1. Phylogenetic Tree Construction and Related Problems Bioinformatics

2. Problems Multiple Sequence Alignment Construction of Phylogenetic Trees Determining Genomic Distance through Rearrangements

3. Alignment of Multiple Sequences We can extend the notion of alignment to multiple strings: -att-c- t-ttac- tat-a-t An alignment of strings S1…Sn is described by strings S1’…Sn’ Each Si’ contains the characters of Si in order interspersed with spaces (-) No position exists that contain spaces for all Si’

4. Scoring a Multiple Alignment Sum of pairs Consider each pair in the set of sequences, determine the similarity score (using gap, match, and mismatch weights) for each pair, and add all pair-wise scores Distance from consensus Consider all columns and count the total number of differences from the consensus character Variations: just count characters that differ from consensus or have a difference score for each differing character

5. Multiple Alignment Problem Formulation: Given sequences S1…Sn, obtain an optimal multiple alignment of the sequences “Optimal” depends on multiple alignment scoring method No known (correct) efficient algorithms for this problem

6. Strategies Brute force algorithm: consider all possible alignments, then determine the one that results in a best score (time complexity?) Common Heuristic: Use regular (two-string) alignment and then repeatedly add a string to a growing alignment O(nm 2 ) – where m is the the max string length Does not always produce an optimal alignment

7. Phylogenetic Tree Construction Multiple alignments are often performed on similar species Next step: Construct an evolutionary tree of these species Input to problem: evolutionary distances between each pair Not the same as similarity score Example: edit distance--count number of indels needed to transform one string to the other

8. Problem Formulation Given a set of species and evolutionary distance between each pair (2d matrix of numbers), construct a phylogenetic tree consistent with the distances Details needed: Characteristics/constraints of a tree Notion of “consistent”

9. Phylogenetic Tree Rooted Leaves are species Internal nodes are speculated ancestors Distances associated with each edge

10. Example ancestor cat bat dog 14 23

11. Minimizing Distance Deviation Phylogenetic tree implies pairwise distances (sum all edge distances in path) Compare with input distances to assess consistency Sum of Differences Alternative computations for deviation (least squares)

12. Example ancestor cat bat dog 14 23 Suppose Input: D(dog,cat): 4 D(dog,bat): 7 D(cat,bat): 10 Implied by Tree: D(dog,cat): 5 D(dog,bat): 7 D(cat,bat): 8 Difference = (5- 4) + (7-7) + (10-8) = 3

13. Character-based Tree Construction Input: a set C of m characters possible values for each character a set S of n species where each element of S is associated with a value for each character in C Output: a phylogenetic tree T such that Elements in S are the leaves of T Each internal node of T has an assigned value per character Each character value induces a connected subtree of T

14. Mutations on Genomes Mutations can occur at the gene level (indels of nucleotides) or at the genome level (operations on genes) At the genome level, for similar species, the operations are rearrangements Reversals Transpositions Translocations

15. Genomes, Chromosomes, and Permutations Genome: set of chromosomes Chromosome: sequence of genes Genes: unique across entire genome Can simplify the representation of a genome by arbitrarily concatenating the chromosomes Result: a permutation of a set of unique genes

16. Determining Genomic Distance Recall: Phylogenetic tree construction need pair-wise distances For similar species with the same set of genes: Can use number of rearrangement operations for distance Common distance model: obtain the fewest (optimal) rearrangements necessary to transform genome X to genome Y

17. Some Assumptions Made Can limit allowed rearrangements to some reasonable subset of possible arrangements e.g., reversals apparently most common for plant species Sometimes, gene-blocks instead of individual genes are the elements that are permuted

18. Problem Formulations Given permutations (species) p and q of the set {1…n}, find the shortest sequence of rearrangements (mutations) that transform p to q p.r 1. r 2 … r s = q Given a permutation p, find the shortest sequence of rearrangements that sort p p.r 1. r 2 … r s = 1 2 3 … n