Guy Grebla 1

Allegro,A new computer program for linkage analysisGuy Grebla

Guy Grebla 2

Overview

What is Allegro Allegro vs. Genehunter Reduced inheritance vectors Founder couple reduction Fast tree traversal

Formalization Calculation of Spairs

Single locus probability calculation (if time permits)

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What is Allegro

Allegro is based on Genehunter.

Allegro runs faster than Genehunter due to algorithmic improvements.

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Allegro vs. Genehunter(1)

Allegro runs much faster than Genehunter, typically the speedup is 20-40 fold, and in many cases as high as 100 fold.

If necessary, Allegro is capable, at a cost of 10-30% in run time, to cut down the memory requirements by a factor of 20-60 compared with Genehunter.

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Allegro vs. Genehunter(2)

Recall that the time complexity of Genehunter is exponential in the pedigree’s size, therefore it is infeasible to run Genehunter with large pedigree’s size.

Due to the algorithmic improvements, Allegro is capable of handling significantly larger pedigrees (even though its time complexity is still exponential in the pedigree’s size).

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Reduced inheritance vectors – the idea The idea is based on symmetry that exists

between the two alleles of a founder.

1 0

V=(0,1,1,0)

0 1

0 0

V=(1,1,0,0)

1 1

n1 n2

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Reduced inheritance vectors

For male (female) founder, the corresponding paternal (maternal) bit of his (her) first child is set to 0 and not expressed in the reduced vector (it is called hidden).

Result: let m be the number of non-founders, f the number of founders, the vector size is reduced to 2m-f

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Reduced inheritance vectors (Cont.)n1 n2 n3 n4

n5 n6

a / b

[0 0]

a / b

1 1

a / b

1] 0[

a / c

b / c0 1

І

ІІ

ІІІ

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Founder couple reduction

Consider a couple of founders which: Have at least one grandchild Both not genotyped Aren’t married twice

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Founder couple reduction (Cont.) v* is like v but :

Invert the corresponding bit of each of the grandchildren.

The paternal and maternal bit of each child are switched

n1 n2 n3 n4

1] 0[

a / c

0 1

Corresponding bit

v and v* has the same probability

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Founder couple reduction - results With the founder couple reduction, the

effective number of bits is 2m-f-c where c is the number of founder couples satisfying the stated conditions.

Therefore, we’ve improved by a factor of 2c over the previous reduction.

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Fast tree traversal

The basic structure of the algorithms implemented in the Genehunter program loops over inheritance vectors in the outermost loop and over people in the pedigree in an inner loop

Drawback: for vectors that only differ for branches of the pedigree, part of the calculation will be duplicated.

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Fast tree traversal (Cont.)

Idea: changing the order of looping to avoid the repeated calculations.

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Fast tree traversal – naïve example Say we want to calculate for each vector v of

length n, the number of 1’s in v.

“Genehunter” method: for each vector calculate the number of 1’s.(add each bit of the vector to the sum)

“Allegro” method: pass the vectors and save calculations along the way.

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naïve example – Allegro method

0

0

0 1

1

1 2

0

0 1 0 1

1

Less additions!

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Fast tree traversal - formalization For each inheritance vector v, S(v) is known. We traverse the pedigree from the top down. When a child is born:

If it has i hidden bits – 22-i possibilities for its bits For each possibility the inheritance vector is appropriately

updated and the branch is descended We add a bit b to update vector v to v+ D(v) is a collection of data N=22m-f - number of possible inheritance vectors

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Fast tree traversal - formalization(2)

Recursive algorithm:

addbit(v, D, b):for b = 0, 1 do

set v+ = (v,b) and calculate D+ = D(v+)if there are more bits, addbit(v+,D+, next bit) ,

else D+ contains data for s(v+)If the calculation of D+ and s are both O(1) then the total time complexity of the calculation is O(N)

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Example – calculation of Spairs

Øij(p,q)= 1 if allele i of p and allele j of q are IBD and 0 otherwise

Spq(v) = ∑1i=0∑1

j=0Øij(p,q)

Spairs(v) = ∑(p,q) is a pair of affecteds Spq(v)

ki- the number of times founder allele i turns up among the affected.

s – the value of Spairs for the traversed portion

D = (s,k1,k2,…,k2f)

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Example (Cont.)

When an unaffected person is added, do nothing (s+=s, ki

+=ki , kj

+=kj) When an affected person is added, perform:

s+ s + ki + kj

ki+ ki + 1

kj+ kj + 1

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Example (Cont.)n1 n2 n3 n4

n5 n6

a / b

[0 0]

a / b

1 1

a / b

1] 0[

a / c

b / c

V=(0,1,1,1,1)

Init (no vector bits)

s=1, k1=1, k3=2, k4=1

ІІІ1 is added

s=2, k1=1, k3=2, k4=2, k5=1

ІІІ2 is added

s=4, k1=1, k3=2, k4=3, k5=1,k6=1

0 1

І

ІІ

ІІІ

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Spairs calculation – Genehunter vs. Allegro Genehunter calculates Spairs by calculating Spq

for each affected pair, and add it to Spairs

This process requires O(Nα2) where α is the number of affected.

We saved a factor of α2 (!)

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Additional improvements

Allegro use FFT for matrices multiplication, some classical computational techniques have been used to speed the FFT by a factor of three or four.

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References

“Fast multipoint linkage analysis and the program Allegro”, Daniel F.Gudbjartsson, Kristjan Jonasson, Michael L.Frigge, Augustine Kong

"Allegro, a new computer program for linkage analysis,"Gudbjartsson DF, Jonasson K, Frigge ML, Kong A. Nat Genet. 2000 May;25(1):12-3.

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BACKUP

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Single locus probability calculation Goal: compute Pr[ml | vl], at locus l for every

vector vl

marker data at this locus (evidence).

A certain inheritance vector.

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Single locus probability calculation(Cont.) In general: p(ml | vl) = ∑aєP∏2f

i=1p(ai)where P is the set of possible allele assignments a=(a1,…a2f) to (n1,…,n2f)

This probability may be calculated for each vl using Fast tree traversal.

Denote p(ml | vl) as q(v)

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Single locus probability - notations

n1 n2 n3 n4

n5 n6

a / b

[0 0]

a / b

1 1

a / b

1] 0[

a / c

b / c0 1

І

ІІ

ІІІ

Founder nodes

Assume our founder nodes are numbered, node ni is numbered i

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Single locus probability – notations(2) Founder nodes are classified to 3 disjoint sets:

A – assigned nodes. E – contains edges – each edge is labeled with 2

distinct alleles. U – unassigned nodes.

ai – allele assigned to i (i єA)

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Single locus probability - initialization Init:

E nodes of genotyped founders (edges). U rest of the founder nodes. A nil (empty) q(v) 0

Goal: build a founder graph. From the graph we can calculate q(v)

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Single locus probability – algorithm When a person genotyped a / b is added:

The value of v (so far) determines the sources of the alleles of the person among the founders.

Denote the corresponding founders by i and j, and consider the edge (i,j).

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Single locus probability – algorithm (2) 6 options for edge (i,j):

123

4

5

6

A U

E

ii

i

i

i

j

j

j

j

j

i

j

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Single locus probability – case by case Case 1:

Put (i,j) in E, remove i,j from U

Case 2: check whether {a,b} = {ai,aj}

Case 3: Check if ai is one of a and b, and if it

is, assign the other to aj , and move j from U to A

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Single locus probability – case by case(2) Case 4:

Check if ai is one of a and b Check if the other one is consistent

with the labeling of an edge (j,k) in E and if it’s consistent force the assignment

Cases 5,6: May need another loop. Set ai=a, aj=b, check and handle

consistency Set ai=b, aj=a, check and handle

consistency

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Single locus probability – algorithm(3) After the last bit of the vector was added, for

the probability calculation a product over the edges in E is needed:

Let (ae,be)єE

q(v) is updated by adding to it:∏i єA ∏ e єE2p(ae)p(be)