Network, parameters used often

The structure of a perfect network may be defined by two variables, the cycle rank and the average junction functionality (f>. Cycle rank is defined as the number of chains that must be cut to reduce the network to a tree. The three other parameters used often in defining a network are (i) the number of network chains (chains between junctions) v, (ii) the number of junctions p, and (iii) the molecular weight Mc of chains between two junctions. They may be obtained from and using the relations... 

For fitting a neural network, it is often recommended to optimize the values of A via C V. An important issue for the number of parameters is the choice of the number of hidden units, i.e., the number of variables that are used in the hidden layer (see Section 4.8.3). Typically, 5-100 hidden units are used, with the number increasing with the number of training data variables. We will demonstrate in a simple example how the results change for different numbers of hidden units and different values of A. 

The second method relies on the experimental determination of the kinetic parameters using techniques from biophysics or enzymology. Also in this case problems exist (1) the kinetic parameters are often determined under conditions different from the conditions in the cytoplasm (2) an enormous number of experiments need to be done, even for a network of moderate size, to determine all kinetic parameters experimentally. When the second method is used to parameterize a kinetic model then the resulting model is considered a silicon cell model. A number of silicon cell models exist [25-27, 29, 75-77]. 

A considerable improvement over purely graph-based approaches is the analysis of metabolic networks in terms of their stoichiometric matrix. Stoichiometric analysis has a long history in chemical and biochemical sciences [59 62], considerably pre-dating the recent interest in the topology of large-scale cellular networks. In particular, the stoichiometry of a metabolic network is often available, even when detailed information about kinetic parameters or rate equations is lacking. Exploiting the flux balance equation, stoichiometric analysis makes explicit use of the specific structural properties of metabolic networks and allows us to put constraints on the functional capabilities of metabolic networks [61,63 69]. 

Although many of the proposed applications for these gels requires that they operate under an applied pressure or generate some kind of mechanical force, a detailed understanding of these relationships does not currently exist. There is data available on the effect of the load on the rate of work and stroke, or generated force vs time, for example, but this is often presented on an empirical basis. Furthermore, much of the work has been carried out under conditions where the stimulus is rate-limiting, rather than the polymer network [66, 67], The development of a mathematical description of these phenomena using independently obtainable polymer parameters is needed. 

In practice, a gray-box model is developed in steps. One early step is to decide which variables and interactions to include. This is often done by the sketching of an interaction-graph. It must then be decided if a variable should be a state or a dependent variable, and how the interactions should be formulated. In the case of metabolic reactions, the expression forms for the reactions have often been characterized in in-vitro experiments. If this has been done, there are also often in-vitro estimates of the kinetic parameters. For enzymatic networks, however, such in-vitro studies are much more rare, and it is hence typically less known which expression to choose for the reaction rates, and what a good estimate for the kinetic parameters is. In any case, the standard method of combining reaction rates, r,-, and an interaction graph into a set of differential equations is to use the stoichiometric coefficients, Sij... 

Very often a mixture of these two approaches is used to determine the values of the parameters. Good examples are Dano et al. [78] and Chassagnole et al. [79]. In these studies many parameters were taken from the literature and, in a parameter estimation approach, were allowed to vary within experimental error to fit the unknown parameters. When considering dynamics, the boundary conditions of the network have to be supplied as explicit functions of time, and therefore they have to be measured in order to give good values for parameter with parameter estimation [74, 79]. Many detailed and core models can be interrogated online at JWS online (www.jjj.bio.vu.nl) [80]. 

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