Abstract
Background
Position Weight Matrices (PWMs) are probabilistic representations of signals in sequences. They are widely used to model approximate patterns in DNA or in protein sequences. The usage of PWMs needs as a prerequisite to knowing the statistical significance of a word according to its score. This is done by defining the Pvalue of a score, which is the probability that the background model can achieve a score larger than or equal to the observed value. This gives rise to the following problem: Given a Pvalue, find the corresponding score threshold. Existing methods rely on dynamic programming or probability generating functions. For many examples of PWMs, they fail to give accurate results in a reasonable amount of time.
Results
The contribution of this paper is two fold. First, we study the theoretical complexity of the problem, and we prove that it is NPhard. Then, we describe a novel algorithm that solves the Pvalue problem efficiently. The main idea is to use a series of discretized score distributions that improves the final result step by step until some convergence criterion is met. Moreover, the algorithm is capable of calculating the exact Pvalue without any error, even for matrices with noninteger coefficient values. The same approach is also used to devise an accurate algorithm for the reverse problem: finding the Pvalue for a given score. Both methods are implemented in a software called TFMPVALUE, that is freely available.
Conclusion
We have tested TFMPVALUE on a large set of PWMs representing transcription factor binding sites. Experimental results show that it achieves better performance in terms of computational time and precision than existing tools.
Background
A key problem in the understanding of gene regulation is the identification of transcription factor binding sites. Transcription factor binding sites are often modeled by Position Weighted Matrices (PWMs for short), also known as Position Specific Scoring Matrices (PSSMs for short), or simply matrices. Examples are to be found in the Jaspar [1] or Transfac [2] databases. The usage of such matrices goes with global bioinformatics strategies that help to elucidate regulation mechanisms: comparative genomics, identification of overrepresented motifs, identification of correlation between binding sites, ... Similar matrixbased models also serve to represent splice sites in messenger RNAs [3] or signatures in amino acid sequences [4].
Matrices are probabilistic descriptions of approximate patterns. Given a finite alphabet Σ and a positive integer m, a matrix M is a function from Σ^{m }to ℝ that associates a score to each word of Σ^{m}. More precisely, it is indexed by {1,...,m} × Σ. Each column corresponds to a position in the motif and each row to a letter in the alphabet Σ. The coefficient M (i, x) gives the score at position i in [1, m] for the letter x in Σ. Given a string u in Σ^{m}, the score of M on u is defined as the sum of the scores of each character symbol of u:
where u_{i }denotes the character symbol at position i in u.
Searching for occurrences of a matrix in a sequence requires to choose an appropriate score threshold to decide whether a position is relevant or not. Let α be such a score. We say that the matrix M has an occurrence in the sequence S at position i if Score(S_{i }... S_{i+m1}, M) ≥ α. The problem of efficiently finding occurrences of a matrix in a text has recently attracted a lot of interest [57]. Here we address the problem of computing the score threshold α. To determine such a score threshold, the standard method is to use a Pvalue function, which gives the statistical significance of an occurrence according to its score. The Pvalue Pvalue(M, α) is the probability that the background model can achieve a score equal to or greater than α. In other words, the Pvalue is the proportion of strings (with respect to the background model) whose score is greater than the threshold α for M. In [8], the authors introduce a generic approach to Pvalue computation for nonparametric models. In the context of matrices, the computation can be carried out using probability generating functions or dynamic programming [912]. In both cases, the time complexity is proportional to the product of the length of the matrix and the number of possible different scores. If the matrix has nonnegative integer coefficient values, then the number of possible different scores is bounded by . It follows that known algorithms are pseudopolynomial. In real life, matrices have actually real coefficient values, such as logratio matrices, or entropy matrices. In this context, the number of different scores that the matrix can achieve is significantly larger.
Theoretically, it can be as high as Σ^{m}. The usual way to deal with real matrices is to round them at a given precision, such as a given number of digits after the decimal point. In this context, the number of scores depends strongly on the chosen precision. Figure 1 displays such an example. It shows the number of distinct scores obtained with the matrix MA0041 from the Jaspar database for a variety of rounding values. With a precision set to 10^{6}, we get more than one million distinct scores. Existing algorithms have difficulties to deal with such a large number of scores. An alternative consists in using a rough estimation, such a 10^{3}. In this context, the estimated distribution induced by the round matrix is likely to give larger error rates. For example, Figure 2 shows the logo [13] of the matrix MA0045 of length 16 from the Jaspar database. We chose 5 as a score threshold, which corresponds approximately to a Pvalue equal to 10^{3}. The number of words whose score is greater than or equal to 5 is 4045101 onto the original matrix, compared to 4034054 for the round matrix with a precision of 10^{3}. This makes a difference of 11047 words. This error naturally affects the accuracy to the Pvalue. To estimate this, we conducted a large scale experiment on all Jaspar matrices (123 matrices) for a variety of precisions and a uniform Pvalue set to 10^{3}. We compared the number of words whose score is larger than the threshold when the Pvalue is computed from the corresponding round matrix to the correct number of words that is observed with the true matrix, without discretization. In each case, we indicate the percentage of matrices for which the number of words is different. Results are reported in Table 1. With a rounding at the third digits after the decimal point, 55 percent of matrices give false results. Even with a rounding at the sixth digits after the decimal point, there exist matrices for which the discretization gives a false result. This demonstrates that it may be necessary to use high precision scores to obtain accurrate results. The choice of the precision is a difficult compromise between accuracy and tractability. To the best of our knowledge, this question is passed over in silence by existing algorithms.
Figure 1. Number of scores for a round matrix. The matrix MA0041 of length 12 from the Jaspar database has been round with a number of digits after the decimal point from 1 to 8. The results are presented by a histogram showing the number of distinct scores that the round matrix can achieve. The number of scores is in log scale. The grey bar shows the number of distinct words (that is 4^{12}).
Figure 2. The MA0045 Jaspar matrix logo. The logo of the matrix MA0045 from the Jaspar database on which experiments in the Background section have been done.
Table 1. Error with round matrices. We report the percentage of Jaspar matrices for which the Pvalue computed from a round matrix leads to a different number of words as for the Pvalue computed with the original matrix. The rounding ranges from 10^{2 }to 10^{6}, and the Pvalue is 10^{3 }for a multinomial background model.
In this paper, we study the theoretical complexity of the Pvalue problem and prove that it is intrinsically difficult. It is actually NPhard. We then introduce a novel algorithm that achieves significant speed up compared to existing algorithms when we allow for some errors like other methods do. This algorithm is also capable to solve the Pvalue problem without error within a reasonable amount of time.
Complexity of the Pvalue problem
We begin by introducing formally the Pvalue problem. We actually define two complementary problems, depending on what is given and what is searched for. In both cases, we are given a finite alphabet Σ, a matrix M of length m and a probability distribution on Σ^{m}. We say that s in ℝ is an accessible score if there exists a word u in Σ^{m }such that Score(u, M) = s.
Pvalue problem – from score to Pvalue: Given a score value α, find the probability of the set {u ∈ Σ^{m}, Score(u, M) ≥ α}. This probability is denoted Pvalue(M, α).
Threshold problem – from Pvalue to score: Given a Pvalue P (0 ≤ P ≤ 1), find the highest accessible score α such that Pvalue(M, α) ≥ P. We write Threshold(M, P) for α.
As we will see later on in this paper, they are closely related problems. We show here that neither of them admits a polynomial algorithm, unless P = NP. For that, we first define the decision problem ACCESSIBLE SCORE as follows.
Instance: a finite alphabet Σ, a matrix M of length m whose coefficients are natural numbers, a natural number t
Question: does there exist a string u of Σ^{m }such that Score(u, M) = t?
Theorem 1 ACCESSIBLE SCORE is NPhard.
The proof of Theorem 1 is by reduction of the SUBSET SUM problem, which is a pseudopolynomial NPcomplete problem [14].
Instance: a set of positive integers A = {a_{0},...,a_{n}} and a positive integer s
Question: does there exist a subset A' of A such that the sum of the elements of A' equals exactly s?
Lemma 1 There exists a polynomial reduction from the SUBSET SUM problem to the ACCESSIBLE SCORE problem.
Proof. Let A = {a_{0},...,a_{n}} be a set of positive integers, and let s be the target integer. We define the matrix M of length n + 1 on the two letter alphabet Σ = {x, y} as follows: M (i, x) = a_{i }and M (i, y) = 0 for each i, 0 ≤ i ≤ n. The set A has 2^{n+1 }different subsets. So we can define a bijection φ from the set of subsets of A onto Σ^{n+1}. For each subset A', the word φ (A') is such as the ith letter is x if and only if a_{i }∈ A', otherwise the ith letter is y. It is easy to see that Score(φ (A'), M) = s if, and only if, ∑_{a∈A' }a = s.
It remains to prove that the ACCESSIBLE SCORE problem polynomially reduces to instances of the From score to Pvalue and From Pvalue to score. problems. We are now given a finite alphabet Σ, a matrice M of length m, and a score value t.
Reduction to the From score to Pvalue problem
We assume that the probability of each nonempty word of Σ^{m }is non null. Under this hypothesis, the ACCESSIBLE SCORE problem admits a solution if, and only if, Pvalue(M, t) ≠ Pvalue(M, t + 1).
Reduction to the From Pvalue to score problem
We assume that the background model for Σ* is provided with a multinomial model. In this context, all words of length m have the same probability: and all Pvalues are of the form . Solving the ACCESSIBLE SCORE problem amounts to decide whether there exists an integer k, 0 ≤ k ≤ Σ^{m}, such that Threshold(M, ) = t. The existence of such k can be decided with iterative computations of From Pvalue to Score for different values of k. This search can be performed within O(log_{2 }(Σ^{m})) steps using binary search, because k decreases monotonically in t and there are at most Σ^{m }different values for k.
Algorithms for the Pvalue problems
From now on, we assume that the positions in the sequence are independently distributed. We denote p(x) the background probability associated to the letter x of the alphabet Σ. By extension, we write p(u) for the probability of the word u = u_{1 }... u_{m}: p(u) = p(u_{1}) × ⋯ × p(u_{m}).
Definition of the score distribution
The computation of the Pvalue is done through the computation of the score distribution. This concept is the core of the large majority of existing algorithms [911,15]. Given a matrix M of length m and a score α, we define Q(M, α) as the probability that the background model can achieve a score equal to α. In other words, Q (M, α) is the probability of the set {u ∈ Σ^{m } Score(u, M) = α}. In the case where s is not an accessible score, then Q(M, s) = 0.
The computation of Q is easily performed by dynamic programming. For that purpose, we need some preliminary notation. Given two integers i, j satisfying 0 ≤ i, j ≤ m, M [i..j] denotes the submatrix of M obtained by selecting only columns from i to j for all character symbols. M [i..j] is called a slice of M. By convention, if i > j, then M [i..j] is an empty matrix.
The score distribution for the slice M [1..i] is expressed from the sore distribution of the previous slice M [1..i  1] as follows.
The time complexity is in O(mΣS), and the space complexity in O(S), where S is the number of scores that have to be visited. If coefficients of M are natural numbers, then S is bounded by m × max {M (i, x)  x ∈ Σ, 1 ≤ i ≤ m}. Equation 1 enables to solve the From score to Pvalue and From Pvalue to score problems. Given a score α, the Pvalue is obtained with the relation:
Conversely, given P, Threshold (M, P) is computed from Q by searching for the greatest accessible score until the required Pvalue is reached.
Computing the score distribution for a range of scores
Formula 1 does not explicitly state which score ranges should be taken into account in intermediate steps of the calculation of Q. To this end, we introduce the best score and the worst score of a matrix slice.
Definition 1 (Best and worst scores) Let M be a matrix. The best score of the slice M [i..j] is defined as
Similarly, the worst score of the slice M [i..j] is defined as
The notion of best scores is already present in [16], where it is used to speed up the search for occurrences of a matrix in a text. It gives rise to look ahead scoring. Best scores allow to stop the calculation of Score(u, M) in advance as soon as it is guaranteed that the score threshold cannot be achieved, because we know the maximal remaining score. It has been exploited in [5,6] in the same context. Here we adapt it to the score distribution problem. Let α and β be two scores such that α ≤ β. If one wants to compute the score distribution Q for the range [α, β], then given an intermediate score s and a matrix position i, we say that Q(M [1..i], s) is useful if there exists a word v of length m  i such that α ≤ s + Score(v, M [i + 1..m]) ≤ β. Lemma 2 characterizes useful intermediate scores.
Lemma 2 Let M be a matrix of length m, let α and β be two score bounds defining a score range for which we want to compute the score distribution Q. Q(M [1..i], s) is useful if, and only if,
Proof. This is a straightforward consequence of Definition 1.
This result is implemented in Algorithm SCOREDISTRIBUTION, displayed in Figure 3. The algorithm ensures that only accessible scores are visited. In practice, this is done by using a hash table for storing values of Q.
Figure 3. Algorithm ScoreDistribution.
If one wants only to calculate the Pvalue of a given score without knowing the score distribution, Algorithm SCOREDISTRIBUTION can be further improved. We introduce a complementary optimization that leads to a significant speed up. The idea is that for good words, we can anticipate that the final score will be above the given threshold without calculating it.
Definition 2 (Good words) Let α be a score and i be a position of M. Given u = u_{1 }... u_{i }a word of Σ^{i}, we say that u is good for α if the following conditions are fulfilled:
1. Score(u, M [1..i]) ≥ α  WS(M [i + 1..m])
2. Score(u_{1 }... u_{i1}, M [1..i  1]) <α  WS(M [i..m])
Lemma 3 Let u be a good word for α. Then for all v in uΣ^{mu}, we have Score(v, M) ≥ α.
Proof. Let w in Σ^{mu }such that v = uw and let i be the length of u. We have
Lemma 4 Let u be a string of Σ^{m }such that Score(u, M) ≥ α. Then there exists a unique prefix v of u such that v is good for α.
Proof. We first remark that if Score(u, M) ≥ α, then Score(u, M) ≥ α  WS(M[m + 1..m]). So there exists at least one prefix of u satisfying the first condition of Definition 2: u itself. Now, consider a prefix v of length i such that Score(v, M[1..i]) ≥ α  WS(M[i + 1..m]). Then for each letter x of Σ, we have Score(vx, M[1..i + 1]) ≥ α  WS(M[i + 2..m]): It comes from the fact that M(i + 1, x) ≥ WS(M[i + 1..m])  WS(M[i + 2..m]). This property implies that if a prefix v of u satisfies the first condition of Definition 2, then all longer prefixes also do. According to the second condition of Definition 2, it follows that only the shortest prefix v such that Score(v, M[1..i]) ≥ α  WS(M[i + 1..m]) is a good word.
Lemma 5 Let M be a matrix of length m.
Proof. We consider the set (α) of words whose score is greater than or equal to α: (α) = {w ∈ Σ^{m}Score(w, M) ≥ α}. According to Lemma 4, each word of (α) has a unique prefix that is good for α. Conversely, Lemma 3 ensures that each word whose prefix is good for α belongs to (α). (α) can thus be expressed as a union of disjoint sets.
It follows that
where p(u) denotes the probability of the string u in the background model. By definition of Q, we can deduce the expected result from Formula 3.
Lemma 5 shows that it is not necessary to build the entire dynamic programming table for Q. Only values for Q(M[1..i], s) such that s <α  WS(M[i + 1..m]) are to be computed. This gives rise to the FASTPVALUE algorithm, described in Figure 4.
Figure 4. Algorithm FastPvalue.
Permuting columns of the matrix
Algorithms 1 and 2 can also be used in combination with permutated lookahead scoring [16]. The matrix M can be transformed by permuting columns without modifying the overall score distribution. This is possible because the columns of the matrix are supposed to be independent. We show that it is also relevant for Pvalue calculation.
Lemma 6 Let M and N be two matrices of length m such that there exists a permutation π on {1,..., m} satisfying, for each letter x of Σ, M(i, x) = N(π_{i}, x). Then for any α, Q(M, α) = Q(N, α).
Proof. Let u be a word of Σ^{m }and let . By construction of N, we have Score(u, M) = Score(v, N). Since the background model is multinomial, we have p(u) = p(v). This completes the proof.
The question is how to permute the columns of a given matrix to enhance the performances of the algorithms. In [6], it is suggested to sort columns by decreasing information content. We refine this rule of thumb and propose to minimize the total size of all score ranges involved in the dynamic programming decomposition for Q in Algorithm SCOREDISTRIBUTION. For each i, 1 ≤ i ≤ m, define δ_{i }as δ_{i }= BS(M[i..i])  WS(M[i..i]).
Lemma 7 Let M be a matrix such that δ_{1 }≥ ... ≥ δ_{m}. Then M minimizes the total size of all score ranges amongst all matrices that can be obtained by permutation of M.
Proof. We write SR(M) for the total size of all score ranges of the matrix M. We have
Since permutation of matrices induces a permutation of the sequence δ_{2},..., δ_{m}, the value is minimal when δ_{1 }≥ δ_{2 }≥ ... ≥ δ_{m}.
In the remaining of this paper, we shall always assume that the matrix M has been permuted so that it fulfills the condition on (δ_{i})_{1≤i≤m }of Lemma 7. This is simply a preprocessing of the matrix that does not affect the course of the algorithms.
Efficient algorithms for computing the Pvalue without error
We now come to the presentation of two exact algorithms, which is are the main algorithms of this paper. In Algorithms SCOREDISTRIBUTION and FASTPVALUE, the number of accessible scores plays an essential role in the time and space complexity. As mentioned in the Background section, this number can be as large as Σ^{m}. In practice, it strongly depends on the involved matrix and on the way the score distribution is approximated by round matrices. The choice of the precision is critical. Algorithms SCOREDISTRIBUTION and FASTPVALUE should compromise between accuracy, with faithful approximation, and efficiency, with rough approximation.
To overcome this problem, we propose to define successive discretized score distributions with growing accuracy. The key idea is to take advantage of the shape of the score distribution Q, and to use small granularity values only in the portions of the distribution where it is required. This is a kind of selective zooming process. Discretized score distributions are built from round matrices.
Definition 3 (Round matrix) Let M be a matrix of real coefficient values of length m and let ε be a positive real number. We denote M_{ε }the round matrix deduced from M by rounding each value by ε:
ε is called the granularity. Given ε, we can define E, the maximal error induced by M_{ε}.
Lemma 8 Let M be a matrix, ε the granularity, and E the maximal error associated. For each word u of Σ^{m}, we have 0 ≤ Score(u, M)  Score(u, M_{ε}) ≤ E.
Proof. This is a straightforward consequence of Definition 3 for M_{ε }and E.
Lemma 9 Let M, N and N' be three matrices of length m, E, E' be two nonnegative real numbers, α, β be two scores such that α ≤ β, satisfying the following hypotheses:
(i) for each word u in Σ^{m}, Score(u, N) ≤ Score(u, M) ≤ Score(u, N) + E,
(ii) for each word u in Σ^{m}, Score(u, N') ≤ Score(u, N) ≤ Score(u, M) ≤ Score(u, N') + E',
(iii) Pvalue(N, α  E) = Pvalue(N, α),
(iv) Pvalue(N', β  E') = Pvalue(N', β),
then
Proof. Let u be a string in Σ^{m}. It is enough to establish that α ≤ Score(u, N) <β if, and only if, α ≤ Score(u, M) <β. The proof is divided into four parts.
 If α ≤ Score(u, N), then α ≤ Score(u, M): This is a consequence of Score(u, N) ≤ Score(u, M) in (i).
 If α ≤ Score(u, M), then α ≤ Score(u, N): By hypothesis (i) on E, α ≤ Score(u, M) implies α  E ≤ Score(u, N). Since Pvalue(N, α  E) = Pvalue(N, α) with (iii), it follows that α ≤ Score(u, N).
 If Score(u, N) <β, then Score(u, M) <β: By hypothesis (ii), Score(u, N) <β implies that Score(u, N') <β. According to (iv), this ensures that Score(u, N') <β  E', which with (ii) guarantees Score(u, M) <β
 If Score(u, M) <β, then Score(u, N) <β: This is a consequence of Score(u, N) ≤ Score(u, M) in (i).
What does this statement tell us ? It provides a sufficient condition for the distribution score Q computed with a round matrix to be valid for the initial matrix M. Assume that you can observe two plateaux ending respectively at α and β in the score distribution of M_{ε}. Then the approximation of the total probability for the score range [α, β[obtained with the round matrix is indeed the exact probability. In other words, there is no need to use smaller granularity values in this region to improve the result.
From score to Pvalue
Lemma 9 is used through a stepwise algorithm to compute the Pvalue of a score threshold. Let α be the score for which we want to determine the associated Pvalue. We estimate the score distribution Q iteratively. For that, we consider a series of round matrices M_{ε }for decreasing values of ε, and calculate successive values Pvalue (M_{ε}, α). The efficiency of the method is guaranteed by two properties. First, we introduce a stop condition that allows us to stop as soon as it is guaranteed that the exact value of the Pvalue is reached. Second, we carefully select relevant portions of the score distribution for which the computation should go on. This tends to restrain the score range to inspect at each step. The algorithm is displayed in Figure 5.
Figure 5. Algorithm From Score to Pvalue.
The correctness of the algorithm comes from the two next Lemmas. The first Lemma establishes that the loop invariants hold.
Lemma 10 Throughout Algorithm 3, the variables β and P satisfy the invariant relation P = Pvalue(M, β).
Proof. This is a consequence of invariant 1 and invariant 2 in Algorithm 3. Both invariants are valid for initial conditions. When P = 0 and β = BS(M) + 1: Pvalue(M, BS(M) + 1) = 0. Regarding N', choose N' = M_{ε}.
There are two cases to consider for invariant 1.
 If s does not exist. P and β remain unchanged, so we still have P = Pvalue(M, β). Regarding invariant 2, if there exists such a matrix N' at the former step for M_{kε}, then it is still suitable for M_{ε}.
 If s actually exists. invariant 1 implies that P is updated to Pvalue(M, β) + ∑_{s≤t<β }Q(M_{ε}, t).
According to Lemma 9 and invariant 2, we have ∑_{s≤t<β }Q(M_{ε}, t) = ∑_{s≤t<β }Q(M, t). Hence P = Pvalue(M, s). Since β is updated to s, it follows that P = Pvalue(M, β). Regarding invariant 2, take N' = M_{ε}.
The second Lemma shows that when the stop condition is met, the final value of the variable P is indeed the expected result Pvalue(M, α).
Lemma 11 At the end of Algorithm 3, P = Pvalue(M, α).
Proof. When s = α  E, then β = α. According to Lemma 10, it implies P = Pvalue(M_{ε}, α). Since the stop condition implies that Pvalue(M_{ε}, α  E) = Pvalue(M_{ε}, α), Lemma 9 ensures that Pvalue(M_{ε}, α) = Pvalue(M, α).
From Pvalue to score
Similarly, Lemma 9 is used to design an algorithm to compute the score threshold associated to a given Pvalue. We first show that the score threshold obtained with a round matrix for a Pvalue gives some insight about the potential score interval for the initial matrix M.
Lemma 12 Let M be a matrix, ε a granularity and E the maximal error associated. Given P, 0 ≤ P ≤ 1, we have
Proof. Let β = Threshold(M_{ε}, P). According to Lemma 8, Pvalue(M_{ε}, β) ≥ P implies Pvalue(M, β) ≥ P, which yields β ≤ Threshold(M, P). So it remains to establish that Threshold(M, P) ≤ β + E. If Pvalue(M, β + E) = 0, then the highest accessible score for M is smaller than β + E. In this case, the expected result is straightforward. Otherwise, there exists β' such that β' is the lowest accessible score for M that is strictly greater than β + E. Since s → Pvalue(M, s) is a decreasing function in s, we have to verify that Pvalue(M, β') <P to complete the proof of the Lemma. Assume that Pvalue(M, β') ≥ P. Let γ = min {Score(u, M_{ε})u ∈ Σ^{m }∧ Score(u, M) ≥ β'}. On the one hand, the definition of γ implies that
On the other hand, γ is an accessible score for M_{ε }that satisfies γ ≥ β'  E > β. By hypothesis of β, it follows that
Equations 5 and 6 contradict the assumption that Pvalue(M, β') ≥ P. Thus Pvalue(M, β') <P.
The sketch of the algorithm is as follows. Let P be the desired Pvalue. We compute iteratively the associated score threshold for successive decreasing values of ε. At each step, we use Lemma 12 to speed the calculation for the matrix M_{ε}. This Lemma allows us to restrain the computation of the detailed score distribution Q to a small interval of length 2 × E. For the remaining of the distribution, we can use the FASTPVALUE algorithm. Lemma 13 ensures that when Pvalue(M_{ε}, α  E) = Pvalue(M_{ε}, α), then α is the required score value for M. The algorithm is displayed in more details in Figure 6.
Figure 6. Algorithm From Pvalue to Score.
Lemma 13 Let M be a matrix, ε the granularity and E the maximal error associated. If Pvalue(M_{ε}, α  E) = Pvalue(M_{ε}, α), then Pvalue(M, α) = Pvalue(M_{ε}, α).
Proof. This is a corollary of Lemma 9 with M_{ε }in the role of N and N', and BS(M) + E in the role of β.
Experimental Results
The ideas presented in this paper have been incorporated in a software called TFMPVALUE (TFM stands for Transcription factor matrix). The software is written in C++ and implements the FROM PVALUE TO SCORE and FROM SCORE TO PVALUE algorithms as described in Algorithms 5 and 6, together with permutated lookahead scoring. It is available for download at [17]. In the worst case, TFMPVALUE does not improve the theoretical complexity of the score threshold problem. This was expected from the NPhardness proof provided in the second section. Nevertheless, experimental results show considerable speedups in practice.
Methods
We chose a multinomial background model with identically and independently distributed character symbols on the four letter alphabet {A, C, G, T} to conduct our experiments. The decreasing step (k) in the algorithm was set to 10 and the initial granularity (ε) was set to 0.1. The test set is made of the Jaspar database of transcription factor binding sites [1]. It contains 123 matrices, whose length ranges from 4 to 30. The matrices are transformed into logratio matrices following the technique given in [18]. For each Pvalue P, we report only results for matrices whose length is suitable for P: we requested that the probability of a single word is smaller than P. So a matrix of length m cannot not achieve a Pvalue smaller than . For example, matrices of length 4 have not been considered for a Pvalue equal to 10^{3}, and matrices of length smaller than 10 have not be considered for a Pvalue equal to 10^{6}.
Experimental results are concerned with the error rate depending on the chosen granularity. To estimate the error made at a given granularity, we first computed α_{ε}, the score threshold associated to the Pvalue with the round matrix M_{ε}, and a the score threshold associated to the Pvalue with the original matrix M. We then denumerate the number of words whose score is between α_{ε }and α for M. Concerning the time efficiency, all computation times were measured on a 2.33 GHz Intel Core 2 Duo processor with 2 Go of main memory under Mac OS 10.4.
Concerning FROM PVALUE TO SCORE, We also compared our results with those of algorithm LAZYDISTRIBUTION described in [6]. To the best of our knowledge, this algorithm is the most efficient algorithm today to compute the score associated to a Pvalue. It uses the dynamic programming formulas of Equation 1 in a lazy way and takes advantage of permutated lookahead scoring as presented in the previous Section. We implemented it in C++, like TFMPVALUE.
Computation times for a given granularity
In this first experiment, we study the time performance of TFMPVALUE compared to LAZYDISTRIBUTION when using the same approximation for the distribution score. So in both cases we use round matrices with the same granularity. To set a maximal granularity for TFMPVALUE, we interrupt the loop of decreasing granularities and output the score threshold found at this granularity. We thus obtain exactly the same score threshold as LAZYDISTRIBUTION.
Granularity 10^{3}
We first chose a granularity of 10^{3 }for the two algorithms and computed the score associated to Pvalues equal to 10^{3 }and 10^{6 }for each matrix of the Jaspar database (see Figure 7). The results show that TFMPVALUE outperforms LAZYDISTRIBUTION in both cases. With the Pvalue set to 10^{3}, the average computation time is 0.64 second per matrix for LAZYDISTRIBUTION compared to 0.03 second for TFMPVALUE. Considering each matrix individually, TFMPVALUE is 61 times faster than LAZYDISTRIBUTION. With the Pvalue set to 10^{6}, the average computation time is 0.118 second per matrix for LAZYDISTRIBUTION and 0.019 second for TFMPVALUE. Considering each matrix individually, TFMPVALUE is 15 times faster than LAZYDISTRIBUTION.
Figure 7. Time efficiency for granularity 10^{3}. We compare the running time for the computation of the score threshold associated to a given Pvalue for FROM PVALUE TO SCORE and LAZYDISTRIBUTION onto the Jaspar matrices with a granularity set to 10^{3}. We choose two Pvalue levels: 10^{3 }and 10^{6}. There are 122 matrices (resp. 75 matrices) that can achieve a Pvalue equal to 10^{3 }(resp. 10^{6}). For each algorithm, we classified the matrices into four groups according to the time needed to complete the computation: less than 0.1 second, from 0.1 second to 1 second, from 1 second to 1 minute, and greater than 1 minute. The results are represented by a histogram with four bars. The height of each bar gives the percentage of matrices involved and the number at the top of each bar indicates the corresponding number of matrices.
Granularity 10^{6}
We then repeated the same procedure as above with a smaller granularity, 10^{6 }instead of 10^{3}. Results are reported in Figure 8. When the granularity decreases, the computation time of LAZYDISTRIBUTION dramatically increases. With the Pvalue set to 10^{3}, LAZYDISTRIBUTION needs a running time greater than one minute for 89 percent of the matrices (109 out of 122). TFMPvalue needs less than 0.1 second for 85 percent of the matrices (104 out of 122). With the Pvalue set to Pvalue = 10^{6}, LAZYDISTRIBUTION needs a computation time greater than 1 minute for 62 percent of matrices (47 out of 75). TFMPVALUE needs less than 0.1 second for 89 percent of matrices (67 out of 75). Moreover, if we compare the histogram for TFMPVALUE in Figure 8 with the histogram for LAZYDISTRIBUTION in Figure 7, it appears that TFMPVALUE is still more efficient, whereas the granularity is a thousand fold larger. This demonstrates that we are able to provide more accurate results within the same amount of time. The same conclusion holds for the amount of memory needed to achieve the computation (data not shown).
Figure 8. Time efficiency for granularity 10^{6}. We compare the computation time for the score associated to a Pvalue of 10^{3 }and 10^{6 }onto the Jaspar matrices when the granularity is set to 10^{6 }for TFMPVALUE and LAZYDISTRIBUTION. The histogram has the same meaning as in Figure 3.
Ability to compute accurate thresholds
In the second series of experiments, we tested the ability of TFMPVALUE to get exact score thresholds within a reasonable amount of time. We ran FROM PVALUE TO SCORE and FROM SCORE TO PVALUE without setting a maximal granularity so that the algorithms stop when they reach the correct result. We tried several Pvalues, from 10^{3 }to 10^{6}, for all matrices of suitable length. Runtime is reported in Figure 9 for FROM PVALUE TO SCORE and in Figure 10 for FROM SCORE TO PVALUE. Regarding FROM SCORE TO PVALUE, the time required to compute the score thresholds remains very small for a large majority of matrices: less than 0.01 second for 253 out of the 383 computations for Pvalues from 10^{3 }to 10^{6}, and less than 0.1 second for 337 computations. As expected, results for FROM SCORE TO PVALUE are very similar: less than 0.01 second for 332 out of the 383 computations for Pvalues from 10^{3 }to 10^{6}, and less than 0.1 second for 358 computations.
Figure 9. Runtime of TFMPvalue – From Pvalue to Score without any granularity bound. This histogram shows time measurements for the PVALUE TO SCORE algorithm without any granularity bound. The algorithm stops when it is guaranteed to find the exact Pvalue, without error. We ran tests on a variety of Pvalue parameters: 10^{3}, 10^{4}, 10^{5}, and 10^{6}. As previously, we report the proportion of matrices for which the runtime was less then 0.1 second, between 0.1 second and 1 second, between 1 second and 1 minute and greater than 1 minute.
Figure 10. Runtime of TFMPvalue – From Score to Pvalue without any granularity bound. This histogram shows time measurements for the SCORE TO PVALUE algorithm without any granularity bound. We chose initial scores corresponding to a Pvalue of 10^{3}, 10^{4}, 10^{5 }and 10^{6}.
We display in Table 2 the value of the granularity required to guarantee an exact score threshold in function of the range of Pvalues with FROM PVALUE TO SCORE. The results show that a granularity lower than or equal to 10^{4 }is often needed: more than 63 percent. It is interesting to remark that the granularity does not directly depend on the length of the matrices. In fact, it depends of the shape and density of the score distribution around the score corresponding to the Pvalue required. Nevertheless, as the size of the matrix increases, the number of words greater than a score grows for a given Pvalue and hence the granularity needs to be lower. To illustrate this, all matrices with length less than or equal to 9 need a granularity ranging from 10^{1 }and 10^{5}, whereas all matrices with length greater than or equal to 13 need a granularity ranging from 10^{4 }and 10^{9}.
Table 2. Granularity required for accurate computation with From Pvalue to Score. This table indicates the granularity value that is required for FROM PVALUE TO SCORE to compute the accurate score threshold without any error. Each row of the table corresponds to a Pvalue: 10^{3}, 10^{4}, 10^{5}, and 10^{6}. Each cell gives the percentage of matrices for which FROM PVALUE TO SCORE ends at the granularity of the corresponding column. For example, 52.4% matrices need a granularity larger than or equal to 10^{3 }when computing threshold for Pvalue 10^{5}.
We also evaluated the behavior of FROM SCORE TO PVALUE. For each matrix, for a given score threshold corresponding to a Pvalue of 10^{3}, we computed the largest granularity necessary to obtain an accurate result with a round matrix. Results are summarized in Figure 11. We then compared this granularity with the granularity found with FROM SCORE TO PVALUE. In more than 60 percent of matrices, FROM SCORE TO PVALUE stops as soon as it is possible, with no extra iteration. In more than 90 percent of matrices, it is able to conclude with at most one supplementary step.
Figure 11. Granularity required for accurate computation with From Score to Pvalue. This figure compares the theoretical necessary granularity and the granularity reached by FROM SCORE TO PVALUE. For example, granularity 10^{2 }is necessary for 17 matrices. It means that the round matrix with granularity 10^{1 }gives a wrong Pvalue, whereas the round matrix with granularity 10^{2 }gives an accurate Pvalue. Amongst these 17 matrices, FROM SCORE TO PVALUE stops at 10^{2 }for 11 matrices, performs one supplementary step at 10^{3 }for 5 matrices, and two supplementary steps, at 10^{3 }and 10^{4}, for one matrix.
Discussion and Conclusion
We performed an extensive analysis of the computation of Pvalues for matrices. We gave a simple proof that the From Pvalue to score and From Score to Pvalue problems are NPhard. We then presented two algorithms to solve them efficiently and accurately for reallife examples. As the problem is intrinsically difficult, the worst complexity is not changed and then some matrices may require large computation time and memory. Fortunately, our experiments show that this arises only in very few cases. Our algorithms can be of interest for at least two tasks. First, they can be exploited to obtain significantly faster algorithms than existing ones when a loss of precision is allowed. Indeed, for a same computation time and amount of memory, our algorithms perform better than existing ones. This allows to avoid precomputation of scores associated to fixed Pvalues as done in some software programs [16], and to compute the desired Pvalue on the fly, as specified by the user. Secondly, the algorithms can be used where it is needed to compute a score threshold with high precision, with arbitrary low granularity, in a reasonable amount of space and time. We provided thus a significant improvement to compute scores and Pvalues with high accuracy.
When running experiments on Jaspar database, we chose a value for k, the decreasing step for successive granularities, equal to 10. A different value may be selected. With a lower decreasing step value, the algorithms stop with more accurate granularity and so may avoid useless computations. But this leads to more iterations and then globally to a higher runtime. With a larger decreasing step value, there are less iterations and then the global runtime is lowered. But choosing a very large decreasing step value (more than 10^{3 }for example) amounts to compute almost the complete score distribution and the algorithms become inefficient because they do not take advantage of the reduction of the score range for which exact Pvalues are computed. As the algorithms are mainly based onto the computation of accessible scores, the memory required is almost the same independently of the decreasing step value (until the value is not very large).
When we allowed for some error, such as in the first experiment, this implicitly amounts to calculate the exact score distribution, and thus the exact Pvalue, for the round matrix as described in Definition 3. One can choose an alternative rounding construction for the initial matrix, such as , before running TFMPVALUE. This leaves the course of the algorithms unchanged.
Finally, in the paper, we assumed that the background model is provided with a multinomial model. All results, except permutated lookahead scoring, can be extended to more sophisticate random sources, such as Markov models [19]. The consequence is an increasing of the computation time by a factor Σ^{n}, where n is the order of the Markov model. But the optimization based on successive decreasing granularities still holds.
Authors' contributions
All authors equally contributed to this paper. All authors read and approved the final manuscript.
Acknowledgements
Part of this work was supported by PPF Bioinformatics – University Lille 1. The authors thank Mireille Regnier for fruitful discussions.
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