stiff ordinary differential equations
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The double phosphorylation/dephosphorylation cycle consists of a symmetric network of biochemical reactions of paramount importance in many intracellular mechanisms. From a network perspective, they consist of four enzymatic reactions interconnected in a specular way. The general approach to model enzymatic reactions in a deterministic fashion is by means of stiff Ordinary Differential Equations (ODEs) that are usually hard to integrate according to biologically meaningful parameter settings. Indeed, the quest for model simplification started more than one century ago with the seminal works by Michaelis and Menten, and their Quasi Steady-State Approximation methods are still matter of investigation nowadays. This work proposes an effective algorithm based on Taylor series methods that manages to overcome the problems arising in the integration of stiff ODEs, without settling for model approximations. The double phosphorylation/dephosphorylation cycle is exploited as a benchmark to validate the methodology from a numerical viewpoint.

In this research, we developed a uniform order eleven of eight step Second derivative hybrid block backward differentiation formula for integration of stiff systems in ordinary differential equations. The single continuous formulation developed is evaluated at some grid point of x=x_(n+j),j=0,1,2,3,4,5 and6 and its first derivative was also evaluated at off-grid point x=x_(n+j),j=15/2 and grid point x=x_(n+j),j=8. The method is suitable for the solution of stiff ordinary differential equations and the accuracy and stability properties of the newly constructed method are investigated and are shown to be A-stable. Our numerical results obtained are compared with the theoretical solutions as well as ODE23 solver.

Simulating a physical system in real-time is widely used in equipment design, test, and validation. Though an implicit multistep numerical method excels at solving physical models that are usually composed of stiff ordinary differential equations, it is not suitable for real-time simulation because of state discontinuity and massive iterations for root finding. Thus, a method based on the backward differential formula is presented. It divides the main fixed step of real-time simulation into limited minor steps according to computing cost and accuracy demand. By analyzing and testing its capability, this method shows advantage and efficiency in real-time simulation, especially when the system contains stiff equations. A simulation application will have more flexibility while using this method.