Making the most of DArT data for phylogenetic inference

Barbara Holland&

Michael Woodhams(Maths & Physics)

Dorothy Steane(Plant Science)

Vincent Moulton(Computational Biology)

1: Collect DNA from reference individuals

2. Digest with one 6bp rare cutter (CTGCAG) and one 4bp frequent cutter (TCGA)

3. Only fragments with two rare ends are amplified and retained

4. Create a microarray with these fragments (~2-3% of the genome)

5: Analyse phylogenetic samples by digesting them with the same cutters and running them against the microarray (DNA-DNA hybridisation).

Each fragment is scored 1 (present) or 0 (absent) *

*This is in math fantasy land – in real life you also get ?s

Generating DArTsDDiversityArArrayTTechnologies

Properties of DArT data

Data is binary (fragments are present or absent, 1/0)

A random set of fragments from across the genome.

Fragments are much more likely to be lost in parallel than gained in parallel

Data exhibit an ascertainment bias: We can observe only the fragments on the chip. These fragments were derived from a small set of reference taxa.

The model

Fragment evolution can be modeled as a stochastic Dollo process, i.e. gained once but lost potentially many times

Parallel gains are forbidden

Fragments are lost at a constant rate r (memoryless)

Chance of loss over time t is 1-exp(-rt)

Hamming Horror

1111111111111111

1111111100000000

1100110000000000

1111000000000000

1000100000000000

1100000000000000

Ref

D

C

B

D(Ref,B)=12/16=(12+0)/(4+0+12+0) D(C,D)=2/16=(1+1)/(1+13+1+1)

Hamming distance D = (n10

+n01

)/(n11

+n00

+n10

+n01

)

Hamming simulation

Tree based on Hamming distancesusing A as the reference taxon

Underlying tree used in simulation

A distance correction is required

R

A B

•Let n00 be the number of fragments absent at both A and B

•Let n01 be the number of fragments absent at A and present at B

•Let n10 be the number of fragments present at A and absent at B

•Let n11 be the number of fragments present at both A and B

A distance correction is required

Michael Woodham's key observation was that, due to the Dollo nature of the process, any fragment that is present at the reference taxon R and at taxon A, must also be present at the internal node X.

R

A B

X

A distance correction is required

Recall, chance of survival over time t is exp(-rt)

d(X,B) = -log[probability fragment survives from X to B]

Anything present at A is known to be present at X

=> d(X,B) = -log[n11

/(n11

+n10

)]

d(A,B) = d(A,X) + d(X,B) = -log[n

11/(n

11+n

01)] - log[n

11/(n

11+n

10)]

= log[(1+n01

/n11

)(1+n10

/n11

)]

R

A B

X

A zoo of distances

Hamming: dH=(n

01+n

10)/(n

11+n

10+n

01+n

00)

Log Det: dLD

=log[det[D]]-0.5Σk(log(C

k)+log(R

k))

Jaccard: dJ=(n

01+n

10)/(n

11+n

10+n

01)

Log Jaccard: dLJ

=-log(1-dJ)=-log[n

11/(n

11+n

10+n

01)]

HS: dHS

=-log[2n11

/(2n11

+n10

+n01

)]

Nei Li: F=2n11

/(2n11

+n10

+n01

);F=Q^2/(2-Q) d

NL=-log(Q)

Simulations

Random (yule) topology,Edge lengths chosen from uniform distribution 0.05<l<0.40

Yule tree, subject to minimum edge length 0.01

Simulation details

• Choose an arbitrary node to start the process at. At this node, the number of DArT fragments is taken from a Poisson distribution with mean M. (We use the result from HS 2004 that a stochastic Dollo process is independent of the root).

• Propagate outward from the start point along tree edges, so that each new node acquires some new DArT fragments and inherits some of those from its parent.

• If the edge length is l, then the probability of a given fragment present in the parent still being present at the end of the edge is exp(-l).

• The number of new fragments in the child but not the parent is Poisson distributed with mean (1-exp(-l))M.

Simulations

R

R

S

R

RS

R

S

T

U

V

Selection of Reference Taxa

One ref, included

Two refs, included

One ref, excluded

Two refs, excluded

All taxa are refs

Simulations

All taxa are references, 9 taxa.

Simulations

Single reference, excluded, 9 taxa. Single reference, included, 9 taxa.

Simulations(distance matrix -> tree by FastME)

Simulations

Simulations

11111 11100 0000011111 11100 0000011111 11100 0000011111 11100 00000

sr

SR

BA00001 11110 0000000001 11110 0000000000 00000 0000000000 00000 00000

00000 01111 1000000000 01111 1000000000 01111 1000000000 01111 10000

00000 00111 1111100000 00111 1111100000 00111 1111100000 00111 11111

00000 00000 0000000000 01111 1000000000 01111 1000000000 00000 00000

00001 11110 0000000001 11110 0000000001 11110 0000000001 11110 00000

Multiple References

If R were the only reference, we'd only see the coloured sites.n

10=6, n

01=2, n

11=2, d(A,B)= -log(2/4) - log(2/8)=3

11111 11100 0000011111 11100 0000011111 11100 0000011111 11100 00000

sr

SR

BA00001 11110 0000000001 11110 0000000000 00000 0000000000 00000 00000

00000 01111 1000000000 01111 1000000000 01111 1000000000 01111 10000

00000 00111 1111100000 00111 1111100000 00111 1111100000 00111 11111

00000 00000 0000000000 01111 1000000000 01111 1000000000 00000 00000

00001 11110 0000000001 11110 0000000001 11110 0000000001 11110 00000

Multiple References

With R and S as references n

10=7, n

01=7, n

11=3, d(A,B)= -2log(3/10)=3.474

Generalizing the DArT Distance

• The DArT distance does less well when there is more than one reference taxon.

• Define

dRDa

(A,B;R)=DArT distance between A and B calculated only from sites that are 1 at R.

• Then dGD

(A,B) is a weighted average:

dGD

(A,B)=(ΣRd

RDa(A,B;R)√n

R)/(Σ

R√n

R)

Partitioned DarT distances(under construction)

• When the reference taxa are known (typically the case)

• And it's also known which fragments come from which reference taxon (not always the case)

• You can define a partitioned DArT distance that takes a weighted average of the DArT distance for each partition.

Simulations

All taxa are references, 9 taxa.

Simulations

DArT, Generalized DArTand HS tree (FastME)

94 Eucalcypt taxa8 reference taxa

Norwich

• Why does the Generalised DarT distance perform so well when the reference taxa are included and so poorly when they are not?

R

A

B

C D

Single reference

pr

pa

pbcd

pb

pcd

pc

pd

Pattern proabilities can be computed by rooting the tree at the reference taxon and then only considering loss of fragments.

E.g. the probability of seeing R 1A 0B 0C 1D 1

is(1-p

r)p

a(1-p

bcd)p

b(1-p

cd)(1-p

c)(1-p

d)

Reference unknown

R B D

n01/n11 0.010 0.010 0.010

n10/n11 0.031 0.020 0.010

D(A,C) 0.040 0.030 0.020D(A,C) = log[(1+n01

/n11

)(1+n10

/n11

)]

R

A

B

C D

pr

pa

pbcd

pb

pcd

pc

pd

Set all edge probabilities to 0.01

R

A

B

C S

Multiple reference taxa

pr

pa

pbcs

pb

pcs

pc

ps

In the multiple reference setting you also have to consider gain of fragments down any edge that is above a reference taxon.

E.g. the probability of seeing R 0A 0B 0C 1S 1

Has 4 termsp

rp

a(1-p

bcs)p

b(1-p

cs)(1-p

c)(1-p

s) +

pbcs

pb(1-p

cs)(1-p

c)(1-p

s) +

pcs

(1-pc)(1-p

s) +

ps

* need to renormalise probabilities

R

A

B

pr

pa

pbs

pb

ps

* need to renormalise probabilities

S

Set all edge probabilities to 0.01

D(A,B) = log[(1+n01

/n11

)(1+n10

/n11

)]

R S R or S

n01/n11 0.010 0.020 0.020

n10/n11 0.020 0.010 0.020

D(A,B) 0.030 0.030 0.040

R S R or S

n00 0.0102 0.0102 0.0203

n01 0.0097 0.0195 0.0196

n10 0.0195 0.0097 0.0196

n11 0.9606 0.9606 0.9702

1.0000 1.0000 1.0297

Future ideas

• The small examples we worked through in Norwich suggest two new ideas to be tested by simulation

• In the case of unknown references, compute D(X,Y|R) for each R and take the max.

• In the case of known references, a modification to the Generalised DArT that only averages over the references

Future work - hybridisation

Links to other peoples work

• Gene content evolution with HGT, aka controlling ancestral genome obesiety (Tal Dagan, Bill Martin)

• Language evolution with borrowing (Geoff Nicholls, Russell Gray)

BIG Thanks to Torsten and Shiju

http://www.maths.utas.edu.au/phylomania/phylomania2011.htm