crystal growth theory symmetry symmetry operator symmetry operators space groups

subtilisin
~1mm cellulase
The color you see is
“ birefringence ”, the wavelengthdependent rotation of polarized light.

High-throughput crystallography labs use pipeting robots to explore thousnds of “conditions”. Each condition is a formulation of the crystal drop and the reservoir solution .
Conditions can have different:
•protein concentration
•pH
•precipitant, precipitant concentration
•detergents
•organic co-solvents
•metal ions
•ligands
•concentration gradient

precipitant concentration blue line = saturation of protein red line = supersaturation limit
Arrows indicate different diffusion experients.
A,B,D,F,G. Vapor diffusion.
E. Bulk
C. Microdialysis
L=liquid
S=solid m=metastable state
(supersaturated)
Crystal growth occurs between these two limits. Above the supersaturation limit, proteins form only disordered precipitate.

a Linbro plate
Volatiles (i.e. water) evaporate from one surface and condence on the other.
Drop has higher water concentration than reservoir, so drop slowly evaporates.

Sitting drops
Microdialysis
Gel filtration

A precipitant (r) causes proteins (p) to stick to each other by competing for solvent.
p p p p r r r r r r r
= EtOH, (NH
4
)
2
SO
4
, methylpentanediol, polyethylene glycol, etc

50 of the most successful crystallization conditions: http://www.ccp14.ac.uk/ccp/web-mirrors/llnlrupp/crystal_lab/hampton_screen.htm

R
Nucleation takes higher concentration than crystal growth.
R
R
R slow
R
R slow R fast
R
R
R
After nucleation, the large size of a face makes the weak bond more likely.
RRRRRRRRRRRRRRRR
RRRRRRRRRRRRRRRR
R
RRRRRRRRRRRRRRRR
RRRRRRRRRRRRRRRR not so slow fast
RRRRRRRRRRRRRRRR
RRRRRRRRRRRRRRRR
RRRRRRRRRRRRRRRR

Bonds A,B are stronger than P,Q. Dimensions of crystal at equilibrium are proportional.
More on Periodic Bond Chain theory : http://www.che.utoledo.edu/nadarajah/webpages/PBC.htm

Growth is unfavorable directions increases as the crystal grows.
Weak bonds in Z favor growth in XY, forming “plate” xtal.
Growing crosssection in XY favors growth in Z.
Ratio of cross sections is inverse to ratio of bond strength.

Crystal growth depletes the surrounding solution of protein, while concentrating impurities.
Local depletion...
...prevents nucleations close to a growing crystal
...slows and eventually stops crystal growth
...concentrates impurities on the surface of the crystal
...causes convection currents.
Cobalt impurities in SiO
2
(amethyst) are concentrated in the part of the crystal that formed last (the tip).

Higher concentration of protein = higher density
Differences in densty cause convection currents, which might cause crystal defects. Microgravity eliminates convection currents.
More at: http://science.nasa.gov/ssl/msad/pcg/

Thin-walled glass capillaries
Protein crystals are extremely
(<1mm in diameter) are filled fragile!!! They with “ mother liquor ”( the fluid may break upon in which the crystal was grown ) sudden contact and a crystal is carefully with a solid dropped in. The mother liquor object. Tiny is removed using filter paper pipets are used to cut to fine strips. The crystal pull crystals from sticks to the glass, immobilized.
drops.
The xtal remains in vapor diffusion contact with the mother liquor. If not it will dryout and crack.

If not freezing
Xtal is mounted in a thin-walled glass capillary tube
If freezing (preferred)
Xtal is mounted on a thin film of water in a wire loop. The loop is fixed to a metal or glass rod.
Mounted xtal is attached to a goniometer head for precise adjustment.
Low-melting hard wax is used to ‘glue’ the rod or capillary here.
Small wrenches fit here, here, here and here.
wax
Must freeze immediately or film will dry out.!
eucentric goniometer head
Crystal must be kept at proper humidity and temperature!! Very fragile!
(made by Nonius)

Essentially eliminates X-ray damage to crystal.
Crystals do not decay during data collection.
Why not?
wire-loop crystal catcher
Cryo equipment is expensive.
Ice crystals may form if freezing is not done properly, ruining data.

...to prevent glass->ice transition
Water must be frozen to < –70 ° C very fast to prevent the formation of hexagonal ice. Water glass forms.
How? Crystals, mounted on loops, are flash frozen by dipping in liquid propane or freon at –70 ° , or by instant exposure to N
2 gas at –70 ° C.
hexagonal ice

whoops it’s off center. Fix it!
“ machine center ” is the intersection of the beam and the two goniostat rotation axes.
Must be set by manufacturer!
To place crystal at machine center, rotate  and  and watch the crystal. If it moves from side to side, it is off center.
If it is off-center, we adjust the screws on the goniometer head.

Rotate the crystal until the zero-layer disappears and the
1-layer is centered on the beam.
misaligned aligned h=0 h=-1 h=1 h=1 beam is here
Concentric circles around beam means axis is aligned with beam.

Spacing of spots is used to get unit cell dimensions.
Note symmetrical pattern. Crystal symmetry leads to diffraction pattern symmetry.

An object or function is symmetrical if a spatial transformation of it looks identical to the original.
This is the original
X
X This is rotated by 180 °

A spatial transformation can be expressed as an operator that changes the coordinates of every point in the object the same way. Symmetry operators do not distort the object. In other words, the distance between any two points is the same before and after being moved by the symmetry operation.
Here is the operator for a 180 ° rotation around Z.
 
1 0 0
0

1 0

0 0 1


 x y z


 z x y equivalent positions

 a b d e f c

 x y
 g h i
 z

 ax
 by
 cz dx
 ey
 fz
 gx
 hy
 iz

•Point of inversion
•mirror plane
•glide plane
•rotation (2,3,4 or 6-fold)
•screw axis
•lattice symmetry

The crystallographic coordinate system is defined by the unit cell. The location of a point is defined by fraction of traveled (from 0 to 1) along each unit cell axis.
(0.33,0.25,0.55) c b a
Fractional coordinates are always measured parallel to each axis. The axes are not necessarilly 90 ° apart!

 
1 0 0
0

1 0
  x y
0 0

1
  z

  x
 y
  z centric symmetry


1 0 0
0 1 0

 x y
 

0 0

1
 z x y
  z centric symmetry


1 0 0
0 1 0

0 0

1

 x y
 z
 

1/ 2
0
0

 x

1/ 2 y
 z centric symmetry x

 cos
 sin
  sin

0
 cos

0

 x y
0 0 1
  z
 x '
  y ' z non-centric symmetry

 cos
 sin
0


 sin cos


0
0
0 1





 x y z






0
0

1/ 3





 x ' y '
 z

1/ 3


120 °
1/3 of a unit cell non-centric symmetry

Mirror images and points of inversion cannot be re-created by pure rotations.
Centric operations would change the chirality of chiral centers such as the alpha-carbon of amino acids or the ribosal carbons of RNA or DNA.
R
H
C a
N C
O

A 2-fold (180 ° ) rotation around the Z-axis
 
1 0 0
0

1 0

0 0 1


 x y z

 x
 y z

... the mathematical description of a rotation.
y
(x’,y’) atom starts here...
..rotates by
..
...goes here axis of rotation
 r a
(x,y) x
In polar coordinates, a rotation is the addition of angles .

cos ( a = cos a cos  sin a sin  sin ( a = sin a cos  sin  cos a

Adding angles in Cartesian space converting internal motion to Cartesian motion y x = |r|cos a y = |r|sin a
(x’,y’)
 r a
(x,y) x x'
= |r| cos ( a y'
= |r| sin ( a
= |r|(cos a cos  sin a sin 
= (|r| cos a cos  |r| sin a sin 
= x cos  y sin  in matrix notation...
= |r|(sin a cos  sin  cos a
= (|r| sin a cos  |r| cos a sin 
= y cos  x sin 

 x y '
'


 cos
 sin
  sin
 cos

 


 x y
 rotation matrix

2D rotation using matrix notation

 x y '
'


 cos
 sin
  sin
 cos

 


 x y
 x'
= x cos  y sin  
= (|r| cos a cos  |r| sin a sin 
= |r| cos ( a y'
= y cos  x sin 
= (|r| sin a cos  |r| cos a sin 
= |r| sin ( a
“row times column”

To rotate the opposite direction, flip the matrix about the diagonal.
the “transpose”

 x y '
'



A B
 C D


T


A C
 B D

 cos
  sin
 sin

 cos




 x y
 inverse rotation matrix = transposed rotation matrix.
cos
  sin
 sin

 cos




 cos sin
  sin
 cos

 


 1 0

0 1


...because cos
 cos

+ sin
 sin

= 1

A 3D rotation matrix
Is the product of 2D rotation matrices.
 cos
  sin

0

 sin
 cos

0 0 1
0


  cos

 sin
0 1 0


0
 sin
0 cos

 


 cos

 sin

 cos
  sin
 cos
 sin cos

 cos
0
  sin
 cos
 sin





Rotate v=(1.,2.,3.) around Z by 60 ° , then rotate around Y by -60 °
 cos60
  sin 60

0
 sin 60
 cos60

0



1
2



1(0.5)

2(0.866)

3(0)
1(0.866)

2(0.5)

3(0)
 
1.232
0 0 1
 
3
 cos60

0
 sin 60

0 1 0





1.232
1.866
 
0

0

3(1) 3
 
1.232(0.5)

1.866(0)

3(0.866)

1.232(0)

1.866(1)

3(0)
 
 
3.214
1.866
 sin 60

0 cos 60

3

1.866
 
1.232(0.866)

1.866(0)

3(0.5) 0.433

z
90 ° rotation around
X

1 0 0
0 0

1

0 1 0 y x
Y
0 0 1

0 1 0

 
1 0 0 x z y
Z

0

1 0
1 0 0 y

0 0 1 x z
Helpful hint:
For a R-handed rotation, the minus sine is the one on the “Right.”

3D angle conventions: axis of rotation:
Order of
 cos
 sin
z’’ x’
  sin

0
 cos

0
 
1 0 0
0 cos
  sin
 z



 cos sin a  sin a a cos a
0
0 a
0 0 1
 
0 sin
 cos

0 0 1

3 2 1 rotations: z’’’’
Polar angles,
y y’’’
 cos
 sin
  sin

0
 cos

0


 cos

0
 sin
0 1 0




 cos sin z’’
  sin

0
 cos

0

 cos
-y’

0 sin

0 1 0
-z



 cos
 sin
 sin

0
 cos

0
0 0 1
  sin

0 cos

0 0 1
   sin

0 cos

0 0 1

5 4 3
Net rotation =

2 1

• Square, 2x2 or 3x3
• The product of any two rotation matrices is a rotation matrix
• The inverse equals the transpose, R -1 = R T
• orthogonality
•The dot-product of any row or column with itself is one.
•The dot-product of any row or column with a different row or column is zero.
• | x | equals | Rx |, for any rotation R .
More at http://mathworld.wolfram.com/RotationMatrix.html

 
1 0 0
0

1 0
  x y
0 0 1
  z
  x
   y z
R
2-fold symbol
R
P2
180 ° rotation. Called a 2-fold because doing it twice brings you back to where you started.
Equivalent positions in fractional coordinates: x,y,z -x,-y,z


  cos120 sin120
0
 
 sin 120 cos 120
0


0
0


 x y
1
  z

 

 x cos120
  y sin 120
 x sin 120
  y cos 120


 z
3-fold symbol
P3
In fractional coordinates:

0

1 0
1

1 0


 x y

0 0 1
 z

 


 y x
 y z


R
Equivalent positions : x,y,z -y,x-y,z -x+y,-x,z


 cos 90
  sin 90

0
 sin 90
 cos 90

0


 x y

 
0 0 1
 z

 x cos 90
  y sin 90
 x sin 90
  y cos 90


 z
P4
R
R
4-fold symbol
In fractional coordinates (same as orthogonal coords):

0

1 0
1 0 0


 x y

 


 y x


Equivalent positions:

0 0 1
 z z x,y,z -x, -y,z
-y, x,z y,-x,z


 cos 60
  sin 60

0
 sin 60
 cos 60

0


 x y

 
0 0 1
 z

 x cos 60
  y sin 60
 x sin 60
  y cos 60


 z
P6
In fractional coordinates:



1 1 0

1 0 0


 x y

 


 x
 y
 x


0 0 1
 z z
R
R
6-fold symbol
Equivalent positions: x,y,z -y,x-y,z -x+y,-x,z
-x,-y,z y,-x+y,z x-y,x,z

(a)
Choose a point r =(0.1,0.2,0.3) [ orthogonal coordinates]
Rotate the point by 30 ° in x.
Then rotate it by -90 ° in y.
What are the new coordinates?
(b)
Choose a point r =(0.1,0.2,0.3) [ fractional coordinates]
Multiply by the symmetry operator:



0

1 0
1

1 0
0 0 1



What are the new fractional coordinates?

A crystal lattice must be space-filling and periodic .
This “Penrose tile pattern” is spacefilling but not periodic.

Quasicrystals: 5-fold point group symmetry, but no space group symmetry

The poliovirus crystal structure has 5-fold , 3-fold , and 2-fold point-group symmetry.

Screw symmetry
2
1
3
1
6
1
6
2
3
2
4
1 6
3
6
4
Example: 6-fold in the projection.
Screw moves up and to the right
4/6 units.
4
2
4
3
6
5
Equivalent positions are related by rotation AND translation


  cos120 sin120
0
 
 sin 120 cos 120
0


0
0


 x y
1
  z

 


0

0
1/ 3

 

 x cos120
  y sin 120
 x sin120
  y cos120


 z

1/ 3
P3
1
Symbol for 3-fold screw
A right-handed 3-fold screw
A screw rotation is a rotation of
2 p /n plus a translation along the axis of rotation by 1/n
(right-handed screw) or -1/n
(left-handed screw).

is an example of translational symmetry. Equivalent positions are
(x,y,z) and
(x+1,y+1,z+1), in fractional coordinates.
Space groups that have no other translational symmetry operations are called “ primitive ”. Space group letter “P”
Space groups have letters indicating the type of translational symmetry:
C ( centered )
F ( face-centered )
I ( body-centered )

Centered: “C”
Translational symmetry operator (1/2,1/2,0)
This is “face-centered” but only on one face.

Face-centered: “F”
Translational symmetry operators:
(1/2,1/2,0),(0,1/2,1/2),(1/2,0,1/2)

Body-centered: “I”
Translational symmetry operator:
(1/2,1/2,1/2)

A symmetry “ group ” is a set of symmetry operators that is closed , meaning any two operations, applied in succession, create a third operation that is part of the group. A “space group” is a symmetry group that includes lattice symmetry operators.
All space groups implicitly include lattice operators:
( ± 1, ± 1, ± 1)

The International Tables for Crystallography
Equivalent positions: (x, y, z), (-x,-y,z+1/2)

Cell type P,C,I or F
I =body centered I4
1
22
Secondary axes of
Principle axis of symmetry are proper symmetry is 4-fold
2-folds screw

A space group is a closed set of operators .
If you apply any two operators in succession. the result is another one of the operators in the group.
 
1 0 0
0

1 0
  x y
0 0 1
  z

0
0
 

1
2

 x
 y
 z
 1
2

Equivs for P2
1
(x, y, z),
:
(-x,-y,z+1/2)
 
0
1

0 0
1 0
  x
0 0 1


 y z
 1
2


0


0
1
2


 x y
 z

1

1
x,y,z
-x+1/2,-y,z+1/2
-x,y+1/2,z+1/2 x+1/2,-y+1/2,-z z,x,y z+1/2,-x+1/2,-y
-z+1/2,-x,y+1/2
-z,x+1/2,-y+1/2 y,z,x
-y,z+1/2,-x+1/2 y+1/2,-z+1/2,-x
-y+1/2,-z,x+1/2

In class exercise: Find equivalent positions
In class exercise: Use the Escher web sketch applet to find the equivalent positions for cm , p4mm , and p6 .
Draw a dot at fractional coordinates (0.1, 0.2, 0.3)
What are the fractional coordinates of the equivalent positions?
Write the 2D symmetry operators (matrix and vector)

Point group symbols, etc.

Finding symmetry in an image

Plane groups space group p1