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Polymerase Chain Reaction: A Markov Process Approach

This paper explores the use of a Markov process model to analyze the dynamics of the polymerase chain reaction (PCR). It discusses the binding of dNTPs and provides an analytical solution for the probability distribution of DNA length. The model is extended to multi-cycle PCR runs and numerical results are presented. The paper also highlights the sensitivity of the reaction condition and the optimization of PCR runs.

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Polymerase Chain Reaction: A Markov Process Approach

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  1. Polymerase Chain Reaction: A Markov Process Approach Mikhail V. Velikanov et al. J. theor. Biol. 1999 Summarized by 임희웅 2003.8.12

  2. Contents • Introduction • Markov Process Model for Primer Extension • Model for Multi-Cycle PCR Runs • Numerical Results • Discussion

  3. Introduction • A stochastic approach to PCR • Focus on the microscopic nature of amplification process • Elementary reaction: binding of dNTP • Markov process  master equation • Analytical solution for the probability distribution of DNA length • Main qualitative feature • Sensitivity of the reaction condition • The amplification plateau effect • Optimal duration of amplification for each cycle

  4. l0 l l+1 … L … Markov Process Model for Primer Extension • Amplification process as Markov process • Binding of dNTP occurs randomly, with the probability per unit time determined entirely by the present state of the system. • State: length of primer • Reaction rate: w = k(t) n • n: total number of dNTP in the current system • k(t): the rate coefficient which depends on temperature (time) • l + n = l0+ n0 = m0 constant • n0: initial total number of dNTP

  5. Master Equation • The master equation for the primer extension process

  6. Analytical Solution

  7. Model for Multi-Cycle PCR Runs • Additional feature • Increasing number of DNA molecules • Statistical independence of the extension process • n0: initial number of dNTPs per template strand • np: the number of primers per template strand • Complementarity • Two kinds of strand: +, - • Pi+(l,n): Prob. distribution of + strand in ith cycle

  8. Evolution of Probability Distribution • ηn: duration of the extension phase of each cycle The distribution for the first cycle Consumed dNTP in first cycle

  9. Numerical Result • Simulation for PCR Runs

  10. Optimization of PCR Runs Arrhenius’ law

  11. Discussion • Primary assumption • DNA synthesis occurs independently on each template strand. • Advantage in Markov process approach • The model can be solved exactly by analytical means.  simple calculation • It accounts for the fluctuations inherent in PCR kinetics through a description of their natural microscopic source. • The model is easy to modify and can be used as the basis for constructing dedicated algorithms for numerical simulations of PCR.

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