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

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.