A second-order, unconditionally positive, mass-conserving integration scheme for biochemical systems

Authors:
Jorn Bruggeman;Hans Burchard;Bob W. Kooi;Ben Sommeijer
Affiliations:
Department of Theoretical Biology, Faculty of Earth & Life Sciences, Vrije Universiteit, Amsterdam, The Netherlands;Baltic Sea Research Institute Warnemünde, Rostock-Warnemünde, Germany;Department of Theoretical Biology, Faculty of Earth & Life Sciences, Vrije Universiteit, Amsterdam, The Netherlands;CWI National Research Institute for Mathematics and Computer Science, Amsterdam, The Netherlands
Venue:
Applied Numerical Mathematics
Year:
2007

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Abstract

Biochemical systems are bound by two mathematically-relevant restrictions. First, state variables in such systems represent non-negative quantities, such as concentrations of chemical compounds. Second, biochemical systems conserve mass and energy. Both properties must be reflected in results of an integration scheme applied to biochemical models. This paper first presents a mathematical framework for biochemical problems, which includes an exact definition of biochemical conservation: elements and energy, rather than state variable units, are conserved. We then analyze various fixed-step integration schemes, including traditional Euler-based schemes and the recently published modified Patankar schemes, and conclude that none of these deliver unconditional positivity and biochemical conservation in combination with higher-order accuracy. Finally, we present two new fixed-step integration schemes, one first-order and one second-order accurate, which do guarantee positivity and (biochemical) conservation.