EMBO course Grenoble, June 2007

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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 courseGrenoble, June 2007

Phasing based on

anomalous diffraction

Zbigniew Dauter

Structure factor

FP(h) = Sj fj. exp (2pih•rj)

fj = fºj(q). exp(-B.sin2q/l2)

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

normal scattering

q

q

anomalous(resonant) scattering

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

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 effect

FT = FN + FA + F’A + i.F”A

Anomalous correction i.f”

shifts the phase of

atomic contribution

in positive direction

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)

|FT(h)| ≠ |FT(-h)|

jT(h) ≠ -jT(-h)

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

DF±= |F+| - |F-|

Sinusoidal dependence of DF and FA

DF±≈2.F”A.sin (jT- jA)

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)

If anomalous sites are known

(DF±, FA, F’A, F”A,jA)

there are two possible

phase solutions

Selection of mean phase

FOM = cos[(1/2) (jT1 – jT2)]

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 measurement errors

and inaccurate anomalous

sites the phase indications

are not sharp

Phase probability

Each phase has then

certain probability

The phase probability is

has two maxima

Two solutions not equivalent

Vectors FN have

different lengths

Solution with jT

closer to jA

is more probable

(Sim contribution)

One solution is slightly

more probable

(depending how large

is the substructure)

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

inflection f’ = -10.5 f” = 2.5

peak f’ = -4.0 f” = 8.0

remote f’ = -0.5 f” = 4.5

inflection f’ = -10.5 f” = 2.5

peak f’ = -4.0 f” = 8.0

remote f’ = -0.5 f” = 4.5

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

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

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