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The course of the real-time polymerase chain reaction (PCR) is determined by the temperature dependence of the kinetics of the component reactions, particularly the DNA strand hybridization. To investigate the effect of thermal processes on the reaction behavior, a mathematical model in which the variable rate constant of dissociation of "primer-single strand" complexes depends on temperature was proposed. The reaction medium temperature, which depends on time, was also introduced into the model. The proposed model of real-time PCR makes it possible to analyze different aspects of the reaction, which are important for the development of instruments and reagents for PCR.

Development of methods for obtaining approximate analytical solutions of nonlinear differential equations and their systems is a rapidly developing field of mathematical physics. Earlier, an approximate solution of the simplest system of kinetic enzymatic equations for calculating dynamics of complementary strands of nucleic acids was obtained. In this study, we consider an alternative approach to selecting the basic linear approximation of the used method, which makes it possible to obtain more accurate analytical solutions of the set problem.

Macroscopic kinetic models describing the process of polymerase chain reaction (PCR) are currently solved only by numerical methods, which hampers the development of effective software algorithms for processing the results of the reaction. This paper considers the application of the homotopy perturbation method for obtaining approximate analytical solution of the simplest system of enzymatic kinetic equations describing the synthesis of nucleic acid molecules during PCR. The resulting approximate analytic solution with high accuracy reproduces the results of a numerical solution of the system in a wide range of ratios of enzyme and substrate concentrations both for the case of a large excess of the substrate over the enzyme and vice versa.

The paper reviews different approaches to the mathematical analysis of polymerase chain reaction (PCR) kinetic curves. The basic principles of PCR mathematical analysis are presented. Approximation of PCR kinetic curves and PCR efficiency curves by various functions is described. Several PCR models based on chemical kinetics equations are suggested. Decision criteria for an optimal function to describe PCR efficiency are proposed.