• Top
• Abstract
• Background
• Related work
• Methods
• Results and discussion
• Conclusion
• Authors' contributions
• Acknowledgements
• References

Research

Exact p-value calculation for heterotypic clusters of regulatory motifs and its application in computational annotation of cis-regulatory modules

Valentina Boeva^1,2 , Julien Clément³ , Mireille Régnier² , Mikhail A Roytberg^4,5 and Vsevolod J Makeev^1,6

¹Institute of Genetics and Selection of Industrial Microorganisms, GosNIIGenetika, 117545 Moscow, Russia

²MIGEC, INRIA Rocquencourt, 78153 Le Chesnay, France

³GREYC, CNRS UMR 6072, Laboratoire d'informatique, 14032 Caen, France

⁴Institute of Mathematical Problems of Biology, Russian Academy of Sciences, Puschino, Moscow Region, Russia

⁵Puschino State University, Puschino, Moscow Region, Russia

⁶Engelhardt Institute of Molecular Biology, Russian Academy of Sciences, Moscow, Russia

author email corresponding author email

Algorithms for Molecular Biology 2007, 2:13doi:10.1186/1748-7188-2-13

The electronic version of this article is the complete one and can be found online at: http://www.almob.org/content/2/1/13

Received:	13 July 2007
Accepted:	10 October 2007
Published:	10 October 2007

© 2007 Boeva et al; licensee BioMed Central Ltd.
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Background

cis-Regulatory modules (CRMs) of eukaryotic genes often contain multiple binding sites for transcription factors. The phenomenon that binding sites form clusters in CRMs is exploited in many algorithms to locate CRMs in a genome. This gives rise to the problem of calculating the statistical significance of the event that multiple sites, recognized by different factors, would be found simultaneously in a text of a fixed length. The main difficulty comes from overlapping occurrences of motifs. So far, no tools have been developed allowing the computation of p-values for simultaneous occurrences of different motifs which can overlap.

Results

We developed and implemented an algorithm computing the p-value that s different motifs occur respectively k₁, ..., k_s or more times, possibly overlapping, in a random text. Motifs can be represented with a majority of popular motif models, but in all cases, without indels. Zero or first order Markov chains can be adopted as a model for the random text. The computational tool was tested on the set of cis-regulatory modules involved in D. melanogaster early development, for which there exists an annotation of binding sites for transcription factors. Our test allowed us to correctly identify transcription factors cooperatively/competitively binding to DNA.

Method

The algorithm that precisely computes the probability of simultaneous motif occurrences is inspired by the Aho-Corasick automaton and employs a prefix tree together with a transition function. The algorithm runs with the O(n|Σ|(m|| + K|σ|^K) ∏_i k_i) time complexity, where n is the length of the text, |Σ| is the alphabet size, m is the maximal motif length, || is the total number of words in motifs, K is the order of Markov model, and k_i is the number of occurrences of the ith motif.

Conclusion

The primary objective of the program is to assess the likelihood that a given DNA segment is CRM regulated with a known set of regulatory factors. In addition, the program can also be used to select the appropriate threshold for PWM scanning. Another application is assessing similarity of different motifs.

Availability

Project web page, stand-alone version and documentation can be found at http://bioinform.genetika.ru/AhoPro/ webcite

Background

During the past few years, a number of computational tools have been designed [1-3] for locating potential transcription factor binding sites (TFBSs) in nucleotide sequences, e.g., in compilations of sequences upstream of putative co-regulated genes. In parallel, experimental approaches were developed [4], which allowed identification of binding motifs for many different transcription factors. Experimental [5] and bioinformatical [6] studies demonstrated that sequences of regulatory DNA that bind transcription factors can exhibit many different types of architecture. In eukaryotes TFBSs found in DNA sequences often form rather dense clusters: this was demonstrated both by experimental [5,7] and computational [8,9] methods. Such clusters can contain sites binding the same factor or several different factors [10]. The cis-regulatory module (CRM) in this case contains respectively homotypic or heterotypic clusters of motifs specifically recognized by binding proteins [11].

The particular arrangement of motifs in a homotypic or heterotypic cluster is not random, and it is commonly accepted, that the motif arrangement within a CRM is important for its functionality [12-20]. Bioinformatics studies indicate that antagonistic factors often bind to overlapping sites [21] whereas synergetic factors are often positioned within a fixed distance [20], often close to the multiple of 10.2 bp, the DNA double-helix pitch value [21].

Non-random arrangements of TFBSs within regulatory segments of DNA sequences are exploited in several TFBS identification tools, and it was observed that cooperativity-based discrimination of TFBSs surpasses the performance of models for individual TFBSs [22].

On observing a cluster of TFBSs in some genome segment one can calculate the probability of observing similar site arrangements in a random sequence. This idea of evaluating the statistical significance of heterotypic clusters of sites was implemented in many programs including ClusterDraw [23], ModuleSearcher [24], MCAST [25], eCIS-ANALYST [26], Cister [27], Cluster-Buster [28] and TargetExplorer [29]. At the moment, such programs use empirical procedures like motif counting in biological and simulated sequences to assess the significance of observed site clustering. But it is highly desirable to have a good statistical measure of site clustering, and we believe that the best measure is the p-value of obtaining the observed cluster by chance in a random sequence of a Markov or Bernoulli (common name for Markov chain of order 0) type. In the case of heterotypic clusters one needs to take into account possible overlapping occurrences of different motifs, a problem that was considered difficult until now [30]. In the case of homotypic clusters, an approximate statistical scoring function was constructed [8,31]; this approach has been implemented in algorithms like FLYENHANCER [32], SCORE [33], and CLUSTER [34]. However, this approximation performs poorly for highly overlapping TFBSs. One cannot ignore site overlapping if the motifs are fuzzy (highly degenerate), which is often the case for so-called "shadow sites" [31]. In the case of heterotypic clusters, competing factors can bind even to very well determined motifs that overlap.

Representation of protein binding motifs in nucleotide sequences

Experimental methods on protein binding to DNA usually locate some DNA segment, or word in DNA text, as a probable binding target. Proteins can bind to similar DNA words [4], the whole assembly of which can be called a motif. The simplest motif representation is the enumeration of sequences that can be bound by a transcription factor (TF) [35]. Sometimes, information about binding sites can be found in SELEX [36,37] or Protein Binding Microarray (PBM) experiments [38]. However, it is possible that such experiments do not give the exhaustive list of sequences of binding sites, so one needs to expand the list of putative binding sites using an appropriate criterion, which brings about the problem of the generalization of several known examples.

For instance, several words aligned with mismatches, can be generalized to IUPAC string (like RSTGACTNMNW for AP-1 binding sites [39]) by disregarding correlated substitutions in different motif positions [40]. Another example of generalization is the set of words that can deviate from a consensus word for less than a given number of mismatches.

The most popular way to represent binding sites is a Position Weight Matrix (PWM), which is also called position-specific weight matrix (PSWM) or position-specific scoring matrix (PSSM) [41]. For a text with length D over an alphabet Σ with |Σ| symbols, a PWM is a |Σ| × D matrix: each row corresponding to a symbol of the alphabet Σ, and each column to a position in the motif. For DNA texts, one has Σ = {A, C, G, T}. The PWM score is defined as , where i represents a position in the D-substring, ω(i) the symbol at position i in the substring, and m_{α, i} the score in row α, column i of the matrix. So, given a cutoff value, one gets a list of D-sequences that score higher than this cutoff; thus representing possible DNA binding sites for the protein.

Any of the three motif representations above can be converted to a list of words. The same is true for many other representations of motifs. In this study, we consider only the motifs that can be represented as a set of words.

P-value for clusters of motif occurrences, problem formulation

The objective of this work is to develop a statistical criterion to assess clustering of TFBS. Intuitively, a TFBS cluster is a DNA segment simultaneously containing "too many" TFBSs for given factor proteins; such a segment can often operate as a CRM regulated by these TFs. From a formal point of view, the problem we address here is as follows. Let s sets of words be given. Typically, each set _i is associated to a TF motif. Given a s-tuple of integers (k₁, ..., k_s), we compute the corresponding p-value, that is the probability to find at least k_i occurrences of words from each set _i in a random text of size n. We assume that the texts where motifs are searched are randomly generated by a Bernoulli process or a Markov model of order K. If (k₁, ..., k_s) occurrences of motifs are found in a DNA segment, the p-value can be used to infer if such numbers of occurrences could be found by chance.

Related work

Most previous works address counting problems for one set of several words . In contrast, in this paper we deal with a separate counting for several sets of several words , each set _j represents one TFBS motif.

All methods of solving the problem of p-value calculations for multiple occurrences of words from a set study some basic languages. Let L_n (; k) be the set of texts of length n containing at least k occurrences of . The desired p-value would therefore be the probability P (L_n (; k)). Let be the set of texts of all lengths that contain exactly k words of , the last one occurring as a suffix [42]. For any H_j in , let be the subset of where H_j is a suffix. One observes that a text contains at least k occurrences if and only if it admits a prefix in . One defines (p) as the probability that a text of size p be in set . If no word in is a subword of another word in , the probability P (L_n (; k)) to find at least k occurrences of words from in a random text of length n satisfies

Therefore, one tries to compute the sequence of ( (p)) values.

Linear induction

In the first class of methods [43-46], one computes, implicitly or explicitly, probabilities P (L_n (; k)) up to a given text length n. Such methods are intrinsically linear in n. In [43-46] one relies on a recurrence relation on (n) that extends the one originally given in [47]. Typically, one step will cost O (||m), where is a set of words of length m and || is its cardinality. Time complexity is O (n||m) and, relying on a combinatorial property, [44] achieves optimal space complexity O (|| log ||m). However the authors of [44] do not consider several motifs occurrences and restrict themselves to the Bernoulli model. The authors of [43] consider the Markov model, still using one motif for TFBS.

Algebraic Formulae

In a second class of methods [47-52], a preprocessing computes generating functions

In a second step, probabilities P (L_n (; k)) are either extracted from the generating function or approximated.

In [49,53], (z) are the solutions of a system of equations. To derive these equations, the authors build an automaton that recognizes these languages (one can prove that they are regular).

A language approach [50] or an induction [48] leads to a formal expression that depends on the words overlaps. The main drawback is that these methods need to compute the determinant of a matrix of polynomials with a huge dimension, e.g. O (||). This O (||²) symbolic computation may be more expensive than the extraction step or the linear computation above, that involve arithmetic operations on real numbers.

When the preprocessing step is achievable, the extraction step is amenable to the solution of a linear recurrence of degree m||; therefore, its complexity is O (m||n) and a classical optimization yields O (m|| log n). There exists some good implementations that are numerically stable. One may cite the REGEXPCOUNT [54] or EXCEP [55] programs that rely on Fast Fourier Transform.

Finally, approximations are available, the computation of which is constant with respect to n, but not to . One approach is the compound Poisson approximation [56], but this approximation is not precise enough [57]. Asymptotic results can also be derived from the algebraic formulae above [44,58], not needing an explicit expression for (z), and therefore avoiding the expensive determinant computation. Time complexity, typically, is the one for computing all possible overlaps, that is approximately O (||²). This yields extremely precise results when the expectation of the number of occurrences, nP (H) is very small [59] or close to 1 [51] (the case studied the most often). Case nP (H) ~2 is achieved in [60]. Nevertheless, extension to larger values of k or multioccurrences and multisets is still open.

Methods

Here we consider in detail the approach we suggest.

A motif assigned to a TF is a finite set of words = (H₁, ..., H_r) where each word represents one putative TF binding site in DNA. Note that words in motif can generally be of different lengths. However, no word from can contain another word from as a substring. We consider, as an occurrence of motif in text T, any occurrence of any word _j ∈ in T. Below all texts and words in motifs are sequences on a given alphabet Σ.

Let () be s different motifs. Our objective is to calculate the probability (p-value) that motifs () have respectively at least (k₁, ..., k_s) possibly overlapping occurrences in a random text T_n.

To be more precise, there is a probability distribution defined on the set Σⁿ of all texts of length n in the alphabet Σ; the most widely used models are random Bernoulli trials and a Markov model of order K. Denote as L_n (; k₁, ..., k_s) the set of all texts of length n containing at least k_i possibly overlapping occurrences of each motif _i; i = 1, ..., s. Then the desired p-value is the probability P (L_n (; k₁, ..., k_s)) of the set L_n (; k₁, ..., k_s) with respect to the given probability distribution on Σⁿ.

Our approach to the calculation of this p-value is similar to that published in [61], which was used there to calculate seed sensitivity in local alignment search. The approach exploits the fact that the algorithm of Aho and Corasick [62] can be modified to efficiently determine whether a given text belongs to the set L_n (; k₁, ..., k_s) or not. Ideas published in [61] and [62] can be adopted to compute the probability P (L_n (; k₁, ..., k_s)) that the random text T_n ∈ Σⁿ belongs to the set L_n (; k₁, ..., k_s).

We start from the simplest case of one motif for which we calculate the probability P (L_n (; 1)) that text T_n contains at least one occurrence of the motif with respect to a Bernoulli probability distribution. More complicated cases (arbitrary number of occurrences; arbitrary number of motifs; Markov distribution) will be discussed in the following sections.

Construction of Aho-Corasick traversal

Aho and Corasick [62] have proposed the algorithm determining if a given text T contains an occurrence of a word from a given set . The basic data structure is a prefix tree which is a variant of the classical trie [42] that may be built on the set of words . Let denote the set of prefixes of these words. In the following, we identify a word q ∈ with node Node (q) at the end of the branch labeled by q. In particular, the root is identified with the empty string ε. The length of a prefix is the depth of Node (q).

The classic Aho-Corasick algorithm is a tree traversal determined by a transition function defined as follows. For any pair (p, a) in × Σ, δ (p, a) is the largest suffix of concatenation pa that belongs to . Remark that δ (p, a) = pa iff pa ∈ .

Given a text T read from left to right, let T [i] denote the letter of T at position i. Let q_i be the largest suffix in text T[1] ⋯ T [i] that belongs to . The sequence of nodes visited during the traversal are defined by words q_i that satisfy the inductive relationship

∀i ≥ 0, q_i+1 = δ (q_i, T [i + 1]),

with the initial condition q₀ = ε.

Example: Let be the set {AAA, AAC, ACA, ACA, CCT}. The corresponding tree is depicted in Figure 1. Values of δ function are given in Table 1. Aho-Corasick traversal of tree according to text T = 'ATGCCAACCTT' produces the following sequence of nodes {q_i}_{i ≥ 1} in (the numbers of corresponding nodes in Figure 1 are shown in square brackets): A[1], ε[0], ε[0], C[2], CC[5], A[1], AA[3], AAC[7], ACC[9], CCT[10], ε[0].

and transition function δ can be efficiently constructed with an algorithm proposed by Aho and Corasick [62]. Both time and space of the algorithm is proportional to the sum of lengths of all words from .

The combination of tree and transition function δ allows solving numerous pattern matching problems: search of the first occurrence of a word from a given set, search of all occurrences, word counting, etc.

Bernoulli text model. Probability to find at least one occurrence of a single motif

In this section we consider the simplest case. One computes the p-value for a single motif in a text T_n of length n, assuming that T_n is generated by independent Bernoulli random trials over alphabet Σ. The algorithm computes probabilities P (L_n (; 1)) by induction on n.

To describe the algorithm we divide the set Σⁱ of all texts T_i of length i into classes that do and do not contain occurrences of .

Definition 1 A text T_i belongs to class C_i (0; q) iff

1. Length of T_i is i,

2. T_i does not contain words from ,

3. A traversal AC (, T_i) ends at node q.

A text T_i belongs to class G_i (1) iff

(i) Length of T_i is i,

(ii) T_i does contain at least one occurrence of a word from .

For a given number i larger than m, the union for classes C_i (0; q), where q is in and the class G_i (1) form a partition of the set Σⁱ of all texts of length i, i.e., any texts of length i belongs either to a class C_i (0; q) for some q in , or to a class G_i (1). Indeed, condition 3. means that the largest suffix of T_i in is q. It follows from condition 2. that classes C_i (q; 0) are empty if q is in . A text T_i of length i is in G_i (1) if and only if a node of was visited during the traversal.

Let P (C_n (0; q)) and P (G_n (1)) denote probabilities that a text T_n belongs to class C_n (0; q) and G_n (1), respectively. Then, L_n (; 1) = G_n (1); therefore the desired p-value P (L_n (; 1)) is equal to P (G_n (1)).

The algorithm calculates probabilities P (C_i (0; q)) and P (G_i (1)) using induction on length i. For i = 0, these probabilities obviously comply with: P (C₀ (0; ε)) = 1; P (C₀ (0; q)) = 0, for any q ≠ ε; P (G₀ (1)) = 0.

The values of P (C_i+1 (0; q)) and P (G_i+1 (1)) are calculated using values of P (C_i (0; q)) and P (G_i (1)). Therefore, the needed space is proportional to the size of (see section Extensions and complexity below).

Calculation of values P (C_i+1 (0; q)) and P (G_i+1 (1)) is based on the following observations. Let U be a set of texts of the same length over the alphabet Σ, P (U) the probability of U in the Bernoulli model and a a character in Σ. Let U·a be the set of all possible concatenations, i.e., U·a = {xa|x ∈ U}. And in the case of the Bernoulli model

P (U·a) = P (U) P (a).(1)

Then the following relations hold for any i ∈ {1, ..., n - 1} and Σ:

(i) if the text T_i contains a word from then all its concatenations with characters from Σ would contain a word from ; i.e.,

G_i (1)·a ⊂ G_i+1 (1).(2)

(ii) if the text T_i does not contain a word from and belongs to C_i+1 (0; q), i.e., ends with q ∈ , then its concatenation T_i·a belongs to the class determined by the result of the Aho-Corasick transition function δ (q, a); i.e.,

if δ (q, a) ∈ ,   then C_i (0; q)·a ⊂ C_i+1 (0; δ (q, a))(3)

otherwise   C_i (0; q) ⊂ G_i+1 (1).(4)

Remembering that classes C_i (0; q) for different q and G_i (1) form a partition of Σⁱ, we obtain the following relation for the texts containing words from :

(5)

Similarly, classes of texts that do not contain words from satisfy

(6)

Classes C_i (0; q) for different q in and G_i (1) form a partition of Σⁱ; classes C_i (0; q) are empty if q is in . Relations (5) and (6) with the help of (1) yield the recursive expressions for probabilities P (C_i+i (0; q)) and P (G_i+1 (1)) in the Bernoulli case:

(7)

(8)

The run-time for each step of the computation of C_i+1 (0; q) and G_i+1 (1) is O (||·|Σ|); therefore the total time of all n stages of p-value computation is O (||·|Σ|·n).

The approach described in this section can be readily extended to the case of multiple occurrences of motif . The detailed procedure can be found in Additional file 1.

Bernoulli text model. Probability to find multiple occurrences of multiple motifs

DNA transcription is usually regulated with several factors simultaneously interacting with DNA and specifically recognizing different DNA sites. Individual regulatory segment of DNA can contain many binding sites for several factors, often substantially overlapping with each other [5]. This brings about a problem of studying of co-occurring motifs.

Let () be s different motifs. Our objective is to calculate the probability that motifs () have respectively at least (k₁, ..., k_s) possibly overlapping occurrences in the random text T_n of the length n. This p-value is the probability P (L_n (; k₁, ..., k_s)) to obtain text T_n belonging to the set of texts L_n (; k₁, ..., k_s). In this section, we will suppose that the probability of each text is given by Bernoulli model. The Markov case will be considered in the next subsection. The recursion for multiple occurrences of multiple motifs obtained here is rather tricky. Therefore we suggest the reader to see Additional file 1 where we describe the recursion for the simpler case of multiple occurrences of a single motif

Let us consider the union of individual motifs . It contains all words that belong to any of motifs _i. The tree is constructed for the overall set , its nodes contain all possible prefixes of all motifs from (). A node of the tree q ∈ can belong to some motif _k or simultaneously to several different motifs from {_j}_1≤j≤s. Let each node q ∈ be marked with numbers j of motifs _j to which it belongs. Nodes, corresponding to proper prefixes of , remain unmarked. The transition function is defined as it was defined in the case of a single motif for the unified motif .

All texts T_n of length n are classified into classes depending on occurrences of different _j. In this case it is difficult to introduce the target class G, since when the target number of occurrences k_i is attained for some motif _i, the corresponding value k_j may not yet be attained for another motif _j. Therefore we need to introduce the occurrence index of a set of motifs.

Definition 2 Let the target number of occurrences of motif _i be k_i. Then, the occurrence index (l₁, ..., l_s) of a set of motifs () in the text T_n containing l_i possibly overlapping occurrences of each _i is an s-vector the ith component of which can be calculated as follows:

(9)

Definition 3 A text T_i belongs to class C_i (λ₁, ..., λ_s; q), 0 ≤ λ_i ≤ k_i iff

1. Length of T_i equals i,

2. The occurrence index of motifs () in text T_i is equal to (λ₁, ..., λ_s),

3. A traversal AC (, T_i) ends in node q.

A text T_i belongs to class G_i (k₁, ..., k_s) if it belongs to the union of classes

(10)

The desired p-value P (L_n (; k₁, ..., k_s)) is equal to P (G_n (k₁, ..., k_s)). The value is calculated iteratively. Again, we have a sum over all possible tree nodes q and symbols a. Now, q', the image of the transition function δ (q, a) can belong simultaneously to several motifs {_j}_1≤j≤s. Thus, the resulting probability P (C_i+1 (λ₁, ..., λ_s; q')) that text T_i+1 belongs to class C_i+1 (λ₁, ..., λ_s; q') calculates as

(11)

where the summation in the second sum is performed over all allowed s-tuples of indexes (r₁, ..., r_s) which together make the set of s-tuples J. A s-tuple of indexes (r₁, ..., r_s) belongs to J if it complies with the following conditions:

1. if q' ∉ _j then r_j = λ_j,

2. if q' _j and λ_j <k_j then r_j = λ_j - 1,

3. if q' ∈ _j and λ_j = k_j then r_j = k_j or r_j = k_j - 1.

Implementation details

Our basic data structure is the prefix tree; we use its standard representation [42] [see also Additional files 2 and 3 for Tree construction from PWM motif representation]. Each tree node q ∈ is supplied with several additional variables.

At stage (i + 1) of probability computation the values P (C_i+1 (λ₁, ..., λ_s; q)) become computed from the values P (C_i (λ₁, ..., λ_s; q)) obtained at the previous stage of induction. Therefore, at stage (i + 1), one no longer needs the values calculated at stage (i - 1). Thus, each node is supplied with two k₁ × ⋯ × k_s-arrays of real values C₀ and C₁ for storing P (C_i (λ₁, ..., λ_s; q)) and P (C_i+1 (λ₁, ..., λ_s; q)) for different λ_j. C₀ is used to store probabilities for even text lengths while C₁ for odd.

In implementation the calculation of values P (C_i+1 (λ₁, ..., λ_s; q')) from P (C_i (λ₁, ..., λ_s; q)) for all q', q ∈ and (λ₁, ..., λ_s): 0 ≤ λ_j ≤ k_j, 1 ≤ j ≤ s, is performed in the parallel way. Initially we set all the values P (C_i+1 (λ₁, ..., λ_s; q')) to 0. Then we look over all tuples (r₁, ..., r_s; q), where q ∈ and (r₁, ..., r_s): 0 ≤ r_j ≤ k_j, 1 ≤ j ≤ s. For each tuple (r₁, ..., r_s; q) and all letters a ∈ Σ we find the prefix q' = δ (q, a) and the value P (C_i (r₁, ..., r_s; q))·p(a). Then we add P (C_i (r₁, ..., r_s; q))·p(a) to the value P (C_i+1 (λ₁, ..., λ_s; q')) where (λ₁, ..., λ_s; q') meet the conditions inverse to those of formula (11):

1. if q' ∉ _j then λ_j = r_j,

2. if q' ∈ _j and r_j <k_j then λ_j = r_j + 1,

3. if q' ∈ _j and r_j = k_j then λ_j = r_j.

At the stage i = n the desired p-value is the sum

Markov text model

Tree approach and the recursion (11) can be readily extended to calculate p-values of motif occurrences in random texts generated by the Markov model of order K. Given the order K of the Markov model, the probability p(a) in (11) depends on K previous letters. Thus, if the length |q| of the prefix q is less than K, one cannot calculate p(a) knowing only the prefix q. To overcome this we divide each class C_i (r₁, ..., r_s; q), where |q| = d <min (K, i) into subclasses C_i (r₁, ..., r_s; q, w); each subclass corresponds to a word w of length min (K, i) - d. Then, a text T_i of length i belongs to class C_i (r₁, ..., r_s; q, w) if the suffix of T_i of length min (K, i) equals to w·q.

Figure 2 gives an example for Markov model of order K = 1. The tree is constructed for the set = {AAA, AAC, ACA, ACC, CCT}. The text T = ATGCCAACCTT produces the following sequence of nodes {q_i}_i≥1 (the numbers of the corresponding nodes in Figure 2 are shown in square brackets): A[4], (ε, T)[3], (ε, G)[2], C[5], CC[8], A[4], AA[6], AAC[10], ACC[12], CCT[13], (ε, T)[3].

The recursive equations for probabilities P (L_n (; 1)), P (L_n (; k)), and P (L_n (; k₁, ..., k_s)) can be obtained from the corresponding formulae (7-8), (11–13) and (16) by substituting probabilities p(a) with p(a|t[1] ⋯ t [K]), where

The Markov extension is currently implemented for K = 1.

Complexity

To resume, the computation of P (L_n (; k)) for one set requires a computation of for i ≤ n. For each iteration, the time complexity is O (k|| |Σ|), where |Σ| is the size of the alphabet. One traverses the tree n times. As || is upper bounded by (m||), where m is the maximal length of word in , this yields the overall O (nkm|||Σ|) time complexity and a O (km||) space complexity.

When several sets are involved, the number of nodes in the tree becomes O (m||) with m equal to the maximal length of word in . Additional memory in each node is ∏_i k_i. Therefore, the time complexity is O (nm|Σ|∏_i k_i||) and the space complexity is O (m ∏_i k_i ||). In the Markov model of order K, one memorizes |Σ|^{K - d} predecessors for each node at depth d, 0 = d <K. In other words, the number of classes becomes (m|| + K|Σ|^K). Therefore, the space memory is O ((m|| + K |Σ|^K) ∏_i k_i) and the running time is O (n|Σ|(m|| + K |Σ|^K )∏_i k_i). This additive increment compares favorably to simple induction methods [45,53] that introduce a multiplicative O (K|Σ|^K) factor in time and space complexity for the Markov(K) model.

Results and discussion

We developed an algorithm for precise calculation of the p-value for multiple occurrences of multiple motifs with possible overlaps. The running time is linear in the text length and depends on the alphabet size, the maximal motif length, the number of words in the motifs, and the number of occurrences of each motif. The algorithm was implemented in the AHOPRO software. Below we give examples of how p-values can be used for studying gene regulation in silico, particularly for selecting optimal cutoff values for motifs represented by PWMs. In the subsection 'Comparison with simulation and approximation methods' we compare our p-value computations with the result of Monte Carlo simulations and the Poisson approximation. Our results confirm the accuracy of our algorithm and show in what cases the Poisson approximation [8,11] cannot be employed. In the subsection 'Optimal cutoffs', we apply AHOPRO to choose an appropriate cutoff score for Position Weights Matrices. In the subsection 'Assessment of gene regulation', we show how AHOPRO can be used for studying regulatory regions containing heterotypic clusters of TFBSs to distinguish genes that are regulated by given transcription factors from those that are not.

As a model example, we use in this section data published in [34] on regulatory clusters in D. melanogaster. This compilation includes information on

(i) known binding motifs for transcription factors,

(ii) known CRM regions, and

(iii) known regulatory interactions.

Comparison with simulation and approximation methods

In our first example we use the even-skipped stripe 2 enhancer (eve2) [63] of length 728 bp that is known to contain binding sites for TFs bicoid, kruppel and hunchback. Below we compare p-values calculated by the AHOPRO program and those calculated using compound Poisson approximation with p-values computed through Monte Carlo simulations.

AhoPro and Monte Carlo comparisons

Table 2 displays results of comparison of p-values calculated with AHOPRO and with Monte Carlo simulation assuming the Bernoulli model M0. The corresponding results for the first order Markov model M1 are displayed in Table 3. Letters probabilities for M0 and the transition matrix for M1 were evaluated from eve2 sequence. We used the PWM cutoff values taken from [34], i.e., 5.3, 5.0, and 6.2 for bicoid, kruppel, and hunchback respectively. With these threshold values in sequence eve2 we have found 3, 4, and 2 occurrences of motifs of each type respectively. In Tables 2 and 3 we listed the p-values, i.e, the probabilities to find no less than the observed number of occurrences of motifs in a random text of length L, where L is the length of eve2 enhancer. The number of Monte Carlo simulations was set to 10⁶ everywhere, except for the triplet (bcd&kr&hb), where we did 10⁷ simulations. The probability to find the observed number of occurrences of (bcd&kr&hb) simultaneously in the same simulated sequence is extremely low; thus we increased the number of simulations so that the product of the probability by the number of simulations be greater than 1.

The results of comparison of the AHOPRO computation with those obtained from simulated random sequences presented in Tables 2 and 3 confirm the accuracy of our algorithm.

Poisson approximation

In practical application, compound Poisson distribution [64] is widely used to assess p-values of multiple motif occurrences [2,8,34,65]. Here we apply it to compute the probability to observe the given number of motif occurrences when the probabilities of individual words are calculated adopting the M0 or M1 models described above. The results of the comparison given in corresponding columns in Tables 2 and 3 show that the p-value calculated using Poisson approximation can be significantly underestimated. This happens most probably because the Poisson approximation does not take into account possible overlaps between motif occurrences and considers motif occurrences as independent. The error increases when the p-value is calculated for simultaneous occurrences of several factors, as it is done in the last two rows. In this case, the Poisson approximation p-value for a combination of several TFs is calculated as a product of p-values calculated independently for each TF. Actually, the motif occurrences can overlap especially when the motifs resemble each other, thus there is no independence, which brings about the error.

Optimal cutoffs

Below, we use AHOPRO to determine the optimal cutoff values for PWMs of regulatory factors, given the sequences of regulatory region assumedly interacting with the factors. The distribution of occurrences of TF binding sites in corresponding experimentally confirmed regulatory regions is strongly biased [34]. In CRMs binding sites often tend to occur in clusters, which is not the case for random sequences.

Different cutoff values correspond to different numbers of putative binding sites of different quality. The higher the cutoff value, the closer the motif occurrences are to the consensus and the smaller the number of motif occurrences. Therefore, for a given factor it is reasonable to select a cutoff value that minimizes the probability of finding in the random sequence the number of motif occurrences observed in the sequence of the regulatory region.

As an example, we considered again transcription factors bicoid, kruppel, which are known to regulate the even-skipped stripe 2 (eve2) enhancer. To select the optimal cutoff value we used the following procedure: first, in the sequence of eve2 we counted occurrences of motifs with a score greater than the cutoff with cutoff values varied from 3 to 8.5. Therefore, each pair of cutoff values (S₁, S₂) corresponded to (k₁, k₂) occurrences for motifs of bicoid and kruppel respectively. For each pair (k₁, k₂), we computed p-value P_n (k₁ (S₁), k₂ (S₂)), which is denoted below as P (S₁, S₂). That is the probability to obtain at least k₁ occurrences of bicoid, with scores greater than S₁, and at least k₂ occurrences of kruppel, with scores greater than S₂. In Figure 3, a 3D-surface is shown, where (x, y, z) corresponds to (S₁, S₂, - log₁₀ P (S₁, S₂)), the cutoff value for bicoid motif, the cutoff value for kruppel motif and -logarithm of the corresponding p-value calculated for the M1 model respectively. The view to the surface from the above is shown in Figure 3C. The maximal value for – log₁₀ P (S₁, S₂), 6.3