Phasing based on anomalous diffraction Zbigniew Dauter

Structure factor F P (h) =  j f j. exp (2  ih r j ) f j = f º j (  ). exp(-B. sin 2  / 2 )

Structure factor with heavy atoms F PH (h) =  j f j. exp (2  ihr j ) +  k f k. exp (2  ihr k ) F PH (h) = F P (h) + F H (h) |F PH | ≠ |F P | + |F H |

Anomalous diffraction normal scattering anomalous (resonant) scattering  

Structure factor and anomalous effect F(h) =  j f j. exp (2  ih r j ) f j = f º j (  ) + f’ j ( ) + i. f” j ( ) Anomalous correction f” is proportional to absorption and fluorescence and f’ is its derivative

-dependence of anomalous corrections for selenium

Structure factor and anomalous vectors F T (h) =  j f j. exp (2  ihr j ) +  k (f o k + f’ k + i. f” k ). exp (2  ihr k )

Structure factor and anomalous effect F T = F N + F A + F’ A + i. F” A Anomalous correction i. f” shifts the phase of atomic contribution in positive direction

Friedel pair – F(h) and F(-h) in effect |F T (h)| ≠ |F T (-h)|  T (h) ≠ -  T (-h) Anomalous correction f” causes the positive shift of phase of both F(h) and F(-h)

Friedel pair – F(h) and *F(-h) |F T (h)| ≠ |F T (-h)|  T (h) ≠ -  T (-h)

Friedel pair – more realistically f º (S) = 16 f”(S) = 0.56 for = 1.54 Å f º (Hg) = 82 f”(Hg) ≈ 4.5 for < 1.0 Å

Bijvoet difference  F ± = |F + | - |F - |

 F depends on phase difference  F ± = 0  F ± = 2 F” A

Sinusoidal dependence of  F and F A  F ± ≈ 2. F ” A. sin (  T -  A )

Partial structure of anomalous atoms Anomalous atoms can be located by Patterson or direct methods, since:  F ± ≈ 2. F ” A. sin (  T -  A ) |  F ± | 2 ≈ 4.  A 2. F A 2. sin 2 (  T -  A ) ≈ 2.  A 2. F A  A 2. F A 2. cos[2(  T –  A )] where  A = f” A /f o A sin 2  = ½ - ½. cos2  and anomalous atoms are mutually distant (even low resolution is “atomic”)

Two solutions for a single (SAD) If anomalous sites are known (  F ±, F A, F’ A, F” A,  A ) there are two possible phase solutions

Selection of mean phase |F SAD | = |F T |. cos[½ (  T 1 –  T 2 )]  SAD = ½ (  T 1 +  T 2 ) FOM = cos[ ½ (  T 1 –  T 2 )]

Electron density is then a superposition of correct structure and noise F 1 F 2 F 1 + F 2 iterative solvent flattening indicates correct phase

With errors in measured F + snd F - With measurement errors and inaccurate anomalous sites the phase indications are not sharp

Phase probability Each phase has then certain probability

Symmetric SAD probability The phase probability is symmetric around  SAD has two maxima

Two solutions not equivalent Solution with  T closer to  A is more probable (Sim contribution) Vectors F N have different lengths

Total SAD probability One solution is slightly more probable (depending how large is the substructure)

Excitation spectrum of Se (not fluorescence spectrum) inflection f’ = f” = 2.5 peak f’ = -4.0 f” = 8.0 remote f’ = -0.5 f” = 4.5

Typical MAD wavelengths inflection f’ = f” = 2.5 peak f’ = -4.0 f” = 8.0 remote f’ = -0.5 f” = 4.5

MAD 1-st phase

MAD 2-nd phase

MAD two phases

Contributors in analytical MAD approach

Analytical MAD approach (Karle & Hendrickson) F T (±) 2 = F T 2 + a( ). F A 2 + b( ). F T. F A. cos (  T -  A ) ± c( ). F T. F A. sin (  T -  A ) a( ) = (f’ 2 + f” 2 )/f o2 b( ) = 2. f’/f o c( ) = 2. f”/f o Three unknowns: F T, F A and (  T -  A ) - system can be solved, and F A used for finding anomalous sites  A (and  T ) can then be calculated

MAD treated as MIR Data from different can be treated as separate derivatives and one native and universal programs (SHARP, SOLVE etc) used for phasing - perfect isomorphism (one crystal) - synchrotron necessary (tunable ) - radiation damage (with long exposures)

Comparison MIR MAD SAD home lab or SR synchrotron home lab or SR several crystals one crystal, 2-3 data one data set non-isomorphism perfect isomorphism perfect isomorphism radiation damage ? radiation damage rad. dam. less acute tedious h.a. search easier easy, if works All methods easy thanks to excellent programs SHELXD/E, SnB, SHARP, SOLVE, CNS, PHENIX, CCP4 etc.

First SAD result crambin Hendrickson & Teeter, S among 46 amino acids =1.54 Å, f”(S)=0.56, / =1.4%

SAD vs. MAD Rice, Earnest & Brünger (2000) re-solved 7 SeMAD structures with SAD and recommended collecting first complete peak data set, and then other MAD wavelengths data, as a sort of insurance policy 1.5-wavelength approach (2002) collecting peak data and rapid phasing, if successful, postponement of next (now it may be < 1-wavelength)

David Blow in 2001 David Blow, Methods Enzymol. 374, 3-22 (2003) “How Bijvoet made the difference ?” (written probably in 2001)... The future of SAD It seems likely, however, that the various improvements to analyze MAD data more correctly are fading into insignificance. The MAD technique is losing ground to SAD....

SAD/(SAD+MAD) in PDB % 22% 32% 45% 55%

Proteinase K data 279 amino acids, 1 Ca + 10 S / = 0.44 % BeamlineSER-CAT 22-ID Unit-cell parameters ( Å ) a=67.55, c= Space groupP Wavelength ( Å ) 0.98 Distance (mm)150 Number of images660 Oscillation (°)/exposure time (s)0.5 / 2 Transmission10% Resolution ( Å ) ( ) Number of unique reflections63537 Completeness (%)96.4 (92.7) Overall I/σI106.1 (31.5) Redundancy27.1 (26.3) R merge (%)3.3 (13.0)

Proteinase K - SHELXD Anomalous difference Fourier RankPositionHeight 1Ca Cys Met Met Met Cys Met Cys Cys Met Cys Results of SHELXD

Proteinase K - SHELXE Experimental map after SHELXE Mean phase error 27.5 o

Indicators of anomalous signal - Bijvoet amplitude or intensity ratio - R anom -  2 difference if Friedels merged - list of outliers - measurability - anomalous signal to noise ratio - correlation between data sets - relation between signal in acentrics and centrics

Bijvoet ratio and R anom / = (2 N A /N P ) 1/2. (f A ” /6.7) R anom =  (F + - F - ) /  (F + + F - )/2 Four data sets from glucose isomerase 1 Mn in 375 a.a.

Proteinase K – data redundancy Dataset Label Peak Height (σ) Number of sites SHELXD Ca SO

Merging  2 difference and R merge crystal soaked in Ta 6 Br 12 cluster compound blue –  2 red - R merge when Friedels independent orange –  2 green - R merge when Friedels equivalent

List of outliers If redundancy if high enough, clearly shows anomalous differences

Signal-to-noise ratio (  F ± )/  (  F) for proteinase K requires proper estimation of  ’s (which is not trivial) signal is meaningful, if this ratio is > 1.3

Correlation between data sets corr (  F 1 ±,  F 2 ± ) F 1 and F 2 may be at different MAD or merged partial SAD data If higher than % - meaningful (advocated by George Sheldrick for SHELXD resolution cutoff)

No indicator is fully satisfactory These indicators of anomalous signal do not tell if the signal is sufficient for structure solution e.g. difficulties with Cu-thionein (Vito Calderone) 8 Cu in ~53 a.a. (12 Cys), P eventually solved from extremely redundant data

Perfect indicator only one satisfactory indicator of anomalous signal exists: successful structure solution nowadays the structure can be solved in few minutes, when the crystal is still at the beam line