An improved and generalized second order, unconditionally positive, mass conserving integration scheme for biochemical systems

  • Authors:

  • N. Broekhuizen;Graham J. Rickard;J. Bruggeman;A. Meister

  • Affiliations:

  • National Institute of Water & Atmospheric Research, PO Box 11-115, Gate 10, Silverdale road, Hamilton, New Zealand;National Institute of Water & Atmospheric Research, Private Bag 14901, 301 Evans Bay Parade, Wellington, New Zealand;Department of Theoretical Biology, Faculty of Earth and Life Sciences, Vrije Universiteit, De Boelelaan 1087, 1081 HV Amsterdam, The Netherlands;Department of Mathematics and Computer Science, University of Kassel, Heinrich Plett street 40 (AVZ), D-34132 Kassel, Germany

  • Venue:
  • Applied Numerical Mathematics
  • Year:
  • 2008

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Abstract

Bruggeman et al. [J. Bruggeman, H. Burchard, B. Kooi, B. Sommeijer, A second-order, unconditionally positive, mass-conserving integration scheme for biochemical systems, Applied Numerical Mathematics 57 (1) (2007) 36-58] presented novel first and second-order implicit integration schemes which guarantee both conservation (in a strict biochemical sense) and positive-definite results (hereafter, BBKS1 and BBKS2 respectively). In this paper we show that it is possible to achieve substantially more accurate results by making a minor modification to the BBKS-schemes (hereafter, we refer to the revised first- and second-order schemes as mBBKS1 and mBBKS2). The BBKS and the mBBKS schemes are shown to be special cases of a more generalized scheme (dubbed gBBKS). All operate by automatically slowing the forecaster time-step-averaged reaction rates in order to maintain positivity. The mBBKS scheme induces less slowing than the BBKS one. With a second modification, gBBKS-type schemes can become unusual adaptive-time-step schemes. Unfortunately, for the ODE-systems that we examined, the adaptive-mBBKS variant proves to be substantially less efficient than standard adaptive schemes (in this instance, adaptive time-step second-order explicit Runge-Kutta). Nonetheless, it is possible that the adaptive-mBBKS-scheme would become more competitive when the right-hand sides of a system of ODEs are more expensive to evaluate.