The number of k-mer matches between two DNA sequences as a function of k

Abstract

We study the number N_k of length-k word matches between pairs of evolutionarily related DNA sequences depending on k. We show that the number of substitutions per site that occurred since two sequences evolved from their last common ancestor, can be estimated from the slope of a certain function of N_k. This approach can be generalized from contiguous word matches to so-called spaced-word matches, where mismatches are allowed at certain pre-defined positions. Based on these theoretical results, we implemented a software program for alignment-free sequence comparison called Slope-SpaM. Test runs on simulated sequence data show that Slope-SpaM can accurately estimate phylogenetic distances for distance values up to around 0.5 substitutions per position. The statistical stability of our results is improved if spaced words are used instead of contiguous words. Unlike previous methods that are based on the number of (spaced) word matches, Slope-SpaM produces accurate results, even if sequences share only local homologies.

Availability https://github.com/burkhard-morgenstern/Slope-SpaM

Contact

1 Introduction

Phylogeny reconstruction is a fundamental task in computational biology [15]. Here, a basic step is to estimate pairwise evolutionary distances between protein or nucleic-acid sequences. Under the Jukes-Cantor model of evolution [25], the distance between two evolutionarily related DNA sequences can be defined as the (estimated) number of nucleotide substitutions per site that have occurred since the two sequences have evolved from their last common ancestor. Traditionally, phylogenetic distances are inferred from pairwise or multiple sequence alignments. For the huge amounts of sequence data that are now available, however, sequence alignment has become far too slow. Therefore, considerable efforts have been made in recent years, to develop fast alignment-free approaches that can estimate phylogenetic distances without the need to calculate full alignments of the input sequences, see [20, 49, 56, 5, 26] for recent review articles.

A number of alignment-free approaches are able to use unassembled short sequencing reads as input [53, 2, 14, 40, 34, 27, 4, 46]. Alignment-free approaches are not only used in phylogeny reconstruction, but are also important in metagenomics [11, 40, 33] and in medical applications, for example to identify drug-resistant bacteria [6] or to classify viruses [54, 3]. In all these applications, it is crucial to rapidly estimate the degree of similarity or dissimilarity between large sets of sequence data.

Some alignment-free approaches are based on word frequencies [41, 47] or on the length of common substrings [51, 29, 23]. Other methods use variants of the D₂ distance which is defined as the number of word matches of a pre-defined length between two sequences [42, 52, 50, 3]; a review focusing on these methods is given in [43]. kWIP [38] is a further development of this concept that uses information-theoretical weighting. Most of these approaches calculate heuristic measures of sequence (dis-)similarity that may not be easy to interpret. At the same time, alignment-free methods have been proposed that can accurately estimate phylogenetic distances between sequences based on stochastic models of DNA or protein evolution, using the length of common substrings [22, 36] or so-called micro alignments [53, 21, 31, 30].

Several authors have proposed to estimate phylogenetic distances from the number of k-mer matches between two sequences. The tools Cnidaria [2] and AAF [14] use the Jaccard index between sets of k-mers from two – assembled or unassembled – genomes to estimate the distance between them. In Mash [40], the MinHash [7] technique is used to reduce the input sequences to small ‘sketches’ which can be used to rapidly approximate the Jaccard index. In a previous article, we proposed another way to infer evolutionary distances between DNA sequences based on the number of word matches between them, and we generalized this to so-called spaced-word matches [37]. This distance function is now used by default in the program Spaced [28].

A spaced-word match is a pair of words from two sequences that are identical at certain positions, specified by a pre-defined binary pattern of match and don’t-care positions. Theoretically, the distance measure proposed in [37] is based on a simple model of molecular evolution without insertions or deletions. In particular, we assumed in this previous paper that the compared sequences are homologous to each other over their entire length. In practice, the derived distance values are still reasonably accurate if a limited number of insertions and deletions is allowed, and phylogenetic trees could be obtained from these distance values that are similar to trees obtained with more traditional approaches. Like other methods that are based on the number of common k-mers, however, our previous approach can no longer produce accurate results for sequences that share only local regions of homology.

Recently, Bromberg et al. published an interesting new approach to alignment-free protein sequence comparison that they called Slope Tree [8]. They defined a distance measure using the decay of the number of k-mer matches between two sequences, as a function of k. Trees reconstructed with this distance measure were in accordance to known phylogenetic trees for various sets of prokaryotes. Slope Tree can also correct for horizontal gene transfer and can deal with composition variation and low complexity sequences. From a theoretical point-of-view, however, it is not clear if the distance measure used in Slope Tree is an accurate estimator of evolutionary distances.

In the present paper, we study the number N_k of word or spaced-word matches between two DNA sequences, where k is the word length or the number of match positions of the underlying pattern for spaced words, respectively. Inspired by Bromberg’s Slope Tree approach, we study the decay of N_k as a function of k. More precisely, we define a function F(k) that depends on N_k and that can be approximated – under a simple probabilistic model of DNA evolution, and for a certain range of k – by an affine-linear function of k. The number of substitutions per site can be estimated from the slope of F, we therefore call our implementation Slope-SpaM, where SpaMstands for Spaced-Word Matches. Using simulated DNA sequences, we show that Slope-SpaM can accurately estimate phylogenetic distances. In contrast to other methods that are based on the number of (spaced) word matches, Slope-SpaM produces still accurate results if the compared sequences share only local homologies. We also applied Slope-SpaM to infer phylogenetic trees based on genome sequences from the benchmarking project AFproject [1].

2 The number of k-mer matches as a function of k

We are using standard notation from stringology as used, for example, in [18]. For a string or sequence S over some alphabet , |S| denotes the length of S, and S(i) is the i-th character of S, 1 ≤ i ≤ |S|. S[i..j] is the (contiguous) substring of S from i to j. We consider a pair of DNA sequences S₁ and S₂ that have evolved under the Jukes-Cantor substitution model [25] from some unknown ancestral sequence. That is, we assume that substitution rates are equal for all nucleotides and sequence positions, and that substitution events at different positions are independent of each other. For simplicity, we first assume that there are no insertions and deletions (indels). We call a pair of positions or k-mers from S₁ and S₂, respectively, homologous if they go back to the same position or k-mer in the ancestral sequence. In our model, we have a nucleotide match probability p for homologous positions and a background match probability q for non-homologous nucleotides; the probability of two homologous k-mers to match exactly is p^k.

In our indel-free model, S₁ and S₂ must have the same length L = |S₁| = |S₂|, and positions i₁ and i₂ in S₁ and S₂, respectively, are homologous if and only if i₁ = i₂. Note that, under this model, a pair of k-mers from S₁ and S₂ is either homologous or completely non-homologous, in the sense that none of the corresponding pairs of positions is homologous, and for a pair of non-homologous k-mers from S₁ and S₂, the probability of an exact match is q^k. Let the random variable X_k be defined as the number of k-mer matches between S₁ and S₂. More precisely, X_k is defined as the number of pairs (i₁, i₂) for which holds. There are (L–k+1) possible homologous and (L–k+1)·(L–k) possible background k-mer matches, so the expected total number of k-mer matches is

In [37], we used this expression directly to estimate the match probability p for two observed sequences using a moment-based approach, by replacing the expected number E(X_k) by the empirical number N_k of word matches or spaced-word matches, respectively. Although in equation (1), an indel-free model is assumed, we could show in our previous paper, that this approach gives still reasonable estimates of p for sequences with insertions and deletions, as long as the sequences are globally related, i.e. as long as the insertions and deletions are small compared to the length of the sequences. It is clear, however, that this estimate will become inaccurate in the presence of large insertions and deletions, i.e. if sequences are only locally related.

Herein, we propose a different approach to estimate evolutionary distances from the number N_k of k-mer matches, by considering the decay of N_k if k increases. For simplicity, we first consider an indel-free model as above. From equation (1), we obtain which – for a suitable range of k – is an approximately affine-linear function of k with slope ln p. Substituting the expected value E(X_k) in the right-hand side of (2) with the corresponding empirical number N_k of k-mer matches for two observed sequences, we define

In principle, we can now estimate p as the exponential of the slope of F. Note that, in practice, we will have to restrict ourselves to a certain range of k. If k is too small, N_k will be dominated by background word matches, if k is too large, no word matches will be found at all.

If we want there to be at least as many homologous as background word matches, we have to require N ·p^k ≥ N² · q^k. We therefore obtain a lower bound for k as

If, on the other hand, we want to have at least one expected word match, we have N · p^k ≥ 1, so k would be upper-bounded by

Therfore, if we want to have a range of length K between the lower and the upper bound for k, we must have

It follows that the match probability p must be large enough for our approach to work. As an example, with q = 1/4 and a range of, say, K = 15, we would need p > 0.52, corresponding to a Jukes-Cantor distance of 0.53 substitutions per position.

The above considerations can be generalized to a model of DNA evolution with insertions and deletions. Let us consider two DNA sequences S₁ and S₂ of different lengthsx L₁ and L₂, respectively, that have evolved from a common ancestor under the Jukes-Cantor model, this time with insertions and deletions. Note that, unlike with the indel-free model, it is now possible that a k-mer match involves homologous as well as background nucleotide matches. Instead of deriving exact equations like (1) and (2), we therefore make some simplifications and approximations.

We can decompose S₁ and S₂ into indel-free pairs of ‘homologous’ substrings that are separated by non-homologous segments of the sequences. Let L_H be the total length of the homologous substring pairs in each sequence. As above, we define the random variable X_k as the number of k-mer matches between the two sequences, and p and q are, again, the homologous and background nucleotide match probabilities. We then use as a rough approximation, and we obtain

Similar as in the indel-free case, we define where N_k is, again, the empirical number of k-mer matches. As above, we can estimate p as the exponential of the slope of F. Note that this estimation can still be applied if L_H is small compared to L₂ and L₂, since L_H appears only in an additive constant on the left-hand side of (4). Thus, while the absolute values F(k) do depend on the extent L_H of the homology between S₁ and S₂, the slope of F only depends on p, q, L₁ and L₂, but not on L_H.

3 The number of spaced-word matches

In many fields of biological sequence analysis, k-mers and k-mer matches have been replaced by spaced words or spaced-word matches, respectively. Let us consider a fixed word P over {0,1} representing match positions (‘1’) and don’t-care positions (‘0’). We call such a word a pattern; the number of match positions in P is called its weight. In most applications, the first and the last symbol of a pattern P are assumed to be match positions. A spaced word with respect to P is defined as a string W over the alphabet {A, C, G, T, *} of the same length as P with W(i) = * if and only if P(i) = 0, i.e. if and only if i is a don’t-care position of P. Here, ‘*’ is interpreted as a wildcard symbol. We say that a spaced word W w.r.t P occurs in a sequence S at some position i if W(m) = S(i + m – 1) for all match positions m of P.

We say that there is a spaced-word match between sequences S₁ and S₂ at (i, j) if the same spaced word occurs at position i in S₁ and at position j in S₂, see Fig. 1 for an example. A spaced-word match can therefore be seen as a gap-free alignment of length |P| with matching characters at the match positions and possible mismatches a the don’t-care positions. Spaced-word matches or spaced seeds have been introduced in database searching as an alternative to exact k-mer matches [9, 35, 32]. The main advantage of spaced words compared to contiguous k-mers is the fact that results based on spaced words are statistically more stable than results based on k-mers [24, 13, 11, 10, 37, 39].

Figure 1:

Spaced-word match between two DNA sequences S₁ and S₂ at (2,3) with respect to a pattern P = 1100101 representing match positions (‘1’) and don’t-care positions (‘0’). The same spaced word TA * *A * C occurs at position 2 in S₁ and at position 3 in S₂.

Quite obviously, approximations (3) and (4) remain valid if we define X_k to be the number of spaced-word matches for a given pattern P of weight k, and we can generalize the definition of F(k) accordingly: if we consider a maximum pattern weight k_max and a given set of patterns {P_k, 1 ≤ k ≤ k_max} where k is the weight of pattern P_k, then we can define N_k as the empirical number of spaced-word matches with respect to pattern Pk between two observed DNA sequences. F(k) can then be defined exactly as in (5), and we can estimate the nucleotide match probability p as the exponential of the slope of F.

4 Implementation

It is well known that the set of all k-mer matches between two sequences can be calculated efficiently by lexicographically sorting the list of all k-mers from both sequences, such that identical k-mers appear as contiguous blocks in this list. Note that, to find k-mer matches for all k ≤ k_max, it is sufficient to sort the list of k_max-mers since this sorted list also contains the sorted lists of k-mers for k < k_max.

This standard procedure can be directly generalized to spaced-word matches. In order to calculate the numbers N_k efficiently, we start by generating a suitable set of patterns {P_k, 1 ≤ k ≤ k_max}. We first specify a pattern P = P_kmax of weight k_max, for example by using the tool rasbhari [19]. As usual, we require the first and the last position of P to be match positions, i.e. we have P(1) = P(k_max) = 1. Let ℓ_k be the k-th match position in the pattern P. We then define the k-th pattern as

In other words, for k < k_max, the pattern P_k is obtained by cutting off P after the k-th match position, so each pattern P_k has a weight of k. To find all spaced-word matches with respect to all patterns P_k, 1 ≤ i ≤ k_max, it suffices to lexicographically sort the set of spaced words with respect to P in S₁ and S₂.

To estimate the slope of F, one has to take into account that the values F(k) are statistically stable only for a certain range of k. If k is too small, N_k and F(k) are dominated by the number of background (spaced) word matches. If k is too large, on the other hand, the number of homologous (spaced) word matches becomes small, and for N_k < L₁ · L₂ · q^k, the value F(k) is not even defined. For two input sequences, we therefore need to identify a range k₁,…, k₂ in which F is approximately affine linear. To this end, we consider the differences and we choose a maximal range k₁… k₂ where the difference |δ_k−δ_k−1| is smaller than some threshold T for all k with k₁ < k ≤ k₂. In our implementation, we used a value of T = 0.2. We then estimate the slope of F as the average value of δ_k in this range, so we estimate the match probability as

Finally, we apply the usual Jukes-Cantor correction to estimate the Jukes-Cantor distance between the sequences, e.g. the number of substitutions per position that have occurred since they diverged from their last common ancestor. Examples are given in Fig. 2, 3 and 4.

Figure 2:

Test runs on simulated indel-free DNA sequences of length L = 10, 000, with a Jukes-Cantor distance of 0.2. Top: F(k) = ln (N_k − L² · q^k) plotted against k, where N_k is the number of k-mer matches. Bottom: for P₂₀ = 11001110111001110111000101011011, patterns P_k were generated for k = 1,…, 19 as explained in section 4. F(k) is plotted against k, based on the number N_k of spaced-word matches with respect to P_k. In both cases, δ_k = F(k) − F(k − 1) is stable for k₁ = 5 < k ≤ k₂ = 20 (red dots).

Figure 3:

F(k) plotted against k as in Fig. 2 and for simulated indel-free DNA sequences of length L = 10, 000 with a Jukes-Cantor distance of 0.5 for contiguous k-mers (top) and for spaced-word matches (bottom).

Figure 4:

F(k) plotted against k as in Fig. 2 and 3, for simulated indel-free DNA sequences of length L = 30, 000. In a region of length L_H = 10, 000, the two sequences are related to each other with a Jukes-Cantor distance of 0.2; this region contains around 20 indels at random positions. The ‘homologous’ region is flanked on both sides and in both sequences by unrelated random sequences of length 10,000 each. F(k) is plotted for contiguous k-mers (top) and for spaced-word matches (bottom).

5 Test results

5.1 Distances calculated from simulated DNA sequences

We first simulated pairs of DNA sequences of length L = 10,000 without indels, and with different phylogenetic distances. As a first example, we generated a pair of sequences with a Jukes-Cantor distance of 0.2, i.e with 0.2 substitutions per position. According to the Jukes-Cantor correction formula, this corresponds to a match probability of p = 0.82. In Fig. 2 (top), F(k) is plotted against k for these sequences based on exact k-mer matches; the difference F(k) − F(k − 1) is stable for k₁ = 5 < k ≤ k₂ = 20. With equation (6), we therefore estimate the slope of F from the numbers N_k of k-mer matches as

This way, we obtain an estimated match probability of which is rather close to the correct value p = 0.82. A similar value was obtained when we used spaced-word matches instead of contiguous k-mer matches (Fig. 2, bottom). Results for indel-free sequences of the same length, with a Jukes-Cantor distance of 0.5 are shown in Fig. 3. Here, the slope of F was 0.449 when k-mer matches were used an 0.446 with spaced-word matches, leading to an estimated match probability of which corresponds to a Jukes-Cantor distance of 0.494, again very close to the true value.

Fig. 4 shows the corresponding results for a pair of simulated sequences with insertions and deletions and with only local sequence homology. In a region of length L_H = 10,000, the sequences are homologous to each other, with a Jukes-Cantor distance of 0.2 substitutions per position, as in our first example (Fig. 2) – but this time, we included 20 insertions and deletions of length 50 each at random positions. In addition, the homology region was flanked on both sides and in both sequences by non-related random sequences of length 10,000 each. So both sequences had a length of around 30,000 where one third of the sequences were homologous to each other. Here, Slope-SpaM estimated a nucleotide match frequency of corresponding to a Jukes-Cantor distance of 0.18, somewhat smaller than the true value of 0.2.

For a more systematic evaluation, we simulated pairs of DNA sequences with Jukes-Cantor distances d between 0.05 and 1.0. For each value of d, we generated 20,000 sequence pairs of length 10,000 each and with distance d, and we estimated the distances with Slope-SpaM. In Fig. 5, the average estimated distances are plotted against the real distances. Our estimates are fairly accurate for distances up to around 0.5 substitutions per site, but become statistically less stable for larger distances.

Figure 5:

For Jukes-Cantor distances d between 0.05 and 1.0, we generated pairs of simulated DNA sequences of length L = 10, 000; for each value of d, 20,000 sequences pairs with a distance of d were generated. For each pair, we estimated the distance d with Slope-SpaM, using exact k-mers (top) and spaced-word matches (bottom). Average estimated distances are plotted against the real distances; standard deviations are shown as error bars.

5.2 Distance estimation for sequences with local homologies

As shown theoretically in Section 2, our distance estimator should still work if two sequences share only local regions of homology, in contrast to previous approaches that also estimate distances from the number of word or spaced-word matches. To verify this empirically, we generated semi-artificial sequences consisting of local homologies, concatenated with non-homologous sequence regions of varying length. We then estimated phylogenetic distances between these sequences with Slope-SpaM, as well as with Mash and Spaced, two other alignment-free methods that are also based on the number of word matches. To see how these tools are affected by the presence of non-homologous sequence regions, we compared the distance values estimated from the semi-artificial sequences to the distances obtained from the original sequences.

To generate these sequence sets, we started with a set of 19 homologous gene sequences from different strains of the bacterium Wolbachia, taken from Gerth et al. [17]; the length of these sequences varied between 165kb and 175kb. We then generated 9 additional data sets by concatenating these gene sequences with unrelated i.i.d. random sequences of varying length. Here, the proportion of the original homologous sequences within the concatenated sequences was 1.0 in the first set – the original sequences –, 0.9 in the second set, 0.8, in the third set,…, and 0.1 in the tenth set. Within each of these 10 sequence sets, we estimated all pairwise distances, and we calculated the ratio between each estimated distance value and the distance estimated the respective original sequence pair. For each data set – i.e. for each proportion of non-homologous random sequences –, we then took the average ratio between the distance of the semi-artificial and the original sequences.

The results of these test runs are plotted Fig. 6. As can be seen, distances calculated with Mash are heavily affected by including random sequences to the original homologous sequences. For the tenth data set, where the homologous gene sequences are only 10% of the sequences and 90% are unrelated random sequences, the distances calculated by Mash are, on average, 40 times higher than for the original gene sequences that are homologous over their entire length. Spaced is also affected if non-related random sequences are added, though to a lesser extent. Still, distances calculated by Spaced for the tenth data set are 1.7 times larger than the distances estimated for the original sequences. By contrast, Slope-SpaM was hardly affected at all by the presence of non-homologous random sequences.

Figure 6:

Comparison of distances estimated between homologous gene sequences and between semi-artificial sequences, obtained from the same homologous sequences by adding non-related random sequences of increasing length. The x-axis is the fraction of the homologous sequences within the semi-artificial sequences, the y-axis is the ratio between the distance estimated from the semi-artificial sequences and the distances estimated from the original gene sequences, see the main text for more details. Distances were estimated with three different alignment-free programs.

5.3 Phylogenetic tree reconstruction

In addition to the above artificial and semi-artificial sequence pairs, we used sets of real genome sequences with known reference phylogenies from the AFproject web page [1] as benchmark data for phylogeny reconstruction. AFproject is a collaboration to systematically benchmark and evaluate software tools for alignment-free sequence comparison in different application scenarios [55]. The web page of the project provides a variety benchmark sequence sets, and the results of alignment-free methods on these sequences can be uploaded to the server. Their quality is calculated and can be compared to a large number of other alignment-free methods, among them the state-of-the-art methods.

To evaluate Slope-SpaM, we downloaded four sets of full-genome sequences that are available in the categories genome-based phylogeny and horizontal gene transfer, namely (1) a set of 29 E.coli/Shigella genomes [53], (2) a set of 25 mitochondrial genomes from different fish species of the suborder Labroidei [16], (3) another set of 27 E.coli/Shigella [48] and (4) a set of 8 genomes of different strains of Yersinia [12]. Some of these data sets have been used previously by developers of alignment-free tools to benchmark their methods.

We ran Slope-SpaM on the four sets of genomes and uploaded the obtained distance matrices to the AFproject web server for evaluation. For these categories of benchmark data, the AFproject server calculates phylogenetic trees from the produced distance matrices using Neighbor Joining [45]. It then compares these trees to trusted reference trees of the respective data sets under the normalized Robinson-Foulds (nRF) metric. Here, the standard Robinson-Foulds (RF) distance [44] that measures the dissimilarity between two tree topologies is divided by the maximal possible RF distance of two trees with the same number of leaves; the nRF distances can therefore take values between 0 and 1. On the AFproject server, the benchmark results are then ranked in order of increasing nRF distance, i.e. in order of decreasing quality.

The results of our test runs were as follows: For the first set of E.coli genomes (29 genomes), the nRF distance of the obtained tree to the reference tree was 0.54, corresponding to rank 11 out of 18. For the fish mitochondrial genomes, the tree produced with the Slope-SpaM distances had a nRF distance of 0.32 to the reference three, corresponding to rank 5 out of 10. For the second E.coli/Shigella data (27 genomes), the nRF distance between the Slope-SpaM tree and the reference tree was again 0.54, which was this time position 11 out of 16. For the set of 8 Yersinia genomes, the topology of the Slope-SpaM tree was identical with the reference tree topology, so the nRF distance was 0, corresponding to rank 1 out of 6.

6 Discussion

A number of alignment-free methods have been proposed in the literature to estimate the nucleotide match probability p for a pair of DNA sequences from the number N_k or D₂ of k-mer matches for a fixed value of k. This can be done by setting Nk in relation to the total number of k-mers in the compared sequences [40] or to the length of the sequences [36]. A certain draw-back of these approaches is that, in order to accurately estimate the match probability p, not only N_k and the sequence lengths (or total number of k-mers, respectively), but also the extent of the homology between the compared sequences must be known.

Indeed, if two sequences share only short, local homologies with each other, an observed number N_k of k-mer matches could indicate a high match probability p within those homologous regions – while the same value of N_k would correspond to a lower p if the homology would extend over the entire length of the sequences. Since it is, in general, not known which proportion of the input sequences is homologous to each other, the above methods assume, for simplicity, that the compared sequences are homologous over their entire length. Obviously, this assumption affects the accuracy of these approaches since, in reality, genome sequences often share only local homologies.

In the present paper, we introduced Slope-SpaM, another approach to estimate the match probability p between two DNA sequences – and thereby their Jukes-Cantor distance – from the number N_k of word matches; we generalized this approach to spaced-word matches, based on patterns with k match positions. The main difference between Slope-SpaM and previous methods is that, instead of using only one single word length or pattern weight k, our program calculates N_k, together with an expression F(k) that depends on N_k, for a whole range of values of k. The nucleotide match probability p in an alignment of the input sequences is then estimated from the slope of F. From the definition of F(k), it is easy to see that the slope of F and our estimate of p depend on the values of N_k, but not on the extent of the homology between the compared sequences.

Test results on simulated sequences show that our distance estimates are accurate for distances up to around 0.5 substitutions per sequence position. Our experimental results also confirm that Slope-SpaM still produces accurate distance estimates if sequences contain only local regions of homology. In Fig. 2 and Fig. 4, sequence pairs with a Jukes-Cantor distance of 0.2 were used. While the sequences in Fig. 2 are similar to each other over their entire length, in Fig. 4 a local region of sequence ‘homology’ is embedded in non-related random sequences twice the length of the homology region. Nevertheless, the slope of the function F is almost the same in both figure, leading to similar estimates of the match probability p. In fact, with the values shown in Fig. 4 (top), we would have obtained the same correct estimate for p that we obtained with the data from Fig. 2, if our program had chosen a minimum value k₁ = 11 for the word length k, instead of k₁ = 10. Alternatively, a sufficiently large maximal value k₂ would have led to a more accurate estimate of p.

On the real-world genomes, the performance of Slope-SpaM was in the medium range, compared to the other methods for which results are available on the AFproject server. On one real-world data set that is available on this server – a set of 14 plant genomes –, Slope-SpaM did dot produce distance values for all pairs of sequences, we therefore had to omit this set from the program evaluation. we expect that a better way of selecting the program parameters, in particular the range of word lengths (or pattern weights) k may further improve the accuracy of Slope-SpaM. Since the approach implemented in Slope-SpaM is novel and very different to existing alignment-free approaches, substantial improvements are possible by systematically analyzing the influence of the applied sequence parameters. Also, we used the Jukes-Cantor model, the simplest possible model for nucleotide substitutions. Using more sophisticated substitution models may also contribute to improve future versions of our program.

6.1 Acknowledgements

We thank Fengzhu Sun for his useful comments on an earlier version of this manuscript, Andrzej Zielezinski and Wojchiech Karlowski for making the AFproject server available, Jendrik Schellhorn for the test runs for the semi-artificial sequences with local homologies, Christoph Bleidorn and Micha Gerth for discussions about the Wolbachia genomes and Chris-André Leimeister and Peter Meinicke for general discussions on the project.

Footnotes

• More test results