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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
( , , ) ?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. For ~1.2Å (atomic resolution) data Sake & Bake Hauptman et al. Half baked Sheldrick et al. Acorn Woolfson et al.
2. For ~3Å SIR, OAS and MAD data OASIS Fan et al.
Direct methods in protein crystallography
Direct methods in protein crystallography
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 !
