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 3637 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 ∑i=1Lmω(i),i MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaadaaeWaqaaiabd2gaTnaaBaaaleaaiiGacqWFjpWDcqGGOaakcqWGPbqAcqGGPaqkcqGGSaalcqWGPbqAaeqaaaqaaiabdMgaPjabg2da9iabigdaXaqaaiabdYeambqdcqGHris5aaaa@3BC0@, 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.

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 ℋ1,...,ℋs MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaat0uy0HwzTfgDPnwy1egaryqtHrhAL1wy0L2yHvdaiqaacqWFlecsdaWgaaWcbaGaeGymaedabeaakiabcYcaSiabc6caUiabc6caUiabc6caUiabcYcaSiab=TqiinaaBaaaleaacqWGZbWCaeqaaaaa@3F88@ be given. Typically, each set ℋ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaat0uy0HwzTfgDPnwy1egaryqtHrhAL1wy0L2yHvdaiqaacqWFlecsaaa@3762@_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 ℋ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaat0uy0HwzTfgDPnwy1egaryqtHrhAL1wy0L2yHvdaiqaacqWFlecsaaa@3762@_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 ℋ1,...,ℋs MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaat0uy0HwzTfgDPnwy1egaryqtHrhAL1wy0L2yHvdaiqaacqWFlecsdaWgaaWcbaGaeGymaedabeaakiabcYcaSiabc6caUiabc6caUiabc6caUiabcYcaSiab=TqiinaaBaaaleaacqWGZbWCaeqaaaaa@3F88@ are found in a DNA segment, the p-value can be used to infer if such numbers of occurrences could be found by chance.

In the first class of methods 43444546, one computes, implicitly or explicitly, probabilities P (L_n (ℋ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaat0uy0HwzTfgDPnwy1egaryqtHrhAL1wy0L2yHvdaiqaacqWFlecsaaa@3762@; k)) up to a given text length n. Such methods are intrinsically linear in n. In 43444546 one relies on a recurrence relation on rjk MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGYbGCdaqhaaWcbaGaemOAaOgabaGaem4AaSgaaaaa@3102@ (n) that extends the one originally given in 47. Typically, one step will cost O (|ℋ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaat0uy0HwzTfgDPnwy1egaryqtHrhAL1wy0L2yHvdaiqaacqWFlecsaaa@3762@|m), where ℋ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaat0uy0HwzTfgDPnwy1egaryqtHrhAL1wy0L2yHvdaiqaacqWFlecsaaa@3762@ is a set of words of length m and |ℋ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaat0uy0HwzTfgDPnwy1egaryqtHrhAL1wy0L2yHvdaiqaacqWFlecsaaa@3762@| is its cardinality. Time complexity is O (n|ℋ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaat0uy0HwzTfgDPnwy1egaryqtHrhAL1wy0L2yHvdaiqaacqWFlecsaaa@3762@|m) and, relying on a combinatorial property, 44 achieves optimal space complexity O (|ℋ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaat0uy0HwzTfgDPnwy1egaryqtHrhAL1wy0L2yHvdaiqaacqWFlecsaaa@3762@| log |ℋ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaat0uy0HwzTfgDPnwy1egaryqtHrhAL1wy0L2yHvdaiqaacqWFlecsaaa@3762@|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.

In a second class of methods 474849505152, a preprocessing computes generating functions

r j k ( z ) = ∑ n r j k ( n ) z n . MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGYbGCdaqhaaWcbaGaemOAaOgabaGaem4AaSgaaOGaeiikaGIaemOEaONaeiykaKIaeyypa0ZaaabuaeaacqWGYbGCdaqhaaWcbaGaemOAaOgabaGaem4AaSgaaOGaeiikaGIaemOBa4MaeiykaKIaemOEaO3aaWbaaSqabeaacqWGUbGBaaaabaGaemOBa4gabeqdcqGHris5aOGaeiOla4caaa@4432@

In a second step, probabilities P (L_n (ℋ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaat0uy0HwzTfgDPnwy1egaryqtHrhAL1wy0L2yHvdaiqaacqWFlecsaaa@3762@; k)) are either extracted from the generating function or approximated.

In 4953, rjk MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGYbGCdaqhaaWcbaGaemOAaOgabaGaem4AaSgaaaaa@3102@ (z) are the solutions of a system of equations. To derive these equations, the authors build an automaton that recognizes these languages ℛHjk MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaat0uy0HwzTfgDPnwy1egaryqtHrhAL1wy0L2yHvdaiqaacqWFBeIudaqhaaWcbaGaeeisaG0aaSbaaWqaaiabdQgaQbqabaaaleaacqWGRbWAaaaaaa@3BB4@ (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 (|ℋ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaat0uy0HwzTfgDPnwy1egaryqtHrhAL1wy0L2yHvdaiqaacqWFlecsaaa@3762@|). This O (|ℋ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaat0uy0HwzTfgDPnwy1egaryqtHrhAL1wy0L2yHvdaiqaacqWFlecsaaa@3762@|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|ℋ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaat0uy0HwzTfgDPnwy1egaryqtHrhAL1wy0L2yHvdaiqaacqWFlecsaaa@3762@|; therefore, its complexity is O (m|ℋ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaat0uy0HwzTfgDPnwy1egaryqtHrhAL1wy0L2yHvdaiqaacqWFlecsaaa@3762@|n) and a classical optimization yields O (m|ℋ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaat0uy0HwzTfgDPnwy1egaryqtHrhAL1wy0L2yHvdaiqaacqWFlecsaaa@3762@| 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 ℋ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaat0uy0HwzTfgDPnwy1egaryqtHrhAL1wy0L2yHvdaiqaacqWFlecsaaa@3762@. 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 4458, not needing an explicit expression for rjk MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGYbGCdaqhaaWcbaGaemOAaOgabaGaem4AaSgaaaaa@3102@ (z), and therefore avoiding the expensive determinant computation. Time complexity, typically, is the one for computing all possible overlaps, that is approximately O (|ℋ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaat0uy0HwzTfgDPnwy1egaryqtHrhAL1wy0L2yHvdaiqaacqWFlecsaaa@3762@|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.

Aho and Corasick 62 have proposed the algorithm determining if a given text T contains an occurrence of a word from a given set ℋ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaat0uy0HwzTfgDPnwy1egaryqtHrhAL1wy0L2yHvdaiqaacqWFlecsaaa@3762@. The basic data structure is a prefix tree which is a variant of the classical trie T(ℋ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaat0uy0HwzTfgDPnwy1egaryqtHrhAL1wy0L2yHvdaiqaacqWFtepvcqGGOaakcqWFlecscqGGPaqkaaa@3AF1@42 that may be built on the set of words ℋ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaat0uy0HwzTfgDPnwy1egaryqtHrhAL1wy0L2yHvdaiqaacqWFlecsaaa@3762@. Let Qℋ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGrbqudaWgaaWcbaWenfgDOvwBHrxAJfwnHbqeg0uy0HwzTfgDPnwy1aaceaGae83cHGeabeaaaaa@38B9@ denote the set of prefixes of these words. In the following, we identify a word q ∈ Qℋ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGrbqudaWgaaWcbaWenfgDOvwBHrxAJfwnHbqeg0uy0HwzTfgDPnwy1aaceaGae83cHGeabeaaaaa@38B9@ 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 δ:Qℋ×Σ→Qℋ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaiiGacqWF0oazcqGG6aGocqWGrbqudaWgaaWcbaWenfgDOvwBHrxAJfwnHbqeg0uy0HwzTfgDPnwy1aaceaGae43cHGeabeaakiabgEna0kabfo6atjabgkziUkabdgfarnaaBaaaleaacqGFlecsaeqaaaaa@4341@ defined as follows. For any pair (p, a) in Qℋ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGrbqudaWgaaWcbaWenfgDOvwBHrxAJfwnHbqeg0uy0HwzTfgDPnwy1aaceaGae83cHGeabeaaaaa@38B9@ × Σ, δ (p, a) is the largest suffix of concatenation pa that belongs to Qℋ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGrbqudaWgaaWcbaWenfgDOvwBHrxAJfwnHbqeg0uy0HwzTfgDPnwy1aaceaGae83cHGeabeaaaaa@38B9@. Remark that δ (p, a) = pa iff pa ∈ Qℋ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGrbqudaWgaaWcbaWenfgDOvwBHrxAJfwnHbqeg0uy0HwzTfgDPnwy1aaceaGae83cHGeabeaaaaa@38B9@.

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 Qℋ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGrbqudaWgaaWcbaWenfgDOvwBHrxAJfwnHbqeg0uy0HwzTfgDPnwy1aaceaGae83cHGeabeaaaaa@38B9@. 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 ℋ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaat0uy0HwzTfgDPnwy1egaryqtHrhAL1wy0L2yHvdaiqaacqWFlecsaaa@3762@ be the set {AAA, AAC, ACA, ACA, CCT}. The corresponding tree T(ℋ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaat0uy0HwzTfgDPnwy1egaryqtHrhAL1wy0L2yHvdaiqaacqWFtepvcqGGOaakcqWFlecscqGGPaqkaaa@3AF1@ is depicted in Figure 1. Values of δ function are given in Table 1. Aho-Corasick traversal of tree T(ℋ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaat0uy0HwzTfgDPnwy1egaryqtHrhAL1wy0L2yHvdaiqaacqWFtepvcqGGOaakcqWFlecscqGGPaqkaaa@3AF1@ according to text T = 'ATGCCAACCTT' produces the following sequence of nodes {q_i}_{i ≥ 1} in Qℋ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGrbqudaWgaaWcbaWenfgDOvwBHrxAJfwnHbqeg0uy0HwzTfgDPnwy1aaceaGae83cHGeabeaaaaa@38B9@ (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].

T(ℋ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaat0uy0HwzTfgDPnwy1egaryqtHrhAL1wy0L2yHvdaiqaacqWFtepvcqGGOaakcqWFlecscqGGPaqkaaa@3AF1@ 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 ℋ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaat0uy0HwzTfgDPnwy1egaryqtHrhAL1wy0L2yHvdaiqaacqWFlecsaaa@3762@.

The combination of tree T(ℋ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaat0uy0HwzTfgDPnwy1egaryqtHrhAL1wy0L2yHvdaiqaacqWFtepvcqGGOaakcqWFlecscqGGPaqkaaa@3AF1@ 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.

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 (ℋ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaat0uy0HwzTfgDPnwy1egaryqtHrhAL1wy0L2yHvdaiqaacqWFlecsaaa@3762@; 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 ℋ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaat0uy0HwzTfgDPnwy1egaryqtHrhAL1wy0L2yHvdaiqaacqWFlecsaaa@3762@.

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 ℋ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaat0uy0HwzTfgDPnwy1egaryqtHrhAL1wy0L2yHvdaiqaacqWFlecsaaa@3762@,

3. A traversal AC (T(ℋ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaat0uy0HwzTfgDPnwy1egaryqtHrhAL1wy0L2yHvdaiqaacqWFtepvcqGGOaakcqWFlecscqGGPaqkaaa@3AF1@, 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 ℋ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaat0uy0HwzTfgDPnwy1egaryqtHrhAL1wy0L2yHvdaiqaacqWFlecsaaa@3762@.

For a given number i larger than m, the union for classes C_i (0; q), where q is in Qℋ\ℋ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGrbqudaWgaaWcbaWenfgDOvwBHrxAJfwnHbqeg0uy0HwzTfgDPnwy1aaceaGae83cHGeabeaakiabcYfaCjab=Tqiibaa@3AFC@ 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 Qℋ\ℋ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGrbqudaWgaaWcbaWenfgDOvwBHrxAJfwnHbqeg0uy0HwzTfgDPnwy1aaceaGae83cHGeabeaakiabcYfaCjab=Tqiibaa@3AFC@, or to a class G_i (1). Indeed, condition 3. means that the largest suffix of T_i in Qℋ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGrbqudaWgaaWcbaWenfgDOvwBHrxAJfwnHbqeg0uy0HwzTfgDPnwy1aaceaGae83cHGeabeaaaaa@38B9@ is q. It follows from condition 2. that classes C_i (q; 0) are empty if q is in ℋ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaat0uy0HwzTfgDPnwy1egaryqtHrhAL1wy0L2yHvdaiqaacqWFlecsaaa@3762@. A text T_i of length i is in G_i (1) if and only if a node of ℋ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaat0uy0HwzTfgDPnwy1egaryqtHrhAL1wy0L2yHvdaiqaacqWFlecsaaa@3762@ 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 (ℋ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaat0uy0HwzTfgDPnwy1egaryqtHrhAL1wy0L2yHvdaiqaacqWFlecsaaa@3762@; 1) = G_n (1); therefore the desired p-value P (L_n (ℋ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaat0uy0HwzTfgDPnwy1egaryqtHrhAL1wy0L2yHvdaiqaacqWFlecsaaa@3762@; 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 Qℋ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGrbqudaWgaaWcbaWenfgDOvwBHrxAJfwnHbqeg0uy0HwzTfgDPnwy1aaceaGae83cHGeabeaaaaa@38B9@ (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).

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

(i) if the text T_i contains a word from ℋ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaat0uy0HwzTfgDPnwy1egaryqtHrhAL1wy0L2yHvdaiqaacqWFlecsaaa@3762@ then all its concatenations with characters from Σ would contain a word from ℋ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaat0uy0HwzTfgDPnwy1egaryqtHrhAL1wy0L2yHvdaiqaacqWFlecsaaa@3762@; i.e.,

G_i (1)·a ⊂ G_i+1 (1).

(ii) if the text T_i does not contain a word from ℋ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaat0uy0HwzTfgDPnwy1egaryqtHrhAL1wy0L2yHvdaiqaacqWFlecsaaa@3762@ and belongs to C_i+1 (0; q), i.e., ends with q ∈ Qℋ\ℋ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGrbqudaWgaaWcbaWenfgDOvwBHrxAJfwnHbqeg0uy0HwzTfgDPnwy1aaceaGae83cHGeabeaakiabcYfaCjab=Tqiibaa@3AFC@, 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) ∈ ℋ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaat0uy0HwzTfgDPnwy1egaryqtHrhAL1wy0L2yHvdaiqaacqWFlecsaaa@3762@,   then C_i (0; q)·a ⊂ C_i+1 (0; δ (q, a))

otherwise   C_i (0; q) ⊂ G_i+1 (1).

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 ℋ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaat0uy0HwzTfgDPnwy1egaryqtHrhAL1wy0L2yHvdaiqaacqWFlecsaaa@3762@:

G i + 1 ( 1 ) = { ∪ a ∈ Σ G i ( 1 ) ⋅ a } ∪ { ∪ ( q , a ) ; δ ( q , a ) ∈ ℋ C i ( 0 ; q ) ⋅ a } . MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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@72A7@

Similarly, classes of texts that do not contain words from ℋ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaat0uy0HwzTfgDPnwy1egaryqtHrhAL1wy0L2yHvdaiqaacqWFlecsaaa@3762@ satisfy

∀ q ′ ∈ Q ℋ \ ℋ : C i + 1 ( 0 ; q ′ ) = ∪ ( q , a ) ; δ ( q , a ) = q ′ C i ( 0 ; q ) ⋅ a . MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaafaqabeqacaaabaGaeyiaIiIafmyCaeNbauaacqGHiiIZcqWGrbqudaWgaaWcbaWenfgDOvwBHrxAJfwnHbqeg0uy0HwzTfgDPnwy1aaceaGae83cHGeabeaakiabcYfaCjab=TqiijabcQda6aqaaiabdoeadnaaBaaaleaacqWGPbqAcqGHRaWkcqaIXaqmaeqaaOGaeiikaGIaeGimaaJaei4oaSJafmyCaeNbauaacqGGPaqkcqGH9aqpdaWeqbqaaiabdoeadnaaBaaaleaacqWGPbqAaeqaaOGaeiikaGIaeGimaaJaei4oaSJaemyCaeNaeiykaKIaeyyXICTaemyyaegaleaacqGGOaakcqWGXbqCcqGGSaalcqWGHbqycqGGPaqkcqGG7aWoiiGacqGF0oazcqGGOaakcqWGXbqCcqGGSaalcqWGHbqycqGGPaqkcqGH9aqpcuWGXbqCgaqbaaqab0GaeSOkIufakiabc6caUaaaaaa@67F3@

Classes C_i (0; q) for different q in Qℋ\ℋ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGrbqudaWgaaWcbaWenfgDOvwBHrxAJfwnHbqeg0uy0HwzTfgDPnwy1aaceaGae83cHGeabeaakiabcYfaCjab=Tqiibaa@3AFC@ and G_i (1) form a partition of Σⁱ; classes C_i (0; q) are empty if q is in ℋ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaat0uy0HwzTfgDPnwy1egaryqtHrhAL1wy0L2yHvdaiqaacqWFlecsaaa@3762@. 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:

P ( G i + 1 ( 1 ) ) = P ( G i ( 1 ) ) + ∑ ( q , a ) : δ ( q , a ) ∈ ℋ P ( C i ( 0 ; q ) ) ⋅ p ( a ) , MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaieqacqWFqbaucqGGOaakcqWGhbWrdaWgaaWcbaGaemyAaKMaey4kaSIaeGymaedabeaakiabcIcaOiabigdaXiabcMcaPiabcMcaPiabg2da9iab=bfaqjabcIcaOiabdEeahnaaBaaaleaacqWGPbqAaeqaaOGaeiikaGIaeGymaeJaeiykaKIaeiykaKIaey4kaSYaaabuaeaacqWFqbaucqGGOaakcqWGdbWqdaWgaaWcbaGaemyAaKgabeaakiabcIcaOiabicdaWiabcUda7iabdghaXjabcMcaPiabcMcaPiabgwSixlabdchaWjabcIcaOiabdggaHjabcMcaPaWcbaGaeiikaGIaemyCaeNaeiilaWIaemyyaeMaeiykaKIaeiOoaOdcciGae4hTdqMaeiikaGIaemyCaeNaeiilaWIaemyyaeMaeiykaKIaeyicI48enfgDOvwBHrxAJfwnHbqeg0uy0HwzTfgDPnwy1aaceaGae03cHGeabeqdcqGHris5aOGaeiilaWcaaa@6E5E@

P ( C i + 1 ( 0 ; q ′ ) ) = ∑ ( q , a ) : δ ( q , a ) = q ′ P ( C i ( 0 ; q ) ) ⋅ p ( a ) . MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaieqacqWFqbaucqGGOaakcqWGdbWqdaWgaaWcbaGaemyAaKMaey4kaSIaeGymaedabeaakiabcIcaOiabicdaWiabcUda7iqbdghaXzaafaGaeiykaKIaeiykaKIaeyypa0ZaaabuaeaacqWFqbaucqGGOaakcqWGdbWqdaWgaaWcbaGaemyAaKgabeaakiabcIcaOiabicdaWiabcUda7iabdghaXjabcMcaPiabcMcaPiabgwSixlabdchaWjabcIcaOiabdggaHjabcMcaPaWcbaGaeiikaGIaemyCaeNaeiilaWIaemyyaeMaeiykaKIaeiOoaOdcciGae4hTdqMaeiikaGIaemyCaeNaeiilaWIaemyyaeMaeiykaKIaeyypa0JafmyCaeNbauaaaeqaniabggHiLdGccqGGUaGlaaa@5E0F@

The run-time for each step of the computation of C_i+1 (0; q) and G_i+1 (1) is O (|Qℋ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGrbqudaWgaaWcbaWenfgDOvwBHrxAJfwnHbqeg0uy0HwzTfgDPnwy1aaceaGae83cHGeabeaaaaa@38B9@|·|Σ|); therefore the total time of all n stages of p-value computation is O (|Qℋ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGrbqudaWgaaWcbaWenfgDOvwBHrxAJfwnHbqeg0uy0HwzTfgDPnwy1aaceaGae83cHGeabeaaaaa@38B9@|·|Σ|·n).

The approach described in this section can be readily extended to the case of multiple occurrences of motif ℋ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaat0uy0HwzTfgDPnwy1egaryqtHrhAL1wy0L2yHvdaiqaacqWFlecsaaa@3762@. The detailed procedure can be found in Additional file 1.

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 (ℋ1,...,ℋs MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaat0uy0HwzTfgDPnwy1egaryqtHrhAL1wy0L2yHvdaiqaacqWFlecsdaWgaaWcbaGaeGymaedabeaakiabcYcaSiabc6caUiabc6caUiabc6caUiabcYcaSiab=TqiinaaBaaaleaacqWGZbWCaeqaaaaa@3F88@) be s different motifs. Our objective is to calculate the probability that motifs (ℋ1,...,ℋs MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaat0uy0HwzTfgDPnwy1egaryqtHrhAL1wy0L2yHvdaiqaacqWFlecsdaWgaaWcbaGaeGymaedabeaakiabcYcaSiabc6caUiabc6caUiabc6caUiabcYcaSiab=TqiinaaBaaaleaacqWGZbWCaeqaaaaa@3F88@) 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 (ℋ1,...,ℋs MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaat0uy0HwzTfgDPnwy1egaryqtHrhAL1wy0L2yHvdaiqaacqWFlecsdaWgaaWcbaGaeGymaedabeaakiabcYcaSiabc6caUiabc6caUiabc6caUiabcYcaSiab=TqiinaaBaaaleaacqWGZbWCaeqaaaaa@3F88@; k₁, ..., k_s)) to obtain text T_n belonging to the set of texts L_n (ℋ1,...,ℋs MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaat0uy0HwzTfgDPnwy1egaryqtHrhAL1wy0L2yHvdaiqaacqWFlecsdaWgaaWcbaGaeGymaedabeaakiabcYcaSiabc6caUiabc6caUiabc6caUiabcYcaSiab=TqiinaaBaaaleaacqWGZbWCaeqaaaaa@3F88@; 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 ℋ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaat0uy0HwzTfgDPnwy1egaryqtHrhAL1wy0L2yHvdaiqaacqWFlecsaaa@3762@ of individual motifs ℋ=ℋ1∪⋯∪ℋs MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaat0uy0HwzTfgDPnwy1egaryqtHrhAL1wy0L2yHvdaiqaacqWFlecscqGH9aqpcqWFlecsdaWgaaWcbaGaeGymaedabeaakiabgQIiilabl+UimjabgQIiilab=TqiinaaBaaaleaacqWGZbWCaeqaaaaa@4249@. It contains all words that belong to any of motifs ℋ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaat0uy0HwzTfgDPnwy1egaryqtHrhAL1wy0L2yHvdaiqaacqWFlecsaaa@3762@_i. The tree T(ℋ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaat0uy0HwzTfgDPnwy1egaryqtHrhAL1wy0L2yHvdaiqaacqWFtepvcqGGOaakcqWFlecscqGGPaqkaaa@3AF1@ is constructed for the overall set ℋ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaat0uy0HwzTfgDPnwy1egaryqtHrhAL1wy0L2yHvdaiqaacqWFlecsaaa@3762@, its nodes Qℋ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGrbqudaWgaaWcbaWenfgDOvwBHrxAJfwnHbqeg0uy0HwzTfgDPnwy1aaceaGae83cHGeabeaaaaa@38B9@ contain all possible prefixes of all motifs from (ℋ1,...,ℋs MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaat0uy0HwzTfgDPnwy1egaryqtHrhAL1wy0L2yHvdaiqaacqWFlecsdaWgaaWcbaGaeGymaedabeaakiabcYcaSiabc6caUiabc6caUiabc6caUiabcYcaSiab=TqiinaaBaaaleaacqWGZbWCaeqaaaaa@3F88@). A node of the tree q ∈ Qℋ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGrbqudaWgaaWcbaWenfgDOvwBHrxAJfwnHbqeg0uy0HwzTfgDPnwy1aaceaGae83cHGeabeaaaaa@38B9@ can belong to some motif ℋ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaat0uy0HwzTfgDPnwy1egaryqtHrhAL1wy0L2yHvdaiqaacqWFlecsaaa@3762@_k or simultaneously to several different motifs from {ℋ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaat0uy0HwzTfgDPnwy1egaryqtHrhAL1wy0L2yHvdaiqaacqWFlecsaaa@3762@_j}_1≤j≤s. Let each node q ∈ Qℋ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGrbqudaWgaaWcbaWenfgDOvwBHrxAJfwnHbqeg0uy0HwzTfgDPnwy1aaceaGae83cHGeabeaaaaa@38B9@ be marked with numbers j of motifs ℋ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaat0uy0HwzTfgDPnwy1egaryqtHrhAL1wy0L2yHvdaiqaacqWFlecsaaa@3762@_j to which it belongs. Nodes, corresponding to proper prefixes of ℋ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaat0uy0HwzTfgDPnwy1egaryqtHrhAL1wy0L2yHvdaiqaacqWFlecsaaa@3762@, remain unmarked. The transition function δ:Qℋ×Σ→Qℋ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaiiGacqWF0oazcqGG6aGocqWGrbqudaWgaaWcbaWenfgDOvwBHrxAJfwnHbqeg0uy0HwzTfgDPnwy1aaceaGae43cHGeabeaakiabgEna0kabfo6atjabgkziUkabdgfarnaaBaaaleaacqGFlecsaeqaaaaa@4341@ is defined as it was defined in the case of a single motif for the unified motif ℋ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaat0uy0HwzTfgDPnwy1egaryqtHrhAL1wy0L2yHvdaiqaacqWFlecsaaa@3762@.

All texts T_n of length n are classified into classes depending on occurrences of different ℋ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaat0uy0HwzTfgDPnwy1egaryqtHrhAL1wy0L2yHvdaiqaacqWFlecsaaa@3762@_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 ℋ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaat0uy0HwzTfgDPnwy1egaryqtHrhAL1wy0L2yHvdaiqaacqWFlecsaaa@3762@_i, the corresponding value k_j may not yet be attained for another motif ℋ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaat0uy0HwzTfgDPnwy1egaryqtHrhAL1wy0L2yHvdaiqaacqWFlecsaaa@3762@_j. Therefore we need to introduce the occurrence index of a set of motifs.

Definition 2 Let the target number of occurrences of motif ℋ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaat0uy0HwzTfgDPnwy1egaryqtHrhAL1wy0L2yHvdaiqaacqWFlecsaaa@3762@_i be k_i. Then, the occurrence index Λ(k1,...,ks) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqqHBoatdaWgaaWcbaGaeiikaGIaem4AaS2aaSbaaWqaaiabigdaXaqabaWccqGGSaalcqGGUaGlcqGGUaGlcqGGUaGlcqGGSaalcqWGRbWAdaWgaaadbaGaem4CamhabeaaliabcMcaPaqabaaaaa@39F8@ (l₁, ..., l_s) of a set of motifs (ℋ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaat0uy0HwzTfgDPnwy1egaryqtHrhAL1wy0L2yHvdaiqaacqWFlecsaaa@3762@) in the text T_n containing l_i possibly overlapping occurrences of each ℋ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaat0uy0HwzTfgDPnwy1egaryqtHrhAL1wy0L2yHvdaiqaacqWFlecsaaa@3762@_i is an s-vector the ith component of which can be calculated as follows:

[ Λ ( k 1 , ... , k s ) ( l 1 , ... , l s ) ] i = λ i = { l i i f l i ≤ k i , k i i f l i > k i . MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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@6BCA@

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 (ℋ1,...,ℋs MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaat0uy0HwzTfgDPnwy1egaryqtHrhAL1wy0L2yHvdaiqaacqWFlecsdaWgaaWcbaGaeGymaedabeaakiabcYcaSiabc6caUiabc6caUiabc6caUiabcYcaSiab=TqiinaaBaaaleaacqWGZbWCaeqaaaaa@3F88@) in text T_i is equal to (λ₁, ..., λ_s),

3. A traversal AC (T(ℋ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaat0uy0HwzTfgDPnwy1egaryqtHrhAL1wy0L2yHvdaiqaacqWFtepvcqGGOaakcqWFlecscqGGPaqkaaa@3AF1@, 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

G i ( k 1 , ... , k s ) = ∪ q ∈ ℋ C i ( k 1 , ... , k s ; q ) . MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGhbWrdaWgaaWcbaGaemyAaKgabeaakiabcIcaOiabdUgaRnaaBaaaleaacqaIXaqmaeqaaOGaeiilaWIaeiOla4IaeiOla4IaeiOla4IaeiilaWIaem4AaS2aaSbaaSqaaiabdohaZbqabaGccqGGPaqkcqGH9aqpdaWeqbqaaiabdoeadnaaBaaaleaacqWGPbqAaeqaaOGaeiikaGIaem4AaS2aaSbaaSqaaiabigdaXaqabaGccqGGSaalcqGGUaGlcqGGUaGlcqGGUaGlcqGGSaalcqWGRbWAdaWgaaWcbaGaem4CamhabeaakiabcUda7iabdghaXjabcMcaPaWcbaGaemyCaeNaeyicI48enfgDOvwBHrxAJfwnHbqeg0uy0HwzTfgDPnwy1aaceaGae83cHGeabeqdcqWIQisvaOGaeiOla4caaa@5CF8@

The desired p-value P (L_n (ℋ1,...,ℋs MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaat0uy0HwzTfgDPnwy1egaryqtHrhAL1wy0L2yHvdaiqaacqWFlecsdaWgaaWcbaGaeGymaedabeaakiabcYcaSiabc6caUiabc6caUiabc6caUiabcYcaSiab=TqiinaaBaaaleaacqWGZbWCaeqaaaaa@3F88@; 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 {ℋ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaat0uy0HwzTfgDPnwy1egaryqtHrhAL1wy0L2yHvdaiqaacqWFlecsaaa@3762@_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

P ( C i + 1 ( λ 1 , ... , λ s ; q ′ ) ) = ∑ ( q , a ) : δ ( q , a ) = q ′ ∑ ( r 1 , ... , r s ) ∈ J P ( C i ( r 1 , ... , r s ; q ) ) ⋅ p ( a ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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@8070@

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' ∉ ℋ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaat0uy0HwzTfgDPnwy1egaryqtHrhAL1wy0L2yHvdaiqaacqWFlecsaaa@3762@_j then r_j = λ_j,

2. if q' ℋ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaat0uy0HwzTfgDPnwy1egaryqtHrhAL1wy0L2yHvdaiqaacqWFlecsaaa@3762@_j and λ_j <k_j then r_j = λ_j - 1,

3. if q' ∈ ℋ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaat0uy0HwzTfgDPnwy1egaryqtHrhAL1wy0L2yHvdaiqaacqWFlecsaaa@3762@_j and λ_j = k_j then r_j = k_j or r_j = k_j - 1.

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 ∈ Qℋ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGrbqudaWgaaWcbaWenfgDOvwBHrxAJfwnHbqeg0uy0HwzTfgDPnwy1aaceaGae83cHGeabeaaaaa@38B9@ 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 ∈ Qℋ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGrbqudaWgaaWcbaWenfgDOvwBHrxAJfwnHbqeg0uy0HwzTfgDPnwy1aaceaGae83cHGeabeaaaaa@38B9@ 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 ∈ Qℋ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGrbqudaWgaaWcbaWenfgDOvwBHrxAJfwnHbqeg0uy0HwzTfgDPnwy1aaceaGae83cHGeabeaaaaa@38B9@ 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' ∉ ℋ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaat0uy0HwzTfgDPnwy1egaryqtHrhAL1wy0L2yHvdaiqaacqWFlecsaaa@3762@_j then λ_j = r_j,

2. if q' ∈ ℋ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaat0uy0HwzTfgDPnwy1egaryqtHrhAL1wy0L2yHvdaiqaacqWFlecsaaa@3762@_j and r_j <k_j then λ_j = r_j + 1,

3. if q' ∈ ℋ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaat0uy0HwzTfgDPnwy1egaryqtHrhAL1wy0L2yHvdaiqaacqWFlecsaaa@3762@_j and r_j = k_j then λ_j = r_j.

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

P ( G n ( k 1 , ... , k s ) ) = ∑ q ∈ ℋ P ( C n ( k 1 , ... , k s ; q ) ) . MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaieqacqWFqbaucqGGOaakcqWGhbWrdaWgaaWcbaGaemOBa4gabeaakiabcIcaOiabdUgaRnaaBaaaleaacqaIXaqmaeqaaOGaeiilaWIaeiOla4IaeiOla4IaeiOla4IaeiilaWIaem4AaS2aaSbaaSqaaiabdohaZbqabaGccqGGPaqkcqGGPaqkcqGH9aqpdaaeqbqaaiab=bfaqjabcIcaOiabdoeadnaaBaaaleaacqWGUbGBaeqaaOGaeiikaGIaem4AaS2aaSbaaSqaaiabigdaXaqabaGccqGGSaalcqGGUaGlcqGGUaGlcqGGUaGlcqGGSaalcqWGRbWAdaWgaaWcbaGaem4CamhabeaakiabcUda7iabdghaXjabcMcaPiabcMcaPiabc6caUaWcbaGaemyCaeNaeyicI48enfgDOvwBHrxAJfwnHbqeg0uy0HwzTfgDPnwy1aaceaGae43cHGeabeqdcqGHris5aaaa@6328@

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 ℋ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaat0uy0HwzTfgDPnwy1egaryqtHrhAL1wy0L2yHvdaiqaacqWFlecsaaa@3762@ = {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 (ℋ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaat0uy0HwzTfgDPnwy1egaryqtHrhAL1wy0L2yHvdaiqaacqWFlecsaaa@3762@; 1)), P (L_n (ℋ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaat0uy0HwzTfgDPnwy1egaryqtHrhAL1wy0L2yHvdaiqaacqWFlecsaaa@3762@; k)), and P (L_n (ℋ1,...,ℋs MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaat0uy0HwzTfgDPnwy1egaryqtHrhAL1wy0L2yHvdaiqaacqWFlecsdaWgaaWcbaGaeGymaedabeaakiabcYcaSiabc6caUiabc6caUiabc6caUiabcYcaSiab=TqiinaaBaaaleaacqWGZbWCaeqaaaaa@3F88@; 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

t [ 1 ] ⋯ t [ K ] = { w ⋅ q if  0 ≤ d < K , K -suffix of  q otherwise . MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWG0baDcqGGBbWwcqaIXaqmcqGGDbqxcqWIVlctcqWG0baDcqGGBbWwcqWGlbWscqGGDbqxcqGH9aqpdaGabeqaauaabaqaciaaaeaacqWG3bWDcqGHflY1cqWGXbqCaeaacqqGPbqAcqqGMbGzcqqGGaaicqaIWaamcqGHKjYOcqWGKbazcqGH8aapcqWGlbWscqGGSaalaeaacqWGlbWscqqGTaqlcqqGZbWCcqqG1bqDcqqGMbGzcqqGMbGzcqqGPbqAcqqG4baEcqqGGaaicqqGVbWBcqqGMbGzcqqGGaaicqWGXbqCaeaacqqGVbWBcqqG0baDcqqGObaAcqqGLbqzcqqGYbGCcqqG3bWDcqqGPbqAcqqGZbWCcqqGLbqzcqGGUaGlaaaacaGL7baaaaa@67AD@

The Markov extension is currently implemented for K = 1.

To resume, the computation of P (L_n (ℋ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaat0uy0HwzTfgDPnwy1egaryqtHrhAL1wy0L2yHvdaiqaacqWFlecsaaa@3762@; k)) for one set ℋ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaat0uy0HwzTfgDPnwy1egaryqtHrhAL1wy0L2yHvdaiqaacqWFlecsaaa@3762@ requires a computation of (P(Ci(l,q)))0≤l<k,q∈Qℋ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaadaqadaqaaGqabiab=bfaqjabcIcaOiabdoeadnaaBaaaleaacqWGPbqAaeqaaOGaeiikaGIaemiBaWMaeiilaWIaemyCaeNaeiykaKIaeiykaKcacaGLOaGaayzkaaWaaSbaaSqaaiabicdaWiabgsMiJkabdYgaSjabgYda8iabdUgaRjabcYcaSiabdghaXjabgIGiolabdgfarnaaBaaameaat0uy0HwzTfgDPnwy1egaryqtHrhAL1wy0L2yHvdaiqaacqGFlecsaeqaaaWcbeaaaaa@4F8E@ for i ≤ n. For each iteration, the time complexity is O (k|Qℋ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGrbqudaWgaaWcbaWenfgDOvwBHrxAJfwnHbqeg0uy0HwzTfgDPnwy1aaceaGae83cHGeabeaaaaa@38B9@| |Σ|), where |Σ| is the size of the alphabet. One traverses the tree n times. As |Qℋ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGrbqudaWgaaWcbaWenfgDOvwBHrxAJfwnHbqeg0uy0HwzTfgDPnwy1aaceaGae83cHGeabeaaaaa@38B9@| is upper bounded by (m|ℋ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaat0uy0HwzTfgDPnwy1egaryqtHrhAL1wy0L2yHvdaiqaacqWFlecsaaa@3762@|), where m is the maximal length of word in ℋ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaat0uy0HwzTfgDPnwy1egaryqtHrhAL1wy0L2yHvdaiqaacqWFlecsaaa@3762@, this yields the overall O (nkm|ℋ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaat0uy0HwzTfgDPnwy1egaryqtHrhAL1wy0L2yHvdaiqaacqWFlecsaaa@3762@||Σ|) time complexity and a O (km|ℋ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaat0uy0HwzTfgDPnwy1egaryqtHrhAL1wy0L2yHvdaiqaacqWFlecsaaa@3762@|) space complexity.

When several sets are involved, the number of nodes in the tree T(ℋ1∪⋯∪ℋs) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaat0uy0HwzTfgDPnwy1egaryqtHrhAL1wy0L2yHvdaiqaacqWFtepvcqGGOaakcqWFlecsdaWgaaWcbaGaeGymaedabeaakiabgQIiilabl+UimjabgQIiilab=TqiinaaBaaaleaacqWGZbWCaeqaaOGaeiykaKcaaa@43E3@ becomes O (m|ℋ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaat0uy0HwzTfgDPnwy1egaryqtHrhAL1wy0L2yHvdaiqaacqWFlecsaaa@3762@|) with m equal to the maximal length of word in ℋ=ℋ1∪⋯∪ℋs MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaat0uy0HwzTfgDPnwy1egaryqtHrhAL1wy0L2yHvdaiqaacqWFlecscqGH9aqpcqWFlecsdaWgaaWcbaGaeGymaedabeaakiabgQIiilabl+UimjabgQIiilab=TqiinaaBaaaleaacqWGZbWCaeqaaaaa@4249@. Additional memory in each node is ∏_i k_i. Therefore, the time complexity is O (nm|Σ|∏_i k_i|ℋ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaat0uy0HwzTfgDPnwy1egaryqtHrhAL1wy0L2yHvdaiqaacqWFlecsaaa@3762@|) and the space complexity is O (m ∏_i k_i |ℋ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaat0uy0HwzTfgDPnwy1egaryqtHrhAL1wy0L2yHvdaiqaacqWFlecsaaa@3762@|). 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|ℋ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaat0uy0HwzTfgDPnwy1egaryqtHrhAL1wy0L2yHvdaiqaacqWFlecsaaa@3762@| + K|Σ|^K). Therefore, the space memory is O ((m|ℋ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaat0uy0HwzTfgDPnwy1egaryqtHrhAL1wy0L2yHvdaiqaacqWFlecsaaa@3762@| + K |Σ|^K) ∏_i k_i) and the running time is O (n|Σ|(m|ℋ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaat0uy0HwzTfgDPnwy1egaryqtHrhAL1wy0L2yHvdaiqaacqWFlecsaaa@3762@| + K |Σ|^K )∏_i k_i). This additive increment compares favorably to simple induction methods 4553 that introduce a multiplicative O (K|Σ|^K) factor in time and space complexity for the Markov(K) model.

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.

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), 6.3044, is attained when the bicoid cutoff is equal to S₁ = 5.1 and the kruppel cutoff is equal to S₂ = 5.6. With such cutoff values in the sequence of the eve2 enhancer there are k₁ = 6 and k₂ = 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 (S₁, S₂) = (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(pvalue) for this random sequence is about 1.02 which is 6.17 times smaller than maximal – log(pvalue) 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 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.

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 functional.

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 minimal 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.