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Improved Algorithms for Multiplex PCR Primer Set Selection with

Amplification Length Constraints

Kishori M. Konwar

Ion I. Mandoiu

Alexander C. Russell

Alexander A. Shvartsman

CS&E Dept., Univ. of Connecticut

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Combinatorial Optimization in Bioinformatics

• Fast growing number of applications– Sequence alignment– DNA sequencing– Haplotype inference– Pathogen identification– …– High-throughput assay design

• Microarray probe selection• Microarray quality control• Universal tag arrays• …• This talk: Multiplex PCR primer set selection

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Outline

• Background and problem formulation

• “Potential function” greedy algorithm

• Approximation guarantee

• Experimental results

• Conclusions

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The Polymerase Chain Reaction

Target Sequence Polymerase

Primer 1Primer 2

Primers

Repeat 20-30 cycles

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Primer Pair Selection Problem

• Given:

• Genomic sequence around amplification locus

• Primer length k

• Amplification upperbound L

• Find: Forward and reverse primers of length k that hybridize within a distance of L of each other and optimize amplification efficiency (melting temperature, secondary structure, mis-priming, etc.)

L

Forward primer

Reverse primer

amplification locus

3'

3'

5'

5'

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PCR for SNP Genotyping

• Thousands of SNPs to be genotyped using hybridization methods (e.g., SBE)

• Selective PCR amplification needed to improve accuracy of detection steps– whole-genome amplification not appropriate

• Simultaneous amplification OK Multiplex PCR

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Multiplex PCR• How it works

– Multiple DNA fragments amplified simultaneously

– Each amplified fragment still defined by two primers

– A primer may participate in amplification of multiple targets

• Primer set selection– Currently done by time-consuming trial and error

– An important objective is to minimize number of primers Reduced assay cost Higher effective concentration of primers higher

amplification efficiency Reduced unintended amplification

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Primer Set Selection Problem• Given:

• Genomic sequences around n amplification loci

• Primer length k

• Amplification upper bound L

• Find:

• Minimum size set S of primers of length k such that, for each amplification locus, there are two primers in S hybridizing with the forward and reverse genomic sequences within a distance of L of each other

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Previous Work on Primer Selection

• Well-studied problem: [Pearson et al. 96], [Linhart & Shamir’02], [Souvenir et al.’03], etc.

• Almost all problem formulations decouple selection of forward and reverse primers– To enforce bound of L on amplification length, select only

primers that hybridize within L/2 bases of desired target

– In worst case, this method can increase the number of primers by a factor of O(n) compared to the optimum

• [Pearson et al. 96] Greedy set cover algorithm gives O(ln n) approximation factor for the “decoupled” formulation

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Previous Work (2)

• [Fernandes&Skiena’02] study primer set selection with uniqueness constraints

• Minimum Multi-Colored Subgraph Problem:– Vertices correspond to candidate primers– Edge colored by color i between u and v iff

corresponding primers hybridize within a distance of L of each other around i-th amplification locus

– Goal is to find minimum size set of vertices inducing edges of all colors

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The Set Cover Problem Given:

- Universal set U with n elements- Family of sets (Sx, xX) covering all elements of

U Find:

- Minimum size subset X’ of X s.t. (Sx, xX’) covers all elements of U

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Selection w/ Length Constraints

• “Simultaneous set covering” problem:

- Ground set partitioned into n disjoint sets Si (one for each target), each with 2L elements

- Goal is to select minimum number of sets == primers covering at least 1/2 of the elements in each partition

L L

SNPi

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Greedy Setcover Algorithm

Classical result (Johnson’74, Lovasz’75, Chvatal’79): the greedy setcover algorithm has an approximation factor of H(n)=1+1/2+1/3+…+1/n < 1+ln(n)

- The approximation factor is tight- Cannot be approximated within a factor of (1-)ln(n) unless

NP=DTIME(nloglog(n))

Greedy Algorithm:- Repeatedly pick the set with most uncovered elements

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Potential Functions• Set cover

• = #uncovered elements

• Initially, = n

• For feasible solutions, = 0

• Primer selection with length constraints

• = minimum number of elements that must be covered = i max{0, L - #uncovered elements in Si}

• Initially, = nL

• For feasible solutions, = 0

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General setting

Potential function (X’) 0 ({}) = max

(X’) = 0 for all feasible solutions X’’ X’ (X’’) (X’) If (X’)>0, then there exists x s.t. (X’+x) < (X’) X’’ X’ ∆(x,X’) ∆(x,X’) for every x, where

∆(x,X’) := (X’) - (X’+x) Objective: find minimum size set X’ with (X’)=0

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Generic Greedy Algorithm

• Theorem: The generic greedy algorithm has an approximation factor of 1+ln ∆max

• Corollary: 1+ln(nL) approximation for PCR primer selection

X’ {} While (X’) > 0

Find x with maximum ∆(x,X’) X’ X’ + x

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Proof Sketch (1)

• x1, x2,…,xg be the elements selected by greedy, in the order in which they are chosen

• x*1, x*2,…,x*k be the elements of an optimum solution.

Charging scheme: xi charges to x*j a cost of

where ij = ∆(xi,{x1,…, xi-1}{x*1,…,x*j})

Fact 1: Each x*j gets charged a total cost of at most 1+ln ∆max

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Proof Sketch (2)Fact 2: Each xi charges at least 1 unit of cost

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Experimental Setting• Datasets extracted from NCBI databases, L=1000• Dell PowerEdge 2.8GHz Xeon• Compared algorithms

– G-FIX: greedy primer cover algorithm [Pearson et al.]

– MIPS-PT: iterative beam-search heuristic [Souvenir et al.]

• Restrict primers to L/2 bases around amplification locus

– G-VAR: naïve modification of G-FIX

• First selected primer can be up to L bases away

• Opposite sequence truncated after selecting first primer

– G-POT: potential function driven greedy algorithm

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Experimental Results, NCBI tests

#Targets

k

G-FIX(Pearson et al.)

G-VAR(G-FIX with dynamic

truncation)

MIPS-PT (Souvenir et al.)

G-POT(Potential- function

greedy)

#Primers CPU

sec

#Primers CPU

sec

#Primers CPU

sec

#Primers CPU

sec

20

8 7 0.04 7 0.08 8 10 6 0.10

10 9 0.03 10 0.08 13 15 9 0.08

12 14 0.04 13 0.08 18 26 13 0.11

50

8 13 0.13 15 0.30 21 48 10 0.32

10 23 0.22 24 0.36 30 150 18 0.33

12 31 0.14 32 0.30 41 246 29 0.28

100

8 17 0.49 20 0.89 32 226 14 0.58

10 37 0.37 37 0.72 50 844 31 0.75

12 53 0.59 48 0.84 75 2601 42 0.61

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#primers, as percentage of 2n (l=8)

n

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#primers, as percentage of 2n (l=10)

n

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#primers, as percentage of 2n (l=12)

n

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CPU Seconds (l=10)

n

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Conclusions

• Numerous combinatorial optimization problems arising in the area of high-throughput assay design

• Theoretical insights such as approximation results can lead to significant practical improvements

• Choosing the proper problem model is critical to solution efficiency

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Ongoing Work & Open Problems

• Degenerate primers• Accurate hybridization model (melting temperature,

secondary structure, cross hybridization,…)– In-silico MP-PCR simulator

• Partition into multiple multiplexed PCR reactions (Aumann et al. Wabi’03)

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Acknowledgments

• Financial support from UCONN’s Research Foundation