Modeling and Analysis of Large-Scale Networks
In the previous chapters, we have discussed the mathematical modeling and dynamical analysis of several simple circuits and coupled genetic circuits. Generally, the established models are ordinary differential equations or stochastic differential equation
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Modeling and Analysis of Large-Scale Networks
Abstract In the previous chapters, we have discussed the mathematical modeling and dynamical analysis of several simple circuits and coupled genetic circuits. Generally, the established models are ordinary differential equations or stochastic differential equations, which are always with the Michaelis–Menten or Hill forms. The ordinary or stochastic differential equation models are inappropriate for largescale networks, due to complexity. In this chapter, we introduce some works on large-scale networks. The word “large-scale” is a relative concept. When we discuss discrete or continuous models for bio-molecular networks, we call networks with tens of nodes as large-scale networks. For hybrid discrete and continuous models and the percolating flow model, we discuss their applications in networks with hundreds or thousands of nodes.
5.1 Backgrounds Real-world bio-molecular networks are large-scale ones, which contain hundreds to tens of thousands of nodes. It is often a difficult task to mathematically model these large-scale networks [1–13]. There are three main reasons. Firstly, for large networks, the traditional ODE or SDE models are often too complex, there are too many parameters to be estimated [12], and parameter estimation is a difficult task in biological systems. Secondly, too large ODEs or SDEs are difficult to be analyzed. The ODEs or SDEs always contain the Hill equations, and numerically solving of these equations is time-consuming. Thirdly, for large-scale networks, there are too many state variables (high dimensional), each variable is always affected by many other nodes, and it is difficult to distinguish which nodes are the main elements of the networks for certain behaviors and should be considered in great detail. In the existing works, some large networks have been modeled and analyzed. In the following, we briefly overview some of the frequently explored networks. The first network is the yeast cell cycle network, which contains hundreds of nodes, as shown in Fig. 5.1. The complete yeast cell cycle network is very complex; however, the core network that controls the cell cycle only contains tens of nodes, and © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2020 J. Lü, P. Wang, Modeling and Analysis of Bio-molecular Networks, https://doi.org/10.1007/978-981-15-9144-0_5
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Fig. 5.1 A comprehensive molecular interaction map for the budding yeast cell cycle. A total of 880 species and 732 reactions are included. Copyright ©(2010) Wiley. Used with permission from Ref. [14]
250 5 Modeling and Analysis of Large-Scale Networks
5.1 Backgrounds
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Fig. 5.2 The yeast cell cycle network. (a) The core cell cycle network for the budding yeast. (b) The simplified cell cycle network for the budding yeast with only one checkpoint “cell size”. Reprinted from Ref. [11] (Copyright (2004) National Academy of Sciences, U.S.A.). (c) The cell cycle control system for the fission yeast. This system
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