Transcript Solving Crystal Structures From Two-wavelength X

Making sense of a MAD
experiment.
Chem M230B
Friday, February 3, 2006
12:00-12:50 PM
Michael R. Sawaya
http://www.doe-mbi.ucla.edu/M230B/
Topics Covered
• What is the anomalous scattering
phenomenon?
• How is the anomalous scattering signal
manifested?
• How do we account for anomalous scattering
effects in its form factor fH?
• How does anomalous scattering break the
phase ambiguity in a
– SIRAS experiment?
– MAD experiment?
Q: What is anomalous
scattering?
A: Scattering from an atom under conditions
when the incident radiation has sufficient
energy to promote an electronic transition.
An electronic transition is an e- jump
from one orbital to another –from quantum chemistry.
Orbitals are paths for electrons
around the nucleus.
Orbitals are organized in shells
with principle quantum
number, n=
1= Kshell
2= L shell
3= M shell
4= N shell
5= O shell
6= P shell etc.
And different shapes possible
within each shell. s,p,d,f
Orbitals have quantized energy levels.
Outer shells are higher energy.
Normally, electrons occupy the lowest
energy orbitals –ground state.
Selenium atom
Ground state
But, incident radiation can excite an e- to an
unoccupied outer orbital if the energy of the
radiation (hn) matches the DE between orbitals.
hn=DE
Excites a transition
from the “K” shell
For Se, DE=12.65keV
=l= 0.9795 Å
Selenium atom
Electronic transition
Under these conditions when an electron can transit
between orbitals, an atom will scatter photons
anomalously.
Anomalously
Scattered
X-ray
Incident
X-ray
-90° phase shift
diminished amplitude
Selenium atom
Excited state
If the incident photon has energy different from the
DE between orbitals, then there is little anomalous
scattering (usual case). Scattered x-rays are not phase shifted.
hn<DE
Selenium atom
Ground state
No transition possible
Insufficient energy
usual case
DE is a function of the periodic table.
DE is near 8keV for most
heavy and some light
elements, so anomalous
signal can be measured on
a home X-ray source with
CuKa radiation
( 8kev l=1.54Å).
At a synchrotron, the
energy of the incident
radiation can be tuned to
match DE (accurately).
K shell transitions L shell transitions
For these elements, anomalous scattering
is significant only at a synchrotron.
Importantly, DEs for C,N,O
are out of the X-ray range.
Anomalous scattering from proteins and
nucleic acids is negligible.
Choose an element with DE that matches an
achievable wavelength.
• Green shading represents typical synchrotron radiation range.
• Orange shading indicates CuKa radiation, typically used for
home X-ray sources.
Q: How is the anomalous
scattering signal manifested in
a crystallographic diffraction
experiment?
A: Anomalous scattering causes small but
measurable differences in intensity between
the reflections hkl and –h-k-l not normally
present.
Under normal conditions,
atomic electron clouds are
centrosymmetric.
For each point x,y,z,
there is an
equivalent point at
–x,-y,-z.
Centrosymmetry
relates points
equidistant from the
origin but in opposite
directions.
Centrosymmetry in the scattering atoms is
reflected in the centrosymmetry in the pattern
of scattered x-ray intensities.
The positions of the reflections
hkl and –h-k-l on the reciprocal
lattice are related by a center
of symmetry through the
reciprocal lattice origin (0,0,0).
(-15,0,6)
Pairs of reflections hkl and
–h-k-l are called Friedel pairs.
· (0,0,0)
Friedel’s law is a consequence
of an atom’s centrosymmetry.
I(hkl)=I(-h-k-l)
and
f(hkl)=-f(-h-k-l)
(15,0,-6)
But, under conditions of anomalous
scattering, electrons are perturbed from
their centrosymmetric distributions.
Electrons are jumping
between orbits.
e-
By the same logic as before,
the breakdown of
centrosymmetry in the
scattering atoms should be
reflected in a loss of
centrosymmetry in the
pattern of scattered x-ray
intensities.
A single heavy atom per protein can produce
a small but measurable difference between
FPH(hkl) and FPH(-h-k-l).
Differences between Ihkl and
I-h-k-l are small typically
between 1-3%.
Keep I(hkl) and I(-h-k-l) as
separate measurements. Don’t
average them together.
Example taken from a single
Hg site derivative of proteinase K
(28kDa protein)
h k
l Intensity
sigma
5 3 19 601.8 +/- 15.4
-5 -3 -19 654.8 +/- 15.7
Anomalous difference = 53
Anomalous signal is about 3 times
greater than sigma
In the complex plane, FP(hkl) and FP(-h-k-l) are
reflected across the real axis.
imaginary
FP(hkl)=FP(-h-k-l)
and
f(hkl)=-f(-h-k-l)
real
FP(h,k,l)
fhkl
f-h-k-l
FP(-h,-k,-l)
True for any crystal in the absence of anomalous scattering.
Normally, Ihkl and I-h-k-l are averaged together to improve redundancy
But not FPH(hkl) and FPH(-h-k-l)
imaginary
|FPH(hkl)|≠|FPH(-h-k-l)|
fhkl
f-h-k-l
FPH(h,k,l)
real
FPH(-h,-k,-l)
The heavy atom structure factor is not reflected across the real axis. Hence,
the sum of FH and FP=FPH is not reflected across the real axis. Hence, an
anomalous difference.
Hey! Look at that! We have two phase
triangles now; we only had one before.
FPH(hkl) =FP (hkl) +FH (hkl)
FPH(-h-k-l) =FP (-h-k-l) +FH (-h-kl-)
In isomorphous replacement
method, we get a single phase
triangle, which leaves an
either/or phase ambiguity.
Anomalous scattering
provides the opportunity of
constructing a second triangle
that will break the phase
ambiguity. We just have to be
sure to measure both.
imaginary
fhkl
f-h-k-l
real
|FPH(hkl)| and |FPH(-h-k-l)|
and...
we have to be able to calculate the effect of anomalous scattering on the values of FHhkl
and FH-h-k-l precisely given the heavy atom position. So far we just have a very faint idea
of what the effect of anomalous scattering is. The form factor f, is going to be different.
Q: How do we correct for
anomalous scattering effects
in our calculation of FH?
A: The correction to the atomic scattering
factor is derived from classical physics and
is based on an analogy of the atom to a
forced oscillator under resonance
conditions.
Examples of forced oscillation:
A tuning fork
vibrating when
exposed to
periodic force
of a sound
wave.
The housing of a
motor vibrating
due to periodic
impulses from an
irregularity in the
shaft.
A child on a
swing
The Tacoma Narrows bridge is an example of
an oscillator swaying under the influence of
gusts of wind.
Tacoma Narrows bridge, 1940
But, when the external force is matched the
natural frequency of the oscillator, the bridge
collapsed.
Tacoma Narrows bridge, 1940
An atom can also be viewed as a dipole
oscillator where the electron oscillates
around the nucleus.
The oscillator is
characterized by
+
e-
•mass=m
•position =x,y
•natural circular frequency=nB
•Characteristic of the atom
•Bohr frequency
•From Bohr’s
representation of the atom
nucleus
An incident photon’s electric field can
exert a force on the e-, affecting its
oscillation.
+
e-
+
E=hn
+
e-
What happens when the external force matches the natural frequency of
the oscillator (a.k.a resonance condition)?
In the case of an atom, resonance (n=nB)
leads to an electronic transition
(analogous to the condition hn=DE from quantum chemistry discussed earlier).
The amplitude of the oscillator
(electron) is given by classical physics:
nucleus
2
+
+
+
Incident
photon
with
n=nB
e+
v Eo
e
A=
2
2
2
mc vB  v  ikv
m=mass of oscillator
e=charge of the oscillator
c-=speed of light
Eo= max value of electric vector of incident photon
n=frequency of external force (photon)
nB=natural resonance frequency of oscillator
(electron)
Knowing the amplitude of the e- leads to a
definition of the scattering factor, f.
f=
Amplitude of scattered radiation from the forced eAmplitude of scattered radiation by a free e-
nucleus
The amplitude of the scattered radiation
is defined by the oscillating electron.
+
+
+
Incident
photon
with
n=nB
e-
The oscillating electron is the source of
the scattered electromagnetic wave
which will have the same frequency and
amplitude as the e-.
+
Scattered
photon
Keep in mind, the frequency and
amplitude of the e- is itself strongly
affected by the frequency and amplitude
of the incident photon as indicated on
the previous slide.
We find that the scattering factor is a complex
number, with value dependent on n.
f = fo+ Df’ + iDf”
n=frequency incident photon
nB=Bohr frequency of oscillator (e-)
(corresponding to electronic transition)
2
fo
Normal
scattering
factor
REAL
2
 v 
 vB 
Df ' = g   log e    1
 v 
 vB 
Df " = g
vB2
v2
correction
factor
correction
factor
REAL
IMAGINARY
Physical interpretation of the real and
imaginary correction factors of f.
f = fo+ Df’ + iDf”
real component, Df”
imaginary component, Df”
A small component of the
scattered radiation is 180° out of
phase with the normally scattered
radiation given by fo.
A small component of the
scattered radiation is 90°out of
phase with the normally
scattered radiation given by fo.
Always diminishes fo.
Absorption of x-rays
Bizarre! Any analogy to real
life?
90o phase shift analogy to a child on a swing
Forced Oscillator Analogy
Maximum negative
displacement
Zero force
Maximum positive
displacement
Zero force
Zero displacement
Maximum +/- force
Swing force is 90o out of phase with the displacement.
90o phase shift analogy to a child on a swing
Forced Oscillator Analogy
Maximum negative
displacement
Zero force
Maximum positive
displacement
Zero force
Zero displacement
Maximum +/- force
force : displacement
incident photon : re-emitted photon.
time->
Displacement of block, x, is 90° behind
force applied
Force
Displacement
1
0 (relaxed spring)
-1(max negative
displacement)
2
-1 (max compress)
0
3
0 (relaxed spring)
1(max positive
displacement)
4
1 (max expand)
0
1
0 (relaxed spring)
-1 (max negative
displacement)
-1 0 1
force : displacement
incident photon : re-emitted photon.
1
2
3
4
Construction of FH under conditions of
anomalous scattering
Imaginary axis
FH( H K L)
f
Real axis
FH=[fo + Df’(l) + iDf”(l)] e2i(hxH+kyH+lzH)
scattering factor for H
real
real
imaginary
Positive
number
180° out of phase
90° out of phase
Assume we have located
a heavy atom, H, by
Patterson methods.
Gives f
FH(-h-k-l) is constructed in a similar way as
FH(hkl) except f is negative.
Imaginary axis
Imaginary axis
FH( H K L)
Real axis
f
Real axis
FH(-H-K-L)
f
FH( H K L)=[fo + Df’(l) + iDf”(l)] e2i(+hxH+kyH+lzH)
FH(-H-K-L)=[fo + Df’(l) + iDf”(l)] e2i(-hxH-kyH-lzH)
Again, we see how Friedel’s Law is broken
Imaginary axis
Real axis
fH(+H+K+L)
f
fH(-H-K-L)
FH(H K L)
f
FH(-H-K-L)
fH(-h-k-l) ≠ -fH(-h-k-l)
Q: How can measurements of
|FPH(hkl)|, and |FPH(-h-k-l)| be
combined to solve the phase of
FP in a SIRAS experiment?
A: Analogous to MIR, using:
measured amplitude, |FP|
measured amplitudes |FPH(hkl)| and |FPH(-h-k-l)|
calculated amplitudes & phases of FH(HKL) , FH(-H-K-L),
two phasing triangles
FPH(hkl) = FP (hkl) + FH (hkl) and
FPH(-h-k-l) =FP (-h-k-l) + FH (-h-kl-)
and Friedel’s law.
Begin by graphing the measured amplitude of
FP for (HKL) and (-H-K-L).
Circles have equal radius by Friedel’s law
Imaginary axis
Imaginary axis
|FP(HKL) |
|FP(-H-K-L) |
Real axis
Real axis
Graph FH(hkl) and FH(-h-k-l) using coordinates of H.
Place vector tip at origin.
FH( H K L)=[fo + Df’(l) + iDf”(l)] e2i(hxH+kyH+lzH)
FH(-H-K-L)=[fo + Df’(l) + iDf”(l)] e2i(-hxH-kyH-lzH)
Structure factor amplitudes and phases calculated using equations derived earlier.
Graph measured amplitudes of FPH for
(H K L) and (-H-K-L).
Imaginary axis
Imaginary axis
|FP(HKL) |
|FPH(hkl)|
|FP(-H-K-L) |
|FPH(-h-k-l)|
Real axis
FPH(hkl) = FP (hkl) +FH (hkl)
Real axis
FPH(-h-k-l) =FP (-h-k-l) +FH (-h-kl-)
There are two possible choices for FP(HKL) and
two possible choices for FP(-H-K-L)
To combine phase information from the pair of reflections,
we take the complex conjugate of the –h-k-l reflection.
Imaginary axis
Imaginary axis
|FP(HKL) |
|FPH(hkl)|
|FP(-H-K-L) |
|FPH(-h-k-l)|
Real axis
Real axis
Complex conjugation means amplitudes stay
the same, but phase angles are negated.
FPH(hkl) = FP (hkl) +FH (hkl)
FPH(-h-k-l) =FP (-h-k-l) +FH (-h-kl-)
FPH(-h-k-l)* =FP (-h-k-l)* +FH (-h-kl-)*
Reflection across real axis.
Complex conjugation allows us
to equate FP (-h-k-l)* and FP (hkl) by Friedel’s law
and thus merge the two Harker constructions into one.
Imaginary axis
Imaginary axis
|FP(HKL) |
|FP(-H-K-L) |
Real axis
Real axis
FPH(-h-k-l)* = FP (-h-k-l)* + FH (-h-k-l)*
FPH(hkl) = FP (hkl) +FH (hkl)
Friedel’s law, FP (-h-k-l)*= FP (hkl).
FPH(-h-k-l)* = FP (hkl) + FH (-h-kl-)*
Phase ambiguity is resolved.
Imaginary axis
FP(HKL)
Real axis
Three phasing
circles intersect at
one point.
Now repeat process for 9999
other reflections
FPH(hkl) = FP(hkl) + FH(hkl)
FPH(-h-k-l)* = FP(hkl) + FH(-h-k-l)*
Q: How can measurements of
|FPH(l1)|, |FPH(l2)|, and |FPH(l3)| be
combined to solve the phase of
FP in a MAD experiment?
A: Again, analogous to MIR, using:
measured amplitudes |FPH(l1)|, |FPH(l2)|, and |FPH(l2)|
calculated amplitudes & phases of FH(l1), FH(l2), & FH(l3)
three phasing triangles
FPH(ll) = FP (ll) + FH (ll)
FPH(l2) = FP (l2) + FH (l2)
FPH(l3) = FP (l3) + FH (l3)
and Friedel’s law
but no measured amplitude, |FP|
Correction factors are largest near n=nB .
f = fo+ Df’ + iDf”
2
2
 v 
 vB 
Df ' = g   log e    1
 v 
 vB 
Df " = g
vB2
v2
when n>nB
Else, 0
n=frequency of external force (incident photon)
nB=natural frequency of oscillator (e-)
The REAL COMPONENT
becomes negative near v= nB.
n>
Df’
n=nB
The IMAGINARY COMPONENT
becomes large and positive near n= nB.
Df’
After dampening correction
n=nB
As the energy of the incident radiation approaches
the DE of an electronic transition (absorption edge),
Df’, varies strongly, becoming most negative at DE.
fo
Df’
Df’
Df’
fo
fo
Se
Df’ is the component of scattered radiation 180°
out of phase with the normally scattered
component fo
Df’
fo
v
Df ' = g  B
 v
DE
Df’
fo
2
2
 v 

 log e    1

 vB 
Similarly,
Df”, varies strongly near the absorption edge,
becoming most positive at energies > DE.
Df”
fo
Df”
fo
Df”
fo
Se
Df” is the component of scattered radiation 90°
out of phase with the normally scattered
component fo
Df”
fo
Df " = g
DE
Df”
fo
vB2
v2
when n>nB
Else, 0
Four wavelengths are commonly chosen
to give the largest differences in FH.
FH(low remote)
FH(inflection)
FH(peak)
FH(high remote)
f”
fo
f’
fo
f’
fo
f’
FH( l)=[fo + Df’(l) + iDf”(l)] e2i(hxH+kyH+lzH)
fo
f”
f’
The basis of a MAD experiment is that the amplitude and
phase shift of the scattered radiation depend strongly on
the energy (or wavelength, E=hc/l) of the incident radiation.
Imaginary axis
Imaginary axis
FH (l2)
FH (ll)
Imaginary axis
FH (l3)
FH( l1)=[fo + Df’(l) + iDf”(l)] e2i(hxH+kyH+lzH)
FH( l2)=[fo + Df’(l) + iDf”(l)] e2i(hxH+kyH+lzH)
FH( l3)=[fo + Df’(l) + iDf”(l)] e2i(hxH+kyH+lzH)
Hence, the amplitude and phase of FH varies with wavelength.
Same heavy atom coordinate, but 3 different structure factors depending on the wavelength.
FPH(l) amplitudes are graphed as circles centered at the
beginning of the FH(l) vectors (as in MAD & SIRAS).
Imaginary axis
Imaginary axis
Imaginary axis
FPH(ll) = FP (ll) + FH (ll)
FPH(l2) = FP (l2) + FH (l2)
FPH(l3) = FP (l3) + FH (l3)
No measurement available for FP, but it can be assumed that its value does not
change with wavelength because it contains no anomalous scatterers.
Hence, FP (ll) = FP (l2) = FP (l3) and all three circles intersect at FP.
A three wavelength MAD experiment solves
the phase ambiguity.
Imaginary axis
Real axis
FPH(ll) = FP + FH (ll)
FPH(l2) = FP + FH (l2)
FPH(l3) = FP + FH (l3)
Anomalous differences between reflections
hkl and –h-k-l could also be measured and
used to contribute additional phase circles.
In principle, one could acquire 2 phase
triangles for each wavelength used for data
collection. Let’s examine more closely how
FH changes with wavelength.
Good choice of l
Imaginary axis
Poor choice of l
Imaginary axis
Real axis
Point of intersection clearly defined.
Real axis
Point of intersection poorly defined.
Technological Advances Leading to the
Routine use of MAD phasing
Appearance of
synchrotron stations
capable of protein
crystallography.
Cryo protection to
preserve crystal
diffraction quality during
long 3-wavelength
experiment.
Production of
selenomethionyl
derivatives in ordinary
E.coli strains.
Fast, accurate data
collection software.
Anomalous electrons
Need to mention that length of correction
factors, f’ and f” are 10 at most, compared
to mercury at 80e.
Need perfect isomorphism to see signal.
Anomalous signal is smaller for lighter
elements compared to heavier elements.
Accuracy of measurement is extremely
important to a successful AS experiment.
The anomalous signal from a derivative is sufficient to phase if it
produces a 2-5% difference between Friedel related pairs.
Useful anomalous signals range from a minimum of f”=4e- (for
selenium (requires 1SeMet/100 residues bare minimum to yield a
sufficient signal for phasing) to a maximum of about f”=14e- for
Uranium).
In comparison with isomorphous differences, anomalous differences
are much smaller. For example, the maximal isomorphous difference
for a Hg atom is 80 e-, while its anomalous difference can be no bigger
than 10e-. But the measurement of the anomalous difference does not
suffer from nonisomorphism. Also, the anomalous scattering factors do
not diminish at high resolution as do the normal scattering factors.
Data collection must be highly redundant to improve the accuracy of
the measurements. Anomalous differences are small differences taken
between large measurements.
How to prepare a selenomethionine
derivative
Use minimal media for bacterial growth and
expression.
Use of a methionine auxotroph to express protein.
Supplement with selenomethionine.
OR use of an ordinary bacterial expression strain,
but supress methionine biosynthesis by the
addition of T,K,F,L,I,V. See Van Duyne et al., JMB
(1993), 229, 105-124.
$68 for 1 gram selenomethionine Acros organics.
Source of ideas &
information
•Concept of anomalous scattering
•R.W. James, The Optical Principles of Diffraction of X-rays. 1948.
•Ethan Merrit’s Anomalous scattering website
•http://www.bmsc.washington.edu/scatter/AS_index.html
•And references therein
•Sherwood, Crystals, X-rays and Proteins. 1976. Out of print
•Woolfson, X-ray Crystallography. 1970
•Halliday & Resnick Physics text book
•Todd Yeates
•Crystallographic concepts
•Stout & Jenesen X-ray structure determination
•Glusker, Lewis & Rossi, Crystal Structure Analysis for Chemists & Biologists
•Drenth, Principles of Protein X-Ray crystallography.
•Hendrickson, Science, 1991, vol 254, p51.
•Ramakrishnan & Biou, Methods in Enzymology vol 276, p538.
•Giacavazzo, Fundamentals of Crystallography.
•others
Brief review of MIR method.
Perspective
Reinforce important concepts for understanding MAD
Each point illustrated with a figure
A typical electron density map is plotted on a
3D grid containing of 1000s of grid points.
Each grid point has a value r(x,y,z)
Y
X
Z
Each value r(x,y,z) is the summation of
1000s of structure factors, Fhkl
r(x,y,z)=1/vSSSFhkle -2i(hx+ky+lz)
h k l
Y
X
Z
Each structure factor Fhkl specifies a cosine
wave with a certain amplitude and phase shift
r(x,y,z) =1/v {
|F0,0,1|e
-2i(0x+0y+1z-f001)
r
x
+
|F0,0,2|e -2i(0x+0y+2z-f002) +
|F0,0,3|e -2i(0x+0y+3z-f003) +
r
r
x
x
|F0,0,4|e -2i(0x+0y+4z-f004) +
|F0,0,5|e -2i(0x+0y+5z-f005) +…
|F50,50,50|e -2i(50x+50y+50z-f50 50 50)}
r
r
x
x
The value of the cosine waves at the point
x,y,z sum up to the value r(x,y,z)
r
x
r
r
Y
r
X
Z
r
x
x
x
x
The task of the crystallographer is to
amplitudes and phases of 1000s of Fhkl to
obtain the electron density map r(x,y,z)
r(x,y,z) =1/v {
r
r
r
Y
Z
X
|F0,0,1|e -2i(0x+0y+1z-f001) +
x
|F0,0,2|e -2i(0x+0y+2z-f002) +
x
x
r
x
r
x
|F0,0,3|e -2i(0x+0y+3z-f003) +
|F0,0,4|e -2i(0x+0y+4z-f004) +
|F0,0,5|e -2i(0x+0y+5z-f005) +…}
r(x,y,z)=1/vSSS|Fhkl|e -2i(hx+ky+lz-fhkl)
h k l
Remarkably, the Fhkl amplitudes and phases we
need are encoded in the radiation scattered by the
atoms in the crystal.
|Fh,k,l| is the square root of
the intensity of the
scattered radiation which
can be measured in a
standard diffraction expt.
Fhkl is the phase shift of
the scattered radiation It
cannot be measured
directly, leaving us with
the Phase Problem.
In solving the phase problem by MIR, it is important to know
that each Fhkl, is the sum of individual atomic structure
factors contributed by each atom in the crystal.
Fhkl=Sfje 2i(hxj+kyj+lzj) = fCae 2i(hxca+kyca+lzca) +
j
Here we show a crystal with a single amino acid
containing 12 atoms; In a protein crystal there
would be thousands of atoms;
fCbe 2i(hxcb+kycb+lzcb) +
fCge 2i(hxcg+kycg+lzcg) +
fCd1e 2i(hxcd1+kycd1+lzcd1) +
fCd2e 2i(hxcd2+kycd2+lzcd2) +
fCg1e 2i(hxcg1+kycg1+lzcg1) +
fCg2e 2i(hxcg2+kycg2+lzcg2) +
fCee 2i(hxce+kyce+lzce) +
fC e 2i(hxc+kyc+lzc) +
fOe 2i(hxo+kyo+lzo) +
fOTe 2i(hxot+kyot+lzot) +
fNe 2i(hxn+kyn+lzn)
fj is called the scattering factor and is proportional to the number of
electrons in the atom j.
Each atomic structure factor can be represented as
a vector in the complex plane with length fj and
phase angle e2i(hxj+kyj+lzj) ..
Fhkl=Sfje 2i(hxj+kyj+lzj) = fCae 2i(hxca+kyca+lzca) +
j
imaginary
real
Argand diagram
fCbe 2i(hxcb+kycb+lzcb) +
fCge 2i(hxcg+kycg+lzcg) +
fCd1e 2i(hxcd1+kycd1+lzcd1) +
fCd2e 2i(hxcd2+kycd2+lzcd2) +
fCg1e 2i(hxcg1+kycg1+lzcg1) +
fCg2e 2i(hxcg2+kycg2+lzcg2) +
fCee 2i(hxce+kyce+lzce) +
fC e 2i(hxc+kyc+lzc) +
fOe 2i(hxo+kyo+lzo) +
fOTe 2i(hxot+kyot+lzot) +
fNe 2i(hxn+kyn+lzn)
The resultant of the atomic vectors give the
amplitude and phase of Fhkl for the protein.
Fhkl=Sfje 2i(hxj+kyj+lzj) = fCae 2i(hxca+kyca+lzca) +
j
imaginary
Fhkl
Fhkl real
Argand diagram
fCbe 2i(hxcb+kycb+lzcb) +
fCge 2i(hxcg+kycg+lzcg) +
fCd1e 2i(hxcd1+kycd1+lzcd1) +
fCd2e 2i(hxcd2+kycd2+lzcd2) +
fCg1e 2i(hxcg1+kycg1+lzcg1) +
fCg2e 2i(hxcg2+kycg2+lzcg2) +
fCee 2i(hxce+kyce+lzce) +
fC e 2i(hxc+kyc+lzc) +
fOe 2i(hxo+kyo+lzo) +
fOTe 2i(hxot+kyot+lzot) +
fNe 2i(hxn+kyn+lzn)
MIR method
imaginary
FP
FPH
real
FH
By the same
reasoning, if a heavy
atom is added to a
protein crystal then the
structure factors of the
heavy atom derivative
FPH must equal the sum
of the component
vectors FP+FH.
FPH=FP+FH, forms
the basis for the
MIR method.
MIR method: FPH=FP+FH
Only the amplitude of FP can be
measured, not its phase. The
amplitude is represented by a
circle in the complex plane with
radius= |FP|
|FP|
imaginary
real
FH
Both the phase and amplitude
of FH can be plotted assuming
the heavy atom position
(xH,yH,zH) can be determined by
difference Patterson methods.
FH= fHe 2i(hxH+kyH+lzH)
•The amplitude of FPH can be
measured and is represented
by a circle in the complex
plane with radius= |FPH|.
|FPH|
•The circle is centered at the start of
the FH vector.
•So in effect FP=FPH-FH
There are two possible choices of phase
angle for FP that satisfy: FPH=FP+FH
imaginary
real
•The phasing ambiguity
can be resolved by
soaking in a different
heavy atom and
collecting a new data
set.
FPH2=FP+FH2
The phase ambiguity is resolved by
combining FPH1=FP+FH1 and
FPH2=FP+FH2
imaginary
•All three circle intersect
at only one point.
real
In practice, the phase ambiguity can be
resolved more easily by taking advantage of
anomalous scattering from PH1.
•Screening for a second derivative, PH2,costs
time, money, and nerves for
•expressing protein
•growing crystals
•Soaking heavy atom
•Collecting and analyzing data.
•Anomalous scattering from PH1 can be used
in combination with native data set (SIRAS) or
with other data sets from the same crystal
collected at different wavelengths (MAD).
How many crystallization plates
does it take to find a decent heavy
atom derivative?
•MAD is like “In situ MIR in which physics
rather than chemistry is used to effect the
change in scattering strength at the site”. Hendrickson, (1991).
•Two phasing circles can be drawn with each
new wavelength used for data collection FPH(l).