Principles of rapid polymerase chain reactions: mathematical modeling and experimental verification

Polymerase chain reaction (PCR) is an important diagnostic tool for the amplification of DNA. The PCR process can be treated as a problem in biochemical engineering. This study focuses on the development of a mathematical model of the polymerase chain reaction. The PCR process consists of three steps: denaturation of target DNA, annealing of sequence-specific oligonucleotide primers and the enzyme-catalyzed elongation of the annealed complex (primer:DNA:polymerase). The denaturation step separates the double strands of DNA; this model assumes denaturation is complete. The annealing step describes the formation of a primer-fragment complex followed by the attachment of the polymerase to form a ternary complex. This step is complicated by competitive annealing between primers and incomplete fragments including primer-primer reactions. The elongation step is modeled by a stochastic method. Species that compete during the elongation step are deoxynucleotide triphosphates dCTP, dATP, dTTP, dGTP, dUTP, and pyrophosphate. Thermal deamination of dCTP to form dUTP is included in the model. The probability for a species to arrive at the active site is based on its molar fraction. The number of random insertion events depends on the average processing speed of the polymerase and the elongation time of the simulation. The numerical stochastic experiment is repeated a sufficient number of times to construct a probability density distribution (PDF). The moment of the PDF and the annealing step products provide the product distribution at the end of the elongation step. The overall yield is compared to six experimental values of the yield. In all cases the comparisons are very good.