fast identification and statistical evaluation of segmental homologies in comparative maps peter...

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