Shape space covering

Fig. 2. Shape space covering by a small spherical region of sequence space. In order to find (at least) one sequence for every common structure, it is only necessary to search a relatively small sphere around an arbitrarily chosen reference point in sequence space. For example, the covering radius for RNA molecules of chain length n= 100 was determined to be rc = 15 the covering sphere thus contains about 4 x 1024 sequences compared to 1.6 10 sequences in the entire sequence space.

Shape space covering by small connected regions in sequence space shows that only a relatively small fraction of all sequences has to be searched in order to find any of the common structure. 

Compared to the diameter of sequence space, the correlation length of structure space for RNA folding is relatively small (Fontana et al., 1993). A small correlation length implies that a small sphere around any sequence can sample all possible secondary structures. The ability to sample many structures from any sequence point is a property of the fitness landscape referred to as shape space covering. Equation (32) predicts shape space covering of structures when the connectivity is greater than Ac (Reidys et al., 1997). The radius of the covering sphere rmv is defined as... 

The phase space representation of the Fourier method is of a rectangular shape. The volume in phase space covered by the Fourier representation is calculated as follows The length of the spatial dimension in phase is L, and the maximum momentum is pmax. Therefore, the represented volume becomes Y = 2L-pmax, where the factor of two appears because the momentum range is from -pmax to + pmax. Using the fact that p = ftk, the phase space volume can be expressed as... 

Let a random walk particle start from site 0 at t = 0, and consider the probability P t) that this particle returns to its starting point at time t. We shall show that the shape of Pit) depends on the dimensionality d of the space covered by the walker. We assume that the space is homogeneous, that is, that after a given time t the walker has the same probability of being at any point within a volume V(/). We can thus write... 

Recently, we have also prepared nanosized polymersomes through self-assembly of star-shaped PEG-b-PLLA block copolymers (eight-arm PEG-b-PLLA) using a film hydration technique [233]. The polymersomes can encapsulate FITC-labeled Dex, as model of a water-soluble macromolecular (bug, into the hydrophilic interior space. The eight-arm PEG-b-PLLA polymersomes showed relatively high stability compared to that of polymersomes of linear PEG-b-PLLA copolymers with the equal volume fraction. Furthermore, we have developed a novel type of polymersome of amphiphilic polyrotaxane (PRX) composed of PLLA-b-PEG-b-PLLA triblock copolymer and a-cyclodextrin (a-CD) [234]. These polymersomes possess unique structures the surface is covered by PRX structures with multiple a-CDs threaded onto the PEG chain. Since the a-CDs are not covalently bound to the PEG chain, they can slide and rotate along the PEG chain, which forms the outer shell of the polymersomes [235,236]. Thus, the polymersomes could be a novel functional biomedical nanomaterial having a dynamic surface. 

The reason for this becomes apparent when one compares the shapes of the localized it orbitals with that of the ethylene 7r orbital. All of the former have a positive lobe which extends over at least three atoms. In contrast, the ethylene orbital is strictly limited to two atoms, i.e., the ethylene 7r orbital is considerably more localized than even the maximally localized orbitals occurring in the aromatic systems. This, then, is the origin of the theoretical resonance energy the additional stabilization that is found in aromatic conjugated systems arises from the fact that even the maximally localized it orbitals are still more delocalized than the ethylene orbital. The localized description permits us therefore to be more precise and suggests that resonance stabilization in aromatic molecules be ascribed to a "local delocalization of each localized orbital. One infers that it electrons are more delocalized than a electrons because only half as many orbitals cover the same available space. It is also noteworthy that localized it orbitals situated on joint atoms (n 2, it23, ir l4, n22 ) contribute more stabilization than those located on non-joint atoms, i.e. the joint provides more paths for local delocalization. 

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