Minimum PCR Primer Set Selection with Amplification Length and Uniqueness Constraints

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Minimum PCR Primer Set Selection with
Amplification Length and Uniqueness Constraints

• Ion Mandoiu
• University of Connecticut
• CSE Department

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Combinatorial Optimization Applications in
Bioinformatics

• Fast growing number of applications
• Dynamic Programming Integer Programming in
  sequence alignment
• TSP and Euler paths in DNA sequencing
• Integer Programming in Haplotype inference
• Integer Programming approximation algorithms
  for efficient pathogen identification (string
  barcoding)

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High-Thrughput Assay Design

• New source of combinatorial problems
• Microarray probe selection
• Mask design for Affy arrays
• Universal tag arrays
• Self-assembling microarrays
• Quality control
• This talk Multiplex PCR primer set selection
• Optimization goals
• Improved speed
• High reliability
• Reduced COST

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Outline

• Motivation and problem formulations
• Greedy algorithm for primer set selection with
  amplification length constraints
• LP-rounding algorithm for primer set selection
  with uniqueness constraints
• Experimental results
• Conclusions

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Uniplex PCR

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Primer Pair Selection Problem
3'
5'
Reverse primer
? L
? L
Forward primer
3'
5'
amplification locus

• 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 temperatures, secondary structure, cross
  hybridization, etc.)

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Motivation for Primer Set Selection (1)

• Spotted microarray synthesis Fernandes and
  Skiena02
• Need unique pair for each amplification product,
  but primers can be reused to minimize cost
• Potential to reduce primers from O(n) to O(n1/2)
  for n products

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Motivation for Primer Set Selection (2)

• SNP Genotyping
• Thousands of SNPs that must genotyped using
  hybridization based methods (e.g., SBE)
• Selective PCR amplification needed to improve
  accuracy of detection steps (whole-genome
  amplification not appropriate)
• No need for unique amplification!
• Primer minimization is critical
• Fewer primers to buy
• Fewer multiplex PCR reactions

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Primer Set Selection Problem

• Given
• Genomic sequences around each amplification
  locus
• Primer length k
• Amplification upperbound L
• Find
• Minimum size set of primers S of length k such
  that, for each amplification locus, there are two
  primers in S hybridizing to the forward and
  reverse sequences within a distance of L of each
  other
• For some applications S should contain a unique
  pair of primers amplifying each each locus

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Previous Work (1)

• Pearson et al. 96LinhartShamir02Souvenir
  et al.03
• - Separately select forward and reverse primers
• - To enforce bound of L on amplification length,
  select only primers that are within a distance of
  L/2 of the target SNP
• Ignores half of the feasible primer pairs
• Solution can increase by a factor of O(n) by
  ignoring them!
• Greedy set cover algorithm gives O(ln n)
  approximation factor for this formulation
• Cannot approximate better unless PNP

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Previous Work (2)

• FernandesSkiena02 model primer selection as
  a minimum multicolored subgraph problem
• Vertices of the graph correspond to candidate
  primers
• There is an edge colored by color i between
  primers u and v if they hybridize to i-th forward
  and reverse sequences within a distance of L
• Goal is to find minimum size set of vertices
  inducing edges of all colors
• No non-trivial approximation factor known
  previously

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Selection w/o Uniqueness Constraints

• Can be seen as a simultaneous set covering
  problem
• - The ground set is partitioned into n disjoint
  sets, each with 2L elements
• The goal is to select a minimum number of sets
  ( primers) that cover at least half of the
  elements in each partition
• Naïve modifications of the greedy set cover
  algorithm do not work
• Key idea use potential function ? for a partial
  solution P minium number of elements that are
  not yet covered as measure of infeasibility
• Initially, ? nL
• For feasible solutions, ? 0

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Potential-Function Driven Greedy

• Select a primer that decreases the potential
  function ? by the largest amount (breaking ties
  arbitrarily)
• Repeat until feasibility is achived
• Lemma Each greedy selection reduces ? by a
  factor of at least (1-1/OPT)
• Theorem The number of primers selected by the
  greedy algorithm is at most ln(nL) larger than
  the optimum

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Selection w/ Uniqueness Constraints

• Can be modeled as minimum multicolored subgraph
  problem add edge colored by color i between two
  primers if they amplify i-th SNP and do not
  amplify any other SNP
• Trivial approximation algorithm select 2
  primers for each SNP
• O(n1/2) approximation since at least n1/2
  primers required by every solution
• Non-trivial approximation?

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Integer Program Formulation

• Variable xu for every vertex (candidate primer)
  u
• xu set to 1 if u is selected, and to 0 otherwise
• Variable ye for every edge e
• ye set to 1 if corresponding primer pair
  selected to amplify one of the SNPs
• Objective minimize sum of xus
• Constraints
• for each i, sum of ye e amplifying SNP i ? 1
• ye ? xu for every e incident to u

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LP-Rounding Algorithm

• Solve linear programming relaxation
• Select node u with probability xu
• Theorem With probability of at least 1/3, the
  number of selected nodes is within a factor of
  O(m1/2lnn) of the optimum, where m is the maximum
  number of edges sharing the same color.
• For primer selection, m ? L2 ? approximation
  factor is O(Lln n)

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Experimental Setting

• SNP sets extracted from NCBI databases
  randomly generated
• C/C code run on a 2.8GHz Dell PowerEdge
  running Linux
• Compared algorithms
• G-FIX greedy primer cover algorithm of Pearson
  et al.
• - Primers restricted to be within L/2 of
  amplified SNPs
• G-VAR naïve modification of G-FIX
• For each SNP, first selected primer can be L
  bases away from SNP
• If first selected primer is L1 bases away from
  the SNP, opposite sequence is truncated to a
  length of L- L1
• G-POT potential function driven greedy
  algorithm
• MIPS-PT iterative beam-search heuristic of
  Souvenir et al (WABI03)

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Experimental Results, NCBI tests
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Experimental Results, k8
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Experimental Results, k10
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Experimental Results, k12
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Runtime, k10
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Conclusions

• New combinatorial optimization problems arising
  in the area of high-throughput assay design
• Theoretical insights (such as approximation
  results) give algorithms with significant
  practical improvements
• Choosing the proper problem model is critical to
  solution efficiency

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Ongoing Work Open Problems

• Allow degenerate primers
• Incorporate more biochemical constraints into the
  model (melting temperature, secondary structure,
  cross hybridization, etc.)
• Close gap between O(lnn) inapproximability bound
  and O(L lnn) approximation factor for minimum
  multi-colored subgraph problem
• Approximation algorithms for partition into
  multiple multiplexed PCR reactions (Aumann et al.
  WABI03)

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Acknowledgments

• Kishori Konwar
• Alex Russell
• Alex Shvartsman
• Financial support from UCONN Research Foundation