Open Access Open Badges Methodology article

A specialized ODE integrator for the efficient computation of parameter sensitivities

Pedro Gonnet123, Sotiris Dimopoulos2, Lukas Widmer2 and Jörg Stelling2*

Author Affiliations

1 Mathematical Institute, University of Oxford, Oxford, UK

2 Department of Biosystems Science and Engineering, , ETH Zurich, 8092 Zürich, Switzerland

3 School of Engineering and Computing Sciences, Durham University, UK

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BMC Systems Biology 2012, 6:46  doi:10.1186/1752-0509-6-46

Published: 20 May 2012



Dynamic mathematical models in the form of systems of ordinary differential equations (ODEs) play an important role in systems biology. For any sufficiently complex model, the speed and accuracy of solving the ODEs by numerical integration is critical. This applies especially to systems identification problems where the parameter sensitivities must be integrated alongside the system variables. Although several very good general purpose ODE solvers exist, few of them compute the parameter sensitivities automatically.


We present a novel integration algorithm that is based on second derivatives and contains other unique features such as improved error estimates. These features allow the integrator to take larger time steps than other methods. In practical applications, i.e. systems biology models of different sizes and behaviors, the method competes well with established integrators in solving the system equations, and it outperforms them significantly when local parameter sensitivities are evaluated. For ease-of-use, the solver is embedded in a framework that automatically generates the integrator input from an SBML description of the system of interest.


For future applications, comparatively ‘cheap’ parameter sensitivities will enable advances in solving large, otherwise computationally expensive parameter estimation and optimization problems. More generally, we argue that substantially better computational performance can be achieved by exploiting characteristics specific to the problem domain; elements of our methods such as the error estimation could find broader use in other, more general numerical algorithms.

Dynamical models; ordinary differential equations; parameter sensitivities; integration