The best models of metabolism

Abstract

Biochemical systems are among of the oldest application areas of mathematical modeling. Spanning a time period of over one hundred years, the repertoire of options for structuring a model and for formulating reactions has been constantly growing, and yet, it is still unclear whether or to what degree some models are better than others and how the modeler is to choose among them. In fact, the variety of options has become overwhelming and difficult to maneuver for novices and experts alike. This review outlines the metabolic model design process and discusses the numerous choices for modeling frameworks and mathematical representations. It tries to be inclusive, even though it cannot be complete, and introduces the various modeling options in a manner that is as unbiased as that is feasible. However, the review does end with personal recommendations for the choices of default models. WIREs Syst Biol Med 2017, 9:e1391. doi: 10.1002/wsbm.1391 For further resources related to this article, please visit the WIREs website.

Figure 1. Model Design Process

Top left: Modeling ideas lead to the selection of (blue) metabolites, (red) reactions, and (gold) regulatory signals, as far as they are known. Top right: These components are arranged in the format of a dynamic system, consisting of a static metabolic network of reactions and their regulation. Bottom right: Each reaction in this dynamic system needs to be mathematically formulated, with account of regulatory signals. Bottom left: The identification of the most appropriate model structure and representations faces many difficult choices. The full characterization of all reactions, including the determination of parameter values, completes the model design. Green arrows indicate that several iterations of the model design process may be necessary.

Figure 2. The diagram in the top-left panel was modeled with two slightly differing models (see Text)

The responses to moderate perturbations (top-right: X₂(0) = Y₂(0) = 1.5 and bottom-left (X₁(0) = Y₁(0) = 0.2) are quite similar. However, a bolus of 2 units added to X₁ during the time period t ∈ [3, 8] triggers very different response trends (bottom-right).

Figure 3. Dynamic Flux Estimation (DFE)

Beginning with the diagram of regulated fluxes (Fig. 1, bottom right), DFE separates the linear stoichiometry from the nonlinear fluxes. Specifically, at a series of time points, the distributions of fluxes are given by linear algebraic systems. These systems are solved and flux values are plotted either against time or against the variables that affect them. One may attempt to find explicit functions for these plots or pursue nonparametric modeling. The subsystem of K, L, M, and N is demonstrated here with the model: K̇ = 1 − v_KM; L̇ = 0.2 − L/(4 + L); Ṁ = v_KM + L/(4 + L) − 1.2 M^0.2; Ṅ = 1.2 M^0.2 − 1.2 N^0.8; v_KM = K^0.4 N^-1; (K₀, L₀ M₀, N₀) = (0, 8, 0.1, 10).