Biochemical systems theory

The dynamics of a species is represented by a differential equation with the structure:

${\displaystyle {\frac {dX_{i}}{dt}}=\sum _{j}\mu _{ij}\cdot \gamma _{j}\prod _{k}X_{k}^{f_{jk}}\,}$

where X_i represents one of the n_d variables of the model (metabolite concentrations, protein concentrations or levels of gene expression). j represents the n_f biochemical processes affecting the dynamics of the species. On the other hand, ${\displaystyle \mu }$_ij (stoichiometric coefficient), ${\displaystyle \gamma }$_j (rate constants) and f_jk (kinetic orders) are two different kinds of parameters defining the dynamics of the system.

The principal difference of power-law models with respect to other ODE models used in biochemical systems is that the kinetic orders can be non-integer numbers. A kinetic order can have even negative value when inhibition is modeled. In this way, power-law models have a higher flexibility to reproduce the non-linearity of biochemical systems.

Models using power-law expansions have been used during the last 35 years to model and analyze several kinds of biochemical systems including metabolic networks, genetic networks and recently in cell signalling.

• Dynamical systems
• Ludwig von Bertalanffy
• Systems theory

Books:

• M.A. Savageau, Biochemical systems analysis: a study of function and design in molecular biology, Reading, MA, Addison–Wesley, 1976.
• E.O. Voit (ed), Canonical Nonlinear Modeling. S-System Approach to Understanding Complexity, Van Nostrand Reinhold, NY, 1991.
• E.O. Voit, Computational Analysis of Biochemical Systems. A Practical Guide for Biochemists and Molecular Biologists, Cambridge University Press, Cambridge, U.K., 2000.
• N.V. Torres and E.O. Voit, Pathway Analysis and Optimization in Metabolic Engineering, Cambridge University Press, Cambridge, U.K., 2002.

Scientific articles:

• M.A. Savageau, Biochemical systems analysis: I. Some mathematical properties of the rate law for the component enzymatic reactions in: J. Theor. Biol. 25, pp. 365–369, 1969.
• M.A. Savageau, Development of fractal kinetic theory for enzyme-catalysed reactions and implications for the design of biochemical pathways in: Biosystems 47(1-2), pp. 9–36, 1998.
• M.R. Atkinson et al., Design of gene circuits using power-law models, in: Cell 113, pp. 597–607, 2003.
• F. Alvarez-Vasquez et al., Simulation and validation of modelled sphingolipid metabolism in Saccharomyces cerevisiae, Nature 27, pp. 433(7024), pp. 425–30, 2005.
• J. Vera et al., Power-Law models of signal transduction pathways in: Cellular Signalling doi:10.1016/j.cellsig.2007.01.029), 2007.
• Eberhart O. Voit, Applications of Biochemical Systems Theory, 2006.

• Savageau Lab at UC Davis
• Voit Lab at GA Tech