1. By Fan Hai-fu, Institute of Physics, Beijing
Direct Methods
By Fan Hai-fu, Institute of Physics, Beijing
1. Introduction
2. Sayre’s equation and the tangent formula
3. Further developments in the 1990’s
4. Recent progress in solving proteins 1 2 3 4

2. The Phase Problem

3. Direct methods:
Deriving phases directly from the magnitudes

4. Why it is possible ?

5. D. Harker & J. Kasper
D. Sayre
1950’s J. Karle & H. Hauptman
I. L. Karle & J. Karle
1970’s M. M. Woolfson
Nobel Prize awarded to
H. Hauptman & J. Karle

6. Conditions for the Sayre Equation to be valid
Sayre’s Equation
Conditions for the Sayre Equation to be valid
1. Positivity
2. Atomicity
3. Equal-atom structure

7. Positivity:
Atomicity:

8. Equal-atom structure:

9. Cochran, W. & Woolfson, M. M. (1955). Acta Cryst. 8, 1-12.
Sign relationship ¾
an important outcome of the Sayre equation
Sh  Sh’ Sh - h’ or S-hSh’ Sh - h’  +1
Cochran, W. & Woolfson, M. M. (1955). Acta Cryst. 8, 1-12.

10. Cochran, W. (1955). Acta Cryst. 8, 473-478.
The Probability distribution of three-phase structure invariants ¾ Cochran distribution
Cochran, W. (1955). Acta Cryst. 8,

11. The tangent formula a sinb = å h’ k h, h’ sin (j h’ + j h- h’)
a cosb = å h’ k h, h’ cos (j h’ + j h- h’)

12. The tangent formula (continued)
Maximizing P(jh) Þ
jh=b
a sinb = å h’ k h, h’ sin (j h’ + j h- h’)
a cosb = å h’ k h, h’ cos (j h’ + j h- h’)
tanb

13. Direct methods in the 1990’s
IUCr Newsletters
Volume 4, Number 3, 1996
IUCr Congress Report (pp. 7-18)
(page 9)
The focus of the Microsym. Direct Methods of Phase Determination (2.03) ¼¼ 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

14. in protein crystallography
Direct methods
in protein crystallography
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.

15. Direct-method phasing of anomalous diffraction
1. Resolving OAS phase ambiguity
2. Improving MAD phases

16. OAS distribution Sim distribution Cochran distribution
Mlphare + dm
Oasis + dm
The first example of solving an unknown protein by direct-method phasing of the 2.1Å OAS data
OAS distribution Sim distribution Cochran distribution
Solvent flattening
OAS distribution Sim distribution
Rusticyanin, MW: 16.8 kDa; SG: P21; a=32.43, b=60.68, c=38.01Å ; b=107.82o ;
Anomalous scatterer: Cu

17. Direct-method phasing of anomalous diffraction
1. Resolving OAS phase ambiguity
2. Improving MAD phases

18. Direct-method aided MAD phasing Sample: yeast Hsp40 protein Sis1 (171-352)
Space group: P Unit cell: a = 73.63, c =80.76Å Independent non-H atoms: Number of Se sites in a.s.u: Wavelength (Å):
Resolution: 30 ~ 3.0 Å Unique reflections: 4590

19. Direct-method aided MAD phasing
(yeast Hsp40 protein Sis1: )
4w-MAD
2w-DMAD

20. Direct-method aided MAD phasing
(yeast Hsp40 protein Sis1: )
MAD (4w)
DMAD (2w)

21. Direct-method aided MAD phasing
(yeast Hsp40 protein Sis1: )
MAD (4w)
DMAD (2w)

22. Y.X. Gu1, Q, Hao4, C.D. Zheng1, Y.D. Liu1,
Acknowledgments
Y.X. Gu1, Q, Hao4, C.D. Zheng1, Y.D. Liu1,
F. Jiang1,2 & B.D. Sha3
1 Institute of Physics, CAS, Beijing, China
2 Tsinghua University, Beijing, China
3 University of Alabama at Birmingham, USA
4 Cornell University, USA
Project 973: G
(Department of Science & Technology, China)

23. Thank you !