Direct MethodsBy Fan Hai-fu, Institute of Physics, Beijing

http://cryst.iphy.ac.cn

Direct MethodsBy Fan Hai-fu, Institute of Physics, Beijing

http://cryst.iphy.ac.cn

1. Introduction2. Sayre’s equation and the tangent formula3. Further developments in the 1990’s4. Recent progress in solving proteins

11

22

33

44

11

( , , ) ?h k l

( , , )( , , ) ( , , ) i h k lF h k l F h k l e

2 ( )1( , , ) ( , , ) i hx ky lz

h k l

x y z F h k lV

e 2 ( )1( , , ) ( , , ) i hx ky lz

h k l

x y z F h k lV

e

The Phase ProblemThe Phase Problem

( , , ) ( , , )F h k l a h k l

Direct methods:Direct methods:

Deriving phases directly from the magnitudesDeriving phases directly from the magnitudes

Why it is possible ?Why it is possible ?

2 ( )

1( , , )

N i h k lyx z jj jj

jF h k l f e

1

1

( , , ) cos ( , , ) cos 2 ( )

( , , ) sin ( , , ) sin 2 ( )

N

jj

N

jj

F h k l h k l f h k lyx z jj j

F h k l h k l f h k lyx z jj j

1947 D. Harker & J. Kasper

1952 D. Sayre

1950’s J. Karle & H. Hauptman

1964 I. L. Karle & J. Karle

1970’s M. M. Woolfson

1985 Nobel Prize awarded to H. Hauptman & J. Karle

1947 D. Harker & J. Kasper

1952 D. Sayre

1950’s J. Karle & H. Hauptman

1964 I. L. Karle & J. Karle

1970’s M. M. Woolfson

1985 Nobel Prize awarded to H. Hauptman & J. Karle

Sayre’s Equation

Conditions for the Sayre Equation to be valid

1. Positivity2. Atomicity

3. Equal-atom structure

Positivity:

' ''

1

resembles ( )

1

exp( 2 )

sq

N

j jj

F F FV

F f i

h h h hh

h

r r r r

h r

Atomicity:

1

exp( 2 )N

sq sqj j

j

F f i

h h r

1

1

exp( 2 )

exp( 2 )

N

j jj

Nsq sq

j jj

F f i

F f i

h

h

h r

h r

Equal-atom structure:

' ''

' ''

1 sq sq sq

sq

F F f f F F FV

fF F F

f V

h h h h h h h hh

h h h hh

Sh Sh’ Sh h’ or ShSh’ Sh h’ +1

' ''

sq

fF F F

f V h h h hh

3, ' ' '3/ 2

2

1 1( ) tanh

2 2P s E E E

h h h h h h

Sign relationship an important outcome of the Sayre equationSign relationship an important outcome of the Sayre equation

Cochran, W. & Woolfson, M. M. (1955). Acta Cryst. 8, 1-12.

1

3 0 , ' , ' 3( ) 2 ( ) exp cosP I

h h h h

The Probability distribution of three-phase structure invariants Cochran distribution

The Probability distribution of three-phase structure invariants Cochran distribution

3 ' ' 0 modulo 2h h h h

3/ 2, ' 3 2 ' '

1/ 2, ' ' '

2

2

E E E

N E E E

h h h h h h

h h h h h h

Cochran, W. (1955). Acta Cryst. 8, 473-478.

The tangent formulaThe tangent formula

3

' 3 , ' 3' '

( ) ( ) exp cosP P N

h h' h h'

h h h hh h

sin = h’ h, h’ sin (h’ +hh’)

cos = h’ h, h’ cos (h’ +hh’)

1

0( ) 2 ( ) exp[ cos( )]P I h h

1

3 0 , ' , ' 3( ) 2 ( ) exp cosP I

h h h h

The tangent formula (continued)The tangent formula (continued)

, ' ' ''

, ' ' ''

sin( )tan

cos( )

h h h h hh

hh h h h h

h

1/ 22 2

, ' ' ' , ' ' '' '

sin( ) cos( )

h h h h h h h h h hh h

1

0( ) 2 ( ) exp[ cos( )]P I h h

sin = h’ h, h’ sin (h’ +hh’)

cos = h’ h, h’ cos (h’ +hh’)

tan

Maximizing P(h) h=

IUCr NewslettersVolume 4, Number 3, 1996IUCr Congress Report (pp. 7-18)(page 9) The focus of the Microsym. Direct Methods of Phase Determination (2.03) was the transition of direct methods application to problems outside of their traditional areas from small to large molecules,single to powder crystals, periodic to incommensurate structures, and from X-ray to electron diffraction data. . . . . . Suzanne Fortier

Direct methods in the 1990’s

http://cryst.iphy.ac.cnhttp://cryst.iphy.ac.cn

1. Locating heavy-atoms

2. Ab initio phasing with ~1.2Å (atomic resolution) data Sake & Bake · · · · · · · · · · Hauptman et al. Half baked · · · · · · · · · · · · Sheldrick et al. Acorn · · · · · · · · · · · · · · · · Woolfson et al.

3. SIR and SAD phasing with ~3Å data and MR model completion OASIS · · · · · · · · · · · · · · · · Fan et al.

Direct methods in protein crystallography

Direct methods in protein crystallography

Direct-method phasing of the 2Å experimental SAD data of the protein aPP

Direct-method phasing of the 2Å experimental SAD data of the protein aPP

Avian Pancreatic Polypeptide

Space group: C2 Unit cell: a = 34.18, b = 32.92, c = 28.44Å; = 105.3o

Protein atoms in ASU: 301Resolution limit: 2.0ÅAnomalous scatterer: Hg, Zn (In centric arrangement)Wavelength: 1.542Å (Cu-K) f” = 7.686, 0.678Phasing: direct methods

Acta Cryst. (1990). A46, 935.

Avian Pancreatic Polypeptide

Space group: C2 Unit cell: a = 34.18, b = 32.92, c = 28.44Å; = 105.3o

Protein atoms in ASU: 301Resolution limit: 2.0ÅAnomalous scatterer: Hg, Zn (In centric arrangement)Wavelength: 1.542Å (Cu-K) f” = 7.686, 0.678Phasing: direct methods

Acta Cryst. (1990). A46, 935.

1. Resolving OAS phase ambiguity2. Improving MAD phases1. Resolving OAS phase ambiguity2. Improving MAD phases

Direct-method phasing ofanomalous diffraction

Direct-method phasing ofanomalous diffraction

The first example of solving an unknown protein by direct-method phasing of the 2.1Å OAS data

The first example of solving an unknown protein by direct-method phasing of the 2.1Å OAS data

Rusticyanin, MW: 16.8 kDa; SG: P21; a=32.43, b=60.68, c=38.01Å ; =107.82o ;Anomalous scatterer: Cu

Rusticyanin, MW: 16.8 kDa; SG: P21; a=32.43, b=60.68, c=38.01Å ; =107.82o ;Anomalous scatterer: Cu

Mlphare + dm

Oasis + dmOAS distribution Sim distribution Cochran distribution

Solvent flattening

OAS distribution Sim distribution

Solvent flattening

1. Resolving OAS phase ambiguity2. Improving MAD phases1. Resolving OAS phase ambiguity2. Improving MAD phases

Direct-method phasing ofanomalous diffraction

Direct-method phasing ofanomalous diffraction

Direct-method aided MAD phasing Sample: yeast Hsp40 protein Sis1 (171352)

Direct-method aided MAD phasing Sample: yeast Hsp40 protein Sis1 (171352)

Space group: P41212

Unit cell: a = 73.63, c =80.76Å Independent non-H atoms: 1380 Number of Se sites in a.s.u: 1 Wavelength (Å):

1.0688 0.9794 0.9798 0.9253 Resolution: 30 3.0 Å Unique reflections: 4590

Space group: P41212

Unit cell: a = 73.63, c =80.76Å Independent non-H atoms: 1380 Number of Se sites in a.s.u: 1 Wavelength (Å):

1.0688 0.9794 0.9798 0.9253 Resolution: 30 3.0 Å Unique reflections: 4590

2w-DMAD2w-DMAD4w-MAD4w-MAD

Direct-method aided MAD phasing(yeast Hsp40 protein Sis1: 171352)

DMAD (2w)DMAD (2w)

Direct-method aided MAD phasing(yeast Hsp40 protein Sis1: 171352)

MAD (4w)MAD (4w)

DMAD (2w)DMAD (2w)

Direct-method aided MAD phasing(yeast Hsp40 protein Sis1: 171352)

MAD (4w)MAD (4w)

AcknowledgmentsY.X. Gu1, Q, Hao4, C.D. Zheng1, Y.D. Liu1,

F. Jiang1,2 & B.D. Sha3

1 Institute of Physics, CAS, Beijing, China2 Tsinghua University, Beijing, China

3 University of Alabama at Birmingham, USA4 Cornell University, USA

Project 973: G1999075604(Department of Science & Technology, China)

AcknowledgmentsY.X. Gu1, Q, Hao4, C.D. Zheng1, Y.D. Liu1,

F. Jiang1,2 & B.D. Sha3

1 Institute of Physics, CAS, Beijing, China2 Tsinghua University, Beijing, China

3 University of Alabama at Birmingham, USA4 Cornell University, USA

Project 973: G1999075604(Department of Science & Technology, China)

Thank you !Thank you !