overview of the phase problem proteindatacrystalstructurephases remember we can measure reflection...

Overview of the Phase Problem

Protein DataCrystal StructurePhases

Remember We can measure reflection intensities

We can calculate structure factors from the intensitiesWe can calculate the structure factors from atomic positions

We need phase information to generate the image

x,y.z

X-ray Diffraction Experiment

All phase information is lost

Fhkl[Real Space] [Reciprocal Space]

What is the Phase Problem

In the X-ray diffraction experiment photons are reflected from the crystal lattice (planes) in different directions giving rise to the diffraction pattern.

Using a variety of detectors (film, image plates, CCD area detectors) we can estimate intensities but we loose any information about the relative phase for different reflections.

Let’s define a phase for an individual atom, j

An atom at xj=0.40, yj=0.25, zj=0.10 for plane [213]

For k = 0 (a 2D case) then

For plane [201]

Now to understand what this means….

)(2 jjjj zkyhx l++= πφ

j = 2π[ 2•(0.40) + 1•(0.25) + 3•(0.10)] = 2π

j = 2π[ 2•(0.40) + 1•(0.10)] = 2π

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φ j = 2π (hx j + l z j )

Phases

201 Phases

A

B

G

C

H

D

F

I

E

A

B

G

C

H

D

F

I

E

0°

720°

c0

a

201 planes

360°

1080°

0.4, y, 0.1

D = 2π[ 2•(0.40) + 1•(0.10)] = 2π

0 c

a

dhkl

dhkl

dhkl 6π

4π

2π

Atom (j) at x,y,z

φ

Plane hklRemember: We express any position in the cell as (1) fractional coordinates

pxyz = xja+yjb+zjc(2) the sum of integral multiples of the reciprocal axes

hkl = ha* + kb* + lc*

In General for Any Atom (x, y, z)

( )

2j j

hkl jhkl

proj pp

d

π= = ⋅

rr r

* * *hkl ha kb lc = + +rr r r

1hkl

hkld =

( * * *) ( )hkl j j j jp ha kb lc x a y b z c ⋅ = + + ⋅ + +r rr r r r r r

2j

j j jhx ky lzπ= + +

2 ( )j j j jhx ky lz π= + +

Phase for Any Atom

Why Do We Need the Phase?

Structure Factor

Fourier transform

Inverse Fourier transform

Electron Density

In order to reconstruct the molecular image (electron density) from its diffraction pattern both the intensity and phase, which can assume any value from 0 to 2π, of each of the thousands of measured reflections must be known.

Importance of Phases

Hauptman amplitudeswith Hauptman phases

Hauptman amplitudeswith Karle phases

Karle amplitudeswith Karle phases

Karle amplitudeswith Hauptman phases

Phases dominate the image!Phase estimates need to be accurate

Understanding the Phase ProblemThe phase problem can be best understood from a simple mathematical construct.

The structure factors (Fhkl) are treated in diffraction theory as complex quantities, i.e., they consist of a real part (Ahkl) and an imaginary part (Bhkl).

If the phases, hkl, were available, the values of Ahkl and Bhkl could be calculated from very simple trigonometry:

Ahkl = |Fhkl| cos (hkl)

Bhkl = |Fhkl| sin (hkl)

this leads to the relationship:

(Ahkl)2 + (Bhkl)2 = |Fhkl|2 = Ihkl

Argand Diagram

real

imaginary

Fhkl

hkl

Ahkl

Bhkl

. Figure An Argand diagram of structure factorFhkl with phase

hkl. (The realAhkl) and imaginary(Bhkl) .components are also shown

The above relationships are often illustrated using an Argand diagram (right).

From the Argand diagram, it is obvious that Ahkl and Bhkl may be either positive or negative, depending on the value of the phase angle, hkl.

Note: the units of Ahkl, Bhkl and Fhkl are in electrons.

(Ahkl)2 + (Bhkl)2 = |Fhkl|2 = Ihkl

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hkl = tan−1 Bhkl

A hkl

lll hkhkhk iBAF +=

sinΘ/λ

f0

Here fj is the atomic scattering factor

The scattering factor for each atomtype in the structure is evaluated at the correct sinΘ/λ. That value isthe scattering ability of that atom.

sin 1

2 hkldλΘ =Remember

We now have an atomic scattering vectorwith a magnitude f0 and direction φj .

The Structure Factor

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Fhkl = f je2πi(hx j +ky j +l z j )

j=1

N

∑Atomic scattering factors

)(2 jjjj zkyhx l++= πφ

∑∑==

++ ==N

j

ij

N

j

zkyhxijhk

jjjj efef11

)(2 φπ llF

real

imaginaryIndividualatom fjs

ResultantFhkl

Ahkl

Bhkl

The Structure FactorSum of all individual atom contributions

Electron DensityRemember the electron density (image of the molecule) is the Fourier transform of the structure factor Fhkl. Thus

€

ρx,y,z =1

VFhkle

−2πi[hx +ky +lz ]

hkl

∑ ⎧ ⎨ ⎩

⎫ ⎬ ⎭=

1

VFhkle

−iΦ

hkl

∑ ⎧ ⎨ ⎩

⎫ ⎬ ⎭

e−iΦ = cosΦ + isinΦ

Fhkl = Ahkl + iBhkl

ρ x,y,z =1

VAhkl cosΦ + Bhkl sinΦ

hkl

∑hkl

∑ ⎧ ⎨ ⎩

⎫ ⎬ ⎭

ρ x,y,z =1

VAhkl cos[2π (hx + ky + lz)] + Bhkl sin[2π (hx + ky + lz)]

hkl

∑hkl

∑ ⎧ ⎨ ⎩

⎫ ⎬ ⎭

Here V is the volume of the unit cell

In practice, the electron density for one three-dimensional unit cell is calculated by starting at x, y, z = 0, 0, 0 and stepping incrementally along each axis, summing the terms as shown in the equation above for all hkl (as limited by the resolution of the data) at each point in space.

Solving the Phase ProblemSmall molecules

Direct MethodsPatterson MethodsMolecular Replacement

MacromoleculesMultiple Isomorphous Replacement (MIR)Multi Wavelength Anomalous Dispersion (MAD)

Single Isomorphous Replacement (SIR)Single Wavelength Anomalous Scattering (SAS)

Molecular Replacement

Direct Methods (special cases)

MACROMOLECULES

Solving the Phase Problem

SMALL MOLECULES

The use of Direct Methods has essentially solved the phase problem for well diffracting small molecule crystals.

Today, anomalous scattering techniques such as MAD or SAS are the most common techniques used for de novo structure determination of macromolecules. Both techniques require the presence of one or more anomalous scatterers in the crystal.

SIR and SAS Methods1. Need a heavy atom (lots of electrons) or a anomalous

scatterer (large anomalous scattering signal) in the crystal.

• SIR - heavy atoms usually soaked in.• SAS - anomalous scatterers usually engineered in

as selenomethional labels. Can also be soaked.2. SIR collect a native and a derivative data set (2 sets

total). SAS collect one highly redundant data set and keep anomalous pairs separate during processing.

• SAS - may want to choose a scatterer or wavelength that enhances the anomalous signal.

3. Must find the heavy atoms or anomalous scatterers• can use Patterson analysis or direct methods.

4. Must resolve the bimodal ambiguity. • use solvent flattening or similar technique

Heavy Atom Derivatives

Table II. Protein Residues and Their Affinities for Heavy Metals

Residue: Affinity for: Conditions:

Histidine K2PtCl4, NaAuCl4, EtHgPO4H2 pH>6

Tryptophan Hg(OAc)2, EtHgPO4H2

Glutamic, Aspartic Acids UO2(NO3)2, rare earth cations pH>5

Cysteine Hg,Ir,Pt,Pd,Au cations ph>7

Methionine PtCl42- anion

Heavy atom derivatives are generally prepared by soaking crystals in dilute (2 - 20 mM) solutions of heavy atom salts (see Table II below for some examples).

Crystal cracking is generally a good indication that that heavy atom is interacting with the crystal lattice, and suggests that a good derivative can be obtained by soaking the crystal in a more dilute solution.

Once derivative data has been collected, the merging R factor (Rmerge) between the native and derivative data sets can be used to check for heavy atom incorporation and isomorphism. Rmerge values for isomorphous derivatives range from 0.05 to 0.15. Values below 0.05 indicate that there is little heavy atom incorporation. Values above 0.15 indicate a lack of isomorphism between the two crystals.

Heavy atom derivatives MUST be isomorphous

Finding the Heavy Atomsor Anomalous Scatterers

The Patterson function - a F2 Fourier transform with = 0 - vector map (u,v,w instead of x,y,z) - maps all inter-atomic vectors - get N2 vectors!! (where N= number of atoms)

The Difference Patterson MapSIR - |F|2 = |Fnat - Fder|2

SAS - |F|2 = |Fhkl - F-h-k-l|2

Patterson map is centrosymmetric - see peaks at u,v,w & -u, -v, -w

Peak height proportional to ZiZj

Peak u,v,w’s give heavy atom x,y,z’s - Harker analysis

Origin (0,0,0) maps vector of atom to itself

From Glusker, Lewis and Rossi

€

Puvw =1

V| Fhkl

hkl

∑ |2 cos2π (hu + kv + lv)

Harker AnalysisPatterson symmetry = Space group symmetry minus translations

Example Space group P21

P21 space group symmetry operators x,y,z -x,1/2+y,-z

x,y,z -x,1/2+y,-z

x,y,z [(x,y,z) - (x,y,z)] [(x,y,z) - (-x,1/2+y,-z)]-x,1/2+y,-z [(-x,1/2+y,-z) – (x,y,z)] [(-x,1/2+y,-z) – (-x,1/2+y,-z)]

x,y,z -x,1/2+y,-z

x,y,z 000 2x,-1/2, 2z-x,1/2+y,-z -2x, 1/2,-2z 000

Harker section v = 1/2 where to look for heavy atom vectors±2x, 1/2, ±2z

Automated programs SOLVE, SHELXD, BNP are available

We identify each reflection by an index, hkl.

The hkl also tells us the relative location of that reflection in a reciprocal space coordinate system.

The indexed reflection has correct handedness if a data processing program assigns it correctly.

The identity of the handedness of the molecule in the crystal is related to the assignment of handedness of the data, which may be right or wrong!

Note: not all data processing programs assign handedness correctly!

Be careful with your data processing.

A Note About Handedness

The Phase Triangle Relationship

O

N

M

L

OLM = OLN

Q

€

∠QLM +∠LON = π

€

∠LON = π −α H

FP, FPH, FH and -FH are vectors (have direction)FP <= obtained from native dataFPH <= obtained from derivative or anomalous dataFH <= obtained from Patterson analysis

FPH = FP + FH

Need value of FH


The Phase Triangle Relationship

O

N

M

LQ

In simplest terms, isomorphous replacement finds the orientation of the phase triangle from the orientation of one of its sides. It turns out, however, that there are two possible ways to orient the triangle if we fix the orientation of one of its sides.


Single Isomorphous Replacement

X1 trueor false

X2 trueor false


Note: FP = proteinFH = heavy atomFP1 = heavy atom derivative

The center of the FP1circle is placed at the end of the vector -FH1.

The situation of two possible SIR phases is called the “phase ambiguity” problem, since we obtain both a true and a false phase for each reflection. Both phase solutions are equally probable, i.e. the phase probability distribution is bimodal.

Resolving the Phase Ambugity

X1 trueor false

X2 trueor false


Note: FP = proteinFH = heavy atomFP1 = heavy atom derivative

The center of the FP1circle is placed at the end of the vector -FH1.

Add more information:

(1)Add another derivative (Multiple Isomorphous Replacement)

(2)Use a density modification technique (solvent flattening)

(3)Add anomalous data (SIR with anomalous scattering)

Multiple Isomorphous Replacement

X true

X false

X fals

Exact overlap at X1

dependent on data accuracy dependent on HA accuracy called lack of closure


Note: FP = proteinFH1 = heavy atom #1FH2 = heavy atom #2FP1 = heavy atom derivative FP2 = heavy atom derivative

The center of the FP1 and FP1 circlesare placed at the end of the vector -FH1

and -FH2, respectively.

We still get two solutions, one true and one false for each reflection from the second derivative. The true solutions should be consistent between the two derivatives while the false solution should show a random

variation.

Solvent FlatteningSimilar to noise filtering

Resolve the SIR or SAS phase ambiguity

B.C. Wang, 1985

Electron density can’t be negative Use an iterative process to enhance true phase!


How Does Solvent Flattening Resolve the Phase Ambiguity?

1. Solvent flattening can locate and enhance the protein image – e.g. whatever is not solvent must be protein!

2. From the protein image, the phases of the structure factors of the protein can be calculated

3. These calculated phases are then used to select the true phases from sets of true and false phases

4. Thus, in essence, the phase ambiguity is resolved by the protein image itself!

5.

The solvent flattening process was made practical by the introduction of the ISIR/ISAS program suite (Wang, 1985) and other phasing programs such DM and PHASES are based on this approach.

The ISAS process is carried twice, once with heavy atom site(s) at refined locations (+++), and once in their inverted locations (---).

Data FOM1 Handedness FOM2 R-Factor Corr. Coef RHE 0.54 Correct 0.82 0.26 0.958

0.54 Incorrect 0.80 0.30 0.940

NP With I3 0.54 Correct 0.80 0.27 0.955

0.54 Incorrect 0.76 0.36 0.919

NP With I & S4 0.56 Correct 0.82 0.24 0.964

0.56 Incorrect 0.78 0.35 0.926 1: Figure of merit before solvent flattening 2: Figure of merit after one filter and four cycles of solvent flattening 3: Four Iodine were used for phasing 4: Four Iodine and 56 Sulfur atoms were used for phasing Heavy Atom Handedness and Protein Structure Determination using S ingle-wavelength Anomalous Scattering Data, ACA Annual Meeting, Montreal, July 25, 1995.

Handedness Can be Determined by Solvent Flattening

Does the Correct Hand Make a Difference?

YES!

The wrong hand will give the mirror image!

Anomalous Dispersion MethodsAll elements display an anomalous dispersion (AD) effect in X-ray diffraction

For elements such as e.g. C,N,O, etc., AD effects are negligible

For heavier elements, especially when the X-ray wavelength approaches an atomic absorption edge of the element, these AD effects can be very large.

The scattering power of an atom exhibiting AD effects is:

fAD = fn + f' + if”

fnis the normal scattering power of the atom in absence of AD effectsf' arises from the AD effect and is a real factor (+/- signed) added to fn

f" is an imaginary term which also arises from the AD effect f" is always positive and 90° ahead of (fn + f') in phase angle

The values of f' and f" are highly dependent on the wave-length of the X-radiation.

In the absence AD effects, Ihkl = I-h-k-l (Firedel’s Law).

With AD effects, Ihkl ≠ I-h-k-l (Friedel’s Law breaks down).

Accurate measurement of Friedel pair differences can be used to extract starting phases if the AD effect is large enough.

Breakdown of Friedel’s Law

f”

Fn

Fhkl

real

F-h-k-lΔf”

F-n

F+++

F---

real

(Fhkl Left) Fn represents the total scattering by "normal" atoms without AD effects, f’ represents the sum of the normal and real AD scattering values (fn + f'), f" is the imaginary AD component and appears 90° (at a right angle) ahead of the f’ vector and the total scattering is the vector F+++.

(F-h-k-l Right) F-n is the inverse of Fn (at -hkl) and f’ is the inverse of f’, the f" vector is once again 90° ahead of f’. The resultant vector, F--- in this case, is obviously shorter

than the F+++ vector.

f’

f’

Collecting Anomalous Scattering DataAnomalous scatterers, such as selenium, are generally incorporated into the protein during expression of the protein or are soaked into the crystals in a manner similar to preparing a heavy atom derivative.

Bromine, iodine, xeon and traditional heavy atom compounds are also good anomalous scatterers.

The anomalous signal, the difference between |F+++| and |F---| is generally about one order of magnitude smaller than that between |FPH(hkl)|, and |FP(hkl)|.

Thus, the signal-to-noise (S/n) level in the data plays a critical role in the success of anomalous scattering experiments, i.e. the higher the S/n in the data the greater the probability of producing an interpretable electron density map.

The anomalous signal can be optimized by data collection at or near the absorption edge of the anomalous scatterer. This requires a tunable X-ray source such as a synchrotron.

The S/n of the data can also be increased by collecting redundant data.

The two common anomalous scattering experiments are Multiwavelength Anomalous Dispersion (MAD) and single wavelength anomalous scattering/dfiffraction (SAS or SAD)

The SAS technique is becoming more popular since it does not require a tunable X-ray source.

Increasing Number of SAS Structures

Increasing S/n with Redundancy

Multiwavelength Anomalous Dispersion


Note: FP = proteinFH1 = heavy atomF+

PH = F+++

F-PH = F---

F+H” = f”+++

F-H” = f”---

The center of the F+PH and F-

PH

circles are placed at the end of the vector -F+

H” and -F-H” respectively.

In the MAD experiment a strong anomalous scatterer is introduced into the crystal and data are recorded at several wavelengths (peak, inflection and remote) near the X-ray absorption edge of the anomalous scatterer. The phase ambiguity resolved a manner similar to the use of multiple derivatives in the MIR technique.

Single Wavelength Anomalous ScatteringThe SAS method, which combines the use of SAS data and solvent flattening to resolve phase ambiguity was first introduced in the ISAS program (Wang, 1985). The technique is very similar to resolving the phase ambiguity in SIR data.

The SAS method does not require a tunable source and successful structure determination can be carried out using a home X-ray source on crystals containing anomalous scatterers with sufficiently large f” such as iron, copper, iodine, xenon and many heavy atom salts.

The ultimate goal of the SAS method is the use of S-SAS to phase protein data since most proteins contain sulfur. However sulfur has a very weak anomalous scattering signal with f” = 0.56 e- for Cu X-rays.

The S-SAS method requires careful data collection and crystals that diffract to 2Å resolution.

A high symmetry space group (more internal symmetry equivalents) increases the chance of success.

The use of soft X-rays such as Cr K (λ = 2.2909Å) X-rays doubles the sulfur signal (f” = 1.14 e-).

There over 20 S-SAS structures in the Protein Data Bank.

Electron Density Maps of Rhe by Sulfur-ISAS

(Calculated using simulated data in 1983)

SAS Unresolved ISAS Filter1 Cycle 1 ISAS Filter 3 Cycle 8 Fcal

(Wang (1985), Methods Enzym, 115, 90-112)

What is the Limit of the SAS Method

f” = 0.56e- using Cu K X-rays

Molecular replacement has proven effective for solving macromolecular crystal structures based upon the knowledge of homologous structures.

The method is straightforward and reduces the time and effort required for structure determination because there is no need to prepare heavy atom derivatives and collect their data.

Model building is also simplified, since little or no chain tracing is required.

The 3-dimensional structure of the search model must be very close (< 1.7Å r.m.s.d.) to that of the unknown structure for the technique to work.

Sequence homology between the model and unknown protein is helpful but not strictly required. Success has been observed using search models having as low as 17% sequence similarity.

Several computer programs such as AmoRe, X-PLOR/CNS PHASER are available for MR calculations.

Molecular Replacement

Molecular ReplacementUse a model of the protein to estimate phases

Must be a structural homologue (RMSD < 1.7Å)Two step process

1. find orientation of model (red ==> black)2. find location of orientated model (black ==> blue)

px.cryst.bbk.ac.uk/03/sample/molrep.htm


Need to determine model’s orientation in X1s unit cell

Use a Patterson rotation search (

The coordinate system is rotated by an angle around the original z axis, then by an angle around the new y axis, and then by an angle around the final z axis.

zyz convention


Need to determine orientated model’s location in X1s unit cell

Use an R-factor search

Orientated model is stepped through the X1 unit cellusing small increments in x, y, and z (eg. x => x+ step)

Point where R is lowest represents the correct location

Other faster methods are available e.g. PHASER

overview of the phase problem proteindatacrystalstructurephases remember we can measure reflection...

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