Map, excluded volume

Figure 10. Excluded volume map for active hexoses pictured in Figure 9. Volume map of alloxan and difference map showing small region of alloxan not accommodated by hexose map.

Fig. 4. Molecular modeling of (-)-2l -THC ligands with different substitution in the C-1 side chain position using molecular mechanics/molecular dynamics. CB1/CB2 receptor-excluded volume map (redcontours and essential volume map (white grid for the C-1 subsite in zl -THC series. The red area represents the free space within the receptor region that accommodates high-affinity C-1 -substituted ligands, whereas, C-1 substituents falling within the white grid experience unfavorable or less favorable interactions at the binding site...

Techniques that can improve statistical sampling are often crucial in these approaches. An excluded-volume map can improve efficiency by focusing particle insertions into free volume voids [159,177-179]. Specially designed Monte Carlo moves allow polymer chain fluctuations within... 

Stapleton, M. R., and Panagiotopoulos, A. Z. 1990. J. Chem. Phys. Application of excluded volume map sampling to phase equihbrium calculations in the Gibbs ensemble. 92 1285. 

The essential feature of the AAA is a comparison of active and inactive molecules. A commonly accepted hypothesis to explain the lack of activity of inactive molecules that possess the pharmacophoric conformation is that their molecular volume, when presenting the pharmacophore, exceeds the receptor excluded volume. This additional volume apparently is filled by the receptor and is unavailable for ligand binding this volume is termed the receptor essential volume [3]. Following this approach, the density maps for each of the inactive compounds (in their pharm conformations superimposed with that of active compounds) were constructed the difference between the combined inactive compound density maps and the receptor excluded volume represents the receptor essential volume. These receptor-mapping techniques supplied detailed topographical data that allowed a steric model of the D[ receptor site to be proposed. 

Another interesting version of the MM model considers a variable excluded-volume interaction between same species particles [92]. In the absence of interactions the system is mapped on the standard MM model which has a first-order IPT between A- and B-saturated phases. On increasing the strength of the interaction the first-order transition line, observed for weak interactions, terminates at a tricritical point where two second-order transitions meet. These transitions, which separate the A-saturated, reactive, and B-saturated phases, belong to the same universality class as directed percolation, as follows from the value of critical exponents calculated by means of time-dependent Monte Carlo simulations and series expansions [92]. 

Universality and two-parameter scaling in the general case of finite excluded volume, Be comes about by the much more sophisticated mechanism of renormalization. As will be discussed in later chapters (see Chap. 11, in particular) both the discrete chain model and the continuous chain model can be mapped on the same renormalized theory. The renormalized results superficially look similar to expressions like Eq. (7.13), but the definition of the scaling variables iie, z is more com plica led. Indeed, it is in the definition of R ) and z in terms of the parameters of the original unrenormalized theory, that the difference in microstructure of the continuous or discrete chain models is absorbed. 

Once we neglect corrections of order n 1+ 2, the mapping indeed is independent of n. This is a crucial feature, as stressed above. The mapping orders according to powers of which may be small, and not in powers of z = 3en f2, which becomes large Thus it also makes sense ill the excluded volume limit fie > 0, n. —> oc. 

Thus all seems perfect. We have constructed an RG mapping, wliich indeed shows a fixed point. However, the expression (8.32) for / is not satisfactory. It must be independent of A, otherwise dilatation by A2 does not lead to the same result as repeated dilatation by A. Now Eq. (8.32) is only approximate since in Eq. (8,31) we omitted terms O 0 2. This is justified only if 0 is small. We thus need a parameter which allows us to make. If arbitrarily small, irrespective of A. Only e — 4 — d can take this role. In all our results the dimension of the system occurs oidy in the form of explicit factors of d or It thus can be used formally as a continuous parameter. To make our expansion a consistent theory, we have to introduce the formal trick of expanding in powers of e — 4 — d. 3 vanishes for = 0, consistent with the observation (see Chap, fi) that the excluded volume is negligible above d = 4, not changing the Gaussian chain behavior qualitatively. For e > 0 Eq. (8.32) to first order in yields... 

The representation (11,39), (11.40) of the RG mapping, introducing two parameters z, Rq, is adequate outside the excluded volume limit. In the excluded volume limit u — u > i.e. / —+ 1, the two parameters combine into a single parameter. The limiting form of the RG mapping is best derived by... 

Using Eq (14 8) together with the full form (14.7) of the mapping we can easily map1 out the double crossover from the dilute to the semidilute excluded volume regimes and further to the semidilute <9-regime. Figure 14.1a shows an example, where we plotted R2/Rq as function of the concentration... 

Partial mapping of the molecules on a pharmacophore model is allowed. At this stage, pharmacophore models and alignments can be visualized. Excluded volumes can be added manually after having aligned the inactive molecules on the pharmacophore models. 

Fig. 12.6 Compound 15g (in yellow) mapped on to MOD2. Pharmacophore features are color coded blue for the unsubstituted aromatic nitrogen (UNA), red for aromatic ring (RA1, RA2 and RA3). In black are shown the excluded volume spheres (EV1 and EV2).

This model includes four projected points of H-bond acceptor features, one positive ionizable volume (as required) and one excluded volume. Figure 15.9 shows the mappings of three representative agonists a small and flexible molecule (l-AP4, the most potent agonist known to date), the cyclic glutamate mimic ACPT-I and a phenylglycine, (S)-4-PPG. 

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