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EMBO course Grenoble, June 2007. Phasing based on anomalous diffraction Zbigniew Dauter. Structure factor. F P (h) = S j f j . exp (2 p ih • r j ) f j = f º j ( q ) . exp(-B . sin 2 q / l 2 ). Structure factor with heavy atoms.

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Embo course grenoble june 2007
EMBO courseGrenoble, June 2007

Phasing based on

anomalous diffraction

Zbigniew Dauter


Structure factor
Structure factor

FP(h) = Sj fj. exp (2pih•rj)

fj = fºj(q). exp(-B.sin2q/l2)


Structure factor with heavy atoms
Structure factor with heavy atoms

FPH(h) = Sj fj. exp (2pih•rj) + Sk fk. exp (2pih•rk)

FPH(h) = FP(h) + FH(h)

|FPH|≠ |FP| + |FH|


Anomalous diffraction
Anomalous diffraction

normal scattering

q

q

anomalous(resonant) scattering


Structure factor and anomalous effect
Structure factor and anomalous effect

F(h) = Sj fj.exp(2pih•rj)

fj = fºj(q) + f’j (l) + i.f”j (l)

Anomalous correction f” is proportional to

absorption and fluorescence and f’ is its derivative


L dependence of anomalous corrections for selenium
l-dependence of anomalous correctionsfor selenium


Structure factor and anomalous vectors
Structure factor and anomalous vectors

FT(h) = Sj fj. exp (2pih•rj)

+Sk(fok + f’k + i.f”k). exp (2pih•rk)


Structure factor and anomalous effect1
Structure factor and anomalous effect

FT = FN + FA + F’A + i.F”A

Anomalous correction i.f”

shifts the phase of

atomic contribution

in positive direction


F riedel pair f h and f h
Friedel pair – F(h) and F(-h)

Anomalous correction f”

causes the positive

shift of phase of both

F(h) and F(-h)

in effect

|FT(h)| ≠ |FT(-h)|

jT(h) ≠ -jT(-h)


Friedel pair f h and f h
Friedel pair – F(h) and *F(-h)

|FT(h)| ≠ |FT(-h)|

jT(h) ≠ -jT(-h)


F riedel pair more realistically
Friedel pair – more realistically

fº(S) = 16

f”(S) = 0.56 for l = 1.54 Å

fº(Hg) = 82

f”(Hg) ≈ 4.5 for l < 1.0 Å


Bijvoet difference
Bijvoet difference

DF±= |F+| - |F-|


D f depends on phase difference
DF depends on phase difference

DF±= 2F”A

DF±= 0


Sinusoidal dependence of d f and f a
Sinusoidal dependence of DF and FA

DF±≈2.F”A.sin (jT- jA)


Partial structure of anomalous atoms
Partial structure of anomalous atoms

Anomalous atoms can be located by Patterson

or direct methods, since:

DF±≈2.F”A.sin (jT- jA)

|DF±|2≈ 4.dA2.FA2.sin2 (jT - jA)

≈ 2.dA2.FA2 - 2.dA2.FA2.cos[2(jT – jA)]

where

dA = f”A/foA sin2a = 1/2 - 1/2.cos2a

and anomalous atoms are mutually distant

(even low resolution is “atomic”)


Two solutions for a single l sad
Two solutions for a single l (SAD)

If anomalous sites are known

(DF±, FA, F’A, F”A,jA)

there are two possible

phase solutions


Selection of mean phase
Selection of mean phase

|FSAD| = |FT|.cos[(1/2)(jT1 – jT2)]

jSAD = (1/2)(jT1 + jT2)

FOM = cos[(1/2) (jT1 – jT2)]


Electron density is then a superposition of correct structure and noise
Electron density is then a superpositionof correct structure and noise

F1 F2 F1 + F2

iterative solvent flattening indicates correct phase


With errors in measured f snd f
With errors in measured F+ snd F-

With measurement errors

and inaccurate anomalous

sites the phase indications

are not sharp


Phase probability
Phase probability

Each phase has then

certain probability


Symmetric sad probability
Symmetric SAD probability

The phase probability is

symmetric around jSAD

has two maxima


Two solutions not equivalent
Two solutions not equivalent

Vectors FN have

different lengths

Solution with jT

closer to jA

is more probable

(Sim contribution)


Total sad probability
Total SAD probability

One solution is slightly

more probable

(depending how large

is the substructure)


Excitation spectrum of se not fluorescence spectrum
Excitation spectrum of Se(not fluorescence spectrum)

inflection f’ = -10.5 f” = 2.5

peak f’ = -4.0 f” = 8.0

remote f’ = -0.5 f” = 4.5


Typical mad wavelengths
Typical MAD wavelengths

inflection f’ = -10.5 f” = 2.5

peak f’ = -4.0 f” = 8.0

remote f’ = -0.5 f” = 4.5


Typical mad wavelengths1
Typical MAD wavelengths

inflection f’ = -10.5 f” = 2.5

peak f’ = -4.0 f” = 8.0

remote f’ = -0.5 f” = 4.5






Analytical mad approach karle hendrickson
Analytical MAD approach(Karle & Hendrickson)

FT(±)2 = FT2 + a(l) . FA2

+ b(l) . FT.FA .cos (jT - jA)

± c(l) . FT .FA . sin (jT - jA)

a(l) = (f’2 + f”2)/fo2

b(l) = 2.f’/fo

c(l) = 2.f”/fo

Three unknowns: FT, FA and (jT - jA)

- system can be solved,

and FA used for finding anomalous sites

jA (and jT) can then be calculated


Mad treated as mir
MAD treated as MIR

Data from different l can be treated

as separate derivatives and one native

and universal programs (SHARP, SOLVE etc)

used for phasing

- perfect isomorphism (one crystal)

- synchrotron necessary (tunable l)

- radiation damage (with long exposures)


Summary
Summary

MIR MAD SAD

home lab or SR synchrotron home lab or SR

several crystals one crystal, 2-3 data one data set

non-isomorphism perfect isomorphism perfect isomorphism

radiation damage ? radiation damage rad. dam. less acute

tedious h.a. search easier easy, if works

All methods easy thanks to excellent programs

SHELXD/E, SnB, SHARP, SOLVE, CNS, PHENIX, CCP4 etc.


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