The Double Phospho/Dephosphorylation Cycle as a Benchmark to Validate an Effective Taylor Series Method to Integrate Ordinary Differential Equations

Alessandro Borri,Pasquale Palumbo,Francesco Carravetta

doi:10.3390/sym13091684

Abstract

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.

Highlights

Protein phosphorylation is a ubiquitous regulatory mechanism for cells, generally working to activate or inactivate molecules [1]
In [17] we showed how the numerical integration of a basic enzymatic reaction model may create serious problems even to well established procedures like the ones implemented by Matlab in ode45 and ode15s functions
We proposed a novel approach to the problem of integrating the solution of biological systems expressed by stiff differential equations

Summary

Introduction

Protein phosphorylation is a ubiquitous regulatory mechanism for cells, generally working to activate or inactivate molecules [1]. To this end, in the present work, we employ some recent technical results published in [24,25], according to which ODE systems can be embedded into higher-order quadratic equations which, in spite of their dimensionality, allow for more efficient differentiation, which in turn can be exploited for numerical integration via TSM. In the present work, we employ some recent technical results published in [24,25], according to which ODE systems can be embedded into higher-order quadratic equations which, in spite of their dimensionality, allow for more efficient differentiation, which in turn can be exploited for numerical integration via TSM These results have been already exploited in [17] for the class of simple enzymatic reactions (two differential equations), and are here extended to the more challenging DPDC case (seven differential equations).

Exact Quadratization and Approximate Integration of σπ Differential Equations

Approximate Taylor Series Integration Method

The Double Phosphosphorylation–Dephosphorylation Cycle

Simulation Results

Discussion