Topological rules

Fig. 8,20 First five iterations of an SDCA system starting from a 4-neighbor Euclidean lattice seeded with a single non-zero site at the center. The global transition rule F consists of totalistic-value and restricted totalistic topology rules C — (26,69648,32904)[3 3[ (see text for rule definitions). Solid sites have <7=1.

Fig. 8.24 Evolution of a 35 x 35 lattice whose sites are initially randomly seeded with O = 1 with probability p = 1/2. The development proceeds according to T value and OT topology rules defined by code C = (84,36864,2048). The constraints are = 0, A = 10]. The appearance of localized substructures is evidence of a geometrical self-organization.

Gafvelin, G., Sakaguchi, M., Andersson, H., and von Heijne G. (1997). Topological rules for membrane protein assembly in eukaryotic cells./. Biol. Chem. 272, 6119-6127. 

In the Smale horseshoe and its variants, the repeller is composed of an infinite set of periodic and nonperiodic orbits indefinitely trapped in the region defining the transition complex. All the orbits are unstable of saddle type. The repeller occupies a vanishing volume in phase space and is typically a fractal object. Its construction is based on strict topological rules. All the periodic and nonperiodic orbits turn out to be topological combinations of a finite number of periodic orbits called the fundamental periodic orbits. Symbols are assigned to these fundamental periodic orbits that form an alphabet... 

This topological rule readily explained the reaction product 211 (>90% stereoselectivity) of open-chain nitroolefins 209 with open-chain enamines 210. Seebach and Golinski have further pointed out that several condensation reactions can also be rationalized by using this approach (a) cyclopropane formation from olefin and carbene, (b) Wittig reaction with aldehydes yielding cis olefins, (c) trans-dialkyl oxirane from alkylidene triphenylarsane and aldehydes, (d) ketenes and cyclopentadiene 2+2-addition, le) (E)-silyl-nitronate and aldehydes, (f) syn and anti-Li and B-enolates of ketones, esters, amides and aldehydes, (g) Z-allylboranes and aldehydes, (h) E-alkyl-borane or E-allylchromium derivatives and aldehydes, (i) enamine from cyclohexanone and cinnamic aldehyde, (j) E-enamines and E-nitroolefins and finally, (k) enamines from cycloalkanones and styryl sulfone. 

After having considered the way in which the chirality of the subunits can influence the lattice dimensionality, it is then possible to apply a series of topological rules to define the number of building blocks that are needed to build closed circuits and, hence, extended framework motifs. 

A reaction forming enamine adducts between nitroalkenes and enamines takes place with excellent diastereoselectivity572,573 and enantioselectivity. Seebach and coworkers574-577 have proposed a topological rule for the carbon-carbon bond-forming process between prochiral centres in enamines and nitroalkenes. This reaction is very important in the synthesis of enantiomerically pure organic compounds. 

Reactions of nitroalkenes and enamines take place not only in good chemical yields but also in excellent diastereomeric yields (>90%)118. A topological rule has been formulated for carbon-carbon bond-forming processes between prochiral centres in enamines and nitroalkenes as well as other systems119. 

Andersson, H., and von Heijne, G. (1993). Sec-dependent and Sec -independent assembly of K. colt inner membrane proteins—the topological rules depend on chain length. EMBOJ. 12, 683-691. 

New refinements. Using the MO method to calculate the optimal geometry of these compounds, it was established that the molecules B2H4, B3H9, and B4H8 have vacant orbitals at a boron atom. To describe the unstable boranes with vacant orbitals, the known topological rules (the balance equations) had to be extended. The new equations for the systems BpHp+g take the form... 

In most proteins the sheets pack together in one of a small number of ways. The connections between secondary structures obey a set of empirical topological rules in almost all cases.. . Subsequently, it was argued that these similarities arise from the intrinsic physical and chemical properties of proteins, and a great deal of work was carried out to demonstrate that this is the case [emphasis added]. 

F. Fuster and B. Silvi, Determination of protonation sites in bases from topological rules, Chem. Phys. 252, 279-287 (2000). 

Reactions of nitroalkenes and enamines take place not only in good chemical yields but also in excellent diastereomeric yields (>W%) . A topological rule has been formulated for carbon-carbon bond-forming processes between prochiral centres in enamines and nitroalkenes as well as other systems The reaction of enamines and imines with acrylamide results in aza-annulation ° . Other electrophilic alkenes which have been us to alkylate enamines and the products used in hetero- or carbocyclic synthesis include ethyl /5-nitroacrylate, where reaction occurs beta to the nitro not the ester group, 2- phenylseleno)prop-2-enenitrile [CH2 = C(SePh)CN] , phenyl a-phenylselenovinyl sulphone [CH2 = C(SePh)-S02Ph] and phenyl a-bromovinyl sulphone. An electrophilic allene, phenylsulpho-nylpropadiene, has also been used to alkylate enamines (Scheme 44). 

Seebach and Golinski advanced a topological rule to account for the stereochemistry of enamine Michael additions along with many other... 

Seebach s topological rule suggesting a preference of synclinal donor/acceptor orientations in a Newman projection along the forming bond cf. Figure 5.8, [60]). For ZfOj-enolates, additional stabilization can be had by metal chelation with the ether oxygen cf. Figure 6.4d). 

The rearside attack model became equally important in the discussion of stereoselectivity in the aldol condensation. In 1981, Seebach and Golinski proposed a topological rule for aldol-like reactions which says that the H atom, the smaller substituent on the donor component, adopts an anti position with respect to the acceptor double bond C=A in the preferred transition state, cf. A and B in Scheme 6.28 [35]. This preference of A over B comes about as a result of the increases in steric hindrance when the donor substituent R2 is jammed against Rj and H of the acceptor in the non-perpendicular approach B. 

From the equations of balance it could be derived that s = (ql2)+y, which indicates that s > (q/2), as y > 0. The maximum value of x is q, which is the case when all the extra hydrogen atoms are used for 3c-2e B-H-B bond. Thus, the values of s can range from q/2 to q. In case of more than one styx codes for the same borane, that which is most symmetric and sticks to the topological rules mentioned previously is selected. 

Figure 3-15 Simple topological rules for complete boundaries.

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