Topology in Molecular Biology
152 N. Rivier and J.-F. Sadoc Fig. 8.3.The single collagen chain PPII (Gly−X−Y) is a left-handed helix, that can be drawn on a r ...
8 The Structure of Collagen 153 Fig. 8.4.The base space of the Hopf fibration of polytope{ 3 , 3 , 5 }is an icosahedron (a). Eac ...
154 N. Rivier and J.-F. Sadoc Fig. 8.5.Bouligand’s overlap–gap transformation between two Archimedean lat- tices. Unit cells in ...
8 The Structure of Collagen 155 Fig. 8.6.(a) The unit cell of the square–triangle 3^2. 4. 3 .4 gap structure, drawn on a triangu ...
156 N. Rivier and J.-F. Sadoc Fig. 8.7.The overlap structure 3^2. 4. 3 .4. The main grid is that of the unit cell (a square with ...
8 The Structure of Collagen 157 Fig. 8.9.Two successive twist grain boundaries overlap–gap–overlap. This illus- trates the 5/4 r ...
158 N. Rivier and J.-F. Sadoc Let us now obtain the metric of the two structures. The inflation multiplier 1+ √ 3 must be presen ...
8 The Structure of Collagen 159 The overlap structure is much more symmetrical than the gap. It has: (a) Orthogonal mirror axes ...
160 N. Rivier and J.-F. Sadoc (1) On an axis of 12-fold symmetry. It is given by the lattice vectorsni≈ p √ 3 iin the two, ortho ...
8 The Structure of Collagen 161 q=Aj,m=Cj,n=Aj+2,p=Cj+2. (8.4) The smallest solution (j=1)isq=1,m=2,n=4,p=7.Itcorresponds to the ...
162 N. Rivier and J.-F. Sadoc the overlap structure. The gap structure, rotated around one of its vertices by 27◦, has vertices ...
9 Euler Characteristic, Dehn–Sommerville Characteristics, and Their Applications V.M. Buchstaber Summary.In this chapter we pres ...
164 V.M. Buchstaber α 0 α 0 α 1 0-dim 1-dim Fig. 9.1.Simplex A pointx∈σncan be written inbarycentric coordinatesas x= ∑n j=0 xjα ...
9 Euler, Dehn–Sommerville Characteristics, and Their Applications 165 Fig. 9.2.Barycentric subdivision Example 4.LetK be an (n−1 ...
166 V.M. Buchstaber For example,χ(σn)=1andχ(Sn−^1 )=1+(−1)n−^1. Putf(t)=tn+f 0 tn−^1 +···+fn− 1 andh(t)=h 0 tn+h 1 tn−^1 +···+hn ...
9 Euler, Dehn–Sommerville Characteristics, and Their Applications 167 9.4 Homology Groups and Characteristic Classes Given asimp ...
168 V.M. Buchstaber Theboundaryof an arbitraryk-chainck= ∑ igiσiis then given by ∂ck:= ∑ i gi∂σi. Again, we have∂∂ck=0. Thek-cyc ...
9 Euler, Dehn–Sommerville Characteristics, and Their Applications 169 Hn(Mn;Z)=0,Hn(Mn;Z 2 )=Z 2. A simplicial mapf:K 1 →K 2 ind ...
170 V.M. Buchstaber g=1 g=2 g=3 Fig. 9.4.Sphere withghandles g=1 a a b b g=2 a a a a b b b b 1 1 1 1 2 2 2 2 Fig. 9.5.Mg^2 :4ged ...
9 Euler, Dehn–Sommerville Characteristics, and Their Applications 171 g=1 g=0 g=2 g=1 a a a a a a a a b b b b b b b b c c c c 1 ...
«
4
5
6
7
8
9
10
11
12
13
»
Free download pdf