Conformation Coordination

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.

Fig. 10. Conformational flooding accelerates conformational transitions and makes them accessible for MD simulations. Top left snapshots of the protein backbone of BPTI during a 500 ps-MD simulation. Bottom left a projection of the conformational coordinates contributing most to the atomic motions shows that, on that MD time scale, the system remains in its initial configuration (CS 1). Top right Conformational flooding forces the system into new conformations after crossing high energy barriers (CS 2, CS 3,. . . ). Bottom right The projection visualizes the new conformations they remain stable, even when the applied flooding potentials (dashed contour lines) is switched off.

The free energy profile or potential of mean force along a conformational coordinate may be defined as... 

Fig. 2. Illustration of the definitions of conformational coordinate 7Zn, e.g., 7Zn = ri, r2,. .., rn. The conformational distribution s (7U1) is sampled for the single molecule in the absence of interactions with solvent by suitable simulation procedures using coordinates appropriate for those procedures. The normalization adopted in this development is/sf (7Zn) dn1Z = V, the volume of the system. Thus, the conformational average that corresponds to adding the second brackets in going from Eq. (4) to Eq. (3) is evaluated with the distribution function sf (7Zn) = V.

Figure 1.8 Illustration of the notational definition of conformational coordinates X" = / i.rj,.. The conformational distribution - see Eq. (1.13),...

The information theory approach studied here grew out of earlier studies of formation of atomic sized cavities in molecular liquids (Pohorille and Pratt, 1990 Pratt and Pohorille, 1992 1993). Since we deal with rigid and spherical solutes in the discussion we will drop the explicit indication of conformational coordinates and discuss p n) = Pa n lR ). We emphasize that the overall distribution p(n) is well described by the information theory with the first two moments, (n)o and n n — 1)/2)q. It is the prediction of the extreme member p 0) that makes the differences in these default models significant. Computing thermodynamic properties demands more than merely observing typical behavior. 

Prec-D-MgTP reactant precursor in the docked conformation and the triphosphate in an extended conformation coordinated with a Mg2+ ion... 

Fig. 11 represents the two-dimensional section of the energy landscape with the free energy plotted against the conformational coordinate. The CGU can... 

Fig. 1. Hypothetical potential energy surface for the oxygen-binding processes of hemoglobin to show relationship between chemical coordinate and conformation coordinate. See text.

Fig. 4. Reaction pathway for chymotryptic catalysis showing relationship between chemical and conformation coordinates and self-inhibited species, [ES], and [EPjH],., formed between enzyme and substrate and between enzyme and acid product. See text.

Conformationally, coordination to the Cu(MeCN) unit requires considerable rearrangement from free 18S6 [80, 86]. Coordination to two Cu(MeCN) groups requires half of the C-S bonds to change to anti placement, and all of the C-C bonds to adopt gauche placement. 

Equating the coefficients of this polynomial to zero, we obtain a system of equations for the coefficients of the matrices A and B. Write these equations in isothermic (conformal) coordinates x, y, in which the Riemannian metric takes the form... 

The independent boolean variables in conformal coordinate scales have exactly the same order as in the boolean function argument list, as depicted in Fig. 1.46. Conformal assignment of the independent variables to the K-map coordinate scales makes the catenated position coordinates for a minterm (or maxterm) identical to the minterm (or maxterm) number. Utilization of this identity eliminates the need for the placement of minterm identification numbers in each square or for a separate position identification table. This significantly decreases the time required to construct K-maps and makes their construction less error prone. The minterm number, given by the catenation of the vertical and horizontal coordinate numbers, is obvious if the binary or octal number system is used. 

FIGURE 1.46 Formats for K-maps for functions of 2-6 independent variables with conformal coordinate scales (a) two-variable case f a,b), three-variable case f(fl,b,c), (c) four-variable case i a,b,c,d), (d) five-variable case... 

The computational domain in (9, F) coordinates is presented in Fig. 12.23. For diffusion limited redox reactions at the generator and collector, the boundary conditions in conformal coordinates are... 

Figure 8-3. Physical domain in boundary-conforming coordinates.

Debye length (3.1.10b), mean free path of a gas molecule (3.1.114), filter coefficient (7.2.187), parameter for a dialyzer (8.1.399), parameter for a distillation plate/stage (8.3.38), latent heat of vaporization/condensation molecular conformation coordinate (3.3.89c) electrode spacings (7.3.18) retention parameter for species i (7.3.213), ionic equivalent conductance of ion i (3.1.108r) value of Xi for a cation value of A, for an anion value of A, at infinite dilution (Table 3.A.8) defined by (5.4.100) equivalent conductance of a salt (an electrolyte) (3.1.108s)... 

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