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<title>Alignment-free sequence analysis</title>

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Alignment-free sequence analysis

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In <a href="Bioinformatics" title="Bioinformatics">bioinformatics</a>, alignment-free sequence analysis approaches to molecular sequence and structure data provide alternatives over alignment-based approaches.<a href="#cite_note-Vinga-1">[1]</a>

The emergence and need for the analysis of different types of data generated through biological research has given rise to the field of <a href="Bioinformatics" title="Bioinformatics">bioinformatics</a>.<a href="#cite_note-2">[2]</a> Molecular sequence and structure data of <a href="DNA" title="DNA">DNA</a>, <a href="RNA" title="RNA">RNA</a>, and <a href="Proteins" class="mw-redirect" title="Proteins">proteins</a>, <a href="Gene_expression" title="Gene expression">gene expression</a> profiles or <a href="Microarray" title="Microarray">microarray</a> data, <a href="Metabolic_pathway" title="Metabolic pathway">metabolic pathway</a> data are some of the major types of data being analysed in bioinformatics. Among them sequence data is increasing at the exponential rate due to advent of next-generation sequencing technologies. Since the origin of bioinformatics, <a href="Sequence_analysis" title="Sequence analysis">sequence analysis</a> has remained the major area of research with wide range of applications in database searching, <a href="Genome_annotation" class="mw-redirect" title="Genome annotation">genome annotation</a>, <a href="Comparative_genomics" title="Comparative genomics">comparative genomics</a>, <a href="Molecular_phylogeny" class="mw-redirect" title="Molecular phylogeny">molecular phylogeny</a> and <a href="Gene_prediction" title="Gene prediction">gene prediction</a>. The pioneering approaches for sequence analysis were based on <a href="Sequence_alignment" title="Sequence alignment">sequence alignment</a> either global or local, pairwise or <a href="Multiple_sequence_alignment" title="Multiple sequence alignment">multiple sequence alignment</a>.<a href="#cite_note-3">[3]</a><a href="#cite_note-4">[4]</a> Alignment-based approaches generally give excellent results when the sequences under study are closely related and can be reliably aligned, but when the sequences are divergent, a reliable alignment cannot be obtained and hence the applications of sequence alignment are limited. Another limitation of alignment-based approaches is their computational complexity and are time-consuming and thus, are limited when dealing with large-scale sequence data.<a href="#cite_note-5">[5]</a> The advent of <a href="Next-generation_sequencing" class="mw-redirect" title="Next-generation sequencing">next-generation sequencing</a> technologies has resulted in generation of voluminous sequencing data. The size of this sequence data poses challenges on alignment-based algorithms in their assembly, annotation and comparative studies.

<div class="mw-heading mw-heading2"><h2 id="Alignment-free_methods">Alignment-free methods</h2></div>

Alignment-free methods can broadly be classified into five categories: a) methods based on k-mer/word frequency, b) methods based on the length of common substrings, c) methods based on the number of (spaced) word matches, d) methods based on micro-alignments, e) methods based on information theory and f) methods based on graphical representation. Alignment-free approaches have been used in sequence similarity searches,<a href="#cite_note-Hide-6">[6]</a> clustering and classification of sequences,<a href="#cite_note-Miller-7">[7]</a> and more recently in phylogenetics<a href="#cite_note-Domazet-Lošo_1466–72-8">[8]</a><a href="#cite_note-Ragan-9">[9]</a> (Figure 1).

Such molecular phylogeny analyses employing alignment-free approaches are said to be part of next-generation phylogenomics.<a href="#cite_note-Ragan-9">[9]</a> A number of review articles provide in-depth review of alignment-free methods in sequence analysis.<a href="#cite_note-Vinga-1">[1]</a><a href="#cite_note-Song-10">[10]</a><a href="#cite_note-Haubold-11">[11]</a><a href="#cite_note-BonhamCarter-12">[12]</a><a href="#cite_note-Zielezinski-13">[13]</a><a href="#cite_note-Bernard-14">[14]</a><a href="#cite_note-Ren-15">[15]</a>

The AFproject is an international collaboration to benchmark and compare software tools for alignment-free sequence comparison.<a href="#cite_note-ZielezinskiGirgis2019-16">[16]</a>

<div class="mw-heading mw-heading3"><h3 id="Methods_based_on_k-mer/word_frequency">Methods based on k-mer/word frequency</h3></div>

The popular methods based on k-mer/word frequencies include feature frequency profile (FFP),<a href="#cite_note-FFP-17">[17]</a><a href="#cite_note-18">[18]</a> Composition vector (CV),<a href="#cite_note-19">[19]</a><a href="#cite_note-20">[20]</a> Return time distribution (RTD),<a href="#cite_note-RTD1-21">[21]</a> frequency chaos game representation (FCGR).<a href="#cite_note-22">[22]</a> and Spaced Words.<a href="#cite_note-spaced-23">[23]</a>

<div class="mw-heading mw-heading4"><h4 id="Feature_frequency_profile_(FFP)">Feature frequency profile (FFP)</h4></div>

The methodology involved in FFP based method starts by calculating the count of each possible k-mer (possible number of k-mers for nucleotide sequence: 4k, while that for protein sequence: 20k) in sequences. Each k-mer count in each sequence is then normalized by dividing it by total of all k-mers' count in that sequence. This leads to conversion of each sequence into its feature frequency profile. The pair wise distance between two sequences is then calculated <a href="Jensen%E2%80%93Shannon_divergence" title="Jensen–Shannon divergence">Jensen–Shannon (JS) divergence</a> between their respective FFPs. The <a href="Distance_matrix" title="Distance matrix">distance matrix</a> thus obtained can be used to construct <a href="Phylogenetic_tree" title="Phylogenetic tree">phylogenetic tree</a> using clustering algorithms like <a href="Neighbor-joining" class="mw-redirect" title="Neighbor-joining">neighbor-joining</a>, <a href="UPGMA" title="UPGMA">UPGMA</a> etc.

<div class="mw-heading mw-heading4"><h4 id="Composition_vector_(CV)">Composition vector (CV)</h4></div>

In this method frequency of appearance of each possible k-mer in a given sequence is calculated. The next characteristic step of this method is the subtraction of random background of these frequencies using <a href="Markov_model" title="Markov model">Markov model</a> to reduce the influence of random neutral <a href="Mutations" class="mw-redirect" title="Mutations">mutations</a> to highlight the role of selective evolution. The normalized frequencies are put a fixed order to form the composition vector (CV) of a given sequence. <a href="Cosine_distance" class="mw-redirect" title="Cosine distance">Cosine distance</a> function is then used to compute pairwise distance between CVs of sequences. The distance matrix thus obtained can be used to construct phylogenetic tree using clustering algorithms like <a href="Neighbor-joining" class="mw-redirect" title="Neighbor-joining">neighbor-joining</a>, <a href="UPGMA" title="UPGMA">UPGMA</a> etc. This method can be extended through resort to efficient pattern matching algorithms to include in the computation of the composition vectors: (i) all k-mers for any value of k, (ii) all substrings of any length up to an arbitrarily set maximum k value, (iii) all maximal substrings, where a substring is maximal if extending it by any character would cause a decrease in its occurrence count.<a href="#cite_note-24">[24]</a><a href="#cite_note-25">[25]</a>

<div class="mw-heading mw-heading4"><h4 id="Return_time_distribution_(RTD)">Return time distribution (RTD)</h4></div>

The RTD based method does not calculate the count of k-mers in sequences, instead it computes the time required for the reappearance of

k-mers. The time refers to the number of residues in successive appearance of particular k-mer. Thus the occurrence of each k-mer in a sequence is calculated in the form of RTD, which is then summarised using two statistical parameters <a href="Mean" title="Mean">mean</a> (μ) and <a href="Standard_deviation" title="Standard deviation">standard deviation</a> (σ). Thus each sequence is represented in the form of numeric vector of size 2⋅4k containing μ and σ of 4k RTDs. The pair wise distance between sequences is calculated using <a href="Euclidean_distance" title="Euclidean distance">Euclidean distance</a> measure. The distance matrix thus obtained can be used to construct phylogenetic tree using clustering algorithms like <a href="Neighbor-joining" class="mw-redirect" title="Neighbor-joining">neighbor-joining</a>, <a href="UPGMA" title="UPGMA">UPGMA</a> etc. A recent approach Pattern Extraction through Entropy Retrieval (PEER) provides direct detection of the k-mer length and summarised the occurrence interval using entropy.

<div class="mw-heading mw-heading4"><h4 id="Frequency_chaos_game_representation_(FCGR)">Frequency chaos game representation (FCGR)</h4></div>

The FCGR methods have evolved from chaos game representation (CGR) technique, which provides scale independent representation for genomic sequences.<a href="#cite_note-Jeffrey_2163–70-26">[26]</a> The CGRs can be divided by grid lines where each grid square denotes the occurrence of oligonucleotides of a specific length in the sequence. Such representation of CGRs is termed as Frequency Chaos Game Representation (FCGR). This leads to representation of each sequence into FCGR. The pair wise distance between FCGRs of sequences can be calculated using the Pearson distance, the Hamming distance or the Euclidean distance.<a href="#cite_note-27">[27]</a>

<div class="mw-heading mw-heading4"><h4 id="Spaced-word_frequencies">Spaced-word frequencies</h4></div>

While most alignment-free algorithms compare the word-composition of sequences, Spaced Words uses a pattern of care and don't care positions. The occurrence of a spaced word in a sequence is then defined by the characters at the match positions only, while the characters at the don't care positions are ignored. Instead of comparing the frequencies of contiguous words in the input sequences, this approach compares the frequencies of the spaced words according to the pre-defined pattern.<a href="#cite_note-spaced-23">[23]</a> Note that the pre-defined pattern can be selected by analysis of the <a href="Variance" title="Variance">Variance</a> of the number of matches,<a href="#cite_note-rasbhari-28">[28]</a> the probability of the first occurrence on several models,<a href="#cite_note-bhit-29">[29]</a> or the <a href="Pearson_correlation_coefficient" title="Pearson correlation coefficient">Pearson correlation coefficient</a> between the expected word frequency and the true alignment distance.<a href="#cite_note-Cov-30">[30]</a>

<div class="mw-heading mw-heading3"><h3 id="Methods_based_on_length_of_common_substrings">Methods based on length of common substrings</h3></div>

The methods in this category employ the <a href="Semantic_similarity" title="Semantic similarity">similarity</a> and differences of substrings in a pair of sequences. These algorithms

were mostly used for string processing in <a href="Computer_science" title="Computer science">computer science</a>.<a href="#cite_note-31">[31]</a>

<div class="mw-heading mw-heading4"><h4 id="Average_common_substring_(ACS)">Average common substring (ACS)</h4></div>

In this approach, for a chosen pair of sequences (A and B of lengths n and m respectively), <a href="Longest_common_substring_problem" class="mw-redirect" title="Longest common substring problem">longest substring</a> starting at some position is identified in one sequence (A) which exactly matches in the other sequence (B) at any position. In this way, lengths of longest substrings starting at different positions in sequence A and having exact matches at some positions in sequence B are calculated. All these lengths are averaged to derive a measure <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L(A,B)}">

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<annotation encoding="application/x-tex">{\displaystyle L(A,B)}</annotation>

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</math><img src="./81cec64690ccea5f050e96a23ceb95b07c61432a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.933ex; height:2.843ex;" alt="{\displaystyle L(A,B)}" loading="lazy">. Intuitively, larger the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L(A,B)}">

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<annotation encoding="application/x-tex">{\displaystyle L(A,B)}</annotation>

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</math><img src="./81cec64690ccea5f050e96a23ceb95b07c61432a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.933ex; height:2.843ex;" alt="{\displaystyle L(A,B)}" loading="lazy">, the more similar the two sequences are. To account for the differences in the length of sequences, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L(A,B)}">

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<annotation encoding="application/x-tex">{\displaystyle L(A,B)}</annotation>

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</math><img src="./81cec64690ccea5f050e96a23ceb95b07c61432a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.933ex; height:2.843ex;" alt="{\displaystyle L(A,B)}" loading="lazy"> is normalized [i.e. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L(A,B)/\log(m)}">

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<annotation encoding="application/x-tex">{\displaystyle L(A,B)/\log(m)}</annotation>

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</math><img src="./7b23610d7d935dda7f79c517fa8d58ff21ff225b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.304ex; height:2.843ex;" alt="{\displaystyle L(A,B)/\log(m)}" loading="lazy">]. This gives the similarity measure between the sequences.

In order to derive a distance measure, the inverse of <a href="Similarity_measure" title="Similarity measure">similarity measure</a> is taken and a correction term is subtracted from it to assure that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d(A,A)}">

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<annotation encoding="application/x-tex">{\displaystyle d(A,A)}</annotation>

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</math><img src="./06bda3881136a72e5af29d3b03676b4688d673dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.545ex; height:2.843ex;" alt="{\displaystyle d(A,A)}" loading="lazy"> will be zero. Thus

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<annotation encoding="application/x-tex">{\displaystyle d(A,B)=\left[{\frac {\log m}{L(A,B)}}\right]-\left[{\frac {\log n}{L(A,A)}}\right].}</annotation>

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</math><img src="./e839899b28da174a99bfd5781de6e19e3d03b0cd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:36.967ex; height:6.343ex;" alt="{\displaystyle d(A,B)=\left[{\frac {\log m}{L(A,B)}}\right]-\left[{\frac {\log n}{L(A,A)}}\right].}" loading="lazy"></dd></dl>

This measure <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d(A,B)}">

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<annotation encoding="application/x-tex">{\displaystyle d(A,B)}</annotation>

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</math><img src="./c0648d7bfb101f8b3be00009fbb8f3584e7adb8e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.566ex; height:2.843ex;" alt="{\displaystyle d(A,B)}" loading="lazy"> is not symmetric, so one has to compute <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d_{s}(A,B)=d_{s}(B,A)=(d(A,B)+d(B,A))/2}">

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<annotation encoding="application/x-tex">{\displaystyle d_{s}(A,B)=d_{s}(B,A)=(d(A,B)+d(B,A))/2}</annotation>

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</math><img src="./01e10bb029e51d7880c5d8ab4238434c746087e7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:45.429ex; height:2.843ex;" alt="{\displaystyle d_{s}(A,B)=d_{s}(B,A)=(d(A,B)+d(B,A))/2}" loading="lazy">, which gives final ACS measure between the two strings (A and B).<a href="#cite_note-32">[32]</a> The subsequence/substring search can be efficiently performed by

using <a href="Suffix_tree" title="Suffix tree">suffix trees</a>.<a href="#cite_note-33">[33]</a><a href="#cite_note-34">[34]</a><a href="#cite_note-35">[35]</a>

<div class="mw-heading mw-heading4"><h4 id="k-mismatch_average_common_substring_approach_(kmacs)">k-mismatch average common substring approach (kmacs)</h4></div>

This approach is a generalization of the ACS approach. To define the distance between two DNA or protein sequences, kmacs estimates for each position i of the first sequence the longest substring starting at i and matching a substring of the second sequence with up to k mismatches. It defines the average of these values as a measure of similarity between the sequences and turns this into a symmetric distance measure. Kmacs does not compute exact k-mismatch substrings, since this would be computational too costly, but approximates such substrings.<a href="#cite_note-kmacs-36">[36]</a>

<div class="mw-heading mw-heading4"><h4 id="Mutation_distances_(Kr)">Mutation distances (Kr)</h4></div>

This approach is closely related to the ACS, which calculates the number of substitutions per site between two DNA sequences using the shortest

absent substring (termed as shustring).<a href="#cite_note-37">[37]</a>

<div class="mw-heading mw-heading4"><h4 id="Length_distribution_of_k-mismatch_common_substrings">Length distribution of k-mismatch common substrings</h4></div>

This approach uses the program kmacs<a href="#cite_note-kmacs-36">[36]</a> to calculate <a href="Longest_common_substring" title="Longest common substring">longest common substrings</a> with up to k mismatches for a pair of DNA sequences. The phylogenetic distance between the sequences can then be estimated from a local maximum in the length distribution of the k-mismatch common substrings.<a href="#cite_note-38">[38]</a>

<div class="mw-heading mw-heading3"><h3 id="Methods_based_on_the_number_of_(spaced)_word_matches">Methods based on the number of (spaced) word matches</h3></div>

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<annotation encoding="application/x-tex">{\displaystyle D_{2}^{S}}</annotation>

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</msubsup>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle D_{2}^{*}}</annotation>

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</math><img src="./ab92a8450f2efca7832051fde1d0e34d754ac8a0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.979ex; height:2.843ex;" alt="{\displaystyle D_{2}^{*}}" loading="lazy"></h4></div>

These approachese are variants of the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D_{2}}">

<msub>

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</msub>

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<annotation encoding="application/x-tex">{\displaystyle D_{2}}</annotation>

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</math><img src="./41b3839c40bd06e3dfea10798dfab41a905af256.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.979ex; height:2.509ex;" alt="{\displaystyle D_{2}}" loading="lazy"> statistics that counts the number of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

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<annotation encoding="application/x-tex">{\displaystyle k}</annotation>

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</math><img src="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy">-mer matches between two sequences. They improve the simple <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D_{2}}">

<msub>

</mrow>

</msub>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle D_{2}}</annotation>

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</math><img src="./41b3839c40bd06e3dfea10798dfab41a905af256.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.979ex; height:2.509ex;" alt="{\displaystyle D_{2}}" loading="lazy"> statistics by taking the background distribution of the compared sequences into account.<a href="#cite_note-39">[39]</a>

This is an extremely fast method that uses the MinHash bottom sketch strategy for estimating the <a href="Jaccard_index" title="Jaccard index">Jaccard index</a> of the multi-sets of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle k}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle k}</annotation>

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<div class="mw-heading mw-heading4"><h4 id="Slope-Tree">Slope-Tree</h4></div>

This approach calculates a distance value between two protein sequences based on the decay of the number of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

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<annotation encoding="application/x-tex">{\displaystyle k}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle k}</annotation>

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<div class="mw-heading mw-heading4"><h4 id="Slope-SpaM">Slope-SpaM</h4></div>

This method calculates the number <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N_{k}}">

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<annotation encoding="application/x-tex">{\displaystyle N_{k}}</annotation>

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</math><img src="./7698f968a98115830bcc378e0f849e0375c858c9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.955ex; height:2.509ex;" alt="{\displaystyle N_{k}}" loading="lazy"> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

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<annotation encoding="application/x-tex">{\displaystyle k}</annotation>

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(SpaM) for different values for the word length or number of match positions <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

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<annotation encoding="application/x-tex">{\displaystyle k}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle F}</annotation>

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</math><img src="./545fd099af8541605f7ee55f08225526be88ce57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}" loading="lazy"> that depends on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N_{k}}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle N_{k}}</annotation>

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</math><img src="./7698f968a98115830bcc378e0f849e0375c858c9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.955ex; height:2.509ex;" alt="{\displaystyle N_{k}}" loading="lazy"> is calculated to estimate the Jukes-Cantor distance between the input sequences .<a href="#cite_note-42">[42]</a>

<div class="mw-heading mw-heading4"><h4 id="Skmer">Skmer</h4></div>

Skmer calculates distances between species from unassembled sequencing reads. Similar to MASH, it uses the <a href="Jaccard_index" title="Jaccard index">Jaccard index</a> on the sets of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

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<annotation encoding="application/x-tex">{\displaystyle k}</annotation>

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</math><img src="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy">-mers from the input sequences. In contrast to MASH, the program is still accurate for low sequencing coverage, so it can be used for <a href="Genome_skimming" title="Genome skimming">genome skimming</a>.<a href="#cite_note-43">[43]</a>

<div class="mw-heading mw-heading3"><h3 id="Methods_based_on_micro-alignments">Methods based on micro-alignments</h3></div>

Strictly spoken, these methods are not alignment-free. They are using simple gap-free micro-alignments where sequences are required to match at certain pre-defined positions. The positions aligned at the remaining positions of the micro-alignments where mismatches are allowed, are then used for phylogeny inference.

<div class="mw-heading mw-heading4"><h4 id="Co-phylog">Co-phylog</h4></div>

This method searches for so-called structures that are defined as pairs of k-mer matches between two DNA sequences that are one position apart in both sequences. The two k-mer matches are called the context, the position between them is called the object. Co-phylog then defines the distance between two sequences the fraction of such structures for which the two nucleotides in the object are different. The approach can be applied to unassembled sequencing reads.<a href="#cite_note-Co-phylog:_an_assembly-free_phyloge-44">[44]</a>

andi estimates phylogenetic distances between genomic sequences based on ungapped local alignments that are flanked by maximal exact word matches. Such word matches can be efficiently found using suffix arrays. The gapfree alignments between the exact word matches are then used to estimate phylogenetic distances between genome sequences. The resulting distance estimates are accurate for up to around 0.6 substitutions per position.<a href="#cite_note-45">[45]</a>

<div class="mw-heading mw-heading4"><h4 id="Filtered_Spaced-Word_Matches_(FSWM)">Filtered Spaced-Word Matches (FSWM)</h4></div>

FSWM uses a pre-defined binary pattern P representing so-called match positions and don't-care positions. For a pair of input DNA sequences, it then searches for spaced-word matches w.r.t. P, i.e. for local gap-free alignments with matching nucleotides at the match positions of P and possible mismatches at the don't-care positions. Spurious low-scoring spaced-word matches are discarded, evolutionary distances between the input sequences are estimated based on the nucleotides aligned to each other at the don't-care positions of the remaining, homologous spaced-word matches.<a href="#cite_note-Leimeister_2017_971–979-46">[46]</a> FSWM has been adapted to estimate distances based on unassembled NGS reads, this version of the program is called Read-SpaM.<a href="#cite_note-47">[47]</a>

Prot-SpaM (Proteome-based Spaced-word Matches) is an implementation of the FSWM algorithm for partial or whole proteome sequences.<a href="#cite_note-Leimeister_2018_giy148-48">[48]</a>

<div class="mw-heading mw-heading4"><h4 id="Multi-SpaM">Multi-SpaM</h4></div>

Multi-SpaM (MultipleSpaced-word Matches) is an approach to genome-based phylogeny reconstruction that extends the FSWM idea to multiple sequence comparison.<a href="#cite_note-49">[49]</a> Given a binary pattern P of match positions and don't-care positions, the program searches for P-blocks, i.e. local gap-free four-way alignments with matching nucleotides at the match positions of P and possible mismatches at the don't-care positions. Such four-way alignments are randomly sampled from a set of input genome sequences. For each P-block, an unrooted tree topology is calculated using RAxML.<a href="#cite_note-50">[50]</a> The program Quartet MaxCut is then used to calculate a supertree from these trees.

<div class="mw-heading mw-heading3"><h3 id="Methods_based_on_information_theory">Methods based on information theory</h3></div>

<a href="Information_Theory" class="mw-redirect" title="Information Theory">Information Theory</a> has provided successful methods for alignment-free sequence analysis and comparison. The existing applications of information theory include global and local characterization of DNA, RNA and proteins, estimating genome entropy to motif and region classification. It also holds promise in <a href="Gene_mapping" title="Gene mapping">gene mapping</a>, <a href="Next-generation_sequencing" class="mw-redirect" title="Next-generation sequencing">next-generation sequencing</a> analysis and <a href="Metagenomics" title="Metagenomics">metagenomics</a>.<a href="#cite_note-51">[51]</a>

<div class="mw-heading mw-heading4"><h4 id="Base–base_correlation_(BBC)">Base–base correlation (BBC)</h4></div>

Base–base correlation (BBC) converts the genome sequence into a unique 16-dimensional numeric vector using the following equation,

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<annotation encoding="application/x-tex">{\displaystyle T_{ij}(K)=\sum _{\ell =1}^{K}P_{ij}(\ell )\cdot \log _{2}\left({\frac {P_{ij}(\ell )}{P_{i}P_{j}}}\right)}</annotation>

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</math><img src="./dfa9e0aad4c91b7adcbec3abb92ef821cf713e30.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:35.01ex; height:7.509ex;" alt="{\displaystyle T_{ij}(K)=\sum _{\ell =1}^{K}P_{ij}(\ell )\cdot \log _{2}\left({\frac {P_{ij}(\ell )}{P_{i}P_{j}}}\right)}" loading="lazy"></dd></dl>

The <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{i}}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle P_{i}}</annotation>

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</math><img src="./3ba1396129f7be3c7f828a571b6649e6807d10d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.292ex; height:2.509ex;" alt="{\displaystyle P_{i}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{j}}">

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<annotation encoding="application/x-tex">{\displaystyle P_{j}}</annotation>

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</math><img src="./4a5da6c3564a2129f714ef11acd8ba649d18e604.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.402ex; height:2.843ex;" alt="{\displaystyle P_{j}}" loading="lazy"> denotes the probabilities of bases i and j in the genome. The <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{ij}(\ell )}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle P_{ij}(\ell )}</annotation>

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</math><img src="./36d3a976b93859f6bd749745de9dfa5c057bb4a7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.748ex; height:3.009ex;" alt="{\displaystyle P_{ij}(\ell )}" loading="lazy"> indicates the probability of bases i and j at distance ℓ in the genome. The parameter K indicates the maximum distance between the bases i and j. The variation in the values of 16 parameters reflect variation in the genome content and length.<a href="#cite_note-52">[52]</a><a href="#cite_note-53">[53]</a><a href="#cite_note-54">[54]</a>

<div class="mw-heading mw-heading4"><h4 id="Information_correlation_and_partial_information_correlation_(IC-PIC)">Information correlation and partial information correlation (IC-PIC)</h4></div>

IC-PIC (information correlation and partial information correlation) based method employs the base correlation property of DNA sequence. IC and PIC were calculated using following formulas,

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<annotation encoding="application/x-tex">{\displaystyle IC_{\ell }=-2\sum _{i}P_{i}\log _{2}P_{i}+\sum _{ij}P_{ij}(\ell )\log _{2}P_{ij}(\ell )}</annotation>

</semantics>

</math><img src="./fc1c265fc1c707936f1ef4ac23fa377179029513.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:46.213ex; height:5.843ex;" alt="{\displaystyle IC_{\ell }=-2\sum _{i}P_{i}\log _{2}P_{i}+\sum _{ij}P_{ij}(\ell )\log _{2}P_{ij}(\ell )}" loading="lazy"></dd></dl>

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<annotation encoding="application/x-tex">{\displaystyle PIC_{ij}(\ell )=(P_{ij}(\ell )-P_{i}P_{j}(\ell ))^{2}}</annotation>

</semantics>

</math><img src="./41a72be96e9821dc00475433a39168511895588b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:30.859ex; height:3.343ex;" alt="{\displaystyle PIC_{ij}(\ell )=(P_{ij}(\ell )-P_{i}P_{j}(\ell ))^{2}}" loading="lazy"></dd></dl>

The final vector is obtained as follows:

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<mtext> where </mtext>

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<annotation encoding="application/x-tex">{\displaystyle V={IC_{\ell } \over PIC_{ij}(\ell )}{\text{ where }}\ell \in \left\{\ell _{0},\ell _{0}+1,\ldots ,\ell _{0}+n\right\},}</annotation>

</semantics>

</math><img src="./06de436e52be0a21c91f3d8bb3f02c1e76ed8c8f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:49.356ex; height:6.176ex;" alt="{\displaystyle V={IC_{\ell } \over PIC_{ij}(\ell )}{\text{ where }}\ell \in \left\{\ell _{0},\ell _{0}+1,\ldots ,\ell _{0}+n\right\},}" loading="lazy"></dd></dl>

which defines the range of distance between bases.<a href="#cite_note-55">[55]</a>

The pairwise distance between sequences is calculated using <a href="Euclidean_distance" title="Euclidean distance">Euclidean distance</a> measure. The distance matrix thus obtained can be used to construct phylogenetic tree using clustering algorithms like <a href="Neighbor-joining" class="mw-redirect" title="Neighbor-joining">neighbor-joining</a>, <a href="UPGMA" title="UPGMA">UPGMA</a>, etc..

<div class="mw-heading mw-heading4"><h4 id="Compression">Compression</h4></div>

Examples are effective approximations to <a href="Kolmogorov_complexity" title="Kolmogorov complexity">Kolmogorov complexity</a>, for example <a href="Lempel-Ziv_complexity" class="mw-redirect" title="Lempel-Ziv complexity">Lempel-Ziv complexity</a>. In general compression-based methods use the <a href="Mutual_information" title="Mutual information">mutual information</a> between the sequences. This is expressed in conditional <a href="Kolmogorov_complexity" title="Kolmogorov complexity">Kolmogorov complexity</a>, that is, the length of the shortest self-delimiting program required to generate a string given the prior knowledge of the other string. This measure has a relation to measuring k-words in a sequence, as they can be easily used to generate the sequence. It is sometimes a computationally intensive method. The theoretic basis for the <a href="Kolmogorov_complexity" title="Kolmogorov complexity">Kolmogorov complexity</a> approach was

laid by Bennett, Gacs, Li, Vitanyi, and Zurek (1998) by proposing the <a href="Information_distance" title="Information distance">information distance</a>.<a href="#cite_note-56">[56]</a> The <a href="Kolmogorov_complexity" title="Kolmogorov complexity">Kolmogorov complexity</a> being incomputable it was approximated by compression algorithms. The better they compress the better they are. Li, Badger, Chen, Kwong,, Kearney, and Zhang (2001) used a non-optimal but normalized form of this approach,<a href="#cite_note-57">[57]</a> and the optimal normalized form by Li, Chen, Li, Ma, and Vitanyi (2003) appeared in <a href="#cite_note-58">[58]</a> and more extensively and proven by Cilibrasi and Vitanyi (2005) in.<a href="#cite_note-59">[59]</a>

Otu and Sayood (2003) used the <a href="Lempel-Ziv_complexity" class="mw-redirect" title="Lempel-Ziv complexity">Lempel-Ziv complexity</a> method to construct five different distance measures for <a href="Phylogenetic_tree" title="Phylogenetic tree">phylogenetic tree</a> construction.<a href="#cite_note-60">[60]</a>

<div class="mw-heading mw-heading4"><h4 id="Context_modeling_compression">Context modeling compression</h4></div>

In the context modeling complexity the next-symbol predictions, of one or more statistical models, are combined or competing to yield a prediction that is based on events recorded in the past. The algorithmic information content derived from each symbol prediction can be used to compute algorithmic information profiles with a time proportional to the length of the sequence. The process has been applied to DNA sequence analysis.<a href="#cite_note-61">[61]</a>

<div class="mw-heading mw-heading3"><h3 id="Methods_based_on_graphical_representation">Methods based on graphical representation</h3></div>

<div class="mw-heading mw-heading4"><h4 id="Iterated_maps">Iterated maps</h4></div>

The use of iterated maps for sequence analysis was first introduced by HJ Jefferey in 1990<a href="#cite_note-Jeffrey_2163–70-26">[26]</a> when he proposed to apply the <a href="Chaos_game" title="Chaos game">Chaos Game</a> to map genomic sequences into a unit square. That report coined the procedure as Chaos Game Representation (CGR). However, only 3 years later this approach was first dismissed as a projection of a Markov transition table by N Goldman.<a href="#cite_note-62">[62]</a> This objection was overruled by the end of that decade when the opposite was found to be the case – that CGR bijectively maps Markov transition is into a fractal, order-free (degree-free) representation.<a href="#cite_note-63">[63]</a> The realization that iterated maps provide a bijective map between the symbolic space and numeric space led to the identification of a variety of alignment-free approaches to sequence comparison and characterization. These developments were reviewed in late 2013 by JS Almeida in.<a href="#cite_note-64">[64]</a> A number of web apps such as <a rel="nofollow" class="external free" href="https://github.com/usm/usm.github.com/wiki">https://github.com/usm/usm.github.com/wiki</a>,<a href="#cite_note-Almeida_12-65">[65]</a> are available to demonstrate how to encode and compare arbitrary symbolic sequences in a manner that takes full advantage of modern <a href="MapReduce" title="MapReduce">MapReduce</a> distribution developed for cloud computing.

<div class="mw-heading mw-heading2"><h2 id="Comparison_of_alignment_based_and_alignment-free_methods">Comparison of alignment based and alignment-free methods</h2></div>

<th>Alignment-based methods</th>

<th>Alignment-free methods

</th></tr>

<tr>

<td>These methods assume that homologous regions are contiguous (with gaps)</td>

<td>Does not assume such contiguity of homologous regions

</td></tr>

<tr>

<td>Computes all possible pairwise comparisons of sequences; hence computationally expensive</td>

<td>Based on occurrences of sub-sequences; composition; computationally inexpensive, can be memory-intensive

</td></tr>

<tr>

<td>Well-established approach in phylogenomics</td>

<td>Relatively recent and application in phylogenomics is limited; needs further testing for robustness and scalability

</td></tr>

<tr>

<td>Requires substitution/evolutionary models</td>

<td>Less dependent on substitution/evolutionary models

</td></tr>

<tr>

<td>Sensitive to stochastic sequence variation, recombination, horizontal (or lateral) genetic transfer, rate heterogeneity and sequences of varied lengths, especially when similarity lies in the "twilight zone"</td>

<td>Less sensitive to stochastic sequence variation, recombination, horizontal (or lateral) genetic transfer, rate heterogeneity and sequences of varied lengths

</td></tr>

<tr>

<td>Best practice uses inference algorithms with complexity at least O(n2); less time-efficient</td>

<td>Inference algorithms typically O(n2) or less; more time-efficient

</td></tr>

<tr>

<td>Heuristic in nature; statistical significance of how alignment scores relate to homology is difficult to assess</td>

<td>Exact solutions; statistical significance of the sequence distances (and degree of similarity) can be readily assessed

</td></tr>

<tr>

<td>Relies on dynamic programming (computationally expensive) to find alignment that has optimal score.

</td>

<td>side-steps computational expensive dynamic programming by indexing word counts or positions in fractal space.<a href="#cite_note-66">[66]</a>

</td></tr></tbody></table>

<div class="mw-heading mw-heading2"><h2 id="Applications_of_alignment-free_methods">Applications of alignment-free methods</h2></div>

<ul><li>Genomic rearrangements<a href="#cite_note-rearrang-67">[67]</a><a href="#cite_note-smashpp-68">[68]</a></li>

<li>Molecular phylogenetics<a href="#cite_note-Ragan-9">[9]</a><a href="#cite_note-Bernard-14">[14]</a><a href="#cite_note-69">[69]</a></li>

<li>Metagenomics<a href="#cite_note-NGS-70">[70]</a><a href="#cite_note-Metagenomics2-71">[71]</a><a href="#cite_note-CLARK-S-72">[72]</a><a href="#cite_note-falcon-73">[73]</a><a href="#cite_note-kraken-74">[74]</a></li>

<li>Next generation sequence data analysis<a href="#cite_note-NGS-70">[70]</a><a href="#cite_note-Cov-30">[30]</a></li>

<li>Epigenomics<a href="#cite_note-75">[75]</a></li>

<li>Barcoding of species<a href="#cite_note-76">[76]</a></li>

<li>Population genetics<a href="#cite_note-Haubold-11">[11]</a></li>

<li>Horizontal gene transfer<a href="#cite_note-Domazet-Lošo_1466–72-8">[8]</a></li>

<li>Sero/genotyping of viruses<a href="#cite_note-RTD1-21">[21]</a><a href="#cite_note-RTD3-77">[77]</a><a href="#cite_note-COMET-78">[78]</a></li>

<li>Allergenicity prediction<a href="#cite_note-AllergenFP-79">[79]</a></li>

<li>SNP discovery<a href="#cite_note-SNP-80">[80]</a></li>

<li>Recombination detection<a href="#cite_note-rush-81">[81]</a></li>

<li>Viral Classification <a href="#cite_note-landscape-82">[82]</a></li>

<li>Archaea Taxonomic Identification<a href="#cite_note-83">[83]</a></li>

<li>Taxonomic Classification<a href="#cite_note-:1-84">[84]</a></li>

<li>Temporal Analysis<a href="#cite_note-:2-85">[85]</a></li>

<li>Low-complexity Regions Identification<a href="#cite_note-:0-86">[86]</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="List_of_web_servers/software_for_alignment-free_methods">List of web servers/software for alignment-free methods</h2></div>

<th>Description</th>

<th>Availability</th>

<th>Reference

</th></tr>

<tr>

<td>Protcomp</td>

<td>Most Expressed Features scoring approach</td>

<td><a rel="nofollow" class="external text" href="https://yadamp.unisa.it/PROTCOMP">PROTCOMP</a></td>

</td></tr>

<tr>

<td>kmacs</td>

<td>k-mismatch average common substring approach</td>

<td><a rel="nofollow" class="external text" href="http://kmacs.gobics.de/">kmacs</a></td>

</td></tr>

<tr>

<td>Spaced words</td>

<td>Spaced-word frequencies</td>

<td><a rel="nofollow" class="external text" href="http://spaced.gobics.de/">spaced-words</a></td>

</td></tr>

<tr>

<td>Co-phylog</td>

<td>assembly-free micro-alignment approach</td>

<td><a rel="nofollow" class="external text" href="https://github.com/yhg926/co-phylog">Co-phylog</a></td>

</td></tr>

<tr>

<td>Proteome-based spaced-word matches</td>

</td></tr>

<tr>

<td>Filtered Spaced-Word Matches</td>

</td></tr>

<tr>

<td>Feature frequency profile based phylogeny</td>

</td></tr>

<tr>

<td>CVTree</td>

<td>Composition vector based server for phylogeny</td>

<td><a rel="nofollow" class="external text" href="http://tlife.fudan.edu.cn/cvtree/">CVTree</a></td>

</td></tr>

<tr>

<td>RTD Phylogeny</td>

<td>Return time distribution based server for phylogeny</td>

<td><a rel="nofollow" class="external text" href="http://bioinfo.net.in/RTD/home.html">RTD Phylogeny</a></td>

</td></tr>

<tr>

<td>A multimethods web server for alignment-free genome phylogeny</td>

</td></tr>

<tr>

<td>Alignment-free detection of local similarity among viral and bacterial genomes</td>

</td></tr>

<tr>

<td>decaf+py</td>

<td>DistancE Calculation using Alignment-Free methods in PYthon</td>

<td><a rel="nofollow" class="external text" href="http://acb.qfab.org/acb/decaf+py/">decaf+py</a></td>

</td></tr>

<tr>

<td>Dengue Subtyper</td>

<td>Genotyping of Dengue viruses based on RTD</td>

<td>Dengue Subtyper</td>

</td></tr>

<tr>

<td>WNV Typer</td>

<td>Genotyping of West nile viruses based on RTD</td>

<td>WNV Typer</td>

</td></tr>

<tr>

<td>AllergenFP</td>

<td>Allergenicity prediction by descriptor fingerprints</td>

<td><a rel="nofollow" class="external text" href="http://ddg-pharmfac.net/AllergenFP/">AllergenFP</a></td>

</td></tr>

<tr>

<td>Alignment-Free SNP Discovery</td>

</td></tr>

<tr>

<td>d2Tools</td>

<td>Comparison of Metatranscriptomic Samples Based on k-Tuple Frequencies</td>

<td><a rel="nofollow" class="external text" href="https://code.google.com/p/d2-tools/">d2Tools</a></td>

</td></tr>

<tr>

<td>Recombination detection Using SHustrings</td>

</td></tr>

<tr>

<td>smash</td>

<td>Genomic rearrangements detection and visualisation</td>

<td><a rel="nofollow" class="external text" href="http://bioinformatics.ua.pt/software/smash/">smash</a></td>

</td></tr>

<tr>

<td>Smash++</td>

<td>Finding and visualizing genomic rearrangements</td>

<td><a rel="nofollow" class="external text" href="https://github.com/smortezah/smashpp">Smash++</a></td>

</td></tr>

<tr>

<td>GScompare</td>

<td>Oligonucleotide-based fast clustering of bacterial genomes</td>

<td><a rel="nofollow" class="external text" href="http://gscompare.ehu.eus/">GScompare</a></td>

<td>–

</td></tr>

<tr>

<td>COMET</td>

<td>Alignment-free subtyping of HIV-1, HIV-2 and HCV viral sequences</td>

<td><a rel="nofollow" class="external text" href="https://comet.lih.lu/">COMET</a></td>

</td></tr>

<tr>

<td>USM

</td>

<td>Fractal MapReduce decomposition of sequence alignment

</td>

<td><a rel="nofollow" class="external text" href="https://usm.github.io">usm.github.io</a>

</td>

</td></tr>

<tr>

<td>FALCON</td>

<td>Alignment-free method to infer metagenomic composition of ancient DNA</td>

<td><a rel="nofollow" class="external text" href="https://github.com/pratas/falcon">FALCON</a></td>

</td></tr>

<tr>

<td>Kraken</td>

<td>Taxonomic classification using exact k-mer matches</td>

<td><a rel="nofollow" class="external text" href="https://github.com/DerrickWood/kraken2">Kraken 2</a></td>

</td></tr>

<tr>

<td>EAGLE</td>

<td>An ultra-fast tool to find relative absent words in genomic data</td>

<td><a rel="nofollow" class="external text" href="https://github.com/cobilab/eagle">EAGLE2</a></td>

</td></tr>

<tr>

<td>Phylogenetic trees using reference-free k-mer based matching</td>

<td><a rel="nofollow" class="external text" href="https://www.qiagenbioinformatics.com/products/clc-microbial-genomics-module/">CLC Microbial Genome Module</a></td>

</td></tr>

<tr>

<td>xgTaxonomy

</td>

<td>A tool for metagenomic classification that uses data compression algorithms to classify genomic sequences.

</td>

<td><a rel="nofollow" class="external text" href="https://github.com/ieeta-pt/xgTaxonomy">xgTaxonomy</a>

</td>

</td></tr>

<tr>

<td>AlcoR

</td>

<td>An extremely efficient method for identifying and visualizing low-complexity regions in genomic and proteomic sequences.

</td>

<td><a rel="nofollow" class="external text" href="https://cobilab.github.io/alcor/">AlcoR</a>

</td>

</td></tr>

<tr>

<td>AltaiR

</td>

<td>A C toolkit for alignment-free and temporal analysis of multi-FASTA data.

</td>

<td><a rel="nofollow" class="external text" href="https://github.com/cobilab/altair">AltaiR</a>

</td>

</td></tr></tbody></table>

<ul><li><a href="Sequence_analysis" title="Sequence analysis">Sequence analysis</a></li>

<li><a href="Multiple_sequence_alignment" title="Multiple sequence alignment">Multiple sequence alignment</a></li>

<li><a href="Phylogenomics" title="Phylogenomics">Phylogenomics</a></li>

<li><a href="Bioinformatics" title="Bioinformatics">Bioinformatics</a></li>

<li><a href="Metagenomics" title="Metagenomics">Metagenomics</a></li>

<li><a href="Next-generation_sequencing" class="mw-redirect" title="Next-generation sequencing">Next-generation sequencing</a></li>

<li><a href="Population_genetics" title="Population genetics">Population genetics</a></li>

<li><a href="Recombination_detection_program" title="Recombination detection program">Recombination detection program</a></li>

<li><a href="Genome_skimming" title="Genome skimming">Genome skimming</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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