title: Fast integration-based prediction bands for ordinary differential equation models.
abstract: To gain a deeper understanding of biological processes and their relevance in disease, mathematical models are built upon experimental data. Uncertainty in the data leads to uncertainties of the model's parameters and in turn to uncertainties of predictions. Mechanistic dynamic models of biochemical networks are frequently based on nonlinear differential equation systems and feature a large number of parameters, sparse observations of the model components and lack of information in the available data. Due to the curse of dimensionality, classical and sampling approaches propagating parameter uncertainties to predictions are hardly feasible and insufficient. However, for experimental design and to discriminate between competing models, prediction and confidence bands are essential. To circumvent the hurdles of the former methods, an approach to calculate a profile likelihood on arbitrary observations for a specific time point has been introduced, which provides accurate confidence and prediction intervals for nonlinear models and is computationally feasible for high-dimensional models.
  
  
  
  In this article, reliable and smooth point-wise prediction and confidence bands to assess the model's uncertainty on the whole time-course are achieved via explicit integration with elaborate correction mechanisms. The corresponding system of ordinary differential equations is derived and tested on three established models for cellular signalling. An efficiency analysis is performed to illustrate the computational benefit compared with repeated profile likelihood calculations at multiple time points.
  
  
  
  The integration framework and the examples used in this article are provided with the software package Data2Dynamics, which is based on MATLAB and freely available at http://www.data2dynamics.org
  
  
  
  helge.hass@fdm.uni-freiburg.de
  
  
  
  Supplementary data are available at Bioinformatics online.
authors.affiliation: Institute of Physics, University of Freiburg and.
authors.forename: Helge
authors.initials: H
authors.lastname: Hass
authors.affiliation: Institute of Physics, University of Freiburg and.
authors.forename: Clemens
authors.initials: C
authors.lastname: Kreutz
authors.affiliation: Institute of Physics, University of Freiburg and BIOSS Centre for Biological Signalling Studies, University of Freiburg, Freiburg im Breisgau, Germany.
authors.forename: Jens
authors.initials: J
authors.lastname: Timmer
authors.affiliation: Institute of Physics, University of Freiburg and.
authors.forename: Daniel
authors.initials: D
authors.lastname: Kaschek
mesh_terms: D008954:Models, Biological; D017711:Nonlinear Dynamics; D011336:Probability; D012107:Research Design; D035501:Uncertainty