Mathematical model for PCR: determination of kinetic parameters

JAVIER MARCELO GONZALEZ; ANA M. KOROL

Congreso; XXI Reunión Anual de la Sociedad de Biología de Rosario; 2002

We have recently developed a mathematical model for polymerase chain reaction (PCR); a non-linear system of two differential equations that simulates the amplification process. Let n be the number of reaction cycles, N the growing population of DNA molecules from an initial value N0, K the environment initial capacity (initial amount of primers), and P the probability of primer-target annealing. The system is {N´=uNP; K´=P´ vN´}, where P=1 (N/K)a, and 0<a<1, 0<u<ln(2) and v>0 are parameters which control the process. a measures the goodness of primer design, u/ln(2) equals the target-sequence duplication efficiency, and the inhibitory effect of essential reagents consumption is ac-counted by v. Numerical analysis of the curve n versus N, obtained by optical PCR devices, allows calculation of the curve inflection point (ni;Ni), its slope at inflection N´i, and the plateau value Nmax. In addition to K0, these data suffice to determine para-meters a, u and v. A geometric analysis allows obtaining the posi-tion and slope of K at its inflection, (ni;Ki) and Ki´. A tangent to N at (ni;Ni) passes by points (n1;Nmax) and (n2;0), where n1=ni-Ni/N´i and n2=ni+(Nmax-Ni)/N´i. A straight line that passes by (n1;Nmax) and (n2;K0) is a good approximation of the tangent to K at (ni;Ki). Thus, K´i»N´i(1-K0/Nmax) and Ki»Nmax+K´i(Ni-Nmax)/N´i. While u is the initial slope of ln(N), the other parameters are determined by v» -N´i/(uN2i)-K´i/N´i and a»ln[1-N´i/(uNi)]/ln(Ni/Ki). The model permits a qualitative and quantitaive characterization of PCR reactions, as well as simulation of responses to certain changes.