Chapter 9: How Do We Locate Disease-Causing Mutations?
Yes, the reference human genome is a mosaic of various genomes that does not match the genome of any individual human. Since various human genomes differ by only 0.1%, however, the amalgamation does not cause significant problems.
Huntington's disease is a rare genetic disease in that it is attributable to a single gene, called Huntingtin. This gene includes a trinucleotide repeat "...CAGCAGCAG..." that varies in length. Individuals with fewer than 26 copies of "CAG" in their Huntingtin gene are classified as unaffected by Huntington's disease, whereas individuals with more than 35 copies carry a large risk of the disease, and individuals with more than 40 copies will be afflicted. Moreover, an unaffected person can pass the disease to a child if the normal gene mutates and increases the repeat length. The reason why many repeated copies of "CAG" in Huntingtin leads to disease is that this gene produces a protein with many copies of glutamine ("CAG" codes for glutamine), which increases the decay rate of neurons.
Perhaps in theory, but in practice, biologists still use one reference genome, since comparison against thousands of reference genomes would be time-consuming.
Construct the suffix trie for "papa" and you will see why we have added the "$" sign – without the "$" sign, the suffix "pa" will become a part of the path spelled by the suffix "papa".
The suffix tree for "panamabananas$" reproduced below contains 17 edges with the following labels (note that different edges may have the same labels):
$ a bananas$ mabananas$ na mabananas$ nanas$ s$ s$ bananas$ mabananas$ na mabananas$ nas$ s$ panamabananas$ s$
In addition to storing the nodes and edges of the suffix tree, we also need to store the information at the edge labels. Storing this information takes most of the memory allocated for the suffix tree.
Suffix trees were introduced by Weiner, 1973. However, the original linear-time algorithm for building the suffix tree was extremely complex. Although the Weiner algorithm was greatly simplified by Esko Ukkonen in 1995, it is still non-trivial. Check out this excellent StackOverflow post by Johannes Goller if you are interested in seeing a full explanation.
Our naive approach to constructing BWT(Text) requires constructing the matrix M(Text) of all cyclic rotations, which requires O(|Text|2) time and space. However, there exist algorithms constructing BWT(Text) in linear time. One such algorithm first constructs the suffix array of Text in linear time and then uses this suffix array to construct BWT(Text).
In short, the last column is the only invertible column of the Burrows-Wheeler matrix. In other words, it is the only column from which we are always able to reconstruct the original string Text.
In practice, it is possible to compute the Last-to-First mapping of a given position of BWT(Text) with very low runtime and memory using the array holding the first occurrence of each symbol in the sorted string. Unfortunately, the analysis is beyond the scope of this class. For details, please see Ferragina and Manzini, 2000 (click here for full text).
We indeed do not use FirstColumn in BWMatching. Although it seemingly does not make sense, we prefer this because we use FirstColumn in a modification of of BWMatching in a later section.
Given an index ind in the array LastColumn (varying from 0 to 13 in the example shown in the text), the number of occurrences of symbol before position ind (i.e., in positions with indices less than ind) is defined by Countsymbol(ind, LastColumn). Since the number of occurrences of symbol starting before position ind is equal to Countsymbol(ind, LastColumn), the rank of the first occurrence of symbol starting from position ind is
To be more precise, it is Countsymbol(ind, LastColumn) + 1 if symbol occurs in LastColumn at or after position ind.
Similarly, the rank of the last occurrence of symbol starting before or at position ind is given by
For example, when ind = 5, the rank of the first occurrence of "n" starting at position 5 is Count"n"(5, LastColumn) + 1 = 1 + 1 = 2. On the other hand, the rank of the last occurrence of "p" starting before or at position ind is Count"p"(6, LastColumn) = 1.
The condition "top ≤ bottom" is a loop invariant, or a property that holds before and after each iteration of the loop. In this case, if pattern matches have been found, the number of matches is equal to bottom - top + 1. If pattern matches are not found, then at some point in the loop, bottom becomes equal to top - 1, in which case top ≤ bottom and the loop terminates.
No; however, you can easily modify BetterBWMatching by first checking whether Pattern contains symbols not present in Text and immediately returning 0 in this case.
Try "walking backwards" to find the one pattern match of "ban" in "panamabananas$".
We indeed got rid of the LastToFirst array; however, in the same section we saw how the Count arrays can be used as a substitute for LastToFirst.
To explain how to modify BetterBWMatching for working with checkpoint arrays, we explain how to quickly compute each value in the count array given the checkpoint arrays and LastColumn.
To compute Countsymbol(i, LastColumn), we represent i as t·K + j, where j < K. We can then compute Countsymbol(i, LastColumn) as Countsymbol(t·K, LastColumn) (contained in the checkpoint arrays) plus the number of occurrences of symbol in positions t·K + 1 to i in LastColumn.
Biologists usually set a small threshold for the maximum number of mismatches, since otherwise read mapping becomes too slow.
For example, does Pattern = "TTACTG" match Text = "ACTGCTGCTG" with d = 2 mismatches? Not according to the statement of the Multiple Approximate Pattern Matching Problem, since there is no starting position in Text where Pattern appears as a substring with at most d mismatches.
BLAST does construct an alignment in a narrow band starting from each end of the seed. However, since the band is narrow, the algorithm for constructing this alignment is fast.
The algorithm illustrated in the epilogue would fail to find an approximate match of "nad" because the final symbol of "nad" does not appear in "panamabananas$". To address this complication, we can modify the algorithm for finding a pattern of length m with up to k mismatches as follows.
We first run the algorithm described in the main text to find all approximate instances of a Pattern of length k against Text. However, this algorithm does not actually find all approximate matches of Pattern – since we do not allow mismatched strings in the early stages of BetterBWMatching, we miss those matches where the last letter of Pattern does not match Text. To fix this shortcoming, we can simply find all locations in Text where the prefix of Pattern of length k - 1 has d - 1 mismatches. Yet this algorithm fails to find matches where the last two letters of Pattern do not match Text. Thus, we need to run the algorithm again, finding all locations in Text where the prefix of Pattern of length k - 2 has d - 2 mismatches. We then find all locations in Text where the prefix of Pattern of length k - 3 occurs with d - 3 mismatches, and so on, finally finding all locations in Text where the prefix of Pattern of length k - d occurs exactly.
Yes, this strategy would fail to match a read with an error at the first position. However, as noted in the main text, if we start considering mismatches at the first position, the running time will significantly increase. As is, the running time explodes with the increase in the maximum number of errors. If one wants to alow mismatches at the first position, a more sensible strategy would be to trim the first position of the read.