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# Stability of metabolic networks via Linear-in-Flux-Expressions

• * Corresponding author: Benedetto Piccoli
• The methodology named LIFE (Linear-in-Flux-Expressions) was developed with the purpose of simulating and analyzing large metabolic systems. With LIFE, the number of model parameters is reduced by accounting for correlations among the parameters of the system. Perturbation analysis on LIFE systems results in less overall variability of the system, leading to results that more closely resemble empirical data. These systems can be associated to graphs, and characteristics of the graph give insight into the dynamics of the system.

This work addresses two main problems: 1. for fixed metabolite levels, find all fluxes for which the metabolite levels are an equilibrium, and 2. for fixed fluxes, find all metabolite levels which are equilibria for the system. We characterize the set of solutions for both problems, and show general results relating stability of systems to the structure of the associated graph. We show that there is a structure of the graph necessary for stable dynamics. Along with these general results, we show how stability analysis from the fields of network flows, compartmental systems, control theory and Markov chains apply to LIFE systems.

Mathematics Subject Classification: Primary: 92C42, 05C21; Secondary: 93B99, 34H99.

 Citation: • • Figure 2.  A directed graph $\tilde G = (\tilde{V}, \tilde{E})$ illustrating Proposition 2. Vertices $v_3$ and $v_4$ form a terminal component. There exists a path from $v_0$ to $v_4$ yet there is no path from $v_4$ to $v_5$

Figure 1.  A directed graph $\tilde G = (\tilde{V}, \tilde{E})$ representing a biochemical system. The rectangles indicate virtual vertices and the subgraph of circular vertices and edges connecting them is $G = (V, E)$

Figure 3.  A directed graph where vertices $v_3$ and $v_4$ do not have a path from $v_0$ and also have no path to $v_5$. For an equilibrium, $\bar{x}$, of this system under Assumption (A), $\bar{x}_{v_4} = 0$ and $\bar{x}_{v_3} \geq x_{v_3}(0)$

Figure 4.  A directed cycle graph $G = (V, E)$ with $n$ vertices and no intakes nor excretions. On such a LIFE system one can prescribe any desired dynamics

Figure 5.  Reverse Cholesterol Transport Network from . This network contains 6 vertices which represent metabolites, 10 edges which represent fluxes and 2 virtual vertices $v_0, v_{n+1}$. There are three intake vertices $v_1, v_2, v_3$ and 1 excretion vertex $v_{6}$

Figure 6.  The trajectories of the values of metabolites over 25 hours

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