String Mathematics, BLAST, and FASTA

Abstract

What is an e-value for ungapped and gapped BLAST? What are the Karlin-Altschul parameters that affect e-value calculation? How nucleotide frequencies and match-mismatch matrices affect such parameters? What are the key algorithms for FASTA and BLAST? How do their differences affect sensitivity of sequence search? This chapter addresses these questions and illustrates applications of string matching in genomics, transcriptomics, and proteomics, as well as in drug discovery.

Appendix: Being Colorful Is Not Enough—How Are Stop Codons Decoded?

In the old and not-so-good days of 1965, three stop codons were known to exist. At that time, almost all molecular biologists study bacteriophages (T4 and λ) and their hosts (typically E. coli ). Sometimes a phage would have a mutation from a sense codon to a stop codon, leading to a truncated protein. Such a phage can only grow in certain E. coli strains containing a suppressor mutation (which is typically a tRNA with a mutated anticodon that can base-pair with a stop codon). Such strains were then called suppressor strains. The first set of nonsense mutations were discovered and isolated by Richard Epstein and Charles Steinberg. The resulting stop codon from these nonsense mutations was named after their friend Harris Bernstein whose last name means “amber” in German. The associated suppressor was termed amber suppressor. Any phage with a nonsense mutation that can grow only in this particular set of suppressor strains was said to harbor an amber mutation. The observation that an amber stop codon could be suppressed (i.e., recognized as a sense codon) in suppressor strains but not in other strains suggests some ambiguity in the meaning of the stop codon, i.e., the stop codon is interpreted differently between the suppressor strains and non-suppressor strains.

Phages with another set of nonsense mutations can grow only in a different set of E. coli strains, and the stop codon resulting from these nonsense mutations was named ochre. The associated suppressor is termed ochre suppressor. The third stop codon was named opal. In short, there are three disjoint sets of suppressor strains corresponding to three stop codons, but biologists did not know which stop codon is associated with amber, ochre, or opal.

Martin Weigert and Alan Garen (1965) studied one particular amino acid site in the alkaline phosphatase gene in E. coli . This site is occupied by tryptophan coded by UGG. An amber mutation (i.e., a nonsense mutation that can be suppressed by an amber suppressor) occurred at this site. In several revertants (in which the amber mutation had reversed to a sense codon), the original amino acid site was found to be occupied by seven different amino acids, Glu, Lys, Leu, Gln, Ser, Trp, and Tyr (Fig. 1.5). Can we now match the amber codon to one of the stop codons?

Fig. 1.5

Data for matching the amber mutation to one of the three possible stop codons

Because mutation is rare, we may assume that the revertants differ from the amber mutant by a single nucleotide. These candidate codons are shown in Fig. 1.6 for each of the three stop codons. UGA is the least likely candidate because it cannot generate a codon for Glu, Lys, Gln, and Tyr by a single nucleotide change. UAA can also be excluded because (1) it cannot generate a Trp UGG codon by a single nucleotide change, and (2) it would need to have a double mutation from the original UGG codon. In contrast, UAG can mutate to one of the codons for all seven amino acids, and it can result from the original UGG through a single nucleotide replacement. So the amber codon is UAG. (I have made things easier because Weigert and Garen were not so sure about the sense codons which were only partially decoded in 1965).

Fig. 1.6

Bioinformatic analysis to match the amber codon to UAG. The numbers indicate the number of matched nucleotides because a sense codon and a stop codon. Amino acids are shown next to its codons

This example illustrates several decision-making principles that are useful to us. The first is the KISS principle (“Keep it simple, stupid”). That is, to use simplifying assumptions to make decision-making easier. For example, if we assume that double mutation (i.e., mutations at two sites of the codon) is so rare as to be negligible, then we can immediately reject UAA and UGA because both require double mutations to generate codons for the seven amino acids. In terms of model selection, we state that only the UAG model fits the data given the model condition (the assumption of no double mutation).

The second is the parsimony principle. We have three alternative hypotheses corresponding to the three stop codons. The UAG hypothesis needs a minimum of seven single point mutations to generate the seven revertants each with a different amino acid. It also needs a point mutation to connect the original UGG codon. The UGA hypothesis needs at least two point mutations to change to a Glu, Lys, Gln, or Try codon, and one mutation to each of the other three amino acids. This comes to a minim number of 11 point mutations, plus a point mutation to connect the original UGG codon. The UAA hypothesis needs to have two point mutations to a UGG revertant as well as two point mutations from the original UGG codon, leading to a minimum of ten point mutations. The UAG hypothesis is therefore the most parsimonious and is chosen over the other two alternatives.

The third is the likelihood principle, but we will not go there without first learning substitution models which is covered in a later chapter.