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Mathematical Biosciences and Engineering

2010, Volume 7, Issue 2: 363-384. doi: 10.3934/mbe.2010.7.363
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Mathematically modeling PCR: An asymptotic approximation with potential for optimization

  • 1.
    Department of Mathematics and Statistics, Utah State University, Logan UT 84322
  • 2.
    Idaho Technology Inc., 390 Wakara Way, Salt Lake City UT 84108
  • Received: 01 April 2008 Accepted: 29 June 2018 Published: 01 April 2010

  • MSC : Primary: 74G10, 92C45.

  • 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.

    Citation: Martha Garlick, James Powell, David Eyre, Thomas Robbins. Mathematically modeling PCR: An asymptotic approximation with potential for optimization[J]. Mathematical Biosciences and Engineering, 2010, 7(2): 363-384. doi: 10.3934/mbe.2010.7.363

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  • 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.


  • This article has been cited by:

    1. A. Sverstiuk, Numerical algorithm for optimal control development for annealing stage of polymerase chain reaction, 2019, 93, 25224433, 147, 10.33108/visnyk_tntu2019.01.147
    2. Joel Tellinghuisen, Andrej-Nikolai Spiess, Comparing real-time quantitative polymerase chain reaction analysis methods for precision, linearity, and accuracy of estimating amplification efficiency, 2014, 449, 00032697, 76, 10.1016/j.ab.2013.12.020
    3. Antoon Lievens, S. Van Aelst, M. Van den Bulcke, E. Goetghebeur, Enhanced analysis of real-time PCR data by using a variable efficiency model: FPK-PCR, 2012, 40, 1362-4962, e10, 10.1093/nar/gkr775
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  • © 2010 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
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