Monotone and near-monotone biochemical networks.
Bottom Line:
Monotone subsystems have appealing properties as components of larger networks, since they exhibit robust dynamical stability and predictability of responses to perturbations.This suggests that natural biological systems may have evolved to be, if not monotone, at least close to monotone in the sense of being decomposable into a "small" number of monotone components, In addition, recent research has shown that much insight can be attained from decomposing networks into monotone subsystems and the analysis of the resulting interconnections using tools from control theory.This paper provides an expository introduction to monotone systems and their interconnections, describing the basic concepts and some of the main mathematical results in a largely informal fashion.
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Affiliation: Rutgers University, New Brunswick, NJ, USA, sontag@math.rutgers.edu.
ABSTRACT
Monotone subsystems have appealing properties as components of larger networks, since they exhibit robust dynamical stability and predictability of responses to perturbations. This suggests that natural biological systems may have evolved to be, if not monotone, at least close to monotone in the sense of being decomposable into a "small" number of monotone components, In addition, recent research has shown that much insight can be attained from decomposing networks into monotone subsystems and the analysis of the resulting interconnections using tools from control theory. This paper provides an expository introduction to monotone systems and their interconnections, describing the basic concepts and some of the main mathematical results in a largely informal fashion. No MeSH data available. |
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Mentions: In this example, which in fact was the one whose study initially led to the definition of MIOS, the following conservation laws: (total MAPKK) and (total MAPK) hold true, assuming no protein turn-over. This assumption is standard in most of the literature, because transcription and degradation occur at time scales much slower than signaling. (There is very recent experimental data that suggests that turn-over might be fast for some yeast MAPK species. Adding turn-over would lead to a different mathematical model.) These conservation laws allow us to eliminate variables. The right trick is to eliminate y2 and z2. Once we do this, and write and we are left with the variables For instance, the equations for look like:\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$ \frac{dz_1}{dt}=-\alpha (z_1,y_3) + \beta (z_{\rm tot}-z_1-z_3)\quad\frac{dz_3}{dt}=\gamma(z_{\rm tot}-z_1-z_3,y_3) -\delta(z_3) $$\end{document}for appropriate increasing functions The equations for the remaining variables are similar. The graph, ignoring, as usual, self-loops (diagonal of Jacobian), is shown in Fig. 19b. This graph has no negative undirected loops, showing that the (reduced) system is monotone. A consistent spin assignment (including the top input node and the bottom output node) is shown in Fig. 20. It is also true that this system has a well-defined monostable state space response (characteristic); there is no space to discuss the proof here, so we refer the reader to the original papers (Angeli and Sontag 2003, 2004b).Fig. 20 |
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Affiliation: Rutgers University, New Brunswick, NJ, USA, sontag@math.rutgers.edu.
No MeSH data available.