General Introduction

Master of Crystallography and Crystallization – 2013T01 – Mathematical, Physical and Chemical basis of

Crystallography

Teachers:

Carmelo Giacovazzo wdc wdc Mois Aroyo

Enrique Gutierrez-Puebla wdc wdc Santiago Garcia-Granda

General Introduction to the module I Course 01

Coffee Break

Introduction periodicity

Workshop on Introductory periodicity I

Lunch

Workshop on Introductory periodicity II

Open discussion

07.10.2013

Fundamentals of periodicity I

Coffee Break

Fundamentals of periodicity II

Workshop on periodicity I

Lunch

Workshop on periodicity II

Open discussion

08.10.2012

09.10.2011

Worshop on Space Groups I

Coffee Break

Worshop on Space Groups II

Crystal lattices I

Lunch

Crystal lattices II

Open discussion

Students Meeting

10.10.2011

Crystal lattices III

Workshop on Crystal lattices I

Coffee Break

Workshop on Crystal lattices II

Lunch

Personal or Group Tutorials

Open discussion

11.10.2011

Math Review

Coffee Break

International Tables for Crystallography

Lunch

Workshop on Symmetry. International Tables I

14.10.2011

Workshop on Symmetry. International Tables II

Group-subgroup relations and lattice transformations I

Coffee Break

Group-subgroup relations and lattice transformations II

Lunch

Group-subgroup relations and lattice transformations III

15.10.2012

Introduction to Reciprocal Space

Single Crystals, Powders and Twins

Atoms, ions, molecules and macromolecules

Coffee Break

Bonds and interactions in the solid state:

Conformations and Quirality

Lunch

Closing Remarks, Survey and

Open discussion or Group Tutorials

The first x-ray diffraction pattern (1912)von Laue, was the first person to perform a diffraction experiment with a crystal. He hypothesized that if atoms really existed, and if x-rays really were waves, then the wavelength of x-rays would approximate the distance between atoms in a crystal and diffraction would be observed. His brilliant insight thus proved both atomicity and the wave character of x-rays. This remarkable achievement was recognized by the Nobel prize. Laue did not actually do the experiment himself. Rather, he persuaded a couple of graduate students to do the experiment for him. Laue then set an example that has inspired PIs ever since – he was given all the credit!

W. L. (Lawrence) Bragg realized that von Laue’s diffraction pattern could be modeled as reflection from Miller planes -- i.e. the angle of incidence with Miller planes = the angle of reflection. This is the reason why diffracted x-rays are generally called reflections. Bragg published this work in 1913. His insight further substantiated the wave character of x-rays (thereby contradicting his father’s theory that x-rays are particles). This work also allowed the Bragg’s (father and son) to determine the first atomic resolution structures. The Bragg’s were jointly awarded the Nobel prize in 1915 – at which time W.L. Bragg was 25 years old and fighting in the trenches of world war I.

The first x-ray diffraction

picture, which was taken

from a crystal of copper

sulfate by von Laue’s

students, and dubbed the

“beerstein” pattern.

Crick & Watson 1953, published the double helical model for DNA based upon knowledge of chemical structure and geometry, possible chemical interactions, and fiber diffraction patterns. In outline, this is essentially the experimental approach of structure determination by x-ray crystallography. The structure immediately suggested the basis for replication and spurred, but did not illuminate, efforts to decipher the genetic code and mechanism of protein synthesis.

This reflects a common theme in structural biology. Function is most easily inferred when looking at the structure of a relevant complex.

DNA Crystallography is analogous to fiber diffraction

Photo 51 taken by Raymond Gosling under the supervision of Rosalind Franklin in 1952

Electron density is calculated from the diffraction pattern.

Structure determination by x-ray crystallography is analogous to light microscopy.

object

image

lens

Except that there is no lens that can focus x-rays. So we record the scattered x-rays and do the mathematical equivalent of focusing (electron density equation).

Electron Densityequation.

Crystal = trillions of copies of the object.

Diffraction (scattering) pattern -- can’t be focused.

Data collection

Electron density map (image) and model.

Resolution of a Reflection. Distance between planes (Å)The term resolution is used to describe the details of features that can be seen in an image. It has a precise meaning in crystallography. It is determined by the angle through which x-rays are scattered with respect to the incident beam. The higher the scattering angle the higher the resolution and the more detail that can be visualized. The resolution of a data set is typically defined as the point at which the intensity of half of the reflections falls to less than 2 times their standard deviation.

Detector

High resolution reflection.(usually weaker)

Low resolution reflection.(usually stronger)

The electron density equationThe electron density equation is used to calculate a focused image from the scattered x-rays, or diffraction pattern. This is a summation, in which the thousands of scattered x-rays are added together in a Fourier transform.

(xyz) = (1/V) |F(hkl)| exp[-2i(hx+ky+lz)-(hkl)]

• x, y, z = position in the crystal.

• (xyz) = electron density at the position x, y, z. Has units of e/Å3.

• V = volume of the unit cell

• h, k, l = position in diffraction space -- each of the diffracted rays has a coordinate.

• |F(hkl)| = Structure Factor Amplitude = Square root of the intensity of the diffracted x-ray at

position h, k, l. Has units of electrons (e).

• 2 = Just a constant. No big deal.

• i = square root of -1. Just a mathematical device to indicate direction. Because the electron

density equation is a summation of vectors, we need to indicate the direction of each vector.

No big deal.

• (hkl) = phase of the reflection at position h, k, l.

Amplitude

F

Origin

Low resolution maps can show overall features such as the shape of the molecule and the location of secondary structural elements. The figure shows a 7Å map of tropomyosin.

2.6 Å resolution. Main chain trace is usually fairly clear and many side chains have obvious if imprecisely defined density. An unwritten requirement for funding is crystals that diffract to at least 3.0Å resolution.

1.2 Å resolution.

< 10 Å

40 atoms

Small

Molecules

< 100 Å

2000 atoms

Proteins

200 Å

100000 atoms

Ribosome

700 Å

500000 atoms

Virus

CRYSTALLOGRAPHY

Substance DataCrystals StructurePhases

CRYSTALLOGRAPHY

Substance Data

Crystals

StructurePhases

CRYSTALLOGRAPHY

Basic Procedure: the pure protein is solved in presence of

precipitants at concentrations slightly lower to the precipitation

conditions.

The concentrations of protein and precipitants is slowly increasing to

reach a crystallization conditions.

Crystallization Methods

Vapor dffusionhanging dropsitting dropsandwich

Under oil.

Microdialysis.

Microdiffusion.

Counterdiffusion.

CRYSTALLOGRAPHY

Substance Crystals StructurePhases

Data

X RAY SOURCES

X Ray Tubes 106 fotons

Rotating Anode 109 fotons

Synchrotron 1019 fotons

Old X Ray Tube Synchrotron (European Synchrotron Radiation Facility, ESRF)

DIFFRACTION EXPERIMENT

CRYSTALLOGRAPHY

Substance DataCrystals Structure

Phases

FROM DIFFRACTION DATA TO

ELECTRON DENSITY

F T

F(x,y,z) = f(hkl)e d(hkl)i(xyz)(hkl)

FT

FT

THE PHASE PROBLEM

|Fp(h)|

Acentric Centric

|Fp(h)||Fp(h)|

Argand Diagram – Complex Plane

THE PHASE PROBLEM

CRYSTALLOGRAPHY

Substance DataCrystals Phases

ESTRUCTURA

Jerome KarleBorn New York City18 June 1918

- June 6, 2013, Annandale, Virginia, United States

Herbert A. HauptmanBorn New York City14 February 1917 - October 23, 2011, Buffalo

Nobel Prize in Chemistry (1985)"for their outstanding achievements in the

development of direct methods for the determination of crystal structures”

Herbert A. Hauptman and Jerome Karle

Made structure solution generally accessible to non experts

ONE of the 27 Nobel Prizes Awarded by Crystallographers

Nobel Prize in Chemistry (1988)

"for the determination of the three-dimensional structure of a photosynthetic reaction centre"

Johann Deisenhofer, Robert Huber, Hartmut Michel

Understanding of the photosynthetic mechanism

The first high-resolution structure of a membrane protein and also the most

complex molecular structure which had been solved up to that moment (1984).

Membrane protein crystallisation (1980)

Order and disorder – The master of analogies

Nobel Prize in Physics (1991)"for discovering that methods developed for

studying order phenomena in simple systems can be generalized to more complex forms of matter, in

particular to liquid crystals and polymers"

Pierre-Gilles de Gennes

In very different systems such as liquid crystal, ferromagnet, superconductor or polymer, universal features can be identified and be

explained by simple scaling laws

Order and disorder – The master of analogies

Georges CharpakBorn 1924 Dabrovica, Poland. Died 29 September 2010, Paris, France

Nobel Prize in Physics (1992)"for his invention and development of particle

detectors, in particular the multiwireproportional chamber"

Georges Charpak

In the multiwire proportional chamber each wire acts as a detector. The important breakthrough was mainly

due to the enormous increase in data-taking rate.

“for pioneering contributions to the development of neutron scattering

techniques for studies of condensed matter”

Bertram N. Brockhouse and Clifford G. Shull

Nobel Prize in Physics (1994)

“for pioneering contributions to the development of neutron scattering

techniques for studies of condensed matter”

Bertram N. Brockhouse and Clifford G. Shull

Nobel Prize in Physics (1994)

Neutrons see more than X Rays

“for their elucidation of the enzymatic mechanism underlying the synthesis of adenosine

triphosphate (ATP)” and “ for the first discovery of an ion-transporting enzyme, Na + , K + -ATPase”

Paul D. Boyer, John E. Walker and Jens C. Skou

Nobel Prize in Chemistry (1997)

A molecular machine for ATP synthesis was discovered when the enzyme ATP synthase was crystallized.

“for discoveries concerning channels in cell membranes ”

Peter Agre and Roderick MacKinnon

Nobel Prize in Chemistry (2003)

Unravelling the secrets of cell channels

“for his studies of the molecular basis of

eukaryotic transcription”

Roger D. Kornberg

Nobel Prize in Chemistry (2006)

Structure of an RNA polymerase II transcribing complex

“for studies of the structure and function of the ribosome”

Venkatraman Ramakrishnan, Thomas A. Steitz and Ada E. Yonath

Nobel Prize in Chemistry (2009)

“for the discovery of quasicrystals”

Dan Shechtman

Nobel Prize in Chemistry (2011)

Where are the atoms? A 3 dimensional direct-space approach (tiling models) is usually used in combination with a 6 dimensional reciprocal-space approach.

The IUCr had to change the deffinition of crystal : A material is a crystal if it has essentially a sharp diffraction pattern

Direct methods for phasing diffraction data for periodic structures are not directly transferrable to higher dimensional problem; the development of novel method is needed.

“for studies of G-protein-coupled receptors”

Robert J. Lefkowitz, Brian K. Kobilka

Nobel Prize in Chemistry (2012)

A combination of several biochemical strategies produced a ternary complex suitable for crystallization and its high

resolution structure was finally determined.

G-protein-coupled receptors mediate a wide range of physiological signals from the outside of the cell.

“for studies of G-protein-coupled receptors”

Robert J. Lefkowitz, Brian K. Kobilka

Nobel Prize in Chemistry (2012)

When a hormone, olfactory molecule or a taste molecule couples with a receptor on the cell surface, a

chain of reactions inside the cell is triggered

- Complex bulk systems with interesting physical properties are often inhomogeneous on a nanometer lengthscale

high-temperature superconductors, colossal magnetoresistive materials,

high performance thermoelectric materials …

- Nano-particles, nano-tubes, nano-wires etc. important for applications optoelectronics, nanosensors, programmed release drug delivery systems…

Physical properties often critically depend on the nano-scale structure, rather than the long-range structure!

Complex nanostructured materials

Nanoporous (mesoporous) materials

NanoparticlesNanostructured bulk crystals

Example: Ho2(Ti2-xHox)O7-x/2 ”stuffed spin ice”

300K neutron diffraction patterns (GPPD, IPNS, Argonne)

x=0.00x=0.30x=0.50x=0.67

pyrochlore fluorite

x=0.3

Crystallography challenged: materials w/ disorderC

HA

LLEN

GE

Rietveld approach assumption: crystals are perfectly periodic…

…but this is not always the case!

Crystallography challenged: nano-crystalsC

HA

LLEN

GE

Figures: J.S.O. Evans et al.

and M. Tucker et al.

Bragg peak info ONLY

Rietveld method

in powders

Global ApproachFU

TUR

E

• Add complementary information

– Extra experimental data

– Theoretical constraints

Science, 316, 561 (2007).