Conformation eigenvalues

Fig. 9. Two-dimensional sketch of the 3N-dimensional configuration space of a protein. Shown are two Cartesian coordinates, xi and X2, as well as two conformational coordinates (ci and C2), which have been derived by principle component analysis of an ensemble ( cloud of dots) generated by a conventional MD simulation, which approximates the configurational space density p in this region of configurational space. The width of the two Gaussians describe the size of the fluctuations along the configurational coordinates and are given by the eigenvalues Ai.

From a mathematical point of view, conformations are special subsets of phase space a) invariant sets of MD systems, which correspond to infinite durations of stay (or relaxation times) and contain all subsets associated with different conformations, b) almost invariant sets, which correspond to finite relaxation times and consist of conformational subsets. In order to characterize the dynamics of a system, these subsets are the interesting objects. As already mentioned above, invariant measures are fixed points of the Frobenius-Perron operator or, equivalently, eigenmodes of the Frobenius-Perron operator associated with eigenvalue exactly 1. In view of this property, almost invariant sets will be understood to be connected with eigenmodes associated with (real) eigenvalues close (but not equal) to 1 - an idea recently developed in [6]. 

Figure 2. Evidence of quantum monodromy in the accurately computed eigenvalue spectrum of H2O [33]. The energy points are plotted for the Jjo rotational levels of the (0, V2, 0) progression, with Ka taken equal to J. Crosses indicate levels that have been reassigned with respect to V2. Open circles indicate reported, usually extra, states [20] that do not conform with the pattern. Taken from Ref. [1] with permission of Taylor and Francis, http //www.tandf.co.uk/journals.

The methodology of nD-QSAR adds to the 3D-QSAR methodology by incorporating unique physical characteristics, or a set of characteristics, to the descriptor pool available for the creation of the models. The methods of Eigenvalue Analysis (40) (EVA) and 4D-QSAR (5) are examples of using unique physical characteristics in the creation of a QSAR model. 4D-QSAR uses an ensemble of molecular conformations to aid in the creation of a QSAR. The EVA-QSAR method uses infrared spectra to extract descriptors for the creation of the QSAR model. 

To find the time required until the flow-equilibrium in the transport zone is achieved, the eigenvalue problem of the system of Eqs. (14) with v = 1 must be solved. This was done in Ref. 4) the results show that this time is essentially longer than the time needed for conformational changes in the macromolecules transferred between sol and gel (e.g. = 10 ps for the P = 1082-mer at 15 °C). 

In principle both approaches may differ in the index of the stationary geometries sensed by the eigenvalues of the Hessian matrix. The variational principle implies, in the present electronuclear approach, that any stationary conformation is a minimum since geometric variations must conserve symmetry to do actual computing one should use variations in the an-space. On the other hand, in the... 

To determine the interplay between the spectral properties, both boundary conditions, we return to Weyl s theory [32]. The key quantity in Weyl s extension of the Sturm-Liouville problem to the singular case is the m-function or ra-matrix [32-36]. To define this quantity, we need the so-called Green s formula that essentially relates the volume integral over the product of two general solutions of Eq. (1), u and v with eigenvalue X and the Wronskian between the two solutions for more details, see Appendix C. The formulas are derived so that it immediately conforms to appropriate coordinate separation into the... 

Figure 3.17 Birefringence as a function of the eigenvalue of the velocity gradient tensor, G, for planar flows generated in a four-roll mill, for dilute solutions of polystyrenes of three different molecular weights in polychlorinated biphenyl solvent. Here G is the strain rate and a the flow type parameter. For planar extension, a — 1 and G = is the extension rate for simple shear, a = 0 and G = y is the shear rate. The different symbols correspond to a values of 1.0 (0)> 0.8 (A), 0.5 (-1-), and 0.25 (diamonds). The curves are theoretical predictions from the FENE dumbbell model, including conformation-dependent drag (discussed in Section 3.6.2.2.2). (From Fuller and Leal 1980, reprinted with permission from Steinkopff Publishers.)...

It is instructive to compare the eigenvalues of transition matrices obtained for different lag times t. That is, we do not count transitions on a timescale of O.lps which means to observe transitions from one instance of the time series to the next, but count transitions on a timescale of, say, ps which is between every tenth step in the global Viterbi path. For all time lags. Fig. 5 clearly indicates two dominant eigenvalues after which we find a gap, followed by other gaps after 4, 9, or 16 eigenvalues. This yields 2, respectively 4, 9, or 16 metastable sets. To avoid confusion we call these metastable sets (molecular) conformations. 

Fig. 8. Representatives of the four conformations obtained in the M = 4 analysis and the conditional transition probabilities between them (lag time r = O.lps). Fat numbers indicating the statistical weight of each conformation, numbers in brackets the conditional probability to stay within a conformation. Flexibility in peptide angles is marked with arrows, cf. Fig. 7. Note that the transition matrix relating to this picture is not symmetric but reversible. Top left For the helix conformation the backbone is colored blue for illustrative purpose. It should be obvious from Fig. 5 that for significantly larger lag time r only two eigenvalues will correspond to metastability such that only the helical conformation and a mixed flexible and partially unfolded one remain with significantly high conditional probability to stay within...

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