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Open Access Methodology article

A variational approach to parameter estimation in ordinary differential equations

Daniel Kaschek1* and Jens Timmer1234

Author Affiliations

1 Institute of Physics, Freiburg University, Freiburg, Germany

2 Freiburg Center for Systems Biology (ZBSA), Freiburg University, Freiburg, Germany

3 Freiburg Institute for Advanced Studies (FRIAS), Freiburg University, Freiburg, Germany

4 BIOSS Centre for Biological Signalling Studies, Freiburg University, Freiburg, Germany

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BMC Systems Biology 2012, 6:99  doi:10.1186/1752-0509-6-99

Published: 14 August 2012



Ordinary differential equations are widely-used in the field of systems biology and chemical engineering to model chemical reaction networks. Numerous techniques have been developed to estimate parameters like rate constants, initial conditions or steady state concentrations from time-resolved data. In contrast to this countable set of parameters, the estimation of entire courses of network components corresponds to an innumerable set of parameters.


The approach presented in this work is able to deal with course estimation for extrinsic system inputs or intrinsic reactants, both not being constrained by the reaction network itself. Our method is based on variational calculus which is carried out analytically to derive an augmented system of differential equations including the unconstrained components as ordinary state variables. Finally, conventional parameter estimation is applied to the augmented system resulting in a combined estimation of courses and parameters.


The combined estimation approach takes the uncertainty in input courses correctly into account. This leads to precise parameter estimates and correct confidence intervals. In particular this implies that small motifs of large reaction networks can be analysed independently of the rest. By the use of variational methods, elements from control theory and statistics are combined allowing for future transfer of methods between the two fields.

Parameter estimation; Calculus of variations; Boundary value problem; Optimal control; Reaction networks; Ordinary differential equations; Statistical inference