Chap. 7 Genome Rearrangements Introduction to Computational Molecular Biology Chapter 7.1~7.2.4

7. Genome rearrangements Comparison of corresponding genes in two or more species can yield less information than comparison if larger portions of the genome. Instead of looking at the sequences, we compare the positions of several related genes and try to determine gene rearrangement operations that transform one genome into the other.

7.1 Biological Background Genome sizes Genome rearrangements A piece can be move or copied to another location, or it can leave one chromosome and land in another. The reversal A single type of rearrangement event Gene 10 4 base pairs Chromosome 10 8 base pairs Genome 3x10 9 base pairs

7.1 Biological Background Operations TypeExample Deletion A certain part is lost ABC : AC Insertion A part is added AC : ABC Duplication Can be copy ABC : ABBC Reversal A part is turned around ABCc1c2c3D : ABCc3c2c1D

7.1 Biological Background As an example, consider the chloroplast genomes of alfafa and pea, two related plant spccies, depicted in Figure 7.1. There each labeled arrow denotes a block. A block here is a section of the genome possibly containing more than one gene, which is transcribed as a unit.

7.1 Biological Background The arrow denotes the fact that blocks have orientation. Relation between the chloroplast genomes of alfafa and garden pea.

7.1 Biological Background minimum # of reversals Series of reversals What is the minimum number of reversals that leads from one genome to the other ? How do we know that no shorter series of reversals exists to accomplish the required transformation? Want a shortest possible series of reversals Using lower bound { A, B }=related species  r i =series of reversals A  BA  r1r2r3r4r5 = B

7.1 Biological Background The reversal operation for oriented blocks A solution to the problem of Figure 7.1

7.1 Biological Background The reversal operation for unoriented blocks A solution to the same problem, but with unoriented blocks.

7.2 Oriented blocks Reversals by using breakpoint A solution to the problem of the previous figure. Each line is obtained from the previous one by a reversal involving the underlined elements. L R

7.2 Definitions L: set of n labels set of oriented labels from L For, is a without the arrow. Oriented permutation over L: such that for any label there is exactly one with. Reversal :

L = {1,2,3,4,5,6} 7.2 Definitions Example A reversal is an operation that transforms one permutation into Another by reversing the order of a contiguous portion of it, And at the same time flipping the signs of the elements involved.

7.2 Definitions The reversal distance Given two oriented permutations a and b over the same label set L, we seek the minimum number of reversals that will transform a into b.  = reversal t = the # of reversals d = distance a  1  2  3 …  t b t=d b (a) d b (a  )  d b (a) d b (a  ) = d b (a) – 1

7.2 Breakpoints A breakpoint of  with respect to  is a pair x, y of elements of L 0 such that xy appears in the extended version of  but neither xy nor the reverse pair xy appear in extended . Extended version of  The number of breakpoints of a permutation a is denoted by b  (  ) or just by b(  ) if  is clear from the context.

7.2 Breakpoints A Lower Bound Using breakpoints b(  ) - b(  )  2 (0,1,2) b(  )  2t  b(  ) / 2  d(  ) ( ∵ d(  ) = t ) For figure 7.5, the lower bound is 2, but d(  ) =3

7.2 The Diagram of Reality & Desire Oriented label as battery. The positive terminal is always at The tip of the arrow, while the negative terminal is at the tail.

7.2 The Diagram of Reality & Desire Reality and desire lines in breakpoint removal.

7.2 The Diagram of Reality & Desire Construction of a diagram of reality and desire Extended  : Replace labels By terminals : Reality edges : Reality and Desire edges :

7.2 The Diagram of Reality & Desire Diagram of reality and desire

7.2 Theorem 7.1 Let (s,t) and (u,v) be two reality edges characterizing a reversal  with (s,t) preceding (u,v) in the permutation . Then RD(  ) differs from RD(  ) as follows. Reality edges (s,t) and (u,v) are replaced by (s,u) and (t,v). Desire edges remain unchanged. The section of the circle going from node t to node u, including these extremities, in counterclockwise direction, is reversed.

7.2 Theorem 7.1 Reality and desire diagram before reversal indicated by two reality lines.

7.2 Theorem 7.1 Reality and desire diagram of permutation obtained from Figure 7.11 after the indicated reversal.

7.2 Theorem 7.2 Let  be a reversal acting on two reality edges e and f of RD(  ). Then 1.If e and f belong to different cycles, c(  )=c(  )-1 2.If e and f belong to the same cycle and converge, then c(  )=c(  ) 3.If e and f belong to the same cycle and diverge, then c(  )=c(  )+1

7.2 Theorem 7.2 The effect of a reversal in the cycles of a reality-desire graph : (a) edges from different cycles, (b) convergent edges, and (c) divergent edges.

7.2 Theorem 7.2

7.2 Interleaving Graph Good cycle Bad cycle Good comp. => Bad comp. Good component

7.2 Interleaving Graph a b c d e f Reversal by using two divergent edges