1
Q
Why use X-rays
A
- microscopes only provide images of objects the same size as the wavelength of visible light (500 nm limit)
- cannot make x-ray microscope
2
Q
Principle of Crystallography
A
- crystal of protein scatters waves and forms specific dot pattern at detector
- we form an electron density map from this that allows us to build the protein structure
- must relate the spot pattern to crystal structure
- to see structure we need to determine electron positions by exploiting their interactions with x-rays
3
Q
X-Ray Waves
A
- travelling electromagnetic wave
- electromagnetic wave is an oscillating electric and magnetic fields at right angles to each other and the travel direction
- we only examine E (electric field)
- E field describes electrostatic force felt by charged particle due to motions of other charged particles
4
Q
Single Electron Scattering
A
- if you have an oscillating electric charge it radiates electromagnetic radiation at same frequency of oscillation
- electron response by oscillating antiphase with incoming electric field
- oscillating electron emits x-rays over wide angle
- similar to waves rippling out
- electrons are charged particles and experience a force so will oscillate and reradiate waves
5
Q
Two Electron Scattering
A
- each electron in structure becomes x-ray source
- scattering pattern is resultant of adding scattered waves
- diffraction pattern depends on structure
6
Q
Molecular scattering
A
- waves emitted interfere to give constant diffraction pattern with peaks and troughs
7
Q
Constructive Interference
A
- two waves of amplitude A that are perfectly in phase
- phase difference is whole numbers of pi
8
Q
Destructive Interference
A
- two waves of amplitude A that are perfectly out of phase
- resultant amplitude is zero
- phase different is odd numbers of pi
9
Q
Wave Phase
A
- at given position in space the phase indicates position in wave cycle
- varies with position
10
Q
Phase Difference (shift)
A
- between waves scattered from 2 electrons depends on their positions relative to one another in the structure
- incident waves are in phase but the top ray will travel further to hit the electrons, introducing a path different between scattered rays which results in a phase difference
- shift due to structure, ie. structure affects amplitude and phase of resultant waves in direction 2theta
11
Q
Scattering Vector (S)
A
- magnitude and direction of S vector contain information about wavelength and scattering angle
- at a fixed wavelength the magnitude varies with the angle
- vector doesn’t point in direction of scattering but is used to work out 2theta
12
Q
Phase Shift Equation
A
phase shift = 2pi x r x S
- r is the relative positions of the 2 electrons
- rewrite with complex numbers
13
Q
Resultant Wave equation
A
add wave scattered from origin and wave scattered from r
- structures that have more than 2 electrons
- structure factor is the addition of all phase shifts in the waves
- essentially adding up all waves each of which has a phase shift dependent on the position of electron causing the scattering
- amplitude and phase of the resultant wave is dependent on the amplitude and phase of the components (amplitudes all the same so phase is changed depending on position of each electron it scattered from
14
Q
Structure Factor f(S)
A
- description of diffraction pattern (amplitude of the wave scattered in each direction encoded by S from the sample)
- describes how diffracted waves in each direction, f(S), are related to positions of electrons (r), which is the structure
- it is a wave with phase and amplitude terms
- it is a complex number
15
Q
Molecule Electron Density
A
- describe distribution of electrons by a 3D electron density function
p(r)=p(xyz) says that the position of a function of a position vector - mathematical way of describing the structure
- at position r you multiply by a infinitesimally small cube of density
16
Q
Molecular Scattering
A
- considering all scattering in just one direction:
- each scattered wave has amplitude (number of electrons at position r) and phase
- varying electron density affects scattering probability
- add all terms of scattering in direction associated with S
- scatter function for all scattering from position r
- phase of each direction is dependent on direction of scatter
17
Q
Fourier Transform
A
- scatter pattern is the FT of the structure
- notation means we add each wave scattered from every part of the atom
- total scattering in direction associated with S from an object described by p(r) is the sum of the waves scattered from every point in the object
18
Q
Inverse Fourier Transform
A
- measure all f(S) values to calculate p(r) which is the structure
- measure scatter at all S values to work out structure
19
Q
Molecular Structure Factor
A
- describes total scattering by molecular as a function of direction
- each direction associated with different value of S
20
Q
Use of Crystals
A
- amplify scattering to make it detectable
- ordered arrays in 3D space
- each molecule is a unit cell
- properties of crystals restrict scattering to specific directions (spots)
- higher signal to noice retio
21
Q
Reflection from Molecule Layers
A
- only 2% of waves are reflected
- planes are semi transparent
- every plane contributes more of less equally to scattering
- add up all reflected waves from each layer in same direction
- diffraction is a series of partial reflections from crystal planes
22
Q
Braggs Law
A
- constructive interference if the path difference between adjacent waves is a whole number of wavelengths
2dsin0 = n x wavelength - strong diffraction only in directions for which law holds (exact whole number of wavelengths)
- destructive interference if there is phase shift introduced by the path difference
odd number of half-wavelengths
23
Q
Crystalline diffraction
A
- scattering only occurs if path difference between waves from adjacent layers is n x wavelength
- crystals amplify scattered intensity making it measurable in allowed directions
- positions observed rays tells us about orientation and spacing of planes
- each spot is a ray emerging in a direction allowed by the law
24
Q
Crystalline Diffraction Equations
A
- F crystal (S) = Nfmol (S) where N is the number of molecules in the crystal
- only discrete S values allowed
- express structure factors as F crystal (hkl) where h,k,l are integer indices denoting discrete allowed S values
- F xtal is the total scattering in a crystalline sample
25
Experimental Set-up
- crystal rotated about horizontal axis perpendicular to beam
- all sets. of planes give diffraction spot when rotated into Bragg condition
- X ray deflected through 20
- for a given d and wavelength only certain angles allow constructive interference between places
26
Good Crystals
- good crystals give good data
- ordered and correct orientation
- need diffraction to high angles which correspond to high resolution data
27
Producing Crystals
- natural sources used for big complexes
- recombinant sources can overexpress protein, are easy to handle, can use tag purification, and it is easy to know the protein sequence
28
E. Coli
- grows rapidly to yield large amount of product
Cons:
- difficult to make proteins with disulphide bridges
- may have unfolding (lower temp)
- may be toxic (reduce leaky expression)
29
Optimise Crystallisability
- want single domain rigid proteins
- avoid multidomain proteins with flexible linkers or with flexible termini
- remove tags
- increase solubility (mutate surface residues to increase hydrophilicity
- add ligand
30
Limited Proteolysis
- identify stable protein fragments
- digest with non specific proteases
- determine termini of fragments
- clone fragments for expression as you are left with a rigid core domain
31
Growing Crystals
- precipitate protein slowly
- can't use open evaporation
- use methods allowing slow alteration of solvent the protein is dissolved in for precipiation
- drives protein into insoluble state
32
Vapor Diffusion
- place precipitant solution in reservoir
- mix equal volumes of protein and precipitant solution in well
- well concentration is 1/2 that of reservoir
- seal chamber and system reaches equilibrium
- water extracted from well increasing concentrations and favouring precipitation
33
Theory of Vapor Diffusion
- energetic barrier to spontaneous nucleation
- need high enough concentrations to drive into spontaneous nucleation zone, then concentration of protein in solution lowers as crystal grows
- cannot have too high protein or too low precipitant
34
Protein Crystal Properties
- soft and wet with large solvent channels
- fragile
- solvent presence may replicate native state
- can add ligand or cocrystallize to visualise activity or binding
35
Bragg's Law and Resolution
- link between good crystal and high resolution atomic level data
- maximum theoretical diffraction angle is 180 degrees (20)
- inversely proportional to dmin of 0.5 Angstroms
- dmin defines resolution limit
- highest resolution data obtained at highest diffraction angles (edge of detector)
- planes with small d spacing need good long range order and these planes give higher resolution data
- protein crystals usually have a maximum 20max of 30-40 and dmin of 1.9 Angstroms
36
Crystal Issues
- growth defects like missing molecules
- inherent flexibility
- non perfect alignment
- may not necessarily have the long range order to probe small dmin
37
Producing X-Rays
Rotating anode generator
- accelerates electrons into copper target
- fixed wavelength
Particle accelerator
- electrons accelerated around a circle and emit radiation
- tuneable wavelength
38
Cryo-cooled crystals
- need rotation through 180 degrees so all Bragg planes end up in Bragg defracting condition
- add cryo protectant solvent to prevent ice crystals in solvent channels
- prevents drying out and protects against radiation damage
39
Radation Damage
Photoelectric effect: release of electrons
Free radicals: diffuse in channels and attack surface
- loss of long range order and resolution
- cooling gives higher x-ray tolerance allowing us to use smaller crystals
40
Phase Problem
- F(S) is a wave with amplitude and phase
- but we only measure intensity of waves not the phase
- need phase to apply inverse FT
- need to solve phase problem to solve structure
41
Electron Density Map
- calculate magnitude of F(hkl) and phase to calculate density map
- set of structure factors, phases, and amplitudes for each value
- Inverse fourier transform used for this
- errors propogate to electron density maps
- shows density variation in space
- plotted with contours to show areas of higher density where electrons are expected
- fit amino acid sequence to density (match to known sequences)
- need understanding of stereochemistry and structure
42
Model Refinement
- refinement adjusts atomic model to give a better fit to observed data
- ensures model conforms to ideal stereochemistry
- calculate improved phases from improved models to give better maps for more model building
43
Observations to Parameters Ratio
- observations are the amplitude measurements from the data
- parameters have 4 per atom (x,y,z,B)
- want a high obs/parameter ratio
- like fitting a line to a series of points
- parameters improved by knowledge of stereochemistry and known atomic arrangements
44
Refinement Process
- minimise target function expressing discrepancy between Fobs and F calc (observed amplitude vs calculated)
- discrepancy between model and ideal geometry
45
Positional Refinment
- computer makes incremental improvements in the model
- on each cycle program determines direction each atom moves to improve it
- when we calculate structure factors they are close to observed f(S)
46
B factor refinement
- temperature factor
- models static disorder and vibrational motions of atom in the structure
- peripheral atoms have degrees of freedom and rotational freedom about carbon bonds
- electron density smears out and disappears from map
- lowest in center and increase at surface
- can potentially indicate errors in model
47
Adding bound water
- high res structures show a shell of bound waters
| - water placed in peaks in map close to hydrogen bonding partners
48
R factor
- assesses progress of minimisation after each cycle
- should reduce
- fractional different between observed and calculated diffraction
- may over fit data reducing R but not improving model (like a false positive)
49
How to calculate Obs/Parameters
number of independent reflections / (number of non hydrogen atoms x 4)
