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.

References

Abrahams, J.P. Acta Crystallogr. (1997) 053, 371-376. Abrahams, J.P., Leslie, AG.W., Lutter, R, and Walker, J.E. Nature (1994) 370,

621-628. Agarwal, RC. Acta Crystallogr. (1978) A34, 791-809. Amemiya, Y. Methods in Enzymol. (1997) 276,233-243. Antson, AA, Otridge, J., Brzozowski, AM., Dodson, E.J., Dodson, G.G., Wilson,

K.S., Smith, T.M., Yang, M., Kurecki, T., and Gollnick, P. Nature (1995) 374, 693-700.

Arndt, U.W. and Wonacott, AJ., eds. The Rotation Method in Crystallography. North-Holland, Amsterdam, 1977.

Asselt, E.J. van, Perrakis, A., Kalk, K.H., Lamzin, V.S., and Dijkstra, B.W. Acta Crystallogr. (1998) 054, 58-73.

Azaroff, L. Acta Crystallogr. (1955) 8, 701-704. Badger, J. Methods in Enzymol. (1997) 277, 344-352. Bella, J. and Rossmann, M.G. Acta Crystallogr. (1998) 054, 159-174. Bentley, G.A and Houdusse, A. Acta Crystallogr. (1992) A48, 312-322. Bhat, T.N. and Cohen, G.H. J. Appl. Cryst. (1984) 17,244-248. Bhat, T.N. J. Appl. Cryst. (1988) 21,279-281. Blessing, RH., Guo, D.Y., and Langs, D.A Acta Crystallogr. (1996) 052, 257-266. Blow, D.M. and Crick, F.H.C. Acta Crystallogr. (1959) 12,794-802. Blow, D.M. and Matthews, B.W. Acta Crystallogr. (1973) A29, 56-62. Bowie, J.U., LUthy, R, and Eisenberg, D. Science (1991) 253,164-170. Bricogne, G. Acta Crystallogr. (1974) A30, 395-405. Bricogne, G. Acta Crystallogr. (1976) A32, 832-847. Bricogne, G. Acta Crystallogr. (1988) A44, 517-545. Bricogne, G. in: Isomorphous Replacement and Anomalous Scattering, Proc.

Daresbury Study Weekend, pp. 60-68; Wolf, W., Evans, P.R, and Leslie, AG.W., eds. SERC Daresbury Laboratory, Warrington, UK, 1991.

Bricogne, G. Acta Crystallogr. (1993) 049, 37-60.

323

324

Bricogne, G. Methods in Enzymol. (1997) 276, 361-423. Brtinger, AT: Acta Crystallogr. (1990) A46, 46-57. Brtinger, AT. Acta Crystallogr. (1991a) A47, 195-204. Brunger, AT. Ann. Rev. Phys. Chern. (1991b) 42, 197-223. Brunger, AT. Nature (London) (1992) 355, 472-475. Brtinger, AT. Acta Crystallogr. (1993) 049, 24-36. Brtinger, AT. Methods in Enzymol. (1997a) 276, 558-580. Brtinger, AT. Methods in Enzymol. (1997b) 277, 366-396.

References

Brtinger, AT., Kuriyan, J., and Karplus, M. Science (1987) 235,458-460. Brtinger, AT. and Nilges, M. Quarterly Rev. Biophys. (1993) 26, 49-125. Brzozowski, A., Derewenda, Z.S., Dodson, E.J., Dodson, G.G., and Turkenburg,

J.P. Acta Crystallogr. (1992) B48, 307-319. Carter, C.W. Methods: Companion to Methods Enzymol. (1990) 1, 12-24. Castellano, E.E., Oliva, G., and Navaza, J. J. Appl. Cryst. (1992) 25, 281-284. CCP4, Collaborative Computational Project, Number 4 Acta Crystallogr. (1994)

050,760-763. Web page: http://www.dl.ac.uk/ccp/ccp4/main.html Chayen, N.E. Structure (1997) 5, 1269-1274. Chayen, N.E. Acta Crystallogr. (1998) 054, 8-15. Chayen, N.E., Shaw Stewart, P.D., Maeder, D.L., and Blow, D.M. J. Appl.

Crystallogr. (1990) 23, 297-302. Chayen, N.E., Show Stewart, P.D., and Blow, D.M. J. Crystal Growth (1992) 122,

176-180. Cohen, S.L. Structure (1996) 4, 1013-1016. Cowtan, K.D. and Main, P. Acta Crystallogr. (1996) 052, 43-48. Crick, F.H.C. and Magdoff, B.S. Acta Crystallogr. (1956) 9, 901-908. Crowther, RA Acta Crystallogr. (1969) B25, 2571-2580. Crowther, RA in: The Molecular Replacement Method, pp. 173-178; Rossmann,

M.G., ed. Gordon & Breach, New York, 1972. Crowther, RA and Blow, D.M. Acta Crystallogr. (1967) 23, 544-548. Cruickshank, D.W.J. in Computing Methods in Crystallography, p. 113. J.S. Rollett,

ed, Pergamon Press, Oxford, 1965. Cruickshank, D.W.J., Helliwell, J.R, and Moffat, K. Acta Crystallogr. (1987) A43,

656-647. Cruickshank, D.W.J., Helliwell, J.R, and Johnson, L.N.: "Time resolved macro­

molecular crystallography," Oxford Science Publications (Oxford University Press) 1992.

Cudney, B., Patel, S., Weisgraber, K., Newhouse, Y., and McPherson, A Acta Crystallogr. (1994) 050,414-423.

Cygler, M. and Desrochers, M. Acta Crystallogr. (1989) A45, 563-572. Daopin, S., Davies, D.R, Schlunegger, M.P., and Grtitter, M.G. Acta Crystallogr.

(1994) 050, 85-92. Dauter, Z. Methods Enzymol. (1997) 276, 326-344. Debaerdemaeker, T. and Woolfson, M.M. Acta Crystallogr. (1983) A39, 193-

196. Derewenda, U., Swenson, L., Green, R, Wei, Y., Morosoli, R., Sharec, F., Kluepfel,

D., and Derewenda, Z.S. J. BioI. Chern. (1994) 269, 20811-20814. DeTitta, G.T., Weeks, C.M., Thuman, P., Miller, R, and Hauptman, H.A Acta

Crystallogr. (1994) A50, 203-210. Diamond, R Acta Crystallogr. (1971) 27A, 436-452.

References 325

Dickerson, RE., Weinzierl, J.E., and Palmer, RA Acta Crystallogr. (1968) B24, 997-1003.

Dijkstra, B.W., Kalk, K.H., HoI, W.G.J., and Drenth, J. J. Mol. BioI. (1981) 147,97-123.

Dodson, E. and Vijayan, M. Acta Crystallogr. (1971) B27, 2402-241l. Driessen, H.P.C., Bax, B., Slingsby, c., Lindley, P.F., Mahadevan, D., Moss, D.S.,

and Tickle, 1.1. Acta Crystallogr. (1991) B47, 987-997. Ducruix, A and Giege, R Crystallization of Nucleic Acids and Proteins, a Practical

Approach. IRL Press, Oxford, 1992. Eagles, P.AM., Johnson, L.N., Joynson, M.A, McMurray, C.H., and Gutfreund, H.

J. Mol. BioI. (1969) 45, 533-544. Ealick, S.E. Structure (1997) 5, 469-472. Engh, RA and Huber, R Acta Crystallogr. (1991) A47, 392-400. Epp, 0., Steigemann, W., Formanek, H., and Huber, R Eur. J. Biochem. (1971) 20,

432-437. Ferre-d'Amare, AR and Burley, S.K. Structure (1994) 2, 357-359. Fisher, RA Mess. of Math. (1912) 41, 155-160. Freund, A.K. Structure (1996) 4,121-125. Fujinaga, M. and Read, RJ. J. Appl. Crystallogr. (1987) 20, 517-52l. Furey, W. and Swaminathan, S. Methods Enzymol. (1997) 277, 590-620. Garcia, AE., Krumhansl, J.A, and Frauenfelder, H. PROTEINS, Structure,

Function and Genetics (1997) 29, 153-160. Garcfa-Ruiz, J.M. and Moreno, A Acta Crystallogr. (1994) 050,484-490. Garman, E.F. and Schneider, T.R J. Appl. Cryst. (1997) 30, 211-237. George, A and Wilson, W.W. Acta Crystallogr. (1994) 050, 361-365. Gewirth, D., The HKL manual. A description of the programs DENZO,

XDISPLA YF, and SCALEPACK: An Oscillation Data Processing Suite for Macromolecular Crystallography (1997).

Gilliland, G.L. and Ladner, J.E. Curro Opin. Struct. BioI. (1996) 6, 595-603. Gilliland, G.L. Methods EnzymoI. (1997) 277, 546-556. Gonzalez, A, Denny, R, and Nave, C. Acta Crystallogr. (1994) 050, 276--282. Gouaux, E. Structure (1998) 6, 5-10. Gros, P., van Gunsteren, W.F., and HoI, W.G.J. Science (1990) 249, 1149-1153. Gruner, S.M. Curro Opin. Struct. BioI. (1994) 4, 765-769. Guss, J.M., Merritt, E.A, Phizackerley, RP., Hedman, B., Murata, M., Hodgson,

K.O., and Freeman, H.C. Science (1988) 241, 806--81l. Hahn, T. ed., International Tables for Crystallography, Vol. A D. Reidel,

Dordrecht, 1993. Hajdu, J. and Johnson, L.N. Biochemistry (1990) 29,1669-1678. Harada, Y., Lifchitz, A, Berthou, J., and Jolles, P. Acta Crystallogr. (1981) A37,

398-406. Harker, D. Acta Crystallogr. (1956) 9, 1-9. Harp, J.M., Timm, D.E., and Bunick, G.J. American Crystallographic Association,

pp. 66 (WeBol) and 148 (P205) July 19-25, St. Louis, Missouri, 1997. Harp, J.M., Timm, D.E., and Bunick, G.J. Acta Crystallogr. (1998) 054,

622--628. Hauptman, H.A Acta Crystallogr. (1982) A38, 289-294. Hauptman, H.A Methods Enzymol. (1997a) 277, 3-13. Hauptman, H.A Current Opin. Structural Biology (1997b) 7, 672--680.

326 References

Heitler, W.G. The Quantum theory of radiation, 3rd ed. Oxford University Press, London, 1966.

Helliwell, J.R Macromolecular Crystallography with Synchrotron Radiation. Cambridge University Press, Cambridge, 1992.

Helliwell, J.R, Ealick, S., Doing, P., Irving, T., and Szebenyi, M. Acta Crystallogr. (1993) D49, 120-128.

Hendrickson, W.A and Lattman, E.E. Acta Crystallogr. (1970) B26, 136-143. Hendrickson, W.A and Ward, K.B. Acta Crystallogr. (1976) A32, 778-780. Hendrickson, W.A Methods Enzymol. (1985) 115, 252-270. Hendrickson, W.A, Love, W.E., and Murray, G.c. J. Mol. BioI. (1968) 33, 829-

842. Hendrickson, W.A, Smith J.L., Phizackerley, RP., and Merritt, E.A Proteins

(1988) 4, 77---88. Hendrickson, W.A, Horton, J.R, and LeMaster, D.M. EMBO J. (1990) 5, 1665-

1672. Hiremath, C.N., Munshi, S.K., and Murthy, M.RN. Acta Crystallogr. (1990) B46,

562-567. Hodel, A, Kim, S.-H., and Brunger, AT. Acta Crystallogr. (1992) A48, 851-858. Honl, H. Annalen der Physik, 5. Folge (1933) 18,625--655. Hooft, RW.W., Vriend, G., Sander. c., and Abola, E.E. Nature (1996) 381, 272. Hoppe, W.Z. Elektrochemie (1957) 61, 1076-1083. Huber, R in: Molecular Replacement. Proceedings of the Daresbury Study Week­

end, 15-16 February, pp. 58--61; Machin, P.A, ed. SERC Daresbury Laboratory, Warrington, 1985.

Huber, R and Schneider, M.J. Appl. Cryst. (1985) 18, 165-169. International Tables for Crystallography, published for the International Union of

Crystallography by Kluwer Academic Publishers, Dordrecht 1995. Jack, A and Levitt, M. Acta Crystallogr. (1978) A34, 931-935. Jacobson, RH., Zhang, X-J., DuBose, RF., and Matthews, B.W. Nature (1994)

369,761-766. James, RW. The Optical Principles of the Diffraction of X-Rays, Chapter 4.

London, Bell, 1965. Jancarik, J. and Kim, S.-H. J. Appl. Cryst. (1991) 24, 409-41l. Johnson, J.E. Acta Crystallogr. (1978) B34, 576-577. Jones, T.A, Zou, J.-Y., and Cowan, S.W. Acta Crystallogr. (1991) A47, 110-119. Kahn, R, Fourme, R, Gadet, A, Janin, J., Dumas, c., and Andre, D. J. Appl.

Cryst. (1982) 15, 330-337. Kam, Y.J.Z. and Eisenberg, D. Protein Science (1994) 3, 687--695. Karle, J. Int. J. Quant. Chern. Symp. (1980) 7, 357-367. Karle, J. Acta Crystallogr. (1989) A45, 765-78l. Kartha, G. and Parthasarathy, R Acta Crystallogr. (1965) 18,749-753. Kauzmann, W., Quantum Chemistry. Academic Press, New York, 1957. Klein, o. and Nishina, Y. Z. fUr Physik (1929) 52, 853-868. Kleywegt, G.J., Bergfors, T., Senn, H., Le Motte, P., Gsell, B., Shudo, K., and Jones,

T.A Structure (1994) 2, 1241-1258. Kleywegt, G.J. and Jones, T.A Structure (1995) 3, 535-540. Kleywegt, G.J. and Brunger, AT. Structure (1996) 4, 897-904. Kleywegt, G.J., Hoier, H., and Jones, T.A Acta Crystallogr. (1996) D52, 858---863. Kleywegt, G.J. and Read, RJ. Structure (1997) 5, 1557-1569.

References 327

Krishna Murthy, H.M., Hendrickson, W.A, Orme-Johnson, W.H., Merritt, E.A, and Phizackerley, RP. J. BioI. Chern. (1988) 263, 18430--18436.

La Fortelle, E. de and Bricogne, G. Trans. of the ACA Meeting, Likelihood, Bayesian Inference and their Application to the Solution of New Structures, Bricogne, G. and Carter, e., eds., 1994.

La Fortelle, E. de and Bricogne, G. Methods EnzymoI. (1997) 276, 472-494. Lamzin, V.S. and Wilson, KS. Acta Crystallogr. (1993) D49, 129-147. Lamzin, V.S. and Wilson, KS. Methods in EnzymoI. (1997) 277, 269-305. Landau, E.M. and Rosenbusch, J.P. Proc. NatI. Acad. Sci. (1996) 93,14532-14535. Landau, E.M., Rummel, G., Cowan-Jacob, S.W., and Rosenbusch, J.P. J. Phys.

Chern. (1997) 101, 1935-1937. Laskowski, RA, MacArthur, M.W., Moss, D.S., and Thornton, J.M. J. AppI.

Crystallogr. (1993a) 26, 283-29l. Laskowski, RA, Moss, D.S., and Thornton, J.M. J. Mol. BioI. (1993b) 231,1049-

1067. Leahy, D.J., Hendrickson, W.A, Aukhil, I., and Erickson, H.P. Science (1992) 258,

987-99l. Leslie, AG.W. Acta Crystallogr. (1987) A43, 134-136. Lindblom, G. and Rilfors, L. Biochim. Biophys. Acta (1989) 988,221-256. Lunin, V.Y. Acta Crystallogr. (1988) A44, 144-150. Lunin, V.Y. and Skovoroda, T.P. Acta Crystallogr. (1991) A47, 45-52. Lunin, V.Y., Urzhumtsev, AG., and Skovoroda, T.P. Acta Crystallogr. (1990) A46,

540--544. Lunin, V.Y. and Vernoslova, E.A Acta Crystallogr. (1991) A47, 238-243. LUthy, R, Bowie, J.D., and Eisenberg, D. Nature (London) (1992) 356, 83-85. Lutter, R, Abrahams, J.P., van Raaij, M.J., Todd, RJ., Lundqvist, T., Buchanan,

S.K, Leslie, AG.W., and Walker, J.E. J. Mol. BioI. (1993) 229, 787-790. Luzzati, V. Acta Crystallogr. (1952) 5, 802-810. MacArthur, M.W., Laskowski, RA, and Thornton, J.M. Curro Opin. Struct. BioI.

(1994) 4, 731-737. Machin, P.A ed., Molecular Replacement. Proceedings of the Daresbury Study

Weekend, 15-16 February. SERC Daresbury Laboratory, Warrington, 1985. Main, P. Acta Crystallogr. (1967) 23, 50--54. Matthews, B.W. J. Mol. BioI. (1968) 33,491-497. Matthews, B.W. and Czerwinski, E.W. Acta Crystallogr. (1975) A31, 480--487. McKenna, R, Di Xia, Willingmann, P., Ilag, L.L. Krishnaswamy, S., Rossmann,

M.G., Olson, N.H., Baker, T.S., and Incardona, N.L. Nature (London) (1992) 355,137-143.

McPherson, A, Jr. Methods Biochem. Anal. (1976) 23, 249-345. McPherson, A, Jr. J. Crystal Growth (1992) 122, 161-167. McPherson, A, Malkin, AJ., and Kuznetsov, Y.G. Structure (1995) 3, 759-768. Michel, H. Crystallization of Membrane Proteins. CRC Press, London, 1990. Miller, R, Gallo, S.M., Khalak, H.G., and Weeks, e.M. J. Appl. Cryst. (1994) 27,

613-62l. Moews, P.e. and Kretsinger, RH. J. Mol. BioI. (1975) 91, 201-228. Moffat, K Ann. Rev. Biophys. Biophys. Chern. (1989) 18, 309-332. Moffat, K Methods in Enzymol. (1997) 277,433-447. Morris, AL., MacArthur, M.W., Hutchinson, E.G., and Thornton, J.M. Proteins

(1992) 12, 345-364.

328 References

Moss, D.S. Acta Crystallogr. (1985) A41, 470-475. Murshudov, G.N., Vagin, AA, and Dodson, E.J., Acta Crystallogr. (1997) 053,

240-255. Nagai, K, Oubridge, e., Jessen, T.-H., Li, J., and Evans, P.R Nature (London)

(1990) 348, 515-520. Nagai, K, Evans, P.R, Li, J., and Oubridge, Ch. Proceedings of the CCP4 Study

Weekend, January, pp. 141-149. Daresbury Laboratory Warrington, 1991. Narasinga Rao, S., Jyh-Hwang Jih, and Hartsuck, J.A Acta Crystallogr. (1980)

A36, 878-884. Narten, AH. and Levy, H.A in: Water, a comprehensive treatise, Vol. I (Frank, F.,

ed.) p. 311-332; Plenum Press, New York, 1972. Navaza, J. Acta Crystallogr. (1994) A50, 157-163. Navaza, J. and Vernoslova, E. Acta Crystallogr. (1995) A51, 445-449. Navaza, J. and Soludjian, P. Methods Enzymol. (1997) 276, 581-594. Ohlendorf, D.H. Acta Crystallogr. (1994) 050, 805-812. Otwinowski, Z. Proceedings of the CCP4 Study Weekend, 25-26 January, pp. 80-

86, Wolf, W., Evans, P.R, and Leslie, AG.W., eds., 1991. Pahler, A, Smith, J.L., and Hendrickson, W.A Acta Crystallogr. (1990) A46, 537-

540. Pai, E.F. Curro Opin. Struct. BioI. (1992) 2, 821-827. Pannu, N.S. and Read, RJ. Acta Crystallogr. (1996) A52, 659-668. Perrakis, A, Sixma, T.K, Wilson, KS., and Lamzin, V.S. Acta Crystallogr. (1997)

053, 448-455. Pflugrath, J.W. Curro Opin. Struct. BioI. (1992) 2, 811-815. Phillips, W.e. Methods Enzymol. (1985) 114,300-329. Podjarny, AD., Bhat, T.N., and Zwick, M. Ann. Rev. Biophys. Biophys. Chern.

(1987) 16, 351-373. Polikarpov, I. and Sawyer, L. Joint CCP4 and ESF-EACBM Newsletter on Protein

Crystallography, Daresbury Laboratory, (1995) 31, 5-1l. Polikarpov, I., Teplyakov, A, and Oliva, G. Acta Crystallogr. (1997) 053, 734-737. Rabinovich, D. and Shakked, Z. Acta Crystallogr. (1984) A40, 195-200. Ramachandran, G.N., Ramakrishnan, e., and Sasisekharan, V.J. Mol. BioI. (1963)

7,95-99. Rayment, I. Acta Crystallogr. (1983) A39, 102-116. Read, RJ. Acta Crystallogr. (1986) A42, 140-149. Read, RJ. Crystallographic Computing School, Bischenberg, 1990a. Read, RJ. Acta Crystallogr. (1990b) A46, 900-912. Read, RJ. Methods in Enzymol. (1997) 277, 110-128. Read, RJ. and Schierbeek, AJ. J. Appl. Crystallogr. (1988) 21, 490-495. Rees, D.e. Acta Crystallogr. (1983) A39, 916-920. Ren, Z. and Moffat, K J. Appl. Cryst. (1995) 28,482-493. Rice, L.M. and Briinger, AT. Proteins (1994) 19, 277-290. Rosenbaum, G., Holmes, KG., and Witz, J. Nature (London) (1971) 230, 434-

437. Rossmann, M.G. Acta Crystallogr. (1990) A46, 73-82. Rossmann, M.G. Structure (1995) 5, 650-655. Rossmann, M.G. and Blow, D.M. Acta Crystallogr. (1961) 14,641-647. Rossmann, M.G. and Blow, D.M. Acta Crystallogr. (1962) 15, 24-3l. Rossmann, M.G. and Blow, D.M. Acta Crystallogr. (1963) 16,39-45.

References 329

Rossmann, M.G., McKenna, R, Liang Tong, Di Xia, Jin-Bi Dai, Hao Wu, Hok-Kin Choi, and Lynch, RE. J. AppI. Crystallogr. (1992) 25, 166-180.

Rould, M.A, Perona, J.J., and Steitz, T.A Acta Crystallogr. (1992) A48, 751-756. Sakabe, N. J. AppI. Crystallogr. (1983) 16,542-547. Sakabe, N. Nuclear Instruments and Methods in Physics Research (1991) A103,

448-463. Sauer, 0., Schmidt, A, and Kratky, e. J. AppI. Cryst. (1997) 30, 476-486. Shah, S.A, Shen, B.w., and Brunger, AT. Structure (1997) 5, 1067-1075. Sheldrick, G.M., Dauter, Z., Wilson, KS., Hope, H., and Sieker, L.e. Acta

Crystallogr. (1993) 049, 18-23. Sheldrick, G.M. and Gould, RO. Acta Crystallogr. (1995) B51, 423-43l. Sheldrick, G.M. and Schneider, T.R Methods EnzymoI. (1997) 277, 319-343. Sigler, P.B. Biochemistry (1970) 9, 3609-3617. Sim, G.A Acta Crystallogr. (1959) 12, 813-815. Sim, G.A Acta crystallogr. (1960) 13, 511-512. SippI, M.J. Proteins (1993) 17,355-362. Skilling, J. in: Maximum Entropy and Bayesian Methods, pp. 45-52. Skilling, J., ed.

Kluwer Academic Publishers, Dordrecht, 1988. Smith, G.D., Nagar, B., Rini, J.M., Hauptman, H.A, and Blessing, RH. Acta

Crystallogr. (1998) 054, 799-804. Sowadski, J.M. Curr. Opin. Struct. BioI. (1994) 4, 761-764. Spurlino, J.e., Smallwood, AM., Carlton, D.D., Banks, T.M., Vavra, KJ., Johnson,

J.S., Cook, E.R, Falvo, J., Wahl, Re., Pulvino, T.A, Wendoloski, J.J., and Smith, D.L. Proteins (1994) 19,98-109.

Srajer, V., Teng, T.-Y., Ursby, T., Pradervand, e., Ren, Z., Adachi, S., Bourgeois, D., Wulff, M., and Moffat, K Science (1996) 274, 1726-1729.

Srinivasan, R Acta Crystallogr. (1966) 20, 143-144. Srinivasan, Rand Chandrasekaran, R Indian J. Pure AppI. Phys. (1966) 4, 178-

186. Srinivasan, R and Ramachandran, G.N. Acta Crystallogr. (1965) 19, 1008-1014. Stec, B., Zhou, R, and Teeter, M.M. Acta Crystallogr. (1995) 051, 663-68l. Stewart, J.M. and Karle, J. Acta Crystallogr. (1976) A32, 1005-1007. Stoddard, B.L., Dean, A, and Bash, P.A Nature Struct. BioI. (1996) 3, 590--595. Strijtveen, B. and Kellogg, RM. Tetrahedron (1987) 43(21), 5045. Stuart, D.I. and Jones, E.Y. Curr Opin. Struct. BioI. (1993) 3, 737-740. Stubbs, M.T., Nar, H., Lowe, J., Huber, R, Ladenstein, R, Spangfort, M.D., and

Svensson, L.A Acta Crystallogr. (1996) 052, 447-452. Sussman, J.L. Methods EnzymoI. (1985) 115B, 271-303. Sweet, RM., Singer, P.T., and Smalas, A Acta Crystallogr. (1993) 049, 305-307. Templeton, D.H. and Templeton, L.K Acta Crystallogr. (1991) A47, 414-420. Ten Eyck, L.F. Acta Crystallogr. (1973) A29, 183-19l. Teng, T.Y. J. AppI. Cryst. (1990) 23, 387-39l. Terwilliger, T.e. Acta Crystallogr. (1994a) 050, 11-16. Terwilliger, T.e. Acta Crystallogr. (1994b) 050, 17-23. Terwilliger, T.C. and Eisenberg, D. Acta Crystallogr. (1983) A39, 813-817. Tickle, 1.1., Laskowski, RA, and Moss, D.S. Acta Crystallogr. (1998) 054, 243-

252. Tollin, P. and Rossmann, M.G. Acta Crystallogr. (1966) 21, 872-876. Tollin, P., Main, P., and Rossmann, M.G. Acta Crystallogr. (1966) 20, 404-407.

330

Tong, L. and Rossmann, M.G. Acta Crystallogr. (1990) A46, 783-792. Tronrud, D.E. Acta Crystallogr. (1992) A48, 912-916.

References

Tronrud, D.E., Ten Eyck, L.F., and Matthews, B.W. Acta Crystallogr. (1987) A43, 489-50l.

Tucker, AD., Baty, D., Parker, M.W., Pattus, F., Lazdunski, e., and Tsernoglou, D. Protein Eng. (1989) 2, 399-405.

Turner, M.A, Yuan, e.-S., Borchardt, RT., Hershfield, M.S., Smith, G.D., and Howell, P.L. Nature Struct. BioI. (1998) 5, 369-376.

Vellieux, F.M.D. and Read, RJ. Methods in Enzymoi. (1997) 277,18-53. Walter, RL., Thiel, D.J., Barna, S.L., Tate, M.W., Wall, M.E., Eikenberry, E.F.,

Gruner, S.M., and Ealick, S.E. Structure (1995) 3, 835-844. Wang, B.-e. Methods Enzymoi. (1985) 115, 90-112 Weeks, e.M., DeTitta, G.T., Hauptman, H.A, Thuman, P., and Miller, R Acta

Crystallogr. (1994) A50, 210-220. Westbrook, E.M. Methods Enzymoi. (1985) 114, 187-196. Wilmanns, M. and Eisenberg, D. Proc. Nati. Acad. Sci. U.S.A (1993) 90, 1379-

1383. Wilson, AJ.e. Nature (1942) 150, 151-152. Wilson, AJ.e. Acta Crystallogr. (1950) 3, 397-398. Wodak, S.J., Pontius, J., Vaguine, A, and Richelle, J. Proc. CCP4 Study Weekend

6-7 January, Hunter, W.N., Thornton, J.M., and Bailey, S., eds., 1995. Xiang, S., Carter, e.W., Bricogne, G., and Gilmore, C.J. Acta Crystallogr. (1993)

D49, 193-212. Zhang, K.Y.J. and Main, P. Acta Crystallogr. (1990a) A46, 41-46. Zhang, K.Y.J. and Main, P. Acta Crystallogr. (1990b) A46, 377-38l. Zhang, K.Y.J. Cowtan, K., and Main, P. Methods in Enzymoi. (1997) 277, 53-64. Zhang, x.-J. and Matthews, B.W. Acta Crystallogr. (1994) D50, 675-686.

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