Optimization Methods for Reliable Genomic-Based Pathogen Detection Systems

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Optimization Methods for Reliable Genomic-Based
Pathogen Detection Systems

• K.M. Konwar, I.I. Mandoiu, A.C. Russell, and A.A.
  Shvartsman
• Computer Science Engineering Department
• University of Connecticut, Storrs, CT 06269

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Abstract
Recent advances in genomic technologies have
opened the way for the development of
Genomic-based Pathogen Detection Systems (GPDSs)
that can provide early warning in case of rapidly
proliferating outbreaks of new natural pathogens
such as the SARS corona-virus or bio-terrorist
attacks. A critical step of all GPDS
architectures proposed to date is DNA
amplification by Multiplexed Polymerase Chain
Reaction (MP-PCR). In this poster we present
ongoing theoretical and practical research on the
minimum primer set selection problem for MP-PCR.
We give algorithms with improved approximation
guarantees for this problem and report results of
empirical experiments on both synthetic and
public genomic database test cases showing that
our algorithms are highly scalable and produce
better results compared to previous heuristics.
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GPDS Components and Requirements

• Key GPDS components
• Selection of distinguishing DNA oligonucleotides
  based on available genomic sequences for the
  pathogens
• Selective amplification of collected genetic
  material
• Hybridization-based detection of present
  distinguishers
• Pathogen identification by comparison with stored
  signatures/barcodes of known pathogens
• GPDS design requirements
• High specificity and sensitivity of detection
• Discrimination between pathogens and
  non-pathogenic organisms
• Ability to work with trace amounts of genetic
  material, and to detect multiple pathogens at the
  same time
• Fully automated operation (should require
  minimal human intervention

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Pathogen Detection System Architecture
Set of (degenerate) primers
PCR Machine
Multiplex PCR
Mixture of (degenerate) primers
Sample possibly containing minute traces of
several pathogens
Multiplex PCR
Mixture of (degenerate) primers


Mixture of (degenerate) primers
Multiplex PCR
Amplified DNA sequences from sample
Fluorescent nucleotides
Set of probes combining barcode distinguishers
with tags
Distinguishing signature of pathogens present
in sample
Universal DNA Tag Array
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The Polymerase Chain Reaction
Polymerase
Primers
Repeat 20-30 cycles
Invented by Mullis in 1980s, PCR uses short
oligonucleotide primers and the DNA polymerase in
a cyclic reaction to produce millions of copies
of a target sequence of DNA
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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 temperatures, secondary structure, cross
  hybridization, etc.)

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Multiplex PCR

• Multiplex PCR (MP-PCR)
• Multiple DNA fragments amplified simultaneously
• Boundaries of each amplification fragment still
  defined by two oligonucleotide primers
• A primer may participate in the amplification of
  multiple targets

• Primer set selection
• Typically done by time-consuming trial and error
• An important objective is to minimize the total
  number of primers
• Reduced assay cost
• Higher effective concentration of primers ?
  higher amplification efficiency
• Reduced unintended amplification

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Other Applications of Multiplex PCR

• Spotted microarray synthesis FernandesSkiena02
  
• Need unique pair for each one of the n
  amplification product, but primers can be used
  multiple times
• Potential to reduce primers from O(n) to O(n1/2)
• SNP Genotyping
• Thousands of SNPs that must genotyped using
  hybridization based methods (e.g., single-base
  extension)
• Selective PCR amplification needed to improve
  accuracy of detection steps (whole-genome
  amplification less appropriate)
• No need for unique amplification!
• Primer minimization is critical
• Reduced cost
• Fewer multiplex PCR reactions

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Primer Set Selection Problem

• Given
• Genomic sequences around each amplification
  locus
• Primer length k
• Amplification upper bound 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 on Primer Set Selection

• All previous works, e.g., Pearson et al.
  96Linhart Shamir02 Souvenir et al.03,
  use problem formulations that decouple selection
  of forward and reverse primers, and hence cannot
  directly enforce constraints on amplification
  product length
• To enforce bound of L on amplification length,
  select only primers that hybridize within L/2
  bases of desired target
• Ignores half of the feasible primer pairs!
• In worst case, this method can increase the
  number of primers by a factor of O(n) compared to
  the optimum
• Greedy set cover algorithm gives O(ln n)
  approximation factor for the decoupled
  formulation
• Cannot find better approximation unless PNP

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Previous Work (contd.)

• 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 algo
  do not work
• Key idea use potential function ? to measure
  progress towards fasibility. For primer
  selection, potential function counts the total
  number of elements that remain to be covered
• Initially, ? nL
• For feasible solutions, ? 0

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Greedy Approximation Algorithm

• Potential-Function Driven Greedy Algorithm
• Select a primer that decreases potential function
  ? by the largest amount (breaking ties
  arbitrarily)
• Repeat until feasibility is achieved

• Theorem The greedy algorithm in returns a
  feasible primer set whose size is at most 1ln ?
  times larger than the optimum, where ? is the
  maximum potential value decrease caused by a
  single primer
• For primer selection ? is equal to nL in the
  worst case, and is much smaller in practice
• 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 sub-graph
  problem add edge colored by color i between two
  primers if they amplify i-th target but do not
  amplify any other genomic sequence
• Trivial approximation algorithm select 2
  primers for each amplification target
• O(n1/2) approximation since at least n1/2
  primers required by every feasible solution
• No non-trivial approximation known previously

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Integer Program Formulation

• Variable xv for every graph node (candidate
  primer) v? V xv set to 1 if v is selected, and
  to 0 otherwise
• Variable ye for every graph edge e ? E ye set
  to 1 if corresponding primer pair selected to
  amplify one of the targets, ? ? ?

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LP-Rounding Approximation

• 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 and n is
  the number of nodes (candidate primers).
• For primer selection, m ? L2 ? approximation
  factor is O(Llnn)

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Experimental Setting

• SNP genotyping datasets
• Extracted from NCBI databases
• Randomly generated using uniform distribution
• C/C code, 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 bases of
  amplified SNPs
• G-VAR naïve modification of G-FIX
• For each SNP, first selected primer can be up to
  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
• MIPS-PT iterative beam-search heuristic of
  Souvenir et al.
• G-POT potential function driven greedy algorithm

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Experimental Results, NCBI tests
Targets k G-FIX (Pearson et al.) G-FIX (Pearson et al.) G-VAR (G-FIX with dynamic truncation) G-VAR (G-FIX with dynamic truncation) MIPS-PT (Souvenir et al.) MIPS-PT (Souvenir et al.) G-POT (Potential- function greedy) G-POT (Potential- function greedy)
Targets k 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
20 10 9 0.03 10 0.08 13 15 9 0.08
20 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
50 10 23 0.22 24 0.36 30 150 18 0.33
50 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
100 10 37 0.37 37 0.72 50 844 31 0.75
100 12 53 0.59 48 0.84 75 2601 42 0.61
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Experimental Results, k8
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Experimental Results, k12
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Runtime, k10
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Ongoing Work on Primer Selection

• Extending the greedy algorithm to degenerate
  primer selection
• Huge number of feasible candidate primers ?
  impractical to find primer with largest reduction
  in potential function
• The greedy algorithm remains provably good if
  only near-optimal choices are made in each step
• Incorporating improved hybridization models
• Allow hybridization with mismatches, enforce
  constraints on melting temperature, secondary
  structure, cross hybridization, etc.
• Closing gap between O(lnn) inapproximability
  bound and O(m1/2lnn) approximation factor for the
  minimum multi-colored subgraph problem
• Finding approximation algorithms and practical
  heuristics for partitioning into multiple
  multiplexed PCR reactions (Aumann et al. WABI03)

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The String Barcoding Problem
String barcoding is a pathogen identification
technique recently proposed by Rash and Gusfield,
and Bornemann et al. In this technique, a number
of short oligos called distinguishers are spotted
or synthesized on a microarray and hybridized
with the fluorescently labeled DNA of unknown
pathogens. The hybridization pattern can be
viewed as a string of 0's and 1s. The unknown
pathogen can be identified by comparing this 0/1
pattern (its barcode'') with a set of
pre-computed patterns for the pathogens. The
main objective is to minimize the number of
distinguishers needed to uniquely identify the
pathogens.
Given Genomic sequences g1,, gn Find Minimum
number of strings t1,,tk Such that For every
gi ? gj, there exists a string tl which is the
Watson-Crick complement for a substring of gi or
gj, but not of both
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Ongoing Work on String Barcoding

• The greedy setcover algorithm, in which pairs of
  pathogens are viewed as elements to be covered,
  and candidate distinguishers are viewed as sets,
  is known to guarantee an approximation factor of
  2lnn
• An information content greedy algorithm was
  recently shown by Berman et al. to have an
  approximation factor of 1lnn
• In ongoing work we explore heuristics for the
  following important extensions of the string
  barcoding problem
• Probe mixtures as distinguishers. In spotted
  microarrays, it is feasible to spot a mixture
  consisting of a limited number of probes at any
  given array location. Using probe mixtures can
  reduce the number of spots on the array - hence
  barcode length - close to the information
  theoretical lower-bound of log2n
• Robust barcodes. Practical application of string
  barcoding is complicated by imperfect
  hybridization, experimental errors, and
  variability in pathogen genomic sequence. We are
  exploring robust barcodes using redundant
  distinguishers and error correcting schemes

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References

• R.J. Fernandes and S.S. Skiena. Microarray
  synthesis through multiple-use PCR primer design.
  Bioinformatics, 18S128S135, 2002.
• M.T. Hajiaghayi, K. Jain, K.M. Konwar, L.C. Lau,
  I.I. Mandoiu, A.C. Russell, A.A. Shvartsman, and
  V.V. Vazirani. The Minimum k-colored subgraph
  problem in haplotyping and DNA primer selection,
  submitted to ACM Symp. on Discrete Algorithms.
• K.M. Konwar, I.I. Mandoiu, A.C. Russell, and A.A.
  Shvartsman, Improved Algorithms for Minimum PCR
  Primer Set Selection with Amplification Length
  Constraints, submitted to 3rd Asia Pacific
  Bioinformatics Conference.
• K.M. Konwar, I.I. Mandoiu, A.C. Russell, and A.A.
  Shvartsman, Approximation algorithms for minimum
  PCR primer set selection with amplification
  length and uniqueness constraints. ACM Computing
  Research Repository, Technical Report
  cs.DS/0406053, 2004.
• W.R. Pearson, G. Robins, D.E. Wrege, and T.
  Zhang. On the primer selection problem for
  polymerase chain reaction experiments. Discrete
  and Applied Mathematics, 71231246, 1996.
• R. Souvenir, J. Buhler, G. Stormo, and W. Zhang.
  Selecting degenerate multiplex PCR primers. In
  Proc. 3rd Intl. Workshop on Algorithms in
  Bioinformatics (WABI), pages 512526, 2003.