Word statistics is a traditional field of genome research. For word-length 1, GC-content is a basic characteristic noted for each organism, and dinucleotide relative abundance profiles provide a reliable genomic signature 1. Dinucleotide content also distinguishes natural RNA from random sequences 2. Trinucleotide (codon) usage can reliably predict bacterial genes 3 even in the presence of horizontal gene transfer. Short palindromic words mark the characteristic sites of restriction enzymes in bacteria, and are therefore under represented in bacterial genomes 4. A theory of over- as well as under-represented words has been laid out in 56.

Unique words are of particular interest. They provide sequence signatures, and microarray probes are often designed to match them. Unique sequences from several genomes exhibiting a perfect match serve as reliable anchors in a multiple genome alignment 7. Recently, Haubold et al. 8 addressed the problem of efficiently computing shortest unique substrings (using their terminology) in a sequence, and provided a program called SHUSTRING for this purpose. Using this program, they found that there is typically much more unique sequence in a genome than one would expect in a random sequence of the same length. While this observation by itself is not a surprise, given the repetitive nature of genomes, their approach and software allows to quantify this fact. Furthermore, they found unique words to be significantly clustered in upstream regions of genes in human and mouse.

One may take such investigations farther and investigate words that do not occur in a genome. We suggest the term "unwords" for shortest words from the underlying alphabet that do not show up in a given sequence.

A first approach at the unwords problem was recently presented by Hampikian and Andersen 9. Their motivation was to "discover the constraints on natural DNA and protein sequences". However, there is no evidence that such constraints exist. The absence of certain shortest words in a sequence data base, no matter what (finite) size it has, is a mathematical necessity. Speculations about negative selection against certain words have been refuted convincingly in 10. There, it is shown that human unwords computed in 9 can be explained by a mutational bias rather than negative selection.

Still, there is twofold interest in the capability of efficiently computing unwords.(1) Statistically, it is interesting to see how length and number of unwords in a given genome deviates from expectation in random sequences. (2) Practically, it is useful to know all the unwords when a genome or chromosome is to be extended by insertion of foreign DNA. Combinations of unwords can directly serve as tags that are guaranteed to be unique in the modified DNA sequence.

Unfortunately, the software presented in 9 is slow and difficult to use: It reads Genbank files rather than the more space efficient Fasta format – and space matters a lot when dealing with genomes as large as human and mouse. It runs an internal conversion routine for over 50 minutes before starting unwords computation. The program generates an excessive number of files that may break your file systems. The C code is platform dependent and internal constants must be adapted. Finally, the human unwords data computed with the program according to 9 appear to be incomplete (and hence incorrect).

In order to make unwords computation possible in an efficient and reliable way, we present here a new algorithm and the software implementing it. Efficient computation of unwords can be done from an index data structure such as a suffix tree or an (enhanced) suffix array 11. For example, in 8 a suffix tree was used to compute unique substrings. In fact, our first unwords-program was an extension to the VMATCH software 12, which is based on enhanced suffix arrays. However, index data structures must be built in memory and are space-consuming. Hence, we developed a direct approach that works more efficiently, because the overall sequence need not be kept in main memory. Computing the unwords of the human genome, for example, takes about 10 minutes computation time on a Linux PC with a single 2.4 MHz CPU. The space requirement is 2.5 megabytes.

In this article, we describe the new program UNWORDS and report its application to the genomes of human, mouse, and other organisms, covering a genome size range from 10⁹ down to 10⁵ bp.

Let Σ be a finite alphabet of at least two letters. Let |Σ| denote the cardinality of Σ. In genome analysis, Σ = {a, c, g, t} and |Σ| = 4. A word is a sequence of letters from the alphabet. The terms "word" and "sequence" are equivalent, but are used here to indicate that a word is short and a sequence is long. |w| denotes the length of a word. If |w| = q, we speak of a q-word.

A word w over Σ is an unword of a sequence G if (1) it does not occur as a substring of G, and (2) all words over Σ shorter than w do occur in G. Note that the unword length is uniquely defined for a given genome G.

The built-in minimality requirement in this definition is motivated by the fact that when w is an unword of length q in G, it has 2|Σ| one-letter extensions that also do not occur in G. Therefore, asking for missing words longer than q would introduce a substantial proportion of redundant results.

Similar to shortest unique substrings, the length of unwords is expected to increase with genome size. For fixed unword length, the number of unwords is expected to decrease while |G| increases. Given G, let q be the unword length. It is easy to see that 1 ≤ q. To derive an upper bound on q, let w be a shortest unique substring in G and let ℓ = |w|. Consider the following cases:

• If |w| = |G|, then for any a ∈ Σ, wa is an unword. Hence q ≤ |wa| = ℓ + 1.

• If |w| < |G| and w is not a suffix of G, then wa occurs in G for exactly one letter a. Hence wb for any b ∈ Σ\{a} is an unword. This implies q ≤ |wb| = ℓ + 1.

• If |w| < |G| and w is not a prefix of G, then aw occurs in G for exactly one letter a. Hence bw for any b ∈ Σ\{a} is an unword. This implies q ≤ |wb| = ℓ + 1.

Thus we conclude 1 ≤ q ≤ ℓ + 1.

The problem of unword analysis of a given sequence G (typically a complete genome) is to determine all unwords of G. The double-stranded nature of DNA lets unwords always show up in complementary pairs, as each word present implies the presence of its Watson-Crick complement on the opposite strand. Sometimes, however, an unword is self-complementary, and hence a "pair" represents only a single word. Therefore, we report unword numbers rather than numbers of pairs (in contrast to 8).

Computation of q-word statistics for small q is straightforward. Efficient computation of unwords when q is unknown, however, requires more advanced techniques. Our unword analysis algorithm is described in the section on computational methods.

The unword analysis problem is mathematically well defined. Unwords must exist for any sequence. The interesting question is their size and number, compared to what one would expect given the alphabet size and the length of G.

Let w be a word of length |w|, w [i] the i-th letter in w, G a genomic sequence and ℙ[w [i]] the relative frequency of nucleotide w [i] in G. The probability for w to occur by chance (i.e. at a fixed position in a random sequence s of the same composition and length as G) is then ℙ[w]=∏i=1|w|ℙ[w[i]] MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaWefv3ySLgznfgDOjdaryqr1ngBPrginfgDObcv39gaiqaacqWFzecucqGGBbWwcqWG3bWDcqGGDbqxcqGH9aqpdaqeWaqaaiab=LriqjabcUfaBjabdEha3jabcUfaBjabdMgaPjabc2faDjabc2faDbWcbaGaemyAaKMaeyypa0JaeGymaedabaWaaqWaaeaacqWG3bWDaiaawEa7caGLiWoaa0Gaey4dIunaaaa@4E3F@. The expectation value for (the number of occurrences of) w in s is E MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaWefv3ySLgznfgDOjdaryqr1ngBPrginfgDObcv39gaiqaacqWFecFraaa@37B5@[w in s] ≈ ℙ[w]·|G|.

Calculating the probability for a word not to occur in a specific sequence is quite difficult and not much literature is available. Following Rahmann et al. 13, a good approximation of the probability can be given using the expectation value. A Poisson Distribution is expected for word counts in a genomic sequence, which is ℙ[Xw=k]=λ(w)kk!⋅e−λ(w) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaWefv3ySLgznfgDOjdaryqr1ngBPrginfgDObcv39gaiqaacqWFzecucqGGBbWwcqWGybawdaWgaaWcbaGaem4DaChabeaakiabg2da9iabdUgaRjabc2faDjabg2da9KqbaoaalaaabaGaeq4UdWMaeiikaGIaem4DaCNaeiykaKYaaWbaaeqabaGaem4AaSgaaaqaaiabdUgaRjabcgcaHaaakiabgwSixlabdwgaLnaaCaaaleqabaGaeyOeI0Iaeq4UdWMaeiikaGIaem4DaCNaeiykaKcaaaaa@521A@ with λ(w) = E MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaWefv3ySLgznfgDOjdaryqr1ngBPrginfgDObcv39gaiqaacqWFecFraaa@37B5@[w in s], and k the number of occurrences of the word w. Now let k = 0. Then

ℙ [ X w = 0 ] = 1 ⋅ e − λ ( w ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaWefv3ySLgznfgDOjdaryqr1ngBPrginfgDObcv39gaiqaacqWFzecucqGGBbWwcqWGybawdaWgaaWcbaGaem4DaChabeaakiabg2da9iabicdaWiabc2faDjabg2da9iabigdaXiabgwSixlabdwgaLnaaCaaaleqabaGaeyOeI0Iaeq4UdWMaeiikaGIaem4DaCNaeiykaKcaaaaa@49B8@

The expected number N of q-words that do not occur is therefore

N ≈ |Σ|^qe^-λ(w)

As an example, for a random sequence G of length 3.1·10⁹ and an unword w of length 14 and typical composition, we obtain a probability of 1.40082·10^-5 for w not occurring in G. Still, the expected number of unwords of length 14 is 2590.798, while for length 13, it is only 5.823108·10^-13. For even shorter unwords, it is practically zero.

For convenience, we map each of the four letters of the DNA-alphabet to an integer in the range 0 to 3 as follows: ā = 0, c¯ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafm4yamMbaebaaaa@2D3C@ = 1, g¯ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafm4zaCMbaebaaaa@2D44@ = 2, t¯ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmiDaqNbaebaaaa@2D5E@ = 3. Moreover, for any fixed value q, we use a standard method to map each possible q-word to a number in the range [0, 4^q - 1]. That is, we define ϕq(w)=∑i=1qw[i]¯⋅4q−i MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaeqy1dO2aaSbaaSqaaiabdghaXbqabaGccqGGOaakcqWG3bWDcqGGPaqkcqGH9aqpdaaeWaqaamaanaaabaGaem4DaCNaei4waSLaemyAaKMaeiyxa0faaaWcbaGaemyAaKMaeyypa0JaeGymaedabaGaemyCaehaniabggHiLdGccqGHflY1cqaI0aandaahaaWcbeqaaiabdghaXjabgkHiTiabdMgaPbaaaaa@46BC@ for any q-word w. In other words, q-words are mapped to their rank in the corresponding lexicographic order. Substrings in G containing at least one wildcard (e.g. N) are ignored. The integer value φ_q (w) serves as an index into a bit table Ω_q such that for all sequences w of length q we have: Ω_q [φ_q (w)] = 1 if and only if w occurs as a substring in the genome G. Let |Ω_q| denote the number of 1-entries in Ω_q.

Initially we set all bits in Ω_q to 0. This requires O(4qω) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaem4ta80aaeWaaKqbagaadaWcaaqaaiabisda0maaCaaabeqaaiabdghaXbaaaeaacqaHjpWDaaaakiaawIcacaGLPaaaaaa@337D@ time, where w is the computer word size. Then we sweep a window of width q over G from left to right. For the first window G [1..q] we determine the integer code φ_q (G [1..q]) as defined above in O(q) time. For each of the remaining n - q windows, say at start position i + 1, we compute φ_q (G[i + 1..i + q]) in constant time from φ_q (G[i..i + q - 1]) according to the following equation:

ϕ q ( G [ i + 1 .. i + q ] ) = ( ϕ q ( G [ i .. i + q − 1 ] ) − 4 q − 1 ⋅ G [ i ] ¯ ) ⋅ 4 + G [ i + q ] ¯ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@69AD@

Thus the computation of the n - q + 1 integer code requires O(n) time. The multiplication and addition in can be implemented by fast bit-shift and bit-or operations. If j is the current integer code and Ω_q [j] is 0, then we set Ω_q [j] to 1 and increment a counter of the number of 1-entries in Ω_q. This can be done in constant time. Note that once |Ω_q| = 4^q, we can stop scanning G. While the time requirement of this algorithm is O(n+4qω) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaem4ta80aaeWaaeaacqWGUbGBcqGHRaWkjuaGdaWcaaqaaiabisda0maaCaaabeqaaiabdghaXbaaaeaacqaHjpWDaaaakiaawIcacaGLPaaaaaa@35C4@ it uses O(1) + 2q + 4^q bits of space, as only q consecutive letters in G need to be stored in memory.

If |Ω_q| = 4^q, i.e. all 4^q entries in Ω_q are 1, then we know that all possible q-words occur in G. Hence there is no unword of length q in G. On the other hand, if after processing all q-words in G, |Ω_q| < 4^q, there are some unwords of length q. If additionally |Ω_q-1| = 4^q-1, then we know that q is the smallest value such that unwords of length q exist. The unwords can easily be computed by determining all j such that Ω_q [j] = 0. Given j, one determines the corresponding q-word w satisfying φ_q (w) = j in O(q) time. Thus the unwords are enumerated in O(4¹ + qz) time where z is the number of unwords.

Let q* be the smallest value such that there are unwords of length q*. Consider the possible range of values for q for a given genome length n. Let q^max = ⌈log₄ (n + 1)⌉. Then 4qmax⁡=4⌈log⁡4(n+1)⌉≥n+1>n≥n−qmax⁡+1 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaeGinaqZaaWbaaSqabeaacqWGXbqCdaahaaadbeqaaiGbc2gaTjabcggaHjabcIha4baaaaGccqGH9aqpcqaI0aandaahaaWcbeqaamaahmaabaGagiiBaWMaei4Ba8Maei4zaC2aaSbaaWqaaiabisda0aqabaWccqGGOaakcqWGUbGBcqGHRaWkcqaIXaqmcqGGPaqkaiaaw6o+caGL5JpaaaGccqGHLjYScqWGUbGBcqGHRaWkcqaIXaqmcqGH+aGpcqWGUbGBcqGHLjYScqWGUbGBcqGHsislcqWGXbqCdaahaaWcbeqaaiGbc2gaTjabcggaHjabcIha4baakiabgUcaRiabigdaXaaa@575A@. Note that G contains n - q^max + 1 substrings of length q^max. Hence G is too short to accommodate all possible q^max-words and therefore there are some unwords of length q^max. Thus q* ≤ q^max, i.e. we can restrict the search for q* to the range [1, q^max].

There are basically two strategies to determine q*. The first strategy (linear search) starts with q = 1 and increments q until |Ω_q| < 4^q. Then q* = q. The space requirement is O(1) + 2q* + 4^q* and the running time is

O ( 4 q ∗ + q ∗ z ) + ∑ q = 1 q ∗ O ( n + 4 q ω ) = O ( 4 q ∗ + q ∗ z ) + O ( q ∗ n ) + O ( 4 q ∗ + 1 ω ) , MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@6B29@

where z is the number of unwords. Note that we have n≥4q∗−1=4q∗+142≥4q∗+1ω MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemOBa4MaeyyzImRaeGinaqZaaWbaaSqabeaacqWGXbqCdaahaaadbeqaaiabgEHiQaaaliabgkHiTiabigdaXaaakiabg2da9KqbaoaalaaabaGaeGinaqZaaWbaaeqabaGaemyCae3aaWbaaeqabaGaey4fIOcaaiabgUcaRiabigdaXaaaaeaacqaI0aandaahaaqabeaacqaIYaGmaaaaaOGaeyyzImBcfa4aaSaaaeaacqaI0aandaahaaqabeaacqWGXbqCdaahaaqabeaacqGHxiIkaaGaey4kaSIaeGymaedaaaqaaiabeM8a3baaaaa@4752@ under the realistic assumption that the machine word size ω is at least 4². Hence n dominates the last term in (4). Thus the overall running time for the linear search is O(4^q* + q* (n + z)).

The second strategy determines q* by a binary search in the range [1, q^max], as described in Table 1. The strategy is optimal in the sense that it tests a minimal number of possible values of q before it arrives at q*. Unfortunately, a value q' determined in line 8 of Table 1, may or may not be modified later in the loop, which means that one has to store the corresponding table Ω_q' or recompute it later. The running time of the binary search is O(4q∗+q∗z)+log⁡2qmax⁡(n+4qmax⁡−1ω) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaem4ta8KaeiikaGIaeGinaqZaaWbaaSqabeaacqWGXbqCdaahaaadbeqaaiabgEHiQaaaaaGccqGHRaWkcqWGXbqCdaahaaWcbeqaaiabgEHiQaaakiabdQha6jabcMcaPiabgUcaRiGbcYgaSjabc+gaVjabcEgaNnaaBaaaleaacqaIYaGmaeqaaOGaemyCae3aaWbaaSqabeaacyGGTbqBcqGGHbqycqGG4baEaaGccqGGOaakcqWGUbGBcqGHRaWkjuaGdaWcaaqaaiabisda0maaCaaabeqaaiabdghaXnaaCaaabeqaaiGbc2gaTjabcggaHjabcIha4baacqGHsislcqaIXaqmaaaabaGaeqyYdChaaOGaeiykaKcaaa@5259@. Its space requirement is O(1) + 2q^max + 4qmax⁡ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaeGinaqZaaWbaaSqabeaacqWGXbqCdaahaaadbeqaaiGbc2gaTjabcggaHjabcIha4baaaaaaaa@32B7@.

We used our first implementation (based on suffix-arrays) of an unwords algorithm to cross-validate the program presented here. Applied to the human genome, both algorithms (which are completely independent) produce the same set of unwords. This makes us sure that our set of 104 human unwords is indeed complete, in contrast to the 80 unwords reported in 9. (If a smaller genome assembly or repeat masked sequences were used in this earlier study, more rather than less unwords should have been detected.) We computed unwords for six eucaryotic genomes: Homo sapiens, Release NCBI36 14, Mus musculus, Release NCBIm36 15, Drosophila melanogaster, Release 5.1 16, Caenorhabditis elegans, Release WS170 17, Neurospora crassa 18 and Saccharomyces cerevisiae, Release SGD1.01 19, including nonchromosomal sequences which could not be assigned to a chromosome. Additionally, unwords for two bacterial genomes were calculated: Staphylococcus aureus subsp. aureus strain MSSA476, Refseq number NC_002953 and Mycoplasma genitalium, Refseq number NC_000908, as well as for two Archaea genomes:

Thermococcus kodakarensis, Release KOD1 20 and Methanocaldococcus jannaschii, Release DSM 2661 21. Table 2 gives a summary of genome sizes and unword lengths and numbers. In Table 3, we show the unwords computed from the human genome. We also indicate the number of occurrences expected for each unword – if the genome was a random sequence, which of course is not the case. Deviation of GC content in unwords is summarized in Table 4. Unwords for the other genomes are given in Tables 5, 6, 7, 8, 9, 10, 11, 12.