Making sense of a MAD experiment. Chem M230B Friday, February 3, 2006 12:00-12:50 PM Michael R. Sawaya

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/

• What is the anomalous scattering What is the anomalous scattering phenomenon?phenomenon?

• How is the anomalous scattering signal How is the anomalous scattering signal manifested?manifested?

• How do we account for anomalous scattering How do we account for anomalous scattering effects in its form factor feffects in its form factor fHH??

• How does anomalous scattering break the How does anomalous scattering break the phase ambiguity in a phase ambiguity in a – SIRAS experiment?SIRAS experiment?– MAD experiment?MAD experiment?

Topics CoveredTopics Covered

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 shell3= M shell4= 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 atomGround state

But, incident radiation can excite an e- to an unoccupied outer orbital if the energy of the

radiation (h) matches the E between orbitals.

hEExcites a transition from the “K” shell

For Se, E=12.65keV == 0.9795 Å

Selenium atomElectronic transition

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

anomalously.

Selenium atomExcited state

IncidentX-ray

AnomalouslyScattered

X-ray-90° phase shift

diminished amplitude

If the incident photon has energy different from the E between orbitals, then there is little anomalous

scattering (usual case). Scattered x-rays are not phase shifted.

Selenium atomGround state

hE

No transition possible, Insufficient energy

usual case

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

E is a function of the periodic table.

E 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 =1.54Å).

At a synchrotron, the energy of the incident radiation can be tuned to match E (accurately).

Importantly, Es 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

Choose an element with E 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 How is the anomalous scattering signal manifested in scattering signal manifested in a crystallographic diffraction a crystallographic diffraction

experiment?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).

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 hkl)=--h-k-l)

(15,0,-6)

(-15,0,6)

· (0,0,0)

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.

e-

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 singleHg 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.

real

imaginary

hkl

|FP h,k

,l|

FP(h,k,l)

-h-k-l

|FP-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

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

and hkl)=--h-k-l)

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

real

imaginary

-h-k-l

|FP-h,-k,-l |

hkl

|FP h,k

,l|

FPH(h,k,l)

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.

|FPH(hkl)|≠|FPH(-h-k-l)|

Hey! Look at that! We have two phase triangles now; we only had one before.

real

imaginary

-h-k-l

|FP-h,-k,-l |

hkl

|FP h,k

,l|

FPH h,

k,l

FPH-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.

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.

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

FFPHPH(hkl)(hkl) =F =FPP (hkl)(hkl) +F +FHH (hkl)(hkl)

FFPHPH(-h-k-l)(-h-k-l) =F =FPP (-h-k-l)(-h-k-l) +F +FHH (-h-kl-)(-h-kl-)

Q: How do we correct for How do we correct for anomalous scattering effects anomalous scattering effects

in our calculation of Fin our calculation of FHH??

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.

nucleus

e-+

The oscillator is characterized by

•mass=m•position =x,y•natural circular frequency=B

•Characteristic of the atom•Bohr frequency•From Bohr’s representation of the atom

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

oscillation.

e-e-+ E=h+

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 (B) leads to an electronic transition

(analogous to the condition hE from quantum chemistry discussed earlier).

The amplitude of the oscillator (electron) is given by classical physics:

m=mass of oscillatore=charge of the oscillatorc-=speed of lightEo= max value of electric vector of incident photon=frequency of external force (photon)B=natural resonance frequency of oscillator (electron)

+

nucleus

+

++

ikvvv

Ev

mc

eA

B

o

22

2

2+

nucleus

+

e-++

Incident photon

with

B

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

Scattered photon

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.

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

+

nucleus

+

e-++

Incident photon

with

B

We find that the scattering factor is a complex number, with value dependent on .

1log'22

Be

B

v

v

v

vgf

=frequency incident photonB=Bohr frequency of oscillator (e-) (corresponding to electronic transition)

f = fo+ f’ + if”

2

2

"v

vgf B

fo

Normal scattering

factorREAL

correctionfactor

REAL

correction factor

IMAGINARY

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

imaginary component, f”

A small component of the scattered radiation is 90°out of

phase with the normally scattered radiation given by fo.

Bizarre! Any analogy to real life?

f = fo+ f’ + if”

real component, f”

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

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

Maximum positive displacement

Zero force

Maximum negative displacement

Zero force

Zero displacementMaximum +/- force

Swing force is 90o out of phase with the displacement.

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

Maximum positive displacement

Zero force

Maximum negative displacement

Zero force

Zero displacementMaximum +/- force

time->force : displacementincident photon : re-emitted photon.

Displacement of block, x, is 90° behindforce applied

Force Displace-ment

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 : displacementincident photon : re-emitted photon.

1 2 3 4

Real axis

Imaginary axis

Construction of FH under conditions of anomalous scattering

FH=[fo + f’() + if”()] e2i(hxH

+kyH

+lzH

)

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

scattering factor for H

realPositivenumber

real180° out of phase

imaginary90° out of phase

fof’

f”FH( H K L)

Real axis

Imaginary axis

fo

f’

f”

FH(-H-K-L)

FH(-h-k-l) is constructed in a similar way as FH(hkl) except is negative.

Real axis

Imaginary axis

f’

f”FH( H K L)

FH( H K L)=[fo + f’() + if”()] e2i(+hxH

+kyH

+lzH

)

FH(-H-K-L)=[fo + f’() + if”()] e2i(-hxH

-kyH

-lzH

)

FH(-H-K-L)fo

Again, we see howFriedel’s Law is broken

Real axis

Imaginary axis

FH(-H-K-L)

fo

f’

f”

fo

f”

FH(H K L)

H(-H-K-L)

H(+H+K+L)

f’

H(-h-k-l) ≠ -H(-h-k-l)

Q: How can measurements of How can measurements of |FPH|FPH(hkl)(hkl)||,, and and |FPH|FPH(-h-k-l)(-h-k-l)|| be be

combined to solve the phase of combined to solve the phase of FPFP in a SIRAS experiment? 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

FPHFPH(hkl)(hkl) = FP = FP (hkl)(hkl) + FH + FH (hkl)(hkl) and

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

and Friedel’s law.

Imaginary axis

Begin by graphing the measured amplitude of FP for (HKL) and (-H-K-L).

Circles have equal radius by Friedel’s law

Real axis

FH(H

KL)

Imaginary axis

Real axis

|FP(-H-K-L) ||FP(HKL) |

FH( H K L)=[fo + f’() + if”()] e2i(hxH

+kyH

+lzH

) FH(-H-K-L)=[fo + f’() + if”()] e2i(-hx

H-ky

H-lz

H)

Structure factor amplitudes and phases calculated using equations derived earlier.

Graph FH(hkl) and FH(-h-k-l) using coordinates of H. Place

vector tip at origin.

FH(-H-K-L)

FH(-H-K-L)

Imaginary axis

Graph measured amplitudes of FPH for (H K L) and (-H-K-L).

Real axis

FH(H

KL)

Imaginary axis

Real axis

|FP(-H-K-L) ||FP(HKL) |

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

FPHFPH(hkl)(hkl) = FP = FP (hkl)(hkl) +FH +FH (hkl)(hkl) FPHFPH(-h-k-l)(-h-k-l) =FP =FP (-h-k-l)(-h-k-l) +FH +FH (-h-kl-)(-h-kl-)

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

FH(-H-K-L)

Imaginary axis

To combine phase information from the pair of reflections, we take the complex conjugate of the –h-k-l reflection.

Real axis

FH(H

KL)

Imaginary axis

Real axis

|FP(-H-K-L) ||FP(HKL) |

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

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

FPHFPH(hkl)(hkl) = FP = FP (hkl)(hkl) +FH +FH (hkl)(hkl) FPHFPH(-h-k-l)(-h-k-l) =FP =FP (-h-k-l)(-h-k-l) +FH +FH (-h-kl-)(-h-kl-)

FPHFPH(-h-k-l)(-h-k-l)* =FP * =FP (-h-k-l)(-h-k-l)* +FH * +FH (-h-kl-)(-h-kl-)**Reflection across real axis.

Imaginary axis

Complex conjugation allows us to equate FP FP (-h-k-l)(-h-k-l)* * and FP FP (hkl)(hkl) by Friedel’s law and

thus merge the two Harker constructions into one.

Real axis

FH(H

KL)

|FP(HKL) |

FH (-H-K-L)*

Imaginary axis

Real axis

|FP(-H-K-L) |

FPHFPH(hkl)(hkl) = FP = FP (hkl)(hkl) +FH +FH (hkl)(hkl)

FPHFPH(-h-k-l)(-h-k-l)* = FP * = FP (-h-k-l)(-h-k-l)* + FH * + FH (-h-k-l)(-h-k-l)**

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

Friedel’s law, FP FP (-h-k-l)(-h-k-l)**= FP FP (hkl)(hkl).

Imaginary axis

Phase ambiguity is resolved.

Real axis

FH(H

KL)

FH (-H-K-L)

FP(HKL)

Three phasing circles intersect at one point.

Now repeat process for 9999 other reflections

FPHFPH(hkl)(hkl) = FP = FP(hkl)(hkl) + FH + FH(hkl)(hkl)

FPHFPH(-h-k-l)(-h-k-l)* * == FPFP(hkl)(hkl) + FH + FH(-h-k-l)(-h-k-l)**

Q: How can measurements of How can measurements of |FPH|FPH((1)1)||,, |FPH|FPH((2)2)||, , and and |FPH|FPH((3)3)|| be be

combined to solve the phase of combined to solve the phase of FPFP in a MAD experiment? in a MAD experiment?

A: Again, analogous to MIR, using:measured amplitudes |FPH|FPH((1)1)||,, |FPH|FPH((2)2)||, , and |FPH|FPH((2)2)||calculated amplitudes & phases of FHFH((1)1),, FHFH((2)2), , & FHFH((3)3)

three phasing triangles

FPHFPH((l)l) = FP = FP ((l)l) + FH + FH ((l)l)

FPHFPH((2)2) = FP = FP ((2)2) + FH + FH ((2)2)

FPHFPH((3)3) = FP = FP ((3)3) + FH + FH ((3)3)

and Friedel’s law

but no measured amplitude, |FP|

Correction factors are largest near =B .

The IMAGINARY COMPONENT becomes large and positive near = B.

1log'22

Be

B

v

v

v

vgf

=frequency of external force (incident photon)B=natural frequency of oscillator (e-)

The REAL COMPONENT becomes negative near v= B.

f = fo+ f’ + if”

2

2

"v

vgf B

when >B

Else, 0

f’

B B

f’After dampening correction

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

1log'22

Be

B

v

v

v

vgf

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

f’

fo

f’

fo

f’

fo

f’

fo

f’

fo

E

SeSe

Similarly,f”, varies strongly near the absorption edge,

becoming most positive at energies > E.

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

fo

f” f”

fo

f”fo

f”

fo

f”

fo

E

SeSe2

2

"v

vgf B

when >B

Else, 0

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

f”

f’fo

FH(peak)FH(inflection)

FHFH( ( )=[fo + f’() + if”()] e2i(hxH

+kyH

+lzH

)

f”

f’fof’fof’fo

FH(high remote)FH(low remote)

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/) of the incident radiation.

Imaginary axis Imaginary axis Imaginary axis

FH FH ((l)l)

FH FH ((2)2) FH FH ((3)3)

FHFH( ( 11)=[fo + f’() + if”()] e2i(hxH

+kyH

+lzH

)

FHFH( ( 22)=[fo + f’() + if”()] e2i(hxH

+kyH

+lzH

)

FHFH((33)=[fo + f’() + if”()] e2i(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() amplitudes are graphed as circles centered at the beginning of the FH() vectors (as in MAD & SIRAS).

Imaginary axis Imaginary axis Imaginary axis

FPHFPH((l)l) = FP = FP ((l)l) + FH + FH ((l)l)

FPHFPH((2)2) = FP = FP ((2)2) + FH + FH ((2)2)

FPHFPH((3)3) = FP = FP ((3)3) + FH + FH ((3)3)

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 FP ((l)l) = = FP FP ((2)2) = = FP FP ((3)3) and all three circles intersect at FP.

Imaginary axis

A three wavelength MAD experiment solves the phase ambiguity.

Real axisFPHFPH((l)l) = = FPFP + FH + FH ((l)l)

FPHFPH((2)2) = = FPFP + FH + FH ((2)2)

FPHFPH((3)3) = = FPFP + FH + FH ((3)3)

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 Poor choice of

Imaginary axis

Real axis

Imaginary axis

Real axis

Point of intersection poorly defined.Point of intersection clearly defined.

Technological Advances Leading to the Technological Advances Leading to the Routine use of MAD phasingRoutine use of MAD phasing

Appearance of Appearance of synchrotron stations synchrotron stations capable of protein capable of protein crystallography.crystallography.Cryo protection to Cryo protection to preserve crystal preserve crystal diffraction quality during diffraction quality during long 3-wavelength long 3-wavelength experiment.experiment. Production of Production of selenomethionyl selenomethionyl derivatives in ordinary derivatives in ordinary E.coliE.coli strains. strains.Fast, accurate data Fast, accurate data collection software.collection software.

Anomalous electronsAnomalous electrons

Need to mention that length of correction Need to mention that length of correction factors, f’ and f” are 10 at most, compared factors, f’ and f” are 10 at most, compared to mercury at 80e. to mercury at 80e.

Need perfect isomorphism to see signal.Need perfect isomorphism to see signal.

Anomalous signal is smaller for lighter Anomalous signal is smaller for lighter elements compared to heavier elements.elements compared to heavier elements.

Accuracy of measurement is extremely Accuracy of measurement is extremely important to a successful AS experiment.important to a successful AS experiment.

The anomalous signal from a derivative is sufficient to phase if it The anomalous signal from a derivative is sufficient to phase if it produces a 2-5% difference between Friedel related pairs.produces a 2-5% difference between Friedel related pairs.

Useful anomalous signals range from a minimum of f”=4e- (for selenium Useful anomalous signals range from a minimum of f”=4e- (for selenium (requires 1SeMet/100 residues bare minimum to yield a sufficient signal (requires 1SeMet/100 residues bare minimum to yield a sufficient signal for phasing) to a maximum of about f”=14e- for Uranium). for phasing) to a maximum of about f”=14e- for Uranium).

In comparison with isomorphous differences, anomalous differences In comparison with isomorphous differences, anomalous differences are much smaller. For example, the maximal isomorphous difference are much smaller. For example, the maximal isomorphous difference for a Hg atom is 80 e-, while its anomalous difference can be no bigger 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 than 10e-. But the measurement of the anomalous difference does not suffer from nonisomorphism. Also, the anomalous scattering factors do suffer from nonisomorphism. Also, the anomalous scattering factors do not diminish at high resolution as do the normal scattering factors.not diminish at high resolution as do the normal scattering factors.

Data collection must be highly redundant to improve the accuracy of the Data collection must be highly redundant to improve the accuracy of the measurements. Anomalous differences are small differences taken measurements. Anomalous differences are small differences taken between large measurements.between large measurements.

How to prepare a selenomethionine How to prepare a selenomethionine derivativederivative

Use minimal media for bacterial growth and Use minimal media for bacterial growth and expression.expression.

Use of a methionine auxotroph to express protein. Use of a methionine auxotroph to express protein. Supplement with selenomethionine.Supplement with selenomethionine.

OR use of an ordinary bacterial expression strain, OR use of an ordinary bacterial expression strain, but supress methionine biosynthesis by the but supress methionine biosynthesis by the addition of T,K,F,L,I,V. addition of T,K,F,L,I,V. SeeSee Van Duyne et al., JMB Van Duyne et al., JMB (1993), 229, 105-124.(1993), 229, 105-124.

$68 for 1 gram selenomethionine Acros organics.$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.Brief review of MIR method.

PerspectivePerspectiveReinforce important concepts for understanding MADReinforce important concepts for understanding MADEach point illustrated with a figureEach point illustrated with a figure

A typical electron density map is plotted on a A typical electron density map is plotted on a

3D grid containing of 1000s of grid points. 3D grid containing of 1000s of grid points.

Y

XZ

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

Each value Each value (x,y,z)(x,y,z) is the summation of is the summation of 1000s of structure factors, F1000s of structure factors, Fhklhkl

Y

XZ

(x,y,z)=1/vFhkle -2i(hx+ky+lz)

h k l

Each structure factor FEach structure factor Fhklhkl specifies a cosine specifies a cosine

wave with a certain wave with a certain amplitudeamplitude and and phase shiftphase shift

(x,y,z1/v

|F0,0,1|e -2i(0x+0y+1z-001

) +

|F0,0,2|e -2i(0x+0y+2z-002

) +

|F0,0,3|e -2i(0x+0y+3z-003

) +

|F0,0,4|e -2i(0x+0y+4z-004

) +

|F0,0,5|e -2i(0x+0y+5z-005

) +…

|F50,50,50|e -2i(50x+50y+50z-50 50 50

)}

x

x

x

x

x

The value of the cosine waves at the point The value of the cosine waves at the point x,y,z sum up to the value x,y,z sum up to the value (x,y,z)(x,y,z)

Y

XZ

x

x

x

x

x

The task of the crystallographer is to The task of the crystallographer is to amplitudesamplitudes andand phases phases of 1000s of Fof 1000s of Fhklhkl to to

obtain the electron density map obtain the electron density map (x,y,z)(x,y,z)

Y

XZ

x

x

x

x

x

(x,y,z1/v

|F0,0,1|e -2i(0x+0y+1z-001

) +

|F0,0,2|e -2i(0x+0y+2z-002

) +

|F0,0,3|e -2i(0x+0y+3z-003

) +

|F0,0,4|e -2i(0x+0y+4z-004

) +

|F0,0,5|e -2i(0x+0y+5z-005

) +…}

(x,y,z)=1/v|Fhkl|e -2i(hx+ky+lz-hkl

)

h k l

Remarkably, the FRemarkably, the Fhklhkl amplitudesamplitudes andand phases phases we we

needneed are encoded in the radiation scattered by the are encoded in the radiation scattered by the atoms in the crystal.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. hkl 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 In solving the phase problem by MIR, it is important to know that each Fthat each Fhklhkl, is the sum of individual atomic structure , is the sum of individual atomic structure

factors contributed by each atom in the crystal.factors contributed by each atom in the crystal.

Fhkl=fje 2i(hxj+ky

j+lz

j)

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

would be thousands of atoms;

= fCe 2i(hxc

+kyc

+lzc

) +fCe 2i(hx

c+ky

c+lz

c) +

fCe 2i(hxc

+kyc

+lzc

) +fCe 2i(hx

c+ky

c+lz

c) +

fCe 2i(hxc

+kyc

+lzc

) +fCe 2i(hx

c+ky

c+lz

c) +

fCe 2i(hxc

+kyc

+lzc

) +fCe 2i(hx

c+ky

c+lz

c) +

fCe 2i(hxc+ky

c+lz

c) +

fOe 2i(hxo+ky

o+lz

o) +

fOTe 2i(hxot

+kyot

+lzot

) +fNe 2i(hx

n+ky

n+lz

n)

fj is called the scattering factor and is proportional to the number of

electrons in the atom j.

j

Each atomic structure factor can be represented as Each atomic structure factor can be represented as a vector in the complex plane with length a vector in the complex plane with length ffjj and and

phase angle phase angle ee22i(hxi(hxjj+ky+ky

jj+lz+lz

jj)) . ...

Fhkl=fje 2i(hxj+ky

j+lz

j)

j

= fCe 2i(hxc

+kyc

+lzc

) +fCe 2i(hx

c+ky

c+lz

c) +

fCe 2i(hxc

+kyc

+lzc

) +fCe 2i(hx

c+ky

c+lz

c) +

fCe 2i(hxc

+kyc

+lzc

) +fCe 2i(hx

c+ky

c+lz

c) +

fCe 2i(hxc

+kyc

+lzc

) +fCe 2i(hx

c+ky

c+lz

c) +

fCe 2i(hxc+ky

c+lz

c) +

fOe 2i(hxo+ky

o+lz

o) +

fOTe 2i(hxot

+kyot

+lzot

) +fNe 2i(hx

n+ky

n+lz

n)

real

imaginary

Argand diagram

The resultant of the atomic vectors give the The resultant of the atomic vectors give the amplitude and phase of Famplitude and phase of Fhklhkl for the protein. for the protein.

Fhkl=fje 2i(hxj+ky

j+lz

j)

j

= fCe 2i(hxc

+kyc

+lzc

) +fCe 2i(hx

c+ky

c+lz

c) +

fCe 2i(hxc

+kyc

+lzc

) +fCe 2i(hx

c+ky

c+lz

c) +

fCe 2i(hxc

+kyc

+lzc

) +fCe 2i(hx

c+ky

c+lz

c) +

fCe 2i(hxc

+kyc

+lzc

) +fCe 2i(hx

c+ky

c+lz

c) +

fCe 2i(hxc+ky

c+lz

c) +

fOe 2i(hxo+ky

o+lz

o) +

fOTe 2i(hxot

+kyot

+lzot

) +fNe 2i(hx

n+ky

n+lz

n)

real

imaginary

Argand diagram

Fhkl

hkl

|F h,k,l|

MIR methodMIR method

By the same reasoning, By the same reasoning, if a heavy atom is if a heavy atom is added to a protein added to a protein crystal then the crystal then the structure factors of the structure factors of the heavy atom derivative heavy atom derivative FFPHPH must equal the sum must equal the sum

of the component of the component vectors vectors FFPP++FFHH..

FFPHPH==FFPP++FFHH, forms , forms

the basis for the the basis for the MIR method.MIR method.

real

imaginary

FFHH

FFPPFFPHPH

MIR method:MIR method: F FPHPH==FFPP++FFHH

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|

real

imaginary

FFHH

|F|FPP||

|F|FPHPH||

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 2i(hx

H+ky

H+lz

H)

•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 effect FP=FPH-FH

There are two possible choices of phase There are two possible choices of phase angle for angle for FFPP that satisfy: that satisfy: F FPHPH==FFPP++FFHH

•The phasing ambiguity can be resolved by soaking in a different heavy atom and collecting a new data

set. FPH2=FP+FH2

real

imaginary

The phase ambiguity is resolved by The phase ambiguity is resolved by combining combining FFPHPH11

==FFPP++FFHH11 andand

FFPHPH22==FFPP++FFHH22

•All three circle intersect at only one point.

real

imaginary

In practice, the phase ambiguity can be In practice, the phase ambiguity can be resolved more easily by taking advantage of resolved more easily by taking advantage of

anomalous scattering from anomalous scattering from PH1PH1..

•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().

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

atom derivative?