See short article by W. Hendrickson
A. Diffraction Patterns / Fourier Transforms of CrystalsSee "Diffraction and Fourier Transforms" from March 12, 1998)
B. Procedure
C. Resolution  Higher resolution allows more accurate positioning of atoms
A. Protein Solubility  how to make protein less soluble > crystal growth; make a supersaturated protein solution in which crystal nuclei will form and grow into crystals until the solution is no longer supersaturated
1. Salting In  many proteins are not very soluble in pure water; dissolve protein at moderate salt concentration and then dialyze against distilled water
2. Salting Out  most proteins become insoluble at very high salt concentrations (ions bind most H_{2}O molecules); dialyze protein solution against high [salt]; commonly use (NH_{4})SO_{4}
3. Isoelectric pH  minimal solubility because net charges on protein molecules is 0 ==> no repulsive electrostatic forces ==> salt in or salt out at pI.
4. Organic Precipitants
a. alcohols such as 2,4methylpentanediol
b. polyethylene glycol (PEG)  very frequently used
B. Methods
1. Batch  add precipitant dropwise to small beaker of stirred protein solution; when it becomes faintly turbid, set aside for several weeks. Requires large amounts of protein and very good eyesight.
2. Dialysis  dialyze in small glass tubes with 1 end covered with dialysis membrane
3. Vapor Diffusion  seal droplet of protein solution in chamber with large volume of solution containing precipitant. If precipitant is not volatile then must include it at lower concentration in protein solution; will become more concentrated as water is drawn out.
a. depression slides sealed in plastic box
b. hanging drop method  drop hanging from glass coverslip sealed over well of tissue culture plate. Uses very small volumes of protein (10  20 µl) and permits organization and easy observation.
C. Crystal Composition: typically 30  60% liquid which fills in the spaces between protein molecules==> very fragile with the consistency of jello. Must keep them from drying out.
III. Crystal Classes and Symmetries
A. Symmetry Principles
1. Point Groups  Symmetry about a point
a. Nomenclature:
 Schoenflies  e.g. D4  used by spectroscopists
 HermannManguin  e.g. 422  used by crystallographers
b. Cyclic Groups
 single axis of rotation  nfold > rotation by 360º/n
 1 or more planes of reflections or mirrors (not allowed in chiral biomolecules)
 examples: C4 = 4; C5 = 5; C6 = 6; C1 = 1 (no symmetry)
E. coli Glutamine Synthetase has 622 symmetry or D6 c. Dihedral Point Groups (the first two are always present)
 nfold rotational symmetry axis
 perpendicular 2fold symmetry axes
 may mave mirror planes and inversion centers (but not for chiral biomolecules)
 examples: D2 = 222; D3 = 322; D4 = 422; D6 = 622 (glutamine synthetase from E. coli)
d. Cubic Point Groups
 four 3fold axes along body diagonals of a cube plus..
 three 2fold axes connecting opposite edges ==> tetrahedral, symbol = T = 23
CH_{4} has tetrahedral symmetry or all rot. axes of cube (4, 3, and 2fold) ==> Octahedral, symbol = O = 432
 many transition state metal ions form ligand complexes having octahedral symmetry
 or all rot. axes of an icosahedron, a regular 12sided solid (6 5fold; 10 3fold; 15 2fold) ==> icosahedral, symbol = I = 532. This is the point group symmetry of spherical viruses. Basic icosahedron contains 60 subunits; viruses contain multiples of 60 subunits.
e. Asymmetric Unit  the smallest part of an object from which the whole can be generated by symmetry operations.
B. Lattices
1. 2dimensional  5types characterized by lattice vectors a and b and the angle between them, g.
a. Primitive ==> P
b. Centered Rectangle ==> C  a = b ; g = 90°
2.. Plane Groups: 2dimensional lattice type + motif point group symmetry
combinations.
a. This is the basis of all regular 2dimensional patterns
b. Not all point groups are compatible with every lattice type; there are 17 possible combinations, five are compatible with biological structures which cannot have a mirror plane or inversion center.
3. 3Dimensional Lattice Types: 14 Bravais Lattices; characterized by 3 lattice vectors (a, b, c) and 3 angles, a (between b and c), b (bet. a and c), and g (bet. a and b); see figure 1616 in Eisenberg and Crowthers Physical Chemistry With Applications to the Life Sciences BenjaminCummings 1979




































4. Space Groups: 3dimensional lattice + 3 dimensional motif point group symmetry; symbol gives lattice type (P, C, I, F etc.) followed by point group symmetry (e.g. P422, P2_{1}2_{1}2, I4, F222 etc.). Note that when some point groups are combined with a lattice a new type of symmetry operater is produced, a screw symmetry operator designated by two integers, n & m. Thus, if the structure is rotated about an axis by 360° / n and translated by m / n of a unit cell, the result is identical to the starting condition.
a. 230 possible combinations of 14 Bravais lattices + 3dimensional point group symmetries (cyclic, dihedral, cubic and screw axes)
b. Only 65 out of 230 are possible space groups for crystals of chiral biological molecules where motif symmetry cannot include mirror planes, inversion centers etc.
IV. Two Descriptions of diffraction
A. von Laue  described in an earlier lecture
1. Incident beam and diffracted beams described by unit vectors, s and so respectively
2. Diffracted ray described by vector, S = s  so = 2 sin q / l according to the Laue Equation.
B. W.L. Bragg > Bragg's Law  useful for describing diffraction from crystals
1. Diffraction is described as reflection of incident Xray beam from "imaginary" sets of parallel planes with different angles and spacings, d.
a. for a 3dimensional crystal each set of planes has indices (h,k,l); remember our 2dimensional example
b. h,k, and l are integers equal to:
 unit cell axis length divided by distance along each unit cell axis at which the first plane intersects each unit cell axis (e.g. Fig. 177,9 in Eisenberg and Crowthers, EC).
h = a / a'; k = b / b'; l = c / c' the number of planes intersecting each axis of a unit cell. This is like the number of times a wave oscillates along each unit cell axis in our previous example.
2. Diffraction occurs when the pathlength of a ray reflected off of 2 planes of the same set is equal to the wavelength (1.54 A for Cu Ka )
a. Bragg's Law ==> 2 d sin q = l
b. the angle of reflection is = 2 q
3. The Intensity of the diffracted ray for a set of (h,k,l) is larger if many atoms lie on the set of planes described by h,k,l.
4. Resolution
a. fine spacings are described by diffraction from sets of planes which are very close together (d = small; h,k,l = large)
b. If d is small, then from Bragg's law q must be large for diffraction to occur ==> high angle diffraction (large h,k,l) contains high resolution information
c. This looks very similar to the Laue Equation derived previously; S = (2 sin q)/l. S is perpendicular to the (h,k,l) planes and has a length, S = 1/d. Laue and Bragg had 2 different ways of describing the same phenomenon > both get the same result.
C. Sphere of Reflection (Ewald Sphere)  The Sphere of Reflection is an imaginary sphere with a radius of 1 / l drawn around the crystal Fig. 1710. Each set of hkl planes must be properly oriented in order for diffraction ("reflection" of the xray beam) to occur. The proper orientation can be described by the relationship between the sphere of reflection and the reciprocal lattice.
1. Where is the Reciprocal Lattice?
a. The S vector for each (h,k,l), Shkl = 1/d_{hkl} and is perpendicular to the (h,k,l) planes. The position of the (h,k,l) reflection in the reciprocal lattice is given by the tip of S_{hkl} when the tail of the vector is placed at position 0 on the Sphere of Reflection as shown in the diagram.
b. (h,k,l) are the cartesian coordinants of a reflection on the Reciprocal Lattice.
c. a* and b* are reciprocal space lattice vectors corresponding to a and b respectively.
d. from Fig. 179  a longer b ==> shorter b* in reciprocal space (ditto for a and c).
e. b* is perpendicular to a and a* is perpendicular to b.
f. if g = 90 ==> a* is parallel to a and b* to b.
g. diffraction occurs when the crystal is tilted (tilting the reciprocal lattice also) so that the tip of S_{hkl} (the point (h,k,l) of the reicprocal lattice) lies on the surface of the Sphere of Reflection ==> must move the crystal to record diffraction patterns ==> this is when the Bragg equation is satisfied.
D. The Structure Factor, F_{hkl}
1. Each point of the reciprocal latt. represents a diffracted wave described by a complex number, F_{hkl}.
2. Complex numbers have 2 parts, a real part and an imaginary part (although there is nothing imaginary about either).
complex numbers are represented by a vector in the complex plane (Fig. 1712a) by:
a. Cartesian coordinants, A and B ==> A + iB
b. Polar coordinants, F and a, which are called the Amplitude and phase of the complex number
c. A = F cos a; B = F sin a; a = Tan^{1} B / A
3. Fhkl can also be written as an exponential: F_{hkl} = F_{hkl} e^{i }^{a(hkl)} = F_{hkl} (cos a + i sina)
4. F_{hkl }describes a wave diffracted by the crystal unit cell
a. h,k,l ==> direction and frequency of the wave
b. F ==> amplitude of the wave
c. a ==> phase of the wave
5. How is the structure factor related to positions of the atoms?
a.
where N = number of atoms, j = jth atom, f_{j} is the diffraction potential of the jth atom
(x_{j},y_{j},z_{j}) are the coordinants of the jth atomb. Each F_{hkl} is determined by all atoms in a unit cell
c. has the form of a Fourier Transform but has no integral ==> Fourier Series because N is finite.
6. How is Fhkl related to the structure in terms of an electron density distribution, r(x,y,z)?
a.
b. this is an inverse Fourier Transform ==> Fourier Synthesis; simplifies to:
E. Phase Problem  When Fhkl is recorded on film or area detector all phase information is lost; density of diffraction spot is the Intensity, I_{hkl}, of F_{hkl}: I_{hkl} = F_{hkl}^{2}
1. Patterson Function  Eqn. in 6.a. using
I_{hkl}'s and no phases ==> autocorrelation
function which is a map of vectors between atoms. Can be used to
solve simple structures, but not proteins
==> phases contain information on where things are.
2. Direct Methodsnormally used for molecules smaller than proteins
3. Multiple Isomorphous Replacement  collect data from "native" crystal and crystals which have had 1 or more heavy atoms (e.g. Hg, Pt, Au) added to each unit cell
a. added heavy atom should not change structure of protein part of the crystal ==> Isomorphous
b. heavy atom should be in same position in each unit cell; position of heavy atom calculated from Patterson function.
c. generally need at least 2 heavy atom derivatives ==> multiple isomorphous replacement.
d. since each F_{hkl }is the sum of f_{hkl} for each atom in the crystal, F_{H} = F_{native} + f_{H} ==> f_{H} can be calculated from the position of the heavy atom in the unit cell.
e. this allows calculation of 2 possible phases for F_{hkl}; a second heavy atom derivative is needed to choose which one is correct.
f. Once all phases are determined, r(x,y,z) can be calculated from equation 6.a.
4. Molecular Replacement  use known structure of a very similar molecule to calculate the approximate phases of a new molecule; determine structure of new molecule from calculated phases and measured F_{hkl}'s, then...
F. Refinement: various methods to make atomic model most consistent with the experimental data, e.g. F_{hkl}'s
V. Experimental Approaches
A. Generation of Xrays
1. Electron bombardment of a metal target (Anode) in a vacuum
a. sealed tubes  choice of target determines the wavelength of peak radiation on a background of low intensity continuous white radiation. Select one wavelength with metal foil filters, e.g. copper Ka ==> l = 1.54 Å. Relatively low intensity source.
b. rotating anode  anode (target) is a rotating wheel so area of electron bombardment is constantly changing; allows cooling of parts not being irradiated ==> much higher intensity possible.
2. Synchrotron sources  very expensive electron storage rings originally used primarily for high energy physics research; a number have now been constructed for radiation sources.
a. electrons accelerated around a large diameter circle; their path is bent by large electromagnets.
b. when electrons change their path in going around the circle, they experience a large change in angular momentum and the energy associated with this change is given off as very intense electromagnetic radiation over a wide range of wavelengths
c. select a single wavelength with a crystal Xray monochromator (Xrays diffract off some crystals at angles determined by wavelength). One can also use several different wavelengths to help calculate phases of diffracted beams; i.e. to locate particular types of atoms which will diffract anomalously when irradiated with Xrays near their absorption edges.
d. extremely high Xray photon flux ==> very short exposures.
B. Collimation  how to make a small diameter beam (0.10.5 mm) without lenses
1. Pairs of slits
2. Mirrors  Xrays will reflect off goldcoated mirrors at very low angles; this allow focusing of Xrays by appropriately bent mirrors
3. Monochromators  Xrays diffract off of crystal surfaces  can be designed to focus Xrays
C. Detection  how diffraction patterns can be recorded
1. Film  oldest method. relatively inefficient (unlike electron microscopy) in photon detection but can collect large amounts of data (large numbers of reflections, Fhkl's simultaneously. Difficult to quantitate; must digitize films. One must also be careful to develope all films in exactly the same way.
2. Xray Diffractometer  position crystal and Xray detector precisely
a. measure 1 diffracted ray at a time; but with improved sensitivity WRT film
b. can be automated; computer collects data for a reflection and then calculates movements of crystal and detector for next reflection and so on. Once started, all data can be collected without operator intervention (Indeed, for small molecules, whole structure can be determined with relatively little input from the experimentor)
3. Area Detector  combines the best features of film and diffractometers
a. detector is sensitive to photons and their positions over a large area
b. data is quantitated (digitized) as it is collected
c. very sensitive
d. collects many (hundreds) of reflections simultaneously
D. Geometry  from discussion of Sphere of Reflection we saw that the crystal must be tilted at particular angles in order for diffraction for each Fhkl to occur. Cameras for recording Xray diffraction patterns manage precise orientation of crystals and film in a variety of geometries; 2 are popularly used for protein crystallography.
1. Precession Camera  both crystal and film are caused to precess about different axes.
a. circular screen lets through only on layer of the reciprocal lattice ==> a 2dimensional slice of the reciprocal lattice; the orientation of this slice is indicated by which indices are nonzero for the particular slice ==> h,k,0; h,0,l; 0,k,l etc. zones
b. F_{hkl}'s are recorded on film in undistorted representation of the reciprocal lattice ==> easy to analyze to determine (i) unit cell axes from spaciangs between Fhkl's, (ii) space group from symmetry of intensities of F_{hkl}'s.
c. relatively inefficient for data collection as the circular screen blocks many of the diffracted waves (F_{hkl}'s) for each orientation of crystal and film. Precession cameras are used to characterize newly grown crystals.
2. Oscillation/Rotation Camera  crystal is gently rocked (oscillated) over very small angle (1  3 degrees) while the film is held stationary perpendicular to the incident beam
a. very efficient in data collection:
 no F_{hkl}'s are blocked
 can automate  complete data set requires successive oscillations around increasing angles between 2 unit cell axes.
b. gives a distorted (and somewhat confusing) record of reciprocal lattice; this is not a severe problem as films are scanned by computercontrolled densitometers which calculate (h,k,l) for each spot on the film.
3. If an area detector is used, it replaces the film in camera geometry.
VI. Other Types of Diffraction Experiments
A. Helical Diffraction  from fibers of aligned helical molecules
1. Helices found in biology are discontinuous  described by:
a. Helical Lattice  diameter of helical path, helix pitch (vertical rise per turn), vertical rise per repeating unit (motif), angular twist around helix axis between adjacent motifs
b. Helically repeating unit or motif (e.g. 1 base pair in doublestranded DNA)
2. Helical Diffraction Pattern (Fourier Transform) is Sampled on:
a. layer planes in 3dimensions
b. layerlines in 2dimensions
c. has characteristic Xshape from angle of incline of helix perpendicular to helix axis
3. Measaurements of diffraction patterns give:
a. helical pitch  vertical rise per turn of the helix from distance between layerlines (reciprocal relationship)
b. arrangement of helically repeating units from variation in intensity along layerlines.
c. can compare diffraction pattern with calculated diffraction patterns from model structures to see which one best fits the data  This is how the structure of double helical DNA was determined by Watson and Crick..
4. 3Dimensional structure
a. possible if molecules are exceptionally wellaligned in fiber
b. some helical viruses (e.g. Tobacco Mosaic Virus; fd bacteriophage) have had structure determined by helical fiber diffraction.
B. Solution XRay Scattering
1. Fourier Transform is not sampled (i.e. has finite value everywhere rather than just at points of reciprocal lattice or on layerlines)
a. diffraction is from single molecules in random arrangements
b. diffraction pattern is very weak compared to diffraction patterns from crystal or helices
2. Diffraction pattern is of single molecule averaged over all possible orientations > the best one can do is determine the rotationally averaged sructure.
3. Can determine the radius of gyration, R_{G}, from Guinier Plot
a. I(q) = intensity of the diffraction patt. at angle q
b. R_{G} interpreted to give molecular size and shape
4. Diffraction at higher angle can be used to confirm or disprove model structures
C. Neutron Diffraction  beam of neutrons has wave properties with wavelength determined by their velocity
1. Scattering Length  a measure of the scattering intensity from different types of atoms (we'll use scattering and diffraction interchangeably)
a. for Xrays, the scattering length is proportional to the number of electrons
b. for neutrons, the scattering length is fairly constant for all atoms
 D (Deuterium) has scattering length similar to C,P,O
 H has a negative scattering length > scatters neutrons 180° out of phase with other atoms > negative density in reconstruction
2. Structures derived from neutron diffraction show positions of H's (or D's) very clearly. This is very difficult to determined from Xray diffraction because H's have only 1 electron and diffract Xrays very weakly.
3. Can determine which H's exchange readily with D_{2}O  these parts of the molecule are accessible to solvent ==> molecular dynamics
4. Solution Scattering
a. adjust H_{2}O/D_{2}O to match scattering length of one component (protein or nucleic acid in ribosome or nucleosome) > neutron diffraction shows only effects from other component  this is called contrast matching.and has been used to determine the relative positions of protein and nucleic acid in ribosomes and other protein/nucleic acid complexes
b. label 2 subunits with D ==> analysis of neutron diffraction gives distance between the 2 subunits; has been used to construct model of distances between ribosome subunits.
5. Neutron diffraction difficult to do  only a few nuclear reactors produce beams of neutrons with enough flux to do neutron diffraction. Beam is large and and requires large samples such as protein crystals approximately 1 cm^{3}.