Molecules in Wonderland of CrystallographyOverview Molecules in Wonderland of Crystallography...

Overview

Molecules in Wonderlandof Crystallography

Dedicated to the 70th anniversary of Peter Wyder, Emeritus

1. Introduction

2. Growth form lattices Snow flakes

3. Molecular form lattices Proteins

4. Polygonal lattices and polygrams Pentagonal case

5. Pentagonal proteins Cyclophilin, D-aminopeptidase

6. Decagonal DNA B-DNA

7. Integral lattices Frank’s ’cubic’ hexagonal lattice

8. Conclusions

Stuttgart, Max-Planck Institute, 16.03.04 A. Janner

. – p.1/35

alice

Alice’s adventures in wonderland

. – p.2/35

magnet

Peter’s adventures in wonderland

. – p.3/35

mignon

(Sci.Am. 2)

Kennst du das Wunderland,

Wo Moleküle die Schneeflockengleichen,

Kennst du es wohl?

. – p.4/35

Cryst+Growt latt. (SA) [SAa]

Ice and Snow Flake: Microscopic and Macroscopic Lattices

Crystal lattice Growth lattice

Ch. and N. Knight, Scientific American (1973) 100-107 . – p.5/35

Sa1+Sa2[SAb]

Hexagrammal Scaling Symmetry of Snow Crystals

Facet-like snow flake Dendric-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.6/35

Indexed starhex. vert.

(ls5/decacg33)

Indexed Star Hexagon

Vertex Scaling Relations

Scaling: Rotation:1 / √3 π / 6

3 -3

6 3

3 6

-3 3

-6 -3

-3 -6

1 -1

2 1

1 2

-1 1

-2 -1

-1 -2

3 0

3 30 3

-3 0

-3 -3 0 -3

1 0

1 10 1

-1 0

-1 -1 0 -1

(2 -1)

(1 1)(-1 2)

(-2 1)

(-1 -1) (1 -2)

(2 -1)

(1 1)(-1 2)

(-2 1)

(-1 -1) (1 -2)

(1 0)

(0 1)

(1 -1)

(-1 0)

(0 -1)

(1 -1)

. – p.7/35

Indexed hexagram

Indexed Star Hexagon

mid-edge scaling relations

8,0

8,8

0,8

-8,0

-8,-8

0,-8

4,0

4,4

0,4

-4,0

-4,-4

0,-4

2,0

2,2

0,2

-2,0

-2,-2

0,-2

1,0

1,10,1

-1,0

-1,-10,-1

(1,0)

(0,1)(-1,1)

(-1,0)

(0,-1) (1,-1)

(1,0)

(0,1)

(1,-1)

. – p.8/35

Olovsson

A Sample of Snow Flakes

An hyperbolic hexagon as growth form?

. – p.9/35

olov-46a (fig1a)

Snow Crystals with Flat and Hyperbolic Boundaries

Olovsson, Bild der Wissenschaft, 12-1985, 50-59

(Olovsson)

. – p.10/35

olov-46a (fig1a)


Hexagon, hyperbolic hexagon, star hexagon and hexagonal lattice

(Olovsson)

(snow46a,n=5)

. – p.10/35

bh109.7-73b (fig1c)


Bentely & Humphreys, Snow Crystals, Dover, 1962 (109.7)

. – p.11/35

bh109.7-73b (fig1c)


Growth lattice-sublattice in Hexagrammal vertex relation

BH 109.7

. – p.11/35

bh109.7-73b (fig1d)



. – p.12/35

bh109.7-73b (fig1d)


Growth lattice-sublattice in Hexagrammal mid-edge relation

BH 27.3. – p.12/35

bh167.8-84 (fig5)

Dendritic Snow Crystal with Degenerated Hyperbolic Boundaries


. – p.13/35

bh167.8-84 (fig5)

Dendritic Snow Crystal with Degenerated Hyperbolic Boundaries

Hyperbolic branching sites at points of the growth lattice

. – p.13/35

rG-quadruplex

Cubic Form Lattice of RNA Guanosine QuadruplexLattice-Sublattice relation: Central channel, Guanine and Sugar-phosphate

Zimmerman, J. Mol. Biol. 106 (1976) 663-672

Cubic Form Lattice of RNA Guanosine QuadruplexBackbone subsystem: Sugar-phosphate


Cubic Form Lattice of RNA Guanosine QuadruplexBases subsystem: Guanine


. – p.14/35

rG-quadruplex





. – p.14/35

rG-quadruplex




Zimmerman, J. Mol. Biol. 106 (1976) 663-672 . – p.14/35

R-phycoerythrin 1-hex.

Hexameric R-phycoerythrin (Trigonal 32)

Isometric hexagonal form lattice

a

b

x

y

a = 4r°r°

x

z

4r°

Chang et al., J.Mol.Biol 262 (1996) 721-731 (PDB 1lia) . – p.15/35

Bacteriorhodopsin sqrt(3)-hex

Bacteriorhodopsin trimer√

3-hexagonal form lattice Cubic host lattice (Lipid)

a

re

x

y

a√3r0

z

x

c = a√3

Edman et al., Nature 401 (1999) 822-826 (PDB 1qko)

a = re = 3 r0 = 1√

3c

. – p.16/35

Penta. Lattice

Pentagonal Case

(sgk1a)

x

y1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

-1 -1 -1 -1

Polygonal Lattice

Basis vectors:ak = a(cos kφ, sin kφ)

φ = 2π5

, k = 1, 2, 3, 4

Euler ϕ-function: ϕ(5) = 4

Note:a0 = −a1 − a2 − a3 − a4

Lattice points:P = (n1, n2, n3, n4)

Indices: n1, n2, n3, n4

(integers)

Only small indices are relevant!. – p.17/35

Star Pentagon

Pentagonal Case

(sgk1b)

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.18/35

Linear Scaling

Pentagonal Case

(sgk1c)

x

y

A

P

Q

B

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

1 -1 2 -1

1 2 0 3

-3 -2 -1 -3

3 0 1 2

-2 1 -2 -1

-1 2 -1 1

-1 -1 1 -2

2 1 0 3

-3 -1 -2 -3

3 0 2 1

Pentagonal Case

(sgk1d)

τ

τ

1 x

y

A

P

Q

B

1 0 0 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

1 2 0 3

-3 -2 -1 -3

3 0 1 2

-2 1 -2 -1

-1 -1 1 -2

2 1 0 3

-3 -1 -2 -3

3 0 2 1

Linear Scaling

Scaling transformation:Yλ(x, y) = (x, λy)

Scaling matrix:

0 1 1̄ 1

1 1̄ 2 1̄

1̄ 2 1̄ 1

1 1̄ 1 0

Scaling factor:1/τ3 = 0.2361...

(τ = 1+√

5

2= 1.618...)

Linear Scaling

The linear scaling Y1/τ3

appears alongthe pentagonal edgewith the scaling ratios

τ : 1 : τ

. – p.19/35

Linear Scaling

Pentagonal Case

(sgk1d)

τ

τ

1 x

y

A

P

Q

B

1 0 0 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

1 2 0 3

-3 -2 -1 -3

3 0 1 2

-2 1 -2 -1

-1 -1 1 -2

2 1 0 3

-3 -1 -2 -3

3 0 2 1

Linear Scaling

Scaling transformation:Yλ(x, y) = (x, λy)

Scaling matrix:

0 1 1̄ 1

1 1̄ 2 1̄

1̄ 2 1̄ 1

1 1̄ 1 0

Scaling factor:1/τ3 = 0.2361...

(τ = 1+√

5

2= 1.618...)

Linear Scaling

The linear scaling Y1/τ3

appears alongthe pentagonal edgewith the scaling ratios

τ : 1 : τ

. – p.19/35

Cyclophylin A

Cyclophilin A - Cyclosporin A Decamer Complex

Ribbond diagram viewed down the five-fold axis

Ke et al., Current Biology Structure, 2 (1994) 33-44. – p.20/35

Cyclo. Tau

Cyclophilin A (Decamer)

Ke and Mayrose (PDB 2rma)

τ = 1.61803... the Golden Ratio

GLY(14)

τ 1 τ

. – p.21/35

Cyclo. Tau

Cyclophilin A (Pentamer)

Pentamer: Pentagrammal scaled form

GLY(14)

τ 1 τ

. – p.21/35

Cyclo. Iso-decagonal

Cyclophilin: 3D Form Lattice

Isometric decagonal lattice: r0 = a = c

(cy21a)

r°

τ

τ

1 x

y

C

P

Q

x

z

4r°

2r°

. – p.22/35

DppA Iso-pentagonal

D-aminopeptidase: 3D Form LatticeIsometric pentagonal lattice: Pentamer: τ3r0 = a = c Decamer: τ4r0 = a = c

x

yτ τ1

a

r°

Zn ion

x

z

τ4r°

τ3r°

Remaut et al., Nature Struct. Biol. 8 (2001) 674-678 (PDB 1hi9) . – p.23/35

B-DNA

Side and Axial Views of B-DNA

S. Arnott and R. Chandrasekaran (1981)

Decagonal right-handed helix 10122 . – p.24/35

TAu B-DNA

The Golden Ratio in B-DNA(AGTCAGTCAG) Courtesy Maria van Dongen, Nijmegen (τ = 1.61803...)

(asca6a)

Projected P-atomic positions

G10C T6S

C1S A5C

A1C

C5S

G7S

A9C

G6C

T10S

T2S

C4C

A8S

C8C

G2C

A4S

T7C

C9S

G3S

T3C

τ 1 τ

. – p.25/35

B-DNA: Polygr. scal.

Star Decagons

Schäfli symbol { P/Q }

{ 10/2 }

λ (10/2) = 0.8506...

{ 10/3 }

λ (10/3) = 0.6180...

{ 10/4 }

λ (10/4) = 0.3249...

Polygrammal scalings with scaling factor λ(P/Q)

. – p.26/35

B-DNA:{10/2},{10/3}

Polygrammal Scaling in B-DNA

Star decagons {10/3} and {10/2}

Backbone Region

Code-Dependent Region

Intermediate Region

. – p.27/35

B-DNA 3D form

B-DNA (GC): 3D Form Lattice

Envelope: re Rise unit: p0 = re/2τ

(GC43f)

r e /2τp 0

r e

3D form lattice: 1

2-decagonal . – p.28/35

evidence

Crystal’s evidence

. – p.29/35

nt.hex.lattices

ca-Distribution of Hexagonal Crystals

Crystal Data Determinative TablesVol.2, Inorganic compounds, Donnay & Ondik, 1973

(24.000 entries)

B. Constant and P.J. Shlichta, Acta Cryst. A59 (2003) 281-282. – p.30/35

nt.tetr.lattices

ca-Distribution of Hexagonal Crystals

√

32

1√

3√

2

√

2

√

8√

3

√

6

√

15√

22

√

8√

33

√

8√

3

. – p.30/35

nt.tetr.lattices

ca-Distribution of Tetragonal Crystals

Crystal Data Determinative TablesVol.2, Inorganic compounds, Donnay & Ondik, 1973

(24.000 entries)

B. Constant and P.J. Shlichta, Acta Cryst. A59 (2003) 281-282 . – p.31/35

nt.tetr.lattices

ca-Distribution of Tetragonal Crystals

12√

21√

31√

2

√

32

32√

2

√

3√

2

√

2

√

5√

22(?)

3√

2

√

6 2√

2√

107√

23

. – p.31/35

Zn-Mg-Sn sqrt(8/3)

Z-Phase Zn6Mg3Sm:√

3

8-Hexagonal (P63/mmc)

Zone F: [211] ∼ [101̄1] Zone G: [212] ∼ [101̄2]

Singh, Abe and Tsai, Phil.Mag. Lett. 77 (1998) 95-103

Ranganathan, Singh and Tsai, Phil. Mag. Lett. 82 (2002) 13-19

. – p.32/35

Latt.-Sublatt. sqrt(3/2)

Frank’s ’Cubic’ Hexagonal Lattice: c

a=

√

3

2

3D Hexagonal lattice projection of a 4D Cubic latticeHexagonal basis: a1, a2, a3

Sublattice basis: b1 = [0 3 0], b2 = [2̄ 1̄ 2], b3 = [2 1 1]

Metric tensors:

g(a) =

1 1̄

20

1̄

21 0

0 0 3

2

g(b) =

1 0 0

0 1 0

0 0 1

2

Transformation matrix: Sba =

0 2̄ 2

3 1̄ 1

0 2 1

Lattice-Sublattice transformation: S̃ba g(a) Sba = 9 g(b)

√

3

2-hexagonal

Sba

−→ 1√

2-tetragonal (Scaling factor 3)

F.C. Frank, Acta Cryst. 18 (1965) 862-866. – p.33/35

Conclusions

Conclusions

The crystallographic laws are amazingly general

One learns crystallography from single molecules also

Crystallographic scaling relations between outer andinner envelope of axial bio-macromolecules

Challenging relevance of integral lattices (multimetrical)

To understand the physics of crystallographic scaling isnow a priority

. – p.34/35

mignon

Fig. 5

4 -4 -4

4 4 -4

-4 4 -4

-4 -4 -4

4 -4 4

4 4 4

-4 4 4

-4 -4 4

4 -4 -1

4 4 -1

-4 4 -1

-4 -4 -1

4 -4 1

4 4 -1

-4 4 1

-4 -4 1

1 -1 -41 1 -4-1 1 -4

-1 -1 -4

1 -1 41 1 4-1 1 4

-1 -1 4

Kennst du das Land,

Wo Moleküle die Kristalle gleichen,

Kennst du es wohl?

Dahin! Dahin,

Geht unser Weg!

. – p.35/35

Molecules in Wonderland of CrystallographyOverview Molecules in Wonderland of Crystallography...

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