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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?

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making sense of a mad experiment
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/

slide2

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?
slide3

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
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

normally electrons occupy the lowest energy orbitals ground state
Normally, electrons occupy the lowest energy orbitals –ground state.

Selenium atom

Ground state

slide7

hn=DE

Excites a transition

from the “K” shell

For Se, DE=12.65keV

=l= 0.9795 Å

But, incident radiation can excite an e- to an unoccupied outer orbital if the energy of the radiation (hn) matches the DE between orbitals.

Selenium atom

Electronic transition

slide8
Under these conditions when an electron can transit between orbitals, an atom will scatter photons anomalously.

Anomalously

Scattered

X-ray

-90° phase shift

diminished amplitude

Incident

X-ray

Selenium atom

Excited state

slide9

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

No transition possible,

Insufficient energy

usual case

Selenium atom

Ground state

d e is a function of the periodic table
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).

Importantly, DEs for C,N,O are out of the X-ray range.

Anomalous scattering from proteins and nucleic acids is negligible.

K shell transitions L shell transitions

For these elements, anomalous scattering is significant only at a synchrotron.

slide11

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.
slide12

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
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.

slide14

(-15,0,6)

· (0,0,0)

(15,0,-6)

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).

Pairs of reflections hkl and

–h-k-l are called Friedel pairs.

Friedel’s law is a consequence of an atom’s centrosymmetry.

I(hkl)=I(-h-k-l)

and

f(hkl)=-f(-h-k-l)

slide15

e-

But, under conditions of anomalous scattering, electrons are perturbed from their centrosymmetric distributions.

Electrons are jumping between orbits.

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.

slide16
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 f p hkl and f p h k l are reflected across the real axis

f-h-k-l

|FP-h,-k,-l|

FP(-h,-k,-l)

In the complex plane, FP(hkl) and FP(-h-k-l) are reflected across the real axis.

FP(h,k,l)

imaginary

|FPh,k,l|

FP(hkl)=FP(-h-k-l)

and

f(hkl)=-f(-h-k-l)

fhkl

real

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 f ph hkl and f ph h k l

|FPh,k,l|

fhkl

f-h-k-l

|FP-h,-k,-l|

But not FPH(hkl) and FPH(-h-k-l)

FPH(h,k,l)

imaginary

|FPH(hkl)|≠|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
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.

|FPH(hkl)|and |FPH(-h-k-l)| and...

imaginary

FPHh,k,l

|FPh,k,l|

fhkl

real

f-h-k-l

FPH-h,-k,-l

|FP-h,-k,-l|

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.

slide20

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
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

slide22
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
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
An atom can also be viewed as a dipole oscillator where the electron oscillates around the nucleus.
  • The oscillator is characterized by
    • mass=m
    • position =x,y
    • natural circular frequency=nB
          • Characteristic of the atom
          • Bohr frequency
          • From Bohr’s representation of the atom

e-

+

nucleus

an incident photon s electric field can exert a force on the e affecting its oscillation
An incident photon’s electric field can exert a force on the e-, affecting its oscillation.

e-

e-

+

+

E=hn

+

What happens when the external force matches the natural frequency of the oscillator (a.k.a resonance condition)?

slide26

nucleus

+

e-

+

+

+

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

+

+

+

+

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)

Incident

photon

with

n=nB

knowing the amplitude of the e leads to a definition of the scattering factor f

nucleus

f=

Amplitude of scattered radiation from the forced e- Amplitude of scattered radiation by a free e-

+

e-

+

+

+

Knowing the amplitude of the e- leads to a definition of the scattering factor, f.

The amplitude of the scattered radiation is defined by the oscillating electron.

The oscillating electron is the source of the scattered electromagnetic wave which will have the same frequency and amplitude as the e-.

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.

Incident

photon

with

n=nB

Scattered photon

we find that the scattering factor is a complex number with value dependent on n
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)

fo

correction

factor

IMAGINARY

correction

factor

REAL

Normal

scattering

factor

REAL

physical interpretation of the real and imaginary correction factors of f
Physical interpretation of the real and imaginary correction factors of f.

f = fo+ Df’ + iDf”

real component, Df”

A small component of the scattered radiation is 180° out of phase with the normally scattered radiation given by fo.

Always diminishes fo.

Absorption of x-rays

imaginary component, Df”

A small componentof thescattered radiation is 90°out of phase with the normally scattered radiation given by fo.

Bizarre! Any analogy to real life?

90 o phase shift analogy to a child on a swing forced oscillator analogy
90o phase shift analogy to a child on a swingForced Oscillator Analogy

Maximum negative displacement

Zero force

Maximum positive displacement

Zero force

Zero displacement

Maximum +/- force

Swing forceis 90o out of phase with the displacement.

90 o phase shift analogy to a child on a swing forced oscillator analogy31
90o phase shift analogy to a child on a swingForced Oscillator Analogy

Maximum negative displacement

Zero force

Maximum positive displacement

Zero force

Zero displacement

Maximum +/- force

time->

force : displacement

incident photon : re-emitted photon.

slide32

Displacement of block, x, is 90° behind

force applied

-1 0 1

force : displacement

incident photon : re-emitted photon.

2

4

1

3

construction of f h under conditions of anomalous scattering

f”

f’

Construction of FHunder conditions of anomalous scattering

Imaginary axis

FH( H K L)

f

fo

Real axis

FH=[fo + Df’(l) + iDf”(l)] e2pi(hxH+kyH+lzH)

scattering factor for H

Assume we have located a heavy atom, H, by Patterson methods.

Gives f

real

Positive

number

imaginary

90° out of phase

real

180° out of phase

fh h k l is constructed in a similar way as fh hkl except f is negative

f”

f”

f’

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)

FH(-H-K-L)

FH(-H-K-L)

f

fo

Real axis

Real axis

-f

fo

FH( H K L)=[fo + Df’(l) + iDf”(l)] e2pi(+hxH+kyH+lzH)

FH(-H-K-L)=[fo + Df’(l) + iDf”(l)] e2pi(-hxH-kyH-lzH)

again we see how friedel s law is broken

f”

f”

fo

f

f’

f’

-f

fo

fH(+H+K+L)

fH(-H-K-L)

Again, we see howFriedel’s Law is broken

Imaginary axis

FH(H K L)

Real axis

FH(-H-K-L)

fH(-h-k-l)≠ -fH(-h-k-l)

slide36
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.

slide37
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) |

FH(-H-K-L)

Real axis

Real axis

FH(HKL)

Graph FH(hkl)andFH(-h-k-l)using coordinates of H. Place vector tip at origin.

FH(-H-K-L)=[fo + Df’(l) + iDf”(l)] e2pi(-hxH-kyH-lzH)

FH( H K L)=[fo + Df’(l) + iDf”(l)] e2pi(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
Graph measured amplitudes of FPH for (H K L)and (-H-K-L).

Imaginary axis

Imaginary axis

|FP(HKL) |

|FP(-H-K-L) |

|FPH(-h-k-l)|

|FPH(hkl)|

FH(-H-K-L)

Real axis

Real axis

FH(HKL)

FPH(-h-k-l) =FP (-h-k-l) +FH (-h-kl-)

FPH(hkl) = FP (hkl) +FH (hkl)

There are two possible choices for FP(HKL) and two possible choices for FP(-H-K-L)

slide39

FPH(-h-k-l)* =FP (-h-k-l)* +FH (-h-kl-)*

Reflection across real axis.

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) |

|FP(-H-K-L) |

|FPH(-h-k-l)|

|FPH(hkl)|

FH(-H-K-L)

Real axis

Real axis

FH(HKL)

Complex conjugation means amplitudes stay the same, but phase angles are negated.

FPH(-h-k-l) =FP (-h-k-l) +FH (-h-kl-)

FPH(hkl) = FP (hkl) +FH (hkl)

slide40

Imaginary axis

|FP(-H-K-L) |

FH(-H-K-L)*

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

|FP(HKL) |

Real axis

FH(HKL)

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
Phase ambiguity is resolved.

Imaginary axis

FP(HKL)

Three phasing circles intersect at one point.

Now repeat process for 9999 other reflections

FH(-H-K-L)

Real axis

FH(HKL)

FPH(hkl) = FP(hkl) + FH(hkl)

FPH(-h-k-l)* =FP(hkl) + FH(-h-k-l)*

slide42

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 n b

After dampening correction

Correction factors are largest near n=nB .

f = fo+ Df’ + iDf”

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.

The IMAGINARY COMPONENT becomes large and positive near n= nB.

n->

Df’

Df’

n=nB

n=nB

slide44

Df’

Df’

Df’

Df’

Df’

fo

fo

fo

fo

fo

As the energy of the incident radiation approaches the DE of an electronic transition (absorption edge),Df’, varies strongly, becoming most negative at DE.

Se

Df’ is the component of scattered radiation 180° out of phase with the normally scattered component fo

DE

similarly d f varies strongly near the absorption edge becoming most positive at energies d e

Df”

Df”

Df”

Df”

Df”

fo

fo

fo

fo

fo

Similarly,Df”, varies strongly near the absorption edge, becoming most positive at energies > DE.

Se

when n>nB

Else, 0

Df” is the component of scattered radiation 90° out of phase with the normally scattered component fo

DE

four wavelengths are commonly chosen to give the largest differences in f h
Four wavelengths are commonly chosen to give the largest differences in FH.

FH(low remote)

FH(high remote)

FH(inflection)

FH(peak)

f”

f”

f’

fo

f’

fo

f’

f’

fo

fo

FH( l)=[fo + Df’(l) + iDf”(l)] e2pi(hxH+kyH+lzH)

slide47

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

Imaginary axis

FH (l3)

FH (l2)

FH (ll)

FH( l1)=[fo + Df’(l) + iDf”(l)] e2pi(hxH+kyH+lzH)

FH( l2)=[fo + Df’(l) + iDf”(l)] e2pi(hxH+kyH+lzH)

FH( l3)=[fo + Df’(l) + iDf”(l)] e2pi(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.

slide48
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
A three wavelength MAD experiment solves the phase ambiguity.

Imaginary axis

FPH(ll) = FP + FH (ll)

FPH(l2) = FP + FH (l2)

FPH(l3) = FP + FH (l3)

Real axis

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 poor choice of l
Good choice of l Poor choice of l

Imaginary axis

Imaginary axis

Real axis

Real axis

Point of intersection clearly defined.

Point of intersection poorly defined.

technological advances leading to the routine use of mad phasing
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
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
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
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. SeeVan Duyne et al., JMB (1993), 229, 105-124.
  • $68 for 1 gram selenomethionine Acros organics.
source of ideas information
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
Brief review of MIR method.

PerspectiveReinforce 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
A typical electron density map is plotted on a 3D grid containing of 1000s of grid points.
slide58

Y

X

Z

Each grid point has a value r(x,y,z)

each value r x y z is the summation of 1000s of structure factors f hkl

Y

X

Z

Each value r(x,y,z) is the summation of 1000s of structure factors, Fhkl

r(x,y,z)=1/vSSSFhkle -2pi(hx+ky+lz)

h k l

each structure factor f hkl specifies a cosine wave with a certain amplitude and phase shift
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 -2pi(0x+0y+1z-f001) +

|F0,0,2|e -2pi(0x+0y+2z-f002) +

|F0,0,3|e -2pi(0x+0y+3z-f003) +

|F0,0,4|e -2pi(0x+0y+4z-f004) +

|F0,0,5|e -2pi(0x+0y+5z-f005) +…

|F50,50,50|e -2pi(50x+50y+50z-f50 50 50)}

r

x

r

x

r

x

r

x

r

x

slide62

r(x,y,z)=1/vSSS|Fhkl|e -2pi(hx+ky+lz-fhkl)

h k l

The task of the crystallographer is to amplitudesandphases of 1000s of Fhklto obtain the electron density map r(x,y,z)

r(x,y,z) =1/v {

|F0,0,1|e -2pi(0x+0y+1z-f001) +

|F0,0,2|e -2pi(0x+0y+2z-f002) +

|F0,0,3|e -2pi(0x+0y+3z-f003) +

|F0,0,4|e -2pi(0x+0y+4z-f004) +

|F0,0,5|e -2pi(0x+0y+5z-f005) +…}

r

x

r

x

r

x

Y

r

x

r

X

x

Z

slide63
Remarkably, the Fhklamplitudesandphases we needare 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.

slide64

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 2pi(hxj+kyj+lzj)

= fCae 2pi(hxca+kyca+lzca) +

fCbe 2pi(hxcb+kycb+lzcb) +

fCge 2pi(hxcg+kycg+lzcg) +

fCd1e 2pi(hxcd1+kycd1+lzcd1) +

fCd2e 2pi(hxcd2+kycd2+lzcd2) +

fCg1e 2pi(hxcg1+kycg1+lzcg1) +

fCg2e 2pi(hxcg2+kycg2+lzcg2) +

fCee 2pi(hxce+kyce+lzce) +

fCe 2pi(hxc+kyc+lzc) +

fOe 2pi(hxo+kyo+lzo) +

fOTe 2pi(hxot+kyot+lzot) +

fNe 2pi(hxn+kyn+lzn)

j

Here we show a crystal with a single amino acid containing 12 atoms; In a protein crystal there would be thousands of atoms;

fj is called the scattering factor and is proportional to the number of electrons in the atom j.

slide65

Each atomic structure factor can be represented as a vector in the complex plane with length fj and phase angle e2pi(hxj+kyj+lzj) ..

Fhkl=Sfje 2pi(hxj+kyj+lzj)

j

= fCae 2pi(hxca+kyca+lzca)+

fCbe 2pi(hxcb+kycb+lzcb)+

fCge 2pi(hxcg+kycg+lzcg)+

fCd1e 2pi(hxcd1+kycd1+lzcd1)+

fCd2e 2pi(hxcd2+kycd2+lzcd2)+

fCg1e 2pi(hxcg1+kycg1+lzcg1)+

fCg2e 2pi(hxcg2+kycg2+lzcg2)+

fCee 2pi(hxce+kyce+lzce)+

fCe 2pi(hxc+kyc+lzc)+

fOe 2pi(hxo+kyo+lzo)+

fOTe 2pi(hxot+kyot+lzot)+

fNe2pi(hxn+kyn+lzn)

imaginary

real

Argand diagram

the resultant of the atomic vectors give the amplitude and phase of f hkl for the protein
The resultant of the atomic vectors give the amplitude and phase of Fhkl for the protein.

Fhkl=Sfje 2pi(hxj+kyj+lzj)

j

= fCae 2pi(hxca+kyca+lzca)+

fCbe 2pi(hxcb+kycb+lzcb)+

fCge 2pi(hxcg+kycg+lzcg)+

fCd1e 2pi(hxcd1+kycd1+lzcd1)+

fCd2e 2pi(hxcd2+kycd2+lzcd2)+

fCg1e 2pi(hxcg1+kycg1+lzcg1)+

fCg2e 2pi(hxcg2+kycg2+lzcg2)+

fCee 2pi(hxce+kyce+lzce)+

fCe 2pi(hxc+kyc+lzc)+

fOe 2pi(hxo+kyo+lzo)+

fOTe 2pi(hxot+kyot+lzot)+

fNe2pi(hxn+kyn+lzn)

imaginary

|Fh,k,l|

Fhkl

Fhkl

real

Argand diagram

mir method
MIR method

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.

imaginary

FP

FPH

real

FH

mir method f ph f p f h
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

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 2pi(hxH+kyH+lzH)

real

FH

  • The amplitude of FPH can be measured and is represented by a circle in the complex plane with radius= |FPH|.
  • The circle is centered at the start of the FH vector.
  • So in effectFP=FPH-FH

|FPH|

there are two possible choices of phase angle for f p that satisfy f ph f p f h
There are two possible choices of phase angle for FP that satisfy: FPH=FP+FH

imaginary

  • The phasing ambiguity can be resolved by soaking in a different heavy atom and collecting a new data set. FPH2=FP+FH2

real

the phase ambiguity is resolved by combining f ph 1 f p f h 1 and f ph 2 f p f h 2
The phase ambiguity is resolved by combining FPH1=FP+FH1andFPH2=FP+FH2

imaginary

  • All three circle intersect at only one point.

real

slide71
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).
  • 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).

How many crystallization plates does it take to find a decent heavy atom derivative?