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Optimization Methods for Reliable Genomic-Based Pathogen Detection Systems

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Title: Optimization Methods for Reliable Genomic-Based Pathogen Detection Systems


1
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

2
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.
3
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

4
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
5
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
6
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.)

7
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

8
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

9
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

10
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

11
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

12
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

13
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

14
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

15
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, ? ? ?

16
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)

17
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

18
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
19
Experimental Results, k8
20
Experimental Results, k12
21
Runtime, k10
22
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)

23
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
24
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

25
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.
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