Algorithm 1: Compute (see Equation (10) for a proper definition of ) in the case of a heterogeneous model. The workspace complexity is O(n × L) and since all matrix vector products exploit the sparse structure of the matrices, the time complexity is O(ℓ × n × || × L) where || × L corresponds to the maximum number of non-zero terms in T_i+d = P_i+d + Q_i+d.

Require: The starting distributions the matrices , , for all 1 ≤ j ≤ r, 1 ≤ i ≤ ℓ_j - d, a O(n × L) workspace to keep the current values of E(y), and a dimension L polynomial row-vector of degree n + 1.

   // Initialization

   E(y) ← 1

   // Loop on sequences

   for j = 1, ..., r do

      E(y) ← (E(y)1^T) ×

      // Loop on positions within the sequence

      for i = 1, ... ℓ_j-d do

Output: return (G_N (y)) = E(y)1^T

When working with heterogeneous models, there is very little room for optimization in the computation of Equation (8). Indeed, since all terms and may differ for each combination of position i and sequence j, there is no choice but to compute the individual contribution of each of these combinations. This may be done recursively by taking advantage of the sparsity of matrices and . Note that, so as to speed up the computation, it is not necessary to keep track of the polynomial terms of degrees greater than n + 1. This may be done by using the polynomial truncation function defined by

This function also applies to vector or matrix polynomials. This approach results in Algorithm 1 whose time complexity is O(ℓ × n × || × L). In particular, one observes that the time complexity remains linear with n, which is a unique feature of this algorithm, while an individual computation of each (y) would obviously result in a final O(r × n²) complexity to perform the polynomial product . It is also interesting to point out that the number r of considered sequences does not appear explicitly in the complexity of Algorithm 1 but only through the total length .

Algorithm 2: Compute the (G_N(y)) in the case of a homogeneous model. The workspace complexity is O(n × L) and since all matrix vector products exploit the sparse structure of the matrices, the time complexity to compute all ((y)) is O(ℓ_r × n × || × L) where || × L corresponds to the maximum number of non-zero terms in T = P + Q. The product updates of U(y) result in a additional time complexity of O(r × n²).

Require: The matrices P and Q, for all 1 ≤ j ≤ r, the starting distributions , the length ℓ_j (assuming ), a O(n × L) workspace to keep the current values of E(y) (a dimension L polynomial row-vector of degree n + 1) and U(y) (a polynomial of degree n + 1).

   // Initialization

   U(y) ← 1 and E(y) ← 1

   // Loop on sequences

   for j = 1, ..., r do

      for i = 1, ..., ℓ_j - ℓ_j-1 do

         E(y)^T← ((P + yQ)E(y)^T)

      optionally return

Output: return (G_N (y)) = U (y)

If we now consider a homogeneous model, we can dramatically speed up the computation of Equation (9) by recycling intermediate results in order to compute efficiently all (y). Without loss of generality, we assume that the sequences are ordered by increasing lengths: ℓ₁≤ ...≤ ℓ_r. If one stores the value of in some polynomial vector E(y)^T, it is clear that . By repeating this trick for all ℓ_j, it is then possible to adapt Algorithm 1 to compute all with a complexity O(ℓ_r × n × || × L) (ℓ_r being the length of the longest sequence), which is a dramatic improvement. Unfortunately, it is then necessary to compute the product , which results in a complexity O(r × n²) to get all polynomial terms of degree smaller that n + 1 in G_N(y). This additional complexity therefore limits the interest of this algorithm in comparison to Algorithm 1, especially when one observes a large number n of pattern occurrences. However, it is clear that Algorithm 2 remains the best option when considering a huge data set where we typically have ℓ_r << ℓ = ℓ₁ + ... + ℓ_r.

Algorithm 3: Compute the (G_N(y)) in the case of a homogeneous model using power computations. The workspace complexity is O(n × K × L²) with K = log₂(max{ℓ₁- d, ℓ₂ - ℓ₁, ..., ℓ_r- ℓ_r-1}). The precomputation time complexity is O(n² × K × L³). All ((y)) are computed with a total time complexity O(r × n² × K × L³). The product updates of U(y) result in an additional time complexity of O(r × n²).

Require: The matrices P and Q, for all 1 ≤ j ≤ r, the starting distributions , the length ℓ_j (assuming ), a O(n × L) workspace to keep the current values of E(y) (a dimension L polynomial row-vector of degree n + 1) and U(y) (a polynomial of degree n + 1), and a O(n × K × L²) workspace to store the values of (y) with 0 ≤ k ≤ K = log₂(max{ℓ₁ - d, ℓ₂ - ℓ₁, ..., ℓ_r - ℓ_r-1}).

   // Precompute all (y)

    (y) ← P + yQ

   for k = 1, ..., K do

   // Initialization

   U(y) ← 1 and E(y) ← 1

   // Loop on sequences

   for j = 1, ..., r do

      compute (y) using a binary decomposition and set E(y) ← ((y)E(y)^T)

      optionally return

      U(y) ← (U(y) × E(y)^T)

Output: return (G_N (y)) = U (y)

We now consider the case where ℓ_r is large (ex: ℓ_r = 100, 000 or 1, 000, 000 or more). With Algorithm 2, the time complexity is linear with ℓ_rand may then result in an unacceptable running time. It is however possible to turn this into a logarithmic complexity by computing directly the powers of (P + yQ). This particular idea is not new in itself and has already been used in the context of pattern problems by several authors 5051. The novelty here is to apply this approach to a data set of multiple sequences.

If we denote by , it is clear that all (y) can be computed (and stored) for 0 ≤ k ≤ K with a space complexity O(n × K × L²) and a time complexity O(n² × K × L³). It is therefore possible to compute all (y) using the same approach as in Algorithm 2 except that all recursive updates of E(y) are replaced by direct power computations. This results in Algorithm 3 whose total complexities are O(n × K × L³) in space and O(r × n² × K × L³) in time with K = log₂(max{ℓ₁ - d, ℓ₂ - ℓ₁, ..., ℓ_r - ℓ_r-1}). The key feature of this algorithm is that we have replaced ℓ_r by the quantity K, which is typically dramatically smaller when we consider large ℓ_r. The drawback of this approach is that the space complexity is now quadratic with the pattern complexity L, and that the time complexity is cubic with L. As a consequence, it is not suitable to use Algorithm 3 for a pattern of high complexity.

If we now consider a moderate or high complexity pattern, we cannot accept either a cubic complexity with L or even a quadratic complexity. Hence only Algorithms 1 or 2 are appropriate. However, if we assume that our data set contains at least one long sequence, it may be difficult to perform the computations. This is why we introduce an approach that allows computing G_N (y) = m_d(P + yQ)^ℓ-d1^T for large ℓ and L. The technique is directly inspired from the partial recursion introduced in 51 to compute g(y) = m_d(P + Q + yQ)^ℓ-d1^T.

In this particular section, we assume that P is an irreducible and aperiodic matrix. We denote by λ the largest magnitude of the eigenvalues of P, and by ν the second largest magnitude of the eigenvalues of P/λ. For all i ≥ 0 we consider the polynomial vector , where and , and hence we have G_N (y) = λ^ℓ-dm_dF_ℓ-d(y).

Like in 51, the idea is then to recursively compute finite differences of F_i(y) up to the point where these differences asymptotically converge at a rate related to νⁱ. We then derive an approximated expression for F_ℓ-d(y) using only terms such as i ≤ α. Unfortunately, this approach through partial recursion suffers the same numerical instabilities as in 51 when computations are performed in floating point arithmetic. For this reason, we chose here not to go further in that direction until a more extensive study has been conducted.

In this part we give a simple example to illustrate the techniques and algorithms presented above. We consider the pattern = {abab, abaab, abbab} over the binary alphabet = {a, b}. The minimal DFA that recognizes the language ℒ = * (which is the set of all texts over ending with occurrence of ) is then given in Figure 1.

Let us now consider the following set of r = 3 sequences:

We process these sequences to the DFA of Figure 1 (starting each sequence in the initial state 0) to get the observed state sequences , and :

Therefore, Sequence x¹ contains n₁ = 2 occurrences of the pattern (ending in positions 5 and 8), Sequence x² contains n₂ = 1 occurrence (ending in position 5) and Sequence x³ contains n₃ = 1 occurrence (ending in position 8).

Let us now consider X¹, X² and X³, three homogeneous order d = 1 Markov chains of respective lengths ℓ₁, ℓ₂ and ℓ₃ such that X¹ and X³ start with a, and X² starts with b, and the transition matrix of which is given by:

The corresponding state sequences , and are hence order 1 homogeneous Markov chains defined over ' = {0, 1,2, 3, 4, 5, 6} with the starting distributions = = (0 1 0 0 0 0 0), = (1 0 0 0 0 0 0) (since starting from 0 in the DFA of Figure 1, a leads to state 1 and b to state 0) and with the following transition matrix (please note that transitions belonging to Q are marked with a '*'. The others ones belong to P):

A direct application of Corollary 3 therefore gives (y) = 0.743104 + 0.208944y + 0.0450490y² + 0.0029030y³ for the moment generating function of N₁ (the number of pattern occurrences in X¹);

(y) = 0.94816 + 0.05184y for the moment generating function of N₂ (the number of pattern occurrences in X²); and (y) = 0.7761376 + 0.1880064y + 0.0353376y² + 0.0005184y³ for the moment generating function of N₃ (the number of pattern occurrences in X³). One should note that occurrences of are strongly disfavored in Sequence X² since it starts with b. We then derive from these expressions the value of the moment generating function G_N (y) of N = N₁ + N₂ + N₃:

Since we observe a total of n = n₁ + n₂ + n₃ = 4 occurrences of Pattern , the P-value of over-representation is given by

Let us finally compare the exact distribution of N', the number of pattern occurrences over X = X₁... X_ℓ with ℓ = ℓ₁ + ℓ₂ + ℓ₃ - 2 × offset, and a homogeneous order 1 Markov chain with transition matrix π:

As contains both words of lengths 4 and 5, offset should be set either to 3 or 4. However, for both these values, 10² × ℙ(N' ≥ 4) (either when X₁ = a or when X₁ = b) differs from the reference exact P-value 10² × ℙ(N ≥ 4) = 0.333.

Protein structures are classically described in terms of secondary structures: α-helices, β-strands and loops. Structural alphabets are an innovative tool that allows describing any three-dimensional (3D) structure by a succession of prototype structural fragments. We here use HMM-27, an alphabet composed of 27 structural letters (it consists in a set of average protein fragments of four residues, called structural letters, which is used to approximate the local backbone of protein structures through a HMM): 4 correspond to the alpha-helices, 5 to the beta-strands and the 18 remaining ones to the loops (see Figure 2) 53. Each 3D structure of ℓ residues is encoded into a linear sequence of HMM-27 structural letters and results in a sequence of ℓ - 3 structural letters since each overlapping fragment of four consecutive residues corresponds to one structural letter.

We consider a set of 3D structures of proteins presenting less than 80% identity and convert them into sequences of structural letters. Like in 54, we then make the choice to focus only on the loop structures which are known to be the most variable ones, and hence the more challenging to study. The resulting loop structure data set is made of 78,799 sequences with length ranging from 4 to 127 structural letters.

In order to study the interest of the single sequence approximation described in the "Single sequence approximation" section, we first perform a simple experiment. We fit an order 1 homogeneous Markov model on the original data set, and then simulate a random data set with the same characteristics (loop lengths and starting structural letters). We then compute the z-score - these quantities are far easier to compute than the exact P-values and they are known to perform well for pattern problems as long as we consider events in the center of the distribution, and such events are precisely the ones expected to occur with a simulated data set - of the 77, 068 structural words of size 4 that we observe in the data, using simulated data sets under the single sequence approximation. We observe that high z-scores are strongly over-represented in the simulated data set: for example, we observed 264 z-scores of magnitude greater than 4, which is much larger than the expected number of 4.88 under H₀. This observation clearly demonstrates that the single sequence approximation completely fails to capture the distribution of structural motifs in this data set. Indeed this experiment initially motivated the present work by putting emphasis on the need for taking into account fragmented structure of the data set.

We further investigate the edge effects in the data set by comparing the exact P-values obtained under the single sequence approximation. Table 1 gives the results for a selected set of 14 motifs whose occurrences range from 4 to 282. We can see that the single sequence approximation with an offset of 0 clearly differs from the exact value: e. g., Pattern ODZR has an exact P-value of 5.78 × 10^-5 and an approximate one of 2.81 × 10^-4; Pattern BZOU has an exact P-value of 2.56 × 10^-11 and an approximate one of 4.49 × 10^-5.

As explained in the Methods section, these differences may be caused by the overlapping positions in the artificial single sequence where the pattern cannot occur in the fragmented data set. Since we consider patterns of size 4, a canonical choice of offset is 4 - 1 = 3. We can see in Table 1 the effects of this correction. For most patterns, this approach improves the reliability of the approximations, even if we still see noticeable differences. For instance we get an approximated P-value larger than the exact one for Pattern BZOU, and an approximated P-value smaller than the exact one for Pattern UOEI. For other patterns, this correction is ineffective and gives even worse results than with an offset of 0. For example, Pattern DRPI has an exact P-value of 7.26 × 10^-167 and an approximate P-value with an offset of 3 equal to 3.56 × 10^-222, while the approximation with no offset gives a P-value of 9.08 × 10^-174.

Hence it is clear that the forbidden overlapping positions alone cannot explain the differences between the exact results and the single sequence approximation. Indeed, there is another source of edge effects which is connected to the background model. Since each sequence of the data set starts with a particular letter, the marginal distribution differs from the stationary one for a number of positions that depends on the spectral properties of the transition matrix. It is well known that the magnitude μ of the second eigenvalue of the transition matrix plays here a key role since the absolute difference between the marginal distribution at position i and the stationary distribution is O(μⁱ). In our example, μ = 0.33, which is very large, leads to a slow convergence toward the stationary distribution: we need at least 30 positions to observe a difference below machine precision between the two distributions. Such an effect is usually negligible for long sequences where 30 << ℓ, but is critical when considering a data set of multiple short sequences.

However, this effect might be attenuated on the average if the distribution of the first letter in the data set is close to the stationary distribution. Figure 3 compares these two distributions. Unfortunately in the case of structural letters, there is a drastic difference between these distributions.

The example of structural motifs in protein loop structures illustrates the importance of explicitly taking into account the exact characteristics of the data set (number and lengths of sequences) when the single sequence approximation appears to be completely unreliable. As explained above, this may be due both to the great differences between the starting and the stationary distributions, as well as to a slow convergence and to the problem of forbidden positions.

We consider the release 20.44 of PROSITE (03-Mar-2009) which encompasses 1, 313 different patterns described by regular expressions of various complexity 9. PROSITE currently contains patterns and specific profiles for more than a thousand protein families or domains. Each of these signatures comes with documentation providing background information on the structure and function of these proteins. The shortest regular expression is for pattern PS00016: RGD, i. e., an exact succession of arginine, glycine and aspartate residues. This pattern is involved in cell adhesion. The longest regular expression, on the opposite, is for pattern PS00041:

[KRQ][LIVMA].(2)[GSTALIV]FYWPGDN.(2)[LIVMSA].(4, 9)[LIVMF].{PLH}[LIVMSTA][GSTACIL]GPKF.[GANQRF][LIVMFY].(4, 5)[LFY].(3)[FYIVA]{FYWHCM}{PGVI}.(2)[GSADENQKR].[NSTAPKL][PARL] (note that X means "any aminoacid", brackets denote a set of possible letters, braces a set of forbidden letters, and parentheses repetitions -fixed number of times or on a given range). This is the signature of the DNA-binding domain of the araC family of bacterial regulatory proteins.

This data set is useful to explore one of the key points of our optimal Markov chain embedding method using DFAs: the impact of the pattern complexity L. For this purpose, we first build 1-unambiguous (since we want to work with an order 1 Markov model) associated DFAs for 1,276 PROSITE patterns (37 patterns requiring a prohibiting computation time and/or memory were not computed). The repartition of the resulting pattern complexities is shown in Figure 4. There is a peak in the distribution at 2, meaning that many DFAs have ≃ 100 states. The smallest DFA is obtained for the RGD pattern (22 states), and the largest is for APPLE (PS00495) which is represented by the regular expression C.(3)[LIVMFY].(5)[LIVMFY].(3)[DENQ][LIVMFY].(10)C.(3)CT.(4)C.[LIVMFY]F.[FY].(13, 14)C.[LIVMFY][RK].[ST].(14, 15)SG.[ST][LIVMFY].(2)C which has 837, 507 states. The mean computing time of the DFA is 3 minutes, but 50% of the DFA could be computed in less than 0.01s, and 95% in less than 9s.

In Table 2, we can see that if short regular expressions usually lead to low complexity patterns, it is difficult to predict the result for longer regular expressions. For instance, the PROSITE signatures PUR_PYR_PR_TRANSFER and ADH_ZINC have the same size, but the former has a complexity of L = 102 while the latter has a complexity of L = 478. Indeed, we know from the theory of language and automata 55 that the minimal DFA corresponding to a regular expression of size R has a size L verifying L ≤ 2^R. Fortunately, in practice, L is usually dramatically smaller than this upper bound.

We now consider the complete proteome of the bacteria Escherichia coli (File NC_000913.faa, retrieved at ftp://ftp.ncbi.nih.gov/genomes/Bacteria/Escherichia_coli_K_12_substr__MG1655/). This data set encompasses a total of 4, 131 protein sequences with lengths ranging from 14 to 2, 358 aminoacids. We fit on this data set a homogeneous order 1 Markov model which is used to derive over-representation P-values of patterns.

Like for structural letters, we compare the exact P-values to the ones obtained using the single sequence approximation, see Table 3. Unlike in Table 1, we see here that the single sequence approximation performs already well with no offset, but that the use of the appropriate offset further improves this approximation.

This result is surprising, since, in this case, the starting distribution of the model strongly differs from the stationary distribution. Indeed, it is a biological fact that all protein sequences start with a methionine (M). As a consequence, it is hence clear that the starting distribution and the stationary distribution of the model strongly differ. This observation obviously does not favor the single sequence approximation. But in this example, this effect is corrected by the rapid convergence of the marginal distribution toward the stationary distribution ensured by a very low second magnitude eigenvalue of the matrix: μ = 0.049. We expect the same kind of behavior for the high complexity patterns of Table 4 but because of the numerical instabilities in the partial recursion approach suggested in the "Long sequences and high complexity pattern" section, unfortunately it was impossible to perform the computations for the single sequence approximation for such pattern in a reasonable time. However, it is possible to perform the exact computation for these high complexity patterns using Algorithm 2.

Considering the multi-testing problem of this study (we consider a total of 1, 276 PROSITE signatures), we can set a significance threshold of 7.84 × 10^-7 at level 0.1% using a Bonferonni correction. Even at this stringent level, it is clear that many of the considered PROSITE signatures (2 out of 8 in Table 3, and 9 out of 11 in Table 4) are over-represented compared to our homogeneous order 1 Markov background model. However, this result is not a surprise since these patterns actually correspond to very precise functional signatures which are therefore expected to be strongly maintained through evolution in order keep their functional activities.

Transcription factors regulate the expression of genes by activating or repressing the RNA polymerase. This is done by specific binding of the transcription factors (TFs) onto DNA, in proximity to the target genes, usually in the upstream regions. The transcription binding signatures on DNA are thus biologically important patterns.

We retrieved the sequence of transcription factor binding sites of Saccharomyces cerevisiae on the YEASTRACT website http://www.yeastract.com/consensuslist.php and searched for a subset of these transcription factor binding sites in the upstream regions of yeast genes, retrieved on the Regulatory Sequence Analysis Tools website 56http://rsat.ulb.ac.be/rsat/. This data set comprises a total of 1,371 upstream sequences between positions -800 and -1 (the length is hence ℓ = 800 for each sequence).

On these data, we first fit an order 1 homogeneous Markov model. Since there is little difference between the starting distribution observed in the data set over = {A, C, G, T}(0.30 0.16 0.19 0.35) and the stationary distribution (0.32 0.18 0.18 0.32), and since the magnitude of the second eigenvalue of the transition matrix is fairly low (μ = 0.092), we do not expect a great difference between the exact computations and the single sequence approximation. However, since exact computations are easily tractable, we do not further consider the single sequence approach for this particular problem.

We can see in Table 5 the P-values (column "homogeneous") of a selection of known TFs (marked with a star) as well as arbitrary candidate patterns. Several known TFs appear to be highly significant (e.g., TF AAGAAAAA with a P-value of 1.31 × 10^-99) while others are not (e.g., TF WWWTTTGCTCR with a P-value of 4.15 × 10^-1). It is the same for arbitrary candidate patterns. These results are difficult to interpret since these variations may be due either to statistical problems (e.g., insufficient Markov order) or real functional activities. Moreover, it is obviously impossible to distinguish a significant pattern which is a real TF of the organism from a significant pattern which is directly or indirectly implicated in another biochemical process.

We now want to get rid of the homogeneous assumption of the model in an attempt to get a better fitting on the data. A simple way to achieve this is to perform a point-wise estimation of our transition function at position i by fitting the model on a window of size w centered around i. Small values of w lead to better fitting, while large values lead to better smoothing (resulting in a homogeneous model if w ≥ ℓ, the length of the sequence). In this example, we achieve a satisfactory trade-off between the two extremes with an arbitrary choice of w = 200. We can see in Figure 5, that the model gives a unique profile for each transition probability (e.g., π_i(A, G) or π_i(G, G)), and these profiles are both quantitatively and qualitatively different from each other. In Figure 6 we consider the model in a more global way with the marginal distributions of the four nucleotides. According to this graph, it is clear that the upstream region has a bias in GC content that depends on the position. In particular, we observe a smaller GC content in the region [-200, -1] (positions 601 to 800) than in the region [-800, -201] (positions 1 to 599).

Thanks to Algorithm 1, it is possible to compute the P-values of DNA patterns in our heterogeneous model. The results are given in Table 5 (column "heterogeneous"). For most patterns, we can see that the P-values obtained with this heterogeneous model are in fact very close to the ones obtained with the homogeneous one. There are however several patterns for which a ratio factor of 10 may appear between these two P-values (e.g., Pattern GCGCGCGC or CGGNNNNNNNNCGG).