Evaluating the Significance of Max-gap Clusters Rose Hoberman David Sankoff Dannie Durand

Evaluating the Significance of

Max-gap Clusters

Rose Hoberman

David Sankoff

Dannie Durand

Gene content and order are preserved

rearrangement, mutation

Similarity in gene content

large scale duplication

or speciation event

original genome

TBOX

Cryb, Lhx, Nos, Tbx, Tcf, Prkar

MHC Abc, C3/4/5, Col, Hsp, Notch, Pbx, Psmb, Rxr, Ten

HOX Achr, Ccnd, Cdc, Cdk, Dlx, En, Evx, Gli, Hh, Hox, If, Inhb, Nhr, Npy/Ppy, Wnt

FGR Adr, Ank, Egr, Fgfr, Pa, Vmat, Lpl

MATN

Eya, Hck, Matn, Myb, Myc, Sdc, Src

… …

Local Spatial Evidence of Duplication

Ruvinsky and Silver, Gene, 97

Chromosome5

Chromosome11

60

65

70

Crybb1/2/3Cryba4

Nos1 50

40

45Tcf1

Tbx3/5

Cryba1, Nos2

Tbx2/4

Tcf2

Lhx1

Lhx5

5575

Gene Clustering for Functional Inference in Bacterial Genomes

The Use of Gene Clusters to Infer Functional Coupling, Overbeek et al., PNAS 96: 2896-2901, 1999.

Clusters are Commonly Used in Genomic Analyses

Genomic Analyses

Blanc et al 2003, recent polyploidy in Arabidopsis

Venter et al 2001, sequence of the human genome

Overbeek et al 1999, inferring functional coupling of genes in bacteria

Vandepoele et al 2002, duplications in Arabidopsis through comparison with rice

Vision et al 2000, duplications in Eukaryotes

Lawrence and Roth 1996, identification of horizontal transfers

Tamames 2001, evolution of gene order conservaiton in prokaryotes

Wolfe and Shields 1997, ancient yeast duplication

McLysaght02, genomic duplication during early chordate evolution

Coghlan and Wolfe 2002, comparing rates of rearrangements

Seoighe and Wolfe 1998, genome rearrangements after duplication in yeast

Chen et al 2004, operon prediction in newly sequenced bacteria

Blanchette et al 1999, breakpoints as phylogenetic features...

Clusters are Commonly Used in Genomic Analyses

Genomic Analyses

Blanc et al 2003, recent polyploidy in ArabidopsisVenter et al 2001, sequence of the human genome





Tamames 2001, evolution of gene order conservaiton in prokaryotes







Need to assess cluster significance

Given: a genome: G = 1, …, n unique genes

a set of m special genes

Can we find a significant cluster of (a subset of) the

m homologs?

Can we find a significant cluster of the red genes?

How do we formally define a cluster?

Given: a genome: G = 1, …, n unique genes

a set of m special genes

Possible Cluster Parameters– size: number of red genes in the cluster

• Example: cluster size ≥ 3

size = 3 genes



– length: number of genes between first and last red gene

• Example: cluster length ≤ 6

length = 6



– length: number of genes between first and last red gene


length = 6



– length: number of genes between first and last red genes


– density: proportion of red genes (size/length)• Example: density ≥ 0.5

density = 6/11





– density: proportion of red genes (size/length)• Example: density ≥ 0.5

density = 6/11





– density: proportion of red genes (size/length)– compactness: maximum gap between adjacent red

genes

gap ≤ 4 genes

Max-Gap Cluster

• Commonly used in genomic analyses

• Expandable, ensures minimum local and global density

• Efficient algorithm to find them (Bergeron, Corteel, Raffinot 2002)

gap g

Max-Gap Clusters are Commonly Used in Genomic Analyses

Genomic Analyses

Blanc et al 2003, recent polyploidy in ArabidopsisVenter et al 2001, sequence of the human genome





Tamames 2001, evolution of gene order conservation in prokaryotes







Statistical models

Statistical Models

Calabrese03

Danchin03

Durand03

Ehrlich97

Trachtulec01

HumanGenomeScience

A gap between statistical tests and cluster definitions used in practice

Genomic AnalysesBlanc et al 2003 Venter et al 2001

Overbeek et al 1999

Vandepoele et al 2002

Vision et al 2000

Lawrence and Roth 1996

Tamames 2001

Wolfe and Shields 1997 McLysaght02

Coghlan and Wolfe 2002

Seoighe and Wolfe 1998

Chen et al 2004

Blanchette et al 1999...

Statistical Models

Calabrese03

Danchin03

Durand03b

Ehrlich97

Trachtulec01

HumanGenomeScience•no analytical statistical

model for max-gap clusters•statistical significance

assessed through Monte Carlo randomization

A gap between statistical tests and cluster definitions used in practice

Genomic AnalysesBlanc et al 2003 Venter et al 2001

Overbeek et al 1999

Vandepoele et al 2002

Vision et al 2000

Lawrence and Roth 1996

Tamames 2001

Wolfe and Shields 1997 McLysaght02

Coghlan and Wolfe 2002

Seoighe and Wolfe 1998

Chen et al 2004

Blanchette et al 1999...

Statistical Models

Calabrese03

Danchin03

Durand03b

Ehrlich97

Trachtulec01

HumanGenomeScience

This Talk

• Analytical statistical tests– reference region model with m genes of interest– complete and incomplete clusters

• Relationship between cluster parameters and significance

• Future extension to whole-genome comparison

Outline

• Introduction

• Complete Clusters

• Incomplete Clusters

• Genome Comparison

• Open problems

Complete Clusters• Given– a genome: G = 1, …, n unique genes – a set of m special genes (the “red” genes)– a maximum-gap size g

• Null hypothesis: Random gene order

• Alternate hypotheses:Evolutionary historyFunctional selection

• Test statistic– the probability of observing all m genes in a max-gap

cluster in G ?

Probability of a Complete Cluster

Count how many of the permutations of m red genes and n-m blue genes contain a max-gap cluster

1. At how many positions in the genome can we locate a cluster (e.g. place the leftmost red gene)?

2. Given the location of the first red gene, how many ways are there to place the remaining m-1 red genes so that they form a max-gap cluster?

w = (m-1)g + m

ways to choose

m-1 gapsways to place the

first gene and still

have w-1 slots left

edge

effects

w-1

For clusters at the end of the genome:– Length of the cluster is constrained– Sum of the gaps is constrained:

Counting clusters at the end of the genome

How many ways can we form a max-gap cluster of size m with length ≤ l ?

g1 g2 g3 gm-1

l < w

g1 g2 g3 gm-1

l < w

=

Number of ways of choosing m-1 gapsbetween 0 and g so sum ≤ l-m

Number of ways of rolling m-1 dice with faces labeled 0 to g so faces total ≤ l-m


g1 g2 g3 gm-1

=

Number of ways of choosing m-1 gapsbetween 0 and g so sum ≤ l-m

Number of ways of rolling m-1 dice with faces labeled 0 to g so faces total ≤ l-m


Accounting for edge effects is the least efficient part of calculation– Eliminate for faster approximation when w << n

Now we can calculate probabilities

for various parameter values

Final Probability

Probability of Observing a Complete Cluster

g

m n = 500

Significant Parameter Values (α = 0.0001)

n = 500

Significant Parameter Values (α = 0.0001)

n = 500

Outline

• Motivation




• Open problems

Incomplete clusters

• Is it significant to find a max-gap cluster of size h < m?

• Test statistic: probability of finding at least one cluster

Given: m genes of interest

h=3

m=5, g=1

Enumerating clusters by starting position will lead to overcounting of permutations that include more than one cluster

Probability of incomplete gene clusters

m = 6, h = 3, g = 1

Alternative Approach

• Enumerate all permutations that do not contain any clusters of size h or larger

• Dynamic programming – Iteratively place “red” or “blue” genes making sure

not to create any cluster of size h or larger by judicious placement of “blue genes”

k = 4c = 0, j =

8

r = 6

Dynamic programming Algorithmr total number of genes to place c size of cluster so far

k number of special genes j distance to previous cluster

n = 6, m = 4, h = 3, g=1

k = 4c = 0, j =

8

r = 6

c = 1, j = 0 c = 0, j =

8 k = 4r = 5k = 3

r = 5



n = 6, m = 4, h = 3, g=1

k = 2r = 4

k = 4c = 0, j =

8

r = 6

c = 1, j = 0 c = 0, j =

8 k = 4r = 5k = 3

r = 5

c = 1, j = 1 k = 3r = 4

c = 1, j = 0



n = 6, m = 4, h = 3, g=1

k = 2r = 4

k = 4c = 0, j =

8

r = 6

c = 1, j = 0 c = 0, j =

8 k = 4r = 5k = 3

r = 5

c = 1, j = 1 k = 3r = 4

c = 1, j = 0

k = 2r = 3

c = 2, j = 0



X

n = 6, m = 4, h = 3, g=1

Incomplete cluster significance

h g

m ( )

significant region of parameter space shown in paper

n = 1000

m = 50

n = 500

Outline

• Motivation



• Extensions


Possible Extensions

• Tandem duplications

• Gene families

• Extensions for prokaryotic genomes– Gene orientation– Circular genomes– Physical distance between genes

• Whole genome comparison

Whole genome comparison

Assuming identical gene content …

What is the probability that at least k genes form a max-gap cluster

in both genomes?

g 30

g 30

The probability of finding a max-gap cluster of size at least k is always one

An Odd Property

Example: g =1


Example: g =1

An Odd Property


Example: g =1

gap gap

An Odd Property

A cluster of size k does not necessarily

contain a cluster of size k-1!

gap

The probability of finding a max-gap cluster of size at least k is always one There will always be a cluster of size n

Example: g =1

An Odd Property

Conclusions

Presented statistical tests for max-gap clusters– Evaluate the significance of clusters of a pre-

specified set of genes– Choose parameters effectively– Understand trends

Thank You

Documents

Evaluating the Significance of Max-gap Clusters Rose Hoberman David Sankoff Dannie Durand