 Methodology article
 Open Access
Efficient computation of absent words in genomic sequences
 Julia Herold^{1},
 Stefan Kurtz^{2} and
 Robert Giegerich^{1}Email author
https://doi.org/10.1186/147121059167
© Herold et al; licensee BioMed Central Ltd. 2008
 Received: 08 November 2007
 Accepted: 26 March 2008
 Published: 26 March 2008
Abstract
Background
Analysis of sequence composition is a routine task in genome research. Organisms are characterized by their base composition, dinucleotide relative abundance, codon usage, and so on. Unique subsequences are markers of special interest in genome comparison, expression profiling, and genetic engineering. Relative to a random sequence of the same length, unique subsequences are overrepresented in real genomes. Shortest words absent from a genome have been addressed in two recent studies.
Results
We describe a new algorithm and software for the computation of absent words. It is more efficient than previous algorithms and easier to use. It directly computes unwords without the need to specify a length estimate. Moreover, it avoids the space requirements of index structures such as suffix trees and suffix arrays. Our implementation is available as an open source package. We compute unwords of human and mouse as well as some other organisms, covering a genome size range from 10^{9} down to 10^{5} bp.
Conclusion
The new algorithm computes absent words for the human genome in 10 minutes on standard hardware, using only 2.5 Mb of space. This enables us to perform this type of analysis not only for the largest genomes available so far, but also for the emerging pan and metagenome data.
Keywords
 Random Sequence
 Suffix Tree
 Short Word
 Suffix Array
 Integer Code
Background
Sequence statistics and unique substrings
Word statistics is a traditional field of genome research. For wordlength 1, GCcontent 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 underrepresented words has been laid out in [5, 6].
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.
Absent words
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.
Software for unwords computation
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 unwordsprogram 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 spaceconsuming. 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 10^{9} down to 10^{5} bp.
Results
Problem statement
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 qword.
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 builtin minimality requirement in this definition is motivated by the fact that when w is an unword of length q in G, it has 2Σ oneletter 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 doublestranded nature of DNA lets unwords always show up in complementary pairs, as each word present implies the presence of its WatsonCrick complement on the opposite strand. Sometimes, however, an unword is selfcomplementary, 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 qword 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.
Unword statistics
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 ith 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 $\mathbb{P}[w]={\displaystyle {\prod}_{i=1}^{\leftw\right}\mathbb{P}[w[i]]}$. The expectation value for (the number of occurrences of) w in s is $\mathbb{E}$[w in s] ≈ ℙ[w]·G.
The expected number N of qwords that do not occur is thereforeN ≈ Σ^{ q }e^{λ(w)}
As an example, for a random sequence G of length 3.1·10^{9} 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.
Unwords algorithm
For convenience, we map each of the four letters of the DNAalphabet to an integer in the range 0 to 3 as follows: ā = 0, $\overline{c}$ = 1, $\overline{g}$ = 2, $\overline{t}$ = 3. Moreover, for any fixed value q, we use a standard method to map each possible qword to a number in the range [0, 4^{ q } 1]. That is, we define ${\varphi}_{q}(w)={\displaystyle {\sum}_{i=1}^{q}\overline{w[i]}}\cdot {4}^{qi}$ for any qword w. In other words, qwords 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 1entries in Ω_{ q }.
Thus the computation of the n  q + 1 integer code requires O(n) time. The multiplication and addition in can be implemented by fast bitshift and bitor 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 1entries 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 $O\left(n+\frac{{4}^{q}}{\omega}\right)$ 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 qwords occur in G. Hence there is no unword of length q in G. On the other hand, if after processing all qwords in G, Ω_{ q } < 4^{ q }, there are some unwords of length q. If additionally Ω_{q1} = 4^{q1}, 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 qword w satisfying φ_{ q }(w) = j in O(q) time. Thus the unwords are enumerated in O(4^{1} + 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} = ⌈log_{4} (n + 1)⌉. Then ${4}^{{q}^{\mathrm{max}}}={4}^{\lceil {\mathrm{log}}_{4}(n+1)\rceil}\ge n+1>n\ge n{q}^{\mathrm{max}}+1$. 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}].
where z is the number of unwords. Note that we have $n\ge {4}^{{q}^{\ast}1}=\frac{{4}^{{q}^{\ast}+1}}{{4}^{2}}\ge \frac{{4}^{{q}^{\ast}+1}}{\omega}$ under the realistic assumption that the machine word size ω is at least 4^{2}. Hence n dominates the last term in (4). Thus the overall running time for the linear search is O(4^{ q }* + q* (n + z)).
Algorithm for computing q* by a binary search strategy.
1: determine sequence length n 

2: l ← 1 
3: r ← log_{4} (n + 1) 
4: while l ≤ r do 
5: q ← (l + r)/2 
6: compute Ω_{ q } 
7: if Ω_{ q } < 4^{ q }then 
8: q' ← q 
9: Ω_{ q' }← Ω_{ q } 
10: r ← q  1 
11: else 
12: l ← q + 1 
13: end if 
14: end while 
15: q* ← q' 
16: Ω_{q*}← Ω_{ q' } 
17: for all j ∈ [0, 4^{ q }*  1] do 
18: if Ω_{q*}[j] = 0 then 
19: print w such that φ_{q*}(w) = j 
20: end if 
21: end for 
Testing
We used our first implementation (based on suffixarrays) of an unwords algorithm to crossvalidate 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:
Genome sizes (including sequences not assigned to a chromosome), the logarithm of the genome size to the base of 10, length and number of unwords of the analyzed genomes
Organism  Genome size  ⌊log_{10} G⌋  ⌊log_{4} G⌋  #unwords  length 

H. sapiens  ≈ 3.1 Gb  9  15.8  104  11 
M. musculus  ≈ 2.7 Gb  9  15.7  192  11 
D. melanogaster  ≈ 132 Mb  8  13.5  104  11 
C. elegans  ≈ 100 Mb  8  13.3  2  10 
N. crassa  ≈ 34 Mb  7  12.5  2262  11 
S. cerevisiae  ≈ 12 Mb  7  11.8  4  9 
S. aureus  ≈ 2.79 Mb  6  10.7  248  8 
T. kodakarensis  ≈ 2.08 Mb  6  10.5  1  8 
M. jannaschii  ≈ 1.66 Mb  6  10.3  3  6 
M. genitalium  ≈ 0.58 Mb  5  9.6  5  6 
Unwords for the human genome and their expected number of occurrences. The four words which are also unwords for the mouse genome are shown in a box.
accgatacgcg  153  accgttcgtcg  153  acgaccgttcg  153  acgatcgtcgg  153 

acgcgcgatat  221  acggtacgtcg  153  agcgtcgtacg  153  atatcgcgcgg  153 
atatcgcgcgt  221  atcgtcgacga  221  atgtcgcgcga  153  catatcgcgcg  153 
ccgaatacgcg  153  ccgacgatcga  153  ccgacgatcgt  153  ccgatacgtcg  153 
ccgcgcgatat  153  ccgtcgaacgc  106  ccgttacgtcg  153  cgaacggtcgt  153 
cgaatcgacga  221  cgaatcgcgta  221  cgaccgatacg  153  cgacgaacgag  153 
cgacgaacggt  153  cgacgcgatac  153  cgacgcgtata  221  cgacggacgta  153 
cgacgtaacgg  153  cgacgtaccgt  153  cgacgtatcgg  153  cgatcgtgcga  153 
cgattacgcga  221  cgattcggcga  153  cgcgacgcata  153  cgcgacgttaa  221 
cgcgcataata  319  cgcgcgatatg  153  cgcgctatacg  153  cgcgtaacgcg  106 
cgcgtaatacg  221  cgcgtaatcga  221  cgcgtatcggt  153  cgcgtattcgg  153 
cgcgttacgcg  106  cgctcgacgta  153  cggtcgtacga  153  cgtacgaaacg  221 
cgtacgacgct  153  cgtatacgcga  221  cgtatagcgcg  153  cgtatcggtcg  153 
cgtattacgcg  221  cgtcgactatc  221  cgtcgctcgaa  153  cgtcgttcgac  153 
cgttacgcgtc  153  cgtttcgtacg  222  ctacgcgtcga  153  ctcgttcgtcg  153 
gacgcgtaacg  153  gatagtcgacg  221  gcgcgacgtta  153  gcgcgtaccga  106 
gcgttcgacgg  106  ggtacgcgtaa  221  gtatcgcgtcg  153  gtccgagcgta  153 
gtcgaacgacg  153  taacgtcgcgc  153  tacgcgattcg  221  tacgcgcgaca  153 
tacgctcggac  153  tacggtcgcga  153  tacgtccgtcg  153  tacgtcgagcg  153 
tagcgtaccga  221  tatacgcgtcg  221  tatcgcgtcga  221  tatgcgtcgcg  153 
tattatgcgcg  321  tattcgcgcga  221  tcgacgcgata  221  tcgacgcgtag  153 
tcgatcgtcgg  153  tcgattacgcg  221  tcgcacgatcg  153  tcgccgaatcg  153 
tcgcgaccgta  153  tcgcgacgtaa  221  tcgcgcgaata  221  tcgcgcgacat  153 
tcgcgtaatcg  221  tcgcgtatacg  221  tcggtacgcgc  106  tcggtacgcta  221 
tcgtacgaccg  153  tcgtcgacgat  221  tcgtcgattcg  222  tgtcgcgcgta  153 
ttaacgtcgcg  221  ttacgcgtacc  221  ttacgtcgcga  221  ttcgagcgacg  153 
GC content of Human, Mouse, Drosophila melanogaster, Caenorhabditis elegans, Saccharomyces cerevisiae, Staphylococcus aureus and Mycoplasma genitalium as well as the GC content of the associated unwords.
Organism  Genome GC%  Unword GC% 

H. sapiens  ≈ 38  ≈ 45–72 
M. musculus  ≈ 40  ≈ 54–72 
D. melanogaster  ≈ 40  ≈ 45–90 
C. elegans  ≈ 35  ≈ 80 
S. cerevisiae  ≈ 38  ≈ 89–100 
S. aureus  ≈ 33  ≈ 50–100 
M. genitalium  ≈ 32  ≈ 66–100 
Unwords for the Mouse genome.
aacgcgtatcg  aatcgcgcgat  acccgcgtacg  accgcgatacg  acgaacgtcga  acgacgcgata 

acgacgtacgg  acgattcgacg  acgattcgcgt  acgcgaaacga  acgcgaatcgt  acgcgtcgaaa 
acgcgtcgcga  acgcgtcgcta  acggtcgtcga  acgttcgaacg  acgttcgaccg  actcgtcgcga 
atcgacgcgcg  atcgcgcgatt  atcgcggtacg  atcgtaccgcg  atcgtacgccg  atcgtcgaccg 
attacgcgcga  attacgcgcgg  attacgtcgcg  attcgcgcgta  attgcgtcgcg  cccgatacgcg 
ccgatacgcgc  ccgcgatacga  ccgcgcgataa  ccgcgcgtaat  ccgcgcgtata  ccggtcgtacg 
ccgtacgtcgt  ccgtcgaatcg  cgaatttcgcg  cgacgagcgta  cgacgcgataa  cgacgcgatac 
cgacgcgtaac  cgacggatacg  cgacgtaacgc  cgacgttaacg  cgactaacgcg  cgatacgacga 
cgatacgccga  cgatacgcgtt  cgatagtcgcg  cgatcgacgcg  cgatcgcgtaa  cgatcgtacga 
cgatcgtcgca  cgattcgacgg  cgattgacgcg  cgcatatcgcg  cgccgattacg  cgcgaaattcg 
cgcgaccgata  cgcgacgcaat  cgcgacgtaat  cgcgactatcg  cgcgatacgaa  cgcgatacgac 
cgcgatatcac  cgcgatatccg  cgcgatatgcg  cgcgatcggta  cgcgcgtaacg  cgcgcgtcgat 
cgcggtacgat  cgcgtaacgta  cgcgtatcggg  cgcgtcaatcg  cgcgtcacgta  cgcgtcgatcg 
cgcgtcgatta  cgcgttagtcg  cgctcgacgta  cggacgtcgta  cggatatcgcg  cggcgtacgat 
cggcgtcgtaa  cgggcgtaacg  cggtcgaacgt  cggtcgacgat  cgtaatcgcga  cgtaatcggcg 
cgtaccgcgat  cgtacgaccgg  cgtacgatcgc  cgtacgcgggt  cgtatccgtcg  cgtatcgcgag 
cgtatcgcggt  cgtccgatcga  cgtcgaatcgt  cgtcgacgagc  cgtcgcgttaa  cgtcgcgttag 
cgtcgttacgc  cgttaacgtcg  cgttacgcccg  cgttacgcgcg  cgttcgaacgt  cgttcgaccga 
cgttgcgcgaa  cgttgcgtcga  ctaacgcgacg  ctcgcgatacg  ctcgcgtacga  gcgatcgtacg 
gcgcgatacga  gcgcgtacgac  gcgcgtatcgg  gcgtaacgacg  gcgttacgtcg  gctcgtcgacg 
gtatcgcgtcg  gtcgcgaacta  gtcgcgcgata  gtcgtacgcga  gtcgtacgcgc  gtcgtatcgcg 
gtgatatcgcg  gttacgcgtcg  taaccgcgcga  taatcgacgcg  taccgatcgcg  tacgacgtccg 
tacgcgcgaat  tacgctcgtcg  tacggacgcga  tacgtcgagcg  tacgtgacgcg  tacgttacgcg 
tagcgacgcgt  tagttcgcgac  tatacgcgcgg  tatcgcgcgaa  tatcgcgcgac  tatcgcgtcgt 
tatcggcgcga  tatcggtcgcg  tcatcgcgcga  tcgacgaccgt  tcgacgcaacg  tcgacgcgtaa 
tcgacgttcgt  tcgatcggacg  tcgcgacgaaa  tcgcgacgagt  tcgcgacgcgt  tcgcgattacg 
tcgcgccgata  tcgcgcgatga  tcgcgcggtta  tcgcgcgtaat  tcgcgtaccga  tcgcgtacgaa 
tcgcgtacgac  tcgcgtccgta  tcggcgtatcg  tcggtacgcga  tcggtcgaacg  tcgtacgatcg 
tcgtacgcgag  tcgtatcgcgc  tcgtatcgcgg  tcgtcgaacga  tcgtcgtatcg  tcgttcgacga 
tcgtttcgcgt  tgcgacgatcg  ttaacgcgacg  ttacgacgccg  ttacgcgatcg  ttacgcgcgaa 
ttacgcgtcga  ttatcgcgcgg  ttatcgcgtcg  ttcgcgcaacg  ttcgcgcgata  ttcgcgcgtaa 
ttcgtacgcga  ttcgtatcgcg  tttcgacgcgt  tttcgtcgcga 
Unwords for the C. elegans genome.
acccccccag  ctgggggggt 
Unwords for the D. melanogaster genome.
acccctaggga  acccctctacg  acccggtaggg  accctaccggg 

acctagcgcgc  acctagcgcgt  acctagcgtga  acctaggtctg 
acgcgctaggt  acggccgtacc  acgggaggttc  acgtcccgcta 
actaggtaccg  aggcccgcgcg  aggcccgctat  agggtacgccg 
agtataggccg  atagcgggcct  cacgcgtgggg  cagacctaggt 
ccccacgcgtg  ccccggcctag  ccccgtagggc  cccgcgttaag 
cccggtagggt  cccggtctagg  cccgtacgcgc  ccctaccgggt 
ccctacggggc  ccctaggcacg  ccggtagctag  ccggtagggta 
cctacgcgtca  cctacgtagag  cctagaccggg  cctagggtccg 
cctataggccg  cgcgcgggcct  cgcgctagcgc  cgcgctaggcc 
cgcggggtacc  cgcgtagtcta  cgctagggccg  cggaccctagg 
cggccctagcg  cggcctatact  cggcctatagg  cggcgtaccct 
cggggcccgac  cgggtagactc  cgggtcgctag  cggtacctagt 
cggtcctatcc  cgtagaggggt  cgtccgtagca  cgtgagggacc 
cgtgcctaggg  ctagcgacccg  ctagctaccgg  ctaggccgggg 
ctctacgtagg  cttaacgcggg  gaacctcccgt  gacctactaga 
gacctaggtac  gacgctagggc  gagtctacccg  gccccgtaggg 
gccctacgggg  gccctagcgtc  gcgcgctaggt  gcgcgtacccc 
gcgcgtacggg  gcgctagcgcg  gcggccctacc  gcgggtacccc 
gctagggtacc  ggataggaccg  ggcctagcgcg  gggacgttaga 
ggggtacccgc  ggggtacgcgc  ggtaccccgcg  ggtaccctagc 
ggtacggccgt  ggtagggccgc  ggtccctcacg  ggtccgcgcta 
gtaacgcggac  gtacctaggtc  gtccgcgttac  gtcgggccccg 
gtcggtcccta  taccctaccgg  tagactacgcg  tagcgcggacc 
tagcgggacgt  tagggaccgac  tcacgctaggt  tccctaggggt 
tctaacgtccc  tctagtaggtc  tgacgcgtagg  tgctacggacg 
Unwords for the S. cerevisiae genome.
ccccgggga  cgccccccg  cggggggcg  tccccgggg 
Unwords for the S. aureus genome (strain MSSA476).
aacccccc  acacgggg  accccgcg  acccgggc  acccgggg  accggcgg 

acgccggg  acgcgggc  acggcccg  acgggacc  acgggccc  acgggggg 
actccggg  actcgggc  agcccggg  agccgagg  aggccccc  aggccccg 
aggcccgg  aggggggg  atccgggg  cacggaga  cacggggc  cacggggg 
cagcgggg  caggccgc  caggccgg  cagggccg  ccacggag  cccacgga 
cccagggg  cccccccc  ccccccct  ccccccgc  ccccccgt  cccccggg 
cccccgtg  ccccgagg  ccccgcgc  ccccgctg  ccccggag  ccccggat 
ccccggcc  ccccggcg  ccccgggc  ccccgggt  ccccgtgt  cccctggg 
cccgaggg  cccgcagg  cccgcggg  cccggagc  cccggagt  cccggcgt 
cccgggag  cccgggcc  cccgggct  cccggggg  ccctaggg  ccctccgc 
ccctcggg  ccgagagc  ccgccccg  ccgccggt  ccgcgccc  ccgcgcgg 
ccgcgggc  ccggaccg  ccggcccg  ccggccga  ccggccgg  ccggcctg 
ccggcggc  ccgggagc  ccgggccg  ccgggcct  ccggggag  ccgggggc 
ccggtcag  cctcagcg  cctccgcg  cctccgga  cctcgccg  cctcggag 
cctcggct  cctcgggg  cctgcggg  cgaccccc  cgagcccc  cgagcctc 
cgagctcg  cgccccga  cgccccgc  cgcccgcg  cgccgggc  cgccgggg 
cgcgcgga  cgcgcggc  cgcggagg  cgcggccg  cgcgggca  cgcgggcg 
cgcggggt  cgctcccg  cgctgagg  cggacccc  cggacccg  cggagacc 
cggagccg  cggagggc  cggccccc  cggccccg  cggcccga  cggcccgc 
cggcccgg  cggccctc  cggccctg  cggccgac  cggccgcg  cggcgagg 
cggcgccc  cggcgccg  cggcgggc  cggctccc  cggctccg  cgggaccc 
cgggagag  cgggagcc  cgggagcg  cgggcccg  cgggccgg  cgggccgt 
cggggcac  cggggccg  cggggcct  cggggcgg  cgggggcc  cgggtccg 
cggtccgg  ctaccccc  ctccccgg  ctcccggg  ctccgacc  ctccgagg 
ctccgcgc  ctccggag  ctccgggg  ctccgtgg  ctcggccc  ctcgggac 
ctcgggcc  ctctcccg  ctgaccgg  ctggcccc  gaggctcg  gagggccg 
gatcccta  gccccccc  gcccccgg  gccccgtg  gcccgagt  gcccgccc 
gcccgccg  gcccgcgc  gcccgcgg  gcccgcgt  gcccggcg  gcccgggc 
gcccgggg  gcccgggt  gccctccg  gccgccgg  gccgcgcg  gccggccc 
gcgagccc  gcgcggag  gcgcgggc  gcgcgggg  gcggaggg  gcggcccc 
gcggccgc  gcggcctg  gcggctcc  gcgggccg  gcggggcg  gcgggggg 
gcggtccc  gctcccgg  gctccggg  gctctcgg  ggactccc  ggagccgc 
ggccagga  ggcccccg  ggcccgag  ggcccgga  ggcccggg  ggccggga 
ggccgggg  ggctcccg  gggaccgc  gggagccg  gggagtcc  gggatccc 
gggcccgt  gggccgag  gggccgca  gggccggc  gggcgccg  gggcgcgg 
gggcgggc  gggctcgc  ggggccag  ggggccgc  ggggctcg  gggggccg 
gggggcct  gggggggc  gggggggg  ggggggtt  gggggtag  gggggtcg 
ggggtccg  gggtaccc  gggtcccg  gggtccga  ggtcccgt  ggtcggag 
ggtctccg  gtcccgag  gtcggccg  gtgccccg  tagggatc  tcccggcc 
tccgcgcg  tccgcgga  tccggagg  tccgggcc  tccgtggg  tcctggcc 
tcggaccc  tcggccga  tcggccgg  tcgggccg  tcggggcg  tctccgtg 
tgcccgcg  tgcggccc 
Unwords for the M. jannaschii genome.
cgatcg  gcgcgc  gtcgac 
Unwords for the T. kodakarensis genome.
tactagta 
Unwords for the M. genitalium genome.
ccggcc  cgcgcg  ctcgga  ggccgg  tccgag 
Conclusion
Genomic unwords may not have a functional meaning, but they do have relevance in practice and in theory. When planning experiments such as large scale mutagenesis [22], a high number of markers is to be included in the inserted DNA. Such markers should be disjoint from each other and from the original genome. Given (say) 100 unwords of length 11, we can directly compose 10,000 markers of length 22 which have a guaranteed Hamming distance from the genome of at least 2. From this supply of candidates, markers can be selected according to other criteria such as melting temperature.
Unwords analysis is fast enough to be applied to the large mammalian genomes. and even to larger data sets resulting from ultrafast sequencing projects. The fact that the genome sequence need not be kept in main memory makes the program applicable to even larger data volumes in pan or metagenome projects. For demonstration, we have applied our program to a recent version of the NTdatabase (all nonredundant GenBank+EMBL+DDBJ+PDB sequences, 21,789,632,349 bp). It requires 136 minutes and 40 MB of main memory to compute all 15,560 unwords of length 14. A further interesting application would be for genomic fragment data. In metagenome projects based on ultrafast sequencing technology, unwords analysis may prove useful in monitoring coverage.
Unwords, by definition, always have a fixed length (say k) in a given genome. Longer absent words may also be of interest. They are easily determined with our program: Adding all unwords as additional sequences to the genome and rerunning the program, it will produce all absent words of length k + 1, since they are the unwords of the extended genome.
No evidence has been collected for selection against specific words in a genomewide fashion. Naturally, unwords tend to have atypical CG content in the ATrich genomes we studied (see Table 4). CpG methylation and subsequent mutation favors unwords containing CG dinucleotides, and leads to an overabundance of their mutated variants [10]. These observations suggest that length and number of unwords, and in particular their deviation from expectation in random sequences, are statistical footprints of the process of real genome evolution. Mathematical models or reconstructions of genome evolution should be tested whether they produce a similar footprint.
The program UNWORDS is available from the Bielefeld University Bioinformatics Server [23]. While online use is restricted to sequence uploads of at most 5 Mb, the UNWORDS source code is available at [24], which has no such restriction.
Declarations
Acknowledgements
We are grateful to the anonymous referee who pointed us to the recent work of [9] and [10]. We thank Sven Rahmann and Ellen Baake for a discussion on unword statistics, and Jens Stoye for helpful discussions and his support for JH when the study was started. We appreciate the help of Jan Krüger and Daniel Hagemeier in composing the unwords website at BiBiServ.
Authors’ Affiliations
References
 Wang Y, Hill K, Singh S, Kari L: The spectrum of genomic signatures; from dinucleotides to chaps game representation. Gene 2005, 346: 173–185.View ArticlePubMedGoogle Scholar
 Workman C, Krogh A: No evidence that mRNAs have lower folding free energies than random sequences with the same dinucleotide distribution. Nucleic Acids Res 1999, 27(24):4816–4822.PubMed CentralView ArticlePubMedGoogle Scholar
 Krause L, McHardy A, Nattkemper T, Pühler A, Stoye J, Meyer F: GISMO – gene identification using a support vector machine for ORF classification. Nucleic Acids Res 2007, 35(2):540–549.PubMed CentralView ArticlePubMedGoogle Scholar
 Pingoud A, Jeltsch A: Structure and function of type II restriction endonucleases. Nucleic Acids Res 2001, 29: 3705–3727.PubMed CentralView ArticlePubMedGoogle Scholar
 Apostolico A, Bock ME, Lonardi S: Monotony of Surprise And LargeScale Quest for Unusual Words. Proceedings of the Sixth Annual International Conference on Computional Biology (RECOMB 2002) 2002, 22–31.View ArticleGoogle Scholar
 Apostolico A, Gong F, Lonardi S: Verbumculus and the Discovery of Unusual Words. Journal of Computer and Science Technology 2004, 19: 22–41.View ArticleGoogle Scholar
 Darling A, Mau B, Blattner F, Perna N: Mauve: multiple alignment of conserved genomic sequence with rearrangements. Genome Res 2004, 14(7):1394–403.PubMed CentralView ArticlePubMedGoogle Scholar
 Haubold B, Pierstorff N, Möller F, Wiehe T: Genome comparison without alignment using shortest unique substrings. BMC Bioinformatics 2005, 6: 123.PubMed CentralView ArticlePubMedGoogle Scholar
 Hampikian G, Andersen T: Absent sequences: nullomers and primes. Pacific Symposium on Biocomputing 2007, 12: 355–366.Google Scholar
 Acquisti C, Poste G, Curtiss D, Kumar S: Nullomers: really a matter of natural selection. PLoS ONE 2007., 2(10):Google Scholar
 Abouelhoda M, Kurtz S, Ohlebusch E: Replacing Suffix Trees with Enhanced Suffix Arrays. Journal of Discrete Algorithms 2004, 2: 53–86.View ArticleGoogle Scholar
 Vmatch[http://www.vmatch.de]
 Rahmann S, Rivals E: On the distribution of the number of missing words in random texts. Combinatorics, Probability and Computing 2003, 12: 73–87.View ArticleGoogle Scholar
 Human Genome[http://www.ensembl.org/Homo_sapiens]
 Mouse Genome[http://www.ensembl.org/Mus_musculus]
 Drosophila Genomes[http://www.fruitfly.org/sequence/release5genomic.shtml]
 C. elegans Genome[http://www.ensembl.org/Caenorhabditis_elegans]
 Galagan J, Calvo S, Borkovich K, Selker E, Read N, Jaffe D, FitzHugh W, Ma L, Smirnov S, Purcell S, Rehman B, Elkins T, Engels R, Wang S, Nielsen C, Butler J, Endrizzi M, Qui D, Ianakiev P, BellPedersen D, Nelson M, WernerWashburne M, Selitrennikoff C, Kinsey J, Braun E, Zelter A, Schulte U, Kothe G, Jedd G, Mewes W, Staben C, Marcotte E, Greenberg D, Roy A, Foley K, Naylor J, StangeThomann N, Barrett R, Gnerre S, Kamal M, Kamvysselis M, Mauceli E, Bielke C, Rudd S, Frishman D, Krystofova S, Rasmussen C, Metzenberg R, Perkins D, Kroken S, Cogoni C, Macino G, Catcheside D, Li W, Pratt R, Osmani S, DeSouza C, Glass L, Orbach M, Berglund J, Voelker R, Yarden O, Plamann M, Seiler S, Dunlap J, Radford A, Aramayo R, Natvig D, Alex L, Mannhaupt G, Ebbole D, Freitag M, Paulsen I, Sachs M, Lander E, Nusbaum C, Birren B: The genome sequence of the filamentous fungus Neurospora crassa. Nature 2003, 6934: 821–2.Google Scholar
 S. cerevisiae Genome[http://www.ensembl.org/Saccharomyces_cerevisiae]
 Fukui T, Atomi H, Kanai T, Matsumi R, Fujiwara S, Imanaka T: Complete genome sequence of the hyperthermophilic archaeon Thermococcus kodakaraensis KOD1 and comparison with Pyrococcus genomes. Genome Res 2005, 15(3):352–63.PubMed CentralView ArticlePubMedGoogle Scholar
 Bult CJ, White O, Olsen GJ, Zhou L, Fleischmann RD, Sutton GG, Blake JA, FitzGerald LM, Clayton RA, Gocayne JD, Kerlavage AR, Dougherty BA, Tomb JF, Adams MD, Reich CI, Overbeek R, Kirkness EF, Weinstock KG, Merrick JM, Glodek A, Scott JL, Geoghagen NS, Venter JC: Complete genome sequence of the methanogenic archaeon, Methanococcus jannaschii. Science 1996, 273(5278):1058–73.View ArticlePubMedGoogle Scholar
 Pobigaylo N, Wetter D, Szymczak S, Schiller U, Kurtz S, Meyer F, Nattkemper T, Becker A: Construction of a large signaturetagged miniTn5 transposon library and its application to mutagenesis of Sinorhizobium meliloti. Appl Environ Microbiol 2006, 72(6):4329–4337.PubMed CentralView ArticlePubMedGoogle Scholar
 Computing Unwords on BibiServ[http://bibiserv.techfak.unibielefeld.de/unwords]
 Unwords[http://www.zbh.unihamburg.de/unwords]
Copyright
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.