By Fan Hai-fu, Institute of Physics, Beijing Direct Methods By Fan Hai-fu, Institute of Physics, Beijing http://cryst.iphy.ac.cn 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

The Phase Problem

Direct methods: Deriving phases directly from the magnitudes

Why it is possible ?

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

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

Positivity: Atomicity:

Equal-atom structure:

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.

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, 473-478.

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’)

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

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 http://cryst.iphy.ac.cn

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.

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

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

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

Direct-method aided MAD phasing Sample: yeast Hsp40 protein Sis1 (171-352) 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

Direct-method aided MAD phasing (yeast Hsp40 protein Sis1: 171-352) 4w-MAD 2w-DMAD

Direct-method aided MAD phasing (yeast Hsp40 protein Sis1: 171-352) MAD (4w) DMAD (2w)

Direct-method aided MAD phasing (yeast Hsp40 protein Sis1: 171-352) MAD (4w) DMAD (2w)

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: G1999075604 (Department of Science & Technology, China)

Thank you !