Simplifying biochemical models with intermediate species

Elisenda Feliu Carsten Wiuf Articles on similar topics can be found in the following collections Receive free email alerts when new articles cite this article - sign up in the box at the top right-hand corner of the article or click here References Subject collections Email alerting service rsif.royalsocietypublishing.org Research Electronic supplementary material is available at http://dx.doi.org/10.1098/rsif.2013.0484 or via http://rsif.royalsocietypublishing.org. Simplifying biochemical models with intermediate species Elisenda Feliu and Carsten Wiuf Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, 2100 Copenhagen, Denmark Mathematical models are increasingly being used to understand complex biochemical systems, to analyse experimental data and make predictions about unobserved quantities. However, we rarely know how robust our conclusions are with respect to the choice and uncertainties of the model. Using algebraic techniques, we study systematically the effects of intermediate, or transient, species in biochemical systems and provide a simple, yet rigorous mathematical classification of all models obtained from a core model by including intermediates. Main examples include enzymatic and post-translational modification systems, where intermediates often are considered insignificant and neglected in a model, or they are not included because we are unaware of their existence. All possible models obtained from the core model are classified into a finite number of classes. Each class is defined by a mathematically simple canonical model that characterizes crucial dynamical properties, such as mono- and multistationarity and stability of steady states, of all models in the class. We show that if the core model does not have conservation laws, then the introduction of intermediates does not change the steady-state concentrations of the species in the core model, after suitable matching of parameters. Importantly, our results provide guidelines to the modeller in choosing between models and in distinguishing their properties. Further, our work provides a formal way of comparing models that share a common skeleton. 1. Introduction Systems biology aims to understand complex systems and to build mathematical models that are useful for inference and prediction. However, model building is rarely straightforward, and we typically seek a compromise between the simple and the accurate, shaped by our current knowledge of the system. Two models of the same system, potentially differing in the number of species and the form of reactions, might have different qualitative properties and the conclusions we draw from analysing the models might be strongly modeldependent. The predictive value and biological validity of the conclusions might thus be questioned. It is therefore important to understand the role and consequences of model choice and model uncertainty in modelling biochemical systems. Transient, or intermediate, species in biochemical reaction pathways are often ignored in models or grouped into a single or few components, either for reasons of simplicity or conceptual clarification, or because of lack of knowledge. For example, models of the multiple phosphorylation systems vary considerably in the details of intermediates [1,2], and intermediates are often ignored in models of phosphorelays and two-component systems [3,4]. Typically, intermediate species are protein complexes such as a kinase substrate protein complex. It has been shown that sequestration of intermediates can cause ultrasensitive behaviour in some systems [5,6]. Therefore, the inclusion/exclusion of intermediates is a matter of considerable concern. As an example, consider the transfer of a modifier molecule, such as a phosphate group in a two-component system, from one molecule to another: A B O . . . O A B , where A, B are unmodified forms (without the modifier group), A*, B* are modified forms (with the modifier), O indicate & 2013 The Author(s) Published by the Royal Society. All rights reserved. reversible reactions, and . . . are potential transient reaction steps. Two-component systems are ubiquitous in nature and vary considerably in architecture and mechanistic details across species and functionality [7]. Whether or not the specifics are known beforehand, it is customary to use a reduced scheme such as A B O A B [3,4]. We use chemical reaction network theory (CRNT) to model a system of biochemical reactions and assume that the reaction rates follow mass-action kinetics. The polynomial form of the reaction rates has made it possible to apply algebraic techniques to learn about qualitative properties of models, without resorting to numerical approaches [8 14]. Building on previous works [15 17], we propose a mathematical framework to compare different models and to study the dynamical properties of models that differ in how intermediates are included. The most fundamental and crucial dynamical features are the number and stability of steady states. We assume that the kinetic parameters are unknown and study the capacity of each model to exhibit different steady-state features. The paper is organized in the following way. We first introduce the concepts of a core model and an extension model. An extension model is constructed from the core model by including intermediates. Next, we discuss how the steady-state equations of different models are related and illustrate the findings with an example. We proceed to discuss the number of steady states of core and extension models. After that, we introduce the steady state classes, a key concept of this paper. Extension models in the same steady-state classes have the same properties at steady state ( provided the parameter sets of the two models can be matched, in some sense). Using these ideas, we build a decision tree to guide the modeller in choosing a model and in understanding the consequences of choosing a particular model. Finally, we illustrate our approach with an example based on two-component systems. All proofs and mathematical details are in the electronic supplementary material. 2. The core model and its extensions We use the notation and formalism of CRNT [18,19]. A reaction network is defined as a set of species, denoted by capital letters (e.g. A, B, C ), a set of complexes and a set of reactions between complexes. Each complex is a combination of species, for example y1 A B or y2 2C (not to be confused with a protein complex). A potential reaction could be A B ! 2C, or also written simply y1 ! y2. A reaction is not necessarily reversible, that is, we can have A B ! 2C without having the reverse reaction 2C ! A B. Whenever a reaction is reversible, we model it as two separate irreversible reactions. We assume that each reaction occurs according to mass-action kinetics, that is, at a rate proportional to the product of the species concentrations in the reactant or source comp (...truncated)