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Appendix 1
A Compilation of Equations for Calculating Electron Density Maps
Straightforward Electron Density Map
p(x y z) = 1. 2:IF(h k 1)1 exp[ -21ti(hx + ky + Iz) + iu (h k I)] V hkl
= 1. 2:IF(h k 1)1 cos[21t(hx + ky + Iz) - u(h k I)] V hkl
The IF(h k I)ls are the structure factor amplitudes of the reflections (h k I).
Difference Electron Density Map
dp (x y z) = -VI 2:dIF(h k 1)liso exp[ - 21ti(hx + ky + lz) + iUp(h k l)] hkl
1 = - 2:dIF(h k l)liso cos[21t(hx + ky + lz) - up(h k I)]
V hkl
dlF(h k l)liso is the difference between the structure factor amplitudes for the protein and some isomorphous derivative of that protein. The phase angles up(h k I) are those of the native protein. The map shows the electron density, which is extra (or which is missing) in the derivative at half the actual height.
A 2Fobs - Fcalc Map
p(x y z) = -VI 2: (2lFobsl - iFealcl) exp[ -21ti(hx + ky + lz) + iUealc] hkl
311
312 Appendix 1
This map can be regarded as the sum of the electron density of a model and of a difference electron density map at double height. It shows, besides the electron density of the model, the difference between the actual structure and the model at normal height. The phase angles are those calculated for the model.
A Residual, or Double Difference, Electron Density Map
1 ~p(x Y z) = -VL{IFobsl - IFnative + Fattachedl}
hkl
X exp[ -21ti(hx + ky + lz) + iUcalcJ
IFobsl is the structure factor amplitude for the derivative. Fattached is the structure factor contribution by those attached atoms or groups of atoms for which the parameters are already known. The phase angles Ucalc are for the native protein and the attached heavy atoms. This is a useful Fourier summation for the detection of extra attached atoms or groups of atoms.
An OMIT Map
~p(x y z) = -Vi LCiFobsl - IFcalcl) exp[ -21ti(hx + ky + lz) + iUcalcJ hkl
Fcalc is the structure factor of a partial model, that is a model from which a fragment has been deleted. The phase angles Ucalc are for the model with fragment deleted. It is a difference Fourier summation that is often used if part of the electron density map cannot be interpreted satisfactorily. This part is then deleted in the model and does not contribute to the phase angle calculation. The map should show the density corresponding to the fragment, at half the height. Alternatively one can calculate
p(x y z) = -Vi LIFobsl exp[ -21ti(hx + ky + lz) + iUcalcJ hkl
This map should show the entire model with the deleted fragment at half height. Or with coefficients 2IFObsi - I Fcalc I , which also shows the entire model but the deleted fragment at full height.
A Simulated Annealing OMIT Map
This is an OMIT map with simulated annealing applied to the "known" part of the structure. It is also called: Composite Annealed OMIT Map.
Appendix 1 313
An OMIT Map with Sim Weighting
Llp(x y z) = -VI 2:m(i Fobs I - IFealcl) exp[-2rci(hx + ky + lz) + iUealc] hkl
m = 11(X)/lo(X) for noncentric reflections and m = tanh (X/2) for centric reflections, where
10 are II are modified Bessel functions of order zero and one, respectively, lFobsl is the observed structure factor amplitude, and IFKI is the amplitude for the known part of the structure. The fis are the atomic scattering factors for the n missing atoms. It is assumed that the partial structure is error-free. In practice this will not be true and then X must be taken as
X = 2crAI Eobs I x IEKI 1 - cri
lEI is the normalized structure factor amplitude. cr A is defined in Section 15.6.
A Weighted Electron Density Map Calculated with Phase Angles Ucalc from the Partial Structure
p(x y z) = -VI 2: (2mlFobsl - IFealel) exp[ - 2rci(hx + ky + /z) + iUeale] hkl
for noncentric reflections and
p(x y z) = -VI 2:mlFobsl exp [-2rci(hx + ky + lz) + iUealc] hkl
for centric reflections. This map is an improvement over the (2Fobs Fealc) map because it applies Sim weighting to the observed structure factor amplitudes (see above). Possible missing parts in the structure will show up more clearly in the electron density map than without Sim weighting. Sim assumed the partial structure to be error-free, but in practice this is never true. The effect of these errors is taken care of by defining X differently:
X = 2crAIEobsi ~ IEKI 1 - crA
314 Appendix 1
and by weighting down I FcaIc I :
p(x y Z) = -VI 2: (2mlFobsl - DIFcaIci) exp[ - 2rri(hx + ky + lz) + iUcaIc] hkl
with D = (jA(~pILN)-1/2 (Section 15.6). lEI is the normalized structure factor amplitude.
Appendix 2
A Compilation of Reliability Indices
Common Crystallographic R-Factor for Indicating the Correctness of a Model Structure
2: IIFobsl - kl FcaIc 1 1 R = h=k,,-l ----,;=-:-_:---_
2:IFobsl hkl
The Free R-Factor
L I !Fobs I - k!Fcalcll R - =hk=lc"""-T--;:;--__ _
free - L lFobsl hklcT
where h k 1 c T means all reflections belonging to test set T of unique reflections. The refinement is carried out with the remaining reflections, the working set W. The advantage of using this R-factor over the regular crystallographic R-factor is that it is unbiased by the refinement process.
R-Factor for Comparing the Intensity of Symmetry-Related Reflections
2: 2:IJi (h k I) - J(h k 1)1 R (J) = =hk::::...l..:..i ______ _
sym 2: 2: Ji(h k I) hkl i
315
316 Appendix 2
for n independent reflections and i observations of a given reflection. I(h k I) is the average intensity of the i observations.
R-Factor for Comparing the Structure Factor Amplitude for Symmetry-Related Reflections
L LllFi(h k 1)1 - IF(h k 1)11 R (F) - .:..::hk:::..l....:.i _______ _
sym - L LIFi(h k 1)1 hkl i
for i observations each of n independent reflections. IF(h k 1)lis the average value for the structure factor amplitude of the i observations of a given reflection.
R-Factor for the Comparison of N Data Sets after Merging
On IFhkll:
N
L L IlFhkll - IFhkl(j)11 R = h:..;,;.k:.:..<IJ_O=~l =-____ _
merge ~ I I LIN X Fhkl hkl
IFhkll is the final value of the structure factor amplitude.
N
L L I/hkl - Ihkl(j)I R = :.::hk:.:..<IJe.-°=..:..l ____ _
merge ~N I LI X hkl hkl
Real Space R-Factor
R - L I Pobs - Peale I real spaee - ~I + I LI Pobs Peale
The function is calculated per residue for either all atoms, or the main chain atoms only, or the side chain atoms. The summation is over all grid points for which Peale has a nonzero value for a particular residue. The
Appendix 2 317
function shows how good the fit is between the model and the electron density map.
Rcu IIis(i so)
L//Ji.H ± Ji./- FH(calc)/
RCullis(isO) = hkl '" ~/Ji.H ± Ji./ hkl
for centric reflections only. Fp, FPH, and FH include their sign (+ or -): FPH + Fp if the signs of FPH and Fp are opposite and FpH - Fp if they are equal.
RCullis(ano)
LII~F:H(obs)I-I~F:H(calc)11 RCullis(ano) = !.!!hk""-t ________ _
L~F:H(obs) hkt
where 6.F~H (obs) is the structure factor amplitude difference between f" Bijvoet pairs and ~F:dcalc) = 2-/FH/sin(aPH - aH) (Note that an al-~f
ternative nomenclature for 6.f is f').
RCullis(l.·)
hkt
where Fp(~) is the structure factor of the protein at ~, and Fp(~) at the parent A.; F(~)(calc) is the calculated contribution by the anomalous scatterer with respect to the parent.
hkl
This R-factor is used in isomorphous replacement methods to check the heavy atom refinement.
318 Appendix 2
LIIFtH(obs)I-IFtH(calc)11 + IIFpH(obs)I-IFpH(calc)11 Rkraut(anO) = !..!!hk~e ______________ _
L(iFtH(obs)1 + IFpH(obs)i) hkt
Derivative R-Factor
LllFderiv(h k 1)1 - IFnative(h k 1)11 Rderiv = Riso = .!..!!hk,,--l ----=-----
LlFnativel hkl
This R-factor is used for checking the quality of an isomorphous derivative.
Ranomalous
LIIFriIl-IFrHII R - !..!!!hk~e ___ _
anomalous - L IFtHI + IFrHI
hkt 2
Rlambda
LLIIFr(l'i)l- IFr(Ao)11 R - !..!!!hk~e~i-=~ ___ _
lambda - LIFr(Ao)1 hke
where IFp(Ai) I is the structure factor amplitude of the protein at Ai, and IF p("-o) I at the parent A.
Standard Linear Correlation Coefficient Between Observed and Calculated Structure Factor Amplitudes
2:(IFh(obs)1 - IFh(obs)l) x (I Fh(calc) I - IFh(calc)l) c= h
[2:(IFh(obs)1 - IFh(obs)I)2 x 2: CI FhCcalc) I - IFh(calc)I)2]1I2 h h
The same but in a different form:
c = IFhCobs)1 x IFhCcalc)I - I FhCobs) I x I FhCcalc) I
[{IFh(obs)j2 - CIFh(obs)1)2} x {IFh(calc)j2 - (IFhCcalc)lf}]1I2
Appendix 2 319
Standard Linear Correlation Coefficient Between two Electron Density Maps, Pl (xyz) and P2(XYZ)
c = I,(Pl(XYZ) - ~) x (Pz(xyz) - Pz(xyz))
[ I, (Pl(XYZ) - Pl(XYZ))z x I,(pz(xyz) - pz(XYZ))ztz
The same but in a different form:
c = Pl(XYZ) x pz(xyz) - Pl(XYZ) x pz(xyz)
[{ z - z} { 2 2 }J1/2 (Pl(XYZ)) - (Pl(XYZ)) X (P2(XYZ)) - (P2(XYZ))
The Phasing Power of Heavy Atoms in an Isomorphous derivative
I, liiI ( calc)1 [
2]1/2 Isomorphous phasing power: hke with
I,IEI2 hkt
2 2 I, lEI = I, {IFPH(obs)1 - IFPH(calc)l} hkt hke
I,liiI(calc)1 An alternative expression is: hkf ~
"'"' lEI hke
The Phasi ng Power of Anomalously Scatteri ng Atoms
[ 2 l~ I, I iiI (imag·)1
Anomalous phasing power: hkt 2
I,11~F1H(obs)I-I~F1H(calc)1I hke
I,lfH(imag·)1 or alternatively: __ .-!.h~kt,----____ _
I,11~FP'k(obs)I-I~FP'k(calc)1I hke
320 Appendix 2
where I':::.F± is the structure factor amplitude difference between Bijvoet pairs, and FH (imag.) is the imaginary component of the calculated structure factor contribution by the anomalously scattering atoms.
[ LIFHCreal)12 ]1/2
Dispersive phasing power: ___ -'.h!!!k"-t -------
LII~Fp"HCobs)I-I~Fp±HCcalc)112 hkt
LIFHCreal)1 or alternatively: ___ ..!..!h""kt'---_____ _
LII~Fp"HCobs)I-I~Fp±HCcalc)11 hkt
where ~F± is the structure factor amplitude difference between Bijvoet pairs, and FH(real) is the real component of the calculated structure factor contribution by the anomalously scattering atoms.
Figure of Merit
The figure of merit for a given reflection (h k I) is defined as
IF(h k I)bestl m = IF(h k 1)1
where
LP(a)FhkICa)
F(h k I)best = a '" LJP(a) a
It can be shown that the figure of merit is the weighted mean of the
cosine of the deviation of the phase angle from abest: m = cos{ a - a(best)}.
It is also equal to 11 (X) for acentric reflections and to tanh (X) for lo(X) 2
centric reflactions. For the definition of X see Section 8.2.
Appendix 3
The Variation in the Intensity of X-ray Radiation
When the anode of an X-ray tube is bombarded by electrons their deceleration causes the emission of photons. One electron impact gives rise to a photon with a certain amount of eI}ergy. There is no relation between the photons, either in time or in energy. Therefore, the number of emitted photons with the same energy, if measured during a time t, is not a fixed number (Figure App. 3.1). If that number is measured n times (where n is very large) with an average value of No, the probability of measuring N photons is
peN) = ~!NS'exp[-No] (Poisson distribution) (App.3.1)
For sufficiently large No (No ~ 9) this distribution can be replaced by the Gauss distribution:
1 [(N - No)2] peN) = Y2nNo exp - 2No (App.3.2)
The general form of the Gauss distribution is
1 [(x - x?] f(x) = crY2n exp - 2cr2 (App.3.3)
The spread of the curve is usually expressed in the variance of x, which is defined as
+00
~ = J (x - x)2f(x) dx (App.3.4) x=-oo
321
322 Appendix 3
anode
Figure App. 3.1. Photons with the same energy leaving the anode as a function of time.
By comparing Eqs. (App. 3.2) and (App. 3.3) it is found that the standard deviation cr for the X-ray photon emission is '\!'No.
In practice the number of photons is usually measured only once and the value of N found in that single measurement is taken as the best value with a standard deviation cr = VN. This is also true for synchrotron radiation, because we do not have perfect control over the physical state of the charged particles.
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Index
A Absolute
scale, 96, 117, 120-122, 126, 127 configuration, 99, 152, 159,207-
209 Absorption, 18,31,34, 109, 110, 113,
115,150,179,216-218 coefficient
atomic mass, 109, 155, 216 linear, 109, 155
correction, 37, 115 edge, 32, 155, 156, 158,215,216 factor, 109 gap,40 photoelectric, 110
Accelerating voltage, 24 Accessible region, 254 Accuracy of the structural model,
298 Acentric reflections, 145 Acupuncture method,S Addition strategy, 235, 238 Ammonium sulfate, 145 A~oFle,238-241,277
Amorphous substances, 11 Amplification gap, 40 Amplitude of electromagnetic wave,
72, 73 Angstrom unit, 20 Annealing, 18
Anode copper, 26 in multiple wire proportional
counter, 39, 40 in X-ray tube, 24, 25
Anomalous differences, 273 diffraction, 274 Patterson, 157, 159-161, 179 scattering, 32, 68, 99,129,142,154-
159,179,193,203-218 Antifreeze, 17 Antioxydant, 144 Area detector, 35, 39-41
calibration, 41 geometric distortion, 41
Argand diagram, 73, 74, 93, 102, 112, 217
Arndt, 42 Asymmetric unit, 63
in the rotation function, 226 Atomic
absorption curve, 216 cross-section, 155 fluctuations, 289 force microscope, 139 movement, 288
anharmonic, 288 anisotropic, 288 isotropic, 288
331
332
positions/coordinates, 167 mean error, 154
scattering factor, 78, 79, 89, 120, 123, 156
of carbon, 79, 89 Attachment of new molecules, 2 Averaging, 199,201
B Babinet's principle, 263 Background, 34,47 Basis set of reflections, 194 Batch and microbatch crystallization, 4 Bayes' theorem, 269 Bayesian statistics, 249, 270 Beam divergence, 31 Beamlines, 29 Beamstop, 15, 34 Becquerel (Bq), 48 Bending magnet, 27, 29, 31 Berylium foil, 25 Bessel function, 182, 185, 273, 290 Bias, 190 Bijvoet differences, 216 Bijvoet pair, 99,157,204,207,209,214 Biological macromolecule
crystallization database, 8 Blind region, 42, 43, 113 Blow and Crick procedure, 174, 184 Body-centered cell, 54, 55 Boltzmann's constant, 285 Bolstzmann's law, 285, 286 Bond
angle, 261,278, 280, 282, 284, 289 302
bending, 284 distance, 289 length, 261,278, 280, 282, 302 stretching, 284
Booster synchrotron, 27, 28, 29 B-parameter, 90, 283, 288 Bragg reflections, 110 Bragg's law, 67, 83, 84, 88, 92, 107 Bremsstrahlung, 24 Brilliance, 27 Brownian motion, 114, 121 Buried area, 303 B-value
average, 282 correlations, 282
Index
C CAD4 diffractometer, 35 Cambridge crystallographic data file,
302 Cathode
in multiple wire proportional counter, 39, 40
in X-ray tube, 23, 25 CCD, See Charge coupled device CCP4,198 Center of symmetry, 57, 58, 99 Centric reflections, 124, 145, 181, 182,
186,307 Centroid, 171, 172, 175 Centrosymmetric
distribution of errors, 307 protein structure, 179
Centrosymmetry in atom, 78 in Patterson map, 134
Characteristic lines, 26 Charge coupled device, 41 Chemical modification, 144 x-axis, 35 Chiral
center, 280 volume, 281
Chirality, 281 Cocrystallization, 140, 141 Collimator, 34 Color centers, 37, 38 Combination of phase information,
292-297 Common origin, 161, 162 Completeness, 282 Complex plane, 73 Compton scattering, 110 Conjugate complex, 137, 139, 232 Conjugate gradient method, 266, 267 Connectivity, 298 Constrained refinement, 261 Constraints, 248, 283 Continuous spectrum, 26 Convergence, 264-267, 274, 283,289 Convolution, 135-139, 190 Cooling, 16 Coordinate triplets, 62 Copper Ka, 113 CORELS, 277, 278, 283 Correction factors, 115, 129
Index
Correlation, 300 coefficient, 194,229,235,239,307,
standard linear, 230, 235-237, 318, 319
Counting statistics, 120 Crick and Magdoff, 146 Critical photon energy, 29, 30 Cross-Patterson
function, 162,241 peaks, 162, 163 vectors, 220, 231
Cross-rotation function, 220, 221, 228, 229
Cross-validation, 300 Crowther and Blow translation
function, 230-235, 239 Crowther's fast rotation function, 220 Cryocooling, 16-18 Cryocrystallography, advantages, 18 Cryoloop, 18 Cryoprotectant, 17, 18 Cryotemperature, 142 Crystal, 50-69
anisotropy, 11 birefringence, 11 characterization, 11 form, 50 imperfect, 104 perfect, 110, 111 refraction indices, 11 scattering by, 81 system, 64-69
Crystallization, 1-11, 18 Cubic system, 64 Curie (Ci), 47, 48 Cut-off edge, 24 Cysteine, 144
D Defects, 2 Delft instruments, 35, 45 dlFlano, 157 dlFlano, 151 Delta function, 93-95, 138 Densitometer, 36 Density modification, 181, 192, 195,
197,200-202,248,272 DENZO,67 Detectors, 35-41
position sensitive, 39
333
single photon counting, 39 Detergent, 2, 4 Diagonal
elements, 266 matrix,266
Dialysis and micro dialysis, 6,7, 19 Diamond plates as mirrors, 33 Dielectric constant, 286 Difference
electron density map, 279, 311 Fourier, 152-155,208
Difference Patterson map, 152, 154, 207
anomalous, 215 dispersive, 215
Diffraction experiment, 12-16 Diffraction pattern
generation, 16 of lysozyme, 15 of papain, 140, 142
Diffractometer, 34 Diffusion in protein crystals, 12 Dihedral angles, 261, 277, 278, 280,
282,284,300 Direct lattice, 85, 86 Direct methods, 129, 152,214,244-252 Disorder
static, 67, 91 dynamic,91
Dispersion difference, 211, 214 Dithiothreitol, 144 Divergence of the X-ray beam, 44 DM,198 Double difference electron density
map, 312 Double monochromator, 33 Double spot, 255-257 Dynamic range, 37, 41
E Eclipsed conformation, 282 Edge, 211 Eisenberg profile method, 303 Elastic scattering, 104 Electromagnetic
field strength, 71 radiation, 22, 71 wave, 72
Electron density, 92-98, 129 average, 196
334
best, 171, 191 frequency distribution, 198 map, 279, 311
from partial structure, 313 mean square error in, 177, 181 most probable, 171 projection, 97 smoothed, 194
Electron microscope, 11 Electrostatic
interaction, 145 term, 286
Enantiomer,280 Energy refinement, 284, 286, 287 Entropy maximization, 194 Envelope, 191, 195,201
molecular, 188, 194, 251 EREF, See Energy refinement Errors in
best Fourier map, 174-178 coordinates, 274, 305-310 difference Fourier map, 177, 178 electron density, 181, 183 heavy-atom model, 272 the structural model, 289-310
ESRF, See European Synchrotron Radiation Facility
Ethyleneglycol, 17 Eularian angles, 225 European Synchrotron Radiation
Facility, 29, 32 Ewald construction, 84 Ewald sphere, 21, 43, 84, 86, 87, 106,
255,256 Exponential terms, properties, 74 Exposure time, 36 Extinction, 109, 110, 115
F
primary, 111 secondary, 111
F (000),95 Face-centered cell, 54, 55 Factor fe, 127 Factorial method, 8 Fast area detector, 45 Fast Fourier
technique, 239, 289 transform, 189,267,275-277,279
Index
Figure of merit, 167, 173, 187, 191, 237, 307,320
Film mode, 35 FITING,239 Flash freezing, 17, 18 Fluorescence, 216 Fluorescent
radiation, 216 screen, 20, 39, 113
Focal spot, 24, 27 Four circle diffractometer, 36 Fourier
inversion, 97, 276 summation, 129, 233, 235, 250, 277 transform, 93, 277
continuous, 239 of a convolution, 137, 190
transformation, 92 Fractional coordinates, 62, 92 Free electron, 104, 105, 112, 155, 156,
217,218 Fresnel zone, 111, 112 Friedel mate, 212, 213 Friedel pair, 99,157,204 Friedrich, 22
G G-function, 192,224 Gauss
distribution function, 117-120, 123 error function, 117-120 probability distribution, 206
Gaussian distribution, 263, 272, 273, 294, 299,
321 errors, 174 function, 271, 273, 308 probability distribution, 169, 268
Genetic engineering, 144 Global minimum, 285 Glycerol, 17 Glycine, 301 Goniometerhead, 14, 18 Gray (Gy), 48
H H-bonding restraints, 283 Hampton research, 8 Hanging drop method, 5, 6,10
Index
Harker diagram for phase angle
determination, 168, 204, 272 lines, 132 peaks, 135, 137 planes, 134 section, 137,241
Harmonics, 33, 252, 255 Heavy atom, 203
attachment, 130, 140-145 contribution, 150 coordinates, 149,274 derivative, 139, 140 effect on X-ray intensities, 145-
149 parameters, 139, 150, 151,274 positions, 209 reagents, 9
Hexagonal axes, 64,65 cell,65,66 system, 64
Histidine, 142 Histogram matching, 181, 198, 199 Homologous protein, 219 Huber's rotation method, 226
I Image intensifier, 39, 41 Image plates, 35, 37, 42
disadvantages, 45 Imaginary
axis, 73 component, 73, 123 contribution to the structure factor,
214 part of the atomic scattering factor,
216,217 Incomplete factorial method, 8 Indices, 52 Injection of beam, 30 Insertion devices in storage rings, 27,
28 Integrated intensity, 103-107 Intensity
average, 117, 120-122 variation, 321, 322
Interaction between protein molecules, 3
Inverse beam geometry, 216 Fourier summation, 250 matrix, 265-167 structure, 135, 137
Inversion axis, 57 center, 57, 58, 62, 68
Ionic strength, 145
335
Isomorphism, 139, 140 Isomorphous replacement, 139-155,
194,203,206 for processing MAD data, 214,
215 Iteration, 265 Iterative
J
process, 263-266 steps, 267
Joint probability, 187
K Ka radiation, 24, 32 lC -axis, 35, 36 lC-construction, 45 K-edge,210 K-shell,24 K/3 radiation, 24, 33 Knipping, 22 Kramers-Kronig transformation, 216
L Lack of closure, 166, 167,206,271 Lasso technique, 17 Lattice
distance, 53 plane, 52, 83, 84,245,247
Laue conditions, 82, 83, 93, 94 diffraction, 31, 253-259
Least squares criterion, 278 function, 150 matrix, 283 method, 165, 263-268 minimum, 166 plane, 280 principle, 166
336
refinement, 168, 257 conditions, 267 full matrix, 268 stereochemically restrained, 279
strategy, 283 Leslie's method, 192, 195 Lifetime of
Crystal, 20 synchrotron beam, 32
Likelihood, 270-274 Linear accelerator, 27 Lipidic cubic phases, 4 Liquid-liquid diffusion, 4, 5, 19 Local minimum, 264, 285 Local scaling, 150 Locked cross-rotation function, 228 Locked rotation function, 227 Locked self-rotation function, 227 Lorentz
correction, 257 factor, 105, 107
L-shell,24 Luzzati plot, 305-307, 310 Lysozyme
M
crystallization, 8-11 diffraction experiment, 13-16
MAD, See Multiple wavelength anomalous scattering
Map inversion, 189 Mar research, 37, 38 Mass spectrometry, 2 Matrix notation, 221 Matthews number (VM)' 68 Maximum likelihood, 165, 168, 200,
249-251,263,268-275,289 Maxwellian distribution, 286 Membrane proteins, 2, 4 Mercury atom, 148 Mercury-containing compounds, 142 Methionine, 142 Microscope cover slips, 9 Miller indices, 246 MIR. See Multiple isomorphous
replacement MIRAS. See Multiple isomorphous
replacement with anomalous scattering
Mirror plane, 57, 58, 62
Mirrors, 33 MLPHARE, 215, 271, 272 Molecular
averaging, 194-197 dynamics, 285-289
Index
replacement, 129, 130, 180,219-243, 261,277
Monochromator, 31, 32, 33, 107 germanium, 33 graphite, 33 silicium, 33
Monoclinic system, 64 Monodisperse,19 Morphology, 50 Mosaic
blocks, 67, 103, 104, 105 crystals
ideal, 110 nonideal, 110
spread, 46 Mosaicity, 17, 18,44,67,259 Mounting protein crystals
in a loop, 17 in an X-ray capillary, 13, 19
MPD. See 2-methyl-2,4-pentanediol M-shell,24 Multibunch mode, 32 Multiple isomorphous replacement,
169,174,193,251,271 MUltiple isomorphous replacement
with anomalous scattering, 207
Multiple wavelength anomalous scattering, 31, 129, 130, 144, 174, 203,209-217,248,251,271
Anomalous, 214 Multiple wire proportional counter, 39,
44 Multiples, 253-257 Multiplicity, 255
N NCS operator, 196 Newtonian mechanics, 286 Nickel filter, 33 Nitrogen, liquid, 17, 18 Non-centric reflections, 124, 147, 181,
182,307 Nonbonded
contacts, 280
Index
interactions, 122 repulsion, 282
Noncrystallographic axis, 241, 242 symmetry (NCS), 179, 180, 194-197,
220,227,261,283,300 Nonisomorphism, 139, 141, 143, 166,
272,274,275 Normal
equations, 264-266 matrix, 265
Normalization, 118, 165, 169, 184,206, 270,290,292,294
Normalized structure factor, 124-128, 187,235
amplitude, 248 Nuclei,2
o
formation studied by light scattering, 2 electron microscopy, 2 fluorescens depolarization, 2
C>ccupancy, 141, 167, 179,274 C>ff-diagonal elements, 266 C>ligonucleotides, 142 w-axis,35 C>MIT map, 181,279,287,312,313 C>rganic solvent, 2 C>rigin, 245
local,231 shift,77
C>rthorhombic system, 64 c>scillating range, 34 C>scillation
angle for the area detector, 47 for the rotation camera, 44,
46 camera, 41-47
C>scillator, 156 C>verdetermination, 261
p
Packing analysis, 230 considerations, 239
Partials, 45 Patterson
cell, 131
correlation refinement, 235 function, 130-137, 140 map, 131, 132, 134, 149,214,230 peaks, 132, 137 summation, 149, 151
337
p-Chloromercuriphenylsulphonate, 9, 10
PCMS. Seep Chloromercuriphenylsulphonate
PDB. See Protein Data Bank PEG. See Polyethyleneglycol Permittivity constant, 286 Phase
angle, 129, 130, 139, 140, 152, 168-172,204-207,271,272
best, 191 error in, 175
combination, 192, 195 extension, 194, 197 problem, 129 refinement, 250 relations, 244 triplet, 245-247
Phase information from anomalous scattering, 295 isomorphous replacement, 293 MAD,296 molecular averaging, 297 partial structure data, 297 solvent flattening, 297
Phased translation function, 235, 237, 239
Phasing power anomalous, 319 dispersive, 320 isomorphous, 319
PHASIT,215 <j>-axis, 35, 36 Phosphate buffer, 145 Phosphorescent material, 38 Photographic film, 36 Photon
energy, 20 factory, 47
Pixel size, 37 Planarity, 302 Planck's constant, 24 Platinum compounds, 142 Point
atoms, 126
338
group, 64, 227 symmetry, 64
symmetry, 62 Poisson distribution, 321 Polar
angles, 225 fraction, 303
Polarization correction, 257 factor, 32, 105, 107, 108 of synchrotron beam, 32
Polarizing microscope, 11 Polyethylene glycol, 3, 17, 145 Potential energy, 284-286
minimum, 287 Precession camera, 34 Precipitation of the protein by
ammonium sulfate, 3 polyethylene glycol, 3 salt (see salting-out), 3
Preconditioning, 267 Primitive cell, 54, 55 Probability, 268
combination, 170 conditional, 269, 290, 305 curve for protein phase angles, 172-
174,191 distribution, 186, 248, 273, 291, 307 function, 166,214 theory, 244
PROCHECK, 302, 303 Profile fitting, 47, 257 Projection, centrosymmetric, 150 PROLSQ, 279, 283 Propane as organic cryogen, 17, 18 PROSA,303 Protein Data Bank, 302 Proteolytic fragments, 19 Purity, 2, 3
Q Quasicrystals, 58
R Ranomalous, 318 Rcullis(ano),317 Rcullis(iso),317 Rcullis(A), 317 Rderiv., 318
Rfree , 282, 284, 300, 315 Rkraut(ano),318 Rkraut(iso),317 Rlambda, 318 Rmerge, 282, 316 Rrea1 space, 299, 316 Rsym, 315, 316 Rworking set> 300 Radiation
damage, 16,32,66, 144,216 protection, 47
Index
Radiation absorbed dose (rad), 48 Radicals, 66, 144 Radius of convergence, 287 Raleigh scattering, 110 Ramachandran plot, 300, 301 Rare earth ions, 142 Real
axis, 73, 98 component, 73, 123 contribution to the structure factor,
214 part of the atomic scattering factor,
216 unit cell, 85
Real-space filtering, 250 Reciprocal
lattice, 20, 21, 43, 84-89 weighted, 86
matrix, 265 space, 255 unit cell, 85
Reciprocity, 12,84 Redundancy, 41 Refinement, 200, 236, 260-291
anisotropic, 283 of anomalous scatterers, 271-275 of the heavy atom parameters, 140,
150, 165-168, 271-275 Reflecting
angle, 75 plane, 75,83,107,108,111,245
Reflection, acentric, 150 centric, 120 conditions,
integral, 102, 103 serial,102
h k e and Ii k l, 88, 98
Index
low order, 262 non-centric, 120
REFMAC,289 Relative biological effect (RBE), 48 Relaxation time, 288 Reliability indices, 315-318 Rem (X-ray equivalent man), 48 Repulsive term, 281 Residual
electron density map, 312 Fourier, 154
Resolution, 40, 41, 67, 199 atomic, 96 of the diffraction pattern, 92 of the electron microscope, 11 limit, 256 sphere, 43, 255
Restrained refinement, 261 Restraints, 251, 278, 279, 283, 287, 290 R factor, 229, 235, 260, 264, 282, 289,
315 Rhombohedral
axes, 64, 65 cell, 65, 66
Rice distribution, 273 Right-handed coordinate system, 52,
159 Rigid body refinement, 239-241, 277-
279 Ring at 3-4 A resolution, 115 Robotic workstations, 8,19 Rocking curve, 67 Roentgen, 22 Rossmann and Blow procedure, 220 Rotation, 219
camera, 34,41-47 function, 220-229 matrix, 236
ROTING,239
S Salting-in, 3 Salting-out, 3 Scalar product, 76 Scale factor, 121, 126, 127, 150,230,
272,274,299 Scaling, 179 Scattering
by atomic nuclei, 105
elastic, 104, 110 by a free electron, 105 inelastic, 110 by a plane of atoms, 111, 112 power, 23 by a proton, 105
Scintillation counter, 34 Screening reagents, 9 Second virial coefficient, 19 Seed model, 199 Selenium, 248 Selenomethionine, 144, 157, 210 Self-Patterson
peaks, 162, 163,226 vectors, 220, 233
339
Self-rotation function, 194, 220, 221, 226-228
Shake-and-Bake,244-250 SHARP, 215, 272-275 SHELXL,283 SHELXS, 139, 152 Shock cooling, 17 Siemens, 44 Sievert (Sv), 48 GA,187 GA plot, 306-310 Signal to noise ratio, 46 Sim weighting, 181,248,287,313 Simulated annealing, 187m 285-287,
289 OMIT map, 287, 312
Single bunch mode, 32 Single isomorphous replacement, 178 Single isomorphous replacement with
anomalous scattering, 197,205, 207,208
Single photon counter, 35 Singles, 253-256 SIR. See Single isomorphous
replacement SIRAS. See Single isomorphous
replacement with anomalous scattering
Sitting drop method, 6 SnB,249 Soaking, 140, 141 Solubility, 3 Solvent
content, 68
340
diffraction by, 262 disordered, 262, 263 flattening, 188-194, 198, 199, 201,
202,261,272 flipping, 189-191 ordered, 263
Space groups, 58 Sparse matrix, 8 Spatial overlap, 257 Spatula technique, 17 Special position, 62, 63 Specific volume, 69 Sphere of reflection, 21 Spherical polar rotation, 226 Square matrix, 265 SQUASH, 198 Staggered conformation, 282 Standard deviation, 118, 120,264,267,
279,283 estimated, 119, 120, 294
Standard matrix, 304, 305 Static light scattering, 19 Stationary crystal, 254 Statistics of
data collection, 282 refinement, 282
Stereochemical information, 261 Stereochemistry check, 302 Storage ring, 23, 27, 28, 29 Structure factor, 82, 92
amplitude, 96, 129 distribution, 123
distribution, 122, 123 Subtraction strategy, 235, 238 Sulfhydryl group, 142 Supersaturation, 2,4 Surface tension, 2, 6, 17 SURVOL, 302, 303 Symmetry, 55-62
axes, 134 2-fold, 56, 57, 62, 99 2-fold screw, 56, 57, 100, 102 3-fold, 56, 57 3-fold screw, 56, 57 4-fold,15 5-fold, 57, 58, 64 7-fold, 57, 58, 64
in the diffraction pattern, 68, 99 elements, graphic symbols, 59, 60, 61
in the reciprocal lattice, 45 Symmetry-related reflections, 150 Synchrotron, 20, 23, 253
Index
radiation, 27-32, 107, 130,209,259
T TABLING, 239 Take-off angle, 25, 27 Target function, 287 Taylor expansion, 264, 266 Temperature factor, 89, 117, 120, 121,
126,167,274 anisotropic, 90 isotropic, 90, 283
Test set, 300 Tetragonal system, 64 Thermal
motion, 120, 126, 139 parameter B, 90 vibrations, 67
Thomson, 104, 105 Threading, 303 3D-1D profile method, 303-305 3D-1D score, 304, 305 Time-averaged molecular dynamics,
287-289 Time-resolved
data collection, 259 X-ray structure determination, 254
Time structure of synchrotron beam, 32
TNT, 277, 279,282, 283 Torroidal mirror, 33 Torsion
angle refinement, 289 angles, 282, 283, 302 energy, 284 potential, 284
TRAING,239 Transform of a sphere, 223 Translation, 219
function, 194,229-243 Translational symmetry, 56 Transmission factor, 105, 109 Transmission for X-rays, 20 Triclinic
space group, 86 system, 64
Trigonal system, 64, 65
Index
Triple spot, 257 Triplet
relation, 244, 247 sum, 248
Truncation, 155, 265 Tunability, 31 Twinned crystals, 283 Two-electron system, 74, 75, 154
U
summation, 154 2-methyl-2,4-pentanediol, 2, 17, 143 2Fobs-FcaJc map, 311 2a-axis,35
Undulator, 29, 30, 31 spectral profile, 31
Unique reflections, 282 Unit
matrix, 265-267 translation, 56
Unit cell choice of, 53, 159 definition, 51, 52 effect of size on diffraction intensity,
113 real,131 scattering by, 79, 81, 82 volume, 68
Unitary structure factors, 125, 126 Unscrambling, 256
V Van der Waals
contacts, 280 distances, 278 interaction, 2845 potential, 281
Vapor diffusion, 5. See also Hanging drop method; Sitting drop method
Variance, 118, 119, 123, 150, 165,272-275,287
Vector matrix, 266 product, 76
Vibration anisotropic, 90 atomic, 90 ellipsoid of, 90 harmonic, 90 isotropic, 89, 90
Video camera, 40, 41 Viruses, 196 Von Laue, 22
W Wang method, 188, 192 wARP, 199,200 Wavelength, 71, 72
choice of, 113 normalization, 257 shifter, 27, 30
Weighting factor/function, 181-183,
341
186,236,241,248,264,272,283 Weissenberg camera, 47 WHALESQ, 174, 215 WHAT_CHECK, 302, 303 White line, 211 Wiggler, 29, 30, 31 Wilson plot, 120-122, 149 Working set, 300
X X-ray
capillary, 13 detectors, 35-48 generator, 23
with sealed tube, 23, 32 with rotating anode, 23, 36
pattern, 20 photons, 24 sources, 22-32 tube current, 26 wavelength, 21
Xenon, 142 XPLOR, 277, 285
Z Zeppezauer method, 7 Zone, 245