Limit-Cycle-Preserving Simulation of Gene Regulatory Oscillators

In order to simulate gene regulatory oscillators more effectively, Runge-Kutta (RK) integrators are adapted to the limit-cycle structure of the system. Taking into account the oscillatory feature of the gene regulatory oscillators, phase-fitted and amplification-fitted Runge-Kutta (FRK) methods are designed. New FRK methods with phase-fitted and amplification-fitted updated are also considered. The error coefficients and the error constant for each of new FRK methods are obtained. In the numerical simulation of the two-gene regulatory system, the new methods are shown to be more accurate and more efficient than their prototype RK methods in the long-term integration. It is a new discovery that the best fitting frequency not only depends on the problem to be solved, but also depends on the method.