Abstract
Large-scale kinetic models of metabolism are becoming increasingly comprehensive and accurate. A key challenge is to understand the biochemical basis of the dynamic properties of these models. Linear analysis methods are well-established as useful tools for characterizing the dynamic response of metabolic networks. Central to linear analysis methods are two key matrices: the Jacobian matrix (J) and the modal matrix (M-1) arising from its eigendecomposition. The modal matrix M-1 contains dynamically independent motions of the kinetic model near a reference state, and it is sparse in practice for metabolic networks. However, connecting the structure of M-1 to the kinetic properties of the underlying reactions is non-trivial. In this study, we analyze the relationship between J, M-1, and the kinetic properties of the underlying network for kinetic models of metabolism. Specifically, we describe the origin of mode sparsity structure based on features of the network stoichiometric matrix S and the reaction kinetic gradient matrix G. First, we show that due to the scaling of kinetic parameters in real networks, diagonal dominance occurs in a substantial fraction of the rows of J, resulting in simple modal structures with clear biological interpretations. Then, we show that more complicated modes originate from topologically-connected reactions that have similar reaction elasticities in G. These elasticities represent dynamic equilibrium balances within reactions and are key determinants of modal structure. The work presented should prove useful towards obtaining an understanding of the dynamics of kinetic models of metabolism, which are rooted in the network structure and the kinetic properties of reactions.
Highlights
In recent years, kinetic models of metabolism have become increasingly detailed, comprehensive, and consistent with the underlying biochemistry and genetics [1,2,3,4,5,6]
To aid in later discussions on mode sparsity structure, we first introduce the key concept of half-reaction equilibria, which appear in G due to linearization of mass action reactions
Dependence of diagonal dominance on the parameters of the metabolic network Having established that diagonal dominance is an important property of kinetic models of metabolism for real networks, we describe the origin of diagonal dominance in terms of the kinetic and physiological parameters of the system
Summary
Kinetic models of metabolism have become increasingly detailed, comprehensive, and consistent with the underlying biochemistry and genetics [1,2,3,4,5,6] These models can address a number of questions that are difficult to analyze directly with constraint-based or statistical models [7,8,9]. A number of studies have made theoretical and practical headway in this regard by analyzing the linear properties of the dynamic system around a steady state. These linear analysis methods have helped to provide insight into metabolic flux control [18, 19], elucidate the temporal hierarchy of dynamic events [20], and describe the fundamental dynamic structure of the network [21]
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