1. Genome Rearrangement and Duplication Distance
Crystal L. Kahn
9/18/08

2. Genome Rearrangement
Over course of evolution, genomes undergo large structural changes
Chromosomal fissions, fusions, inversions, transpositions
Genome rearrangement is an area of computational biology that uses parsimony* methods to compute “distances” between pairs of genomes
Characterize similarity between genomes by quantifying number of operations required to transform one into another
Not interested in point mutations (SNPs) -- different than edit distance
* Maximum likelihood methods can also be used

3. Genome Rearrangements
Humans and mice have similar genomes, but their genes are ordered differently
~245 rearrangements
~ 300 large synteny blocks

4. History of Chromosome X
Rat Consortium, Nature, 2004
Rearrangement Events:
Reversals
Fusions
Fissions
Translocation

5. Genome Rearrangement Models
Types of rearrangement operations that have been considered:
Reversal (Inversion) [HP, STOC95], [Bader et al., WADS01]
Translocation [Hannenhalli, DAM95]
Duplication transposition [El-Mabrouk, JCSS02]
Ultimate goal: generic genome rearrangement model that allows any type of rearrangement G1 G1 G2
Duplications
common
in cancer G2

6. Duplication Distance: DX(Z,Y)
Input strings X, Y, Z (X non-ambiguous)
Def: duplication operation, Z°s,t,p(X) X Z
s t p
Problem: Compute DX(Z,Y) = min number duplication operations to transform Z into Y
Theorem: O(n4) algorithm, n = |Y|

7. Definitions T = abcdefg  = bcd  = ace String: sequence of characters
Substring: contiguous sequence of characters
Subsequence: sequence of characters, not necessarily contiguous
Note: a substring is a subsequence, but not necessarily vice versa
T = abcdefg
 = bcd
 = ace

8. Key Insight W.L.O.G., let Z = ØX
a b c d e f g h i j k l m n o p q r s
“overlapping” Y
a b c d j k c d e f l o p q
a b c d c d j e k f l o p q
Observation: overlapping subsequences interfere with each other
Lemma: a set of subsequences that are substrings of X and that cover all the characters of Y can be converted into a sequence of duplicate operations iff they are mutually non-overlapping
“Feasible set”

9. Finding min-cardinality feasible set for Ys,t
Let  be element of feasible set that includes index s
2 Cases:
 includes index t
 does not include index t Y s t 
Ys,t Y s t 
Ys,t

10. Let d(Ys,t) = DX(Ø,Ys,t)
where
Case 1
Ys,t
and
Case 2
Ys,t

11. Assume, by induction, already computed
Ys,t
Assume, by induction, already computed
Substring of X
“internal substrings” of 
placements of Xs,t in Ys,t
Xs,t = abcd
Ys,t = abcbccabcd
Ys,t \ 
Ys,t = abcbccabcd
Ys,t = abcbccabcd
Ys,t = abcbccabcd
Ys,t = abcbccabcd
Possibly exponential number of “placements”
as,t computed with second recurrence in O(n2) time

12. Assume, by induction, already computed
Ys,t
Assume, by induction, already computed
bs,t computed in O(n) time

13. Running Time n = |Y| For a substring Ys,t:
Computing as,t takes O(n2) time
Computing bs,t takes O(n) time
Total of O(n2) substrings of Y
Total running time: O(n4)

14. Duplication Transposition vs. Duplication
s t p n
G ° s,t,p
s t (p-1) p n G
Duplication transposition:
“paste” into same string
s < t < p
s t n
G ° s,t,p(G)
1 s t (p-1) p n G
p n
Duplication:
“paste” into another string

15. Duplication can be more complicated…
s t n G
p n G
s (p-1) p t n
G ° s,t,p(G)
s < p < t

16. Duplication Transposition Distance in Semi-Ambiguous Genomes
[El-Mabrouk, JCSS02] incorrectly computes duplication transposition distance
Implication in paper is that:
Given X non-ambiguous and Y semi-ambiguous, DT(X,Y) = # maximal repeated segments of Y
Counterexample:
X = abcdefg
Y = abdecdbcefg
Y0 = abcdefg
Y1 = abcdbcefg
Y2 = abdcdbcefg
Y3 = abdecdbcefg

17. A Lower Bound for Duplication Transposition Distance
Lemma: If Y has at most 2 copies of every character, X is non-ambiguous, and X is a subsequence of Y, then DX(X,Y)  DT(X,Y)
There is still no known algorithm for duplication transposition distance

18. Conclusions
Duplication Distance is a simple model for genome rearrangement and can be computed efficiently.
In a special case, it provides a lower bound to duplication transposition distance
Thank you!
Questions?

19. New Model for Cancer Mutation: Amplisomes
Can show that minimum amplisome distance can be reframed as:
min [DG(A,Ø) + DA(T,A)]
where min is taken over all possible choices of A A
Duplication
Distance
is subproblem

20. Tumor Amplisomes (Maurer, et al. 1987; Wahl, 1989…) Other terms:
Episome
Amplicon
Double-minute 20

21. DX(X,Y) ≤ DT(X,Y) when Y is semi-ambiguous Why is semi-ambiguity necessary?
Semi-ambiguity ensures that all copied substrings are substrings of original X (not some intermediate) -- so for every DT operation, there exists a duplicate operation that produces the same result
Example:
X = A
Y = AAAAAAAA
DT(X,Y) = 3
DX(X,Y) = 7