Information Theory of DNA Sequencing

David Tse Dept. of EECSU.C. Berkeley

LIDS Student ConferenceMIT

Feb. 2, 2012

Research supported by NSF Center for Science of Information.

Guy Bresler Abolfazl Motahari

DNA sequencing

DNA: the blueprint of life

Problem: to obtain the sequence of nucleotides.

…ACGTGACTGAGGACCGTGCGACTGAGACTGACTGGGTCTAGCTAGACTACGTTTTATATATATATACGTCGTCGTACTGATGACTAGATTACAGACTGATTTAGATACCTGACTGATTTTAAAAAAATATT…

courtesy: Batzoglou

Impetus: Human Genome Project

1990: Start

2001: Draft

2003: Finished3 billion basepairs

courtesy: Batzoglou

Sequencing Gets Cheaper and Faster

Cost of one human genome• HGP: $ 3 billion• 2004: $30,000,000• 2008: $100,000• 2010: $10,000• 2011: $4,000 • 2012-13: $1,000• ???: $300

courtesy: Batzoglou

But many genomes to sequence

100 million species(e.g. phylogeny)

7 billion individuals (SNP, personal genomics)

1013 cells in a human(e.g. somatic mutations

such as HIV, cancer) courtesy: Batzoglou

Whole Genome Shotgun Sequencing

Reads are assembled to reconstruct the original DNA sequence.

Sequencing Technologies • HGP era: single technology

(Sanger)

• Current: multiple “next generation” technologies (eg. Illumina, SoLiD, Pac Bio, Ion Torrent, etc.)

• All provide massively parallel sequencing.

• Each technology has different read lengths, noise profiles, etc

Assembly Algorithms

• Many proposed algorithms.

• Different algorithms tailored to different technologies.

• Each algorithm deals with the full complexity of the problem while trying to scale well with the massive amount of data.

• Lots of heuristics used in the design.

A Basic Question

• What is the minimum number of reads needed to reconstruct with a given reliability?

• A benchmark for comparing different algorithms.

• An algorithm-independent basis for comparing different technologies and designing new ones.

Coverage Analysis

• Pioneered by Lander-Waterman

• What is the minimum number of reads to ensure there is no gap between the reads with a desired prob.?

• Only provides a lower bound.

• Can one get a tight lower bound?

Communication and Sequencing: An Analogy

Communication:

Sequencing:

Communication: Fundamental Limits

Cchannel = channel capacityHsource = sourceentropy rate

Shannon 48

Asymptotically reliable communication at rate R source symbols per channel output symbol if and only if:

R < C channelH source

Given statistical models for source and channel:

DNA Sequencing: Fundamental Limits?

S1;S2; : : : ;SG R 1;R 2; : : : ;R N

• Define: sequencing rate R = G/N basepairs per read

• Question: can one define a sequencing capacity C such that:

asymptotically reliable reconstruction is possible if and only if R < C?

A Simple Model

• DNA sequence: i.i.d. with distribution p.

• Starting positions of reads are i.i.d. uniform on the DNA sequence.

• Read process is noiseless.

Will extend to more complex source model and noisy read process later.

The read channel

• Capacity depends on

– read length: L

– DNA length: G

• Normalized read length:

• Eg. L = 100, G = 3 £ 109 :

read channelAGCTTATAGGTCCGCATTACC AGGTCC

¹L := LlogG

L ") C "

G ") C #

¹L = 4:6

Result: Sequencing Capacity

H2(p) = ¡ log4X

i=1p2i

Renyi entropy of order 2

C = 0

C = ¹L

Coverage Constraint

TL

Starting positions of reads ~ Poisson(1/R)

E [# of gaps]= N ¢P [T > L]= Ne¡ LR

R = GN

E [# of gaps] ! 0, R < ¹L

G

N reads

No-Duplication Constraint

E [# of duplicated pairs]¼N 2 ¢Ã 4X

i=1p2i

! L

L L L L

E [# of duplicated pairs] ! 0, ¹L > 2

H2(p)

= N 2e¡ L H 2(p)

The two possibilities have the same set of length L subsequences.

Achievability

coverage constraint

no-duplication constraint

achievable?

Greedy Algorithm

Input: the set of N reads of length L

1. Set the initial set of contigs as the reads.

2. Find two contigs with largest overlap and merge them into a new contig.

3. Repeat step 2 until only one contig remains or no more merging can be done.

Algorithm progresses in stages: at stage merge reads at overlap

`= L ¡ 1;L ¡ 2;: : : ;0`

Greedy algorithm: the beginning

gap

Most reads have large overlap with neighbors

Expected # of errors in stage L-1:probability two disjoint

reads are equal

Very small since no-duplication constraint is satisfied.

Greedy algorithm: stage

probability two disjoint reads appear to overlap

Expected # of errors at stage

This may get larger, but no larger than when¡Ne¡ L =R ¢2

Very small since coverage constraint is satisfied.

`= 0

Summary: Two Regimes

coverage-limitedregime

duplication-limitedregime

Relation to Earlier Works

• Coverage constraint: Lander-Waterman 88

• No-duplication constraint: Arratia et al 96

• Arratia et al focused on a model where all length L subsequences are given (seq. by hybridization)

• Our result: the two constraints together are necessary and sufficient for shotgun sequencing.

Rest of Talk

• Impact of read noise.

• Impact of repeats in DNA sequence

Read Noise

Model:

discrete memoryless channel defined by transition probabilities

ACGTCCTATGCGTATGCGTAATGCCACATATTGCTATGCGTAATGCGTTATACTTA

Modified Greedy algorithm

YX

This is a hypothesis testing problem!

We observe two strings: X and Y.

Are they noisy versions of the same DNA subsequence?

Or from two different locations?

Y

Do we merge the two reads at overlap ?

(merge)

(do not merge)

Impact on Sequencing Rate

MAP rule: declare H0 if

• Hypothesis test:

H0: noisy versions of the same DNA subsequence (merge)

H1: from disjoint DNA subsequences (do not merge)

Y

X Y

Two types of error:

• missed detection (new type of error)

• false positive (same as before)

coverage constraint

no-duplication constraint

obtained by optimizing MAP threshold

Impact on Sequencing Rate

More Complex DNA Statistics

• i.i.d. is not a very good model for the DNA sequence.

• More generally, we may want to model it as a correlated random process.

• For short-scale correlation, H2(p) can be replaced by the Renyi entropy rate of the process.

• But for higher mammals, DNA contains long repeats, repeat length comparable or longer than reads.

• This is handled by paired-end reads in practice.

H2 = lim`! 1

¡ 1` logP (x` = y`)

A Simple Model for Repeats

K

Model: M repeats of length K placed uniformly into DNA sequence

If repeat length K>> read length L, how to reconstruct sequence?

Use paired-end reads:

J reads come in pairs with known separationThese reads can bridge the repeats

coverage constraint

no-duplication constraint

Impact on Sequencing Rate

coverage of repeatsconstraint

If J > 2d + Kthen capacityis the same aswithout repeats

constant indep of K

K= repeat lengthJ = paired-end

separation

Conclusion

• DNA sequencing is an important problem.

• Many new technologies and new applications.

• An analogy between sequencing and communication is drawn.

• A notion of sequencing capacity is formulated.

• A principled design framework?