Torsion in biochemical reaction networks

Abstract

This article starts in Part I with a simple example of two biochemical reaction networks that are indistinguishable at the macroscopic level but are different at the molecular level and are shown to have significantly different kinetic properties. So, if one completely ignores the fact that reactions advance in discrete steps at the molecular level, then one can fail to distinguish between networks with widely different kinetics. In part II biochemical reaction networks are treated in a general way to discover what property of a network, only seen at the molecular level, affects its kinetics. It is shown that every such network has a unique torsion group which can be described numerically and readily determined by a programmable computation. If the group is found to be the singleton {0} (as is most often the case in practice), then the network is said to be torsion-free and its kinetic properties unaffected by ignoring its discrete character. A chemical reaction network has to be represented algebraically to calculate its torsion group. If the network is to be understood only at the macroscopic level, it can be placed in the context of real vector spaces, but to recognize its discrete character and its torsion group, each vector space is replaced by a discrete subset of that space, where each molecule can be recognized as a distinct and indivisible entity. Next, the process of calculating a torsion group is shown in several cases, including the example in part I. In this particular case it is shown to have the torsion group with 2 elements, reflecting the fact that the substrate molecules become product molecules 2 at a time, with the result that the overall macroscopic reaction is R ⇔ T, whereas at the molecular level it is 2R ⇔ 2T. In general, however, the torsion group of a biochemical reaction network can be any finite additive group, which is a property of the network that can only be seen at the molecular level. Finally, this fact is demonstrated by showing how to construct a hypothetical, but plausible, biochemical reaction network that has any given finite additive group as its torsion group.