Statistical network models

Price ND, Shmulevich I. Biochemical and statistical network models for systems biology. Curr. Opin. Biotechnol. 2007. Shmulevich I, et al. Probabilistic Boolean Networks a rule-based uncertainty model for gene regulatory networks. Bioinformatics. 2002 18 261-274... 

Statistical network models were first developed by Flory (Flory and Rehner, 1943, Flory, 1953) and Stockmayer (1943, 1944), who developed a gelation theory (sometimes referred to as mean-field theory of network formation) that is used to determine the gel-point conversions in systems with relatively low crosslink densities, by the use of probability to determine network parameters. They developed their classical theory of network development by considering the build-up of thermoset networks following this random, percolation theory. 

A challenging task in material science as well as in pharmaceutical research is to custom tailor a compound s properties. George S. Hammond stated that the most fundamental and lasting objective of synthesis is not production of new compounds, but production of properties (Norris Award Lecture, 1968). The molecular structure of an organic or inorganic compound determines its properties. Nevertheless, methods for the direct prediction of a compound s properties based on its molecular structure are usually not available (Figure 8-1). Therefore, the establishment of Quantitative Structure-Property Relationships (QSPRs) and Quantitative Structure-Activity Relationships (QSARs) uses an indirect approach in order to tackle this problem. In the first step, numerical descriptors encoding information about the molecular structure are calculated for a set of compounds. Secondly, statistical and artificial neural network models are used to predict the property or activity of interest based on these descriptors or a suitable subset. 

Clearly, any measurement that differentiates between the properties of high and low temperature forms of H20(as), and/or delineates the relationship between H20(as) and liquid H20, can be used to test the hypotheses advanced vis a vis their structures. These and the experimental tests suggested, together with the construction of continuous random network models more sophisticated than that for Ge(as), the increased use of computer simulation, and exploitation of the available experimental information to guide the choice of appproximations in a statistical mechanical theory should increase our understanding of H20(as) and, uitimately, liquid H20. 

Recently, Jung et al. [42] developed two artificial neural network models to discriminate intestinal barrier-permeable heptapeptides identified by the peroral phage display experiments from randomly generated heptapeptides. There are two kinds of descriptors one is binary code of amino acid types (each position used 20 bits) and the other, which is called VHSE, is a property descriptor that characterizes the hydrophobic, steric, and electronic properties of 20 coded amino acids. Both types of descriptors produced statistically significant models and the predictive accuracy was about 70%. 

Table 3 lists the major advanced computational software tools that are currently used for data analysis, visualization, modeling, simulation, and statistical computing, especially for microbial metabolic networks, models, and omics experiments. The given selection while intended to cover currently available software in this field is subjective, and the reader should consider available literature to focus on the specialized aspects and specific applications of the listed databases and software tools. 

Several extensions of this statistical formalism are possible and would allow taking into account more physiological and neuroanatomic features. For example, most of neural network models consider that every neuron receives exactly the same input current from the electrode, however in the case of DBS neurons do not receive the same current depending of their position with respect to the electrode [12]. This consideration is the first element in favor of adding a spatial state variable to the population density. Another example that goes in favor of this idea is the fact that peculiar features of small-world networks should be included. This class of complex networks was recently formalized [66] and some recent studies... 

It is well known that the elasticity of polymer networks with constrained chains in the rubbery state is proportional to the number of elastically active chains. The statistical (topological) model of epoxy-aromatic amine networks (see Sect. 2) allows to calculate the number of elastically active chains1 and finally the equilibrium modulus of elasticity Eca,c for a network of given topological structure 9 10). The following Equation 9) was used for the calculations of E, c ... 

It is actually possible within the framework of the statistical theory of elasticity to deduce an expression similar to Eq, (3.33) that considers the experimentally observed decrease in modulus. This is done by using a model different from the affine deformation model, known as the phantom network model. In the phantom network the nodes fluctuate around mean... 

In this section some general considerations of interest regarding non-Gaussian statistical theory are made with the aim of bringing the simple network model discussed in Section 3,2 closer to a real network (2,4). 

Universal Statistics and Control of Fluctuations. Statistics of the number fluctuations of each molecular species is studied. We have found (i) power-law distribution of fast switching molecules, (ii) suppression of fluctuation in the core hypercycle species, and (iii) ubiquity of log-normal distribution for most other molecular species. The origin of log-normal distribution is generally due to multiplicative stochastic process in the catalytic reaction dynamics, as is confirmed in several other reaction network models. On the other hand, suppression of the number fluctuations of the core hypercycle is due to high connections in reaction paths with other molecules. In particular, reduced is the number fluctuations of the minority molecular species that has high catalytic connections with others. This suppression of fluctuation further reinforces the... 

Models can be generated using stepwise addition multiple linear regression as the descriptor selection criterion. Leaps-and-bounds regression [10] and simulated annealing (ANNUN) can be used to find a subset of descriptors that yield a statistically sound model. The best descriptor subset found with multiple linear regression can also be used to build a computational neural network model. The root mean square (rms) errors and the predictive power of the neural network model are usually improved due to the higher number of adjustable parameters and nonlinear behavior of the computational neural network model. 

The multiple linear regression models are validated using standard statistical techniques. These techniques include inspection of residual plots, standard deviation, and multiple correlation coefficient. Both regression and computational neural network models are validated using external prediction. The prediction set is not used for descriptor selection, descriptor reduction, or model development, and it therefore represents a true unknown data set. In order to ascertain the predictive power of a model the rms error is computed for the prediction set. 

Equations similar to = -x+ /Jtanhx arise in statistical mechanical models of magnets and neural networks (see Exercise 3.6.7 and Palmer 1989). Show that this equation undergoes a supercritical pitchfork bifurcation as P is varied. Then give a numerically accurate plot of the fixed points for each p. 

Reeves, P.C., and M.A. Celia. 1996. A functional relationships between capillary pressure, saturation, and interfacial area as revealed by a pore-scale network model. Water Resour. Res. 32 2345-2358. Rice, J.A. 1995. Mathematical statistics and data analysis. 2nd ed. Duxbury Press, Belmont, CA. Skopp, J. 1985. Oxygen uptake and transport in soils Analysis of the air-water interfacial area. Soil Sci. Soc. Am. J. 49 1327-1331. 

The statistics by which the phases are related depend on the theoretical model used. In general, three theoretical models are considered and these are the homogeneous lamellar model, the network model, and the lamellar stacking model. In the following paragraphs, the particular distribution relationships are presented depending on the index of the distance. 

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