Introduction to mathematical methods in bioinformatics

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Systems Biology , Genetic and Biochemical Network. Opresko, Julie M. Gephart, Michaela B. Keener, James Sneyd, , Springer Verlag. Shoemaker, Simon M. Simon, Edward L. Korn, Lisa M.

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Reconstruction of additive trees is complicated by the fact that, unlike in an ultrametric tree, sister taxa need not be equidistant from their most recent common ancestor. Instead, branchlengths must be inferred by solving a system of simultaneous linear equations. Larger trees can be handled by solving for three sequences at a time, collapsing the two closest sequences into their common vertex, calculating the average distance from that vertex to each remaining sequence in the distance matrix, adding the next sequence to the remaining two and solving again. A Genetic distance matrix for amino acid sequences generated by EvolSeq; B Corresponding phylogenetic tree illustrating the additive criterion.

Thus far, we have measured genetic distance as the number of nucleotide or amino acid differences between sequences. However, this simple metric does not consider that multiple substitutions may have occurred at the same position, and thus tends to underestimate actual amount of divergence. Therefore, the next logical refinement is the Jukes—Cantor distance metric, which includes a correction for this phenomenon.

We also discuss distance metrics used to analyze amino acid sequences, such as the BLOSUM80 matrix for closely related sequences and the PAM matrix for more distant relationships. The workbook begins with a single random sequence, and then follows that sequence through time as it reproduces and mutates.

Eventually, up to 20 related sequences are generated. EvolSeq then calculates the ultrametric genetic distances between each pair of DNA sequences and also the additive distances between the associated amino acid sequences. The central lesson for students is that simple algebraic methods can yield substantial insight into phylogenetic methods, and that additional refinements based on the evolutionary models appropriate to a particular biological system can improve those methods still further.

As described earlier in the text, genetic distance matrices for groups of four or more taxa can seldom be fitted exactly to a phylogenetic tree. In cases where this mismatch is not the focus of study, best-fit approaches may be used to infer an approximate tree. However, substantial deviation from purely tree-like structure may itself contain valuable biological information, such as episodes of convergent evolution or violation of the evolutionary model used to infer the tree.

Therefore, split decomposition may be used to resolve and display the full set of taxonomic groupings supported by the data, rather than only those groupings inferred to result from shared ancestry [ 21 ]. Split decomposition begins by dividing the set of taxa into two partitions J and K such that each partition contains at least two taxa.

This statistic simply measures the strength of the inequality described by the four-point condition.

Introduction to Mathematical Methods in Bioinformatics

See Figure 4 B for a graphical depiction of this formula. A Split decomposition results as produced by SplitDecomp. B A graphical interpretation of an isolation index. Addition and subtracting appropriate pairwise distances allows estimation of the internal branchlength, which is equivalent to determining which tree topology best separates one pair of sequences from the other pair. The user can type in the sequences or paste them in from a text file. Split decomposition results as produced by SplitsTree [ 22 ].

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Data set modified from [ 23 ], with taxon names updated. Long-branch attraction. The left-hand figure shows the actual phylogeny for four sequences; the right-hand figure shows the phylogeny reconstructed using ClustalW in Biology Workbench. Sequences B and C, at the end of the two longest branches, have been mistakenly clustered together. Long-branch attraction is a well-known problem for several phylogenetic methods, notably parsimony [ 24 ]. By contrast, there is nearly as much evidence for a Euglena -heterokont clade 0. SplitDecomp can be used to address a number of overarching issues in phylogenetics.

For example, students can explore the relationship between branchlength and phylogenetic confidence by generating random DNA sequences of given length and using split indices to determine the length of the inferred central branch. Because the sequences are random, there is no central branch, but phylogenetic methods will still infer one based on chance resemblances among sequences.

This can be shown using Biology Workbench [ 25 ], EvolSeq, or any of the numerous other phylogenetic tools available online. This exercise can reinforce the concept of phylogenetic trees as hypotheses to be tested. A similar approach can illuminate other important topics such as long-branch attraction see Figure 6 and recombination.

SplitDecomp can also be used to explore and analyze a specific phylogenetic question.

For example, a study [ 26 ] examined whether an HIV-infected dentist had inadvertently transmitted the virus to several of his patients during invasive dental surgery. This case had enormous implications for health care practice and received international attention. The study focused on sequence data from the viruses infecting the dentist, the HIV-positive patients and a number of HIV-positive local controls with no known epidemiological link to the dentist. Curricular materials already exist [ 27 ] for introducing students to phylogenetic concepts and methods using the data from this study.

Students can then further analyze their findings using SplitDecomp to estimate the lengths of individual branches and determine whether they are well supported by the data Figure 7. Phylogeny of HIV sequences from the dentist, three patients and two local controls.

How would you interpret this phylogeny? Can split decomposition be used to quantify the statistical support for your conclusions? Modified from [ 27 ]. Phylogenetic inferences rely on the assumption that the characters under study are homologous, representing descent from a shared ancestor rather than convergent evolution from different starting points. For example, when analyzing morphological data, the presence of eyespots on the wings of two different butterflies of the same or different species may result from those butterflies both having inherited the relevant genetic signals from the same ancestor [ 28 ].

Similarly, if two individuals have a thymine residue at the same position in the same gene, this may also reflect common ancestry. The process of posing hypotheses about which residues in one sequence are homologous to specific residues in another is called sequence alignment. An example of sequence alignment.