Tables for
Volume F
Crystallography of biological macromolecules
Edited by E. Arnold, D. M. Himmel and M. G. Rossmann

International Tables for Crystallography (2012). Vol. F, ch. 15.3, pp. 407-412   | 1 | 2 |

Chapter 15.3. DM/DMMULTI software for phase improvement by density modification

K. D. Cowtan,a* K. Y. J. Zhangb and P. Mainc

aDepartment of Chemistry, University of York, York YO1 5DD, England,bDivision of Basic Sciences, Fred Hutchinson Cancer Research Center, 1100 Fairview Ave N., Seattle, WA 90109, USA, and cDepartment of Physics, University of York, York YO1 5DD, England
Correspondence e-mail:

The DM/DMMULTI software for phase improvement by density modification is described.

15.3.1. Introduction

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DM is an automated procedure for phase improvement by iterated density modification. It is used to obtain a set of improved phases and figures of merit, using as a starting point the observed diffraction amplitudes and some initial poor estimates for the phases and figures of merit. DM improves the phases through an alternate application of two processes: real-space electron-density modification and reciprocal-space phase combination. DM can perform solvent flattening, histogram matching, multi-resolution modification, averaging, skeletonization and Sayre refinement, as well as conventional or reflection-omit phase combination. Solvent and averaging masks may be input by the user or calculated automatically. Averaging operators may be refined within the program. Multiple averaging domains may be averaged using different operators.

DMMULTI is a modified version of the DM software that can perform density modification simultaneously across multiple crystal forms. The procedure is general, handling an arbitrary number of domains appearing in an arbitrary number of crystal forms. Initial phases may be provided for one or more crystal forms; however, improved phases are calculated in every crystal form.

DM and DMMULTI are distributed as a part of the CCP4 suite of software for protein crystallography (Collaborative Computational Project, Number 4, 1994[link]). The theoretical and algorithmic bases for the DM and DMMULTI software suites are reviewed in Chapter 15.1[link] . In this chapter, some specific issues concerning the programs are described, including program oper­ation, data preparation, choices of modes and code description.

15.3.2. Program operation

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DM and DMMULTI are largely automatic; in order to perform a phase-improvement calculation only two tasks are required of the user:

  • (1) Provide the input data. These must include the reflection data and solvent content, and may also include averaging operators, solvent mask and averaging domain masks.

  • (2) Select the appropriate density modifications and the phase-combination mode to be used in the calculation.

DM and DMMULTI can run with the minimum input above, since the optimum choices for a whole range of parameters are set in the program defaults. For some special problems it may be useful to control the program behaviour in more detail; this is possible through a wide range of keywords to override the defaults. These are all detailed in the documentation supplied with the software.

15.3.3. Preparation of input data

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Input data are provided by two routes: numerical parameters, such as solvent content and averaging operators, are included in the command file using appropriate keywords, whereas reflections and masks are referenced by giving their file names on the command line. In the simplest case; for example a solvent-flattening and histogram-matching calculation, all that is required is an initial reflection file and an estimate of the solvent content.

Use all available data: The reflection file must be in CCP4 `MTZ' format, and contain at least the structure-factor amplitudes, phase estimates and figures of merit. If the phase estimates are obtained from a homologous structure by molecular replacement, the figures of merit can be generated by the SIGMAA program (Read, 1986[link]). When the phases are estimated using a single isomorphous derivative (SIR), it is recommended that Hendrickson–Lattman coefficients (Hendrickson & Lattman, 1970[link]) are used to represent the phase estimate instead of the figure of merit. Hendrickson–Lattman coefficients can represent the bimodal distribution of the SIR phases, whereas the figure of merit can only represent the unimodal distribution of the average of two equally probable phase choices. It is recommended that a reflection file containing every possible reflection is used. The low-resolution data should be included since they provide a significant amount of information on the protein–solvent boundary. The high-resolution data without phase estimates should also be included since their phases can be estimated by DM. Phase extension can usually improve the original phases further compared to phase refinement only. Unobserved reflections are marked by a missing number flag. This is important for the preservation of the free-R reflections. It also enables DM to extrapolate missing reflections from density constraints and increases the phase improvement power.

The estimation of solvent content: The solvent content, [C_{\rm solv}], can be obtained by various experimental methods, such as the solvent dehydration method and the deuterium exchange method (Matthews, 1974[link]). It can also be estimated through [C_{\rm solv} = 1 - (NV_{a}ML/V). \eqno(]Here, N is the total number of atoms, including hydrogen atoms, in one protein molecule. Va is the average volume occupied by each atom, which is estimated to be approximately 10 Å3 (Matthews, 1968[link]). M is the number of molecules per asymmetric unit. L is the number of asymmetric units in the cell. V is the unit-cell volume. The correctly estimated solvent content should be entered in the program with the SOLC keyword, since this will be used not only to find the solvent–protein boundary but also to scale the input structure-factor amplitudes. If it is desirable to use a more conservative solvent mask in order to prevent clipping of protein densities, especially in the flexible loop regions, different solvent and protein fractions should be specified using the SOLMASK keyword.

Solvent mask: A solvent mask may be supplied; it may be used for the entire calculation or updated after several cycles. The solvent mask usually divides the cell into protein and solvent regions; however it is also possible to specify excluded regions which are unknown. If no solvent mask is supplied, it will be calculated by a modified Wang–Leslie procedure (Wang, 1985[link]; Leslie, 1987[link]) and updated as the phase-improvement calculation progresses.

Averaging operators: In an averaging calculation, the averaging operators must be supplied; these are typically obtained by rotation and translation searches using a program such as AMoRe (Navaza, 1994[link]) or X-PLOR (Brünger, 1992[link]). If the coordinates of several heavy atoms are known, they can be used to calculate the noncrystallographic symmetry (NCS) operators. If a partial model can be built into the density, structure-superposition programs, such as LSQKAB (Kabsch, 1976[link]), can be used to obtain the rotation and translation matrices that relate different molecules in the asymmetric unit. This can also be achieved through the program O using the `lsq_explicit' command (Jones et al., 1991[link]). The averaging operators can be further refined in DM by minimizing the residual between NCS related densities.

Averaging mask: An averaging mask may be supplied; this is distinct from the solvent mask, allowing for parts of the protein to remain unaveraged if required. If no averaging mask is supplied, the mask will be calculated by a local-correlation approach (Cowtan & Main, 1998[link]; Vellieux et al., 1995[link]). If multiple domains are to be averaged with different averaging operators (Schuller, 1996[link]), then one mask must be specified for each averaging domain. When averaging molecules related by improper NCS operations, the averaging mask must be in accord with the NCS operators provided. For example, if the supplied NCS matrix maps molecule A to molecule B, then the averaging mask must cover the volume occupied by molecule A rather than molecule B.

Multi-crystal averaging: In the case of a multi-crystal averaging calculation, one reflection file is provided for each crystal form (however, initial phases are not required in every crystal form), and one reflection file will be output for each crystal form containing the improved phases. One mask is required per averaging domain; thus, in general, only a single mask is required. This may be defined for any crystal form or in an arbitrary crystal space of its own. Averaging operators are then provided to map the mask into each of the crystal forms.

Solvent and averaging masks that are calculated within the program may be output for subsequent analysis. Refined averaging operators are also output. The input and output data for a simple DM calculation, a DM averaging calculation and a DMMULTI multi-crystal averaging calculation are shown in Figs.[link], (b)[link] and (c)[link], respectively.


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(a) Input and output data for a DM calculation with no averaging. Light outlines indicate optional information. (b) Input and output data for a DM averaging calculation: for a single averaging domain, the averaging mask may be calculated automatically. For multi-domain averaging, all domain masks must be given. (c) Input and output data for DMMULTI. An averaging mask (or masks, for multiple domains) must be provided.

15.3.4. Choice of modes

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Two major choices have to be made in a DM run. They are the real-space density-modification modes and reciprocal-space phase-combination modes. Moreover, the phase-extension schemes can be selected if needed. This can also be left to the program, which uses its default automatic mode for phase extension. The choices of various modes are described in the following sections. Density-modification modes

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The following density-modification modes (specified by the MODE keyword) are provided by DM:

  • (1) Solvent flattening: This is the most common density-modification technique and is powerful for improving phases at fixed resolution, but weaker at extending phases to higher resolution. Its phasing power is highly dependent on the solvent content. Solvent flattening can be applied at com­paratively low resolutions, down to around 5.0 Å.

  • (2) Histogram matching: This method is applied only to the density in the protein region. This method is weaker than solvent flattening for improving phases, but is much more powerful at extending phases to higher resolutions. This is due to a unique feature of histogram matching which uses a resolution-dependent target for phase improvement. The phasing power of histogram matching is inversely related to the solvent content. Therefore, histogram matching plays a more important role in phase improvement when the solvent content is low. Histogram matching works to as low as 4.0 Å, but does no harm below that. Histogram matching should probably be applied as a matter of course in any case where the structure is not dominated by a large proportion of heavy-metal atoms. Even in this case, histogram matching may be applied by defining a solvent mask with solvent, protein and excluded regions.

  • (3) Multi-resolution modification: This method controls the level of detail in the map as a function of resolution by applying histogram matching and solvent flattening at multiple resolutions. This strengthens phase improvement at fixed resolution, although it generally improves phase-extension calculations too.

  • (4) Noncrystallographic symmetry averaging: Averaging is one of the most powerful techniques available for improving phases and is applicable even at very low resolutions. In extreme cases, averaging may be used to achieve an ab initio structure solution (Chapman et al., 1992[link]; Tsao et al., 1992[link]). It should therefore be applied whenever it is present and the operators can be determined.

  • (5) Skeletonization: Iterative skeletonization is the process of tracing a `skeleton' of connected densities through the map and then building a new map by filling density around this skeleton. The implementation in DM is adapted for use on poor maps, where it is sometimes but not always of use. To bring out side chains and missing loops, the ARP program (Lamzin & Wilson, 1997[link]) is more suitable.

  • (6) Sayre's equation: This method is more widely used in small-molecule calculations, and is very powerful at better than 2.0 Å resolution and when there are no heavy atoms in the structure. However, its phasing power is lost quickly as resolution decreases beyond 2.0 Å. The calculation takes significantly longer than other density-modification modes.

The most commonly used modes are solvent flattening and histogram matching – these give a good first map in most cases. Recently, multi-resolution modification has been added to this list. Averaging is applied whenever possible. Skeletonization and Sayre's equation are generally only applied in special situations. Phase-combination modes

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Density-modification calculations are somewhat prone to producing grossly overestimated figures of merit (Cowtan & Main, 1996[link]). Users should be aware of this. In general the phases and figures of merit produced by density-modification calculations should only be used for the calculation of weighted [F_{o}] maps. They should not be used for the calculation of difference maps or used in refinement or other calculations (the REFMAC program is an exception, containing a mechanism to deal with this form of bias). The use of [2F_{o} - F_{c}]-type maps should be avoided when the calculated phases are from density modification, since they are dependent on two assumptions, neither of which hold for density modification: that the current phases are very close to being correct and that the calculated amplitudes may only approach the observed values as the phase error approaches zero.

To limit the problems of overestimation, three phase-combination modes are provided (controlled by the COMBINE keyword):

  • (1) Free-Sim weighting: This is the simplest mode to use. Although convergence is weaker than the reflection-omit mode, the calculation never overshoots the best map. If there is averaging information, then convergence is much stronger and the phase-combination scheme is much less important. In addition, phase relationships in reciprocal space limit the effectiveness of the reflection-omit scheme. Therefore, the free-Sim weighting scheme should usually be used when there is averaging.

  • (2) Reflection-omit: The combination of a reciprocal-space omit procedure with SIGMAA phase combination (Read, 1986[link]) leads to much better maps when applying solvent flattening and histogram matching. However, the omit calculation is computationally costly and introduces a small amount of noise into the maps, thus the phases can get worse if the calculation is run for too many cycles. A real-space free-R indicator (Abrahams & Leslie, 1996[link]) is therefore used to stop the calculation at an appropriate point.

  • (3) Perturbation-γ correction: This new approach is an extension of the γ correction of Abrahams (1997)[link] to arbitrary density-modification methods. The results are a good approximation to a perfect reflection-omit scheme and required considerably less computation. This is therefore the preferred mode for all calculations.

In the case of a molecular-replacement calculation or high noncrystallographic symmetry, it may be desirable only to weight the modified phases and not to recombine them back with the initial phases so that any initial bias may be overcome. In the case of high noncrystallographic symmetry, it may also be possible to restore missing reflections in both amplitude and phase. Options are available for both these situations. Phase-extension schemes

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When performing phase extension, the order in which the structure factors are included will affect the final accuracy of the extended phases. The phases obtained from previous cycles of phase extension will be included in the calculation of new phases for the unphased structure factors in the next cycle. A reflection with more accurately determined phases might enhance the phase-extension power of the original set of reflections, whereas a reflection with less accurately determined phases might corrupt the phase-extension power of the original set of reflections and make the phase extension deteriorate quickly. The factors that might affect phase extension are the structure-factor amplitudes, the resolution shell and the figure of merit. Based on the above considerations, the following phase-extension schemes are provided in DM:

  • (1) Extension by resolution shell: This performs phase extension in resolution steps, starting from the low-resolution data, and extends the phase to the high-resolution limit of the data or that specified by the user. Structure factors are related by the reciprocal-space density-modification function that is dominated by low-resolution terms, as shown by equation ([link] ) and Fig.[link] in Chapter 15.1. This means that only structure factors in a small region of reciprocal space are related. Thus, when initial phases are only available at low resolution, phase extension is performed by inclusion of successive resolution shells. In the case of fourfold or higher NCS, this can allow extension to 2 Å starting from initial phasing at 6 Å or worse.

  • (2) Extension in structure-factor amplitude steps: In this mode, those reflections with larger amplitudes are added first, gradually extending to those reflections with smaller amplitudes in many steps. The contribution of a reflection to the electron density is proportional to the square of its structure-factor amplitude according to Parseval's theorem, as shown in equation ([link]. This favours the protocol of extending the stronger reflections first so that they can be more reliably estimated. These stronger reflections will be used to phase relatively weaker subsequent reflections.

  • (3) Extension in figure-of-merit step: To extend phases for those structure factors with experimentally measured, albeit less accurate, phases and figures of merit, the reflections can be added in order of their figure of merit, starting from the highest to the lowest. It is advantageous to use the more reliably estimated phases with higher figure of merit to phase those reflections with lower figure of merit. This can be useful when working with initial phasing from multiwavelength anomalous diffraction (MAD) or molecular replacement (MR) sources.

  • (4) Automatic mode: This combines the previous three extension schemes. The program automatically works out the optimum combination of the above three schemes according to the density-modification mode, the phase-combination mode and the nature of the input reflection data. The automatic mode is the default and is the recommended mode of choice unless specific circumstances warrant a different choice.

  • (5) All reflection mode: One advantage of the reflection-omit and perturbation-γ methods is that the strength of extrapolation of a structure-factor amplitude is a good indicator of the reliability of its corresponding phase. As a result, a phase-extension scheme is unnecessary in reflection-omit calculations; all reflections may be included from the first cycle.

15.3.5. Code description

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The program was designed to be run largely automatically with minimal user intervention. This is achieved by using extensive default settings and by automatic selection of options based on the data used. The program is also modular by design so that additional density-modification methods can be incorporated easily.

A simplified flow diagram for DM is shown in Fig.[link]. When a reflection-omit calculation is performed, an additional loop is introduced, shown in Fig.[link]. The Sayre's equation calculation adds another level of complexity, described in Zhang & Main (1990b)[link]. Skeletonization imposes the protein histogram and solvent flatness implicitly and so is performed, if necessary, every second or third cycle in place of solvent flattening and histogram matching. Simplified conceptual and actual flow diagrams for DMMULTI are shown in Figs.[link] and (b)[link].


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(a) Flow chart for a simple DM calculation with free-Sim phase combination. (b) Flow chart for a simple DM calculation with reflection-omit phase combination.


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(a) Conceptual flowchart for a DMMULTI multi-crystal calculation. (b) Actual flow chart for a DMMULTI multi-crystal calculation.

Many of the basic approaches used in DM and DMMULTI are described in Chapter 15.1[link] . Some practical aspects of the application and combination of these approaches are described here. Scaling

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All forms of map modification are affected by the overall temperature factor of the data, and histogram matching in particular is critically dependent on the accurate determination of the scale factor. Wilson statistics have been found inadequate for scaling in this case, especially when the data resolution is worse than 3 Å, because of the dip in scattering below 5 Å.

More accurate estimates of the scale and temperature factors may be achieved by fitting the data to a semi-empirical scattering curve (Cowtan & Main, 1998[link]). This curve is prepared using Parseval's theorem, which relates the sum of the intensities to the variance of the map: [\sigma_{\rho}^{2} = {1\over V^2}\sum\limits_{{\bf h} \neq 000}\displaystyle |F({\bf h})|^{2}. \eqno(]Thus, the sum of the intensities in a particular resolution shell is proportional to the difference in variance of maps calculated with and without that shell of data. The empirical curve is therefore calculated from the variance in the protein regions of a group of known structures, calculated as a function of resolution. The curve is scaled to the protein volume of the current structure, and a correction is made for the solvent, which is assumed to be flat.

The overall temperature factor is removed, and an absolute scale is imposed by fitting the data to this curve. The use of sharpened F's (with no overall temperature factor) is necessary for histogram matching and often increases the power of averaging for phase extension.

Since the solvent content is used in scaling the data, it is important that this value be entered correctly. However, the volume of the solvent mask may be varied independently of the true solvent content, as discussed in Section 15.3.3[link]. Solvent-mask determination

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If the user does not supply a solvent mask, the solvent mask is calculated by Wang's (1985) method, using the reciprocal-space approach of Leslie (1987)[link]. A number of variants on this algorithm are implemented; however, the parameter that affects the quality of the solvent mask most dramatically is the radius of the smoothing function (Chapter 15.1[link] ). This parameter may be estimated empirically by [r_{\rm Wang} = 2r_{\max} \overline{w}^{1/3}, \eqno(]where [r_{\max}] is the resolution limit of the observed amplitudes, and [\overline{w}] is the mean figure of merit over the same reflections (with w = 0 for unphased reflections).

Once the smoothed map has been determined, cutoff values are chosen to divide the map into protein and solvent regions. If the protein boundary is poorly defined, the user may specify protein, solvent and excluded volumes, in which case two cutoffs are specified and the intermediate region is marked as neither protein nor solvent. Averaging-mask determination

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If the user does not supply an averaging mask, it is determined by a local correlation method (Vellieux et al., 1995[link]). A large region covering 27 unit cells is selected, and the local correlation between the maps before and after rotation by one of the noncrystallographic symmetry operators is calculated. The largest contiguous region that is in agreement among different NCS operators is isolated from the local correlation map, and a finer local correlation map is calculated over this volume. This process is iterated until a good mask with a detailed boundary is found.

This approach is fully automatic, except in the case where a noncrystallographic symmetry operator intersects a crystallographic symmetry operator, in which case the mask is not uniquely defined, and some user intervention may be required. The method is robust, and by increasing the radius of the sphere within which the local correlation is calculated, it may be used with very poor maps (Cowtan & Main, 1998[link]). The method is easily extended to include information from multiple averaging operators. Fourier transforms

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For simplicity of coding, all Fourier transforms are performed in core using real-to-Hermitian and Hermitian-to-real fast Fourier transforms (FFTs). The data are expanded to space group P1 before calculating a map and averaged back to a reciprocal asymmetric unit after inverse transformation. Most of the map modifications preserve crystallographic symmetry, so restricted phases are not constrained except during phase combination. Histogram matching

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The target histograms are calculated from the protein regions of several stationary-atom structures at resolutions from 6 to 1.5 Å, according to the method described by Zhang & Main (1990a)[link]. The histogram variances should be consistent with the map variances used in scaling the data. The resolution of the target histogram can be accurately matched to the data resolution by averaging the target histograms on either side of the current resolution. Averaging

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Averaging is performed using a single-step approach (Rossmann et al., 1992[link]), in which every copy of the molecule in a `virtual' asymmetric unit is averaged with every other copy. Density values are obtained at non-grid positions using a 27-point quadratic spectral spline interpolation. A sharpened map is first calculated by dividing by the Fourier transform of the quadratic spline function. The same spline function is then convoluted with the sharpened map to obtain the density value at an arbitrary coordinate (Cowtan & Main, 1998[link]). This approach gives very accurate interpolation from a coarse grid map with relatively little computation and additionally provides gradient information for the refinement of averaging operators. Multi-crystal averaging

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The multi-crystal averaging calculation in DMMULTI is equivalent to several single-crystal averaging calculations running simultaneously, with the exception that during the averaging step, the molecule density is averaged across every copy in every crystal form. This average is weighted by the mean figure of merit of each crystal form; this allows the inclusion of unphased crystal forms, since in the first cycle they will have zero weight and therefore not disrupt the phasing that is already present. In subsequent cycles, the unphased form contains phase information from the back-transformed density.

This technique can be extremely useful, since adding a new crystal form usually provides considerably more phase information than adding a new derivative if the cross-rotation and translation functions can be solved.

In the multi-crystal case, averaging is performed using a two-step approach, first building an averaged molecule from all the copies in all crystal forms, then replacing the density in each crystal form with the averaged values. This approach is computationally more efficient when there are many copies of the molecule.

The conceptual flow chart of simultaneous density-modification calculations across multiple crystal forms is shown in Fig.[link]; in practice, this scheme is implemented using a single process and looping over every crystal form at each stage (Fig.[link]. Maps are reconstructed from a large data object containing all the reflection data in every crystal form. Averaging is performed using a second data object containing maps of each averaging domain. By this means, an arbitrary number of domains may be averaged across an arbitrary number of crystal forms.

Multi-crystal averaging has been particularly successful in solving structures from very weak initial phasing, since the data redundancy is usually higher than for single-crystal problems.


KDC acknowledges the support of the UK BBSRC (grant No. 87/B03785). KYJZ acknowledges the National Institutes of Health for grant support (GM55663).


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