1 Genome Rearrangements (Lecture for CS498-CXZ Algorithms in Bioinformatics) Dec. 6, 2005 ChengXiang Zhai Department of Computer Science University of Illinois, Urbana-Champaign 2 Outline The Problem of Genome Rearrangements Sorting By Reversals Greedy Algorithm for Sorting by Reversals Position-based Breakpoint-based 3 Genome Rearrangements Example I: Turnip vs Cabbage Although cabbages and turnips share a recent common ancestor, they look and taste different 4 Turnip vs Cabbage: Comparing Gene Sequences Yields No Evolutionary Information 5 Turnip vs Cabbage: Almost Identical mtDNA gene sequences In 1980s Jeffrey Palmer studied evolution of plant organelles by comparing mitochondrial genomes of the cabbage and turnip 99% similarity between genes These surprisingly identical gene sequences differed in gene order This study helped pave the way to analyzing genome rearrangements in molecular evolution 6 Turnip vs Cabbage: Different mtDNA Gene Order Gene order comparison: 7 Turnip vs Cabbage: Different mtDNA Gene Order Gene order comparison: 8 Turnip vs Cabbage: Different mtDNA Gene Order Gene order comparison: 9 Turnip vs Cabbage: Different mtDNA Gene Order Gene order comparison: 10 Turnip vs Cabbage: Different mtDNA Gene Order Gene order comparison: Before After Evolution is manifested as the divergence in gene order 11 Transforming Cabbage into Turnip 12 What are the similarity blocks and how to find them? What is the architecture of the ancestral genome? What is the evolutionary scenario for transforming one genome into the other? Unknown ancestor ~ 75 million years ago Mouse (X chrom.) Human (X chrom.) Genome Rearrangements Example II: Human vs. Mouse 13 History of Chromosome X Rat Consortium, Nature, 2004 14 Mouse vs Human Genome Humans and mice have similar genomes, but their genes are ordered differently ~245 rearrangements Reversal, fusion, fission, translocation Reversal: flipping a block of genes within a genomic sequence 15 Types of Rearrangements Reversal Translocation Fusion Fission Chromosome 1: Chromosome 2: Chromosome 1: Chromosome 2: 16 Reversals Blocks represent conserved genes. In the course of evolution blocks 1,,10 could be misread as 1, 2, 3, - 8, -7, -6, -5, -4, 9, 10. Rearrangements occurred about once-twice every million years on the evolutionary path between human and mouse , 2, 3, 4, 5, 6, 7, 8, 9, 10 17 Reversals , 2, 3, -8, -7, -6, -5, -4, 9, 10 Blocks represent conserved genes. In the course of evolution or in a clinical context, blocks 1,,10 could be misread as 1, 2, 3, -8, -7, -6, -5, -4, 9, 10. Reversals occurred one-two times every million years on the evolutionary path between human and mouse. 18 Reversals , 2, 3, -8, -7, -6, -5, -4, 9, 10 The reversion introduced two breakpoints (disruptions in order). 19 Reversals Lets first assume that genes in genome do not have direction 1 i-1 i i j-1 j j n ` 1 i-1 j j i+1 i j n Reversal ( i, j ) reverses the elements from i to j in and transforms into ` ,j) 20 Reversals: Example Example: = (3,5) = 21 Reversal Distance Problem Goal: Given two permutations, find the shortest series of reversals that transforms one into another Input: Permutations and Output: A series of reversals 1, t transforming into such that t is minimum t - reversal distance between and d( , ) - smallest possible value of t, given and 22 Sorting By Reversals Problem [ (1 2 n )] Goal: Given a permutation, find a shortest series of reversals that transforms it into the identity permutation (1 2 n ) Input: Permutation Output: A series of reversals 1, t transforming into the identity permutation such that t is minimum 23 Sorting By Reversals: Example t =d( ) - reversal distance between and Example : input: = output: So d( ) = 3 24 Sorting by reversals 25 Sorting by reversals Most parsimonious scenarios The reversal distance is the minimum number of reversals required to transform into . Here, the reversal distance is d=4. 26 Sorting By Reversals: A Greedy Algorithm If sorting permutation = , the first three numbers are already in order so it does not make any sense to break them. These already sorted numbers of will be defined as prefix( ) prefix( ) = 3 This results in an idea for a greedy algorithm: increase prefix( ) at every step 27 Doing so, can be sorted d( ) = 2 Number of steps to sort permutation of length n is at most (n 1) Greedy Algorithm: An Example 28 Greedy Algorithm: Pseudocode SimpleReversalSort( ) 1 for i 1 to n 1 2 j position of element i in (i.e., j = i) 3 if j != i 4 * (i, j) 5 output 6 if is the identity permutation 7 return Progress is ensured by moving forward in the position 29 Analyzing SimpleReversalSort Greedy algorithm; does not guarantee the smallest number of reversals For example, let = SimpleReversalSort( ) takes five steps: Step 1: Step 2: Step 3: Step 4: Step 5: 30 But it can be done in two steps: = Step 1: Step 2: So, SimpleReversalSort( ) is not optimal Analyzing SimpleReversalSort (cont d) 31 = 2 3 n-1 n A pair of elements i and i + 1 are adjacent if i+1 = i + 1 For example: = (3, 4) or (7, 8) and (6,5) are adjacent pairs Breakpoints 32 There is a breakpoint between any pair of non- adjacent elements: = Pairs (1,9), (9,3), (4,7), (8,2) and (2,5) form breakpoints of permutation Breakpoints: An Example 33 We put two elements 0 and n + 1 at the ends of 0 = 0 and n + 1 = n + 1 This gives us the goal to sort the elements between the end blocks to the identity permutation Example: Extending with 0 and 10 Note: A new breakpoint was created after extending Extending Permutations = = 34 b( ) = number of breakpoints Each reversal eliminates at most 2 breakpoints. This implies that d( ) >= b( ) / 2 = b( ) = b( ) = b( ) = b( ) = 0 Reversal Distance and Breakpoints 35 Sorting By Reversals: A Better Greedy Algorithm BreakPointReversalSort( ) 1 while b( ) > 0 2 Among all possible reversals, choose minimizing b( ) 3 (i, j) 4 output 5 return 36 Strip: an interval between two consecutive breakpoints in Decreasing strips: strips that are in decreasing order (e.g. 6 5 and 3 2 ). Increasing strips: strips that are in increasing order (e.g. 7 8) A single-element strip can be declared either increasing or decreasing. We will choose to declare them as decreasing with possible exception of the strips with 0 and n+1 Strips 37 Strips: An Example For permutation : There are 7 strips: 38 Things To Consider Fact 1: If permutation contains at least one decreasing strip, then there exists a reversal which decreases the number of breakpoints (i.e. b( ) < b( ) ) 39 Things To Consider (contd) For = b( ) = 5 Choose decreasing strip with the smallest element k in ( k = 2 in this case) Find k 1 in the permutation, reverse the segment between k and k-1: b( ) = 5 b( ) = 4 40 Things To Consider (contd) Fact 2: If there is no decreasing strip, there may be no reversal that reduces the number of breakpoints (i.e. b( ) = b( ) ). By reversing an increasing strip ( # of breakpoints stay unchanged ), we will create a decreasing strip at the next step. Then (fact 1) the number of breakpoints will be reduced in the next step. 41 Things To Consider (contd) There are no decreasing strips in , for: = b( ) = 3 (3,4) = b( ) = 3 (3,4) does not change the # of breakpoints (3,4) creates a decreasing strip, guaranteeing that the next step will decrease the # of breakpoints. 42 ImprovedBreakpointReversalSort ImprovedBreakpointReversalSort( ) 1 while b( ) > 0 2 if has a decreasing strip 3 Among all possible reversals, choose that minimizes b( ) 4 else 5 Choose a reversal that flips an increasing strip in 6 7 output 8 return 43 ImprovedBreakPointReversalSort is an approximation algorithm with a performance guarantee of at most 4 It eliminates at least one breakpoint in every two steps; at most 2b( ) steps Approximation ratio: 2b( ) / d( ) Optimal algorithm eliminates 2 breakpoints in every step: d( ) b( ) / 2 Performance guarantee: ( 2b( ) / d( ) ) [ 2b( ) / (b( ) / 2) ] = 4 ImprovedBreakpointReversalSort: Performance Guarantee 44 Signed Permutations Up to this point, all permutations to sort were unsigned But genes have directions so we should consider signed permutations 55 33 = 45 GRIMM Web Server Real genome architectures are represented by signed permutations Efficient algorithms to sort signed permutations have been developed GRIMM web server computes the reversal distances between signed permutations: 46 GRIMM Web Server 47 What You Should Know What is genome rearrangement? How does the problem of sorting by reversal capture genome rearrangements? How do the two greedy algorithms work?