lecture_notes:04-22-2011
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lecture_notes:04-22-2011 [2011/04/25 22:01] eyliaw |
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| - | ====== Burrows Wheeler Transform ====== | + | ====== Burrows Wheeler Aligner ====== |
| - | Incomplete. | + | We discussed the [[bioinformatic_tools:bwa]]. It uses the Burrows Wheeler Transform to represent a prefix trie, allowing for short read alignment with mismatches and gaps. |
| ===== The prefix trie ====== | ===== The prefix trie ====== | ||
| - | The prefix trie is a tree built from possible prefixes, starting at the end of a branch and going backwards to the root. Each node is the range in the suffix array where the | + | The prefix trie is a tree built from possible prefixes, starting at the end of a branch and going backwards to the root. Each node is the range in the suffix array where that prefix may be found. Each edge is a character in the prefix. A string can be found--allowing for mismatches and gaps--in the prefix tree by going up each branch and comparing characters to the prefix until the all characters have been matched or the mismatch limit is met. |
| ===== Suffix array ====== | ===== Suffix array ====== | ||
| - | The suffix array is found by sorting the cyclic transformation lexicographically and taking their original indices. | + | The suffix array is found by appending an end character and sorting the cyclic transformation of the resulting string lexicographically and storing the original indices. |
| - | Range in the suffix array can be found by the R_ and R- | + | |
| + | Example: | ||
| + | GOOGOL | ||
| + | Append end character: | ||
| + | X = GOOGOL$ | ||
| + | Cyclic transformation: | ||
| + | 0 GOOGOL$ | ||
| + | 1 OOGOL$G | ||
| + | 2 OGOL$GO | ||
| + | 3 GOL$GOO | ||
| + | 4 OL$GOOG | ||
| + | 5 L$GOOGO | ||
| + | 6 $GOOGOL | ||
| + | Sort: | ||
| + | 6 $GOOGOL | ||
| + | 3 GOL$GOO | ||
| + | 0 GOOGOL$ | ||
| + | 5 L$GOOGO | ||
| + | 2 OGOL$GO | ||
| + | 4 OL$GOOG | ||
| + | 1 OOGOL$G | ||
| + | |||
| + | S(i) = [6,3,0,5,2,4,1] | ||
| + | The Burrows Wheeler transform takes the last character of the sorted cyclic strings: | ||
| + | B(i) = LO$OOGG | ||
| + | ===== Searching ===== | ||
| + | We can use the FM-index to find the upper and lower bounds of a substring. | ||
| + | |||
| + | C_X(a) is the number of characters lexicographically before a in X. | ||
| + | G 0 | ||
| + | L 2 | ||
| + | O 3 | ||
| + | O_X(a,i) is the number of occurrences of a in B_X[0,i]. | ||
| + | ^ i ^ B_X[0,i] ^ O_X(G,i) ^ O_X(L,i) ^ O_X(O,i) ^ | ||
| + | | 1 | L | 0 | 1 | 0 | | ||
| + | | 2 | LO | 0 | 1 | 1 | | ||
| + | | 3 | LO$ | 0 | 1 | 1 | | ||
| + | | 4 | LO$O | 0 | 1 | 2 | | ||
| + | | 5 | LO$OO | 0 | 1 | 3 | | ||
| + | | 6 | LO$OOG | 1 | 1 | 3 | | ||
| + | | 7 | LO$OOGG | 2 | 1 | 3 | | ||
| + | |||
| + | There are then two recursive formulas to find the start and end positions of a substring. | ||
| + | End R_(aW) = C(a) + O(a,R_(W) - 1) + 1 or 1 if W is the empty string | ||
| + | Start R-(aW) = C(a) + O(a,R-(W)) or (n - 1) if W is the empty string | ||
| + | Where a is the first character and W is the word. | ||
| + | |||
| + | To find the end position of the substring GO in | ||
lecture_notes/04-22-2011.txt · Last modified: 2015/08/09 23:06 by 212.129.31.47
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