Genome Rearrangement SORTING BY REVERSALS Ankur Jain Hoda Mokhtar CS290I – SPRING 2003

Comparative Genomics The practice of analyzing and comparing the genetic material of different species for the purpose of studying evolution, the function of genes and inherited diseases. Chromosome breakage and mistakes in repair, along with a number of other processes, give rise to changes in gene order. These have important consequences for the evolution of species.

Problem Definition During biological evolution, inter- and intra- chromosomal exchanges of chromosomal fragments disrupt the order of genes on a chromosome. The genome rearrangements approach, is the use of combinatorial optimization techniques, to infer a sequence of rearrangement events to account for the differences among the genomes.

Outline Problem definition Genome Comparison Possible chromosomal changes Sorting by reversals : - Previous work - Definitions - Duality Theorem Our technique :- Bit Vector Method - Experimental results : - Synthetic datasets - Real datasets - Breakpoints Technique Conclusions and Future work

Genome Comparison In the late 1980 was discovered remarkable and novel pattern of evolutionary change in plant organelles. Jeffrey Palmer and his collegues compared the mitochondrial genomes of cabbage and turnip, which are very closely related. Molecules which are almost identical in gene sequences, differ dramatically in gene order. {Sridhar, Pevzner 1995} This discovery and many other studies proved that genome rearrangements represent a common mode of molecular evolution.

Cabbage and Turnip Gene orientation

Single Chromosome Operations Reversal: A section of a chromosome is excised, reversed in orientation, and re-inserted. (abc 1 c 2 c 3 c 4 de -> ab-c 4 -c 3 -c 2 -c 1 de) Transposition: A section of a chromosome is excised and inserted at new position in the chromosome, without changing orientation. (abcd -> cdab) Inverted transposition: Exactly like transposition, except that the transposed segment changes orientation. (abcd -> -c-dab) Gene duplication: A section of a chromosome is duplicated, so that multiple copies exist of every gene in that section. (abc -> abcb, abc -> abbc) Gene loss: A section of a chromosome is excised and lost. (abc->ac )

Operations on 2 Chromosomes Translocation: The end of one chromosome is broken and attached to the end of another chromosome. Fusion: two chromosomes merge. Fission: one chromosome splits up into two chromosomes.

Genomic Sorting Problem Given genomes the genomic sorting problem is to find a series of reversals where and t is minimal. We call t the genomic distance between and

Sorting by Reversals Genome rearrangements can be modelled by a combinatorical problem of sorting by reversals. Break and Invert A T G C C T G T A C T A A T G A T G T C C C T A Reversal

Sorting by Reversals (Cont.) Minimum Sorting by Reversals: Given a permutation , what is the shortest sequence (  1  2….  t ) of reversals that sorts ?  Complexity remains open. (NP-Hard) {Caprara ‘97} Minimum Signed Sorting by Reversals: Given a signed permutation , what is the shortest sequence (  1  2….  t ) of reversals that sorts  ?  Solvable in polynomial time.

Sorting of Signed Permutations Transforming cabbage into turnip. {Hannenhalli, S., and Pevzner, P. ‘95} - Polynomial algorithm for sorting signed permutations by reversals A Very Elementray Presentation of the Hannenhalli-Pevzner Theory, {A. Bergeron’95} – Polynomial algorithm for sorting signed permutations, efficiently implemented using bit vectors. Experiments in Computing Sequences of Reversals, {A. Bergeron and F. Strasbourg’95} – Polynomial algorithm for sorting signed permutations. Fast Sorting by Reversal, {Berman, P., Hannenhalli, S. ‘96. }- exploit a few combinatorial properties of the cycle graph of a permutation and provided a polynomial algorithm. A Faster and Simpler Algorithm for Sorting Signed Permutations by Reversals, {Kaplan, H., Shamir, R., and Tarjan, R. ‘99.} – O(n 2 ) using hurdles, cycles and fortress. A Linear-Time Algorithm for Computing Inversion Distance between Signed Permutations with an Experimental Study, {Moret, and Yan’ 00} - Computes reversal distance (without actually sorting) in O(n) time. Computes the connected components using stack rather than Union-Find. {Hannenhalli- Pevzner ’96} (GRAPPA program) A Very Elementray Presentation of the Hannenhalli-Pevzner Theory, {A. Bergeron’95} – Polynomial algorithm for sorting signed permutations, efficiently implemented using bit vectors.

Outline Problem definition Genome Comparison Possible chromosomal changes Sorting by reversals : - Previous work - Definitions Our technique :- Bit Vector Method Experimental results : - Synthetic datasets - Real datasets - Breakpoints Technique Conclusions and Future work

What is a Permutation? Permutation (  ) : an ordered arrangement of the set { 1,2,…,n} Signed Permutation (  ): a permutation where the elements are oriented a reversal switches element orientation { }  (7,-5) = { }

Let i ~ j if | i – j | = 1. Extend permutation by adding = 0 and = n + 1. We call pair of elements, 0 ≤ i ≤ n, of an adjacency if ~ and a breakpoint if is not ( ~ ) BreakPoint ~ =0=n+1 ~

~ The breakpoint graph of a permutationis a edge-colored graphwith 2n+2 vertices We join vertices and by a black edge We join vertices and by a gray edge if What is breakpoint graph?

Breakpoint graph – signed case Straight edges – every other pair of consecutive elements Curved edges - every other pair of consecutive integers Every connected component of the graph is a cycle

Correlation between the breakpoints and reversal distance Correlations exists between the reversal distance and the number of breakpoints Sorting by reversals corresponds to eliminating breakpoints Every resersal can eliminate at most 2 breakpoints {Shamir, 95}

Outline Problem definition Genome Comparison Possible chromosomal changes Sorting by reversals : - Previous work - Definitions - Duality Theorem (Hurdles !!) Our technique :- Vector-Method Experimental results : - Synthetic datasets - Real datasets -Breakpoints Technique Conclusions and Future work

Hurdle Hurdle - an unoriented component whose elements are consecutive Simple hurdle - a hurdle whose deletion decreases the number of hurdles Super hurdles - hurdles that are not simple

Duality Theorem for Sorting Signed Permutations Hannenhalli and Pevzner, For every signed permutation = if is a fortress otherwise

Safe reversal For an arbitary reversal Reversal is safe if C=3, h=1 C = 5, h= 2

Outline Problem definition Genome Comparison Possible chromosomal changes Sorting by reversals : - Previous work - Definitions - Duality Theorem (Hurdles !!) Our technique :- Bit Vector Method Experimental results : - Synthetic datasets - Real datasets - Breakpoints Technique Conclusions and Future work

Our Approach Finding hurdles and fortresses in a graph are difficult and expensive {Kaplan, H., Shamir, R., and Tarjan, R. ‘99.} Use oriented sort to remove the oriented components in a graph and then apply the breakpoint approach to perform the remaining reversals We used the bit-vector approach to perform the oriented sort

Oriented Sort Choose among the several candidates, a safe reversal, that is a reversal that decreases the reversal distance. Theorem : The reversal that maximizes the number of oriented vertices is safe {A. Bergeron’95}

Basic Sorting – oriented pair An oriented pair is a pair of consecutive integers, that is with opposite signs Example: ( ) Oriented pairs are: (1,-2), (3, -4)

Reversalscore Reversal score The number of oriented pairs in the resulting permutation as a result of a reversal Example: ( ) (3, -2) (1, -2) ( ) ( ) Score 4Score 2 ( )

Algorithm As long as has an oriented pair choose the oriented reversal that has maximal score ( –2 4 7) ( ) (-3, 4) ( ) (-1,2) ( ) (-6,7) ( ) (-5,6) ( )

Orientededge Oriented edge Letbe a gray edge incident toblack edges andThen is oriented if and only if i – k = j - l Edge is oriented (contains 3 [odd] number of vertices). I= 20, j=21, k=22, l=23 I-k = -2 = j-l = -2 Bergeron Pevzner.

Oriented reversals, if, and, if Reversals that create consecutive integers are always induced by oriented pairs. Such reversals are called oriented reversal. Reversals induces by an oriented pair will be: Example: The pair (1, -2) induces the reversal: ( –2 4 7) ( –5 –6 4 7)

Interleaving Graph Every 2 components are adjacent if there is an overlap between them but neither of them contains the other. C

Constructing the Bit Matrix Consider the sequence P = –2 4 7 Represent P i by 2i-1, 2i if P i is +ve and 2i-1, 2i if P i is +ve and 2i, 2i-1 otherwise P i is -ve 2i, 2i-1 otherwise P i is -ve Bit Matrix Parity Scores

The Algorithm Step 1. Select the vertex v i with the maximum score and perform the these operations until we reach a situation when parity of all the vertices is zero Step 2. If the sequence is not sorted completely apply the breakpoint technique to complete the sorting

Outline Problem definition Genome Comparison Possible chromosomal changes Sorting by reversals : - Previous work - Definitions - Duality Theorem (Hurdles !!) Our technique :- Bit Vector Method Experimental results : - Synthetic datasets - Real datasets - Breakpoints Technique Conclusions and Future work

Experimental Settings 1- Synthetic Datasets:  generated random signed permutation of different lengths and evolution rate using GRIMM permutation generation module 2- Real Datasets:  Used GRAPPA test sets for different species of “Campanulaceae” (flower plant)  MGR (multiple genome rearrangement) human-mouse gene order data  Genome.org Herpes Virus that affects human

Experiment 1 - Synthetic 1- Generated files of random permutations of different lengths (50, 100, 200, 400, 800, 1600) each file with 50 permutations. 2- We computed the number of correctly sorted permutations. 3- Evolution rate varies : 20,30,40

Experiment 2 - Synthetic 1- Generated files of random permutations of different lengths (50, 100, 200, 400, 800, 1600) each file with 50 permutations. 2- We computed the time needed to obtain the correctly sorted permutations. 3- Evolution rate varies : 20,30,40

Experiment 3 - Synthetic 1- Generated files of random permutations of length We computed the time needed to obtain the correctly sorted permutations. 3- Evolution rate varies in increments of 100. Observation: Saturation state is reached as evolution rate approaches 1000

Experiment 1 - Real Considered Herpes simplex virus (HSV), Epstein-Barr virus (EBV), and Cytomegalovirus (CMV) gene orders (Hannenhalli et al. 1995) as well as the identity gene order (A) Observations:Our reversal results matched those obtained in optimal evolutionary scenario recovered by MGR-MEDIAN.

Experiment 2 - Real 1- Considered Campanulaceae species 2- Obtained reversals for Cyanathus (11 reversals), Triodanus (13 reversals), and Symphanra (12 reversals) versus Tobacco but failed to sort Platyncodon, Legousia and Codonopsis Observation: The ones we sorted were sorted with same number of reversals as GRIMM

Experiment 3 - Real 1- Considered Human-Mouse gene order from MGR (mouse genome and human is identity) 40 reversals reversals Identity GRIMM sorts the permutation in 41 reversals

Conclusions We implemented a technique that integrates the bit-matrix oriented sorting technique together with the greedy breakpoint reversal technique. The technique proposed was tested on both real and synthetic data and was able to sort signed permutations in a fair number of the test data We think that such integration can yield good results beside being a simple and relatively fast technique However, the oriented sort algorithm fails to sort permutations that have hurdles, in those cases we have to apply the breakpoint approach

Future Work We really think that the technique we implemented can provide good results, we think that further experiments can strengthen our claim We started implementing the algorithm proposed in Kaplan, H., Shamir, R., and Tarjan R. ’99} but didn’t succeed to complete the implementation. We think that having this technique implemented under that same conditions as ours can provide a good source of comparative results, and can give a better confidence about what we propose. Applying the technique in different datasets including exon order rather than gene order Considering different species and trying to compute reversal distance and use it to confirm phylogenetic trees

Oriented Pairs An oriented pair (, ) is a pair of consecutive integers, that isAn oriented pair (, ) is a pair of consecutive integers, that is with opposite signs Example: ( –2 4 7) Oriented pairs are Oriented pairs are = (0 … ) (1,-2)(3, 2)

Reversal Distance Estimation This reversal distance is very in-accurate. Bafna and Pevzner, 1996 showed that another hidden parameter ”hurdles” estimated reversal distance with much greater accuracy.

Proper reversal For every permutation and reversal Given an arbitary reversal denote (increase in the size of cycle decomposition) Then for every permutationand reversal We call reversal proper if = 1

Orientedpairs Oriented pairs Oriented pairs are useful because they indicate reversals that create consecutive elements of the permutation. Example: The pair (1, -2) induces the reversal: ( –2 4 7) ( –5 –6 4 7)