Berlin Direct Methods

workshop Facets of Electron Crystallography
Berlin 7-9 july 2010
Direct Methods
Carmelo Giacovazzo
Istituto di Cristallografia, CNR,
Bari University, Italy
carmelo.giacovazzo@ic.cnr.it
Let us answer the following questions:
crystal structure   F ?
2
 F  2
crystal structure ?
N N
Fh   f j exp( 2ihrj ) f j exp( 2ihrj )
2
j 1 j 1
N
 f f
i , j 1
i j exp[ 2ih( ri  rj )]
N N
  f   f i f j exp[ 2 ih( ri  rj )]
j
2
j 1 i  j 1
 
As a 2consequence :
F   r 
 ?
O
A third question: structure
Xo O’
rj
rj’
Fh   f j exp 2ihrj    f j exp 2ih( X 0  rj' ) 

N N
j 1 j 1
 exp 2ihX 0  f j exp 2ihrj' 

N
j 1
 exp 2ihX 0 Fh'

 Fh'  Fh exp  2ihX 0   Fh exp i  h  2hX 0 
A fourth basic question
How can we derive the phases from the diffraction moduli ? This
seems contradictory: indeed
The phase values depend on the origin chosen by the user,

the moduli are independent of the user .
The moduli are structure invariants ,

the phases are not structure invariants.
Evidently, from the moduli we can derive information only on

those combinations of phases ( if they exist) which are structure
invariants.
The simplest invariant : the triplet invariant
Use the relation

F’h = Fh exp ( -2ihX0)
to check that the invariant Fh FkF-h-k does not depend on the origin.
Fh' Fk' F' h  k | Fh | exp i h  2hX 0  | Fk | exp i h  2kX 0 

| F h  k | exp i[  h  k  2 (h  k ) X 0 ]
| Fh || Fk || F h  k | exp i ( h   k    h  k )
The sum (h +  k+ -h-k ) is called triplet phase invariant
.
Structure invariants
Any invariant satisfies the condition that the sum of the

indices is zero:
doublet invariant : Fh F-h = | Fh|2
triplet invariant : Fh Fk F-h-k
quartet invariant :Fh Fk Fl F-h-k-l
quintet invariant : Fh Fk Fl Fm F-h-k-l-m

……………….
The prior information we can use for deriving the phase
estimates may be so summarised:
1) atomicity: the electron density is concentrated in atoms:
r2
r1
a2
a1
 r    aj r  rj 
N
j 1
2) positivity of the electron density:
( r ) > 0  f > 0
3) uniform distribution of the atoms in the unit cell.

The Wilson statistics
• Under the above conditions Wilson ( 1942,1949)
derived the structure factor statistics. The main results
where:
• | Fh |2   j 1 f j2
N (1)
• Eq.(1) is :
• a) resolution dependent (fj varies with θ ),
• b) temperature dependent: f j  f j0 exp( B j sin 2  / 2 )
• From eq.(1) the concept of normalized structure factor
arises:
Eh  Fh /( j 1 f j2 )1 / 2
N
The Wilson Statistics
• |E|-distributions:
P1 (| E |)  2 | E | exp(  | E |2 )
2
P1 (| E |)  exp( | E |2 / 2)

and
| E |2  1
in both the cases.

The statistics may be
used to evaluate the
average themel factor
and the absolute scale
factor.
The Wilson plot
N  sin 2 


Fh   f j exp 2ihr j   f j exp  B j 2  exp 2ihr j
0
j 1   
A  sin 2  
exp   B 2  f j0 exp 2ihr j
  
s2 Fh0
2
Fh obs  K Fh
2
 K Fh 0 2

exp  2 Bs 2 
A
 Fh obs  K  F 02
 exp  2 Bs 2   K0s exp  2 Bs 2 
2
h
 F 2 
ln
h obs 
 ln K  2 Bs 2
 0  x
 s 
y
The Cochran formula
h,k =h + k + -h-k = h + k - h+k
P(hk) [2 I0]-1exp(G cos hk)
where G = 2 | Eh Ek Eh+k |/N1/2
Accordingly:
h + k - h+k  0  G = 2 | Eh Ek Eh+k |/N1/2

h - k - h-k  0  G = 2 | Eh Ek Eh-k |/N1/2
h  k - h-k  G = 2 | Eh Ek Eh-k |/N1/2
The tangent formula
A reflection can enter into several triplets.Accordingly
h  k1 + h-k1 = 1 with P1(h)  G1 = 2| Eh Ek1 Eh-k1 |/N1/2
h  k2 + h-k2 = 2 with P2(h)  G2 = 2| Eh Ek2 Eh-k2 |/N1/2

……………………………………………………………………………………………………….
h  kn + h-kn = n with Pn(h)  Gn = 2| Eh Ekn Eh-kn |/N1/2

Then
P(h)  j Pj(h)  L-1 j exp [Gj cos (h - j )]
= L-1 exp [ cos (h - h )]

where
tan  h 
 G j sin  jT
 ,  h  T  B
2

2 1/ 2
G j cos  j B
A geometric interpretation of 
The random starting approach
To apply the tangent formula we need to know one
or more pairs ( k + h-k ). Where to find such an
information?
The most simple approach is the random starting

approach. Random phases are associated to a
chosen set of reflections. The tangent formula
should drive these phases to the correct values. The
procedure is cyclic ( up to convergence).
How to recognize the correct solution?

Figures of merit can or cannot be applied
Tangent cycles
• φ1 φ’1 φ’’1 ……………. φc1
• φ2 φ’2 φ’’2 ……………. φc2
• φ3 φ’3 φ’’3 …………….. φc3

• ……………………………………………………………………..
• φn φ’n φ’’n………………. φcn

•
• Ab initio phasing
• SIR2009 is able to solve
-small size structures (up to 80 atoms in the a.u.);
-medium-size structures ( up to 200);
-large size (no upper limit)
• It uses
• Patterson deconvolution techniques
• ( multiple implication transformations)
• as well as
• Direct methods
• to obtain a starting set of phases. They are extended and
refined via
• electron density modification techniques
•
• Direct methods limits for proteins:
• 1) the large size ( proteins range from300

atoms in the asymmetric unit to several
thousands). The G factor in the Cochran
formula are very small.
• 2) data resolution
• To overcome the limits one is obliged to :
• -increase the number of direct methods trials .
The cost to pay concerns the computing time.
• - improve and extend the the poor phases
available by DM by exploiting some specific
features of the proteins ( e.g., the solvent , etc. ).
About the data resolution limit
Atomic resolution at length was considered a necessary ( and
not sufficient ) condition for ab initio phasing ( Sheldrick
rule) , condition relaxed later on ( up to 1.2 Å). If it is not
satisfied:
• - the atomicity condition is violated;
• - the number of reliable triplet invariants exploitable by
the tangent procedure is small.
• - Patterson and EDM procedures are less effective;
• - the small ratio
• number of observations/ number of parameters
• make least squares unreliable.

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workshop Facets of Electron Crystallography

Berlin 7-9 july 2010

Fh   f j exp 2ihrj    f j exp 2ih( X 0  rj' ) 

 exp 2ihX 0  f j exp 2ihrj' 

 exp 2ihX 0 Fh'

The phase values depend on the origin chosen by the user,

The moduli are structure invariants ,

Evidently, from the moduli we can derive information only on

Use the relation

Fh' Fk' F' h  k | Fh | exp i h  2hX 0  | Fk | exp i h  2kX 0 

Any invariant satisfies the condition that the sum of the

doublet invariant : Fh F-h = | Fh|2

triplet invariant : Fh Fk F-h-k

quartet invariant :Fh Fk Fl F-h-k-l

quintet invariant : Fh Fk Fl Fm F-h-k-l-m

1) atomicity: the electron density is concentrated in atoms:

3) uniform distribution of the atoms in the unit cell.

in both the cases.

where G = 2 | Eh Ek Eh+k |/N1/2

h + k - h+k  0  G = 2 | Eh Ek Eh+k |/N1/2

h  k2 + h-k2 = 2 with P2(h)  G2 = 2| Eh Ek2 Eh-k2 |/N1/2

h  kn + h-kn = n with Pn(h)  Gn = 2| Eh Ekn Eh-kn |/N1/2

= L-1 exp [ cos (h - h )]

The most simple approach is the random starting

How to recognize the correct solution?

• φ2 φ’2 φ’’2 ……………. φc2

• φ3 φ’3 φ’’3 …………….. φc3

• φn φ’n φ’’n………………. φcn

• 1) the large size ( proteins range from300

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