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 Results.

Given that the maximum entropy map is the most uniform map consistent with the data, it is surprising how much more informative can be from the conventional Fourier synthesis. The following figure makes the point : The MAXENT map corresponding to the 15Å synthesis, shows a level of detail which can only be seen clearly in the 11Å conventional Fourier synthesis19.

Projections of two lysozyme molecules.

There are several things to note in these maps : The first is that the MAXENT map is always positive with almost no detail in the background. This is clearly not the case with the Fourier synthesis, which have negative regions (dashed lines) and fine detail in the background which arise not only from the absence of the F00 term, but also from the series termination errors20. The second observation is that the peaks on the MAXENT map are better resolved, even when compared with the 11Å Fourier synthesis. This is not too surprising given that the Fourier transform of the maximum entropy map has non-zero amplitudes all the way to physical limits of the transform (the Nyquist critical frequency).

The second example is with real data. The top figure on the next page shows the asymmetric unit of the theoretical [001] Patterson projection function for the TMLAc derivative of EcAmOx (4 heavy atom sites, plane group pgg). Below it, are shown the observed conventional and MAXENT 10-3Å Patterson projections.


TMLAc derivative of EcAmOx, Courtesy Drs Maire Convery & Mark Parsons, University of Leeds.

Although both the conventional and MAXENT syntheses are noisy, there are several important differences between them. The highest peaks in the conventional synthesis (marked as 1 and 2), are close to noise level in the MAXENT map and (reassuringly) absent from the theoretical Patterson : These peaks are artifacts not required by the data and not predicted from the correct heavy atom structure. Peak 4 shows exactly the same behaviour : it is a significant feature in the conventional synthesis, but is absent from both the MAXENT and theoretical Pattersons. Peak 3 on the other hand, was the third highest peak in the conventional synthesis, but is the most significant feature in both the MAXENT and theoretical syntheses : its presence is required by both the data and the postulated heavy atom structure. Peak 5 shows a similar behaviour. Finally, there are several other peaks (for example 6, 7, 9) which are close to the noise level in the conventional synthesis, but are significant features in both the MAXENT and theoretical Patterson.

It should be stressed that the method is not a panacea : if, for example, the signal in the TMLAc data was coming not from the underlying heavy atom structure, but from some sort of non-isomorphism, the MAXENT map would still show only these features required by the data, but, of course, the final map could not be interpreted in terms of a heavy atom structure.

The final example is based again on real data and shows the importance of having reasonably accurate estimates of the errors present in the data. The figure on the left hand side is the conventional 30Å projection of photosystem II. The other three figures show MAXENT maps which were calculated with standard deviations ranging from grossly overestimated (second figure from the left) to seriously underestimated (right hand side figure). Clearly, overestimating the standard deviations is no harm : although the final map will not be the best that can be done with the data, it will not be possible to misinterpret it. Underestimating the standard deviations, on the other hand, can lead to serious problems : the MAXENT algorithm will be ``fitting'' noise instead of real signal and the final map will contain fine structure not required by the data. Misinterpreting such a MAXENT map should present no problems.

Photosystem II, Courtesy Dr Andreas Holzenburg, University of Leeds.

It is worth noting that most data processing programmes will produce raw data with underestimated standard deviations (especially for weak reflections). The solution is, of course, to calculate a normal probability plot of the form (Iobs - < I > )/$ \sigma$(I) and confirm that it has mean 0 and variance 1 (the programme agrovata from the CCP4 suite produces such a plot by default).



Footnotes

... synthesis19
The asymmetric unit of the pmg plane group consists of the projections of two lysozyme molecules related by a simple translation.
... errors20
In all maps shown, contours are plotted every 10% of the maximum density.

next up previous contents
Next:  Using the programme. Up:  Old (obsolete) version's documentation Previous:  The method.   Contents
NMG, March 2000