th } order Markov chain can effectively model strong signals such as those embedded in highly conserved positions where the probability of a certain base occurring is almost one. In addition, positions where base composition contributes little or nothing to motif function need no more complex model than a 0^{ th } order Markov model. On the other hand, a higher order Markov model is necessary for detecting subtle dependency signals that can be essential for distinguishing true motifs from false ones.
Motif dissection
To apply the mixture of Markov models to a motif, the first step is to dissect the motif into several independent submotifs, each of which is modeled as a Markov chain. For a given set of sequences of a motif, we employ chisquare tests to find significant pairwise dependencies between positions within the motif (see also [21]). Based on pairwise dependencies, motif positions are grouped into independent sets, each forming a Markov chain. The outline of our procedure for grouping motif positions is described in the following steps.
1. Calculate base frequencies for each position, and find highly conserved positions where the observed frequency of a certain base (almost) equals 1. These conserved positions then are put into set H as defined below.
where f(i, x) is the frequency of base x at position i, and B is the set of bases.
2. Place remaining positions in the set M, and calculate pairwise chisquare values for every pair of positions in M.
where B_{ i }and B_{ j }are the sets of bases observed in positions i and j, respectively; O(x_{ i }, x_{ j }) and E(x_{ i }, x_{ j }) are the observed and expected counts of pair (x_{ i }, x_{ j }), respectively. E(x_{ i }, x_{ j }) is the product of observed base frequencies x_{ i }and x_{ j }. The degrees of freedom of this test is (B_{ i }  1) × (B_{ j }  1), where B_{ i } and B_{ j } are the number of different bases in sets B_{ i }and B_{ j }, respectively.
3. Based on the above χ^{2} tests, find all positions that show little dependence on any other positions in M, and move them to the set I, as defined by
Here p_{ i,j }is the pvalue corresponding to ${\chi}_{i,j}^{2}$, and α is the significance level, e.g., 0.05.
 (a)
Set s = 1.
 (b)
Calculate θ_{ i }= ∑_{j∈M,j≠i}δ(p_{ i,j }<α) for each position i in M, where δ is a 0/1 indicator function. Find the largest θ_{ i }, and move position i and all positions j that p_{ i,j }<α from M into a new set C_{ s }.
 (c)
For each remaining position, check if it significantly depends on any position in C_{ s }. If it does, then move it from M into C_{ s }.
 (d)
If M is not empty, update s = s+1 and go back to step (b).
Step 4 above essentially groups positions into independent subsets, each potentially forming a functional unit. For the special 0k mixture model, we simply set M = C_{1} at this step.
Markov chain optimization
The next step is to arrange the positions in each subset into a Markov chain. Since the positions in sets H and I are independent of each other, they can be arranged in their natural order to form a 0^{ th } order Markov chain. The positions in H can also be treated differently from those in set I in motif identification by requiring a perfect match for a true site. Sets C_{ s }are different. The position arrangement for each set C_{ s }needs to be optimized so that the Markov model can account for most dependencies while minimizing the Markov order. For a given set C_{ s }, we use the median (K_{ s }) of θ (θ = {θ_{ j }, j ∈ C_{ s }}) as the maximum order of its potential Markov model. We then optimize position arrangement for the k^{ th } order Markov chain (k = 0,..., K_{ s }) by the Directed NeighborJoining (DNJ) method described below.
 1.
For a given set C_{ s }, put each position in the set into a different vector. Here a vector is represented by a letter, an arrow at the top of the letter may be used to indicate the direction of a vector, e.g., a stands for either $\overrightarrow{a}$ or $\overleftarrow{a}$. If $\overrightarrow{a}$ = (1, 2, 3), then $\overleftarrow{a}$ = (3, 2, 1), $\overrightarrow{a}$$\overleftarrow{a}$ = (1, 2, 3, 3, 2, 1), and $\overrightarrow{a}$$\overrightarrow{a}$ = (1, 2, 3, 1, 2, 3). Initially, each vector has only one position.
 2.
Create an initial distance matrix (d) whose elements are d(u, v) = p_{ i,j }, where i is the position in vector u, j is the position in vector v, and p_{ i,j }is the pvalue of chisquare test described above.
 3.
Convert the distance matrix d to the transformed distance matrix D, whose elements are D(u, v), by the following conversion function (see [26]):
 4.
Find the minimum D(u, v) in D. Then a new vector x is formed by joining vector u and v according to Algorithm 1 [see Additional file 1] for a k^{ th } order Markov chain.
 5.
Update the matrix d by replacing u and v by x. The distance of x to other remaining vector y is defined by:
 6.
Go back to step 3 if the number of vectors in C_{ s }is larger than 2, otherwise join the last two vectors according to Algorithm 1.
The order of positions in the final vector is the optimized linear chain for Markov model. Joining the first position to the last position in the vector forms a circular chain. A linear chain could be further optimized by forming a circular chain first from the final vector, then breaking the circular chain between positions with the weakest dependency, e.g., between positions i and j where p_{ i,j }is the largest or the loglikelihood of the corresponding linear chain model is maximized. DNJ not only optimizes position order for linear chain models but also improves circular chain models, particularly when the order of Markov model is low, e.g., 1^{ st } or 2^{ nd } order Markov models.
Model selection

2·loglik + λ·DF
where λ = 2 for AIC and λ = log(N) for BIC (N is the number of sequences); loglik is the maximized log likelihood of data given the model; DF is the degrees of freedom (number of free parameters). We replace DF with the effective degrees of freedom (EDF) in calculating AIC or BIC of the mixture of Markov models, which enables an appropriate model to be selected (see subsection Effective degrees of freedom). There is no clear better choice between AIC and BIC for model selection. AIC tends to choose a model too complex as N → ∞, while BIC tends to choose a model too simple when N is small. In our test on 61 different TFBS datasets, whose sample sizes range from 20 to 130, we preferred AIC to BIC for picking an appropriate model.
Effective degrees of freedom
Let B be the set of bases (B denotes the number of different bases in B), e.g., for DNA sequences B = {A, C, G, T} (B = 4). For a motif of length w, the DF for a k order Markov model is (B^{ k }  1) × (w  k) for a linear Markov chain; and (B^{ k }  1) × w for a circular chain model. That is, the DF increases exponentially as the order of Markov chain increases. As a result, AIC or BIC often pick a simpler mixture model than the best model, especially when B is large. Tested on 61 human regulatory motifs from the Transfac database (ver. 7.4) [29], we found that both AIC and BIC selected the 0^{ th } order Markov models for all 61 DNA regulatory motifs when using the DF. To avoid picking overly simple models, we used the EDF described below to calculate AIC and BIC.
Generally, only a subset of bases from B appears in a particular position of a set of biological motifs. The more conserved a position, the fewer bases are in the subset. The EDF for a model is related to the observed bases in training samples. For example, suppose that one would like to estimate nucleotide frequencies occurring in a position in a set of DNA training motifs. If only base A is observed in the position, then one needs to estimate only the frequency of A, the remaining parameters, i.e., the frequencies of C, G, T, can be derived from any prior information. Therefore, the actual DF is one in this case. For our mixture of Markov models, the EDF is defined as the number of parameters that are direct estimates of the observed bases in a training motif set. Let b_{ i }be the base set observed in a position i of a training set of motifs. Additionally, let h^{ k } be the sequence of motif positions in the k^{ th } order Markov chain, ${h}_{i}^{k}$ be the motif position in the i^{ th } element of h^{ k }, and ∑h^{ k }  = w ( h^{ k }  is the number of positions in h^{ k }), then we define the EDF for the k^{ th } Markov chain as
where ${h}_{ik}^{k}={h}_{\left{h}^{k}\righti+k}^{k}$ if i  k ≤ 0; ${\text{EDF}}_{{\text{k}}_{\text{L}}}$ and ${\text{EDF}}_{{\text{k}}_{\text{C}}}$ are for linear and circular chains, respectively. The total EDF for a mixture Markov model is just the sum of EDFs of all individual chains. For example, the total EDF for the special 0k mixture model equals to the EDF sum of the 0^{ th } and the k^{ th } order chains.
Performance assessment
We test the effectiveness of our method on TFBS data and the donor splice sites, where training data for OMiMa are a set of sequences of a motif. For prediction results, we use the following abbreviations for empirical quantities: TP (# true positives), TN (# true negatives), FP (# false positives), FN (# false negatives), Ac (Accuracy), Sn (sensitivity), Sp (specificity), and Mc (Matthews correlation coefficient). Sn, Sp, Ac, and Mc are defined as:
Matthews correlation coefficient [30], also called Phi (correlation) coefficient, has a value between 1 and 1, with 1, 0, and < 0 indicating a perfect prediction, a random prediction, and a worse than random prediction, respectively.
OMiMa can use two ways to score a motif site x: loglikelihood and loglikelihood ratio, which are defined by
where M_{ s }is the signal model trained by true motif sites, and M_{ b }is the background model or false signal model trained by background sequences or false motif sites. A sequence x is predicted as a positive site if the score of x is larger than a certain threshold. We select a cutoff threshold using one of the following three criteria: balanced sensitivity and specificity, the maximum prediction accuracy, and the maximum Matthews correlation coefficient. Each potential threshold yields an estimated true positive rate and a false positive rate. The plot of true positive rates against false positive rates generates a Receiver Operating Characteristic (ROC) curve, which can be used for comparing and selecting the best model.
We used a threesymbol notation 'kms' to distinguish different models, where 'k' stands for a 0k mixture Markov model, 'm' is either 'L' or 'C' to indicate whether the k^{ th } order chain is linear ('L') or circular ('C'), and 's' is either 0 or 1 to indicate whether log likelihood score (0) or loglikelihood ratio score (1) is used. For example, '1Ll' stands for a 0–1 mixture of linear Markov models that uses loglikelihood ratio to score a motif site.
Effectiveness of DNJ method for optimization
To assess the ability of our DNJ method for optimizing a Markov chain, we compared the DNJ method with random permutation method. In this evaluation, we used a 0k mixture model (k = 0, 1, 2) (Figure 2) to model transcription factor binding sites from the Transfac database. For each TFBS, we first fitted a 0k mixture model (denoted as M_{ DNJ }) with its k^{ th } order Markov chain optimized by the DNJ method. We calculated the loglikelihood of the data given the model M_{ DNJ }(log Pr(dataM_{ DNJ })). Second, with the same data, we fitted a new 0k mixture model (denoted as M_{ R }), which is the same as M_{ DNJ }except that the positions in its k^{ th } order chain are ordered by random permutation, and calculated logPr(dataM_{ R }). This step was repeated 1,000 times, so we have 1,000 loglikelihoods of the randomly permuted models $({M}_{{R}_{1}},\dots ,{M}_{{R}_{1000}})$. We then calculated the empirical p_value of the DNJoptimized model as follows:
where δ is an indicator function with value 1 if condition is true, and 0 otherwise. The smaller the p_value, the better the DNJ optimization is; and p_value = 0 means the DNJoptimized model performs better than any one of the 1,000 randomly permuted models. The p_value approximates the probability of observing log Pr(date${M}_{{R}_{i}}$) larger than (log Pr(dataM_{ DNJ }).
The optimized 1^{ st } order Markov chains for TFBS. The optimized arrangement of dependent positions within TFBS for the 1^{ st } order Markov model. N and N_{ D }are total number of motif positions and the number of positions significantly dependent, respectively.
ID#  Name  N  N _{ D }  Position order 

1  V$AP1_Q4_01  8  8  73120654 
2  V$AP1_Q6_01  9  8  23145786 
3  V$AP1_Q2_01  12  9  43567101191 
4  V$CDPCR1_01  10  9  342965781 
5  V$ATF_01  14  8  101091121312 
6  V$CHOP_01  13  10  5467910081112 
7  V$CDPCR3_01  15  10  30189134625 
8  V$CDPCR3HD_01  10  5  18927 
9  V$CREB_Q2_01  14  8  1111202398 
10  V$CREB_Q4_01  11  6  7618910 
11  V$CREB_Q3  6  4  4510 
12  V$CEBP_Q3  12  9  89564113210 
13  V$CEBPB_01  14  4  013113 
14  V$E2F_Q4_01  11  4  1870 
15  V$E2F_Q6_01  12  8  8370211910 
16  V$E2F1DP1_01  8  5  34067 
17  V$E2F1DP2_01  8  5  56734 
18  V$E2F4DP1_01  8  4  3401 
19  V$E2F4DP2_01  8  5  43710 
20  V$ETS_Q4  12  8  1125104301 
21  V$ELK1_02  14  4  101123 
22  V$FAC1_01  14  12  1261011134985107 
23  V$FOXD3_01  12  11  1387910112046 
24  V$FOXO1_02  14  11  89101276201113 
25  V$HNF4_Q6  9  7  4326817 
26  V$HNF1_Q6  18  15  311121481359061614210 
27  V$HNF3_Q6  13  11  1107534129028 
28  V$E2F1DP1RB_01  8  5  17304 
29  V$IRF7_01  18  13  320161517176814912 
30  V$LUN1_01  17  8  8910712111413 
31  V$MZF1_01  8  4  0145 
32  V$MYC_Q2  7  4  4531 
33  V$NFAT_Q4_01  10  4  6895 
34  V$NFKAPPAB_01  10  4  5792 
35  V$NKX22_01  10  6  986107 
36  V$OCT_Q6  11  10  82010539647 
37  V$PAX_Q6  11  10  10670931542 
38  V$PAX6_01  21  21  15171618681913113210207104951412 
39  V$PBX1_02  15  10  61220311113144 
40  V$RSRFC4_Q2  17  6  6701323 
41  V$RSRFC4_01  16  8  6791213128 
42  V$STAT5A_01  15  7  812113045 
43  V$SOX9_B1  14  9  113021153104 
44  V$SRY_01  7  4  4601 
45  V$SRY_02  12  4  13114 
46  V$STAT5A_02  24  16  712201516171819221211356923 
47  V$SP1_Q2_01  10  7  7380495 
48  V$SP1_Q4_01  13  13  0211126131098745 
49  V$SP1_Q6_01  10  10  3589074261 
50  V$USF_Q6_01  12  8  311457218 
51  V$XBP1_01  17  9  13534151110120 
52  V$ZID_01  13  8  67481210911 
53  I$DRI_01  10  7  6987012 
Results suggest that DNJ method performed remarkably well in optimizing the 1^{ st } order linear Markov chains, that in 49 out 53 cases, the DNJ optimized models were the best or close to the best (Figure 4a). The optimization for the 2^{ nd } order linear chains was slightly worse than that for the 1^{ st } order linear chains, partially because the DNJ method relies only on the pairwise dependencies between two single positions. Nevertheless, most of the DNJ optimized models were still close to the best [see Additional file 1 Figure 1a]. Although our DNJ method was designed for optimizing linear Markov chains, it still worked well in optimizing the 1^{ st } order circular Markov chains (Figure 4b). However, the DNJ method did not perform well in optimizing the 2^{ nd } order circular Markov chains [see Additional file 1 Figure 1b].
Theoretically, the optimal model can be found by exhaustively searching through all possible models. An exhaustive search is not always possible in practice, however, as the search space can be very large. The number of possible Markov chains is the factorial of the length of the Markov chain and dramatically increases as the length of chain increases. For example, the computational time for a motif of 15 bases (15! = 1.307674e + 12) can be practically unacceptable. Our DNJ method can deal with such long motifs because of its computational efficiency.
TFBS identification
One interesting application of our mixture model is TFBS identification. In this assessment, we used a couple of examples to show how OMiMa can improve prediction accuracy when there are position dependencies within a TFBS. We first tested our method on simulated data where the exact dependency structure of a TFBS is known. We tested whether OMiMa can capture such dependency and optimize the Markov model accordingly. Next, we tested our method on real motif data for AP1. In both examples, we compared OMiMa performance to PWM, PVLMM, and the 1^{ st } order Markov model (1stMM) with its motif positions in the natural order. PVLMM, run on Microsoft Windows, is based on the variable length Markov model (VLMM) [31, 32]. Except for its order and depth parameters, PVLMM was run under its default settings in all comparisons.
Simulated TFBS prediction
Simulation of two palindromic TFBS. Simulation of two palindromic TFBS A and B. The first 2 columns are the complementary positions of the palindromic TFBS. The 3^{ rd } and 4^{ th } columns are simulation parameters, which specify the probabilities of forming a complementary base pair. The last 2 columns are the pvalues of OMiMa's pairwise χ^{2} tests of position dependency for the simulated data.
Position pair  Complementary Prob.  pvalue  

1^{ st }  2^{ nd }  A  B  A  B 
0  11  0.99  0.90  4.88e88  3.84e63 
1  10  0.95  0.85  6.62e72  2.66e56 
2  9  0.90  0.75  3.84e69  2.25e35 
3  8  0.65  0.65  1.44e19  5.89e24 
4  7  0.50  0.50  2.00e07  3.05e03 
5  6  0.25  0.25  3.35e01  1.43e01 
Performance evaluation using simulated palindromic TFBS. Performance comparison of OMiMa (1L0) with PWM, 1stMM, and PVLMM (order 1 and depth 1) for predicting two simulated TFBS A and B. The performance was measured as the maximum Mc achieved by each model.
Motif  PWM  1stMM  PVLMM  OMiMa 

A  0.306  0.414  0.807  0.914 
B  0.253  0.428  0.647  0.794 
AP1 TFBS prediction
TFBS V$AP1_Q4_01 prediction. Comparison of OMiMa (1L0/1C0), PWM, 1stMM, and PVLMM (order 1 and depth 1) for AP1 TFBS prediction. The performance results are the average values of 10fold cross validation.
Model  Sn  Sp  Mc 

PWM  0.857  0.997  0.860 
1stMM  0.839  0.998  0.870 
PVLMM  0.789  0.999  0.847 
1L0  0.866  0.998  0.882 
1C0  0.874  0.998  0.884 
Donor splice site recognition
The transcription of most higher eukaryotic genes involves RNA splicing, in which primary transcripts become mature mRNA by removing introns. The donor or 5' splice sites and the acceptor or 3' splice sites on the boundaries of exons and introns provides critical signals for precise splicing. Therefore, splice site recognition has been widely used by gene finding tools such as GENESCAN [2] and GENIE [33] for gene prediction. The splicing process starts with U1 snRNP binding to the donor site via basepairing of U1 snRNA. The base pairing between U1 snRNA and the donor site, however, need not be perfectly complementary in all positions [34, 35]. Both experimental and computational evidence suggest that there are mutually dependent positions within the donor site: a mismatched pair of U1 snRNA and the donor sites at one position can be compensated for by a matching base pair at another position, and vice versa [2, 24, 36, 37]. Modeling such dependency structure within the donor site has been used to improve donor site prediction [2, 23, 24, 33]. We used two independent datasets of human donor sites to assess the performance of OMiMa in comparison with leading competitors.
Comparison with NNSplice and PVLMM
The test dataset of human donor splice sites (Reese data) was from [38]. This dataset has 6246 donor sites (1324 real and 4922 false) of length 15 bases from 7 to +8 around the conserved 'GT' dinucleotide. The dataset consists of a training set (containing $\frac{5}{6}$ of data) and a testing set (the remainder), which were previously used to assess the performance of NNSplice [33]. We used the same partitions for training and testing in the following comparisons.
Comparison OMiMa with NNSplice and PVLMM for donor site prediction. Comparing two OMiMa models (1L1 and 1C1) with NNSplice's neural network model and PVLMM (order 1 and depth 1) for donor splice site prediction.
Network  PVLMM  1L1  1C1  

Ac maximized  Ac  0.951  0.927  0.955  0.954 
Sn  0.904  0.793  0.928  0.947  
Sp  0.963  0.963  0.962  0.955  
Mc maximized  Mc  0.857  0.786  0.869  0.869 
Sn  0.942  0.889  0.938  0.952  
Sp  0.951  0.934  0.959  0.954 
Comparison with MEM and PVLMM
Given enough training data, we can use more complicated models than the 0–1 mixture model to improve prediction accuracy. In this evaluation, we test whether 0k mixture models can compete with the MEM on a much larger dataset. This large donor site dataset (Yeo data), used to assess performance of the MEM, was from [40]. The dataset, extracted from 1821 nonredundant human transcripts, has 8,415 real and 179,438 decoy sites in the training set, and 4,208 real and 89,717 in decoy sites in the testing set. Each real site has length 9 bases from 3 to +6 around the conserved 'GT' of donor splice sites recognized by U2 type spliceosome. The decoy sites are any other sequence segments in the exons and introns matching the pattern N_{3}GTN_{4}. So a decoy site can have the exactly same sequence as a real site. We applied this original training and testing sets to assess performance of OMiMa, where we used only loglikelihood ratio scoring. In addition, we ran a 3fold crossvalidation, in which the number of sites in new training and testing sets are roughly the same as those in the original ones [see Additional file 1 Table 2]. The top 4 models selected by AIC are 3C1, 3L1, 2C1 and 2L1, respectively, consistent with the ROC analysis. To find the top 4 submodels of PVLMM by ROC analysis, we used a series of Markov orders and/or depths (1 ≤ order ≤ 4 and order ≥ depth) to predict the same data sets. For convenience, we use notation "P:kd" to denote a PVLMM of order k and depth d. We adopt notation in [23] for submodels of MEM.
Briefly, the notation has the form "meK sD" or "meK xD", where "me" stands for maximum entropy; "K" is a number for the marginal order or the maximum length of an oligomer in consideration; "D" is the skip number or maximum skip number determining which positions the bases of an oligomer are from; "s" stands for skip number and "x" for the maximum skip. For example, model "me5s0" considers all marginal distributions of p(x_{ i }), p(x_{ i }, x_{i+1}), p(x_{ i }, x_{i+1}, x_{i+2}), p(x_{ i }, x_{i+1}, x_{i+2}, x_{i+3}), p(x_{ i }, x_{i+1}, x_{i+2}, x_{i+3}, x_{i+4}).
Comparison of OMiMa PVLMM and MEM for donor site prediction. Comparing OMiMa with PVLMM and MEM for donor splice site prediction. The table shows Matthews correlation efficients (Mc) of top 4 models from each model class. The splice site data and results of MEM models were from Yeo and Burge [23]. In the 3fold cross validation, the sample sizes for both training and testing sets are approximately equal to those of the original partition by Yeo and Burge.
MEM  PVLMM  OMiMa  

submodel  Mc  submodel  Mc (Org./3CV)  submodel  Mc (Org./3CV) 
me2x5  0.659  P:22  0.629/0.631  3C1  0.658/0.663 
me2x4  0.655  P:32  0.626/0.632  3L1  0.654/0.657 
me2x3  0.653  P:42  0.625/0.630  2C1  0.647/0.657 
me5s0  0.653  P:43  0.622/0.628  2L1  0.643/0.653 
Table 7
TFBS  0^{ th } chain  1^{ st } chain 

A  6  7411038501192 
B  5  2961018374110 
Biological explanation

2 5 1 3 4 3 7 6 5 4 7 6 2
We found that this position arrangement is supported by the following biological evidence of basepairing between U1 snRNA and the donor site: (a) 5'/3' compensation effect: a base pair at position 1 can prevent an aberrant splicing caused by a mismatched pair at position 5 [37]; (b) Adjacent basepair effect: a matching base pair at position 3 is rare in the absence of a matching base pair at position 4 [2, 41]; (c) A matching basepair at the nonconserved positions 6 and 7 can compensate for a mismatched pair at position 2 [42]. Interestingly, the model also arranged nonconserved positions (4, 5, 6, 7, 6, 7) together as it did for the other more conserved positions. Second, based on 8,415 real donor sites of the Yeo original training sites, the 0–3 mixture model was the best model. The optimized position order of its 3^{ rd } order chain was:
2 5 1 4 2 3 3
We can see that this model is consistent with the above evidence (a) and (b). In addition, it is well supported by experimentally verified position dependencies of position 4 on the positions 1, 2, 3 and 5 [37], and the computationally confirmed dependency of position 3 on position 2 due to the adjacent basepair effect [2].
Discussion
The prediction accuracy of a probabilistic model is largely determined by the effectiveness of the model in characterizing a biological motif. Since there is large variation of the signals embedded in biological motifs, an effective model can be as simple as a consensus sequence or as complex as a fully connected network model. In this paper, we described a mixture of Markov models to allow adjustment of model complexity for different motifs. Also, we extended the traditional linear chain Markov model to the circular chain Markov model, which can better represent position dependencies within a motif in some cases. We presented a novel method, DNJ, for efficiently optimizing position arrangement of a non 0^{ th } order Markov chain to incorporate most dependencies. We described methods for calculating the EDF and for selecting the best mixture Markov model. We implemented these methods in our motif finding OMiMa system, which is freely available. Finally, we demonstrated from different aspects in several examples that OMiMa can improve motif prediction accuracy in biological sequences.
The interaction of biological macromolecules, such as transcription factors bound to DNA sites, usually involves several highly dependent positions functioning as a unit. Many methods including Markov chains, Bayesian trees, and neutral networks have been used to model dependency structures within a motif. The Markov model is the simplest yet can be very powerful when it is optimized. Our results showed that the optimized Markov models performed better than the neural network model and PVLMM, and comparably with MEM for splice site prediction. The optimized Markov model can incorporate both local and nonlocal dependencies into the model, which enables it to compete with tree or network models in predicting short biological motifs. We also showed that the optimized Markov model can be an excellent motif predictor. Moreover, it is also computationally efficient due to its simplicity.
Model complexity, measured by parameter number, is an important issue in motif modeling. The more complex a model, the more data are needed for adequate training. For many biological motifs, however, the number of known (experimentally determined) sites is small. This limits the usage of complex models, such as higher order Markov models, Bayesian trees, network models or MEM, even though these models in some cases can perform better than the simpler models given enough training data. For a standard Markov model, the number of its parameters increases exponentially as its Markov order increases. Without sufficient training data, it is difficult to accurately estimate all model parameters, even using more robust methods (e.g. interpolated Markov chains [43, 44]). As a result, lack of sufficient training data often makes it impractical to train a higher order Markov model. On the other hand, a low order Markov model may perform poorly by failing to incorporate more distant dependencies. Several motif models and methods have been developed to address this issue. One of these models is the variable length Markov model (VLMM), whose Markov orders (also called context lengths) can vary among different positions. VLMM can effectively reduce Markov model complexity when the variation of actual context lengths is large. VLMM, however, is not the best choice to incorporate longrange dependencies. The position optimized Markov model (POMM) [21] is able to incorporate important distant dependencies without increasing Markov chain order. However, the effectiveness of this model largely depends on the optimization routine.
More recently, Zhao et al. [24] described the PVLMM in an attempt to combine advantages of both VLMM and POMM. The disadvantage of PVLMM is that the number of possible permutations is the factorial of motif length, which makes it more computationally expensive. In addition, the random permutation method used by PVLMM for optimization is more likely to overfit the model, e.g., incorporating nonsignificant dependencies into the PVLMM model that can reduce its prediction power. The optimized mixture of Markov models we presented here tries to inherit advantages of these existing models while avoiding their disadvantages. In OMiMa, we replace VLMM with a mixture of several lower order Markov models, which are subsequently optimized to account for longrange dependencies.
In comparison with other leading methods, OMiMa can incorporate more than the NNSplice's pairwise dependencies; OMiMa avoids model overfitting better than the PVLMM; and OMiMa requires smaller training samples than the MEM. These are primarily reasons that OMiMa showed superior performance, in terms of prediction accuracy, required size of training data or computational time, over other leadingmethods in our results.
With any model selection procedure, the possibility of choosing a model that drastically over or underfits is a concern. OMiMa employs AIC and BIC, two standard criteria, that are widely used because they tend to avoid extreme over or underfitting. Both have theoretical support [27, 28]. In our application, neither criterion worked well when using the DF (results not shown); but both, particularly AIC, performed well when using EDF. We found that models selected by AIC using EDF were consistent with models selected by crossvalidation and by ROC analysis.
Our OMiMa approach has two features that can be limitations when the size of the training data is small. First, the chisquare test that partitions motif positions into those with dependencies and those without dependencies will, like any statistical test, make mistakes, and its statistical power to detect dependencies will suffer with small training samples. Although the test will not always provide a correct partition, our approach should adapt to strong or weak dependencies overall and improve prediction when dependencies are strong. In addition, weakly dependent positions mistakenly placed in the set with no dependencies are often adequately modeled by a 0^{ th } order chain, whereas independent positions mistakenly assigned to the set with dependencies will be placed by the DNJ algorithm in locations with the least impact on the k^{ th } order chain. Second, the EDF that we used in model selection is an estimate based on the training data. For degenerate sites, the estimate should be accurate with even small training samples; whereas for conserved sites a larger training sample might reveal additional bases and change the EDF. Still, such additions should be minimal and would generally induce small changes in the EDF, so we expect little impact on model selection. Any methods that employ chisquare techniques to test for dependent sites face similar limitations. Nevertheless, OMiMa with its relatively small parameter space should adapt to small training datasets better than many competitors. Of course, any motif finding algorithm would do better with larger training samples.
OMiMa places no limit on the length of sequences that it can scan, and it could be used to find TFBS in any sequenced organism as long as a training motif set is available. The larger the genome evaluated, the more false positives are likely to be declared. Although OMiMa's prediction accuracy will help, other approaches to reducing false positives will be needed. Crossspecies comparisons and relative location compared to transcription start sites have been used to reduce false positives and could be used with OMiMa too. Furthermore, OMiMa's ability to accurately and quickly identify splice sites should be easy to incorporate into probabilistic geneprediction programs where correct prediction of splice sites is critical.
Conclusion
Our optimized mixture of Markov models represents an alternative to the existing methods for modeling dependent structures within a biological motif. Unlike existing methods, our model is conceptually simple and effective, which has advantages in a large scale motif prediction. In particular, with its ability to minimize model complexity, our method can work effectively even with limited training data. The optimized mixture of Markov models is implemented in our computational tool OMiMa, which can use a variety of mixture models for motif prediction. OMiMa, in which most parameters are configurable, is freely available to all users.
Declarations
Acknowledgements
We thank Drs Bruce Weir and Jeffrey Thorne for critically reading the manuscript, and Drs Clarice Weinberg and Joseph Nevins for helpful comments. This research was supported by Intramural Research Programs of the NIH, National Institute of Environmental Health Sciences.
Authors’ Affiliations
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