Direct Methods 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

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 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

Structure Factors

Unitary Structure Factors

Normalized Structure Factors

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

Positivity: Atomicity:

Equal-atom structure:

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

The Probability distribution of three-phase structure invariants  Cochran distribution Cochran, W. (1955). Acta Cryst. 8,

Central limit theorem

The tangent formula  sin  =  h’  h, h’ sin (  h’ +  h  h’ )  cos  =  h’  h, h’ cos (  h’ +  h  h’ )

The tangent formula (continued)  sin  =  h’  h, h’ sin (  h’ +  h  h’ )  cos  =  h’  h, h’ cos (  h’ +  h  h’ ) tan  Maximizing P(  h )   h = 

Saytan

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)  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

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

The first example of solving an unknown protein by direct- method phasing of the 2.1 Å OAS data Rusticyanin, MW: 16.8 kDa; SG: P2 1 ; a=32.43, b=60.68, c=38.01Å ;  = o ; Anomalous scatterer: Cu Rusticyanin, MW: 16.8 kDa; SG: P2 1 ; a=32.43, b=60.68, c=38.01Å ;  = o ; Anomalous scatterer: Cu Mlphare + dm Oasis + dm OAS distribution Sim distribution Cochran distribution Solvent flattening OAS distribution Sim distribution Solvent flattening

OASIS (CCP4: Supported Program) DESCRIPTION OASIS is a computer program for breaking phase ambiguity in One-wavelength Anomalous Scattering or Single Isomorphous Replacement (Substitution) protein data. The phase problem is reduced to a sign problem once the anomalous-scatterer or the replacing-heavy-atom sites are located. OASIS applies a direct method procedure to break the phase ambiguity intrinsic to OAS or SIR data. REFERENCES Fan, H. F. and Gu, Y. X. (1985) Combining direct methods with isomorphous replacement or anomalous scattering data III. The incorporation of partial structure information, Acta Cryst. A41, Fan H. F., Hao, Q., Gu, Y. X., Qian, J. Z., Zheng, C. D. and Ke, H. (1990) Combining direct methods with isomorphous replacement or anomalous scattering data VII. Ab initio phasing of the OAS data from a small protein, Acta Cryst. A46, Y. -D. Liu, I. Harvey, Y. -X. Gu, C. -D. Zheng, Y. -Z. He, H. -F. Fan, S. S. Hasnain and Q. Hao (1999) Is single-wavelength anomalous scattering sufficient for solving phases? A comparison of different methods for a 2.1 A structure solution, Acta Cryst. D55, AUTHORS Q. Hao (1, 2), Y. X. Gu, C. D. Zheng & H. F. Fan (2) (1) School of Applied Sciences, De Montfort University, Leicester LE1 9BH, England. (2) Institute of Physics, Chinese Academy of Sciences, Beijing , P. R. China. or

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

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: 1380 Number of Se sites in a.s.u: 1 Wavelength (Å): Resolution: 30  3.0 Å Unique reflections: 4590 Space group: P Unit cell: a = 73.63, c =80.76Å Independent non-H atoms: 1380 Number of Se sites in a.s.u: 1 Wavelength (Å): Resolution: 30  3.0 Å Unique reflections: 4590

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

DMAD (2w) 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)

Direct phasing of the 2 Å SIR data of aPP with the replacing atoms (Hg) in centrosymmetric arrangement Space group: C2 Unit cell: a=34.18, b=32.92, c=28.22Å;  =105.3 o with the replacing atoms (Hg) in centrosymmetric arrangement Space group: C2 Unit cell: a=34.18, b=32.92, c=28.22Å;  =105.3 o SIR D-SIR

Recent progress on Direct-methods in Protein Crystallography is available in Direct-method_SAD_phasing.ppt Recent progress on Direct-methods in Protein Crystallography is available in Direct-method_SAD_phasing.ppt

Acknowledgments Y.X. Gu 1, Q, Hao 4, C.D. Zheng 1, Y.D. Liu 1, F. Jiang 1,2 & B.D. Sha 3 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) Acknowledgments Y.X. Gu 1, Q, Hao 4, C.D. Zheng 1, Y.D. Liu 1, F. Jiang 1,2 & B.D. Sha 3 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)

Thank you !