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/??