Abstract
The time-scale hierarchies of a very general class of models in differential equations is analyzed. Classical methods for model reduction and time-scale analysis have been adapted to this formalism and a complementary method is proposed. A unified theoretical treatment shows how the structure of the system can be much better understood by inspection of two sets of singular values: one related to the stoichiometric structure of the system and another to its kinetics. The methods are exemplified first through a toy model, then a large synthetic network and finally with numeric simulations of three classical benchmark models of real biological systems.
Highlights
Biochemical systems are amenable to be modeled using differential equations but, due to the great diversity of mechanisms involved, the resulting models lack a defined structure
Control engineering using linear systems is a case in point, where chemical plants, steam engines, and electric systems can all be treated within the same framework
An analysis of the first case need only take into account the subsystem that corresponds to the right time-scale, while the second case would better be analyzed by focusing on interactions between a fast and a slow subsystem
Summary
Biochemical systems are amenable to be modeled using differential equations but, due to the great diversity of mechanisms involved, the resulting models lack a defined structure. Models with a well defined structure enable a great level of abstraction and generality. The analysis of ad-hoc biological models is often restricted to the numerical integration of a few scenarios. The intervening processes often progress at different time-scales, the resulting models tend to be stiff and difficult to analyze. An analysis of the first case need only take into account the subsystem that corresponds to the right time-scale, while the second case would better be analyzed by focusing on interactions between a fast and a slow subsystem. Separating time scales reduces the stiffness of the system, and the computing power needed for numerical integration of the models
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