Segmental Q-coordinates

Each axis of the segmental Q-coordinates is given by the fractional native contact Qs i,j) between a pair of different segments i and j when j > i, or within segment i when i = j ... 

The advantage of the use of the segmental Q-coordinate system becomes apparent when we compare the unfolding trajectories represented in the conformational hypers pace of the segmental Q-coordinates and those in the hyperspace of the Cartesian coordinates (Fig. 2.11) [25]. Because unfolded conformations are very widely distributed in the hyperspace of the Cartesian... 

Fig. 2.11. Two-dimensional representations of the structural ensemble observed in the unfolding trajectories, which were mapped onto the two largest principal components in the 17-dimensional segmental Q-coordinates (a) and in the hyperspace of the Cartesian coordinates (b) [25]. Reproduced with permission from [25]...

By fc-mcans cluster analysis with Euclidean distance in the segmental Q-coordinates, we divided the structure ensemble of the MD unfolding trajectories into nine clusters [25]. The clustering was performed using all data obtained for the authentic and recombinant proteins, and the clusters were numbered in the order of the distance from the native structure. Figure 2.10(c)-(f) shows protein structures in four representative clusters (Clusters 1, 4, 5, and 9), in which Cluster 1 is almost identical to the native structure with all of the 17 Q-coordinates close to unity, whereas Cluster 9, which lost 84% of its native contacts, represents the unfolded state. 

We carried out unfolding MD simulations of goat a-lactalbumin at 498 K. The protein structure was represented in the segmental Q-coordinate, and cluster analyses and multiple-trajectory alignments were carried out to obtain the transition-state structure solely from the MD simulation. The structure obtained by this approach was very close to that obtained experimentally, and hence the results of the kinetic unfolding experiments were well reproduced by the simulations. 

Straight lines plotted in the coordinates iph — < sc or iph —

Mott-Schottky plots C 2 —

depletion layer thickness Lsc on the potential. The straight line segments intersect at a single point. For materials studied in the works cited below, this point coincides, within 0.2 V, with the value of q>tb measured independently by the differential capacity technique. 

For an A-dimensional paraboloid in the space Q= Q 1 .. . , QN let us order the times to reach the point Q, , = tuj 1 cosh 1(<2/ IQ)), as follows t1>t2> - >tn. Then the tunneling path consists of segments separated by takeoff points. At the A th segment, classical motion in the -dimensional space of slow coordinates occurs, while the other N — k degrees of freedom are frozen. The genuine classical motion takes place only at the last segment, when all the times r, are equalized. 

Here subscripts p and s represent polymer and solvent, respectively, Q is the coordinate of all polymers with configurations inherited and q is that of all solvent molecules, V, t(Q) is the intrinsic energy function. If all the potential terms in addition to the intrinsic energy of r connected segments are taken into account, the real Helmholtz energy functional... 

For numerical computation, these results are recast in discrete form. To be concrete, our objective is to represent the dislocation via a series of generalized coordinates of the form q with the intervening segments represented according to the interpolation scheme of eqn (12.79). In particular, if we substitute... 

Hare, z is a coordination number (usually set equal to 10). the, are molecular segment fractions, aed the 0j are molecular area fractions, Quantifies aed 0, are related to pure-component molecular-slruotare constants r,- and q, by the prescriptions... 

Equation (8.6.2) expresses S(q, t) in terms of the molecular segmental positions in a laboratory-fixed coordinate system. It is convenient for calculations to express these positions in terms of the position of the molecular center of mass R(r) and some vector giving the position of the segment relative to the center of mass. Thus,... 

The angles (6t, i) represent the orientation angles of segment / in a laboratory-fixed spherical polar coordinate system. Note that in Eq. (8.9.5) the last term contains a cross term in Pa. For large stiff molecules this term does not, in general, vanish. In the limit q - 0, however, the Pa term vanishes and Eqs. (8.9.1) and (8.9.2) are valid. 

Fig. 2.4. Two-dimensional lattice model for a bundle or a lamella composed of parallel chain-molecules (or chain-segments) with a coordination number of q = 6. The angle between the parallel chains and the lattice plane need not necessarily be 90°...

It is left to the reader to apply Lagrange s equation (7.11), using the two generalized coordinates of the two-segment system, = 0, and q2 = 02, to work through the mathematics involved in solving Eq. (7.11) to yield the equations of motion. For the first generalized coordinate, q, = the equation of motion is solved to be... 

The constant serves to maintain dimensionality, and is, therefore, purely a factor of convenience (in the subsequent mathematical treatment it is canceled out). It is then assumed, in fact, that the coordinates of each individual segment change in the same proportion as the external coordinates of the test body. Thus if the external coordinates are extended by an elongation ratio a, the coordinate in the x direction of the i segment should be a times as large after elongation as the initial unextended value (x,- q), etc. ... 

The crossover from the Gaussian chain behavior on large length scales to the rod-Hke behavior on the length scale of the statistical segment length can be described by the worm-like chain model [18]. In this case, the end-segment distribution q depends both on the spatial coordinate r as well as on... 

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