Implementation A

In implementation A, the algorithm operates on the set of the complete unit cell den- sitiesfn(x).

5.4.1.1 Reciprocal space projection

The reciprocal space projection, denotedPM, makes the smallest change in the current estimate offn(x), such that the Fourier intensities are equal to the experimental inten- sities In. This projection was defined in Chapter 1 (equation (1.26)) and is the usual Fourier space projection for reconstruction of a single object and is repeated here for convenience: PMfn(x) =F−1 ( p In(h) |Fn(h)| Fn(h) ) , (5.7)

whereFn(h) is the Fourier transform of fn(x).

5.4.1.2 Real space projection

The real space projection, denoted PS, enforces the following two constraints: (i) the iteratefn(x) is such that the component partsgnk(x) from which it is built are identical for each unit celln, i.e. thegnk(R_nk−1(x−tnk)) are identical for allnandk, and (ii)fn(x) is restricted to the support regions(x). Both of these requirements can be satisfied, in the least distance sense, by averaging the components, applying a support constraint

s(x), and then rebuilding the unit cells. The real space projection can then be written in two steps as g0(x) = s(x) P N X n=1 Kn X k=1 Snkfn(x), (5.8) PSfn(x) = Kn X k=1 g0(Rnkx+tnk), (5.9) whereP =P

nKn is the total number of asymmetric units in all the crystals, and the operatorSnk extracts an estimate ofg(x) from the kth asymmetric unit infn(x).

If the support regions in a unit cell do not overlap, then the operation Snk can be achieved by simply repositioning fn(x) within the corresponding support region, and

Snk is given by

If the support regions overlap, then one can in principle still develop a distance minimising projection, but the implementation depends on the nature of all the over- laps in the unit cell, and its effectiveness depends on the degree of overlap. Such an implementation would be computationally intensive in practice.

5.4.1.3 Two-dimensional simulations of approach A

Simulations were conducted to illustrate implementation A ofaiMR described in the previous sections. For each simulation, a pair of different crystal forms of a 32×32 Lena electron density, the “protein”, were generated. The support,s(x) was defined by the 32×32 logical mask and the Fourier intensity dataIn(h) calculated using the DFT. The aiMR real and reciprocal space projections were implemented and incorporated into the difference map algorithm.

(a)

(b)

(c)

Figure 5.3 Reconstruction results foraiMR phase retrieval using approach A. Orig- inal pairs of crystal forms (left) and corresponding reconstructions (right). (a) Two

p1 crystal forms with different shape of the same volume, (b) a p1 crystal and a pm

crystal, and (c) a p1 crystal and ap2 crystal.

The three unit cell pairs used in simulation are shown in Fig. 5.3. The first pair consists of twop1 unit cells with different shapes of the same volume (Fig. 5.3(a)). The unit cells have dimensions 37×43 and 43×37 pixels. The corresponding solvent content is 36% in both crystals. The second pair is composed of a p1 unit cell of dimensions 37×37 and a pm unit cell of 74×37 pixels (Fig. 5.3(b)). Both have 25% solvent content. The third cell is similar to the previous case but with a p2 unit cell rather thanpm(Fig. 5.3(c)). Computation of Ωc= 1/2pfor these crystals indicates no unique solution can be expected if taken alone. However, usingε= 0.8 and equation (5.4) the

aiMR constraint ratio is calculated as (a) ΩaiM R = 1.25, (b) ΩaiM R = 1.06 and (c) ΩaiM R = 1.6 indicating a unique solution may be obtained usingaiMR. Note that, for the third case, the second crystal is centric so that Ωc= 1/p.

Figure 5.4 Real space (blue) and reciprocal space (red) errorseandE as a function of iteration for the unit cell and reconstruction shown in Fig. 5.3(a).

For each simulation, the reciprocal and real space errors were computed with the usual error metrics given in Section 1.4.5. Representative error plots are shown in Fig. 5.4, Fig. 5.5 and Fig. 5.6 for case (a), (b) and (c) respectively. Successful recon- structions were obtained in all cases, in accordance with the values of ΩaiM Rabove. The ease of reconstruction is correlated to the values of ΩaiM R. Case (b) is the hardest with ΩaiM R = 1.06 as can be seen in Fig. 5.5 with a lengthy search for an attractor, followed by a slow convergence to the solution. Case (c) is the easiest with ΩaiM R = 1.6, con- verging in less than a thousand iterations (Fig. 5.6), while case (a) with ΩaiM R= 1.25 converges in about 1500 iterations (Fig. 5.4). The average number of iterations required for convergence decrease, as ΩaiM R increases, as expected. The smaller final error for case (c) may be due to the problem’s centric nature which constrains the phase to just the two values, 0 and π. These results are consistent with the uniqueness theory described in Section 5.3.

Figure 5.5 Real space (blue) and reciprocal space (red) errorseandE as a function of iteration for the unit cell and reconstruction shown in Fig. 5.3(b).

Figure 5.6 Real space (blue) and reciprocal space (red) errorseandE as a function of iteration for the unit cell and reconstruction shown in Fig. 5.3(c).