Authors: Doug Collins and Jim Stewart

Contact: Jim Stewart, Department of Chemistry,

University of Maryland, College Park, MD 20742, USA

MEDENS takes a low resolution direct space (electron density) map and computes a constrained exponential electron density distribution. The resulting density function is written to output. The output function is a function of maximum entropy and is intended to be used for the extrapolation, interpolation and smoothing of reflection phases. The resulting phases are recovered by the use of RFOURR. MEDENS may be used most easily by first running MERUN to set up the run stream for all the links necessary in a maximum entropy refinement. This program is modelled on a subroutine "MAXENT" written by E. Prince at the National Bureau of Standards, Gaithersburg, Maryland.

Purpose

MEDENS processes an electron density map stored on a scratch bdf of the kind generated by FOURR. This map will usually be an Fo Fourier transform produced using a limited set of phase data. The output file will be the same form as the input file and all the electron density will have been processed to be a function of maximum entropy. This means that new density will conform to an exponential distribution. All negative areas will have been scaled above zero and all large positive regions will have been sharpened. The final sharpness may be controlled by the use of a sharpening parameter under the user's control. Values of the sharpening parameter greater than one enhance the sharpness of the map, while those less damp it. The MEDENS process provides an electron density map, which when processed by RFOURR, will produce phases for more reflections than were used to create the input map. The input map must be for a whole unit cell and calculated at sufficient resolution to allow phase extension if desired.

Method

The input electron density is scanned to establish the maximum, minimum and average electron density. The process that follows is very sensitive to average electron density, and will not work if the F(0,0,0) term has been left out of the original electron density calculation. The process becomes ill-conditioned and will fail as the mean of the electron density approaches zero from the positive. It is also important that the grid of the input electron density be "fine" enough to provide the resolution desired in the output map and in subsequent RFOURR runs, and that the chosen grid sizes in the three crystallographic directions conform to the FFT grid restrictions for that program.

The output electron density with unit sharpening is an exponential representation which satisfies two constraints:

1) The mean of the electron density remains constant.

2) The mean square of the electron density remains constant.

This is accomplished by calculating the new electron density, NED, from the old electron density, OED, by:

NED = exp(Z*B+(1 - Z)*A)

where Z = (OED - min(OED))/(max(OED) - min(OED))

The scale factors A and B are obtained by a Newton-Raphson iteration.

The initial values of A and B are:

A = ln(0.005*max(OED))

B = ln(max(OED))

The values of A and B correspond to ln(rho) minimum and ln(rho) maximum. The sum of the electron density will be F(0,0,0)*TOTPIX/VOL where TOTPIX is the total pixels in the whole cell map and VOL is the volume of the unit cell.

Using the values of A and B at each iteration four summations are made:

1) The sum of all NED (This is constraint 1)

2) The sum of all (This is constraint 2)

3) The sum of all Z*NED

4) The sum of all Z*

During the initial survey the sums of all OED and are saved in order to calculate scale and tolerance factors.

A renormalization scale factor is obtained by:

S = Σ(OED)/Σ(NED)

and two tolerance (agreement) factors are calculated:

T' =|Σ(NED) -Σ(OED)| / Σ(OED)

T'' =|(Σ()* -Σ()| / Σ()

when the larger of these two factors is less than 0.001, the iteration proceedure is stopped. At each iteration A and B are adjusted by:

A' = A + ln(S)

B' = B + ln(S) + Q'

where Q' = (Σ() - ()*(Σ()))/D

where D = E*F

where E = 2.0*S*Σ(OED)*F

where F = Σ(Z*)/Σ(NED) - G

where G = ((Σ()*(Σ(Z*NED))/(Σ(NED))²

The value of Q' is always constrained to lie between ± LL, where LL is 1.0 by default, but may be set by the user. Once convergence has been achieved, the new electron density is computed from the final values of A and B. The new electron density is written to output on the file med in the same format as it came from the map file created by FOURR. The final electron density is calculated by:

Z = (OED - min(OED))/(max(OED) - min(OED))

NED = S*exp(SHARPF*(Z*B + (1.0 - Z)*A)

The parameter SHARPF controls the sharpness of the output map, and defaults to 1.0. However, in many applications, 0.5, (the square root), may be found to be more satisfactory. If the value of SHARPF is set in the MEDENS line, the final density is rescaled to restore the correct mean value, but the original mean square of the density is not recovered.

File Assignments

Reads cell volume from input archive bdf

Reads an electron density map from file map

Writes a constrained exponential electron density map on file med

Example

The MERUN documentation contains an example of how MEDENS is used in conjunction with MEFFIT, RFOURR, and FOURR to carry out a phase refinement run.

Reference

Prince, E. Sjolin, L. and Alenljung, R.. 1988. Phase Extension by Combined Entropy Maximization and Solvent Flattening. Acta Cryst. A44, 216-222.