X-ray CrystallographyPiotr Sliz

BCMP 201

sliz@crystal.harvard.edu

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 Schoolsliz@crystal.harvard.edu

Dr. Meg Bentley (Vice-Chair)

iDB Educational InitiativeHarvard Medical Schoolmeg_bentley@hms.harvard.edu

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 6thmeg_bentley@hms.harvard.edu

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