Through-space vector model

Figure 11. Spherical coordinates used in the "through-space" vector model. 2w is the angle between the two CC bonds carrying the radical centers, and cp the angle between x axis and the rotational plane perpendicular to this CC axis.

Figure 12. The "through-space" vector model. Cartesian components (a-c) of the orientational factors in the angular momentum integrals and (d) the overlap integral (A B).

Figure 13. SOC surfaces for trimethylene (a) total SOC value and (b-d) Cartesian components SOC, SOCy, SOC, comparison of MNDOC-CI results (upper row) with the through-space vector model (lower row).

Most of protein structural information from NMR is obtained in the form of nuclear Overhauser effects or NOEs between pairs of protons that are less than 6 A apart through space. An NOE between a spin pair carries distance information, but only short distances are observed because NOEs have an inverse sixth power dependence on distance. However, the distance cannot be uniquely determined given a measured NOE intensity without making some assumption about the environment of the spin pair and the motion of the vector between them. The simplest model for obtaining a distance from cross peak intensities in a nuclear Overhauser effect spectrum (NOESY) is the isolated ri d spin pair (RRNN - rigid rotor nearest neither) approximation (Jardetzky and Roberts, 1981). In this approximation the observed cross peak intensity, which is proportional to the cross relaxation rate, is related to a sini e intemudear distance, r. 

In the past, the concept of field has been elaborated for modeling an action at a distance, which could explain how some information could be transmitted between entities through space. Nowadays, the concept of exchanged particle is proposed for replacing a field for instance, the photon is the vector of the electromagnetic field. However, the question of the speed of transmission remains a problem to be solved with regard to the (apparent ) immediacy of many phenomena. 

The model protein is used to search the crystal space until an approximate location is found. This is, in a simplistic way, analogous to the child s game of blocks of differing shapes and matching holes. Classical molecular replacement does this in two steps. The first step is a rotation search. Simplistically, the orientation of a molecule can be described by the vectors between the points in the molecule this is known as a Patterson function or map. The vector lengths and directions will be unique to a given orientation, and will be independent of physical location. The rotation search tries to match the vectors of the search model to the vectors of the unknown protein. Once the proper orientation is determined, the second step, the translational search, can be carried out. The translation search moves the properly oriented model through all the 3-D space until it finds the proper hole to fit in. 

It is a bit of a lie to say, as we did in previous chapters, that complex scalar product spaces are state spaces for quantum mechanical systems. Certainly every nonzero vector in a complex scalar product space determines a quantum mechanical state however, the converse is not true. If two vectors differ only by a phase factor, or if two vectors normaUze to the same vector, then they will determine the same physical state. This is one of the fundamental assumptions of quantum mechanics. The quantum model we used in Chapters 2 through 9 ignored this subtlety. However, to understand spin we must face this issue. 

Fig. 23 Bridging biological space. The overlap of epothilone B (cyan carbons) and PTX (green carbons) models derived from EC reveal shared anchors between the exchangeable nucleotide site through H227 and the truncated B9-B10 loop of the beta tubulin site. Perhaps rigidifying this vector across the site is of greater importance to the MT stabilizing effect than picking up interactions within the deep hydrophobic pocket...

Figure 5-2 The plot of the misfit functional value as a function of model parameters tn. The vector of the steepest ascent, l(m ), shows the direction of "climbing on the hill" along the misfit functional surface. The intersection between the vertical plane P drawn through the direction of the steepest descent at point m and the misfit functional surface is shown by a solid parabola-type curve. The steepest descent step begins at a point 0(m ) and ends at a point 0(m +i) at the minimum of this curve. The second parabola-type curve (on the left) is drawn for one of the subsequent iteration points. Repeating the steepest descent iteration, we move along the set of mutually orthogonal segments, as shown by the solid arrows in the space M of the model parameters.

In classic RSM the region for rdiich the simulation model is approximated iteratively moves through the sample space (controlled by a gradient nde) to explore the performance function of the system. Here, the region depends on the performance vector to be tested (y). However, no additional simulation runs shall be performed. Instead, the already evaluated constellations (Xj,yj) can be analysed. In this set, configurations with a performance similar to y can be used to fit a local approximation of H (say H ). H can... 

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