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{{Technical|date=September 2013}}{{Infobox algorithm|name= <p!-- Defaults to article name -->|class= [[File:Needleman-Wunsch pairwise sequence Sequence alignment.png]]|framedimage=|rightcaption=|Figure 1: Needlemandata=|time= <math>O(mn)</math>|best-time=|average-Wunsch pairwise sequence alignmenttime=|space= <math>O(mn)</pmath>}}

The '''Needleman–Wunsch algorithm''' is an [[algorithm]] used in [[bioinformatics]] to [[sequence alignment|align]] [[protein]] or [[nucleotide]] sequences. It was one of the first applications of [[dynamic programming]] to compare biological sequences. The algorithm was developed by Saul B. Needleman and Christian D. Wunsch and published in 1970.<pref name=Needleman>&lt;pre&gt;{{cite journal | journal=Journal of Molecular Biology | volume=48 | issue=3 | pages=443–53 | year=1970 |author1=Needleman, Saul B. |author2=Wunsch, Christian D. |last-author-amp=yes | title=A general method applicable to the search for similarities in the amino acid sequence of two proteins | url=http://linkinghub.elsevier.com/retrieve/pii/0022-2836(70)90057-4 | pmid=5420325 | doi = 10.1016/0022-2836(70)90057-4 }}<br /ref> The algorithm essentially divides a large problem (e.g. the full sequence) into a series of smaller problems and uses the solutions to the smaller problems to reconstruct a solution to the larger problem.<ref>Results{{cite web|title=bioinformatics |url=http://www.britannica.com/EBchecked/topic/1334661/bioinformatics/285871/Goals-of-bioinformatics#ref1115380|accessdate=10 September 2014}}</pref>It is also sometimes referred to as the [[optimal matching]] algorithm and the [[Sequence alignment#Global and local alignments|global alignment]] technique. The Needleman–Wunsch algorithm is still widely used for optimal global alignment, particularly when the quality of the global alignment is of the utmost importance.

<p>Sequences &nbsp; &nbsp;Best alignments<br />[[File:Needleman-Wunsch pairwise sequence alignment.png|framed|right|Figure 1: Needleman-------- &nbsp; &nbsp;----------------------<br />GCATGCU &nbsp; &nbsp; &nbsp;GCATG-CU &nbsp; &nbsp; &nbsp;GCA-TGCU &nbsp; &nbsp; &nbsp;GCAT-GCU<br />GATTACA &nbsp; &nbsp; &nbsp;G-ATTACA &nbsp; &nbsp; &nbsp;G-ATTACA &nbsp; &nbsp; &nbsp;G-ATTACA</p>Wunsch pairwise sequence alignment

<ppre>Interpretation of the initialization stepResults:</p>

<p>AlignmentInterpretation of the initialization step: &nbsp;+GCATGCU<br />&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; +GATTACA<br />Score: &nbsp; &nbsp; &nbsp;0 &nbsp;// Handle matches handle, doesn&#39;t win any score</p>

<p>Alignment: &nbsp;+GCATGCU<br />&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;One can interpret the leftmost column in the above figure like this (putting a "handle" before each sequence, annotated as +GATTACA<br />Scorehere): &nbsp; &nbsp; &nbsp;0 &nbsp;// 1 gap, &nbsp;score -1</p>

<p>Alignment: &nbsp; +GCATGCU<br />&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; +GATTACA<br />Score: &nbsp; &nbsp; &nbsp; 0 &nbsp; // 2 gapsHandle matches handle, doesn't win any score -2</p>

<p>Alignment: &nbsp; +GCATGCU<br />&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; +GATTACA<br />Score: &nbsp; &nbsp; &nbsp; 0 &nbsp; // 3 gaps1 gap, score -3</p>1

<p>Alignment: &nbsp; +GCATGCU<br />&nbsp; &nbsp; &nbsp; &nbsp; +GATTACA<br />Score: &nbsp; &nbsp; &nbsp; 0 &nbsp; // 4 2 gaps, score -4</p>2

<p>...<Alignment: +GCATGCU +GATTACAScore: 0 /p>/ 3 gaps, score -3

<p>The same thing can be done for the uppermost row.<br />Alignment: +GCATGCU +GATTACA&lt;/pre&gt;]]<br Score: 0 />__TOC__<br />{{4 gaps, score -}}</p>4

<p>==Introduction==<br />This algorithm can be used for any two [[String (computer science)|strings]]. This guide will use two small [[DNA sequences]] as examples as shown in the diagram:<br />&nbsp;GCATGCU<br />&nbsp;GATTACA</p>..

<p>===Constructing the grid===<br />First construct a grid such as one shown in Figure 1 above. Start the first string in the top of the third column and start the other string at The same thing can be done for the start of the third uppermost row. Fill out the rest of the column and row headers as in Figure 1. There should be no numbers in the grid yet.</ppre>]]__TOC__{{-}}

<p>{| class=&quot;wikitable&quot;<br />|-<br />!| &nbsp;|| &nbsp; || G || C || A || T || G || C || U<br />|-<br />! scope=&quot;row&quot; |<br />| &amp;nbsp; || &nbsp;|| &nbsp;|| &nbsp;|| &nbsp;|| &nbsp;|| &nbsp;||<br />|-<br />! scope=&quot;row&quot; | G<br />| &nbsp;|| &nbsp; || &nbsp; || &nbsp; || &nbsp; || &nbsp; || &nbsp; ||&nbsp;<br />|-<br />! scope=&quot;row&quot; | A<br />| &nbsp;|| &nbsp; || &nbsp; || &nbsp; || &nbsp; || &nbsp; || &nbsp; ||&nbsp;<br />|-<br />! scope=&quot;row&quot; | T<br />| &nbsp;|| &nbsp; || &nbsp; || &nbsp; || &nbsp; || &nbsp; || &nbsp; || &nbsp;<br />|-<br />! scope=&quot;row&quot; | T<br />| &nbsp;|| &nbsp; || &nbsp; || &nbsp; || &nbsp; || &nbsp; || &nbsp; ||&nbsp;<br />|-<br />! scopeIntroduction=&quot;row&quot; | A<br />| &nbsp;|| &nbsp; || &nbsp; || &nbsp; || &nbsp; || &nbsp; || &nbsp; ||&nbsp;<br />|-<br />! scope=&quot;row&quot; | C<br />| &nbsp;|| &nbsp; || &nbsp; || &nbsp; || &nbsp; || &nbsp; || &nbsp; ||&nbsp;<br />This algorithm can be used for any two [[String (computer science)|-<br />strings]]. This guide will use two small [[DNA sequences]] as examples as shown in the diagram:! scope=&quot;row&quot; | A<br /> GCATGCU| &nbsp;|| &nbsp; || &nbsp; || &nbsp; || &nbsp; || &nbsp; || &nbsp; ||&nbsp;<br />|}</p> GATTACA

<p>===Choosing a scoring systemConstructing the grid===<br />Next, decide how to score each individual pair First construct a grid such as one shown in Figure 1 above. Start the first string in the top of letters. Using the example above, one possible alignment candidate might be:<br />{{DNA sequence|<br />&nbsp;12345678<br />&nbsp;GCATG-CU<br />&nbsp;G-ATTACA<br />}}<br />The letters may match, mismatch, or be matched to a gap (a deletion or insertion ([[indel]]):<br />* Match: The two letters third column and start the other string at the current index are start of the samethird row.<br />* Mismatch: The two letters at Fill out the rest of the current index are differentcolumn and row headers as in Figure 1. &nbsp;<br />* Indel (INsertion or DELetion): The best alignment involves one letter aligning to a gap There should be no numbers in the other stringgrid yet.</p>

<p>Each of these scenarios is assigned a score and the sum of the score of each pairing is the score of the whole alignment candidate. Different systems exist for assigning scores; some have been outlined in the [[#Scoring systems{| class="wikitable"|-!| || || G || C || A || T || G || C || U|-! scope="row" ||Scoring systems]] section below. For now, the system used by Needleman and Wunsch&ltnbsp;ref name|| || || || || || |||-! scope="row" | G| || || || || || || || |-! scope="row" | A| || || || || || || || |-! scope="row" | T| || || || || || || || |-! scope=&quot;Needleman&quot;/&gt; will be used:</p>"row" | T| || || || || || || || |-! scope="row" | A| || || || || || || || |-! scope="row" | C| || || || || || || || |-! scope="row" | A| || || || || || || || |}

<p>===Choosing a scoring system===Next, decide how to score each individual pair of letters. Using the example above, one possible alignment candidate might be:{{DNA sequence| 12345678 GCATG-CU G-ATTACA}}The letters may match, mismatch, or be matched to a gap (a deletion or insertion ([[indel]]):* Match: +1<br />The two letters at the current index are the same.* Mismatch : The two letters at the current index are different. * Indel (INsertion or IndelDELetion): &minus;1</p>The best alignment involves one letter aligning to a gap in the other string.

<p>For Each of these scenarios is assigned a score and the sum of the Example above, score of each pairing is the score of the whole alignment would be 0:<br />{{DNA sequencecandidate. Different systems exist for assigning scores; some have been outlined in the [[#Scoring systems|Scoring systems]] section below. For now, the system used by Needleman and Wunsch<br />&nbsp;GCATG-CU<br />&nbsp;G-ATTACA}}<br />&nbsp;+-++--+- -&gt; -1*4 + 1*4 ref name= 0<"Needleman"/p>will be used:

<p>===Filling in the table===<br />Start with a zero in the second row, second column. Move through the cells row by row, calculating the score for each cell. The score is calculated by comparing the scores of the cells neighboring to the left, top or top-left (diagonal) of the cell and adding the appropriate score for match, mismatch or indel. Calculate the candidate scores for each of the three possibilities* Match:<br />+1* The path from the top Mismatch or left cell represents an indel pairing, so take the score of the left and the top cell, and add the score for indel to each of them.<br />* The diagonal path represents a match/mismatch, so take the score of the top-left diagonal cell and add the score for match if the corresponding bases in the row and column are matching or the score for mismatch if they do not.<br />The resulting score for the cell is the highest of the three candidate scores.</p>Indel: −1

<p>Given there is no &#39;top&#39; or &#39;top-left&#39; cells for For the second row only the existing cell to the left can be used to calculate Example above, the score of each cell. Hence -1 is added for each shift to the right as this represents an indelible from the previous score. This results in the first row being alignment would be 0, :{{DNA sequence| GCATG-1, CU G-2, ATTACA}} +-3, ++-4, -5, +-6, -7. The same applies to the second column as only the existing score above each cell can be used. Thus the resulting table is:</p>-1*4 + 1*4 = 0

<p>{| class=&quot;wikitable&quot;<br />|-<br />!| &nbsp;|| &nbsp; || G || C || A || T || G || C || U<br />|-<br />! scope=&quot;row&quot; |<br />| 0 || -1 || -2 || -3 || -4 || -5 || -6 || -7<br />|-<br />! scope=&quot;row&quot; | G<br />| -1 || &nbsp; || &nbsp; || &nbsp; || &nbsp; || &nbsp; || &nbsp; ||&nbsp;<br />|-<br />! scopeFilling in the table===&quot;row&quot; | A<br />| -2 || &nbsp; || &nbsp; || &nbsp; || &nbsp; || &nbsp; || &nbsp; ||&nbsp;<br />|-<br />! scope=&quot;Start with a zero in the second row&quot; | T<br />| -3 || &nbsp; || &nbsp; || &nbsp; || &nbsp; || &nbsp; || &nbsp; || &nbsp;<br />|-<br />! scope=&quot;, second column. Move through the cells row&quot; | T<br />| -4 || &nbsp; || &nbsp; || &nbsp; || &nbsp; || &nbsp; || &nbsp; ||&nbsp;<br />|-<br />! scope=&quot;by row&quot; | A<br />| -5 || &nbsp; || &nbsp; || &nbsp; || &nbsp; || &nbsp; || &nbsp; ||&nbsp;<br />|, calculating the score for each cell. The score is calculated by comparing the scores of the cells neighboring to the left, top or top-<br />left (diagonal) of the cell and adding the appropriate score for match, mismatch or indel. Calculate the candidate scores for each of the three possibilities:! scope=&quot;row&quot; | C<br />* The path from the top or left cell represents an indel pairing, so take the score of the left and the top cell, and add the score for indel to each of them.| -6 || &nbsp; || &nbsp; || &nbsp; || &nbsp; || &nbsp; || &nbsp; ||&nbsp;<br * The diagonal path represents a match/>|mismatch, so take the score of the top-<br />! scope=&quot;left diagonal cell and add the score for match if the corresponding bases in the row&quot; | A<br />and column are matching or the score for mismatch if they do not.| -7 || &nbsp; || &nbsp; || &nbsp; || &nbsp; || &nbsp; || &nbsp; ||&nbsp;<br />|}</p>The resulting score for the cell is the highest of the three candidate scores.

<p>The first case with Given there is no 'top' or 'top-left' cells for the second row only the existing scores in all 3 directions cell to the left can be used to calculate the score of each cell. Hence -1 is added for each shift to the intersection of our first letters (in right as this case G and G)represents an indelible from the previous score. The surrounding cells are below:<br />{| class=&quot;wikitable&quot;<br />|-<br />!| &nbsp;|| &nbsp; || G<br />|-<br />! scope=&quot;This results in the first row&quot; |<br />| being 0 || , -1<br />|, -2, -3, -4, -5, -<br />! scope=&quot;row&quot; | G<br />| 6, -1 || &#39;&#39;&#39;X&#39;&#39;&#39;<br />|}</p>7. The same applies to the second column as only the existing score above each cell can be used. Thus the resulting table is:

<p>* The diagonal top-left neighbor has score 0. The pairing first case with existing scores in all 3 directions is the intersection of our first letters (in this case G and G is a match, so add the score for match). The surrounding cells are below: 0+1 {| class= 1<br />"wikitable"|-!| || || G* The top neighbor has score |-1 and moving from there represents an indel, so add the score for indel: (! scope="row" || 0 || -1) + (|-1) ! scope= (-2)<br />"row" | G* The left neighbor also has score | -1, represents an indel and also produces (-2).</p>|| '''X'''|}

<p>The highest This cell has three possible candidate is 1 and is entered into the cellsums:</p>

<p>{| class=&quot;wikitable&quot;<br />|* The diagonal top-<br />!| &nbsp;|| &nbsp; || left neighbor has score 0. The pairing of G and G<br />is a match, so add the score for match: 0+1 = 1|* The top neighbor has score -<br />! scope=&quot;row&quot; |<br />| 0 || 1 and moving from there represents an indel, so add the score for indel: (-1<br />|) + (-<br />! scope1) =&quot;row&quot; | G<br />(-2)| * The left neighbor also has score -1 || &#39;&#39;&#39;1&#39;&#39;&#39;<br />|}</p>, represents an indel and also produces (-2).

<p>The cell which gave the highest candidate score must also be recorded. In the completed diagram in figure is 1 above, this and is represented as an arrow from entered into the cell in row and column 3 to the cell in row and column 2.</p>:

<p>{| class=&quot;wikitable&quot;<br />|-<br />!| &nbsp;|| || G || C<br />|-<br />! scope=&quot;row&quot; |<br />| 0 || -The cell which gave the highest candidate score must also be recorded. In the completed diagram in figure 1 || -2<br />|-<br />! scope=&quot;above, this is represented as an arrow from the cell in row&quot; | G<br />| -1 || 1 || &#39;&#39;&#39;X&#39;&#39;&#39;<br />|-<br />! scope=&quot;and column 3 to the cell in row&quot; | A<br />| -and column 2 || &#39;&#39;&#39;Y&#39;&#39;&#39; ||&nbsp;<br />|}</p>.

<p>In the next example, the diagonal step for both Xand Y represents a mismatch:<br />* Top: (-2)+(-1) = (-3)<br />* Left: (+1)+(-1) = (0)<br />* Top-Left: (-1)+(-1) = (-2)</p>

<p>Y:<br />{| class="wikitable"|-!| || || G || C* Top: (1)+(|-1) ! scope= ("row" || 0)<br />* Left: (|| -1 || -2)+(|-1) ! scope= (-3)<br />"row" | G* Top-Left: (| -1)+(|| 1 || '''X'''|-1) ! scope= ("row" | A| -2)</p>|| '''Y''' || |}

<p>For both X and Y, the highest score is zero:</p>* Top: (-2)+(-1) = (-3)* Left: (+1)+(-1) = (0)* Top-Left: (-1)+(-1) = (-2)

<p>{| class=&quot;wikitable&quot;<br />Y:|* Top: (1)+(-<br />!| &nbsp;|| || G || C<br />1) = (0)|* Left: (-<br />! scope=&quot;row&quot; |<br />| 0 || 2)+(-1 || ) = (-2<br />3)|* Top-<br />! scope=&quot;row&quot; | G<br />| Left: (-1 || )+(-1 || &#39;&#39;&#39;0&#39;&#39;&#39;<br />|-<br />! scope) =&quot;row&quot; | A<br />| (-2 || &#39;&#39;&#39;0&#39;&#39;&#39; ||&nbsp;<br />|}</p>)

<p>The For both X and Y, the highest candidate score may be reached by two or all neighboring cellsis zero:</p>

<p>{| class=&quot;"wikitable&quot;<br />"|-<br />!| &nbsp; || T || G<br />|| C|-<br />! scope=&quot;"row&quot; " | T<br />| 0 || -1 || -2|-! scope="row" | G| -1 || 1<br />|| '''0'''|-<br />! scope=&quot;"row&quot; " | A<br />| -2 || '''0 ''' || &#39;&#39;&#39;X&#39;&#39;&#39;<br />|}</p>

<p>* TopThe highest candidate score may be reached by two or all neighboring cells: (1)+(-1) = (0)<br />* Top-Left: (1)+(-1) = (0)<br />* Left: (0)+(-1) = (-1)</p>

<p>In this case, all directions reaching the highest candidate score must be noted as possible origin cells in the finished diagram in figure {| class="wikitable"|-!| || T || G|-! scope="row" | T| 1 || 1, e.g. in the cell in |-! scope="row and column 7.</p>" | A| 0 || '''X'''|}

<p>===Tracing arrows back to origin===<br />Mark a path from the cell on the bottom right back to the cell on the top left by following the direction of the arrows. From In this pathcase, all directions reaching the sequence is constructed by these rules:<br />* A diagonal arrow represents a match or mismatch, so the letters of the column and the letter of the row of the highest candidate score must be noted as possible origin cell will align.<br />* A horizontal or vertical arrow represents an indel. Horizontal arrows will align a gap (&quot;-&quot;) to the letter of the row (cells in the &quot;side&quot; sequence)finished diagram in figure 1, vertical arrows will align a gap to the letter of the column (the &quot;top&quot; sequence)e.<br />* If there are multiple arrows to choose from, they represent a branching of the alignmentsg. If two or more branches all belong to paths from the bottom left to in the top right cell, they are equally viable alignments. In this case, note the paths as separate alignment candidatesin row and column 7.</p>

<p>Following these rulesFilling in the table in this manner gives the scores or all possible alignment candidates, the steps score in the cell on the bottom right represents the alignment score for one possible the best alignment candidate in figure 1 are:</p>.

<p>&nbsp;U &rarr; CU &rarr; GCU &rarr; -GCU &rarr; T-GCU &rarr; AT-GCU &rarr; CAT-GCU &rarr; &#39;&#39;&#39;GCAT-GCU&#39;&#39;&#39;<br />===Tracing arrows back to origin===Mark a path from the cell on the bottom right back to the cell on the top left by following the direction of the arrows. From this path, the sequence is constructed by these rules:* A diagonal arrow represents a match or mismatch, so the letters of the column and the letter of the row of the origin cell will align.&nbsp;* A &rarr; CA &rarr; ACA &rarr; TACA &rarr; TTACA &rarr; ATTACA &rarr; horizontal or vertical arrow represents an indel. Horizontal arrows will align a gap ("-ATTACA &rarr; &#39;&#39;&#39;G-ATTACA&#39;&#39;&#39;<br />&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &darr;<br />&nbsp; &nbsp; &nbsp;") to the letter of the row (the "side" sequence), vertical arrows will align a gap to the letter of the column (branchthe "top" sequence) &rarr; TGCU &rarr; ...<br />&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &rarr; TACA &rarr; * If there are multiple arrows to choose from, they represent a branching of the alignments.If two or more branches all belong to paths from the bottom left to the top right cell, they are equally viable alignments.In this case, note the paths as separate alignment candidates.</p>

<p>===Basic scoring schemes===<br /> U → CU → GCU → -GCU → T-GCU → AT-GCU → CAT-GCU → '''GCAT-GCU'''The simplest scoring schemes simply give a value for each match, mismatch and indel. The step A → CA → ACA → TACA → TTACA → ATTACA → -byATTACA → '''G-step guide above uses match = 1, mismatch &nbsp;= &minus;1, indel = &minus;1ATTACA''' ↓ (branch) → TGCU → . Thus the lower the alignment score the larger the [[Levenshtein distance|edit distance]], for this scoring system one wants a high score. Another scoring system might be:<br />* Match = 0<br />* Indel = 1<br />* Mismatch = 1<br />For this system the alignment score will represent the edit distance between the two strings.<br />Different scoring systems can be devised for different situations, for example if gaps are considered very bad for your alignment you may use a scoring system that penalises gaps heavily, such as:<br />* Match = 0<br />* Mismatch = 1<br />* Indel = 10<br />[https://blievrouw → TACA → .github.io/needleman-wunsch/ Try it out].</p>

<p>===Similarity matrix=Scoring systems==<br />More complicated scoring systems attribute values not only for the type of alteration, but also for the letters that are involved. For example, a match between A and A may be given 1, but a match between T and T may be given 4. Here (assuming the first scoring system) more importance is given to the Ts matching than the As, i.e. the Ts matching is &nbsp;assumed to be more significant to the alignment. This weighting based on letters also applies to mismatches.</p>

<p>In order to represent all the possible combinations of letters ===Basic scoring schemes===The simplest scoring schemes simply give a value for each match, mismatch and their resulting scores &nbsp;a similarity matrix is usedindel. The similarity matrix step-by-step guide above uses match = 1, mismatch = −1, indel = −1. Thus the lower the alignment score the larger the [[Levenshtein distance|edit distance]], for this scoring system one wants a high score. Another scoring system might be:* Match = 0* Indel = 1* Mismatch = 1For this system the alignment score will represent the edit distance between the most basic two strings.Different scoring systems can be devised for different situations, for example if gaps are considered very bad for your alignment you may use a scoring system is represented that penalises gaps heavily, such as:<* Match = 0* Mismatch = 1* Indel = 10[https://blievrouw.github.io/needleman-wunsch/p>Try it out].

<p>{| style=&quot;font-family: monospace; font-size: 150%;&quot; cellpadding=&quot;2&quot; class=&quot;wikitable&quot;<br />! scopeSimilarity matrix=&quot;col&quot; |<br />! scope=&quot;col&quot; | A<br />! scope=&quot;col&quot; | G<br />! scope=&quot;col&quot; | C<br />! scope=&quot;col&quot; | T<br />|- style=&quot;text-align: right;&quot;<br />! scope=&quot;row&quot; | A<br />| 1 || &nbsp;-1|| -1 || -1<br />|- style=&quot;text-align: right;&quot;<br />! scope=&quot;row&quot; | G<br />| -1 || 1 || -1 || -1<br />|- style=&quot;text-align: right;&quot;<br />! scope=&quot;row&quot; | C<br />| -1 || -1 || 1 || -1<br />|- style=&quot;text-align: right;&quot;<br />! scope=&quot;row&quot; | T<br />| -1 || -1 || -1 || &nbsp;1<br />|}<br />Each score represents a switch from one More complicated scoring systems attribute values not only for the type of alteration, but also for the letters the cell matches to the otherthat are involved. Hence this represents all possible matches For example, a match between A and deletions A may be given 1, but a match between T and T may be given 4. Here (for an alphabet of ACGTassuming the first scoring system). Note all more importance is given to the matches go along Ts matching than the diagonalAs, also not all i.e. the table needs Ts matching is assumed to be filled, only this triangle because more significant to the scores are reciprocalalignment.= (Score for A &rarr; C = Score for C &rarr; A)This weighting based on letters also applies to mismatches. If implementing the T-T = 4 rule from above the following similarity matrix is produced:</p>

<p>{| style=&quot;font-familyIn order to represent all the possible combinations of letters and their resulting scores a similarity matrix is used. The similarity matrix for the most basic system is represented as: monospace; font-size: 150%;&quot; cellpadding=&quot;2&quot; class=&quot;wikitable&quot;<br />! scope=&quot;col&quot; |<br />! scope=&quot;col&quot; | A<br />! scope=&quot;col&quot; | G<br />! scope=&quot;col&quot; | C<br />! scope=&quot;col&quot; | T<br />|- style=&quot;text-align: right;&quot;<br />! scope=&quot;row&quot; | A<br />| 1 || -1 || -1 || -1<br />|- style=&quot;text-align: right;&quot;<br />! scope=&quot;row&quot; | G<br />| -1 || 1 || -1 || -1<br />|- style=&quot;text-align: right;&quot;<br />! scope=&quot;row&quot; | C<br />| -1 || -1 || 1 || -1<br />|- style=&quot;text-align: right;&quot;<br />! scope=&quot;row&quot; | T<br />| -1 || -1 || -1 || &nbsp;4<br />|}</p>

<p>Different scoring matrices have been statistically constructed which give weight to different actions appropriate to {| style="font-family: monospace; font-size: 150%;" cellpadding="2" class="wikitable"! scope="col" |! scope="col" | A! scope="col" | G! scope="col" | C! scope="col" | T|- style="text-align: right;"! scope="row" | A| 1 || -1|| -1 || -1|- style="text-align: right;"! scope="row" | G| -1 || 1 || -1 || -1|- style="text-align: right;"! scope="row" | C| -1 || -1 || 1 || -1|- style="text-align: right;"! scope="row" | T| -1 || -1 || -1 || 1|}Each score represents a particular scenario. Having weighted scoring matrices is particularly important in protein sequence alignment due switch from one of the letters the cell matches to the varying frequency other. Hence this represents all possible matches and deletions (for an alphabet of ACGT). Note all the matches go along the diagonal, also not all the table needs to be filled, only this triangle because the different amino acidsscores are reciprocal. There are two broad families of scoring matrices, each with further alterations = (Score for A → C = Score for specific scenariosC → A). If implementing the T-T = 4 rule from above the following similarity matrix is produced:<br />* [[Point accepted mutation|PAM]]<br />* [[BLOSUM]]</p>

<p>{| style="font-family: monospace; font-size: 150%;" cellpadding="2" class="wikitable"! scope="col" |! scope=Gap penalty"col" | A! scope="col" | G! scope="col" | C! scope=<br />"col" | TWhen aligning sequences there are often gaps (i.e. indels), sometimes large ones. Biologically, a large gap is more likely to occur as one large deletion as opposed to multiple single deletions. Hence two small indels should have a worse score than one large one. The simple and common way to do this is via a large gap|- style="text-start score for a new indel and a smaller gapalign: right;"! scope="row" | A| 1 || -extension score for every letter which extends the indel. For example, new1 || -indel may cost 1 || -5 and extend1|-indel may cost style="text-1. In this way an alignment such asalign:<br />&nbspright;GAAAAAAT<br />"&nbsp;! scope="row" | G| -1 || 1 || -A1 || -1|-Astyle="text-T<br />which has multiple equal alignments, some with multiple small alignments will now align as:<br />right;"&nbsp;GAAAAAAT<br />! scope="row" | C&nbsp;GAA| -1 || -1 || 1 || -1|-style="text-align: right;"! scope="row" | T<br />or any alignment with a | -1 || -1 || -1 || 4 long gap in preference over multiple small gaps.</p>|}

<p>==Advanced presentation Different scoring matrices have been statistically constructed which give weight to different actions appropriate to a particular scenario. Having weighted scoring matrices is particularly important in protein sequence alignment due to the varying frequency of the different amino acids. There are two broad families of algorithm==<br />scoring matrices, each with further alterations for specific scenarios:Scores for aligned characters are specified by a * [[similarity matrixPoint accepted mutation|PAM]]. Here, {{math|&#39;&#39;S&#39;&#39;(&#39;&#39;a&#39;&#39;, &#39;&#39;b&#39;&#39;)}} is the similarity of characters &#39;&#39;a&#39;&#39; and &#39;&#39;b&#39;&#39;. It uses a linear * [[gap penaltyBLOSUM]], here called {{mvar|d}}.</p>

<p>For example, if the similarity matrix was<br />{| style=&quot;font-family: monospace; font-size: 150%;&quot; cellpadding=&quot;2&quot; class=&quot;wikitable&quot;<br />! scopeGap penalty=&quot;col&quot; |<br />! scope=&quot;col&quot; | A<br />! scope=&quot;col&quot; | G<br />! scope=&quot;col&quot; | C<br />! scope=&quot;col&quot; | T<br />|When aligning sequences there are often gaps (i.e. indels), sometimes large ones. Biologically, a large gap is more likely to occur as one large deletion as opposed to multiple single deletions. Hence two small indels should have a worse score than one large one. The simple and common way to do this is via a large gap- style=&quot;textstart score for a new indel and a smaller gap-align: right;&quot;<br />! scope=&quot;row&quot; | A<br />| 10 || extension score for every letter which extends the indel. For example, new-1 || indel may cost -3 || -4<br />|5 and extend- style=&quot;textindel may cost -align1. In this way an alignment such as: right;&quot;<br />! scope=&quot;row&quot; | GAAAAAAT G<br />| -1 || 7 || -5 || A-A-3<br />T|- style=&quot;text-which has multiple equal alignments, some with multiple small alignments will now alignas: right;&quot;<br />! scope=&quot;row&quot; | C<br /> GAAAAAAT| GAA-3 || -5 || 9 || 0<br />|- style=&quot;text-align: right;&quot;<br />! scope=&quot;row&quot; | T<br />| -or any alignment with a 4 || -3 || 0 || 8<br />|}</p>long gap in preference over multiple small gaps.

<p>then the alignment:<br />&nbsp;AGACTAGTTAC<br />&nbsp;CGA---GACGT<br />==Advanced presentation of algorithm==with Scores for aligned characters are specified by a gap penalty of -5[[similarity matrix]]. Here, would have the following score:<br />:{{math|&#39;&#39;''S&#39;&#39;''(A''a'',C''b'') + &#39;&#39;S&#39;&#39;(G}} is the similarity of characters ''a'' and ''b''. It uses a linear [[gap penalty]],G) + &#39;&#39;S&#39;&#39;(A,A) + (3 &times; &#39;&#39;here called {{mvar|d&#39;&#39;) + &#39;&#39;S&#39;&#39;(G,G) + &#39;&#39;S&#39;&#39;(T,A) + &#39;&#39;S&#39;&#39;(T,C) + &#39;&#39;S&#39;&#39;(A,G) + &#39;&#39;S&#39;&#39;(C,T)}}<br />:= -3 + 7 + 10 - (3 &times; 5) + 7 + (-4) + 0 + (-1) + 0 = 1</p>.

<p>To find For example, if the alignment with the highest score, a twosimilarity matrix was{| style="font-family: monospace; font-dimensional [[Array data structuresize: 150%;" cellpadding="2" class="wikitable"! scope="col" |array]] (or [[Matrix (mathematics)! scope="col" |matrix]]) &#39A! scope="col" | G! scope="col" | C! scope="col" | T|- style="text-align: right;&#39"! scope="row" | A| 10 || -1 || -3 || -4|- style="text-align: right;F&#39"! scope="row" | G| -1 || 7 || -5 || -3|- style="text-align: right;&#39; is allocated. The entry in "! scope="row &#39;&#39;i&#39;&#39; and column &#39" | C| -3 || -5 || 9 || 0|- style="text-align: right;&#39;j&#39;&#39; is denoted here by<br />"&lt;math&gt;F_{ij}&lt;/math&gt;. There is one ! scope="row for each character in sequence &#39;&#39;A&#39;&#39;, and one column for each character in sequence &#39;&#39;B&#39;&#39;. Thus, if aligning sequences of sizes &#39;&#39;n&#39;&#39; and &#39;&#39;m&#39;&#39;, the amount of memory used is in &lt;math&gt;O(nm)&lt;/math&gt;. [[Hirschberg&#39;s algorithm]] only holds a subset of the array in memory and uses &lt;math&gt;\Theta(\min \{n,m\" | T| -4 || -3 || 0 || 8|})&lt;/math&gt; space, but is otherwise similar to Needleman-Wunsch (and still requires &lt;math&gt;O(nm)&lt;/math&gt; time).</p>

<p>As the algorithm progresses, the &lt;math&gt;F_{ij}&lt;/math&gt; will be assigned to be the optimal score for then the alignment : AGACTAGTTAC CGA---GACGTwith a gap penalty of the first &lt;math&gt;i=0-5,\dotsc,n&lt;/math&gt; characters in &#39;&#39;A&#39;&#39; and would have the first &lt;math&gt;j=0,\dotsc,m&lt;/math&gt; characters in &#39;&#39;B&#39;&#39;. The [[Bellman equation#Bellman&#39;s Principle of Optimality|principle of optimality]] is then applied as followsfollowing score:<br />* Basis:<br />:&lt;math&gt;F_{0j} = d*j&lt;/math&gt;<br />:&lt;math&gt;F_{i0} = d*i&lt;/math&gt;<br />* Recursion|''S''(A,C) + ''S''(G, based on the principle of optimality:<br />:&lt;math&gt;F_{ij} = \maxG) + ''S''(F_{i-1A,j-1} A) + (3 × ''d'') + ''S''(A_{i}G, B_{j}G)+ ''S''(T, \; F_{iA) + ''S''(T,j-1} C) + d''S''(A, \; F_{i-1G) + ''S''(C,jT)} }:= -3 + 7 + 10 - (3 × 5) + d7 + (-4) + 0 + (-1)&lt;/math&gt;</p>+ 0 = 1

<p>The pseudo-code for To find the algorithm to compute alignment with the F matrix therefore looks like this:<br />&nbsp;d &larr; MismatchScore<br />&nbsp;&#39;&#39;&#39;for&#39;&#39;&#39; i=0 &#39;&#39;&#39;to&#39;&#39;&#39; &#39;&#39;&#39;length&#39;&#39;&#39;(A)<br />&nbsp; &nbsp;F(ihighest score,0) &larr; d*i<br />&nbsp;&#39;&#39;&#39;for&#39;&#39;&#39; j=0 &#39;&#39;&#39;to&#39;&#39;&#39; &#39;&#39;&#39;length&#39;&#39;&#39;(B)<br />&nbsp; &nbsp;F(0,j) &larr; d*j<br />&nbsp;&#39;&#39;&#39;for&#39;&#39;&#39; i=1 &#39;&#39;&#39;to&#39;&#39;&#39; &#39;&#39;&#39;length&#39;&#39;&#39;(A)<br />&nbsp; &nbsp;&#39;&#39;&#39;for&#39;&#39;&#39; j=1 &#39;&#39;&#39;to&#39;&#39;&#39; &#39;&#39;&#39;length&#39;&#39;&#39;(B)<br />&nbsp; &nbsp;{<br />&nbsp; &nbsp; &nbsp;Match &larr; F(ia two-1,j-1) + Sdimensional [[Array data structure|array]] (A&lt;sub&gt;i&lt;/sub&gt;, B&lt;sub&gt;j&lt;/sub&gt;)<br />&nbsp; &nbsp; &nbsp;Delete &larr; For [[Matrix (i-1, jmathematics) + d<br />&nbsp; &nbsp; &nbsp;Insert &larr; F(i, j-1|matrix]]) + d<br />&nbsp; &nbsp; &nbsp;''F('' is allocated. The entry in row ''i,'' and column ''j) &larr; &#39;&#39;&#39;max&#39;&#39;&#39;(Match, Insert, Delete)<br />'' is denoted here by&nbsp; &nbsp;}<br /math>Once the &#39;&#39;F&#39;&#39; matrix is computed, the entry &lt;math&gt;F_{nmij}&lt;</math&gt; gives the maximum score among all possible alignments>. To compute an alignment that actually gives this score, you start from the bottom right cellThere is one row for each character in sequence ''A'', and compare the value with the three possible sources (Match, Insert, and Delete above) to see which it came fromone column for each character in sequence ''B''. If Match, then &lt;math&gt;A_i&lt;/math&gt; and &lt;math&gt;B_j&lt;/math&gt; are alignedThus, if Delete, then &lt;math&gt;A_i&lt;/math&gt; is aligned with a gap, aligning sequences of sizes ''n'' and if Insert''m'', then &lt;math&gt;B_j&lt;/math&gt; the amount of memory used is aligned with a gap. (In general, more than one choice may have the same value, leading to alternative optimal alignments.)in <br />&nbsp;AlignmentA &larr; &quot;&quot;<br />&nbsp;AlignmentB &larr; &quot;&quot;<br /math>&nbsp;i &larr; &#39;&#39;&#39;length&#39;&#39;&#39;O(Anm)<br /math>&nbsp;j &larr; &#39;&#39;&#39;length&#39;&#39;&#39;(B). [[Hirschberg's algorithm]] only holds a subset of the array in memory and uses <br /math>&nbsp;&#39;&#39;&#39;while&#39;&#39;&#39; \Theta(i &gt; 0 &#39;&#39;&#39;or&#39;&#39;&#39; j &gt; 0\min \{n,m\})<br /math>&nbsp;{<br />&nbsp; &nbsp;&#39;&#39;&#39;if&#39;&#39;&#39; (i &gt; 0 &#39;&#39;&#39;and&#39;&#39;&#39; j &gt; 0 &#39;&#39;&#39;and&#39;&#39;&#39; F(ispace,j) == F(i-1,jbut is otherwise similar to Needleman-1) + SWunsch (A&lt;sub&gt;i&lt;/sub&gt;, B&lt;sub&gt;j&lt;/sub&gt;))and still requires <br /math>&nbsp; &nbsp;{<br />&nbsp; &nbsp; &nbsp;AlignmentA &larr; A&lt;sub&gt;i&lt;/sub&gt; + AlignmentA<br />&nbsp; &nbsp; &nbsp;AlignmentB &larr; B&lt;sub&gt;j&lt;/sub&gt; + AlignmentB<br />&nbsp; &nbsp; &nbsp;i &larr; i - 1<br />&nbsp; &nbsp; &nbsp;j &larr; j - 1<br />&nbsp; &nbsp;}<br />&nbsp; &nbsp;&#39;&#39;&#39;else&#39;&#39;&#39; &#39;&#39;&#39;if&#39;&#39;&#39; O(i &gt; 0 &#39;&#39;&#39;and&#39;&#39;&#39; F(i,j) == F(i-1,j) + dnm)<br /math>&nbsp; &nbsp;{<br />&nbsp; &nbsp; &nbsp;AlignmentA &larr; A&lt;sub&gt;i&lt;/sub&gt; + AlignmentA<br />&nbsp; &nbsp; &nbsp;AlignmentB &larr; &quot;-&quot; + AlignmentB<br />&nbsp; &nbsp; &nbsp;i &larr; i - 1<br />&nbsp; &nbsp;}<br />&nbsp; &nbsp;&#39;&#39;&#39;else&#39;&#39;&#39;<br />&nbsp; &nbsp;{<br />&nbsp; &nbsp; &nbsp;AlignmentA &larr; &quot;-&quot; + AlignmentA<br />&nbsp; &nbsp; &nbsp;AlignmentB &larr; B&lt;sub&gt;j&lt;/sub&gt; + AlignmentB<br />&nbsp; &nbsp; &nbsp;j &larr; j - 1<br />&nbsp; &nbsp;}<br />&nbsp;}</p>time).

As the algorithm progresses, the <pmath>== Complexity ==<br />Computing the score &lt;math&gt;F_{ij}&lt;</math&gt; > will be assigned to be the optimal score for each cell in the table is an &lt;math&gt;O(1)&lt;/math&gt; operation. Thus the time complexity alignment of the algorithm for two sequences of length &lt;first <math&gt;>i=0,\dotsc,n&lt;</math&gt; > characters in ''A'' and &lt;the first <math&gt;m&lt;/math&gt; is &lt;math&gt;O(mn)&lt;/math&gt;.&lt;ref name>j=&quot;:0&quot;&gt;{{Cite book|url=https://www.worldcat.org/oclc/429634761|title=Algorithms in bioinformatics : a practical introduction|last=Wing-Kin.|first=Sung,|date=2010|publisher=Chapman &amp; Hall/CRC Press|year=|isbn=9781420070330|location=Boca Raton|pages=34&ndash;35|oclc=429634761}}&lt;/ref&gt; It has been shown that it is possible to improve the running time to &lt;math&gt;O(mn/ \log n)&lt;dotsc,m</math&gt; using the > characters in ''B''. The [[Method Bellman equation#Bellman's Principle of Four RussiansOptimality|principle of optimality]].&lt;ref nameis then applied as follows:* Basis::<math>F_{0j} =&quot;d*j</math>:0&quot; /&gt;&lt;ref&gt;<math>F_{{Cite journal|lasti0} =Masek|first=William|last2=Paterson|first2=Michael|date=February 1980|title=A faster algorithm computing string edit distances|url=https:d*i<//www.sciencedirect.com/science/article/pii/0022000080900021|journal=Journal math>* Recursion, based on the principle of Computer and System Sciences|volumeoptimality::<math>F_{ij} =20|pages=18&ndash;31|doi=10.1016/0022\max(F_{i-1,j-00001} + S(80A_{i}, B_{j})90002, \; F_{i,j-1|via=Elsevier Science Direct}}&lt;/ref&gt; Since the algorithm fill an &lt;math&gt;n + d, \times m&lt;/math&gt; table the space complexity is &lt;math&gt;O(mnF_{i-1,j} + d)&lt;</math&gt;.&lt;ref name=&quot;:0&quot; /&gt;</p>

<p>The pseudo-code for the algorithm to compute the F matrix therefore looks like this: d ← MismatchScore '''for''' i=0 '''to''' '''length'''(A) F(i,0) ← d*i '''for''' j=Historical notes and algorithm development0 '''to''' '''length'''(B) F(0,j) ← d*j '''for''' i=1 '''to''' '''length'''(A) '''for''' j=1 '''to''' '''length'''(B) { Match ← F(i-1,j-1) + S(A<sub>i<br /sub>, B_j) Delete ← F(i-1, j) + d Insert ← F(i, j-1) + d F(i,j) ← '''max'''(Match, Insert, Delete) }The original purpose of Once the ''F'' matrix is computed, the entry <math>F_{nm}</math> gives the maximum score among all possible alignments. To compute an alignment that actually gives this score, you start from the bottom right cell, and compare the value with the algorithm described by Needleman three possible sources (Match, Insert, and Wunsch was Delete above) to find similarities in see which it came from. If Match, then <math>A_i</math> and <math>B_j</math> are aligned, if Delete, then <math>A_i</math> is aligned with a gap, and if Insert, then <math>B_j</math> is aligned with a gap. (In general, more than one choice may have the amino acid sequences of two proteinssame value, leading to alternative optimal alignments.&lt;ref name) AlignmentA ← "" AlignmentB ← "" i ← '''length'''(A) j ← '''length'''(B) '''while''' (i > 0 '''or''' j > 0) { '''if''' (i > 0 '''and''' j > 0 '''and''' F(i,j) ==Needleman F(i-1,j-1) + S(A<sub>i</&gt;sub>, B<sub>j</psub>)) { AlignmentA ← A_i + AlignmentA AlignmentB ← B_j + AlignmentB i ← i - 1 j ← j - 1 } '''else''' '''if''' (i > 0 '''and''' F(i,j) == F(i-1,j) + d) { AlignmentA ← A_i + AlignmentA AlignmentB ← "-" + AlignmentB i ← i - 1 } '''else''' { AlignmentA ← "-" + AlignmentA AlignmentB ← B_j + AlignmentB j ← j - 1 } }

== Complexity ==Computing the score <math>F_{ij}<p/math>Needleman and Wunsch describe their algorithm explicitly for each cell in the case when table is an <math>O(1)</math> operation. Thus the alignment is penalized solely by time complexity of the matches algorithm for two sequences of length <math>n</math> and mismatches<math>m</math> is <math>O(mn)</math>.<ref name=":0">{{Cite book|url=https://www.worldcat.org/oclc/429634761|title=Algorithms in bioinformatics : a practical introduction|last=Wing-Kin.|first=Sung, and gaps have no penalty (|date=2010|publisher=Chapman &#39;&#39;d&#39;&#39;Hall/CRC Press|year=|isbn=9781420070330|location=Boca Raton|pages=34–35|oclc=0429634761}}</ref> It has been shown that it is possible to improve the running time to <math>O(mn/ \log n). The original publication from 1970 suggests </math> using the [[recursionMethod of Four Russians]].<br ref name=":0" />&lt;math&gt;F_<ref>{{ij} Cite journal|last=Masek|first=William|last2=Paterson|first2=Michael|date=February 1980|title=A faster algorithm computing string edit distances|url=https://www.sciencedirect.com/science/article/pii/0022000080900021|journal=Journal of Computer and System Sciences|volume=20|pages=18–31|doi= \max_{h&lt;i,k&lt;j} \{ F_{h,j10.1016/0022-1}+S0000(A_{i},B_{j}80), F_{i90002-1,k|via=Elsevier Science Direct}}+S</ref> Since the algorithm fill an <math>n \times m</math> table the space complexity is <math>O(A_i,B_jmn) \}&lt;</math&gt;>.<ref name=":0" /p>

<p>==Historical notes and algorithm development==The corresponding dynamic programming original purpose of the algorithm takes cubic timedescribed by Needleman and Wunsch was to find similarities in the amino acid sequences of two proteins. The paper also points out that the recursion can accommodate arbitrary gap penalization formulas:<ref name=Needleman /p>

<p>&lt;blockquote&gt;<br />A penalty factor, a number subtracted Needleman and Wunsch describe their algorithm explicitly for every gap madethe case when the alignment is penalized solely by the matches and mismatches, may be assessed as a barrier to allowing the gapand gaps have no penalty (''d''=0). The penalty factor could be a function of original publication from 1970 suggests the size and/or direction of the gap. [page 444[recursion]]<br /math>&lt;/blockquote&gt;F_{ij} = \max_{h<i,k<j} \{ F_{h,j-1}+S(A_{i},B_{j}), F_{i-1,k}+S(A_i,B_j) \}</pmath>.

<p>A better The corresponding dynamic programming algorithm with quadratic running takes cubic time for . The paper also points out that the same problem (no recursion can accommodate arbitrary gap penalty) was first introduced&lt;ref name=Sankoff&gt;{{cite journal | doi=10.1073/pnas.69.1.4 | journal=Proceedings of the National Academy of Sciences of the USA | volume=69 | issue=1 | pages=4&ndash;6 | year=1972 &nbsp;| author=Sankoff D | title=Matching sequences under deletion/insertion constraints | url=http://www.pnas.org/content/69/1/4.abstract | pmid=4500555 | pmc=427531}}&lt;/ref&gt; by David Sankoff in 1972.<br />Similar quadratic-time algorithms were discovered independently<br />by T. K. Vintsyuk&lt;ref name=Vintsyuk&gt;{{cite journal | journal=Kibernetika | volume=4 | pages=81&ndash;88 | year=1968 &nbsp;| author=Vintsyuk TK | title=Speech discrimination by dynamic programming|url=httpspenalization formulas://link.springer.com/content/pdf/10.1007/BF01074755.pdf}}&lt;/ref&gt; in 1968 for speech processing<br />([[Dynamic time warping|&quot;time warping&quot;]]), and by Robert A. Wagner and [[Michael J. Fischer]]&lt;ref name=WagnerFischer&gt;{{cite journal |vauthors=Wagner RA, Fischer MJ | journal = [[Journal of the ACM]] | title=The string-to-string correction problem | volume=21 | issue=1 | year=1974 | pages=168&ndash;173 | doi=10.1145/321796.321811}}&lt;/ref&gt; in 1974 for string matching.</p>

<pblockquote>Needleman and Wunsch formulated their problem in terms of maximizing similarity. Another possibility is A penalty factor, a number subtracted for every gap made, may be assessed as a barrier to minimize allowing the [[Levenshtein distance|edit distance]] between sequences, introduced by [[Vladimir Levenshtein]]gap. Peter H. Sellers showed&lt;ref name=Sellers&gt;{{cite journal | doi=10.1137/0126070 | title=On The penalty factor could be a function of the theory size and computation /or direction of evolutionary distances | author=Sellers PH | journal = SIAM Journal on Applied Mathematics | volume = 26 | issue = 4 | pages = 787&ndash;793 | year = 1974}}&lt;/ref&gt; in 1974 that the two problems are equivalentgap.[page 444]</pblockquote>

A better dynamic programming algorithm with quadratic running time for the same problem (no gap penalty) was first introduced<ref name=Sankoff>{{cite journal | doi=10.1073/pnas.69.1.4 | journal=Proceedings of the National Academy of Sciences of the USA | volume=69 | issue=1 | pages=4–6 | year=1972 | author=Sankoff D | title=Matching sequences under deletion/insertion constraints | url=http://www.pnas.org/content/69/1/4.abstract | pmid=4500555 | pmc=427531}}<p/ref>The Needleman&ndash;Wunsch algorithm is still widely used by David Sankoff in 1972.Similar quadratic-time algorithms were discovered independentlyby T. K. Vintsyuk<ref name=Vintsyuk>{{cite journal | journal=Kibernetika | volume=4 | pages=81–88 | year=1968 | author=Vintsyuk TK | title=Speech discrimination by dynamic programming|url=https://link.springer.com/content/pdf/10.1007/BF01074755.pdf}}</ref> in 1968 for optimal speech processing([[Sequence alignment#Global and local alignmentsDynamic time warping|global alignment"time warping"]]), particularly when the quality of the global alignment is of the utmost importanceand by Robert A. However, the algorithm is expensive with respect to time Wagner and space[[Michael J. Fischer]]<ref name=WagnerFischer>{{cite journal |vauthors=Wagner RA, proportional to the product Fischer MJ | journal = [[Journal of the length of two sequences and hence is not suitable for long sequencesACM]] | title=The string-to-string correction problem | volume=21 | issue=1 | year=1974 | pages=168–173 | doi=10.1145/321796.321811}}</pref>in 1974 for string matching.

<p>Recent development has focused on improving the time Needleman and space cost Wunsch formulated their problem in terms of maximizing similarity. Another possibility is to minimize the algorithm while maintaining quality[[Levenshtein distance|edit distance]] between sequences, introduced by [[Vladimir Levenshtein]]. For example, in 2013, a Fast Optimal Global Sequence Alignment Algorithm (FOGSAA),&lt;Peter H. Sellers showed<ref&gt;name=Sellers>{{cite journal|last1doi=Chakraborty10.1137/0126070 |first1title=Angana|last2=BandyopadhyayOn the theory and computation of evolutionary distances |first2author=Sanghamitra|title=FOGSAA: Fast Optimal Global Sequence Alignment AlgorithmSellers PH |journal=Scientific Reports|date=29 April 2013SIAM Journal on Applied Mathematics |volume=326 |doiissue =10.1038/srep017464 |urlpages =http://www.nature.com/srep/2013/130429/srep01746/full/srep01746.html|accessdate=11 September 2014|pmid=23624407787–793 |pmcyear =36381641974}}&lt;</ref&gt; suggested alignment of nucleotide/protein sequences faster than other optimal global alignment methods, including the Needleman&ndash;Wunsch algorithm. The paper claims > in 1974 that when compared to the Needleman&ndash;Wunsch algorithm, FOGSAA achieves a time gain of 70&ndash;90% for highly similar nucleotide sequences (with &gt; 80% similarity), and 54&ndash;70% for sequences having 30&ndash;80% similaritytwo problems are equivalent.</p>

<p>==The Needleman–Wunsch algorithm is still widely used for optimal [[Sequence alignment#Global and local alignments|global alignment]], particularly when the quality of the global alignment tools using is of the utmost importance. However, the Needleman&ndash;Wunsch algorithm==<br />* [http://www.ebi.ac.uk/Tools/psa EMBOSS Needle is expensive with respect to time and EMBOSS Stretcher Global Alignment Tools]<br />* [https://blast.ncbi.nlm.nih.gov/Blast.cgi?PAGE_TYPE=BlastSearch&amp;PROG_DEF=blastn&amp;BLAST_PROG_DEF=blastn&amp;BLAST_SPEC=GlobalAln&amp;LINK_LOC=BlastHomeLink Needleman-Wunsch alignment for space, proportional to the product of the length of two nucleotide sequences]<br />* [http://www.mathworks.com.au/help/bioinfo/ref/nwalign.html MathWorks - Globally align two and hence is not suitable for long sequences using Needleman-Wunsch algorithm]<br />* [https://github.com/bitkeeper-scm/bitkeeper/blob/master/src/libdiff.c#L948 BitKeeper &ndash; Source Control Management Software]</p>

Recent development has focused on improving the time and space cost of the algorithm while maintaining quality. For example, in 2013, a Fast Optimal Global Sequence Alignment Algorithm (FOGSAA),<pref>{{cite journal|last1=Chakraborty|first1=Applications outside bioinformaticsAngana|last2=Bandyopadhyay|first2=Sanghamitra|title=FOGSAA: Fast Optimal Global Sequence Alignment Algorithm|journal=Scientific Reports|date=29 April 2013|volume=3|doi=10.1038/srep01746|url=http://www.nature.com/srep/2013/130429/srep01746/full/srep01746.html|accessdate=11 September 2014|pmid=23624407|pmc=3638164}}</pref> suggested alignment of nucleotide/protein sequences faster than other optimal global alignment methods, including the Needleman–Wunsch algorithm. The paper claims that when compared to the Needleman–Wunsch algorithm, FOGSAA achieves a time gain of 70–90% for highly similar nucleotide sequences (with >80% similarity), and 54–70% for sequences having 30–80% similarity.

<p>==Global alignment tools using the Needleman–Wunsch algorithm==* [[Computer stereo vision]]===<br http://>Stereo matching is an essential step in the process of 3D reconstruction from a pair of stereo imageswww.ebi.ac. When images have been rectified, an analogy can be drawn between aligning nucleotide uk/Tools/psa EMBOSS Needle and protein sequences and matching [[pixelsEMBOSS Stretcher Global Alignment Tools]] belonging to * [[scan lines]], since both tasks aim at establishing optimal correspondence between two strings of characters. The &lsquo;right&rsquo; image of a stereo pair can be seen as a mutated version of the &lsquo;left&rsquo; imagehttps: noise and individual camera sensitivity alter pixel values (i//blast.encbi. character substitutions); and different view angle reveals previously occluded data and introduces new occlusions (inlm.enih. insertion and deletion of characters)gov/Blast. As consequence, minor modifications of the Needlemancgi?PAGE_TYPE=BlastSearch&PROG_DEF=blastn&ndash;Wunsch algorithm make it suitable for stereo matching.BLAST_PROG_DEF=blastn&lt;refBLAST_SPEC=GlobalAln&gt;Dieny R., Thevenon J., MartinezLINK_LOC=BlastHomeLink Needleman-del-Rincon J., Nebel J.-C. (2011) &quot;Wunsch alignment for two nucleotide sequences]* [http://citeseerxwww.istmathworks.psucom.eduau/viewdochelp/summary?doi=10.1.1.220.9699 Bioinformatics inspired algorithm for stereo correspondence]&quot;. International Conference on Computer Vision Theory and Applications, March 5&ndash;7, Vilamoura - Algarve, Portugal.&lt;bioinfo/ref&gt; Although performances in terms of accuracy are not state/nwalign.html MathWorks -ofGlobally align two sequences using Needleman-the-art, the relative simplicity of the Wunsch algorithm allows its implementation on [[embedded systems]].&lt;ref&gt;Madeo S., Pelliccia R., Salvadori C., Martinez-del-Rincon J., Nebel J.-C. (2014) &quot;* [https://puregithub.qub.ac.ukcom/bitkeeper-scm/bitkeeper/portalblob/filesmaster/12586298src/stereovision_endlibdiff.pdf An optimized stereo vision implementation for embedded systems: application to RGB and Infra-Red imagesc#L948 BitKeeper – Source Control Management Software]&quot;. Journal of Real-Time Image Processing.&lt;/ref&gt;</p>

<p>Although in many applications image rectification can be performed, e.g. by [[camera resectioning]] or calibration, it is sometimes impossible or impractical since the computational cost of accurate rectification models prohibit their usage in [[Real-time computing|real-time]] applications. Moreover, none of these models is suitable when a camera lens displays unexpected [[distortions]], such as those generated by raindrops, weatherproof covers or dust. By extending the Needleman&ndash;Wunsch algorithm, a line in the &lsquo;left&rsquo; image can be associated to a curve in the &lsquo;right&rsquo; image by finding the alignment with the highest score in a three-dimensional array (or matrix). Experiments demonstrated that such extension allows dense pixel matching between unrectified or distorted images.&lt;ref&gt;Martinez-del-Rincon J., Thevenon J., Dieny R., Nebel J.-C. (2012) &quot;[https://www.researchgate.net/profile/Jean-Christophe_Nebel/publication/257928290_Dense_pixel_matching_between_unrectified_and_distorted_images_using_dynamic_programming/links/0c9605266f2d98c719000000.pdf Dense Pixel Matching Between Unrectified and Distorted Images Using Dynamic Programming]&quot;. International Conference on Computer Vision Theory and ==Applications, 24&ndash;26 February, Rome, Italy.&lt;/ref&gt;</p>outside bioinformatics==

<p>==See also=[[Computer stereo vision]]===<br />* Stereo matching is an essential step in the process of 3D reconstruction from a pair of stereo images. When images have been rectified, an analogy can be drawn between aligning nucleotide and protein sequences and matching [[Smith&ndash;Waterman algorithmpixels]]<br />* belonging to [[Sequence miningscan lines]], since both tasks aim at establishing optimal correspondence between two strings of characters. The ‘right’ image of a stereo pair can be seen as a mutated version of the ‘left’ image: noise and individual camera sensitivity alter pixel values (i.e. character substitutions); and different view angle reveals previously occluded data and introduces new occlusions (i.e. insertion and deletion of characters). As consequence, minor modifications of the Needleman–Wunsch algorithm make it suitable for stereo matching.<br /ref>* Dieny R., Thevenon J., Martinez-del-Rincon J., Nebel J.-C. (2011) "[[Levenshtein distance]http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.220.9699 Bioinformatics inspired algorithm for stereo correspondence]". International Conference on Computer Vision Theory and Applications, March 5–7, Vilamoura - Algarve, Portugal.<br /ref>* Although performances in terms of accuracy are not state-of-the-art, the relative simplicity of the algorithm allows its implementation on [[Dynamic time warpingembedded systems]].<br /ref>* Madeo S., Pelliccia R., Salvadori C., Martinez-del-Rincon J., Nebel J.-C. (2014) "[[Sequence alignment]https://pure.qub.ac.uk/portal/files/12586298/stereovision_end.pdf An optimized stereo vision implementation for embedded systems: application to RGB and Infra-Red images]". Journal of Real-Time Image Processing.</pref>

Although in many applications image rectification can be performed, e.g. by [[camera resectioning]] or calibration, it is sometimes impossible or impractical since the computational cost of accurate rectification models prohibit their usage in [[Real-time computing|real-time]] applications. Moreover, none of these models is suitable when a camera lens displays unexpected [[distortions]], such as those generated by raindrops, weatherproof covers or dust. By extending the Needleman–Wunsch algorithm, a line in the ‘left’ image can be associated to a curve in the ‘right’ image by finding the alignment with the highest score in a three-dimensional array (or matrix). Experiments demonstrated that such extension allows dense pixel matching between unrectified or distorted images.<pref>==References==<br Martinez-del-Rincon J., Thevenon J., Dieny R., Nebel J.-C. (2012) "[https://www.researchgate.net/profile/Jean-Christophe_Nebel/publication/257928290_Dense_pixel_matching_between_unrectified_and_distorted_images_using_dynamic_programming/links/>{{Reflist}}0c9605266f2d98c719000000.pdf Dense Pixel Matching Between Unrectified and Distorted Images Using Dynamic Programming]". International Conference on Computer Vision Theory and Applications, 24–26 February, Rome, Italy.</pref>

<p>==External linksSee also==<br />{{External links|date=May 2017}}<br />* [http://zhanglab.ccmb.med.umich.edu/NW-align NW-align: A protein sequence-to-sequence alignment program by Needleman-Wunsch [Smith–Waterman algorithm (online server and source code)]<br />]* [[:File:ParallelNeedlemanAlgorithm.pdf | Parallel Needleman-Wunsch Algorithm for GridSequence mining]] - Implementation by Tahir Naveed, Imitaz Saeed Siddiqui and Shaftab Ahmed - Bahria University<br />* [http://chneukirchen.org/blog/archive/2006/03/dynamic-programming-in-haskell.html Needleman-Wunsch Algorithm as Haskell Code[Levenshtein distance]]<br />* [http://ds9a.nl/nwunsch A live Javascript-based demo of Needleman&ndash;Wunsch]<br />* [http://alfehrest.org/sub/nwa/index.html An interactive Javascript-based visual explanation of Needleman-Wunsch AlgorithmDynamic time warping]<br />* [http://baba.sourceforge.net/ B.A.B.A.] &mdash; an applet (with source) which visually explains the algorithm.<br />* [https://web.archive.org/web/20140729133238/http://www.science.marshall.edu/murraye/Clearer%20Matrix%20slide%20show.pdf A clear explanation of NW and its applications to sequence alignment]<br />* [http://technology66.blogspot.com/2008/08/sequence-Sequence alignment-techniques.html Sequence Alignment Techniques at Technology Blog]<br />* [https://web.archive.org/web/20091106224548/http://svitsrv25.epfl.ch/R-doc/library/Biostrings/html/00Index.html Biostrings] R package implementing Needleman&ndash;Wunsch algorithm among others<br />* [https://gist.github.com/jonasmalacofilho/5226596 Needleman-Wunsch Algorithm as Haxe Code]</p>

<p>==References=={{StringsReflist}}</p>

<p>==External links=={{External links|date=May 2017}}* [http://zhanglab.ccmb.med.umich.edu/NW-align NW-align: A protein sequence-to-sequence alignment program by Needleman-Wunsch algorithm (online server and source code)]* [[:File:ParallelNeedlemanAlgorithm.pdf | Parallel Needleman-Wunsch Algorithm for Grid]] - Implementation by Tahir Naveed, Imitaz Saeed Siddiqui and Shaftab Ahmed - Bahria University* [http://chneukirchen.org/blog/archive/2006/03/dynamic-programming-in-haskell.html Needleman-Wunsch Algorithm as Haskell Code]* [http://ds9a.nl/nwunsch A live Javascript-based demo of Needleman–Wunsch]* [http://alfehrest.org/sub/nwa/index.html An interactive Javascript-based visual explanation of Needleman-Wunsch Algorithm]* [http://baba.sourceforge.net/ B.A.B.A.] — an applet (with source) which visually explains the algorithm.* [https://web.archive.org/web/20140729133238/http://www.science.marshall.edu/murraye/Clearer%20Matrix%20slide%20show.pdf A clear explanation of NW and its applications to sequence alignment]* [http://technology66.blogspot.com/2008/08/sequence-alignment-techniques.html Sequence Alignment Techniques at Technology Blog]* [https://web.archive.org/web/20091106224548/http://svitsrv25.epfl.ch/R-doc/library/Biostrings/html/00Index.html Biostrings] R package implementing Needleman–Wunsch algorithm among others* [https://gist.github.com/jonasmalacofilho/5226596 Needleman-Wunsch Algorithm as Haxe Code] {{Strings}} {{DEFAULTSORT:Needleman-Wunsch Algorithm}}<br />[[Category:Bioinformatics algorithms]]<br />[[Category:Sequence alignment algorithms]]<br />[[Category:Computational phylogenetics]]<br />[[Category:Dynamic programming]]<br />[[Category:Articles with example pseudocode]]</p>