Structural Mode Analysis

The application of the Routh-Hurwitz analysis or the direct calculation of the eigenvalues and eigenvectors of the Jacobian Jg of the network of reactors is a formidable task for moderate and large arrays of coupled reactors. There is an alternative approach, a spectral analysis of networks, that works for any size array of coupled reactors, as long as the array is homogeneous, i.e., / -, = /i for all i,i = 1. n, and the coupling is diffusive. In this case, the system (13.4) has a uniform steady state  

In the following we suppress the explicit dependence of the stationary state on the set of parameters fi and write = /o. The Jacobian Jg is given by 

As we saw in Sect. 10.1, the stability properties of the uniform steady state of spatially continuous reaction-diffusion systems can be analyzed in terms of normal modes corresponding to the eigenfunctions of the Laplace operator. Othmer and 

Scriven [334] showed that the stability of spatially discrete homogeneous reaction-diffusion systems can be analyzed in terms of the structural modes of the network, i.e., the eigenvectors of the Laplacian matrix L. We have extended that approach [305], and the eigenvalues and eigenvectors of the matrix 

For simplicity we assume throughout this chapter the generic case that J(r) has a complete set of eigenvectors for all appropriate values of r. We extend the definition 

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The influence of solvent can be incorporated in an implicit fashion to yield so-called langevin modes. Although NMA has been applied to allosteric proteins previously, the predictive power of normal mode analysis is intrinsically limited to the regime of fast structural fluctuations. Slow conformational transitions are dominantly found in the regime of anharmonic protein motion. 

Evidence exists that some of the softest normal modes can be associated with experimentally determined functional motions, and most studies apply normal mode analysis to this purpose. Owing to the veracity of the concept of the normal mode important subspace, normal mode analysis can be used in structural refinement methods to gain dynamic information that is beyond the capability of conventional refinement techniques. 

Vibration spectra of fluoride and oxyfuoride compounds correspond to X Me ratios, especially in the case of island-type structure compounds. Analysis of IR absorption spectra provides additional indication of the coordination number of the central atom. Fig. 45 shows the dependence on the X Me ratio of the most intensive IR bands, which correspond to asymmetric Me-F modes in fluoride complexes, as well as v(Me=0) and v(Me-F) in oxyfluoride complexes. Wave numbers of TaF5, NbF5 and NbOF3 IR spectra were taken from [283-286]. 

Fig. 12a, b. Dynamic structure factor for two polyethylene melts of different molecular mass a Mw = 2 x 103 g/mol b Mw = 4.8 x 103 g/mol. The momentum transfers Q are 0.037, 0.055, 0.077, 0.115 and 0.155 A-1 from top to bottom. The solid lines show the result of mode analysis (see text). (Reprinted with permission from [36]. Copyright 1994 American Chemical Society, Washington)... 

Following the mode analysis approach described in Section 3.2.1, the spectra at different molecular masses were fitted with Eqs. (32) and (33). Figure 13 demonstrates the contribution of different modes to the dynamic structure factor for the specimen with molecular mass 3600. Based on the parameters obtained in a common fit using Eq. (32), S(Q,t) was calculated according to an increasing number of mode contributions. 

D. M. Ferguson, G. L. Seibel, and P. A. Kollman, AMBER, a package of computer programs for applying molecular mechanics, normal mode analysis, molecular dynamics and free energy calculations to simulate the structural and energetic properties of molecules, Comp. Phys. Comm. 91 1 (1995). 

The L-mode analysis of the simulated taxonic data produced a bimodal distribution, while analyses of the simulated nontaxonic data and the actual data revealed only one mode. What this means is unclear. It is possible that depression is actually taxonic, but the simulated taxonic data does not adequately reflect the structure of the actual data set. For example, higher nuisance correlations and lower indicator validities in the actual data may not have allowed a latent bimodality to emerge, even though this was detected in the simulated data. Another possibility, of course, is that the BDI items do not define a taxon. 

Lindahl, E., Azuara, C., Koehl, P. and Delarue, M. (2006) NOMAD-Ref visuahzation, deformation and refinement of macromolecular structures based on all-atom normal mode analysis. Nucleic Acid Res. 34, W52-56. 

Zhao et al. implemented a structure-based docking protocol to narrow down 500 compounds from a database of 57 compounds in their pursuit of FKBPs inhibitors (89). A novel scaffold was designed using the information obtained from the binding mode analysis of a known weak binder. To avoid any scoring function shortcomings, three scoring functions were used to select the 500 compounds. Of these, 43 were synthesized and tested to identify one potent inhibitor in a mouse peripheral synthetic nerve model. 

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