May 25, 20042004 GSU Biotech Symposium1 Minimum PCR Primer Set Selection with Amplification Length and Uniqueness Constraints Ion Mandoiu University of

May 25, 2004 2004 GSU Biotech Symposium 1

Minimum PCR Primer Set Selection with Amplification Length and

Uniqueness Constraints

Ion Mandoiu

University of Connecticut

CS&E Department


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)

– …


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


Uniplex PCR

…


Primer Pair Selection Problem

L

Forward primer

Reverse primer

amplification locus

3'

3'

5'

5'

L

• 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.)


Motivation for Primer Set Selection (1)

• Spotted microarray synthesis [Fernandes and Skiena’02]– 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


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


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


Previous Work (1)• [Pearson et al. 96][Linhart&Shamir’02][Souvenir 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 P=NP


Previous Work (2)• [Fernandes&Skiena’02] 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


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


Potential-Function Driven Greedy

1. Select a primer that decreases the potential function by the largest amount (breaking ties arbitrarily)

2. 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


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?


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 xu’s

• Constraints:

- for each i, sum of {ye : e amplifying SNP i} 1

- ye xu for every e incident to u


LP-Rounding Algorithm

1. Solve linear programming relaxation

2. 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)


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 (WABI’03)


Experimental Results, NCBI tests


Experimental Results, k=8






Runtime, k=10


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


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. WABI’03)


Acknowledgments

• Kishori Konwar

• Alex Russell

• Alex Shvartsman

• Financial support from UCONN Research Foundation

Documents

May 25, 20042004 GSU Biotech Symposium1 Minimum PCR Primer Set Selection with Amplification Length and Uniqueness Constraints Ion Mandoiu University of