SAD/MAD phasing

Erice school, June 2012
SAD/MAD phasing
Zbigniew Dauter
Natl. Cancer Inst. / Argonne Natl. Lab.

Structure factor
F P = Σ j f j
.exp 2πi(h . r j )
Structure factor F P can be expressed in the form of

Structure factor with heavy atoms
F PH = Σ j f j
.exp 2πi(h . r j ) + Σ h f h. exp 2πi(
F PH = F P +
al structure factor F PH as a sum of contributions of norm

Anomalous scattering
normal scattering
θ
θ
anomalous (resonant) scattering
Additional contribution if the energy of X-rays is

Structure factor and anomalous effect
For anomalously scattering atoms
f j = fº j (θ) + f’ j (λ) + i . f” j (λ)
Anomalous correction f” is proportional
to absorption and fluorescence and
dispersive correction f’ is its derivative

Anomalous corrections f’ and f” for Se
black –
theory for single
atom in vacuum
Absorption edge
for Se at 0.979 Å
blue –
measured curve
from real sample
ten the extra EXAFS features (e.g. white line) occurring

nfluence of anomalous effect on phases
i . exp iα
= i . (cos α + i . sin α)
= i . cos α – sin α
= i . sin (90 o +α) + cos (90
= exp i . (90 o +α)
f j = fº j (θ) + f’ j (λ) + i . f” j (λ)
Anomalous correction if” shifts the phase

nfluence of anomalous effect on phases
Anomalous correction if” shifts the phase

Friedel pair: F(h) and F(-h)
Anomalous correction f”
causes the positive
phase shift of both
F(h) and F(-h)
in effect
|F(h)| ≠ |F(-h)|
and
Friedel’s Law
is broken

Friedel pair: F(h) and *F(-h)
|F(h)| ≠ |F(-h
Friedel’s La
does not ho
t is customary to present F(-h) as *F(-h), with its phase

Friedel pair more realistically
f º (S) = 16
f”(S) = 0.56
for λ = 1.5
f º (Hg) = 82
f”(Hg) ≈ 4.
for λ < 1.0
e anomalous differences are 1 – 5 % of the total intensit

Bijvoet difference
e observed anomalous Bijvoet difference ΔF ± = |F + | - |F

Bijvoet difference
ΔF ± = 2 F” sin(ϕ – ϕ )

Bijvoet difference depends on (ϕ T – ϕ A )
The observed Bijvoet difference ΔF ± is largest when
o
o

Locating anomalous atoms from ΔF
Most proper for locating anomalous atoms are F A ’s,
but they are not measurable directly, however
ΔF ± = 2 F ” A sin(ϕ T - ϕ A ) = 2 F A (f”/fo ) sin(ϕ T - ϕ A )
therefore
ΔF ± ≈ F A for large Bijvoet differences
Measured anomalous differences can be used to
locate anomalous atoms by Patterson or direct methods

There are two places wh
the properly oriented pin
vector 2F” corresponds
anomalous difference ΔF
ingle-wavelength Anomalous Diffraction)
f the anomalous substructure is known, F A , F’ A , F” A , ϕ A

SAD phase ambiguity
ϕ SAD = (ϕ 1 + ϕ 2 )
F SAD = F T . FOM
where
figure-of-meri
FOM = cos (α
t is not known which solution is correct, therefore one ca

Electron density map
F good F wrong F good + F wrong
ince Fourier summation is additive, the resulting electro

Solvent flattening
F good F wrong F good + F wrong
e featureless solvent regions of this map can be flattene

With errors in measured |F | and |F |
The measured amplitudes are not perfectly accurate,

With errors in measured |F | and |F |
In effect, the phase probability varies
o
P anom (ϕ Τ )=N exp{-[ΔF ± +2F” A sin(ϕ T -ϕ A

Known partial structure
P part (ϕ Τ )=N exp{2[F T F A )/F N2 ] co
The case where normal and anomalous vectors

With both probabilities combined
When anomalous and partial structure probabilities

emote (high energ
Multi-wavelength anomalous diffraction)
peak (white line)
edge (inflection po
These three wavelengths give largest differences

Multi-wavelength anomalous diffraction)
peak (maximum f”, medium f
edge (maximum f’, medium f
remote (large f”, very small f
These three wavelengths give largest differences

Analytical MAD (after Hendrickson)
|F Τ+ | 2 = |F To | 2 + y 2 – 2 |F To | y cosα
y 2 = |F A ’| 2 + |F A ”| 2 = |F A | 2 [(f’ 2 +f” 2 )/f o2 ]
y cosα = x 1 + x 2 = |F A | (f’/f o ) cosδ + |F A | (f’/f
δ = ϕ A - ϕ T
Law of cosines applied

Analytical MAD (after Hendrickson)
T ± | 2 = |F To | 2 + a(λ) |FA| 2
+ b(λ) |F To | |F A | cos(ϕ T – ϕ A )
± c(λ) |F To | |F A | sin(ϕ T – ϕ A )
a(λ) = (f’ 2 + f” 2 )/f o2
b(λ) = 2 f’/f o
c(λ) = 2 f”/f o
the same for all reflection
at the same wavelength
stem of equations with three unknowns, |F To |, |F A |, (ϕ T -
n be solved analytically with more than one wavelength
en |F A | is used to locate anomalous substructure and
lculated ϕ A gives |F To | and ϕ T necessary for the map

Probabilistic MAD
owadays MAD phasing is based on a probabilistic appro
Maximum Likelihood, taking into account various effects
e errors in measured amplitudes, inaccuracies of the
bstructure model, non-isomorphism, or even radiation
mage and anisotropy of f”.
Excellent and powerful programs are widely available:
SHARP, PHASER, SOLVE, MLPHARE

SAD with peak wavelength

SAD with edge wavelength

MAD with two wavelengths
Only one of possible solutions

MAD with two wavelengths
Probability for the correct solution

Excit