x-ray Crystallography
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Transcript of x-ray Crystallography
X-ray CrystallographyPiotr Sliz
BCMP 201
Computer demos:• Symmetry• Maps
Handouts:• Wave Functions• Symmetry
Reading
• Crystallography Made Crystal Clear, Gale Rhodes
Software
• HKL2000 - data processing
• molrep/BnP - phasing
• CNS - refinement
• COOT - model building
Ar
Gl
structure
OO O
AcHN
OH
Opeptapeptide
OOHO
AcHN
OHO
OOAcHN
OH
Opeptapeptide
O PO
O
O
PO
O OLipid
OHOHO
AcHN
OHO
OOAcHN
OH
Opeptapeptide
O PO
O
O
PO
O OLipid
TG
Membrane
Growing peptidoglycan chainDonor
Lipid IIAcceptor
Model 1
reducing end
elongated chain
Gal
non-reducing end
drug design
mechanism
biology
forcefieldsand folding
http://cmcd.med.harvard.edu
Quo VadisStructural Biology?
5 - 6 MAY | 2008
SPEAKERS:
Axel BrungerNaomi ChayenJames ChouFrank DelaglioBen EisenbraunPaul EmsleyJoachim FrankRachelle GaudetMark GersteinDavid GoharaNikolaus GrigorieffIan LevesqueStephen C HarrisonRalf Grosse-KunstleveMiron LivnyGaël McGillHarry PowellStefan RaunserJason SchnellDavid E ShawPiotr SlizWoody ShermanIan Stokes-Rees
NANOCOURSES:
(symposium registration is required)
• Advanced Crystallography• Structure-based Lead Discovery*• Roadmaps for Structure Determination (EM, NMR, XRAY)• Animating your data*• Mac OSX Development*• Intro to Python*• Bioinformatics*• Sys admin for structural biologists*
Graduate students from Harvard for credit,all others IDB certificate.Organized in Mac Classrooms
(75 workstations provided by Apple.)
Early Registration Deadline:
February 29th, 2008
Registration assistance:
www.sbgrid.org/quovadis
The symposium will focus on computational methods in biology. Data animation, molecular simulation, introduc-tory programming and methods in structure determination will be covered by a diverse group of lecturers. In addition we will review trends in structural biology and share perspectives on its future direction.
Organizers:
Dr. Piotr Sliz (Chair)
SBGrid, Center for Molecular & Cellular DynamicsHarvard Medical [email protected]
Dr. Meg Bentley (Vice-Chair)
iDB Educational InitiativeHarvard Medical [email protected]
Harvard Medical School BOSTON
*
Stanford
Imperial College
Harvard
NIH
SBGrid
Oxford
Wadsworth
Harvard
Yale
Washington U
Brandeis
SBGrid
Harvard
Berkley Lab
U Wisconsin
Digizyme
MRC
Harvard
Harvard
David E Shaw
SBGrid
Schrödinger
SBGrid
www.sbgrid.org/quovadis
May 5 and [email protected]
Phase IA1965 Phillips -determined the first 3-d structure of anenzyme; lysozyme. Second protein structure to be solved.See Nat.struc. Biol. (Nov. 1998). 5(11). pp 942-926. for amore detailed history.
Phase IB1971-74 Application of synchrotron radiation to aprotein crystallography: Rosenbaum, G. Homles, K.C. &Witz, J. Wychoff & Harrison Phillips, J.C. Wlodawer, A.Yevitz, M.M & Hodgson.
Phase IIA1988 Guss et al -first structure solved by the MADmethod/Introduction of second generation synchrotronsources.
Phase IIB1990 Teng describes a method of collecting data from asingle protein crystal mounted in loops by cryofreezing.[Tsu-Yi Teng (1990) j. Appl. Cryst. 23. 387-391.
Phase IIIA1999 – Rosenbaum et al. Third generation protein beamlines
Phase IIIB2003 –better understanding of radiation damage/RIP?.
Macromolecular X-ray crystallography(proteins, viruses, DNA, RNA)
1. Crystals
2. Structure Determination
a. From crystals to diffraction
b. From diffraction to electron density
c. From electron density to models
3. Structure quality and statistics
reciprocal lattice electron density mapcrystal model
Crystal lattice:
Unit Cell: the portion of a space lattice that is repeated in order to form the entire latticeAsymmetric Unit: the smallest structural unit which, when operated upon by the symmetry elements of the space group, yields the total crystal structure.
Rotational symmetry
n=2
n=3
n=4
n=6
Rotational symmetry of order n, also called n-fold rotational symmetry, or discrete rotational symmetry of nth order, with respect to a particular point (in 2D) or axis (in 3D) means that rotation by an angle of 360°/n (180°, 120°, 90°, 60°.) does not
change the object.
monoclinic
tetragonal cubic
trigonalcubic
hexagonal
32 622
422 4 432
triclinic orthorhombicCrystal Systems
2 2221
monoclinic
•A screw axis is a symmetry operation describing how a combination of rotation about an axis and a translation parallel to that axis leaves a crystal unchanged.
•If φ = 360°/n for some positive integer n, then screw axis symmetry implies translational symmetry with a translation vector which is n times that of the screw operation.
•The possibilities are 21, 31, 41, 42, 61, 62, and 63, and the enantiomorphous 32, 43, 64, and 65.
Screw Axis Symmetry and Point Groups
Tables:
P212121: Origin at midpoint off three non-intersecting pairs of parallel 21 axes.
P21: Origin on 21.
Tables:
Protein Crystallization
Full Factorial Incomplete Factorial
Random Sparse Matrix
All elements of the matrix of parameters are sampled.
Factor levels are chosen randomly and then balanced to achieve uniform sampling. All two-factor interactions are sampled as uniformly as possible.
Random sampling of all parameters, but it approximates incomplete factorial designs.
Intentional bias towards combinations of conditions that have worked previously.
Crystallization screens
Rational Screens for Protein Crystallization:
Microfluidics can provide a robust and systematic environment for crystallization screening.
Protein solubility profile
Microfluidics device.
Macromolecular X-ray crystallography(proteins, viruses, DNA, RNA)
1. Crystals
2. Structure Determination
a. From crystals to diffraction
b. From diffraction to electron density
c. From electron density to models
3. Structure quality and statistics
reciprocal lattice electron density mapcrystal model
Images of Microscopic Objects
d = λ/(2n sinθ)
minimumseparation wavelength
The minimum separation (d) that can be resolved by any kind of a microscope is given by the following formula:
Magnification:
Lens Optics (wikipedia):
Obtaining Images of Molecules
Wavelength must be corresponding to the object.
X-rays (~10-10 m or 1Å angstrom)
Suitable radiation:
Crystallographic Analogy of Lens Action:
Bragg’s Law (1/2):
Constructive interference: distance traveled by two waves differs by a multiple of a wavelength.
Lattice planes reflect X-rays.
x
y 010 110
100
210 310
110
Bragg’s Law (2/2):
A
B
C
Difference in path: 2xBC:
Reciprocal Space:
Whenever the crystal is rotated so that a reciprocal-lattice point comes in contact with the circle of radius 1/lambda, Bragg’s law is satisfied and a reflection occurs.
Direction of reflections and number of reflections depend only on unit-cell dimensions, and not contents of the unit cell.
Fig 4.21
The number of measurable reflections:
If the sphere of reflections has a radius of 1/λ, then any reciprocal-lattice point with a distance of 2/λ can be rotated into contact with the sphere of reflections.
Total number of measurable reflections: number of reciprocal unit cells within the limiting sphere.Number of required reflections is limited by diffraction resolution and symmetry.
Wave equation:
f(x) = Fcos2π(hx + α)
vertical height at any horizontal position x along
the wave
amplitude frequency
phase
any simple wave function can be described in three constants:
Simple wave functions: f(x) = Fcos2π(hx + α)
f(x) = 3cos2π(x)
f(x) = cos2π(x) f(x) = cos2π(5x)
f(x) = cos2π(x+1/4)
Complicated wave functions:
Complicated periodic functions can be described as the sum of simple sine and cosine functions.
Fourier terms
Fourierseries
Each Fourier term is a simple sin or cos function.
simple wave
complex number wave (a + i b)
Fourier Series:
unspecified number of waveforms
Fourier Series (exponential form):
equality from complexnumber theory: Fh - intensity
h - frequency
simple diffraction
waves
FourierSynthesis
ReciprocalSpace
RealSpace
Fourier series for each reflection is a sum of contributions from individual atoms in the unit cell.
Each reflection is a contribution from all atoms in the unit cell.
The structure factor is a wave created by the superposition of many individual waves (described as Fourier series).
Fhkl= fA+fB+...+fA’+FB’+..+ fF”
Structure-factor equation:
Data Collection:
1) INDEX (determine unit cell and space group)2) Integrate (measure intensity of reflections)3) Scale (combine partials and scale between frames)
scale, convert I to F
integrate:
structure factor file
Data collection strategy dependson crystal symmetry
typically 180 1° oscillation frames:
Intensities of systematic absences h k l Intensity Sigma I/Sigma
0 0 20 6.1 4.8 1.2 0 0 22 -3.5 7.4 -0.5 0 0 23 5.0 5.5 0.9 0 0 25 -3.8 8.6 -0.4 0 0 26 9.1 6.9 1.3
Data Collection Statistics:
overall/high res bin
Rules of three to determine resolution limit (calculated in high res bin):
1. Completeness > 70% 2. Rsymm < 30% 3. I/sigma > 3
orthorhombic (4 molecules / unit cell)
mosaicity (workable range: 0.1 -1 degree)
resolution model building power
lower than 4Å possible to fit exist structures (rigid body refinement)
3.3-4 Å limited remodeling of existing structures
2.5 - 3 Å de novo model building
higher than 2.5 Åatomic details visible (model waters, detailed hydrogen
bonding network, alternative conformations)
Summary of reflections intensities and R-factors by shells R linear = SUM ( ABS(I - <I>)) / SUM (I)
R square = SUM ( (I - <I>) ** 2) / SUM (I ** 2) Chi**2 = SUM ( (I - <I>) ** 2) / (Error ** 2 * N /
(N-1) ) ) In all sums single measurements are excluded
Shell Lower Upper Average Average Norm. Linear Square limit Angstrom I error stat. Chi**2 R-fac R-fac 30.00 4.31 1521.8 30.8 19.4 15.312 0.108 0.150 4.31 3.42 1753.0 36.3 23.8 16.596 0.118 0.160 3.42 2.99 866.4 20.6 15.6 13.148 0.130 0.168 2.99 2.71 506.4 14.3 11.8 10.336 0.142 0.186 2.71 2.52 363.8 12.1 10.5 8.596 0.150 0.195 2.52 2.37 298.4 11.1 9.9 7.195 0.156 0.202 2.37 2.25 228.2 10.1 9.3 5.693 0.160 0.198 2.25 2.15 188.1 9.6 9.0 5.098 0.171 0.213 2.15 2.07 150.3 9.1 8.7 3.907 0.175 0.207 2.07 2.00 122.2 8.8 8.5 3.186 0.184 0.214 All reflections 605.3 16.4 12.7 8.857 0.130 0.162
Shell Summary of observation redundancies: Lower Upper % of reflections with given No. of observations
limit limit 0 1 2 3 4 5-6 7-8 9-12 13-19 >19 total 30.00 4.31 7.5 4.2 10.3 17.8 19.3 23.9 11.3 5.7 0.0 0.0 92.5 4.31 3.42 2.1 3.8 9.9 23.0 22.5 20.3 13.5 4.8 0.0 0.0 97.9 3.42 2.99 0.8 2.8 6.6 20.6 25.6 27.8 12.7 3.1 0.0 0.0 99.2 2.99 2.71 0.6 1.8 5.3 21.8 25.4 29.6 13.0 2.5 0.0 0.0 99.4 2.71 2.52 0.2 1.4 5.0 21.1 25.5 32.2 12.2 2.4 0.0 0.0 99.8 2.52 2.37 0.0 0.8 4.5 21.4 26.7 32.3 12.3 2.0 0.0 0.0 100.0 2.37 2.25 0.0 0.6 4.3 20.0 27.8 33.7 11.9 1.7 0.0 0.0 100.0 2.25 2.15 0.0 0.3 3.9 21.0 27.9 34.7 10.7 1.5 0.0 0.0 100.0 2.15 2.07 0.0 0.4 3.0 20.9 27.2 36.3 10.3 1.9 0.0 0.0 100.0 2.07 2.00 0.0 0.3 3.4 20.0 28.1 36.9 9.1 2.1 0.0 0.0 100.0 All hkl 1.2 1.7 5.7 20.7 25.5 30.6 11.7 2.8 0.0 0.0 98.8
Macromolecular X-ray crystallography(proteins, viruses, DNA, RNA)
1. Crystals
2. Structure Determination
a. From crystals to diffraction
b. From diffraction to electron density
c. From electron density to models
3. Structure quality and statistics
reciprocal lattice electron density mapcrystal model
Fourier series for electron density is a sum of contributions from individual reflections.
simple diffraction
waves
FourierSynthesis
FourierAnalysis
Fourier Transform:
ReciprocalSpace
RealSpace
Phase Problem:
• amplitudes (can be measured, ~ sq rt of intensity)
• frequency (X-ray source)• phase ??
Fhkl
F
Real
Real
Solving Phase Problem: Molecular Replacement
1) Rotation Function 2) Translation Function
Combining model phases with experimental intensities will reveal details of missing elements.
Typically 30% identity and 1/3 of a structure required.
Homologous or incomplete model:
Self Rotation Function
MOLREP
Rad : 30.00 Resmax : 3.00RF(theta,phi,chi)_max : 0.1137E+05 rms : 771.3
Chi = 180.0
X
Y
RFmax = 0.1137E+05
Chi = 90.0
X
Y
RFmax = 1254.
Chi = 120.0
X
Y
RFmax = 1038.
Chi = 60.0
X
Y
RFmax = 1038.
Polar angles theta, phi, chi define the standard system orientation in the cell. Theta, phi - polar coordinates of Z standard axis. Chi - angle of rotation around theta-phi-axis (Z standard axis) which bring X axis to standard X axis.
Number of RF peaks : 10 theta phi chi alpha beta gamma Rf Rf/sigma
Sol_RF 1 146.34 -139.17 155.19 55.63 65.55 153.97 0.1453E+07 3.69Sol_RF 2 132.29 165.77 54.08 56.82 39.30 265.27 0.1409E+07 3.57Sol_RF 3 126.82 159.77 57.42 51.60 45.23 272.06 0.1395E+07 3.54Sol_RF 4 147.99 -141.48 160.47 49.99 62.99 152.96 0.1387E+07 3.52Sol_RF 5 154.58 -138.22 162.21 51.61 50.18 148.05 0.1285E+07 3.26Sol_RF 6 127.83 148.20 69.52 35.15 53.52 278.74 0.1265E+07 3.21Sol_RF 7 43.33 96.64 95.04 45.10 60.81 31.83 0.1252E+07 3.18Sol_RF 8 61.94 104.17 98.84 42.94 84.17 14.61 0.1213E+07 3.08Sol_RF 9 82.25 169.61 53.05 83.46 52.52 284.24 0.1213E+07 3.08Sol_RF 10 24.69 44.74 106.19 5.17 39.02 95.68 0.1185E+07 3.01
32
2
1
1Finding heavy atom sites using Patterson methods:
-Fh+Fph=Fp
110
310
Harker Construction:
Macromolecular X-ray crystallography(proteins, viruses, DNA, RNA)
1. Crystals
2. Structure Determination
a. From crystals to diffraction
b. From diffraction to electron density
c. From electron density to models
3. Structure quality and statistics
reciprocal lattice electron density mapcrystal model
rebuilding
refinement:
real space
reciprocalspace
annealing
rigid body
minimization
b-factor
Macromolecular X-ray crystallography(proteins, viruses, DNA, RNA)
1. Crystals
2. Structure Determination
a. From crystals to diffraction
b. From diffraction to electron density
c. From electron density to models
3. Structure quality and statistics
reciprocal lattice electron density mapcrystal model
Kleywegt et al. Homo crystallographicus--quo vadis?. Structure (2002) vol. 10 (4) pp. 465-72
Rfree
10 x resolution