Fast identification and statistical evaluation of segmental

homologies in comparative maps

Peter Calabrese1, Sugata Chakravarty2 and Todd Vision3

1Department of Mathematics, University of Southern California; Departments of 2Operations Research and 3Biology

University of North Carolina at Chapel Hill

comparative maps Spaghetti Diagram Crop Circle

Livingstone et al 1999 Genetics 152:1183Gale & Devos 1998 PNAS 95:1972

some terms

• Feature: a gene or some other marker

• Segment: a string or substring of features

• Homology: descent from a common ancestor

• Block: a pair of segments that are putatively homologous. These are what we seek!

local genome alignment

• Consider each chromosome to be a string of features

• Assign common letters to homologous features • Identify segments sharing multiple pairs of

common letters• Differences from DNA/protein alignment

– high frequency of gaps relative to matches– inversions may occur within the alignment

homology matrix

gene 1 2 3 4 5 6 7 8 1 - 0 0 0 1 0 0 0 2 0 - 0 0 0 1 0 0 3 0 0 - 0 0 0 1 0 4 0 0 0 - 0 0 0 1 5 1 0 0 0 - 0 0 0 6 0 1 0 0 0 - 0 0 7 0 0 1 0 0 0 - 0 8 0 0 0 1 0 0 0 -

1 2 3 4

5 6 7 8

duplication and multiplication

there is not necessarily a one-to-one alignment

genome rearrangements

inversion reciprocal translocation

homologous segments may be small

Bancroft (2001) TIG 17, 89 after Ku et al (2000) PNAS 97, 9121

Non-homologous features may be abundant within homologous segments

We must allow some non-colinearityin marker order between segmental

homologs

homology matrix for Arabidopsis

going beyond eyeballing

• LineUp – Hampson et al (2003)– Designed for genetic maps with error

• ADHoRe – Van der Poele (2002)– Designed for unambiguous marker order data

• Both provide automatic detection of blocks• For statistics, both employ permutation tests

– Computationally intensive– p-values are approximate

FISH:Fast Identification of Segmental Homology

• Block identification– Dynamic programming provides speed and

optimality guarantee– Can be generalized to multiple alignments

• Statistical assessment– Null model of duplication and transposition– Closed-form equation for calculating p-

values (i.e. no permutation testing)

from homology matrix to graph

• nodes ()– represent dots in the homology

matrix

from homology matrix to graph

• nodes ()– represent dots in the homology

matrix

• edges ()– connect nodes with nearest

neighbors– are unidirectional– have an associated distance– must be shorter than some

threshold

from homology matrix to graph

• nodes ()– represent dots in the homology matrix

• edges ()– connect nodes with nearest neighbors– are unidirectional– have an associated distance– must be shorter than some threshold

• paths ()– traverse shortest available edges– can be efficiently computed– can be considered candidate blocks

null model

• Within a genome: homologies are due to the duplication of individual features followed by insertion into a (uniformly) random position

• Between genomes: homologies are due to the above process plus the transposition of randomly chosen features into randomly chosen positions.

computing neighborhood size

• h = # nodes / # cells in matrix• n = # cells in neighborhood• Prob(neighborhood has 1 node)

p = 1 – (1-h)n

• Threshold distance for p=T under Manhattan distance (x+y)dT = 0.5 + sqrt[(log(1-p)/log(1-h)+0.25]

• T is analogous to a gap parameter– small T: few false positive edges, short blocks– large T: more false positive edges, longer blocks

neighborhood geometry

blocks of nearest neighbors

block statistics

• Chen-Stein Theorem: number of blocks with k nodes is approximately Poisson

• Expected number of blocks = cpu

• Conservative matrix-wide p-value

Prob(X k) < 1 – e –cpu

where c is the # of cells in the matrix and pu = h(nh)k-1

identifying blocks

• Let edge from i to j have weight wij =1

• Initialize: score of block terminating at i Si = 0

• Recursion for block scores Sj = max(Si + wij) i such that j Ti

• Dynamic programming can be used to find all maximally extended blocks

simulation experiment

k obs stderr upbound lowbound2 45.8 0.06 47.6 40.13 2.28 0.02 2.39 1.784 0.113 0.003 0.120 0.0795 0.006 0.001 0.006 0.0046 0.0003 0.0002 0.0003 0.0002

How often are blocks of size k observed under the null model compared with

expectation?

FISH v.1.0

• http://www.bio.unc.edu/faculty/vision/lab/FISH– source code – compiled executables– documentation– sample data

• Adjustable parameters (e.g. T)• Reports statistics on blocks• Is fast and memory-efficient

applications

• Automated pairwise alignment of genome maps as part of Phytome project

• Prediction of gene content in regions of unsequenced genomes

• Studies of genome evolution, especially duplication and gene order rearrangement

future work

• Biologically motivated neighborhood geometries (ADHoRe)

• Non-discrete marker positions (LineUp)– Genetic versus physical maps– Map uncertainty

• Robustness to deviation from null model (permutation tests)

• Extension to homologies among 3 or more segments

Thanks!

Sugata Chakravarty

Peter Calabrese

U.S. National Science Foundation

http://www.bio.unc.edu/vision/faculty/lab/FISH

Ghosts

Simillion, Vandepoele, Van Montagu, Zabeau, Van de Peer (2002) PNAS 99, 13627