Mathematically modeling PCR: An asymptotic approximation with potential for optimization

Pages: 363 - 384, Volume 7, Issue 2, April 2010

doi:10.3934/mbe.2010.7.363       Abstract        Full Text (225.7K)       Related Articles

Martha Garlick - Department of Mathematics and Statistics, Utah State University, Logan UT 84322, United States (email)
James Powell - Department of Mathematics and Statistics, Utah State University, Logan UT 84322, United States (email)
David Eyre - Idaho Technology Inc., 390 Wakara Way, Salt Lake City UT 84108, United States (email)
Thomas Robbins - Idaho Technology Inc., 390 Wakara Way, Salt Lake City UT 84108, United States (email)

Abstract: A mathematical model for PCR (Polymerase Chain Reaction) is developed using the law of mass action and simplifying assumptions regarding the structure of the reactions. Differential equations are written from the chemical equations, preserving the detail of the complementary DNA single strand being extended one base pair at a time. The equations for the annealing stage are solved analytically. The method of multiple scales is used to approximate solutions for the extension stage, and a map is developed from the solutions to simulate PCR. The map recreates observed PCR well, and gives us the ability to optimize the PCR process. Our results suggest that dynamically optimizing the extension and annealing stages of individual samples may significantly reduce the total time for a PCR run. Moreover, we present a nearly optimal design that functions almost as well and does not depend on the specifics of a single reaction, and so would work for multi sample and multiplex applications.

Keywords:  PCR, polymerase chain reaction, dynamical systems, mathematical model, method of multiple scales, optimization.
Mathematics Subject Classification:  Primary: 74G10, 92C45.

Received: April 2008;      Accepted: February 2009;      Published: April 2010.