Transcript Document

X-ray Diffraction
Stephen J Everse
Fall 2004
Obtaining images of
macromolecules
In order for an object to be visible under magnification, the
wavelength (l) of the light must be, roughly speaking, no
larger than the object.
Visible light (400-700 nm) cannot produce an image of protein molecules,
in which bonded atoms are about 1.5Å apart (0.15 nm). Electromagnetic
radiation of this wavelength falls into the X-ray range.
S. Doublié ‘00
X-rays
X-rays are just another form of electromagnetic radiation:
Electromagnetic Spectrum:
X-rays
Energy:
Ultraviolet
Visible
Light
Infrared
Microwave
Radio
Low
High
Frequency:
High
Low
Wavelength:
Short
~1Å(=0.1nm)
~400nm
Long
Resolving Power: (Ability to see detail)
High
Atomic
Resolution
Low
{Electrons (~500keV, as in electron microscope) are not a form of electromagnetic radiation,
but they still have wave-like character (deBroglie wavelength ~0.01Å).
Unlike photons (EM rad.), electrons are charged --> fry the specimen faster }
M. Rould ‘02
X-ray Crystallography
•
a method for studying the three-dimensional,
atomic structure of molecules. In this course we
will concentrate on applications for biological
macromolecules.
A protein crystal
is placed in the x-ray beam
the x-rays are
diffracted by the
electron clouds
around atoms
the atomic structure
can be deduced from
the data
Why can’t we visualize molecules directly?
A single molecule is a very weak scatterer of X-rays. Most of the
X-rays will pass through the molecule without being diffracted. The
diffracted rays are too weak to be detected.
Solution: Analyzing diffraction from crystals instead of single
molecules. A crystal is made of a three-dimensional repeat of ordered
molecules (1014) whose signals reinforce each other. The resulting
diffracted rays are strong enough to be detected.
Unlike visible light, X-rays cannot be focused by lenses. The refractive
index of X-rays in all materials is very close to 1.0.
Solution: Use a computer to simulate an image-reconstructing lens.
In short, the computer plays the part of the objective lens, computing
the image of the object, then displaying it on a screen.
Sylvie Doublié © 2000
The nature of crystals
Under certain circumstances, macromolecules (protein, DNA, RNA)
can form crystals. The resulting crystal is a three-dimensional array of
ordered molecules held together by noncovalent interactions.
Evidence that solution and crystal structure are similar:
1- NMR and X-ray crystallography have been used to determine the
structure of the same molecule. The two methods produce similar
models.
2-Many macromolecules are still functional
in the crystalline state.
Most protein crystals contain 50-70% solvent.
S. Doublie ‘02
What is a Crystal ?
object formed by stacking a basic unit in all 3-dimensions
Unit Cell
M. Rould ‘02
The Ideal Crystal
the ordered disposition of molecules such that
there exists a regular repetition of a pattern in 3-D space,
where this repetition extends over a distance equal to or
greater than thousands of molecular dimensions.
The Real Crystal
a crystal with less than perfect periodicity,
imperfections are often caused by impurities and the effects
of non-zero temperatures.
The Protein Crystal
the crystal contains a high degree of solvent,
meaning that some molecules present are not in the
crystalline state, but in the liquid state, creating disorder.
Supersaturation
to add more of a substance ( to a solution) than can normally be
dissolved. This is a thermodynamically unstable state, achieved
most often in protein crystallography by vapor diffusion or
slow evaporation techniques.
Zone 1 - Metastable zone.
The solution may not nucleate for a long time
but this zone will sustain growth.
It is frequently necessary to add a seed crystal.
Zone 2 - Nucleation zone.
Protein crystals nucleate and grow.
Zone 3 - Precipitation zone.
Proteins do not nucleate but precipitate out
of solution.
Diagram from the website for The University of Reading, Course FS460
Investigating Protein Structure and Function
Nucleation
phenomenon whereby a “nucleus”, such as a dust particle, a
tiny seed crystal, or commonly in protein crystallography,
a small protein aggregrate, starts a crystallization process.
Common difficulties:
1.
If supersaturation is too high, too many nuclei form,
hence an overabundance of tiny crystals.
2. In supersaturated solutions that don’t experience
spontaneous nucleation, crystal growth often only occurs
in the presence of added nuclei or “seeds”.
Cessation of growth
Caused by the development of growth defects
or the approach of the solution to equilibrium.
Mother liquor
The solution in which the crystal exists - this
is often not the same as the original
crystallization screening solution, but is
instead the solution that exists after some
degree of vapor diffusion, equilibration
through dialysis, or evaporation.
Factors that affect crystallization
1) Purity of proteins
2) Protein concentration
3) Starting conditions (make-up of the protein solution)
4) Precipitating agent (precipitant)
5) Temperature
6) pH
7) Additives: Detergents, reducing agents, substrates, cofactors, etc.
Hanging/Sitting Drop Vapor Diffusion
Most popular method among protein
crystallographers.
1. Crystal screen buffer is the well solution
(0.5 - 1 mL)
2. Drop (on siliconized glass cover slip) is 1/2
protein solution, 1/2 crystal screen buffer (0.54 L). So, the concentration of precipitant in
the drop is 1/2 the concentration in the well.
3. Cover slip is inverted over the top of the
well and sealed
with vacuum grease (airtight).
4. The precipitant concentration in the drop
will equilibrate with the precipitant
concentration in the well via vapor diffusion.
Interpreting the Results of the
Crystallization Experiment
The Hampton Crystal Gallery
http://www.hamptonresearch.com/stuff/gallery.html
Experimental Set- Up
Cryostream
Rigaku rotating copper anode
(in-house source)
Beam Stop
Detector
Crystal
Monochromator
Or Mirrors
X-ray source
X-ray beam
S. Cates ‘02
Goniometer
European Synchrotron
Radiation Facility
Grenoble, France
How are X-rays produced?
X-rays in the useful range for crystallography (around 1 Å) can be
produced by bombarding a metal target (most commonly copper or
molybdenum) with electrons produced by a heated filament and
accelerated by an electric field. A high energy electron collides
with and displaces an electron from a low lying orbital in a
target metal atom. Then an electron from a higher orbital
drops into the resulting vacancy, emitting its excess energy as
an X-ray photon.
S. Doublié © 2000
X-ray Generators - The Rotating Anode
Rigaku rotating copper anode
(in-house source)
X-rays are generated by bombarding a rotating
copper anode with electrons. This creates X-ray
radiation consisting of two wavelenghts characteristic
of copper sources, 1.54 Å (K radiation) and 1.39 Å
(K radiation). Crystallographers usually use
K radiation (the intensity is greater).
X-ray Generators - The Synchrotron
European Synchrotron
Radiation Facility
Grenoble, France
Electrons (or positrons) are released from a particle accelerator into a storage ring.
The trajectory of the particles is determined by their energy and the local magnetic
field. Magnets of various types are used to manipulate the particle trajectory.
When the particle beam is “bent” by the magnets, the electrons (or positrons) are
accelerated toward the center of the ring. Charged particles moving under the
influence of an accelerating field emit electromagnetic radiation, and when they are
moving at close to relativistic speeds, the radiation emitted includes high energy xray radiation.
The oscillation equipment
Rotates the crystal about an axis () perpendicular to the
x-ray beam (and normal to the goniometer). The diffraction
pattern from a crystal is a 3-D pattern, and the crystal must
be rotated in order to observe all the diffraction spots.
Check out Bernhard Rupp’s Crystallography 101 website:
http://www-structure.llnl.gov/Xray/101index.html
Detectors
1- Photographic film
Not much used anymore because of the availability of far
more sensitive detectors. Superior resolution due to its fine
grain, but limited dynamic range.
2- Image plates
Image plates are coated with a layer of inorganic storage
phosphor. X-ray photons excite electrons in the material to
higher energy levels. Part of the energy is emitted as
fluorescence, but an appreciable amount of energy is retained
in the material. The stored energy is released upon
illumination with a red laser. Blue light is emitted and
measured with a photomultiplier. The light emitted is
proportional to the number of photons. Ten times more
sensitive than film, dynamic range (1:104-105)
S. Doublié © 2000
Diffraction
A characteristic of wave phenomena, where whenever
a wavefront encounters an obstruction that alters
the amplitude or phase of a part of the wavefront,
diffraction will occur.
The components of the wavefront, both the
unaffected and the altered, will interfere with one
another, causing an observable energy-density
distribution referred to as the diffraction pattern.
Interactions between X-rays and atoms
X-rays are scattered almost exclusively by the electrons
in the atoms, not by the nuclei.
The incident electromagnetic wave exerts a force on the
electrons. This causes the electrons to oscillate with the
same frequency as the incident radiation. The oscillating
electrons act as radiation scatterers and emit radiation
at the same frequency as the incident radiation.
S. Doublié © 2000
When an incident x-ray beam hits a
scatterer, scattered x-rays are
emitted in all directions. Most of
the scattering wavefronts are out
of phase interfere destructively.
Some sets of wavefronts are in
phase and interfere constructively.
A crystal is composed of many
repeating unit cells in 3-dimensions,
and therefore, acts like a 3dimensional diffraction grating. The
constructive interference from a
diffracting crystal is observed as a
pattern of points on the detector.
The relative positions of these
points are related mathematically to
the crystal’s unit cell dimensions.
Destructive Interference
Constructive Interference
Diffraction gratings
Diffraction patterns
Notice - when the diffraction grating gets smaller, the
pattern spacing gets larger (inverse relationship)
Bragg’s Law
2d sin = nl
where
l = wavelength of incident x-rays
 = angle of incidence
d = lattice spacing
n = integer
Spots are observed when the following conditions are met:
1. The angle of incidence = angle of scattering.
2. The spacing between lattice planes is equal to
an integer number of wavelengths.
The Ewald
Sphere
A tool to visualize the
conditions under which Bragg’s
law is satisfied and
therefore a reflection
(diffraction spot) will be
observable.
This occurs when the surface of
a sphere centered about the
crystal with radius = 1/l
intersects with a point on the
reciprocal lattice.
QuickTime™ and a Vidéo decompressor are needed to see this picture.
Movie downloaded from
An Interactive Course on Symmetry and
Analysis of Crystal Structure by Diffraction
By: Gervais Chapuis and Wes Hardaker
http://perch.cimr.cam.ac.uk/Course/Adv_diff2/Diffraction2.html#Ewald
Unit Cell
A crystal’s unit cell dimensions are defined by six
numbers, the lengths of the 3 axes, a, b, and c, and
the three interaxial angles, ,  and .
The convention for designating the reciprocal lattice
defines its axes as a*, b*, and c*, and its interaxial
angles as *, * and *.
Asymmetric unit
Recall that the unit cell of a crystal is the smallest 3-D
geometric figure that can be stacked without rotation to form
the lattice. The asymmetric unit is the smallest part of a
crystal structure from which the complete structure can be
built using space group symmetry. The asymmetric unit may
consist of only a part of a molecule, or it can contain more than
one molecule, if the molecules not related by symmetry.
Symmetry
"An object has a particular symmetry if the object looks exactly the same
after applying the corresponding symmetry operation."
Types of Symmetry Operations:
• Translational
4-fold rotation
•n-fold Rotation
• Combination symmetries:
• Screw axis (translation + rotation)
• Glide plane (translation + mirror)
• Roto-inversion axis
• Mirror
operation
Note that mirror and
inversion operations
change the hand. I.e., if
an object possesses this
symmetry, either both
enantiomers must be
present, or the object
must be achiral.
Mirror Plane
• Inversion
operation
M. Rould ‘02
Inversion center
Symmetry
Can natural proteins have mirror or inversion symmetry?
x
L-Alanine
No - proteins are chiral -only L-amino acids are present.
D-Alanine
How about nucleic acids (DNA, RNA)?
No, (deoxy)ribose is chiral -only the D- stereoisomers are present.
Of the 232 Crystallographic Space Groups, only 65 are possible for
crystals containing enantiomorphic specimens such as most biological
macromolecules.
M. Rould ‘02
X-Ray Scattering from a Crystal
A typical image of x-rays
scattered by a crystal:
(Dark spots are the
scattered x-rays)
When x-rays scatter
from a crystal we see
discrete spots:
Reflections
Why?
X-Ray Diffraction Pattern
M. Rould ‘02
X-Ray Diffraction from a Crystal
• Electromagnetic radiation is wave-like:
Electric
field
+
+
-
+
-
+
-
+
-
+
-
•Waves can add constructively or destructively:
Electric
field
+
+
Sum
=
M. Rould ‘02
+
-
+
-
+
-
+
-
Direction
of motion
of x-ray
photon
Structure Factor - F(hkl)
Each reflection in the diffraction pattern is the result of
diffractive contributions from all the atoms in the unit cell.
F(hkl) = f1 ei + f2 ei + f3 ei + …
+ fN ei + f1' ei + f2' ei + f3' ei + …
or, F(hkl) = ∑fj ei
The term fj describing the diffractive contributions of each
atom is called the atomic scattering factor of atom j.
The scattering factor essentially describes the amplitude for
the scattering contributed by a particular species of atom.
Structure Factor - F(hkl) cont’d
F(hkl), as a complex number, can be expressed in terms of its real
and imaginary components:
F(hkl) = A(hkl) + i B(hkl),
where A = ∑fj cos j = fresultant cos resultant
and B = ∑fj sin j = fresultant sin resultant,
fj are the atomic scattering factors
and j are the phase angles of the waves
scattered from individual atoms.
This is just an alternate, mathematically equivalent representation
for the structure factor that sometimes proves useful.
Fourier Methods in Diffraction Theory
For each point in a diffraction, there is a corresponding spatial frequency.
Therefore, the distribution of a far-field diffraction pattern is the Fourier
transform of the aperture function. (aperture - an opening, often adjustable,
that controls the amount of light reaching the lens on a camera or other
optical instrument.)
In our case, the aperture function is the regularly periodic (due to the
repetition of the unit cell in the lattice) electron density distribution within
our crystals. The electron density is the inverse Fourier transform of the
diffraction pattern expressed as follows:
(x, y, z) = 1/Vunit cell ∑∑∑F(hkl)
e -2πi(hx+ky+lz),
h k l
where Vunit cell = volume of one unit cell and F(hkl) is called the structure
factor for a particular set of Miller indices h, k and l. We can do a
summation here, instead of integrating, because we know we will only
have reflections at integer values for h, k and l.
Electron Density
Electron density distribution:
(x, y, z) = 1/Vunit cell ∑∑∑F(hkl) e -2πi(hx+ky+lz)
h k l
for convenience, let us substitute  = 2π(hx+ky+lz) in the future
The amplitude of the structure factor F(hkl) for any given
reflection is proportional to the square root of the intensity of
the diffracted beam, or:
|F(hkl)|2  I
Therefore, we can deduce |F(hkl)|, the magnitude of F, directly
from our data, but not its phase.
We have all the
information we need,
except the phase. Why
worry about the phase?
On the top are photographs of Jerome Karle
(left) and Herb Hauptman (right), who won the
Nobel Prize for their work on solving the phase
problem for small molecule crystals. We can
treat the photographs as density maps and
calculate their Fourier transforms, to get
amplitudes and phases. If we combine the
phases from the picture of Hauptman with the
amplitudes from the picture of Karle, we get
the picture on the bottom left. The bottom
right picture combines the phases of Karle with
the amplitudes of Hauptman.
The pursuit of phases
Although |Fhkl| can be derived from the recorded intensities Ihkl, the
phase angle ahkl cannot be derived straightforwardly from the
diffraction pattern. Several methods have been developed to solve
this problem.
Multiple isomorphous replacement (MIR)
Free of model bias, but noisy due to lack of isomorphism
Multiwavelength anomalous diffraction (MAD)
Most reliable source of phases; isomorphism is nearly perfect
Molecular replacement (MR)
Widely used; errors due to model bias are variable and difficult
to detect and correct
MIR
Basic principle:
• Add heavy atom compound (Hg, Pt, Au, etc.) to the crystal.
• Collect diffraction data from this derivatized crystal.
• Hopefully the heavy metal will bind to just a few sites
• It is relatively easy to determine the positions of these few
really big atoms.
-> knowing the positions of the heavy atoms, we can calculate
their effect on the intensity and phase of each reflection.
Caveat:
MIR only works if the heavy atom doe not change the conformation
of the protein or the crystal lattice in any way. The only differences
allowed are the presence of the heavy atom in the crystal and the
resulting change in intensity and phase of the scattered X-rays.
S. Doublié ‘02
Protein phase angles
A- Single isomorphous replacement
Imaginary axis
FPH= FP + FH
H
FP, protein structure factor
FH, heavy atom structure factor
FPH, structure factor for derivatized crystal
FP
O
Real axis
-FH
FPH
S. Doublié ‘02
G
Harker construction for phase
determination by the method of single
isomorphous replacement: the vectors
OH and OG represent two
possibilities for FP.
Harker construction for MIR
Imaginary axis
H
FP
-FH
-FH2
FPH
O
Real axis
G
FPH2
The addition of another derivative breaks the phase ambiguity: FP is given
unequivocally by the vector OH.
S. Doublié ‘02
MAD
MAD depends on the presence of sufficiently strong anomalously
scattering atoms in the protein. Anomalous scattering occurs if the
electrons in an atom cannot be regarded as free electrons.
An anomalous scatterer absorbs X-rays of specified wavelength. As
a result of this absorption, Friedel’s law does not hold, i.e, the
reflections hkl and -h-k-l are not equal in intensity. This inequality
of symmetry related reflections is called anomalous scattering.
Metalloproteins (Fe) structures have been solved with MAD.
Proteins that do not naturally contain anomalous scatterers can be
expressed in E. coli in a defined medium with selenomethionine.
The selenium atoms serve as anomalous scatteringheavy atoms.
Caveat:
MAD requires a tunable wavelength: data collection can only be done
at synchrotron radiation facilities (Brookhaven, Stanford, APS etc.).
S. Doublié ‘02
Molecular replacement
Prerequisite: The protein of interest should have a structural homologue
in the PDB in order to use the related protein as phasing model.
Molecular replacement entails calculating initial phases by placing
the model of a known protein in the unit cell of the unknown protein.
Caveat: Errors due to model bias are variable and difficult to detect
and correct
S. Doublié ‘02
R factor
Measure of the crystallographic residual, indicates
the correctness of a model:
R = ∑ | (|Fobs|-|Fcalc|) |
∑ (|Fobs|
Variations that can prove confusing to the novice:
Rmerge
measurement of the quality of a merged data set
Rsym
measurement of the variation between symmetry-related
reflections
Rfree
R factor for a test set of unique reflections that have
been omitted from the refinement process (unbiased)
Rfree
R factor for a test set of unique
reflections that have been omitted from
the refinement process (unbiased)
R = ∑ | (|Fobs|-|Fcalc|) |
hkl  T
∑ |F obs|
hkl  T
where hkl  T designates all reflections belonging to a
test set T of randomly selected, unique reflections. The
size of the test set is commonly 10% of the data set.
Rmerge
measurement of the quality of
a merged data set
N
R = ∑ ∑ | (|Fhkl|-|Fhkl(j)|) |
hkl j=1
∑ N x (|Fhkl|
hkl
where | Fhkl| is the final value of the structure factor
amplitude for that reflection,
N = total no. of data sets (or images) merged.
Rsym
measurement of the variation between
symmetry-related reflections
R = ∑ ∑ | (|F(i)hkl|-|Fhkl|) |
hkl i
∑ ∑ |F (i) hkl|
hkl i
for i observations of each symmetry-related reflection,
where |Fhkl| is the average value for the structure factor
amplitude of the i observations of a given reflection.
Refinement Target
Refinement searches for a global minimum for a target
energy function similar to the one illustrated below:
E
where
total
= wxray Exray + Econformation + Enonbonded
wxray = weight for the xray energy term
Exray = xray energy term
Econformation = conformational energy terms
(bonds, angles)
and Enonbonded = nonbond energy terms
(van der Waals, electrostatic)
Rigid Body
Refinement
Reduces the conformational
RIGID
freedom within the model to
BODY 1
improve the ratio of observables
to parameters in the early stages
of refinement. The entire model
can be treated as a rigid body, or
it can be regarded as linked, rigid groups.
RIGID
BODY 2
For each group of atoms specified by the user as a rigid body,
the 3 rotational and 3 translational degrees of freedom are
minimized.
Positional Refinement
The atomic position parameters x, y and z are refined for
each atom.
•
•
Difficulties in protein crystallography:
large number of parameters to fit
macromolecular crystals diffract weakly, producing a
poor parameters to observations ratio.
The geometrical constraints introduced by the
conformational energy terms greatly reduces the number
of parameters to be refined. Least-squares optimization
or conjugate gradient minimization techniques are
commonly used for finding the best fit of the model to the
data.
B-factor (temperature factor) refinement
B-factors are indicators of atomic mobility. High values
correspond to low electron density, indicating a dynamic or
disordered region, or a possible error in position.
The B-factor is an exponential expression applied to the
scattering factor that relates to the thermal motion of the
scattering atom and the decrease in scattering intensity that
results from thermal motions.
fe
-B[(sin 2 )/l2 ]
The x-ray energy term is modified in the target energy
function is revised where Fcalc is replaced by Fcalc e -s2 B/4
Occupancy Refinement
The occupancy factor is used to describe disorder
in the model. An atom with a partial occupancy
factor can be thought of as an atom that does not
occupy that position 100% of the time (i.e., ions,
water, cofactors). Some refinement programs do
not require that the occupancy factor be ≤ 1, so it
is up to the crystallographer to remember that 1
is the upper limit on the occupance factor for a
given atom in a given position.
Simulated Annealing
Simulated annealing - MD-refinement technique that involves the
control of the temperature, mathematically related to the kinetic
energy (KE) of the MD simulation by:
Tcurrent = 2 KE/3nkb, for n = degrees of freedom, kb =
Boltzmann constant
Gradient descent minimization and least-squares optimization
methods are prone to get “stuck” in regions of local minima when
applied to the vast problem of solving the structure of a biological
macromolecule. In these cases, it is often necessary to overcome
an energy barrier between the local minimum and the global
minimum. Therefore, to reach the global minimum, an algorithm
must be applied that can go energetically “uphill”.
Model Building
Starting model:
Molecular replacement model
initial model is the search model that has been
positioned in the unit cell by the rotation and translation
function.
MAD/isomorphous model
electron density is calculated using the heavy atom
phases, then the model has to be built into the electron
density.
Maps
Electron density distribution:
(x, y, z) = 1/Vunit cell ∑∑∑F(hkl) e-ø
h k l
The first map is an approximation to the true electron density
derived from the observed structure factor amplitudes (Fobs)
and the estimated phases from the model (MR, MAD, or MIR
phases).
(Remember our illustrations that the correctness of the model
image depends more on having the correct phase information
than on having the correct amplitudes.)
Maps cont’d
Both tryptophans are from the same 1.7 Å crystal structure, but
the map in Figure 1 is the first map calculated using the initial
MR phases and the map in Figure 2 is the final map calculated
using the refined phases.
1
2
Resolution limits
6.0 - 4.5Å
3.0Å
2.5Å
1.8Å
1.2Å
Placement of secondary structures
Chain tracing
Side chain orientation
Alternate side chain orientations
Hydrogen atoms
Map types
2 FO - FC Maps
FO = observed structure factors
FC = calculated structure factor
Subtracting Fc from 2 Fo exaggerates the areas where Fo differs
from Fc. In the case where Fo is greater than Fc, the net
structure factor amplitude is intensified and in the case where Fo
is less than Fc, the net structure factor amplitude is decreased.
FO - FC Maps
(Difference Maps)
Produces “positive” or
“negative” peaks in areas
where Fo differs from
Fc. This map is usually
contoured at a high level
- 3 or 4  - so all the
crystallographer views
are the large difference
peaks (not likely to be
just noise).
Atomic Model Deposition - The Protein Data Bank
You’ve solved your 1.2 Å crystal structure with an R-factor of
15.4% and an R-free of 16.2%. It’s time to share your hardwon scientific knowledge with the rest of the world. When
you publish your paper, most journals will request that you
provide your PDB accession number, indicating you have
deposited your coordinates for the betterment of mankind.
So, you type the following URL into your browser:
http://www.rcsb.org/pdb/
and wind up here:
Welcome to the PDB, the single worldwide repository for the
processing and distribution of 3-D biological macromolecular
structure data.
PDB Validation Suite
PROCHECK, NUCHECK, SFCHECK
PROCHECK
Assesses the geometry of the residues in a given protein structure,
as compared with stereochemical parameters derived from wellrefined, high-resolution structures.
Unusual regions highlighted by PROCHECK are not necessarily errors,
but may be unusual features for which there is a reasonable
explanation (eg distortions due to ligand-binding in the protein's
active site). Nevertheless, they are regions that should be checked
carefully.
The only input required for PROCHECK is the PDB file holding the
coordinates of the structure of interest.
Practical Considerations - generalizations (that means,
of course, that there are always exceptions)
Resolution:
R-factor:
Good:
For sidechain conformations:
≤ 2Å
Good upper limit for ~ 2Å data:
R-free:
20 - 23 %
within 10% of R
(closer for hi res)
< 3Å