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

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.

Projected explicit and implicit Taylor series methods for DAEs

The recently developed new algorithm for computing consistent initial values and Taylor coefficients for DAEs using projector-based constrained optimization opens new possibilities to apply Taylor series integration methods. In this paper, we show how corresponding projected explicit and implicit Taylor series methods can be adapted to DAEs of arbitrary index. Owing to our formulation as a projected optimization problem constrained by the derivative array, no explicit description of the inherent dynamics is necessary, and various Taylor integration schemes can be defined in a general framework. In particular, we address higher-order Padé methods that stand out due to their stability. We further discuss several aspects of our prototype implemented in Python using Automatic Differentiation. The methods have been successfully tested on examples arising from multibody systems simulation and a higher-index DAE benchmark arising from servo-constraint problems.

Adjoint Estimation of the Variation in Model Functional Output due to the Assimilation of Data

A parametric approach to the adjoint estimation of the variation in model functional output due to the assimilation of data is considered as a tool to analyze and develop observation impact measures. The parametric approach is specialized to a linear analysis scheme and it is used to derive various high-order approximation equations. This framework includes the Kalman filter and incremental three-and four-dimensional variational data assimilation schemes implementing a single outer loop iteration. Distinction is made between Taylor series methods and numerical quadrature methods. The novel quadrature approximations require minimal additional software development and are suitable for testing and implementation at operational numerical weather prediction centers where a data assimilation system (DAS) and the associated adjoint DAS are in place. Their potential use as tools for observation impact estimates needs to be further investigated. Preliminary numerical experiments are provided using the fifth-generation NASA Goddard Earth Observing System (GEOS-5) atmospheric DAS.