A growing understanding of complex processes in biology has led to large-scale mechanistic models of pharmacologically relevant processes. These models are increasingly used to study the response of the system to a given input or stimulus, e.g., after drug administration. Understanding the input-response relationship, however, is often a challenging task due to the complexity of the interactions between its constituents as well as the size of the models. An approach that quantifies the importance of the different constituents for a given input-output relationship and allows to reduce the dynamics to its essential features is therefore highly desirable. In this article, we present a novel state- and time-dependent quantity called the input-response index that quantifies the importance of state variables for a given input-response relationship at a particular time. It is based on the concept of time-bounded controllability and observability, and defined with respect to a reference dynamics. In application to the brown snake venom-fibrinogen (Fg) network, the input-response indices give insight into the coordinated action of specific coagulation factors and about those factors that contribute only little to the response. We demonstrate how the indices can be used to reduce large-scale models in a two-step procedure: (i) elimination of states whose dynamics have only minor impact on the input-response relationship, and (ii) proper lumping of the remaining (lower order) model. In application to the brown snake venom-fibrinogen network, this resulted in a reduction from 62 to 8 state variables in the first step, and a further reduction to 5 state variables in the second step. We further illustrate that the sequence, in which a recursive algorithm eliminates and/or lumps state variables, has an impact on the final reduced model. The input-response indices are particularly suited to determine an informed sequence, since they are based on the dynamics of the original system. In summary, the novel measure of importance provides a powerful tool for analysing the complex dynamics of large-scale systems and a means for very efficient model order reduction of nonlinear systems.
Keywords: Blood coagulation network; Control theory; Model order reduction; Nonlinear systems.