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Outline of Today's Lecture. Crystal growth and handlingDiffraction and how it encodes spatial informationCrystal symmetry and how it affects data collection. Power and Limitations of X-ray Diffraction. POWERPicture of molecule with almost no assumptions: r(x,y,z)No size limitations: Mr > 107Rapid structure determination (high throughput) now a reality.

X-ray Crystallography: Diffraction, Symmetry, and Data Collection

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**1. **X-ray Crystallography: Diffraction, Symmetry, and Data Collection MBIC 709
Eric A. Toth, Ph.D.
02/18/10

**2. **Outline of Today’s Lecture Crystal growth and handling
Diffraction and how it encodes spatial information
Crystal symmetry and how it affects data collection

**3. **Power and Limitations of X-ray Diffraction POWER
Picture of molecule with almost no assumptions: r(x,y,z)
No size limitations: Mr > 107
Rapid structure determination (high throughput) now a reality LIMITATIONS
Full detail (X-ray crystallography) requires a crystal. Otherwise, less information (e.g. DNA fibers)
Each picture is static
Resolution often limited

**4. **What can this information tell you? information ranges from broad perspective of macromolecular complex arrangement to important atomic interactions
examples:
hemoglobin and myoglobin
KcsA K+ channel
p53
the ribosome saga

**5. **Getting Diffraction-Quality Crystals still a huge stumbling block in x-ray crystallography
requires three very non-trivial things:
obtain sample of superior quality
induce the sample to form single crystals
manipulate the crystals such that they yield useful data

**6. **A Sample of "Superior Quality" purity is first check - SDS-PAGE, either Coomassie or silver staining
should be a single band when overloaded on Coomassie-stained gel
this is not enough

**7. **Biophysical Characterization not only must be pure, must be monodisperse (i.e. a single, homogenous population of molecules) and at high concentration (10 mg/ml is a typical starting point)
buffer system, salt, reducing agents, ligands, can have drastic effect on protein happiness
gel filtration
native gel
dynamic light scattering

**9. **Example of Good DLS Profile

**10. **Growing Crystals = Voodoo try to induce complex molecules to make a number of weak contacts that force it to form a three-dimensional lattice
crystals grow from a supersaturated solution
nucleation is the initial event in the process
want this to occur a limited number of times
after nucleation, want crystals to grow to a reasonable size (0.1mm at a minimum)
need them to be single crystals

**11. **Crystals alter the solution environment such that proteins are forced to aggregate
mostly leads to junk, but every so often crystal formation occurs
common precipitants:
polyols: typically polyethyleneglycol of varying molecular weights (400-20,000)
salts: ammonium sulfate, NaCl, etc.
alcohols: ethanol, isopropanol, etc.
other: jeffamine, methylpentane diol

**12. **Methods of Crystallization

**13. **Key Considerations during the crystallization process, the protein must remain in its native state and monodisperse
huge number of variables (precipitant concentration, pH, additives, temperature, etc.) to screen)
crystallization space is most often sampled semi-randomly, by "sparse matrix" methods
membrane proteins are trickier because they need a lipid in order to keep them solubilized
nature of lipid as important as conditions screened

**15. **Crystal Handling almost as capricious as crystal growth
must get small crystal from drop into either a capillary or rayon loop for data collection without damaging it
might require solutions vastly different than crystallization conditions (and serial transfers too)
collection at 100K also requires that the solution form a glass and not destroy the crystal

**17. **Cryoprotection idea is to prevent ice formation (ice crystals diffract very well), form a glass
usually glycerol or MPD or ethylene glycol is incorporated into the crystallization conditions
can be done gradually or suddenly, depending on what the crystal will take
some saturated salts also form a glass at 100K
oils, like paratone-N and mineral oil can also work

**18. **Diffraction

**19. **The Phenomenon of Diffraction Diffraction occurs when waves encounter an obstacle

**20. **Waves

**21. **Waves continued basic wave equation:
frequency of light ranges from one cycle every 10-14 to 10-19 seconds
can't detect time-dependent phase shift
diffraction measurements independent of time:

**22. **Diffraction contains information about the obstacle that caused it
observed only if obstacle used not too much larger than wavelength of radiation
x-rays 1.54Å, similar to bond lengths
diffracted waves distinct from incoming radiation
wave vector k contains spatial information

**23. **Information "Encoded" in Diffraction Pattern wave vectors k vary in direction—spatial info

**24. **Encoding, continued diffraction from each volume element:
Since = 0 outside the obstacle, can recast as
Now cast as a Fourier transform :
diffraction pattern is Fourier transform of amplitude function ( ) of obstacle

**25. **varies with k
exponential describes phase relationship between scattering centers and arbitrary origin (very important info)
How to go diffraction pattern ? obstacle?
Fourier transform (T)
T(T(a)) = a

**27. **What Information Resides in a Diffraction Pattern? how to break down info into its components
start with simple objects and expand to diffraction from a 3-D crystal
measure not the complex amplitude, but the intensity
want to reconstruct the obstacle

**28. **Diffraction Summary

**29. **Diffraction from a Crystal infinite crystal = infinitely sharp peaks
position of main peaks determined by lattice
shape of the peaks determined by shape of the obstacle
intensity of the peaks determined by the motif under study

**30. **More Diffraction from a crystal finite crystal equivalent to multiplying infinite lattice by a shape function
crystal contains three types of structural info:
Lattice
Shape of Crystal (not important in practice)
Motif
determine lattice, then motif

**31. **Still More Diffraction from a Crystal if the obstacle of interest is a protein, why a crystal?
crystal acts as an amplifier
S/N for single protein molecule would be too low for detection

**32. **Real lattice = actual crystalline lattice
Reciprocal lattice = T(crystal lattice)
reciprocal relationship between the two
smaller motif, wider spacing of diffraction maxima and vice versa
simple geometric relationship between the two
This relationship helps you determine the lattice and eventually the structure

**33. **Interference most radiation is lost due to destructive interference
diffraction only observed when waves constructively interfere
exactly same pathlength (qincident = qreflected):

**34. **Bragg's Law and Diffraction path difference = multiple of wavelength
nl = 2d x sinq

**35. **Scattering Vector useful construct accounts for difference in angle between incident and scattered waves

**36. **phase of diffracted wave from an electron at position is:
to be in phase, must be an integer
results in phases of 0, 2p, 4p, etc.
otherwise, any phase is possible, leading the integral to go to 0

**37. **Laue Conditions position in 3D:
total scattering from crystal:
Laue conditions -- , , integers

**38. **Ewald Construction (2D)

**39. **More Ewald Construction (2D)

**40. **Limiting Circle Ewald construction can't sweep out area larger than its diameter (2/l)
can get directly from Bragg's Law (sinq=1)
visible light wouldn't give useful data

**41. **Ewald Sphere for any orientation of the crystal, a limited number of lattice points will fall on the surface of the sphere
can determine what the lattice looks like based on a few frames of data
will know where all of the spots should be, then measure their intensities
Example

**43. **Symmetry and Data Collection

**44. **Crystal periodic array of molecules:

**45. **Lattice lattice is conceptual array of points in space that define geometrical relationship between motifs in structure
ƒ(crystal) = ƒ(motif)?ƒ(lattice)
nature of lattice (i.e. its symmetry) tells us where to expect diffraction data
finite number possible
basic components = unit cell and asymmetric unit

**46. **Unit Cell basic repeating unit of crystal (origins of the unit cells form the lattice)
basis vectors , , and describe boundaries

**47. **by convention (usually) lies along x, along y, and along z
a, b, and g are angles between vectors
choice of unit cell arbitrary, so enforce conventions
right-handed coordinate system
unit cell axes coincide with highest symmetry
smallest cell that obeys above

**48. **depending on the situation, one might choose different unit cells encompassing one or more lattice points:

**49. **Types of Cells Primitive (P), Plane-centered (A, B, or C), Body-centered (I), Face-centered (F)

**50. **Asymmetric Unit definition: smallest part of the unit cell which will generate the whole cell if all symmetry operators are applied to it
abbreviated ASU
in protein crystallography, ASU is one or more protein chains

**51. **Symmetry determines the details of how we solve the structure
definition: a symmetry element (or operator) when applied to an object leaves that object unchanged
most trivial is identity operator (I), which does nothing
primarily rotations and translations in protein crystallography
exception is center of inversion inherent in reciprocal lattice
mirrors and glide planes incompatible with chiral molecules

**52. **Basic Symmetry Operators: Rotation 180º about an axis is called a "2-fold" rotation

**54. **Screw Axes combine rotation and translation
mn ("m sub n"): translate n/m of a unit cell, then rotate by an m-fold
21 screw denoted by
other screw axes: 31, 32, 41, 42, 43, 61, 62, 63, 64, 65,

**56. **Rotations compatible with a lattice

**57. **Groups symmetry elements that describe a lattice must make up a group
must contain I
combination of any two elements of a group results in another element of the group (closure)
#elements = #objects repeated = order of group
every element must have an inverse

**58. **Point Groups That Can Exist in Crystals

**59. **Space Groups we're interested in 3D groups (e.g. P2):

**61. **Seven Crystal Systems

**63. **Space Groups 230 possible
65 contain only rotational symmetry elements ? relevant for chiral molecules

**64. **Data Collection seek to measure all available Bragg peaks for the crystal under study
nature of reciprocal lattice needs to be determined
orientation of reciprocal lattice relative to laboratory reference frame also unknown
properties of crystal (including symmetry) determine how data collection proceeds

**65. **Some Factors Affecting Data Collection unit cell size – must resolve spots
dmax – influences where we place detector
mosaic spread
takes into account that neither the beam (divergence) nor the crystal ("mosaic blocks") are perfect
can limit rotation range of each frame

**66. **Mosaic Blocks

**67. **Symmetry and Diffraction Data Friedel's Law:
reflection from back of plane is the same as from the front

**68. **Systematic Absences 2-fold screw axis along y: x,y,z ? -x,y+½,-z

**70. **Laue symmetry the diffraction pattern will contain symmetry that is related to the symmetry of the crystal
rotational symmetry plus Friedel's Law (inversion center)
Laue symmetry plus systematic absences in almost all cases gives an unambiguous space group determination

**71. **Symmetry and Data Collection data are complete if all reflections or their symmetry mates have crossed the Ewald sphere
higher symmetry ? more symmetry mates ? smaller rotation range needed to access all unique reflections

**73. **Why do we care? most crystals are radiation-sensitive
proper orientation can make the difference between complete and incomplete data
incomplete data
"redundancy"—the more times you measure a reflection, the more accurate that measurement is (duh)
low symmetry can make collection of highly redundant data impractical (time) or impossible (radiation damage)

**74. **Evaluation of Data Quality
how well multiple observations of a reflection and/or its symmetry mates agree
unweighted statistic, subject to manipulation

**75. **Evaluation of Data Quality I/s—observed intensity divided by its error
"signal-to-noise" ratio, tells how much information is in a measured reflection or group of reflections (i.e. within a certain resolution range, etc.)
better indicator of data quality than Rmerge

**76. **OK, so I've figured out my lattice and space group, I've measured all of the intensities I am able to measure, and I'm satisfied with their quality. Where's my structure?!?