Simplifying biochemical models with intermediate species

Abstract

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.

Keywords: algebraic methods; model choice; multistationarity; stability; transient species.

Figure 1.

Representation of reaction networks: (a,b) detailed representation; (c–e) schematic representation. (a) and (e) are core models and (b–d) are extended models of (a) and (e). (a) A reaction network with complexes S₀ + E, S₁ + E, S₂ + E (enclosed in dashed boxes). Each reaction is labelled with its rate constant (k or ). (b) An extension model of network (a) with intermediate Y. (c) The complex y₁ is involved in a reversible ‘dead-end’ reaction with one intermediate. (d) The complex y₁ is converted into Y, which splits into y₂ or y₃, respectively (the former reversibly). (e) Schematic of (a).

Figure 2.

The steady-state curve (4.3) for a₁ = 2, a₂ = 0.5 (dashed) together with the curve for the conservation law. The steady states for a fixed conserved amount are the intersection points of the two curves (dashed and solid lines). (a) Core model. Conservation law curves (solid) for different values of . (b) Extension model. Conservation law curves (solid) as in (4.4) for different values of and a₃ = 2. (c) Extension model. Conservation law curves (solid) as in (4.4) for different values of a₃ and . (Online version in colour.)

Figure 3.

We consider the core model of figure 1a (with y₁ = S₀ + E, y₂ = S₁ + E, y₃ = S₂ + E) and its steady-state classes. Each class is characterized by an extension model (the canonical model) with a dead-end reaction added for each class complex (upper right corner). Each class (except the class of the core model) has an infinite number of members and a few of these are shown. Class complexes are source complexes of a reaction with an intermediate as product (marked in bold in the figure). For the number of steady states we consider the model given in figure 1a and dephosphorylation reactions S₂ → S₁ and S₁ → S₀ (not shown in the figure). The number of steady states in each class refers to the maximal number of steady states that a model in the class can have for some choice of rate constants and total amounts. This has been found by direct computation of the steady states (see electronic supplementary material). Alternatively, the CRN Toolbox could have been used [25].

Figure 4.

A decision tree to detect multistationary steady-state classes. ‘All’ means that the model exhibits multistationarity as long as the rate constants of the core model can be realized by the extension model. Q3 must be checked for different canonical models as necessary.