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

Valentina Boeva, Julien Clément, Mireille Régnier, Mikhail Roytberg, Vsevolod Makeev

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Valentina Boeva, Julien Clément, Mireille Régnier, Mikhail Roytberg, Vsevolod Makeev. Exact p-value calculation for heterotypic clusters of regulatory motifs and its application in computational annotation of cis-regulatory modules. Algorithms for Molecular Biology, 2007, 2 (1), pp.13. 10.1186/1748-7188-2-13. hal-00784463

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# 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 Roytberg⁴,⁵ and Vsevolod J Makeev¹,⁶

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

Email: Valentina Boeva* - valeyo@yandex.ru; Julien Clément - Julien.Clement@info.unicaen.fr; Mireille Régnier - Mireille.Regnier@inria.fr; Mikhail A Roytberg - mroytberg@impb.psn.ru; Vsevolod J Makeev - makeev@genetika.ru

Published: 10 October 2007

Received: 13 July 2007

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

Accepted: 10 October 2007

This article is available from: http://www.almob.org/content/2/1/13

© 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 $k_1, ..., k_s$ 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 $O(n|Σ|(m|H| + K|σ|^K) \prod_i k_i)$ time complexity, where n is the length of the text, |Σ| is the alphabet size, m is the maximal motif length, |H| 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

<footnote>* Corresponding author</footnote>
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/

**Background**

During the past few years, a number of computational
tools have been designed [1-3] for locating potential tran-
scription 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 identifica-
tion 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 dem-
onstrated 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-regula-
tory 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 cooperativ-
ity-based discrimination of TFBSs surpasses the perform-
ance 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 evaluat-
ing the statistical significance of heterotypic clusters of
sites was implemented in many programs including Clus-
terDraw [23], ModuleSearcher [24], MCAST [25], eCIS-
ANALYST [26], Cister [27], Cluster-Buster [28] and Targe-
tExplorer [29]. At the moment, such programs use empir-
ical 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 sta-
tistical measure of site clustering, and we believe that the

best measure is the *p*-value of obtaining the observed clus-
ter by chance in a random sequence of a Markov or Ber-
noulli (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 FLY-
enhancer [32], sCOre [33], and CLuSter [34]. How-
ever, 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 usu-
ally 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 enumera-
tion of sequences that can be bound by a transcription fac-
tor (TF) [35]. Sometimes, information about binding sites
can be found in SELEX [36,37] or Protein Binding Micro-
array (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 substi-
tutions 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 mis-
matches.

The most popular way to represent binding sites is a Posi-
tion Weight Matrix (PWM), which is also called position-
specific weight matrix (PSWM) or position-specific scor-
ing 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 $\sum_{i=1}^{L} m_{\omega(i),i}$, where i represents a position in the D-substring, ω(i) the symbol at position i in the substring, and $m_{\alpha, i}$ 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 $H_1, ..., H_s$ be given. Typically, each set $H_i$ is associated to a TF motif. Given a s-tuple of integers $(k_1, ..., k_s)$, we compute the corresponding p-value, that is the probability to find at least $k_i$ occurrences of words from each set $H_i$ 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 $(k_1, ..., k_s)$ occurrences of motifs $H_1, ..., H_s$ 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 $H$. In contrast, in this paper we deal with a separate counting for several sets of several words $H_1, ..., H_s$, each set $H_j$ represents one TFBS motif.

All methods of solving the problem of p-value calculations for multiple occurrences of words from a set $H$ study some basic languages. Let $L_n(H; k)$ be the set of texts of length n containing at least k occurrences of $H$. The desired p-value would therefore be the probability $P(L_n(H; k))$. Let $R_H^k$ be the set of texts of all lengths that

contain exactly k words of $H$, the last one occurring as a suffix [42]. For any $H_j$ in $H$, let $R_{H_j}^k$ be the subset of $R_H^k$ 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 $R_H^k = \cup_{H_j \in H} R_{H_j}^k$. One defines $r_j^k(p)$ as the probability that a text of size p be in set $R_{H_j}^k$. If no word in $H$ is a subword of another word in $H$, the probability $P(L_n(H; k))$ to find at least k occurrences of words from $H$ in a random text of length n satisfies

$$P(L_n(H; k)) = \sum_{p \le n} \sum_{H_j \in H} r_j^k(p)$$

Therefore, one tries to compute the sequence of $(r_j^k(p))$ values.

### Linear induction

In the first class of methods [43-46], one computes, implicitly or explicitly, probabilities $P(L_n(H; k))$ up to a given text length n. Such methods are intrinsically linear in n. In [43-46] one relies on a recurrence relation on $r_j^k(n)$ that extends the one originally given in [47]. Typically, one step will cost $O(|H|m)$, where $H$ is a set of words of length m and $|H|$ is its cardinality. Time complexity is $O(n|H|m)$ and, relying on a combinatorial property, [44] achieves optimal space complexity $O(|H| \log |H|m)$. 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

$$r_j^k(z) = \sum_n r_j^k(n) z^n.$$

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

In [49,53], $r_j^k(z)$ are the solutions of a system of equations. To derive these equations, the authors build an
automaton that recognizes these languages $\mathcal{R}_{H_j}^k$ (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 com-
pute the determinant of a matrix of polynomials with a
huge dimension, e.g. $O(|\mathcal{H}|)$. This $O(|\mathcal{H}|^2)$ symbolic
computation may be more expensive than the extraction
step or the linear computation above, that involve arith-
metic operations on real numbers.

When the preprocessing step is achievable, the extraction
step is amenable to the solution of a linear recurrence of
degree $m|\mathcal{H}|$; therefore, its complexity is $O(m|\mathcal{H}|n)$
and a classical optimization yields $O(m|\mathcal{H}|\log n)$. 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 $\mathcal{H}$. 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 expres-
sion for $r_j^k$ ($z$), and therefore avoiding the expensive
determinant computation. Time complexity, typically, is
the one for computing all possible overlaps, that is
approximately $O(|\mathcal{H}|^2)$. This yields extremely precise
results when the expectation of the number of occur-
rences, $nP(\mathcal{H})$ is very small [59] or close to 1 [51] (the case
studied the most often). Case $nP(\mathcal{H}) \sim 2$ is achieved in
[60]. Nevertheless, extension to larger values of $k$ or mul-
tioccurrences 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 $\mathcal{H} = (H_1, \dots, H_r)$ 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 $\mathcal{H}$ can contain another word from $\mathcal{H}$ as a substring. We consider, as an occurrence of motif $\mathcal{H}$ in text $T$, any occurrence of any word $H_j \in \mathcal{H}$ in $T$. Below all texts and words in motifs are sequences on a given alphabet $\Sigma$.

Let $(\mathcal{H}_1, \dots, \mathcal{H}_s)$ be $s$ different motifs. Our objective is to
calculate the probability ($p$-value) that motifs
$(\mathcal{H}_1, \dots, \mathcal{H}_s)$ have respectively at least $(k_1, \dots, k_s)$ possibly
overlapping occurrences in a random text $T_n$.

To be more precise, there is a probability distribution
defined on the set $\Sigma^n$ of all texts of length $n$ in the alphabet
$\Sigma$; the most widely used models are random Bernoulli tri-
als and a Markov model of order $K$. Denote as $L_n$
$(\mathcal{H}_1, \dots, \mathcal{H}_s; k_1, \dots, k_s)$ the set of all texts of length $n$ con-
taining at least $k_i$ possibly overlapping occurrences of each
motif $\mathcal{H}_i$, $i = 1, \dots, s$. Then the desired $p$-value is the prob-
ability $\mathbf{P}$ ($L_n$ $(\mathcal{H}_1, \dots, \mathcal{H}_s; k_1, \dots, k_s)$) of the set $L_n$
$(\mathcal{H}_1, \dots, \mathcal{H}_s; k_1, \dots, k_s)$ with respect to the given probability
distribution on $\Sigma^n$.

Our approach to the calculation of this $p$-value is similar
to that published in [61], which was used there to calcu-
late 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 $L_n(\mathcal{H}_1, \dots, \mathcal{H}_s; k_1,
\dots, k_s)$ or not. Ideas published in [61] and [62] can be
adopted to compute the probability $\mathbf{P}(L_n(\mathcal{H}_1, \dots, \mathcal{H}_s; k_1,
\dots, k_s))$ that the random text $T_n \in \Sigma^n$ belongs to the set $L_n$
$(\mathcal{H}_1, \dots, \mathcal{H}_s; k_1, \dots, k_s)$.

We start from the simplest case of one motif $\mathcal{H}$ for which
we calculate the probability $\mathbf{P}(L_n(\mathcal{H}; 1))$ that text $T_n$ con-
tains 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 $\mathcal{H}$. The basic data structure is a pre-
fix tree which is a variant of the classical trie $\mathcal{T}(\mathcal{H})$ [42]
that may be built on the set of words $\mathcal{H}$. Let $Q_{\mathcal{H}}$ denote
the set of prefixes of these words. In the following, we
identify a word $q \in Q_{\mathcal{H}}$ with node *Node*($q$) at the end of
the branch labeled by $q$. In particular, the root is identified
with the empty string $\varepsilon$. 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 $\delta: Q_H \times \Sigma \to Q_H$ defined as follows. For any pair $(p, a)$ in $Q_H \times \Sigma$, $\delta(p, a)$ is the largest suffix of concatenation $pa$ that belongs to $Q_H$. Remark that $\delta(p, a) = pa$ iff $pa \in Q_H$.

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 $T[i]$ that belongs to $Q_H$. The sequence of nodes visited during the traversal are defined by words $q_i$ that satisfy the inductive relationship

$$ \forall i \ge 0, q_{i+1} = \delta(q_i, T[i+1]), $$

with the initial condition $q_0 = \varepsilon$.

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

$\mathcal{T}(\mathcal{H})$ and transition function $\delta$ can be efficiently constructed with an algorithm proposed by Aho and Corasick

<table><caption>**Table 1:** Values of $\delta$ function for the set $\mathcal{H} = \{aaa, aac, aca, acc, cct\}$.</caption><thead><tr><th>q\α</th><th>A</th><th>C</th><th>G</th><th>T</th></tr></thead><tbody><tr><td>0</td><td>1</td><td>2</td><td>0</td><td>0</td></tr><tr><td>1</td><td>3</td><td>4</td><td>0</td><td>0</td></tr><tr><td>2</td><td>1</td><td>5</td><td>0</td><td>0</td></tr><tr><td>3</td><td>6</td><td>7</td><td>0</td><td>0</td></tr><tr><td>4</td><td>8</td><td>9</td><td>0</td><td>0</td></tr><tr><td>5</td><td>1</td><td>5</td><td>0</td><td>10</td></tr><tr><td>6</td><td>6</td><td>7</td><td>0</td><td>0</td></tr><tr><td>7</td><td>8</td><td>9</td><td>0</td><td>0</td></tr><tr><td>8</td><td>3</td><td>4</td><td>0</td><td>0</td></tr><tr><td>9</td><td>1</td><td>5</td><td>0</td><td>10</td></tr><tr><td>10</td><td>1</td><td>2</td><td>0</td><td>0</td></tr></tbody></table>

Values of $\delta(q, \alpha)$ function for $q \in Q$ and $\alpha = A, C, G, T$ constructed for the set $\mathcal{H} = \{AAA, AAC, ACA, ACC, CCT\}$.

<figure><img src="image_1.png" /><figcaption>**Figure 1**</figcaption></figure>

Tree $\mathcal{T}(\mathcal{H})$ for the set $\mathcal{H} = \{aaa, aac, aca, acc, cct\}$ with dashed links for $\delta$ function. Tree $\mathcal{T}(\mathcal{H})$ for the set $\mathcal{H} = \{AAA, AAC, ACA, ACC, CCT\}$. Dashed colored links represent $\delta$ function for internal node (5) – in red, and for marked node (7) corresponding to the word AAC $\in \mathcal{H}$ – in purple.

[62]. Both time and space of the algorithm is proportional to the sum of lengths of all words from $\mathcal{H}$.

The combination of tree $\mathcal{T}(\mathcal{H})$ and transition function $\delta$ 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 $\Sigma$. The algorithm computes probabilities $P(L_n(\mathcal{H}; 1))$ by induction on $n$.

To describe the algorithm we divide the set $\Sigma^i$ of all texts $T_i$ of length $i$ into classes that do and do not contain occurrences of $\mathcal{H}$.

**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 $\mathcal{H}$,
### 3. A traversal AC ($\mathcal{T}(\mathcal{H})$, $T_i$) 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 $\mathcal{H}$.

For a given number $i$ larger than $m$, the union for classes $C_i(0; q)$, where $q$ is in $Q_{\mathcal{H}} \setminus \mathcal{H}$ and the class $G_i(1)$ form a partition of the set $\Sigma^i$ 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 $Q_{\mathcal{H}} \setminus \mathcal{H}$, or to a class $G_i(1)$. Indeed, condition 3. means that the largest suffix of $T_i$ in $Q_{\mathcal{H}}$ is $q$. It follows from condition 2. that classes $C_i(q; 0)$ are empty if $q$ is in $\mathcal{H}$. A text $T_i$ of length $i$ is in $G_i(1)$ if and only if a node of $\mathcal{H}$ was visited during the traversal.

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

The algorithm calculates probabilities $\mathbf{P}(C_i(0; q))$ and $\mathbf{P}(G_i(1))$ using induction on length $i$. For $i=0$, these probabilities obviously comply with: $\mathbf{P}(C_0(0; \varepsilon)) = 1$; $\mathbf{P}(C_0(0; q)) = 0$, for any $q \neq \varepsilon$; $\mathbf{P}(G_0(1)) = 0$.

The values of $\mathbf{P}(C_{i+1}(0; q))$ and $\mathbf{P}(G_{i+1}(1))$ are calculated using values of $\mathbf{P}(C_i(0; q))$ and $\mathbf{P}(G_i(1))$. Therefore, the needed space is proportional to the size of $Q_{\mathcal{H}}$ (see section Extensions and complexity below).

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

$$ \mathbf{P}(U \cdot a) = \mathbf{P}(U) \mathbf{P}(a). \quad (1) $$

Then the following relations hold for any $i \in \{1, \dots, n-1\}$ and $\Sigma$:

(i) if the text $T_i$ contains a word from $\mathcal{H}$ then all its concatenations with characters from $\Sigma$ would contain a word from $\mathcal{H}$; i.e.,

$$ G_i(1) \cdot a \subset G_{i+1}(1). \quad (2) $$

(ii) if the text $T_i$ does not contain a word from $\mathcal{H}$ and belongs to $C_{i+1}(0; q)$, i.e., ends with $q \in Q_{\mathcal{H}} \setminus \mathcal{H}$, then its concatenation $T_i \cdot a$ belongs to the class determined by the result of the Aho-Corasick transition function $\delta(q, a)$; i.e.,

$$ \text{if } \delta(q, a) \in \mathcal{H}, \text{ then } C_i(0; q) \cdot a \subset C_{i+1}(0; \delta(q, a)) \quad (3) $$

$$ \text{otherwise } C_i(0; q) \subset C_{i+1}(1). \quad (4) $$

Remembering that classes $C_i(0; q)$ for different $q$ and $G_i(1)$ form a partition of $\Sigma^i$, we obtain the following relation for the texts containing words from $\mathcal{H}$:

$$ G_{i+1}(1) = \{ \bigcup_{a \in \Sigma} G_i(1) \cdot a \} \cup \{ \bigcup_{(q,a); \delta(q,a) \in \mathcal{H}} C_i(0; q) \cdot a \}. \quad (5) $$

Similarly, classes of texts that do not contain words from $\mathcal{H}$ satisfy

$$ \forall q' \in Q_{\mathcal{H}} \setminus \mathcal{H}: C_{i+1}(0; q') = \bigcup_{(q,a); \delta(q,a) = q'} C_i(0; q) \cdot a. \quad (6) $$

Classes $C_i(0; q)$ for different $q$ in $Q_{\mathcal{H}} \setminus \mathcal{H}$ and $G_i(1)$ form a partition of $\Sigma^i$; classes $C_i(0; q)$ are empty if $q$ is in $\mathcal{H}$. Relations (5) and (6) with the help of (1) yield the recursive expressions for probabilities $\mathbf{P}(C_{i+1}(0; q))$ and $\mathbf{P}(G_{i+1}(1))$ in the Bernoulli case:

$$ \mathbf{P}(G_{i+1}(1)) = \mathbf{P}(G_i(1)) + \sum_{(q,a): \delta(q,a) \in \mathcal{H}} \mathbf{P}(C_i(0; q)) \cdot p(a), \quad (7) $$

$$ \mathbf{P}(C_{i+1}(0; q')) = \sum_{(q,a); \delta(q,a) = q'} \mathbf{P}(C_i(0; q)) \cdot p(a). \quad (8) $$

The run-time for each step of the computation of $C_{i+1}(0; q)$ and $G_{i+1}(1)$ is $O(|Q_{\mathcal{H}}| \cdot |\Sigma|)$; therefore the total time of all $n$ stages of p-value computation is $O(|Q_{\mathcal{H}}| \cdot |\Sigma| \cdot n)$.

The approach described in this section can be readily extended to the case of multiple occurrences of motif $\mathcal{H}$. 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 ($\mathcal{H}_1, ..., \mathcal{H}_s$) be s different motifs. Our objective is to calculate the probability that motifs ($\mathcal{H}_1, ..., \mathcal{H}_s$) have respectively at least ($k_1, ..., k_s$) possibly overlapping occurrences in the random text $T_n$ of the length $n$. This p-value is the probability $P(L_n(\mathcal{H}_1, ..., \mathcal{H}_s; k_1, ..., k_s))$ to obtain text $T_n$ belonging to the set of texts $L_n(\mathcal{H}_1, ..., \mathcal{H}_s; k_1, ..., k_s)$. 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 $\mathcal{H}$ of individual motifs $\mathcal{H}=\mathcal{H}_1 \cup \dots \cup \mathcal{H}_s$. It contains all words that belong to any of motifs $\mathcal{H}_i$. The tree $\mathcal{T}(\mathcal{H})$ is constructed for the overall set $\mathcal{H}$, its nodes $Q_{\mathcal{H}}$ contain all possible prefixes of all motifs from ($\mathcal{H}_1, ..., \mathcal{H}_s$). A node of the tree $q \in Q_{\mathcal{H}}$ can belong to some motif $\mathcal{H}_k$ or simultaneously to several different motifs from {$\mathcal{H}_j$}$_{1 \le j \le s}$. Let each node $q \in Q_{\mathcal{H}}$ be marked with numbers $j$ of motifs $\mathcal{H}_j$ to which it belongs. Nodes, corresponding to proper prefixes of $\mathcal{H}$, remain unmarked. The transition function $\delta: Q_{\mathcal{H}} \times \Sigma \to Q_{\mathcal{H}}$ is defined as it was defined in the case of a single motif for the unified motif $\mathcal{H}$.

All texts $T_n$ of length $n$ are classified into classes depending on occurrences of different $\mathcal{H}_j$. 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 $\mathcal{H}_i$, the corresponding value $k_j$ may not yet be attained for another motif $\mathcal{H}_j$. Therefore we need to introduce the occurrence index of a set of motifs.

**Definition 2** Let the target number of occurrences of motif $\mathcal{H}_i$ be $k_i$. Then, the occurrence index $\Lambda(k_1, ..., k_s)$ ($l_1, ..., l_s$) of a set of motifs ($\mathcal{H}$) in the text $T_n$ containing $l_i$ possibly overlapping occurrences of each $\mathcal{H}_i$ is an s-vector the ith component of which can be calculated as follows:

$$[\Lambda(k_1, ..., k_s)(l_1, ..., l_s)]_i = \lambda_i = \begin{cases} l_i & \text{if } l_i \le k_i, \\ k_i & \text{if } l_i > k_i. \end{cases} \quad (9)$$

**Definition 3** A text $T_i$ belongs to class $C_i(\lambda_1, ..., \lambda_s; q)$, $0 \le \lambda_i \le k_i$ iff

1. Length of $T_i$ equals $i$,

2. The occurrence index of motifs ($\mathcal{H}_1, ..., \mathcal{H}_s$) in text $T_i$ is equal to ($\lambda_1, ..., \lambda_s$),

3. A traversal AC ($\mathcal{T}(\mathcal{H}), T_i$) ends in node $q$.

A text $T_i$ belongs to class $G_i(k_1, ..., k_s)$ if it belongs to the union of classes

$$G_i(k_1, ..., k_s) = \bigcup_{q \in \mathcal{H}} C_i(k_1, ..., k_s; q). \quad (10)$$

The desired p-value $P(L_n(\mathcal{H}_1, ..., \mathcal{H}_s; k_1, ..., k_s))$ is equal to $P(G_n(k_1, ..., k_s))$. 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 $\delta(q, a)$ can belong simultaneously to several motifs {$\mathcal{H}_j$}$_{1 \le j \le s}$. Thus, the resulting probability $P(C_{i+1}(\lambda_1, ..., \lambda_s; q'))$ that text $T_{i+1}$ belongs to class $C_{i+1}(\lambda_1, ..., \lambda_s; q')$ calculates as

$$P(C_{i+1}(\lambda_1, ..., \lambda_s; q')) = \sum_{(q,a): \delta(q,a)=q'} \sum_{(r_1, ..., r_s) \in \mathcal{J}} P(C_i(r_1, ..., r_s; q)) \cdot p(a) \quad (11)$$

where the summation in the second sum is performed over all allowed s-tuples of indexes ($r_1, ..., r_s$) which together make the set of s-tuples $\mathcal{J}$. A s-tuple of indexes ($r_1, ..., r_s$) belongs to $\mathcal{J}$ if it complies with the following conditions:

1. if $q' \notin \mathcal{H}_j$ then $r_j = \lambda_j$,

2. if $q' \in \mathcal{H}_j$ and $\lambda_j < k_j$ then $r_j = \lambda_j - 1$,

3. if $q' \in \mathcal{H}_j$ and $\lambda_j = k_j$ then $r_j = k_j$ or $r_j = k_j - 1$.
Implementation details

Our basic data structure is the prefix tree; we use its stand-
ard representation [42] [see also Additional files 2 and 3
for Tree construction from PWM motif representation]. Each
tree node $q \in Q_H$ is supplied with several additional var-
iables.

At stage (i + 1) of probability computation the values P (C<sub>i+1</sub> (λ<sub>1</sub>, ..., λ<sub>s</sub>; q)) become computed from the values P (C<sub>i</sub> (λ<sub>1</sub>, ..., λ<sub>s</sub>; 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 k<sub>1</sub> × ∪ × k<sub>s</sub>-arrays of real values C<sub>0</sub> and C<sub>1</sub> for storing P (C<sub>i</sub> (λ<sub>1</sub>, ..., λ<sub>s</sub>; q)) and P (C<sub>i+1</sub> (λ<sub>1</sub>, ..., λ<sub>s</sub>; q)) for different λ<sub>j</sub>. C<sub>0</sub> is used to store probabilities for even text lengths while C<sub>1</sub> for odd.

In implementation the calculation of values P (C<sub>i+1</sub> (λ<sub>1</sub>, ..., λ<sub>s</sub>; q')) from P (C<sub>i</sub> (λ<sub>1</sub>, ..., λ<sub>s</sub>; q)) for all q', q ∈ Q<sub>H</sub> and (λ<sub>1</sub>, ..., λ<sub>s</sub>): 0 ≤ λ<sub>j</sub> ≤ k<sub>j</sub>, 1 ≤ j ≤ s, is performed in the parallel way.
Initially we set all the values P (C<sub>i+1</sub> (λ<sub>1</sub>, ..., λ<sub>s</sub>; q')) to 0.
Then we look over all tuples (r<sub>1</sub>, ..., r<sub>s</sub>; q), where q ∈ Q<sub>H</sub>
and (r<sub>1</sub>, ..., r<sub>s</sub>): 0 ≤ r<sub>j</sub> ≤ k<sub>j</sub>, 1 ≤ j ≤ s. For each tuple (r<sub>1</sub>, ..., r<sub>s</sub>; q) and all letters a ∈ Σ we find the prefix q' = δ(q, a) and
the value P (C<sub>i</sub> (r<sub>1</sub>, ..., r<sub>s</sub>; q)) ⋅ p(a). Then we add P (C<sub>i</sub> (r<sub>1</sub>, ..., r<sub>s</sub>; q)) ⋅ p(a) to the value P (C<sub>i+1</sub> (λ<sub>1</sub>, ..., λ<sub>s</sub>; q')) where (λ<sub>1</sub>, ..., λ<sub>s</sub>; q') meet the conditions inverse to those of formula
(11):

1. if $q' \notin \mathcal{H}_j$ then $\lambda_j = r_j'$

2. if $q' \in \mathcal{H}_j$ and $r_j < k_j$ then $\lambda_j = r_j + 1$,

3. if $q' \in \mathcal{H}_j$ and $r_j = k_j$ then $\lambda_j = r_j$.

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

$$
P(G_n(k_1, \dots, k_s)) = \sum_{q \in \mathcal{H}} P(C_n(k_1, \dots, k_s; q)).
$$

Markov text model

Tree approach and the recursion (11) can be readily
extended to calculate p-values of motif occurrences in ran-
dom 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 calcu-
late p(a) knowing only the prefix q. To overcome this we
divide each class C<sub>i</sub> (r<sub>1</sub>, ..., r<sub>s</sub>; q), where |q| = d <min (K, i)

into subclasses C<sub>i</sub> (r<sub>1</sub>, ..., r<sub>s</sub>; q, w); each subclass corre-
sponds to a word w of length min (K, i) - d. Then, a text T<sub>i</sub>
of length i belongs to class C<sub>i</sub> (r<sub>1</sub>, ..., r<sub>s</sub>; q, w) if the suffix of
T<sub>i</sub> 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 H = {AAA, AAC, ACA, ACC, CCT}. The text T = ATGCCAACCTT produces the following sequence of nodes {q<sub>i</sub>}<sub>i≥1</sub> (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<sub>n</sub> (H; 1)), P (L<sub>n</sub> (H; k)), and P (L<sub>n</sub> (H<sub>1</sub>,...,H<sub>s</sub>; k<sub>1</sub>, ..., k<sub>s</sub>)) 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

$$
t[1] \cdots t[K] = \begin{cases} w \cdot q & \text{if } 0 \le d < K, \\ K\text{-suffix of } q & \text{otherwise.} \end{cases}
$$

The Markov extension is currently implemented for K = 1.

<figure><img src="image_2.png" /><figcaption>Figure 2</figcaption></figure>

Tree T(H) for the set H = {aaa, aac, aca, acc, cct}
with dashed links for δ function under Markov(1)
model. Tree T(H) for the set H = {AAA, AAC, ACA,
ACC, CCT} under Markov model of order 1. Dashed
colored links represent δ function for internal node (8) – in
red, and for marked node (10) corresponding to the word
AAC ∈ H – in purple.
## Complexity

To resume, the computation of $P(L_n(H; k))$ for one set $H$ requires a computation of $(P(C_i(l,q)))_{0 \le l < k, q \in Q_H}$ for $i \le n$. For each iteration, the time complexity is $O(k|Q_H| + |\Sigma|)$, where $|\Sigma|$ is the size of the alphabet. One traverses the tree $n$ times. As $|Q_H|$ is upper bounded by $(m|H|)$, where $m$ is the maximal length of word in $H$, this yields the overall $O(nkm|H||\Sigma|)$ time complexity and a $O(km|H|)$ space complexity.

When several sets are involved, the number of nodes in the tree $T(H_1 \cup \cdots \cup H_s)$ becomes $O(m|H|)$ with $m$ equal to the maximal length of word in $H=H_1 \cup \cdots \cup H_s$. Additional memory in each node is $\prod_i k_i$. Therefore, the time complexity is $O(nm|\Sigma|\prod_i k_i|H|)$ and the space complexity is $O(m \prod_i k_i|H|)$. In the Markov model of order $K$, one memorizes $|\Sigma|^{K-d}$ predecessors for each node at depth $d$, $0 = d < K$. In other words, the number of classes becomes $(m|H| + K|\Sigma|^K)$. Therefore, the space memory is $O((m|H| + K|\Sigma|^K) \prod_i k_i)$ and the running time is $O(n|\Sigma|(m|H| + K|\Sigma|^K) \prod_i k_i)$. This additive increment compares favorably to simple induction methods [45,53] that introduce a multiplicative $O(K|\Sigma|^K)$ 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

<table><caption>**Table 2:** Comparison of $p$-values calculated by the AHOPRO program, by Monte Carlo simulations and by compound Poisson distribution formula under the $M0$ model</caption><thead><tr><th>MOTIF, CUTOFF</th><th>OCC.</th><th>AHOPRO</th><th>MONTE CARLO</th><th>POISSON</th><th>AHOPRO/MC</th><th>AHOPRO/POISSON</th></tr></thead><tbody><tr><td>bcd, 5.3</td><td>3</td><td>0.012</td><td>0.012</td><td>0.010</td><td>1.00</td><td>1.10</td></tr><tr><td>kr, 5.0</td><td>4</td><td>0.0044</td><td>0.0044</td><td>0.0033</td><td>1.01</td><td>1.34</td></tr><tr><td>hb, 6.2</td><td>2</td><td>0.013</td><td>0.013</td><td>0.012</td><td>0.99</td><td>1.04</td></tr><tr><td>bcd & kr</td><td>3&4</td><td>0.00025</td><td>0.00026</td><td>3.6E-05</td><td>0.99</td><td>7.10</td></tr><tr><td>bcd & kr & hb</td><td>3&4&2</td><td>6.54E-06</td><td>5.8E-06</td><td>4.34E-07</td><td>1.13</td><td>7.13</td></tr></tbody></table>

Comparison of $p$-values calculated for the Markov(0) model by the AHOPRO program with $p$-values calculated by Monte Carlo simulations and by Poisson formula for motifs of *D. melanogaster* developmental transcription factors bicoid, kruppel and hunchback.
<table><caption>Table 3: Comparison of *p*-values calculated by the AHOPRO program, by Monte Carlo simulation and by compound Poisson distribution formula under the M1 model</caption><thead><tr><th>MOTIF, CUTOFF</th><th>OCC.</th><th>AHOPRO</th><th>MONTE CARLO</th><th>POISSON</th><th>AHOPRO/MC</th><th>AHOPRO/POISSON</th></tr></thead><tbody><tr><td>bcd, 5.3</td><td>3</td><td>0.013</td><td>0.014</td><td>0.012</td><td>0.998</td><td>1.11</td></tr><tr><td>kr, 5.0</td><td>4</td><td>0.011</td><td>0.011</td><td>0.008</td><td>1.01</td><td>1.43</td></tr><tr><td>hb, 6.2</td><td>2</td><td>0.14</td><td>0.14</td><td>0.11</td><td>0.9987</td><td>1.25</td></tr><tr><td>bcd & kr</td><td>3&4</td><td>0.00051</td><td>0.00051</td><td>9.62E-05</td><td>0.9991</td><td>5.34</td></tr><tr><td>bcd & kr & hb</td><td>3&4&2</td><td>6.9E-05</td><td>6.97E-05</td><td>1.08E-05</td><td>0.9889</td><td>6.36</td></tr></tbody></table>

Comparison of *p*-values calculated by the AHOPRO program for the Markov(1) model with those calculated by Monte Carlo simulations and by Poisson formula for motifs of *D. melanogaster* developmental transcription factors *bicoid*, *kruppel*, and *hunchback*.

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 $10^6$ everywhere, except for the triplet (*bcd* & *kr* & *hb*), where we did $10^7$ 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 (*S*<sub>1</sub>, *S*<sub>2</sub>) corresponded to (*k*<sub>1</sub>, *k*<sub>2</sub>) occurrences for motifs of *bicoid* and *kruppel* respectively. For each pair (*k*<sub>1</sub>, *k*<sub>2</sub>), we computed *p*-value *P*<sub>n</sub> (*k*<sub>1</sub> (*S*<sub>1</sub>), *k*<sub>2</sub> (*S*<sub>2</sub>)), which is denoted below as *P* (*S*<sub>1</sub>, *S*<sub>2</sub>). That is the probability to obtain at least *k*<sub>1</sub> occurrences of *bicoid*, with scores greater than *S*<sub>1</sub>, and at least *k*<sub>2</sub> occurrences of *kruppel*, with scores greater than *S*<sub>2</sub>. In Figure 3, a 3D-surface is shown, where (*x*, *y*, *z*) corresponds to (*S*<sub>1</sub>, *S*<sub>2</sub>, -log<sub>10</sub> *P* (*S*<sub>1</sub>, *S*<sub>2</sub>)), 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 -log<sub>10</sub> *P* (*S*<sub>1</sub>, *S*<sub>2</sub>), 6.3044, is attained when the *bicoid* cutoff is equal to *S*<sub>1</sub> = 5.1 and the *kruppel* cutoff is equal to *S*<sub>2</sub> = 5.6. With such cutoff values in the sequence of the *eve2* enhancer
<figure><img src="image_3.png" /><figcaption>**Figure 3**
P-value distribution for eve2 and random sequences. Distribution of log₁₀ (Pvalue) calculated for the M1 model as a function of cutoff values for PWMs for BICOID and KRUPPEL in the even-skipped stripe 2 enhancer (A), in a random sequence (B). View from above: eve2 sequence (C), random sequence (D).</figcaption></figure>

there are $k_1 = 6$ and $k_2 = 4$ occurrences of bicoid and kruppel motifs defined by corresponding PWMs. We believe that the sites that are found with this optimal *p*-value are the best candidates for functional TF binding sites.

For comparison, we simulated random sequences with the same length as the *eve2* enhancer and the same dinucleotide probabilities. In most of simulated sequences, for the cutoff values for *bicoid* and *kruppel* equal to ($S_1, S_2$) = (5.1, 5.6) we found no more than one occurrence of each motif. The average number of occurrences is 0.54 for *bicoid* and 0.31 for *kruppel*. The average *p*-value is 0.633. We took one of the random sequences and compared *p*-values calculated for various cutoff values in this random sequence (Figures 3B, 3D) and in the real biological sequence of the *eve2* enhancer (Figures 3A, 3C). One can see that there are two major differences between *p*-value

distributions in really regulated sequences and in the random sequence. First, *p*-values in the random sequence are much greater than those in the enhancer sequence. In particular, maximal – log(*p*-value) for this random sequence is about 1.02 which is 6.17 times smaller than maximal – log(*p*-value) for the enhancer sequence (see also Table 4). Second, the shapes of *p*-value distributions are different. For the enhancer sequence, there are only few distinct peaks (4.3, 5.6), (4.3, 6.8), (5.1, 5.6), (5.1, 6.8) whereas for the random sequence we see ridges between (2.2, 2.0) and (2.2, 4.8), and (2.8, 2.0) and (2.8, 4.8). As we expected, it is impossible to choose the appropriate cutoff for PWMs of factors from the random sequence data (Figures 3B and 3D).

We also would like to address the choice between the M0 and M1 models. We observed, that in almost all cases the
<table><caption>Table 4: Comparison of *p*-values and cutoff for different sets of DNA sequences</caption><thead><tr><th>regulatory regions<br>bicoid regulated</th><th>minimal<br>pvalue</th><th>Cut-off</th><th>regulatory regions not<br>regulated by bicoid</th><th>minimal<br>pvalue</th><th>Cut-off</th><th>random seq.</th><th>minimal<br>pvalue</th><th>Cut-off</th></tr></thead><tbody><tr><td>Btd crm</td><td>3.24E-05</td><td>3.4</td><td>Gt p. enh.</td><td>0.023</td><td>2.7</td><td>seq. 1</td><td>0.16</td><td>2.6</td></tr><tr><td>Hb P2</td><td>4.13E-05</td><td>3.7</td><td>Hb upstream enh.</td><td>0.053</td><td>4.4</td><td>seq. 2</td><td>0.12</td><td>1.7</td></tr><tr><td>Kni cis element</td><td>0.01</td><td>5.3</td><td>Eve stripe 4+6 enh.</td><td>0.41</td><td>3.6</td><td>seq. 3</td><td>0.25</td><td>1.2</td></tr><tr><td>Kr CD-1 enh.</td><td>0.0001</td><td>5.1</td><td>Eve stripe 3+7 enh.</td><td>0.58</td><td>2.5</td><td>seq. 4</td><td>0.065</td><td>1.6</td></tr><tr><td>Otd early enh.</td><td>0.024</td><td>5</td><td>Ftz upstream enh.</td><td>0.037</td><td>5.8</td><td>seq. 5</td><td>0.11</td><td>1</td></tr><tr><td>Sal blastoder. enh.</td><td>8.62E-04</td><td>6.5</td><td>Ftz</td><td>0.28</td><td>3.3</td><td>seq. 6</td><td>0.0087</td><td>3.8</td></tr><tr><td>Tll PD enh.</td><td>0.26</td><td>4.2</td><td>Ubx PBX enh.</td><td>0.196</td><td>6.7</td><td>seq. 7</td><td>0.024</td><td>2.9</td></tr><tr><td>Tll AD+PD enh.</td><td>0.025</td><td>8.1</td><td>Ubx BXD enh.</td><td>0.698</td><td>4.6</td><td>seq. 8</td><td>0.17</td><td>3.4</td></tr><tr><td>Eve stripe 2 enh.</td><td>4.04E-05</td><td>5.1</td><td>Ubx BX enh. (BRE)</td><td>0.05</td><td>7.5</td><td>seq. 9</td><td>0.092</td><td>2.8</td></tr><tr><td>Eve stripe 1 enh.</td><td>8.09E-06</td><td>5.2</td><td>Emx upstream enh.</td><td>0.276</td><td>4.4</td><td>seq. 10</td><td>0.052</td><td>3.6</td></tr><tr><td>Eve stripe 5 enh.</td><td>0.27</td><td>3.8</td><td>En stripe enh. (intr. 1)</td><td>0.049</td><td>5</td><td>seq. 11</td><td>0.13</td><td>1.7</td></tr><tr><td>Median</td><td>8.62E-04</td><td>5.1</td><td>Median</td><td>0.196</td><td>4.4</td><td>Median</td><td>0.1128</td><td>2.6</td></tr></tbody></table>

Comparison of minimal *p*-values and best found cutoffs for bicoid PWM calculated (i) in regulatory regions which are regulated by *bicoid*, (ii) in regulatory regions which are not regulated by *bicoid*, and (iii) in random sequences of the same length and with the same dinucleotide distribution as in the *even-skipped stripe 2* enhancer.

*p*-value calculated for the M0 model is smaller than the *p*-value calculated for the M1 model. This can probably be explained by the fact that using the M1 model we take into account more information about the real sequence than in the M0 model. Nevertheless, the difference is not crucial; for instance, the greatest value of the ratio between *p*-values calculated adopting the M0 and M1 for *bicoid* and *kruppel* is about 3.62 for the *eve2* enhancer. So, the M0 model can be equally used in practical applications.

## Assessment of gene regulation

Enhancers may contain clusters of TF binding sites for gene regulators. In such cases, *p*-value computation can be used to distinguish genes that are regulated by a given transcription factor from those that are not. To illustrate this, we took PWM for TF *bicoid* and calculated *p*-values for different cutoff values in various sets of sequences:

- regulatory regions which are regulated by *bicoid*, the positive set;

- regulatory regions which are not regulated by *bicoid*, the negative set;

- random sequences of the same length as *eve2* enhancer and with the same dinucleotide distribution, the random set.

Minimal *p*-value and the corresponding cutoff value for 11 sequences in each set are presented in Table 4. Comparing the *p*-values we observed that *p*-values calculated for the positive set generally were significantly smaller than those, calculated for the negative and for the random sets.

The median for the *p*-value in the positive set is equal to 8.62E-04. But there are some exceptions, for instance, the *tailless* PD enhancer with a minimal *p*-value that is equal to 0.26 and the *even-skipped stripe 5* enhancer with the minimal *p*-value that is equal to 0.27. Despite the fact that these genes are reported to be regulated by *bicoid* and that there are experimentally confirmed individual *bicoid* binding sites in these sequences, these sequences do not contain clusters of *bicoid* binding sites.

Most *p*-values calculated for the negative set, (second set in Table 4), are significantly higher than *p*-values calculated for the positive set. But we observed rather small *p*-values for sequences of the *giant posterior* enhancer (0.023), the *hunchback* upstream enhancer (0.053), the *fushi tarazu* upstream enhancer (0.037), the *ultrabithorax* BX enhancer (0.05), and the *engrailed* stripe enhancer (0.049). We believe that this can be explained by the fact that these regions contain clusters of binding sites of regulatory factors with motifs that are similar to the *bicoid* motif. Indeed, it was experimentally shown that TF *kruppel* regulates the *giant posterior* enhancer, TF *tailless* regulates the *hunchback* upstream enhancer and the *ultrabithorax* BX enhancer, and TF *fushi tarazu* regulates the *fushi tarazu* upstream enhancer, the *ultrabithorax* BX enhancer and the *engrailed* stripe enhancer. All these motifs of *kruppel*, *tailless* and *fushi tarazu* exhibit some similarity to the *bicoid* motif. This observation shows the necessity to use some sort of conditional *p*-values in order to distinguish between the true *bicoid* clusters and the clusters of weak *bicoid* sites induced by presence of the clusters of other TF sites [67]. Moreover, the apparent false positive hit (*p*-value = 0.05, cutoff = 7.5) in a region that was not reported to be regulated by *bicoid* seems to be related to
the real *bicoid* binding, although not necessarily func-
tional.

For the random set, i.e., sequences simulated with the
same dinucleotide probabilities as in the *even-skipped*
*stripe 2* enhancer, we observe a rather broad range of min-
imal *p*-values, from 0.0087 for the 6th sample to 0.25 for
the 3rd sample. It shows that the predictive power of this
approach is limited to the case of regulatory sequences
containing clusters of motifs.

Conclusion

In this work we have developed an algorithm inspired by
the Aho-Corasick pattern matching algorithm that allows
precise calculation of the probability to find given motif
conformation in a random text. It was implemented in the
AHOPRO software for the Bernoulli model and the
Markov model of order 1 of random sequences. There
would be no difficulty in extending our approach for
Markov models of order *k*, *k* > 1. We compared probabil-
ities computed with AHOPRO with those computed by
compound Poisson distribution and showed that in the
case of multiple occurrences of multiple motifs the Pois-
son approximation often substantially underestimate the
*p*-value.

As we have demonstrated, the statistical significance of
multiple motif occurrence in the text can be efficiently cal-
culated with a simple algorithm. This can give an inde-
pendent criteria to improve the results of site extraction
algorithms, which still performs rather poorly. P-values or
E-values are used in such programs as BLAST and make
quantities to which practicing biologists are used to. Thus,
adopting this measure to motif extraction (for a single or
multiple motif occurrences) would greatly help the users
who use motif extraction analysis as a preliminary stage
for experiments in the lab. On the other hand, our algo-
rithm is not connected with a particular motif extraction
program, and uses a most general motif representation,
the list of the allowed words [35], as input. Thus, it can be
used when the results of several motif extraction algo-
rithms are compared, for instance in the interpretation of
ChIP-chip experiments [5]. In addition, our algorithm
AHOPRO can easily be extended to amino acid sequences
and applied in identification of protein domain signa-
tures.

Authors' contributions

VM initiated the study by pointing at the biological prob-
lem. JC suggested the initial idea of using Aho-Corasick
structure. The final version of the algorithm was devel-
oped in discussions between JC, VB, MR and MAR. JC and
VB developed the implementation. VB obtained results on
simulated and biological sequences. VB designed the web
site. MR, MAR, VB and VM participated in manuscript

writing. MR and VM coordinated the study. All authors
read and approved the final manuscript.

**Additional material**

Additional file 1

Bernoulli text model. Probability to find multiple occurrences of a single motif. The detailed description of the algorithm for the p-value calculation in the case of multiple occurrences of a single motif.

Click here for file

[http://www.biomedcentral.com/content/supplementary/1748-7188-2-13-S1.pdf]

Additional file 2

Tree construction from PWM motif representation. The brief description of the procedure of the prefix tree construction from PWM motif representation.

Click here for file

[http://www.biomedcentral.com/content/supplementary/1748-7188-2-13-S2.pdf]

Additional file 3

Tree construction from PWM motif representation. Steps of the prefix tree construction for a PWM and a given cut-off.

Click here for file

[http://www.biomedcentral.com/content/supplementary/1748-7188-2-13-S3.bmp]

**Acknowledgements**

Thanks to Andrey Mironov, Stephen Small, Dmitri Papatsenko, Bruno Salvy and Philippe Flajolet for helpful comments and suggestions. Thanks to Alexander Favorov for help with the programming. Thanks to Tim Barker for correcting the English in the manuscript. This research was partially supported by INTAS #04-83-3994 and #05-1000008-8028, French Program EcoNet-12635WG, the RFBR grants 07-04-01584 and 06-04-49249, and by Russian Federation Agency in Science and Innovation State Contract 02.531.11.9003.

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