Functional similarities of reaction sets in metabolic pathways

Analyzing metabolic pathways by means of their steady states has proven to be accurate and efficient for practical purposes. The models such as elementary flux modes (EFMs) and extreme pathways(EPs) define the boundaries of the metabolic flux cone that is the set of all steady states of a pathway. However, the contributions of the subsets of pathway components in this flux cone so far has not been characterized mathematically. Also, the functional similarities of different component sets (e.g., sets of reactions) has not been expressed as a function of the steady states of metabolic pathways. Here, we aim to fill this gap by proposing a model that quantifies the impact of a set of components on the steady states of a pathway by using EFMs. At a high level, we model the impact of a given component set as the change in the flux cone when all the elements of that set are inhibited. Furthermore, given two sets of components from different pathways, we measure their functional similarity as the similarity of their impacts on corresponding pathways. Computing this functional similarity is a computationally challenging task as it requires finding the volumes of the intersection and the union of two polyhedral cones in high dimensional space. These volumes cannot be expressed in closed form. In this paper, we develop a novel method that first transforms the polyhedral cones to polytopes and then uses minimum enclosing balls to approximate this intersection efficiently. Our experiments on real metabolic pathways demonstrate that our method is of great use for both measuring the impacts of pathway components and identifying functionally similar component sets.