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2/7/11 Bafna
Detecting structural variations in genomes
The course so far
• Genetic variations create diversity�– Mutations, recombination �
• A population evolving neutrally under these variations might be in equilibrium conditions�– Hardy Weinberg �– Linkage�
• The evolution under neutral conditions can be modeled reasonably and efficiently by coalescent theory�
• Today: structural variations add another source of variation to the mix. Not easily captured via coalescent theory.�
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Fluoroscent in situ hybridization
• (Cancer genomes show extensive structural variation)�• Historically, larger structural variations (easily
observed under a microscope were commonly studied, mostly in the context of diseases…�
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HapMap and other projects
• The development of molecular techniques (sequencing, genotyping) shifted interest onto smaller scale mutations. They were assumed to be the dominant source of genetic variation �
• It turns out that structural variation in normal populations is more common than assumed. �
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Science 2007
Nature 2008
Topics of discussion
• Structural Variation �– Paired-end sequence analysis�– Statistical Design of experiments�– Detecting DNA lesions against a ‘normal’
background �– Analysis of genotypes for structural
variation (inversion polymorphisms)�– Transcript sequencing �
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Can you analyze paired end reads to detect gene fusion? How much sequencing is sufficient to detect AND resolve structural variation?
Can you detect a tumor cell with a DNA variation against a background of normal DNA?
Can you detect structural variation using only genotypes? What about inversions?
Mechanisms for structural variation
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DNA is double stranded
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ACGAGGTACGATGACG!TGCTCCATGCTACTGC!
ACGAAGTACGATGACG!TGCTTCATGCTACTGC!
5’ 3’ 5’
Homologous chromosomes
5’
• Note: sequence can be read from either strand, but is always read in the 5’ to 3’ orientation. • By convention, the reference is read from the telomere of the p (short) arm to the telomere of the q-arm (forward strand). The reverse complement of it is the reverse strand.
ACGAGGTACGATGACG!Reference from the human genome project
GTCAT! Fragment from the reverse strand
Low Copy Repeats (LCRs)
• About 5% of the reference haploid genome is present in two or more copies (Lupski 2010) �– Also known as segmental duplications�
• LCRs can lead to Non-allelic homologous recombinations�
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Homologous recombination
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Non Allelic Homologous Recombination (NAHR)
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Non-homologous end joining (NHEJ)
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Retrotranspositions
• Reetrotransposons and DNA transposons can insert and delete themselves, leading to s.v.�
• DNA transposons work via endonuclease activity, NHEJ �
• Retrotransposons can also act as repeats for NAHR �
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The breakage fusion bridge cycle, and amplicon formation
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SV events
• Caveat: SVs are poorly understood, and none of the following is ‘textbook’ material.�
• We consider the following topics�– Sequence based signatures for SVs�– Fine-mapping of SV breakpoints�– Clustering of reads supporting a single SV �– Signatures of common events�– Design issues�– Pop. Genetic parameter changes.�
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Detecting ‘simple’ SVs
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End Sequence Profiling (ESP)* , or Paired End Mapping (PEM)
1) Pieces of donor/sample genome: clones
Human DNA
2) Sequence ends of clones.
3) Map end sequences to human (reference) genome.
Donor/Sample DNA
Each clone corresponds to pair of end sequences (ES pair) (x,y). Retain clones that correspond to a unique ES pair.
y x
• Colin Colins and S. Volik (UCSF) • Eichler & colleagues
PES signatures of structural variation
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19 OCTOBER 2007 VOL 318 SCIENCE
PEM data
• The paired-ends of a clone help identify deformities/ structural variation in the donor genome.�
• Some SVs are copy neutral (inversions), while others are copy number variant (deletions/duplications).�
• Besides raw detection, there a re a number of problems that we might want to solve computationally.�
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Fine-mapping breakpoints of structural variations
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Paired Ends and Gene Fusions
Tumor Genome
L= (a-xC) + (b-yC)
yC a b xC
Gene V Gene U Normal Genome
Fused Gene Pair
Does this clone indicate a gene fusion event?
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Breakpoints
Tumor Genome
L= (a-xC) + (b-yC)
yC a b xC
Gene V Gene U Normal Genome
• Breakpoint: Pair of points (a,b) in the reference that comes together in the query.�• Paired end mapping does not pinpoint the breakpoint.�
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L= (a-xC) + (b-yC)
yC a b xC
Gene V Gene U Normal Genome
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Genom
ic Position
Clone size = 0
Clone size = LMin
Clone size = LMax
xC
yC
a
b
Gene V
Gene U
L= (a-xC) + (b-yC)
yC a b xC
Gene V Gene U Normal Genome
2/7/11 Bafna Genomic Position
Genom
ic Position
Clone size = 0
Clone size = LMin
Clone size = LMax
xC
yC
a
b
Gene V
Gene U
L= (a-xC) + (b-yC)
yC a b xC
Gene V Gene U Normal Genome
Probability (Gene fusion given aberrant PEM) = fraction of the quadrilateral that lies within the rectangle.
Proof: • We know that the breakpoint must lie within the quadrilateral. Why? • All points are equally likely. • If it fell in the rectangle, we’d have a fusion.
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Reality: Distribution of Sizes
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€
W.l.o.g, a ≥ xC ,b ≥ yCLength of clone lC (a,b) = (a + b) − (xC + yC )
• Let C(a,b) be the event that clone C overlaps breakpoint (a,b). • The event C(a,b) implies the length lC(a,b)
€
Pr(C(a,b )) =1
NC (a + b)Pr(LC = lC (a,b))
2/7/11 Bafna Genomic Position
Genom
ic Position
Clone size = 0
Clone size = LMin
Clone size = LMax
xC
yC
a
b
Gene V
Gene U
LC= (a-xC) + (b-yC)
1 over the length of the clone
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Results: NTNG1-BCAS1
Overlap = 70%
Probability of Fusion = .99
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Results BCAS3-BCAS4
Probability of Fusion = 1
Note: More precise sizing information available for some clones
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• † indicates that a single clone contained more than two chromosomal segments�
Getting CNVs out of sequence data
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Complex rearrangements in tumors
• Research Question: Can you identify the architecture, or gene disruptions?�
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A flow based approach to architecture
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Donor
Reference
• Suppose the donor is sampled to 40X coverage. • Each haploid chromosome is sampled to ~20X. Represented by • How will the counts appear on the reference?
A flow based approach to architecture
• Can we reconstruct the haplotype based architecture?�
• Can we get the true change in copy numbers�2/7/11 Bafna
Donor
Reference
A flow based approach to architecture
• Partition the reference into blocks s.t. there are no discordant reads within blocks, and the copy counts are somewhat uniform.�
• Replace each block by a pair of nodes (s,t)�
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Reference
A flow based approach to architecture
• Replace each block by a pair of nodes (s,t)�• Connect edges (arcs)�
– Within a block (st), adjacent blocks, linked blocks�• Weigh each edge by the strength of the links�
– Most edges have weight ~40, but 3s3t is ~60, and a few have ~20 �
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Reference
1 2 3 4
1s 1t 2t 2s 3s 3t 4t 4s 18
23 21
59
A flow based approach to architecture
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Reference
1 2 3 4
1s 1t 2t 2s 3s 3t 4t 4s 18
23 21
59
• Think of this as pipes (with constraints) �• Add a start and end node, and send a flow of 40 units.�• Can you send a flow so that all constraints are satisfied�
A flow based approach to architecture
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1s 1t 2t 2s 3s 3t 4t 4s 18
23 21
59
1s 1t 2t 2s 3s 3t 4t 4s
1s 1t 2t 2s 3s 3t 4t 4s 18
23 21
Possible decomposition: all edges have a flow of ~20 after the partition
Computing such flows is a standard problem in CS
• In this case, the flows help in correcting the CNV counts, and getting rid of noisy edges�
• They do not unambiguously reconstruct the haplotypes themselves.�
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Flows as LP
• While Flows can be solved as LP, direct, combinatorial techniques exist.�
• It is often worth checking if the optimization formulation can be treated as a flow �
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€
max Fs.t.
A i,e( )e∍i
∑ fe = 0 (i∉{s,t})
A s,e( )e∍s
∑ fe = F
fe ≥ 0fe ≤ be
s t i
Capacity of edge e
-1 0 0 0 1 0 -1 0 0 1 0 1 i nodes
edges
A flow based approach to architecture
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1s 1t 2t 2s 3s 3t 4t 4s 18
23 21
59
1s 1t 2t 2s 3s 3t 4t 4s
1s 1t 2t 2s 3s 3t 4t 4s 18
23 21
Possible decomposition: all edges have a flow of ~20 after the partition
Minimum cost flow problem for genome architecture
• Note that the solution does not give the architecture, only a refined set of counts.�
• Can be used to infer possible architectures.�• Can be used to identify copy number variations�
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€
min fe − cee∑
s.t.
A i,e( )e∍i
∑ fe = 0 (i∉{s,t})
A s,e( )e∍s
∑ fe = F
fe ≥ 0
€
min xee∑
s.t.
A i,e( )e∍i
∑ fe = 0 (i∉{s,t})
A s,e( )e∍s
∑ fe = F
xe ≥ fe − cexe ≥ ce − fexe ≥ 0fe ≥ 0
Design issues for s.v. detection
• In the early days of sequencing, Lander and waterman came up with clean solutions to common sequencing problems�
• In spite of the idealization, their models were widely used in practice. �
• If you wanted to detect SVs, how much sequencing should you do?�
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Lander Waterman Statistics
G
L €
G = Genome LengthL = Clone LengthN = Number of ClonesT = Required Overlapc = Coverage = LN/Gα = N/Gθ = T/Lσ = 1-θ
Island
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The clark-carbon formula
• What is the probability that an arbitrary position is not sequenced?�
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€
Pr[Position x is not sequenced] = 1− LG
⎛
⎝ ⎜
⎞
⎠ ⎟ N
≅ e−NLG = e−c
LW statistics: sample questions
• As the coverage c increases, more and more areas of the genome are likely to be covered. Ideally, you want to see 1 island.�
• Q1: What is the expected number of islands?
• Ans: N exp(-cσ) (why?)
• The number increases at first, and gradually decreases.
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Design related questions for s.v.
• Q: How much sequencing would one need to do in order to uncover most genomic SV events?�
• If you can choose clone lengths�– What is the optimal mix of clone lengths for
detecting and resolving s.v.?�
Breakpoints are key to SV detection
!"#$
%&$%'$"&$$$"'$
(&$!%#$
($
"$ %$
!)#$
*+,-.$
/,0,-,1),$$
!2#$
3$
l!
SVs are typically detected when 1.! Clones span breakpoints
2.! Clone insert-size L << SV length l!
"&$$$"'$ %&$$$$$$$$$$$$$$$$$$$%'$
l!
l!
3$
3$
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Breakpoint detection and resolution
• Detection �– Compute Pr[ ξ is detected | G,N,L)�
• Resolution �– Let Θξ be the set of bp in which ξ can lie�– Resolution is given by |Θξ|�– Compute Exp(|Θξ| | G,N,L)�
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Experiment design for s.v. detection
€
Pξ =1− e−c
• Probability that a breakpoint is covered is given directly by LW statistics.�
• The expected resolution of breakpoint is also computed.�
• Tradeoff with clone length: �– higher clone length implies easier
detection but poorer resolution �
• Explicit formula for probability of covering within a certain resolution �
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€
E(θξ | ξ is covered) =2Lc−
2Lec −1
breakpoint �
• Let A be the right end-point of the left-most clone�
• Pr (A>s and breakpoint is covered)�
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A=s
€
1− sG
⎛
⎝ ⎜
⎞
⎠ ⎟ N
1− 1− L − sG
⎛
⎝ ⎜
⎞
⎠ ⎟ N⎛
⎝ ⎜
⎞
⎠ ⎟ = e
−sNG 1− e
−L−s( )NG
⎛
⎝ ⎜
⎞
⎠ ⎟
= e−sNG − e
−LNG = e
−scL − e−c
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€
Pr(A > s |ξ) =e−scL − e−c
1− e−c
€
Exp(A |ξ) = Pr(A > s |ξ)ds0
L
∫ =e−scL − e−cds
0
L
∫1− e−c
=Lc−
Lec −1
=GN−
L
eNLG −1
Detection resolution tradeoff
0 1 2 3 4 5 6 7 8 9 10
x 107
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
N
P! (" 2
clo
nes)
P! for Paired!End Libraries, Requiring 2 clones spanning !
L = 600
L = 804
L = 1179
L = 1711
L = 2841
L = 3469
L = 2350
0 1 2 3 4 5 6 7 8 9 10
x 107
0
500
1000
1500
2000
2500
N
E(|
!" |
) (#
2 c
lon
es)
E(| !" |) for Paired!End Libraries, Requiring 2 clones spanning "
L = 600
L = 804
L = 1179
L = 1711
L = 2841
L = 3469
L = 2350
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How much sequencing?
Note * indicates that at least two clones span the breakpoint �
DETECTION OF SV UNDER
HETEROGENEITY
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Gene-fusion in tumor genomes
• Fusion observed in leukemia, lymphoma, and sarcomas�
• “Philadelphia Translocation” Drugs target this fusion protein �
• Can we detect fused genes in tumors?�
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Assaying for Rare Variants
• PCR can be used to assay for a given genomic abnormality, even in a heterogenous population of tumor and normal cells�
Extract Genomic DNA
PCR
Distance too large for amplification Tumor cell
Detection
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Polymerase Chain Reaction
• PCR is a technique for amplifying and detecting a specific portion of the genome�
• Amplification takes place if the primers are ‘appropriate’ distance apart (<2kb)�
PCR is DNA-copier
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ACGGATACGATCAGATAAGTAGATCGTGTGTGGG TGCCTATGCTAGTCTATTCATCTAGCACACACCC
ACGGATACGATCAGATAAGTAGATCGTGTGTGGG
TGCCTATGCTAGTCTATTCATCTAGCACACACCC TGTGGG ACGGATACGATCAGATAAGTAGATCGTG
TGCTAGTCTATTCATCTAGCACACACCC
ACGGATACGATCAGATAAGTAGATCGTGTGTGGG
TGCCTATGCTAGTCTATTCATCTAGCACACACCC heat
TGCCTA primer
2000 bp can be amplified
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Variant Variants
• The detection is not as simple when deletion boundaries are uncertain �
Deletion
Deletion
Deletion
Patient A
Patient B
Patient C
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Observed variation in deletion size
Sizes of homozygous deletions in cell lines from different human cancers. (scale is in megabases).
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Primer Approximation Multiplex PCR (PAMP)*
• Multiple primers are optimally spaced, flanking a breakpoint of interest �– Upstream of breakpoint, forward primers�– Downstream of breakpoint, reverse primers�
• The primers are run in a multiplex PCR reaction �– Any pair can form a viable product �
Deletion Deletion
Patient B Patient C
* Liu & Carson, PLoS ONE
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Experimental Design (500Kb region)
• 10 sets of 25 primers: upstream and downstream�– 250 upstream�– 250 downstream�
• Primer-pairs closest to breakpoint amplified�
• Assay by oligo array �
Goal: Computational selection of an ‘optimal’ primer set �
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The computational problem
• Recall that the primers are substrings of the genome that prime the PC Reaction.�
• A primer design refers to a selection (subset) of substrings�
• Can we design the primers so that the reaction fails in very few patients.�• False negatives are worse than false positives! �
• What are the criteria that primers must satisfy so that PAMP succeeds?�
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Primer design criterion I
• Candidate primers must satisfy: �– melting temperature�– GC content �– Uniqueness in the
genome�– Etc�
• Well studied problem, and we can use standard tools to generate a candidate primer set (about 5000 primers in a 500K region).�
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Design criterion II: coverage
• Coverage�– Let 2d be the maximum size of an amplifiable product.�– If the distance between two forward (reverse) primers exceeds d, some
deletion boundaries will not be amplified.�• We measure coverage cost as D-d. This cost must be minimized�
Long product not amplified
deletion
D d
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Design Criterion III: Dimerization
• Primers that cross-dimerize will�1. Not participate in genomic amplification �2. Consume PCR reagent �3. Are ubiquitous: A candidate set of 5000 primers has ~24000
dimerizing pairs�
3'CTCAACAGGTGTTACATGACCTAGCACTGACCCTG- ! ||||||||||||||||||||! 5'-GTTTCCCAGTCACGATCGAGTTGTCCACAATGTACTG-3'!
3'- GGTGTAAGACGCGAACCTATCTAGCACT….! |||||||! 5'-GTTTCCCAGTCACGATCCATTTCCCGCCACCCACATT-3'!
3'-GGTTACACCCTACTCTGATT-5'! ||||x||x|||x|! 5'-AACAAACCCAACGTCGGAAG-3'!
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Primer Set (Size)
Mut: Wild-type
Left Probe Signal?
Right Probe Signal?
Other Probe Signal?
Initial (20) 0: 1 No No No Initial (20) 1 : 49 Yes Yes No Initial (20) 1 : 9 Yes Yes No Initial (20) 1: 0 Yes Yes No Initial+ Repeats (25)
1 : 49 Yes Yes No
Initial+ Repeats (25)
1 : 9 Yes Yes No
Initial+ Repeats (25)
1 : 0 Yes Yes No
Initial+ Primer Dimers (28)
1:9 No No No
Initial+Primer Dimers (28)
1:0 No No No
Left Probe
Right Probe
Amplification using dimerizing primers
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Design Criterion V: cost
• The cost is directly proportional to the number of primers.�
• Define Primer density ρ�– ρ = (# primers)/(length of region in kb)�
• As we need primers to be about 1kb apart, we must choose ρ >=1 �
• Other measures of cost exist, including protocol complexity, and efficiency of reactions. These will be included later.�
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Computational Goal
• Choose (compute) a low-cost, high-coverage, unique, and non-dimerizing subset from a candidate set of physiochemically appropriate primers �
Note on the problem
• In this case the optimization is very important.�– The optimization seeks to minimize the number of
breakpoints that missed�– Each missed breakpoint represents a tumor that we did
not detect �– False negatives are more important than false positives�
• We use general approaches to solving these problems�• We design algorithms based on simulated annealing (fast
convergence, but nor guaranteed optimality), and Integer Linear Programming (guaranteed optimality, but not running time) �
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ILP formulation for PAMP
x1 X2 = 0 x3
P31=1 if (i,j) dimerize
• Also consider other formulations for the problem�
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Simulated annealing
• Start with a current solution {1,2,4,5,7,10,11}, with cost proportional to uncovered region �
• A neighboring solution is obtained by adding a primer, and deleting all its dimerizing partners.�
• Goal is to go through the space of all possible solutions, till we find one of an optimal cost �
1
cost
1 2 3 5 4 7 9 8 11 10 6
The general problem
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1
cost
1 2 3 5 4 7 9 8 11 10 6 1 2 3 5 4 7 9 8 11 10 6
• We are given a very large graph. • Each node has a cost associated with it. • We want to traverse the graph to find the node with minimum cost.
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Simulated annealing
• From current solution, we move to a new solution with probability ~exp(-ΔC)�
• Non favorable solutions are also considered with low probability�
S1 S2 S3 …
Cost
ΔC�
Simulated annealing
• Choose initial Temperature (high)�• Choose an initial state s�• Iterate�
– Choose a neighbor s’ at random�– Define�– Move to s’ with Prob �
– Reduce T �2/7/11 Bafna
€
Δ(s,s') = C(s') −C(s)
€
P =1 (C(s) > C(s'))
e−Δ (s,s' )
T otherwise⎧ ⎨ ⎩
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Results: the CDKN2A region
• The best solution corresponds to 17,846 bp not covered (> 97% coverage)�• Another solution (better than the naïve greedy) corresponds to 103,857 bp
not covered (< 80% coverage)�
PAMP in multiple cell-lines
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21,879,447
21,983,814
CACACACACAC CCCCAAAAGAT
17 bp
(unmapped)
21,879,245 21,983,832
21,878,490 21,983,96121,818,384
21,982,809
ACACCGAAGG ACTTTTGCTG
21,817,319 21,983,088
21,817,281 21,983,17521,817,940
L M R
inversion
21,897,451 21,897,127
21,822,462 22,113,318
CTAGATGAAA CTTTCCAATTAACCCTTACTGTT CTCCTGAGTA21,821,842
22,113,50021,822,123 21,974,311
22,113,527 LM
R
CEM
A549
MOLT4
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Preliminary conclusions
• Prelim. results are very encouraging for small regions (< 1Mb). Many improvements are possible.�
• Clearly, coverage and cost are the two most important criteria.�– Higher coverage implies more patients can be sampled.�
• Cost depends upon the number of primers, protocol complexity, and miniaturization. The final design must optimize cost and protocol complexity as well.�
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Detecting SVs using genotypes
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3. Inversions and HapMap
A A A
A
A
A C C
G G
G
G
T
T T
G
G G A G
A
A A
G A
A G G A
C T C A
• SNP data is seemingly oblivious to Inversions (& other structural polymorphisms)�
• Can we detect a signal for inversion polymorphisms in SNPs?�
reference genome sequence
population
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Fan et al. Nature Reviews Genetics 7, 632–644 (August 2006) | doi:10.1038/nrg1901
SNP-CGH
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Chr. 17 Inversion
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Inversion polymorphisms
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Common inverted haplotype
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Conclusion
• Structural variations are prevalent in normal and disease genomes�
• A variety of techniques can be used to probe for structural variations, �
• Computation helps increase the power�
Stepping back….
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• Variation in DNA mediates phenotypes. • DNA is inherited.
Many sources of variation: • Mutation • Recombination • Structural variation
o HW Equilibrium o Genealogy is given by a tree
o LD
o Coalescent theory o SV detection using paired-end mapping o Design of experiments (LW statistics for SV detection) o Rare SVs
o Combinatorial optimization o LP/ ILP o Simulated annealing
o Flows
o Departure from neutrality
