slides2014 hoa tinh the prof van luc

Crystallography - Molecular architecture • Symmetry molecule, stacking objects point groups, plane groups, space groups • Crystal structure determination using X-ray diffraction • Cam

Crystallography - Molecular architecture

• Symmetry (molecule, stacking objects)

point groups, plane groups, space groups

• Crystal structure determination using X-ray diffraction

• Cambridge Structural Database

What is: crystallography – crystal ?

• Krústallos (Gr Frozen water)

• solid state of matter

• regular external faces

geometrical law - Steno (1669)

angle between two corresponding

faces is constant

• regular internal structure

Keppler (New year’s day 1611)

• Hauy (1784): identical building blocks

• originaly part of geology - mineralogy

Relation to other sciences…

Structure and properties of…

• metals, alloys material science

• (in)organic compounds chemistry, pharmacy

solid-state physics

• biomolecules biochemistry, molecular

biology , medecine

With the help of…

mathematics informatics

Some examples… CHEMISTRY

C60, fullerene Proven by X-ray (derivative)

Relation to other sciences…

Some examples… BIOCHEMISTRY

NP Chemistry 2003 – structure determination water- ion channels

Relation to other sciences…

Some examples… PHARMACY

Relation to other sciences…

Some examples… POLYMER SCIENCES

Polyethylene Importance crystallinity

Relation to other sciences…

Importance symmetry

External order of a crystal reflects

an internal submicroscopic order.

Symmetry is important in almost all disciplines of science, art and

society

Some important realisations crystallography

• theory electrostatic bond in inorganic ion structures

• proof tetrahedral carbon atom

• structure elucidation benzene ring

• proof existence H-bond

•  -helix,  -plate in proteins

• DNA-model

• determination absolute configuration

• Principle : bring the system slowly toward a state of

minimum solubility (supersaturation)

• Driving force ?

• Large number of variables (purity, concentration, pH,

temperature, buffer, additives, precipitants…)

Crystallisation techniques

- Vapour diffusion

- Reactant diffusion (gel)

- Seed crystals

- Sublimation

• Sparse matrix screen

- samples a finite number of variables, e.g pH and [(NH4)2SO4]

- uses very small samples

- usually gives preliminary conditions (to be optimised)

Grease

Precipitant

in reservoir

1-5 L sample 1-5 L precipitant Mix

Invert, seal

• Crystallisation robot

liquid handling arm with tips (0.25-250 µl)

Mosquito

XtalScreens

Database of crystallisation screens

http://xray.bmc.uu.se/markh/php/xtalscreens.php

(a) Description of symmetry

• Schoenflies and Hermann-Mauguin notation

Rotation inversion axis n

Rotation reflection axis Sn

 Multiplication table for NH3

 Overview 32 point groups

 Some examples

http://www.molwave.com/software/3dmolsym/3dmolsym.htm

Chapter 2 Translation

symmetry

• One-dimensional space groups (7)

• Two-dimensional space groups or plane groups

Limited possibilities of

order of a rotation axis!

Possible two-dimensional

lattices (5)

Oblique p Rectangular p

Rectangular c

Square p

Hexagonal p

Rectangular

Square

Hexagonal

Plane group pg

pg

• Crystal systems (7)

triclinic monoclinic

(2 or m) (three 2 and/or m) orthorhombic tetragonal (4)

hexagonal (6) rhombohedral (3) (four 3 and three 2) cubic

• Relation with simple crystal structures of metals

a -Po

CN 6 or 8

PC 0.524 or 0.680

W

HCP

CN 12

PC 0.74

Why cubic? Cubic F lattice

Chapter 3 Space groups

• Glide mirror plane

IN OUT n

• Screw axis

Name order translation symbol

nb: rotation 360°/n + translation b/n

• Space groups (230, listed in International Tables vol A)

Some examples…

 set of symmetry operations that describe the symmetry of a crystal and form a mathematical group

 combination of 32 point groups and 14 Bravais lattices

 space group symbol:

Lijk L = Bravais lattice (P, C, I, F)

ijk = minimal symmetry elements for

different directions

+ +

+ +

) , , ( x y z

Z=1;

) , , 1

) 1

, ,

) , , ( x y z

1

Gridline for a/2

Note the "halfway marks" on each unit cell as visual guide!

Number of equivalent

positions in unit cell or

Non-centrosymmetric

+

2

a

c

Perspective view of a unit cell in P1bar with all centers of

symmetry shown The eight independent centers of symmetry ( i e., those NOT related by any symmetry operation ) are shown in blue

+

Z=2; ( x , y , z )

+

) , , ( x y z

+

3

Monoclinic, combine P with point group 2

Special positions on 2-fold axes:

(0, y , 0) (0, y , ½) (½, y , 0) (½, y , ½)

) , 2 1 ,

( x y  z

Non-centrosymmetric

4

2 1 No P

Change 2-fold axis into 2-fold screw axis

Special positions on 2-fold screw axes?

Z=4;

5

2 No

C

Non-centrosymmetric

) , ,

+

 ) 0 , 0 , 0 (

) , , ( x y z

1 , 2

Change P lattice into C lattice

Only special positions on 2-fold axes

Next combinations for monoclinic system?

) z , y , x

Bovine Pancreatic Trypsin Inhibitor (BPTI)

Four molecules in unit cell related by 2-fold screw axes Space group P212121

19

2 2

2

1 1

+

½-

) z 2

1 , y 2

1 , x

1 , y , x 2

+

) z , y 2

1 , x 2

All possible combinations of symmetry operations result in 230 space groups describing the symmetry

of a crystal

Some conventions for nomenclature:

• Priority for parallel symmetry elements:

m > a > b > c > n > rotation axis > screw axis

• Origin on inversion centre (centrosymmetric) or

point with highest symmetry

• Nomenclature of centering and glide planes

depends on choice axes!

C A a c

For biomolecules only space groups with only rotation axes or screw axes are possible! As a consequence only 65 space groups are possible (known as chiral or Sohncke space groups)

Space groups statistics for:

biomolecules organic molecules

P212121 24%

P21 13%

• Rotating anode

• Microfocus source

solid anode

air cooled ! high intensity (4.5 x rot anode)

microsource 50kV, 0.6mA, 30W rot anode 50kV, 80mA, 4kW sealed tube 50kV, 40mA, 2kW

• Metal jet anode (Ga)

K  = 1.34 Å

high intensity (140 x rot anode)

Metaljet 70kV, max 4.3mA, 200W

• Synchrotron radiation (SR)

 electrons emitted by an electron gun

 accelerated in a linear accelerator (linac) - 200 MeV

 transferred into a circular accelerator (booster) - 6 GeV

 injected into storage ring (844m circumference) where they circulate

at constant energy for many hours at 350.000 revolutions per second

 the life time depends on quality of vacuum (between 10-9 and 10-12

mbar)

 the storage ring includes both straight and curved sections

 Bending magnets: electrons are deflected from their straight path

by several degrees This change in direction causes them to emit SR

 Undulator: • complex array of small magnets

• wavy trajectory

• beams of radiation overlap and

interfere with each other

• much more intense beam!

Advantages…

High brightness (extremely intense, highly collimated)

Wide energy spectrum (choose  with monochromator)

Highly polarised

Very short pulses (less than a nanosecond)

Brillance = Number of photons per seconde

per mm2 (source area) per mrad2 (opening angle)

Example

Calculate how much CuK radiation is absorbed by a thin plate of 0.1 mm NaCl ( is 2.2 g/cm-3 for NaCl, m is 30.1 cm2/g for Na and 106 cm2/g for Cl)

• Scattering by electrons (Rayleigh scattering)

Waves as vectors

A cos[2π(νt-x/λ)] Summation of two waves by vector addition

Chapter 5 X-ray diffraction

Scattering by a row of atoms

When in phase?

x = y or in = out

Scattering by an array of atoms

When in phase? x = y Bragg planes as mirrors!

Use Miller indices

• Miller planes

Families of parallel planes with specific orientations and periodic

spacings can be drawn through a crystal

Miller index = 1/intercept

(110) (011) (120) d-value

dhkl

 -Fe2O3

For orthogonal systems:

First representative of (hkl) intersects:

X- axis at a/h Y-axis at b/k Z-axis at c/l

Note (hkl) versus [hkl] !





Laue – Bragg – 1912 Constructive interference when

2dsin  =n 

Consider only first order diffraction (n=1)

Intensity Ihkl depends on electron density  (xyz) in corresponding Miller plane hkl

• Bragg’s law

d-value ~ -1

Magenta atom scatters out of

phase for black plane

• Reciprocal space (*)

Peter Ewald (1888-1985)

Direct space – family of planes



Reciprocal space – one point

Mathematical definition:

Triclinic system

Rotate crystal to obtain as much

reflections as possible…

Direction given by Bragg’s law! Intensity?

• Scattering by an electron

• Scattering by an atom

atomic form factor or scattering factor fj

(scattering amplitude of an atom expressed

in terms of free electrons)

International Tables Crystallography, volume C

Chapter 6 Intensity of diffracted beams

• Scattering by a unit cell

electron density in unit cell is  sum of spherical atoms

structure factor F is scattering amplitude of the unit cell

expressed in terms of scattering factors fj of free atoms at fractional positions xj, yj, zj)

in two dimensions… OC = x.a OD = y.b

OA = a/h OB = b/k

difference in path length between O and C? difference in path length between O and P? difference in phase between O and P?

in three dimensions…

Structure factor expressed as sum of individual waves:

Intensity of diffracted beam Ihkl = Fhkl

Centrosymmetric case: +Fhkl or –Fhkl

For each atom at position xj,yj,zj, also an atom at –xj,-yj,-zj

B = 0, structure factor is real number, phase is 0 or 180° (or sign +1 or -1)

2

T = exp – (B sin2 / 2)

B is temperature factor (Å-2) isotropic or anisotropic

• Data collection

Relation between position and intensity of reflections and the

electron density in the unit cell

Data collection = measure intensities of all possible reflections

  1Å

Crystal mounted in capillary or loop on goniometer

Crystal Mounting

Typically a single crystal, once chosen, will be mounted on

a glass fiber, a nylon loop or a MiTeGen mount Usually the crystal will be attached to the mount with grease, super glue or 5-minute epoxy We normally cool the crystal to a VERY COLD temperature to immobilize the crystal (100 K)

Glass fiber + epoxy Nylon loop MiTeGen mounts

Intensity Ihkl depends on electron density  (xyz) in corresponding Miller plane hkl

X-ray

Cryocooling

•Procedure

•Advantages

- reduces radiation damage

- improves sometimes resolution

- reduces atomic motion

N2 100K dry air

Nylon loop

cryoprotectant

• Friedel’s law

diffraction pattern always centrosymmetric

symmetry diffraction pattern given by Laue group (= centrosymmetric point groups)

- - -

General expression of structure factor:

Fourier representation of electron density:

Summation is 0, except for h’=-h, k’=-k, l’=-l

Chapter 7 Fourier synthesis

Fourier representation of electron density:

Compare with structure factor…

Relation? Fourier transform of each other! Analogy with microscope

Key formulas

Ihkl ~ F2

hkl

Fhkl =  fj exp 2  i(hxj+kyj+lzj)

(sum over atoms in unit cell, fj scattering factor)

Fhkl = |Fhkl| exp i hkl

(structure factor is complex number)

 (xyz) = 1/V    Fhkl exp -2  i(hx+ky+lz)

(sum over all reflections, Fourier synthesis)

Phase problem: calculation of electron density needs both

amplitude and phase of the structure factor

To know the answer, you need the answer !

Fourier transform

lattice

motif

crystal

Phase problem can not be neglected…

colour = phase

amplitude duck + phase cat

amplitude cat + phase duck

Fourier synthesis

phase is position of wave

One-dimensional example: line Fourier of PbI2

Hexagonal, centrosymmetric structure

c = 6.977 Å

Systematic absences: Consequence of

translation symmetry!

(a) Patterson function (1934)

amplitude is square of structure factor, no phases needed peaks at locations corresponding to vectors between atoms intensity ~ product of number of electrons of both atoms Two-dimensional example:

centro- symmetric

N atoms (N2-N)/2 intramolecular peaks

Chapter 8 Solving the phase problem

Patterson map contains image of original molecule!

Check which pair of peaks, along with an atom at the origin, would

reproduce the Patterson map? Too complex for large number of atoms!

Use unit cell symmetry!

Suppose 21 axis  c For each atom at (x,y,z) also one at (-x,-y,z+1/2) Difference vector (u,v,w) becomes: (2x,2y,1/2) Harker plane

Gives coordinates of heavy atom!

Symmetry of the Patterson function

• centrosymmetric, translation symmetry lost

• 24 Patterson groups

Equivalent positions of P21/c from International Tables

x y z -x –y –z -x ½+y ½-z x ½-y ½+z

Same colors indicate overlap positions, but

four black coordinates are just four different

positions

(b) The heavy atom method (isomorphous replacement)

• add a strong diffractor (so-called heavy atom) to specific

sites in unit cell

• causes slight perturbations in diffraction pattern

• prepare after the normal data collection one or more heavy-atom

derivatives with e.g AgNO3, HgCl2, PtCl42-, AuCl42-

• protein crystals are soaked in solutions of heavy ions

• collect diffraction patterns

• measurable changes in at least a modest number of intensities

• derivative crystal must be isomorphous and diffract to a reasonably high resolution

• locate heavy atom (x,y,z)H by Patterson methods and calculate FH

FPH = FH + FP

Known: |FPH|, |FP|, FH

FPH = FH + FP or FP = FPH - FH

• Draw vector -FH

• Draw circle with radius |FPH| centered on head vector -FH

• FPlies somewhere on this circle

• Add circle with radius |FP|

• Two points of intersection or two possible phases

• Repeat this for second (or more… ) heavy-atom derivative

(should bind at a different site, otherwhise same phase information, as the phase depends only on the atom location and not its identity)

• One should agree better with one of the two solutions from the first derivative

• In general three or more heavy-atom derivatives are necessary to produce enough phase estimates (MIR, multiple isomorphous

(c) Direct methods

• Nobel Prize for Chemistry 1985: J Karle, H Hauptman

• Direct = derive the phases of Fhkl by mathematical means using

only intensity information

• F(hk) = 2  fj cos 2  (hxj + kyj) (summation over N/2 atoms)

s(H) x s(K) x s(-H-K) = +1 Sayre, Cochran, Zachariasen, 1952 Acentric case

 (H) +  (K) +  (-H-K) = 0

 2-relation or triplet relation

centro, 2D

9 Interpretation of electron density maps

Calculate electron density map (= Fourier synthesis) – contour – grid

Chapter 8 Interpretation of electron density maps

Tracing of CA backbone + positioning of side chains

Resolution (Å) of a crystal structure determination

(minimum d-value corresponding to maximum 2-value)

Chapter 10 Refinement techniques

• iterative process of improving agreement between Fo and Fc

• Locate water molecules, ions … from difference Fourier synthesis

 (xyz) = 1/V    (Fo-Fc)hkl exp -2  i(hx+ky+lz)

Criteria: peak height, distances, H-bonds, co-ordination, …

h k l