Breaking the phase ambiguity in protein crystallography

        The philosophy is that, the phase ambiguity is first converted to a sign ambiguity, the latter is then resolved by a direct method. This was proposed in the 1960's [Fan Hai-fu, Acta Phys. Sin. (1965) 21, 1114-1118 (in Chinese). See also Fan, H.F., Chinese Physics (1965) 1429-1435 ]. Further development has been carried out from 1983 till now.

        In SIR case, for a given reciprocal-lattice vector h, we have the structure-factor relation

     .     (1)

Here the subscript N, D and R denotes respectively the native protein, the derivative and the replacing-atom substructure. From experiment we know the magnitude of F(h,D) and F(h,N). Hence we can find the position of the replacing atoms and calculate the vector F(h,R). Then we have two different ways to draw the triangle of equation (1) leading to an enantiomorphous phase doublet for F(h,D) and for F(h,N) as in the following figure.

In OAS case on the other hand, we have

          (2)

The magnitude of F(+h) and F*(–h) can be obtained from experiment, while F"(h,A) can be derived from the known anomalous-scatterer substructure. Hence we have also two ways to draw the triangle of equation (2) leading to an enantiomorphous phase doublet for F(h) as shown below

         To summarize, the phase doublet in both SIR and OAS cases can be expressed as

     .    (3)
The problem of splitting enantiomorphous phase doublets is that of finding the sign of , given the known values of and .
        For SIR case we have and when we are deriving phases for the native protein, or when phases are derived for the derivative.
        For OAS case we have  and the phases are derived for .
        In order to break the phase ambiguity, a tangent formula among  and the probability for   to be positive have been worked out [Fan & Gu, (1985). Acta Cryst. A41, 280-284 ]:

          (4)

                         .    (5)

Here

, Z_j is the atomic number of the j^th atom in the unit cell;

     ;

     ;    (6)

     ;

     ;    (7)

, E_{h, p} is the contribution of the known partial structure (replacing atoms or anomalous scatterers) to the magnitude of normalized strructure factor;
, Z_u is the atomic number of the u^th atom belonging to the unknown part of the unit cell;
, is the phase of E_{h, p}.

        At start, equations (6) and (7) are used with P₊ = 1/2. The results are substituted into (5) to obtain a new set of P₊.This is fedback to (6), (7) and then to (5). Usually two cycles will be enough for breaking the sign ambiguity of . A program OASIS has been written in Fortran for the above calculation.