RECOMBINOMICS : Myth or Reality?

RECOMBINOMICS: Myth or Reality?

Laxmi Parida

IBM Watson Research New York, USA

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IBM Computational Biology Center

1. Motivation

2. Reconstructability (Random Graphs Framework)

3. Reconstruction Algorithm (DSR Algorithm)

4. Conclusion

RoadMap

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www.nationalgeographic.com/genographic

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www.ibm.com/genographic

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Five year study, launched in April 2005 to address anthropological questions on a global scale using genetics as a tool

Although fossil records fix human origins in Africa, little is known about the great journey that took Homo sapiens to the far reaches of the earth. How did we, each of us, end up where we are?

Samples all around the world are being collected and the mtDNA and Y-chromosome are being sequenced and analyzed

phylogeographic question

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DNA material in use under unilinear transmission

58 mill bp

0.38%

16000 bp

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Missing information in unilinear transmissions

past

present

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Table MountainCape Town, South Africa

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Paradigm Shift in Locus & Analysis

Using recombining DNA sequences Why?

Nonrecombining gives a partial story

1. represents only a small part of the genome

2. behaves as a single locus

3. unilinear (exclusively male of female) transmission Recombining towards more complete information

Challenges Computationally very complex How to comprehend complex reticulations?

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1. Motivation

2. Reconstructability (Random Graphs Framework)


4. Conclusion

RoadMap

L Parida, Pedigree History: A Reconstructability Perspective using Random-Graphs Framework,

Under preparation.

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GRAPH DEF:1. Infinite number of vertices

arranged in finite sized rows

2. Edges introduced via a random processacross immediate rows

PROPERTIES:Address some topological questions

1. First, identify a Probability Space2. Then, pose and address specific questions

(such as expected depth of LCA etc..)

The Random Graphs Framework

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1. Infinite number of verticeswith a specific organization

2. Edges introduced via a random processsatisfying specific rules

3. Address some topological questions1. Define a Probability Space2. Pose and answer specific questions

(such as expected depth of LCA etc..)


Wright-Fisher Model

1. Constant population 2. Non-overlapping generations3. Panmictic

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Properties of this Pedigree Graph

1. DAG Directed Acyclic Graph

2. |E| = O(|V|) for any finite fragment; sparse graph…Vertex-centric view..

3. Focus on the flow of genetic material: relevant pedigree graph

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Pedigree Graph: GPG(K,N)

K no of extant units 2N population size/generation

Can the model ignore color of vertex?

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Pedigree Graph: GPG(K,N)

K no of extant units 2N population size/generation

Can the model ignore color of vertex?

Forbidden Structure

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Probability Space

Space is non-enumerable

Uniform probability measure?WF pop

Probability of some event F(h) for a fixed depth, h, & take limit:

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Topological Property of GPG(K,N)

Least Common Ancestor (LCA) of ALL (K) extant vertices ------TMRCA or GMRCA-------

How many LCA’s ?

Expected Depth of the shallowest LCA

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Infinite No. of LCA’s in a GPG(4,3) instance …..

In fact, there exist infinite such instances!

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Topological Property of GPG(K,N)

Least Common Ancestor (LCA)------TMRCA or GMRCA-------

How many LCA’s ?

Expected Depth of the shallowest “LCA” MEASURE OF RECONSTRUCTABILITY

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(Genetic Exchange) Sexual Reproduction vs Graph Model

Ancestor without ancestry

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1. Graph Theoretic (topological): CA common ancestor

LCA Least CA or Shallowest CAMRCA Most Recent CATMRCA The MRCA

2. Graph Theoretic + Biology (Genetic Exchange): CAA common ancestor-&-ancestry

LCAA Least CAAGMRCA Grand MRCA

Unilinear Transmission

Graph Theory vis-à-vis Population Genetics

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Different Models as Subgraphs

mtDNA TreeNRY Tree

Genetic Exchange Model (ARG)

Pedigree Graph GPG(K,N)each vertex has 2 parents

1. Red Subgraph GPTX(K,N)Blue Subgraph GPTY(K,N)each vertex has 1 parent

2. Mixed Subgraph GPGE(K,N,M)No of vertices/row no more than KM

each vertex has 1 OR 2 parentsM is no. of completely linked segs in each extant unit

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Different Models

GPG(4,8) GPTY(4,8) GPGE(4,8,2)

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LCA GMRCA

LCA TMRCA

LCA GMRCA

Pedigree Graph GPG(K,N)

1. Red Subgraph GPTX(K,N)Blue Subgraph GPTY(K,N)

2. Mixed Subgraph GPGE(K,N,M)

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GPGE(K,N,M) ARG

Ancestral Recombinations Graph Griffiths & Marjoram ‘97

Embellish GPGE(K,N,M) with Genetic Exchanges (GE) Each extant unit has M segments No vertex with zero ancestral segments (to extant units)

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1. Plausible GE assignment?2. Can GPGE(K,N,M) go colorless?

Yes....through algorithmic subsampling…

Mixed Subgraph GPGE(K,N,M)

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Algorithm: Embellish GPGE(K,N,M)

1. Assign sequence, s, to an instance eg. s = K, (2K), (2K-7), (2K-15), ……….

2. Construct M sequences si

Each si is monotonically decreasing; si[j] no bigger than s[j]

3. Associate each si with a segment and each element si[j] = k to k randomly selected vertices at depth j

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Algorithm: Constructing seqs…

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“Topological” Defn of LCAAin GPGE(K,N,M)

Input: GPGE(K,N,M) with GE embellishment

LCAA1.CA in all M subgraphs (trees)2.Least such CA

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LCAA GMRCA

LCA TMRCA

LCAA GMRCA




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Probability of Instances with Unique LCA/LCAA




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GMRCA LCAA LCA & lone pair

TMRCA LCA

GMRCA LCAA LCA & lone node




“Topological” Defns of LCAA

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Expected Depth E(D) of LCA/LCAA

O(N2)

O(K)

O(KM)




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RECONSTRUCTABILITY

O(N2)

O(K)

O(KM)




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Summary:History Reconstruction?

1. Mixed Subgraph models recombinations Only fragments of the chromosome

2. In reality, only a minimal structure (HUD) of the GPGE(K,N,M) or ARG can be estimated Forbidden structures ….

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1. Motivation

2. Reconstructability (Random Graph Framework)


4. Conclusion

RoadMap

L Parida, M Mele, F Calafell, J Bertranpetit and Genographic Consortium Estimating the Ancestral Recombinations Graph (ARG) as Compatible Networks of SNP Patterns

Journal of Computational Biology, vol 15(9), pp 1—22, 2008

L Parida, A Javed, M Mele, F Calafell, J Bertranpetit and Genographic Consortium, Minimizing Recombinations in Consensus Networks for Phylogeographic Studies, BMC Bioinformatics 2009

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OUTPUT: Recombinational Landscape (Recotypes)

INPUT: Chromosomes (haplotypes)

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Granularity g

Analyze Results

YES

NO

IRiS

Acceptable p-value?

Our Approach

statistical

statistical

combinatorial

M Mele, A Javed, F Calafell, L Parida, J Bertranpetit and Genographic Consortium Recombination-based genomics: a genetic variation analysis in human populations ,

under submission.

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Preprocess: Dimension reduction via Clustering

11 12 13 14 15 16 0

17 1 18 4

19 6 5

20 8 21 9 10 7 22

23 3 2 24

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Granularity g

Analyze Results

YES

NO

IRiS

Acceptable p-value?

Analysis Flow

statistical

statistical

combinatorial

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p-value Estimation

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Comparison of the Randomization Schemes

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SNP Blocks (granularity g=3)

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Granularity g

Analyze Results

YES

NO

IRiS

Acceptable p-value?

Analysis Flow

statistical

statistical

combinatorial

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Stage Haplotypes: use SNP block patterns

Segment along the length: infer trees

Infer network (ARG)

biological insights

computational insights

IRiS(Identifying Recombinations in Sequences)



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Segmentation

12345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345 11111111111111111111111111111111111111112222222222222222222222222222222222233333333344444444455555555555555----

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Segmentation

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Consensus of Trees

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Algorithm Design

1. Ensure compatibility of component trees

2. Parsimony model: minimize the no. of recombinations

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Algorithm Design

1. Ensure compatibility of component trees

2. Parsimony model: minimize the no. of recombinations

Theorem: The problem is NP-Hard.

“It is impossible to design an algorithm that guarantees optimality.”

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DSR Scheme (Dominant—Subdominant---Recombinant)

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DSR Scheme: Level 1

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DSR Assignment Rules

1. At most one D per row and column;

if no D, at most one S per row and column

2. At most one non-R in the row and column, but not both

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DSR Assignment Rules

1. Each row and each columnhas at most one D

ELSE has at most one S

2. A non-R can have other non-Rs either in its row or its column but NOT both

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DSR Scheme: Level 1

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DSR Scheme: Level 2

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DSR Scheme: Level 2

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DSR Scheme: Level 3

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DSR Scheme: Level 3

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DSR Scheme: Level 4

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DSR Scheme: Level 5

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Mathematical Analysis: Approximation Factor

Greedy DSR Scheme Z and Y are computable functions of the input

L Parida, A Javed, M Mele, F Calafell, J Bertranpetit and Genographic Consortium, Minimizing Recombinations in Consensus Networks for Phylogeographic Studies, BMC Bioinformatics 2009

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Granularity g

Analyze Results

YES

NO

IRiS

Acceptable p-value?

Analysis Flow

statistical

statistical

combinatorial

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IRiS Output: RECOTYPE

Recombination vectorsR1 R2 R3 R4 R5 R6 R7 R8 R9 R10 R11 R12 R13 R14 ……….

s1 1 0 0 0 1 1 1 1 0 0 0 0 1 0 ……….

s2 0 1 0 1 1 1 0 1 0 0 1 0 0 0 ……….

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.

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.

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Quick Sanity Check:Ultrametric Network on RECOTYPES

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Stage Haplotypes: use SNP block patterns

Segment along the length: infer trees

Infer network (ARG)

biological insights

computational insights

IRiS(Identifying Recombinations in Sequences)



IRiS software will be released by the end of summer ’09

Asif Javed

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What’s in a name?

1. Allele-frequency variations between populations is also reflected in the purely recombination-based variations

2. Detects subcontinental divide from short segments based on populations level analysis

3. Detects populations from short segments based on recombination events analysis

RECOMBIN-OMICS Jaume Bertranpetit

RECOMBIN-OMETRICS

Robert Elston

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1. Allele-frequency variations between populations is also reflected in the purely recombination-based variations

2. Detects subcontinental divide from short segments based on populations level analysis

3. Detects populations from short segments based on recombination events analysis

Are we ready for the OMICS / OMETRICS?

o population-specific signals ?o other critical signals ?

o anything we didn’t already know?

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Thank you!!

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Documents

RECOMBINOMICS : Myth or Reality?