# American Institute of Mathematical Sciences

• Previous Article
On the role of tumor heterogeneity for optimal cancer chemotherapy
• NHM Home
• This Issue
• Next Article
A case study of optimal input-output system with sampled-data control: Ding et al. force and fatigue muscular control model
March  2019, 14(1): 101-130. doi: 10.3934/nhm.2019006

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

 1 Center for Computational and Integrative Biology, Rutgers Camden. Camden NJ, USA 2 Researcher, Univ. Grenoble Alpes, Inria, CNRS, Grenoble INP, GIPSA-lab. Grenoble, France 3 Bill & Melinda Gates Medical Research Institute, 245 Main Street Kendall Square. Cambridege, MA 02142, USA 4 Senior scientist, Translational Informatics, Sanofi. Bridgewater NJ, USA 5 Joseph and Loretta Lopez chair professor of Mathematics, Center for Computational and Integrative Biology, Rutgers Camden. Camden NJ, USA

* Corresponding author: Benedetto Piccoli

Received  June 2018 Published  January 2019

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.

Citation: Nathaniel J. Merrill, Zheming An, Sean T. McQuade, Federica Garin, Karim Azer, Ruth E. Abrams, Benedetto Piccoli. Stability of metabolic networks via Linear-in-Flux-Expressions. Networks & Heterogeneous Media, 2019, 14 (1) : 101-130. doi: 10.3934/nhm.2019006
##### References:

show all references

##### References:
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 $v_5$
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 $G = (V, E)$
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 $\bar{x}_{v_3} \geq x_{v_3}(0)$
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
Reverse Cholesterol Transport Network from [19]. 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 $v_{6}$
The trajectories of the values of metabolites over 25 hours
 [1] Tomasz Kapela, Piotr Zgliczyński. A Lohner-type algorithm for control systems and ordinary differential inclusions. Discrete & Continuous Dynamical Systems - B, 2009, 11 (2) : 365-385. doi: 10.3934/dcdsb.2009.11.365 [2] Bin Wang, Arieh Iserles. Dirichlet series for dynamical systems of first-order ordinary differential equations. Discrete & Continuous Dynamical Systems - B, 2014, 19 (1) : 281-298. doi: 10.3934/dcdsb.2014.19.281 [3] Serge Nicaise. Control and stabilization of 2 × 2 hyperbolic systems on graphs. Mathematical Control & Related Fields, 2017, 7 (1) : 53-72. doi: 10.3934/mcrf.2017004 [4] Eugene Kashdan, Dominique Duncan, Andrew Parnell, Heinz Schättler. Mathematical methods in systems biology. Mathematical Biosciences & Engineering, 2016, 13 (6) : i-ii. doi: 10.3934/mbe.201606i [5] Ping Lin, Weihan Wang. Optimal control problems for some ordinary differential equations with behavior of blowup or quenching. Mathematical Control & Related Fields, 2018, 8 (3&4) : 809-828. doi: 10.3934/mcrf.2018036 [6] N. Bellomo, A. Bellouquid. From a class of kinetic models to the macroscopic equations for multicellular systems in biology. Discrete & Continuous Dynamical Systems - B, 2004, 4 (1) : 59-80. doi: 10.3934/dcdsb.2004.4.59 [7] Angelo B. Mingarelli. Nonlinear functionals in oscillation theory of matrix differential systems. Communications on Pure & Applied Analysis, 2004, 3 (1) : 75-84. doi: 10.3934/cpaa.2004.3.75 [8] Santiago Capriotti. Dirac constraints in field theory and exterior differential systems. Journal of Geometric Mechanics, 2010, 2 (1) : 1-50. doi: 10.3934/jgm.2010.2.1 [9] Heiko Enderling, Alexander R.A. Anderson, Mark A.J. Chaplain, Glenn W.A. Rowe. Visualisation of the numerical solution of partial differential equation systems in three space dimensions and its importance for mathematical models in biology. Mathematical Biosciences & Engineering, 2006, 3 (4) : 571-582. doi: 10.3934/mbe.2006.3.571 [10] Monique Chyba, Benedetto Piccoli. Special issue on mathematical methods in systems biology. Networks & Heterogeneous Media, 2019, 14 (1) : ⅰ-ⅱ. doi: 10.3934/nhm.20191i [11] Marissa Condon, Alfredo Deaño, Arieh Iserles. On systems of differential equations with extrinsic oscillation. Discrete & Continuous Dynamical Systems - A, 2010, 28 (4) : 1345-1367. doi: 10.3934/dcds.2010.28.1345 [12] Bernard Dacorogna, Alessandro Ferriero. Regularity and selecting principles for implicit ordinary differential equations. Discrete & Continuous Dynamical Systems - B, 2009, 11 (1) : 87-101. doi: 10.3934/dcdsb.2009.11.87 [13] Zvi Artstein. Averaging of ordinary differential equations with slowly varying averages. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 353-365. doi: 10.3934/dcdsb.2010.14.353 [14] Elena Goncharova, Maxim Staritsyn. On BV-extension of asymptotically constrained control-affine systems and complementarity problem for measure differential equations. Discrete & Continuous Dynamical Systems - S, 2018, 11 (6) : 1061-1070. doi: 10.3934/dcdss.2018061 [15] Fritz Colonius, Alexandre J. Santana. Topological conjugacy for affine-linear flows and control systems. Communications on Pure & Applied Analysis, 2011, 10 (3) : 847-857. doi: 10.3934/cpaa.2011.10.847 [16] Lukáš Adam, Jiří Outrata. On optimal control of a sweeping process coupled with an ordinary differential equation. Discrete & Continuous Dynamical Systems - B, 2014, 19 (9) : 2709-2738. doi: 10.3934/dcdsb.2014.19.2709 [17] Tayel Dabbous. Identification for systems governed by nonlinear interval differential equations. Journal of Industrial & Management Optimization, 2012, 8 (3) : 765-780. doi: 10.3934/jimo.2012.8.765 [18] Judith R. Miller, Huihui Zeng. Stability of traveling waves for systems of nonlinear integral recursions in spatial population biology. Discrete & Continuous Dynamical Systems - B, 2011, 16 (3) : 895-925. doi: 10.3934/dcdsb.2011.16.895 [19] Urszula Ledzewicz, Stanislaw Walczak. Optimal control of systems governed by some elliptic equations. Discrete & Continuous Dynamical Systems - A, 1999, 5 (2) : 279-290. doi: 10.3934/dcds.1999.5.279 [20] Krešimir Burazin, Marko Vrdoljak. Homogenisation theory for Friedrichs systems. Communications on Pure & Applied Analysis, 2014, 13 (3) : 1017-1044. doi: 10.3934/cpaa.2014.13.1017

2018 Impact Factor: 0.871