May 25, GSU Biotech Symposium1 Minimum PCR Primer Set Selection with Amplification Length and Uniqueness Constraints Ion Mandoiu University of Connecticut CS&E Department

May 25, GSU Biotech Symposium2 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) –…

May 25, GSU Biotech Symposium3 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

May 25, GSU Biotech Symposium4 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

May 25, GSU Biotech Symposium5 Uniplex PCR …

May 25, GSU Biotech Symposium6 Primer Pair Selection Problem  L L Forward primer Reverse primer amplification locus 3'3' 3'3' 5'5' 5'5'  L 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.)

May 25, GSU Biotech Symposium7 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(n 1/2 ) for n products

May 25, GSU Biotech Symposium8 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

May 25, GSU Biotech Symposium9 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

May 25, GSU Biotech Symposium10 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

May 25, GSU Biotech Symposium11 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

May 25, GSU Biotech Symposium12 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

May 25, GSU Biotech Symposium13 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

May 25, GSU Biotech Symposium14 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(n 1/2 ) approximation since at least n 1/2 primers required by every solution Non-trivial approximation?

May 25, GSU Biotech Symposium15 Integer Program Formulation Variable x u for every vertex (candidate primer) u - x u set to 1 if u is selected, and to 0 otherwise Variable y e for every edge e - y e set to 1 if corresponding primer pair selected to amplify one of the SNPs Objective: minimize sum of x u ’s Constraints: - for each i, sum of {y e : e amplifying SNP i}  1 - y e  x u for every e incident to u

May 25, GSU Biotech Symposium16 LP-Rounding Algorithm 1.Solve linear programming relaxation 2.Select node u with probability x u Theorem: With probability of at least 1/3, the number of selected nodes is within a factor of O(m 1/2 lnn) of the optimum, where m is the maximum number of edges sharing the same color. For primer selection, m  L 2  approximation factor is O(Lln n)

May 25, GSU Biotech Symposium17 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 L 1 bases away from the SNP, opposite sequence is truncated to a length of L- L 1 G-POT: potential function driven greedy algorithm MIPS-PT: iterative beam-search heuristic of Souvenir et al (WABI’03)

May 25, GSU Biotech Symposium18 Experimental Results, NCBI tests

May 25, GSU Biotech Symposium19 Experimental Results, k=8

May 25, GSU Biotech Symposium20 Experimental Results, k=10

May 25, GSU Biotech Symposium21 Experimental Results, k=12

May 25, GSU Biotech Symposium22 Runtime, k=10

May 25, GSU Biotech Symposium23 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

May 25, GSU Biotech Symposium24 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)

May 25, GSU Biotech Symposium25 Acknowledgments Kishori Konwar Alex Russell Alex Shvartsman Financial support from UCONN Research Foundation