Network stability analysis

One of the usual purposes of GRN modeling is to determine the origins of switching or threshold behaviors. These behaviors are often associated with the stability properties of the system against perturbations. Would initial perturbations of a species or a reaction in the network die out, or would it reverberate throughout the network At least near steady states, the stability of the network is influenced by the network structure to a large extent. 

More often than not, kinetic or other rate parameters are unknown in GRNs. Only the qualitative interactions between species are usually known, e.g., X activates Y or V inhibits W. As mentioned earlier, we can interpret the meaning of these qualitative interactions as follows S(dY/dt)/0X 0 and 0(dW/dt)/0V 0, respectively. A qualitative network can be defined as a set of nodes (species) and a set of qualitative interactions ( activation and 

The coefficients Oi are functions of the elements of and more importantly, the Ki s are functions of cycles in the qualitative network graph. An example of cycles would be the three-cycle ai2 23 3i) and the one-cycle (an). The eigenvalues, and therefore the linear stability of a network, are determined only by cycles in the qualitative network graph. 

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B. L. Clarke, Stability analysis of a model reaction network using graph theory. J. Chem. Phys. 60(4), 1493 1501 (1974). 

I. Stoleriu, Stability analysis of metabolic networks, in M. Kirkilionis, U. Kummer, and I. Stoleriu, eds., 2nd UniNet Workshop Data, Networks and Dynamics, Logos Verlag, Berlin, 2006. 

Many methods have been developed for model analysis for instance, bifurcation and stability analysis [88, 89], parameter sensitivity analysis [90], metabolic control analysis [16, 17, 91] and biochemical systems analysis [18]. One highly important method for model analysis and especially for large models, such as many silicon cell models, is model reduction. Model reduction has a long history in the analysis of biochemical reaction networks and in the analysis of nonlinear dynamics (slow and fast manifolds) [92-104]. In all cases, the aim of model reduction is to derive a simplified model from a larger ancestral model that satisfies a number of criteria. In the following sections we describe a relatively new form of model reduction for biochemical reaction networks, such as metabolic, signaling, or genetic networks. 

We repeat these simple examples from earlier work since they form a major justification for the elaborations we report below, and if they are understood well, then much of what follows should be clear. The dynamics in the differential equations are in qualitative agreement with the dynamics in the synthetic genetic networks in E. coU. One important aspect of biology is to understand the ways in which the organization and structure of the control networks can be used to predict the dynamics. Thus, we would like to develop methods that could be used to predict the dynamic behaviors just demonstrated without integrating or carrying out the stability analysis of the differential equations. 

Wang H. 2000. Hybrid model of discrete fracture network and discrete element method for rock slope stability analysis. J. of Hydrogeology Engineering Geology, Vol. 2, pp. 30-34. 

The linear stability analysis of the homogeneous steady state /> = /o of arbitrary, homogenous networks... 

Mathew, A.P., Packirisamy, S., Thomas, S. Studies on the thermal stability of natural rubber/polystyrene interpenetrating polymer networks thermogravimetric analysis. Polym. Degrad. Stab. 72, 423 39 (2001)... 

Clarke, B. (1974a). Graph theoretic approach to the stability analysis of steady state chemical reaction networks. J. Chem. Phys., 60 (4), 1481-92. 

A still more ambitious approach has been pioneered by Clarke (1976, 1980). His stoichiometric network analysis attempts to solve both the network stability and the stability diagram problems by using sophisticated mathematical techniques to identify critical subnetworks within a complex mechanism that can result in instabihty. The problem can be converted into the geometrical problem of finding the vertices, edges and higher dimensional faces of a convex polyhedron (polytope) in a high-dimensional space. 

CFBs can be used to reaHze inverting and noninverting amplifiers, current-to-voltage and voltage-to-current converters, and summing and differencing amplifiers. Thus, they can be utilized in a variety of active filters such as Sallen-Key active filters. However, careful stability analysis must be performed for circuits in which CFBs have reactive components in their feedback networks. For example, the usual Miller... 

Particularly in the preceding Chapter 6 we have seen the significance of sf ability properties for biological network models. This experience leads us to devote the present chapter to a development of quite general techniques for a stability analysis of networks. Such an analysis can be performed under two different aspects a thermodynamic aspect, which relates stability to thermodynamic properties of the network like entropy production and the second lawi and a nonthermodynamic aspect, which derives the stability properties from the mathematical structure of the differential equations represented by the network on the basis of a topological analysis. Sections 7.1 to 7.4 of this chapter will be devoted to the thermodynamic aspect and in Sections 7.5 to 7.7 we briefly describe a few simple techniques to obtain information on stability from a topological analysis. 

In the absence of other reactive groups, these react with one another by addition-type reaction to aflford crosslinked networks. Thermogravimetric analysis (TGA) showed that the thermal stability of the final networks with respect to the crosslinking groups decreases in the order [S50] ... 

Hikmet RAM, Boots HMJ, Michielsen M (1995) Ferroelectric liquid crystal gels—network stabilized ferroelectric display. Liq Cryst 19 65-74 Inoue T, Higuchi N, Fume H (2008) The effect of pol5mer doping on the formation of helical stmcture in ferroelectric liquid crystals. Ferroelectrics 364 113-120 Jean YC (1990) Positron annihilation spectroscopy for chemical analysis a novel probe for microstmctural analysis of polymers. Microchem J 42 72-102 Jean YC, Nakanishi H, Hao LY, Sandreczki TC (1990) Anisotropy of free-volume hole dimensions in polymers probed by positron annihilation spectroscopy. Phys Rev B 42 9705-9708 Jean YC, Mallon PE, Schrader DM (2003) Principles and application of positron and positronium chemistry. World Scientific, Singapore... 

Philippova and Starodubtzev have also extensively studied the complex-ation behavior of polyacids and PEG, especially, the system of crosslinked of poly(methacrylic acid) and linear poly(ethylene glycol) (Philippova and Starodubtzev, 1995 Philippova et al., 1994). They observed that decreasing the molecular weight of PEG from 6000 to 1500 resulted in its slower diffusion into the swollen network of PMAA, and a drastic decrease in both the stability and equilibrium composition of the intermacromolecular complex. Analysis of dried polymer networks of PMAA with absorbed PEG chains by FT-IR spectroscopy revealed the presence of two types of hydrogen bonded structures (1) dimers of methacrylic acid at absorption frequency of 1700 cm-1 and (2) interpolymer complexes of PMAA and PEG at 1733 cm-1. In addition, they also suggested as a result of their studies, that the hydrogen bonded dimer of PMAA forms preferentially to the intermacromolecular complex between the PMAA network and PEG chains. 

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