American Institute of Mathematical Sciences

Advanced
2010, 7(2): 363-384. doi: 10.3934/mbe.2010.7.363

Mathematically modeling PCR: An asymptotic approximation with potential for optimization

Martha Garlick 1, , James Powell 1, , David Eyre 2, and  Thomas Robbins 2,

1. 

Department of Mathematics and Statistics, Utah State University, Logan UT 84322, United States, United States
2. 

Idaho Technology Inc., 390 Wakara Way, Salt Lake City UT 84108, United States, United States

Received  April 2008 Revised  February 2009 Published  April 2010

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: mathematical model, polymerase chain reaction, PCR, optimization., method of multiple scales, dynamical systems.
Mathematics Subject Classification: Primary: 74G10, 92C4.
Citation: Martha Garlick, James Powell, David Eyre, Thomas Robbins. Mathematically modeling PCR: An asymptotic approximation with potential for optimization. Mathematical Biosciences & Engineering, 2010, 7 (2) : 363-384. doi: 10.3934/mbe.2010.7.363
[1]

Piotr Oprocha. Chain recurrence in multidimensional time discrete dynamical systems. Discrete & Continuous Dynamical Systems - A, 2008, 20 (4) : 1039-1056. doi: 10.3934/dcds.2008.20.1039
[2]

Yancong Xu, Deming Zhu, Xingbo Liu. Bifurcations of multiple homoclinics in general dynamical systems. Discrete & Continuous Dynamical Systems - A, 2011, 30 (3) : 945-963. doi: 10.3934/dcds.2011.30.945
[3]

Andrew J. Majda, John Harlim, Boris Gershgorin. Mathematical strategies for filtering turbulent dynamical systems. Discrete & Continuous Dynamical Systems - A, 2010, 27 (2) : 441-486. doi: 10.3934/dcds.2010.27.441
[4]

Oanh Chau, R. Oujja, Mohamed Rochdi. A mathematical analysis of a dynamical frictional contact model in thermoviscoelasticity. Discrete & Continuous Dynamical Systems - S, 2008, 1 (1) : 61-70. doi: 10.3934/dcdss.2008.1.61
[5]

Nina Yan, Tingting Tong, Hongyan Dai. Capital-constrained supply chain with multiple decision attributes: Decision optimization and coordination analysis. Journal of Industrial & Management Optimization, 2019, 15 (4) : 1831-1856. doi: 10.3934/jimo.2018125
[6]

Liejune Shiau, Roland Glowinski. Operator splitting method for friction constrained dynamical systems. Conference Publications, 2005, 2005 (Special) : 806-815. doi: 10.3934/proc.2005.2005.806
[7]

Alan D. Rendall. Multiple steady states in a mathematical model for interactions between T cells and macrophages. Discrete & Continuous Dynamical Systems - B, 2013, 18 (3) : 769-782. doi: 10.3934/dcdsb.2013.18.769
[8]

Xiangjin Xu. Multiple solutions of super-quadratic second order dynamical systems. Conference Publications, 2003, 2003 (Special) : 926-934. doi: 10.3934/proc.2003.2003.926
[9]

Yunfei Peng, X. Xiang, W. Wei. Backward problems of nonlinear dynamical equations on time scales. Discrete & Continuous Dynamical Systems - S, 2011, 4 (6) : 1553-1564. doi: 10.3934/dcdss.2011.4.1553
[10]

Katherinne Salas Navarro, Jaime Acevedo Chedid, Whady F. Florez, Holman Ospina Mateus, Leopoldo Eduardo Cárdenas-Barrón, Shib Sankar Sana. A collaborative EPQ inventory model for a three-echelon supply chain with multiple products considering the effect of marketing effort on demand. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-14. doi: 10.3934/jimo.2019020
[11]

Wei Feng, Nicole Rocco, Michael Freeze, Xin Lu. Mathematical analysis on an extended Rosenzweig-MacArthur model of tri-trophic food chain. Discrete & Continuous Dynamical Systems - S, 2014, 7 (6) : 1215-1230. doi: 10.3934/dcdss.2014.7.1215
[12]

Wei Feng, C. V. Pao, Xin Lu. Global attractors of reaction-diffusion systems modeling food chain populations with delays. Communications on Pure & Applied Analysis, 2011, 10 (5) : 1463-1478. doi: 10.3934/cpaa.2011.10.1463
[13]

Hyun Geun Lee, Yangjin Kim, Junseok Kim. Mathematical model and its fast numerical method for the tumor growth. Mathematical Biosciences & Engineering, 2015, 12 (6) : 1173-1187. doi: 10.3934/mbe.2015.12.1173
[14]

Elvira Barbera, Giancarlo Consolo, Giovanna Valenti. A two or three compartments hyperbolic reaction-diffusion model for the aquatic food chain. Mathematical Biosciences & Engineering, 2015, 12 (3) : 451-472. doi: 10.3934/mbe.2015.12.451
[15]

Leonid A. Bunimovich. Dynamical systems and operations research: A basic model. Discrete & Continuous Dynamical Systems - B, 2001, 1 (2) : 209-218. doi: 10.3934/dcdsb.2001.1.209
[16]

Mihaela Negreanu, J. Ignacio Tello. On a comparison method to reaction-diffusion systems and its applications to chemotaxis. Discrete & Continuous Dynamical Systems - B, 2013, 18 (10) : 2669-2688. doi: 10.3934/dcdsb.2013.18.2669
[17]

Xin Li, Xingfu Zou. On a reaction-diffusion model for sterile insect release method with release on the boundary. Discrete & Continuous Dynamical Systems - B, 2012, 17 (7) : 2509-2522. doi: 10.3934/dcdsb.2012.17.2509
[18]

Klemens Fellner, Wolfang Prager, Bao Q. Tang. The entropy method for reaction-diffusion systems without detailed balance: First order chemical reaction networks. Kinetic & Related Models, 2017, 10 (4) : 1055-1087. doi: 10.3934/krm.2017042
[19]

Wenrui Hao, Jonathan D. Hauenstein, Bei Hu, Yuan Liu, Andrew J. Sommese, Yong-Tao Zhang. Multiple stable steady states of a reaction-diffusion model on zebrafish dorsal-ventral patterning. Discrete & Continuous Dynamical Systems - S, 2011, 4 (6) : 1413-1428. doi: 10.3934/dcdss.2011.4.1413
[20]

Massimiliano Berti, Philippe Bolle. Fast Arnold diffusion in systems with three time scales. Discrete & Continuous Dynamical Systems - A, 2002, 8 (3) : 795-811. doi: 10.3934/dcds.2002.8.795

2018 Impact Factor: 1.313

Tools

Metrics

  • PDF downloads (8)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences