Mobius loops

The last exception has an unusual twist, literally. The transition state is a strange loop with a half-twist, a Mobius loop. In contrast to a normal loop, Mobius loops are predicted to be stable with An electrons in them. Given enough heat, the cyclobutene sigma bond twists open to form a diene. This and other pericyclic reactions are discussed in more detail in Chapter 12. 

If A transforms to B by an antara-type process (a Mobius four electron reaction), the phase would be preserved in the reaction and in the complete loop (An i2p loop), and no conical intersection is possible for this case. In that case, the only way to equalize the energies of the ground and excited states, is along a trajectory that increases the separation between atoms in the molecule. Indeed, the two are computed to meet only at infinite interatomic distances, that is, upon dissociation [89]. 

So far cyclotides have been discovered in plants from the Violaceae (violet), Rubiaceae (coffee), and Cucurbitaceae (cucumber) families " and have been divided mainly into two structural subfamilies called the Mobius and bracelet cyclotides. These two cyclotide subfamilies are distinguished by the presence of a ar-proline residue in loop 5 for the Mobius subfamily. ° ° On the basis of their trypsin inhibitory activity, the two cyclotides MCoTi-I and MCoTi-II from the seeds of the tropical vine Momordica cochmchinensu form a third subfamily, referred to as the trypsin inhibitor subfamily of cyclotides. No other cyclotides have this activity. 

B was just about to tear a strip off the edge of his pad and demonstrate when A surprised him with Oh, I know what a Mobius strip is. It s made by putting half a twist in a long strip before making it into a loop. It has only one side and only one edge. I once saw that well-known line from the Pervigilium Veneris... 

Fig. 11.17 Topoisomerase 1 Allows Strain Relief Ahead of the Replication Fork. Prokaryotic chromosomes are Mobius strips. Topoisomerase 11 creates gates by nicking both strands of the dsDNA, allowing helices to cross one another (Fig 11.18). Tangles and linked loops can therefore be separated. It also forms part of the eukaryotic nuclear scaffold, for similar reasons.

The FMO analysis is as shown in Figure 15.10 C. The HOMO-LUMO interaction is now favorable and leads naturally to the formation of the two new bonds. Figure 15.10 D shows the aromatic transition state analysis. Using the looped lines, we have designated the full cyclic array of interactions. As shown, there is one node in the system, so this is a Mobius system. Since there are four electrons in the cyclic array, the reaction is allowed. By the generalized orbital symmetry rule, this approach trajectory ([ 2s + is thermally allowed [only the component fits the 4q + 2)s and (4r)a formulas]. In summary, it is incorrect to say that... 

Dewar [16] and Zimmerman [17] proposed selection rules based on the cyclic transition state. A Hiickel transition state is one in which there are zero or an even number of phase transitions perpendicular to the plane of the reaction and is the allowed condition for thermal cases in which there are 4n-n2 electrons. These reactions occur in a suprafacial manner. A Mobius transition state, in which there is one or an odd number of phase transitions perpendicular to the plane of the reaction, predicts thermally allowed reactions when there are An electrons involved. These reactions have an antarafacial component. The advantage of this transition state analysis is that all types of concerted reactions are covered by basically one selection rule. If a continuous loop of overlap through all the... 

The electrocyclic ring closure of hexatriene (Eq. 5.13) can also be considered in terms of its transition state, shown in Figure 5.13a, where a complete loop of overlap is followed with a dashed line from atom to atom through all six. This as drawn with a minimum of nodal zones resembles the lowest molecular orbital of aromatic benzene since there are six electrons and no nodes. This transition state is favored by aromatic stabilization and is reached only by disrotation. Conrotation would have given a transition state with one nodal zone (Fig. 5.13b) that would be part of a Mobius orbital set, but six electrons do not give a closed shell in this set, and the transition state does not have aromatic stabilization and is not allowed. 

The transition state for the closure (or the reverse) of a butadiene is shown in Figure 5.14. The disrotatory closure allows a loop with no nodes (a Hiickel orbital), but four electrons cannot give a closed-shell occupation and the transition state is forbidden. However, the conrotatory closure can be drawn with one nodal zone and therefore belongs to the Mobius group. 

In the same section we give the bifurcation diagrams for the codimension two case with a first zero saddle value and a non-zero first separatrix value (the second term of the Dulac sequence) at the bifurcation point. Leontovich s method is based on the construction of a Poincare map, which allows one to consider homoclinic loops on non-orientable two-dimensional surfaces as well, where a small-neighborhood of the separatrix loop may be a Mobius band. Here, we discuss the bifurcation diagrams for both cases. 

Remark 4. The above proof can be easily adopted to the case of a separatrix loop on a general two-dimensional surface, regardless whether it is orientable or non-orientable. In both cases the map will have the form (13.2.9). Note, however, that if a small neighborhood of the separatrix loop is homeomorphic to an annulus, then A>0 and if a neighborhood of f is a Mobius band, then A < 0 (the latter corresponds, obviously, to the non-orientable case). In the case > 0, the Andronov-Leontovich theorem holds without changes. 

Let us consider next the case where — 1 < A < 0 which corresponds to a separatrix loop F on a non-orientable surface (the case 4 < — 1 follows similarly by a reversion of time). A neighborhood of V is then a Mobius band whose median is F. The Poincare map in this case also has the form (13.3.8) with the function satisfying estimates (13.3.7). However, now we need more smoothness. So we assume that the system is at least C -smooth, i.e. r > 4 in (13.3.7). 

L2 corresponds to a double separatrix loop on the Mobius band. The saddle value (Tq is positive on this curve ... 

Let us describe the essential bifurcations in this system on the path 6 = 2 as fjL increases. On the left of the curve AH, the equilibrium state 0 is stable. It undergoes the super-critical Andronov-Hopf bifurcation on the curve AH. The stable periodic orbit becomes a saddle through the period-doubling bifurcation that occurs on the curve PD. Figure C.6.7 shows the unstable manifold of the saddle periodic orbit homeomorphic to a Mobius band. As a increases further, the saddle periodic orbit becomes the homoclinic loop to the saddle point 0(0,0,0,) at a 5.545. What can one say about the multipliers of the periodic orbit as it gets closer do the loop Can the saddle periodic orbit shown in this figure get pulled apart from the double stable orbit after the fiip bifurcation In other words, in what ways are such paired orbits linked in in R ... 

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