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Overview
Molecular Crystallography
1. Molecular crystallography
2. Snow crystals
3. Crystallographic scaling
4. Axial-symmetric proteins
5. Integral lattices
6. Perspectives
Nijmegen, 21.04.08 A. Janner
– p. 1/??
molecular crystallography
Molecular CrystallographyQuanosine 5’-phosphate tetramer
(Zimmerman, JMB 106 (1976) 663-677)
Cubic Form lattice
[6 0 0][-6 0 0]
[6 0 2][-6 0 2]
[6 0 2]
[0 6 2]
[-6 0 2]
[0 -6 2]
[3 3 2][-3 3 2]
[-3 -3 2] [3 -3 2]
Sugar-Phosphate
Bases:Guanine
CentralHole
Envelope
– p. 2/??
bh167.8-84 (fig5)
Dendritic Snow Crystal with Growth LatticeBentely & Humphreys, Snow Crystals, Dover, 1962 (167.8)
– p. 3/??
bh167.8-84 (fig5)
Dendritic Snow Crystal with Growth LatticeBranching sites at points of the growth lattice
BH 167.8
– p. 3/??
bh114.8-83 (fig4b)
Facet-like Snow Crystal with Growth LatticeBentely & Humphreys, Snow Crystals, Dover, 1962 (114.8)
– p. 4/??
bh114.8-83 (fig4b)
Facet-like Snow Crystal with Growth LatticeRegular hexagons with center and vertices at points of the growth lattice
BH 114.8
– p. 4/??
bh167.8-84 (fig5)
Indexed Snow CrystalBentely & Humphreys, Snow Crystals, Dover, 1962 (53.1)
BH 53.1
– p. 5/??
bh167.8-84 (fig5)
Indexed Snow CrystalHexagrammal scaled form
(bh53.1-86)
1 0
0 1
4 0
4 40 4
-4 0
-4 -4 0 -4
– p. 5/??
Cryst.Scal.
Crystallographic Scaling
Scaling with scaling factor λ
1D (linear) Xλ(x, y, z) = (λx, y, z)
2D (planar) Pλ(x, y, z) = (λx, λy, z)
3D (isotropic) Iλ(x, y, z) = (λx, λy, λz)
Higher dimensional .......
– p. 6/??
Cryst.Scal.
Crystallographic Scaling
Scaling with scaling factor λ
1D (linear) Xλ(x, y, z) = (λx, y, z)
2D (planar) Pλ(x, y, z) = (λx, λy, z)
3D (isotropic) Iλ(x, y, z) = (λx, λy, λz)
Higher dimensional .......
Crystallographic transforming a lattice into a lattice
SλΛ = Λ Sλ integral invertible
in general: SλΛ = Σ Σ ⊆ Λ or Λ ⊆ Σ
Sλ rational invertible– p. 6/??
mid-edge vertex
Hexagrammal Scaling
mid-edge vertex
1 0
0 1
2 0
0 2
1 0
0 12 1
-1 1
S =
2 0
0 2
S =
2 −1
1 1
λ = 2, ϕ = 0 λ =√
3, ϕ = 30o
– p. 7/??
Pentagram
Pentagonal Case
1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 1
-1 -1 -1 -1
-2 0 -1 -1
1 -1 1 0
0 1 -1 1
-1 -1 0 -2
2 1 1 2
Polygrammal Scaling
Star Pentagon:Schäfli Symbol {5/2}
Scaling matrix: (planar scaling)
2̄ 1 0 1̄
0 1̄ 1 1̄
1̄ 1 1̄ 0
1̄ 0 1 2̄
Scaling factor:-1/τ2 = −0.3820...
(τ = 1+√
5
2= 1.618...)
– p. 8/??
Sa1+Sa2[SAb]
Hexagrammal Scaling Symmetry of Snow Crystals
Facet-like snow flake Dendritic-like snow flake
(Sci.Am. 2) (Sci.Am. 1)
Scientific American (1961)
Hexagrammal Scaling Symmetry of Snow CrystalsMid-edge star hexagons: λME = 1/2 Vertex star hexagons: λV E = 1/
√
3
(Sci.Am. 2) (Sci.Am. 1)
– p. 9/??
Sa1+Sa2[SAb]
Hexagrammal Scaling Symmetry of Snow Crystals
Facet-like snow flake Dendritic-like snow flake
(Sci.Am. 2) (Sci.Am. 1)
Scientific American (1961)
Hexagrammal Scaling Symmetry of Snow CrystalsMid-edge star hexagons: λME = 1/2 Vertex star hexagons: λV E = 1/
√
3
(Sci.Am. 2) (Sci.Am. 1)
– p. 9/??
R-phycoerythrin 1-hex.
R-phycoerythrin (trigonal hexamer)
Hexagonal form lattice Hexagrammal mid-edge scaling
x
y
r°re
1 0
0 1
4 0
4 40 4
-4 0
-4 -4 0 -4
Chang et al., J.Mol.Biol 262 (1996) 721-731 (PDB 1lia)
– p. 10/??
creatine kin. square
Mitochondrial creatine kinase (tetragonal octamer)
Square form lattice with scaling relations
1 1-1 1
-1 -1 1 -1
6 6-6 6
-6 -6 6 -6
0 7
-7 0
0 -7
7 0
1 6-1 6
-6 1
-6 -1
-1 -6 1 -6
6 -1
6 1
Gly365
Thr1
Fritz-Wolf, Schnyder, Wallimann & Kabsch, Nature (1996) 341-345 (1crk)
– p. 11/??
Cycloph.decam.pentam.
Cyclophilin A (Decamer)
Ke and Mayrose (PDB 2rma)
τ = 1.61803... the Golden Ratio
GLY(14)
τ 1 τ
– p. 12/??
Cycloph.decam.pentam.
Cyclophilin A (Pentamer)
Pentamer: Pentagrammal scaled form
GLY(14)
τ 1 τ
– p. 12/??
Creatine Kinase cubic
Mitochondrial creatine kinase (tetragonal octamer)
x
y
[6 -6 4][1 -6 4][-1 -6 4][-6 -6 4]
[6 6 4][1 6 4][-1 6 4][-6 6 4]
[6 -1 4]
[6 1 4]
[-6 -1 4]
[-6 1 4]
Gly365
x
z
[-6 -6 4] [-1 -6 4] [1 -6 4] [6 -6 4]
[-6 -6 -4] [-1 -6 -4][1 -6 -4] [6 -6 -4]
Gln366
Cubic indexed form
Fritz-Wolf, Schnyder, Wallimann & Kabsch, Nature (1996) 341-345 (1crk)– p. 13/??
Cyclo. Iso-pentagonal
Cyclophilin: Isometric Decagonal Lattice
r°
τ
τ
1τ3r°
x
y
A
C
P
Q
D
[3 1 1 3, 2]
[-1 -1 -1 -1, 2]
[1 0 0 0, 2]
[0 1 0 0, 2]
[0 0 1 0, 2]
[0 0 0 1, 2]
[1 -1 2 -1, 2]
[1 2 0 3, 2]
[-3 -2 -1 -3, 2]
[3 0 1 2, 2]
[-2 1 -2 -1, 2]
[-1 2 -1 1, 2]
[-1 -1 1 -2, 2]
[2 1 0 3, 2]
[-3 -1 -2 -3, 2]
[3 0 2 1, 2]
Glu15
x
z
4r°
2r°
Glu15
Glu15
[2 1 0 3,-2]
[2 1 0 3, 2]
[-1 2 -1 1,-2]
[-1 2 -1 1, 2]
Ke et al., Current Biology Structure, 2 (1994) 33-44
r0 = a = c
– p. 14/??
R-phycoerythrin 1-hex.
R-phycoerythrin: Isometric hexagonal
x
y
a = 4r°r°
[0 -4 4]
[4 0 4]
[4 4 4]
[0 4 4]
[-4 0 4]
[-4 -4 4]
x
z
4r°
[-4 -4 -4] [0 -4 -4] [4 0 -4]
[-4 -4 4] [0 -4 4] [4 0 4]
– p. 15/??
hexag. inorg.
Distribution of Hexagonal Inorganic Crystals
c/a 0 1 2 3 4 5
0
200
400
600
800
1000
1/√2 1 √2 √(8/3) √6
964 Hexagonal isometric lattices
as for the molecular forms of:
- Hexameric R-Phycoerythrin
- Trimeric Outer Membrane Protein F
Inorganic Crystal Structure Database (ICSD)
12’000 hexagonal entries
(collaboration R. de Gelder)– p. 16/??
pseudo-tetr. inorg.
Distribution of Pseudo-Tetragonal Orthorhombic Crystals
c/a = c/b1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5
0
100
200
300
400
500
600
700
800
900
1 √2 √6 3
Inorganic Crystal Structure Database (ICSD)
4095 pseudo-tetragonal entries
(collaboration R. de Gelder) – p. 17/??
relation - symmetry
From Structural Relations to Symmetry
System Axial proteins, Viral capsid, Holoenzymes (Ferritin, SOR)
Property External envelope - Central hole Same form lattice
Integral latticeInternal - External polygons & polyhedra Indexed vertices
Relation Crystallographic scaling Star polygons, scaled capsidIntegral lattice Rational (c/a)2
Problems Infinite order point group Finite structuresCrystallographic scaling is not a molecular symmetry
Axial ratio: not determined by crystallographic laws
Solution?
Finite Higher-Dimensional crystallographic point groups
– p. 18/??
I4 orbits
4D Symmetry of pentagonal and decagonal star polygons
I4 = {A5 = B4 = 1, [A, B] = A2} ∈ G`(4, Z)
{10/3} = {I4 |[1-100]}Decagram {10/3}
{5/1} = {I4 | [1000]}Pentagon
{5/2} = {I4 | [0-1-10]}Pentagram {5/2}
{5/2}*{5/2} = {I4 | [1221]}Squared Pentagram
BBNWZ, Crystallographic groups of four-dimensional space (isom 20.5, p.242)
– p. 19/??
weber-frank
Frank’s cubic hexagonal lattice
(frank2)
1922: Weber0001
1000
0100
0010
K = (hkl) = (hkil) r = [mnp] = [uvtw]
1965: Frank
[1000]
[0100]
[0010]
- -[2110]/3ϕ
cos ϕ = √(2/3)
Hexagonal four-indicesKr = hm + kn + lp = hu + kv + it + lw
i = - k - l w = p
u = 2m−n3 v = −m+2n
3 t = −m−n3
4D cubic
[001] = [0001]
c2 c2
[100] = 1
3[21̄1̄0]
a2 6
9c2
ca =
√
32
– p. 20/??
thanks
Thanking for your attention!– p. 21/??