Exponentially Fitted Two-Derivative Runge-Kutta Methods for Simulation of Oscillatory Genetic Regulatory Systems

Oscillation is one of the most important phenomena in the chemical reaction systems inliving cells. The general purpose simulation algorithms fail to take into account this specialcharacter and produce unsatisfying results. In order to enhance the accuracy of the integrator,the second-order derivative is incorporated in the scheme. The oscillatory feature of the solutionis captured by the integrators with an exponential fitting property. Three practical exponentiallyfitted TDRK (EFTDRK) methods are derived. To test the effectiveness of the new EFTDRKmethods, the two-gene system with cross-regulation and the circadian oscillation of the periodprotein in Drosophila are simulated. Each EFTDRK method has the best fitting frequencywhich minimizes the global error. The numerical results show that the new EFTDRK methodsare more accurate and more efficient than their prototype TDRK methods or RK methods ofthe same order and the traditional exponentially fitted RK method in the literature.