research papers
FEM: featureenhanced map
^{a}Physical Biosciences Division, Lawrence Berkeley National Laboratory, One Cyclotron Road, MS64R0121, Berkeley, CA 94720, USA, ^{b}Biology and Soft Matter Division, Oak Ridge National Laboratory, PO Box 2008, Oak Ridge, TN 37831, USA, ^{c}Los Alamos National Laboratory, Mail Stop M888, Los Alamos, NM 87545, USA, ^{d}Biochemistry and Molecular and Structural Biology, Jozef Stefan Institute, Jamova 39, SI1000 Ljubljana, Slovenia, ^{e}Centre of Excellence for Integrated Approaches in Chemistry and Biology of Proteins, Jozef Stefan Institute, Jamova 39, SI1000 Ljubljana, Slovenia, ^{f}Centre for Integrative Biology, IGBMC, CNRS–INSERM–UdS, 1 Rue Laurent Fries, BP 10142, 67404 Illkirch, France, ^{g}Département de Physique, Faculté des Sciences et des Technologies, Université de Lorraine, 54506 VandoeuvrelèsNancy, France, and ^{h}Department of Bioengineering, University of California Berkeley, Berkeley, CA 94720, USA
^{*}Correspondence email: pafonine@lbl.gov
A method is presented that modifies a 2mF_{obs} − DF_{model} σ_{A}weighted map such that the resulting map can strengthen a weak signal, if present, and can reduce model bias and noise. The method consists of first randomizing the starting map and filling in missing reflections using multiple methods. This is followed by restricting the map to regions with convincing density and the application of sharpening. The final map is then created by combining a series of histogramequalized intermediate maps. In the test cases shown, the maps produced in this way are found to have increased interpretability and decreased model bias compared with the starting 2mF_{obs} − DF_{model} σ_{A}weighted map.
Keywords: Fourier map; map sharpening; map kurtosis; model bias; map improvement; density modification; PHENIX; cctbx; FEM; featureenhanced map; OMIT.
1. Introduction
An electron (or nuclear) density map is typically an important step in obtaining an atomic representation of a
or the map itself may serve as the model of the contents of the crystal. In either case the quality of the map affects its utility. In this work, we consider a desirable map to be one that accurately represents the actual electron (or nuclear) density in an average of the crystal. There are at least three different factors that affect the quality of crystallographic maps and their interpretation.

A large amount of methodology has been developed to improve crystallographic maps. These methodologies can be roughly divided into four categories.
All of these methods aim to address one or a few specific issues pertinent to map quality, but not all or at once. Some of them are quick to compute (e.g. σ_{A} or kick maps), while the calculation of others (e.g. OMIT, SA OMIT or iterativebuild OMIT maps) is more computationally intensive and includes more complex calculations such as and/or model building that require problemspecific parameterization. Some may rely on information that needs to be either provided by a user or estimated, such as values of the F_{000} reflection for RAPID/END or maximumentropy maps or solvent content for density modification. OMIT maps attempt to address one problem (model bias) while potentially introducing other perturbations into the map. Indeed, the calculation of an OMIT map involves removing parts of the model, so one may expect that the quality of the resulting map might be worse than that of the original map (although the model bias may be reduced). Densitymodification methods are generally tailored to the beginning of and are good at bulk map improvements and eliminating gross errors, while these methods may be harsh on weak local signal (such as densities arising from partially occupied mobile ligands).
This manuscript describes a procedure that improves a 2mF_{obs} − DF_{model} σ_{A}weighted map (Read, 1986). The result is a new map that possesses a reduced level of noise and model bias and that also shows enhancement of weak features, often bringing them onto the same scale as the strong features. We call this map a featureenhanced map (FEM). The new map is expected to contain less noise and have a more easily interpretable available signal both for human interpretation and for modelbuilding software. While in this work we focus on 2mF_{obs} − DF_{model} maps, the procedure can be extended to other kinds of maps. The only inputs to the current procedure are measured data (I_{obs} or F_{obs}), measurement uncertainties [σ(I_{obs}) or σ(F_{obs})] and the current atomic model. Optionally, additional phase information can be used in the form of Hendrickson–Lattman coefficients. The procedure does not involve sophisticated machinery such as model building or Specifically, the novel, new or redesigned methods implemented as part of the FEM calculation procedure include the following.

2. Methods
Conceptually, the FEM method consists of three key stages. Firstly, the starting map is defined and calculated. In the second step, the starting map is randomized in a variety of different ways; these include randomizing the initially calculated map itself and varying the way in which the map is calculated. This procedure generates an ensemble of slightly different maps that are then combined into one single map with the goal of reducing noise and model bias. As expected, combining maps tends to blur density peaks and therefore map sharpening of some form is helpful. The final step involves map modification to equalize the strength of the signal throughout the entire unitcell volume, so the weak features become roughly the same magnitude as the strong features. This step is not selective and could enhance both the signal and noise; therefore, it is essential to eliminate the noise as much as possible in the first two steps.
The implementation of this concept is detailed in Fig. 1, with the individual steps explained in the sections below. The protocol is empirical and is the result of experimentation with a wide range of procedures.
2.1. Calculation of a map
The map that is subjected to the FEM procedure is computed with the following complex Fourier coefficients (given as an amplitude and a phase),
The total model structure factors, F_{model} = k_{total}(F_{calc} + k_{mask}F_{mask}), are calculated as described in Afonine, GrosseKunstleve et al. (2013). Prior to F_{model} calculation, the maximal possible common isotropic B factor, B_{max}, is subtracted from all atoms (B_{max} is defined such that after subtraction all atoms still have nonnegative definite B factors; see §§2.6.1 and 3 for further details). After overall scaling has been performed, B_{max} becomes part of an overall anisotropic scale factor k_{total} (see Afonine, GrosseKunstleve et al., 2013 for definitions). Dividing the map coefficients F_{map} by k_{total} sharpens the map and removes anisotropy. Unit weights w are used by default. For acentric and centric reflections (the first two rows in equation 1), σ_{A}weighted (Read, 1986) anisotropyremoved and sharpened Fourier map coefficients corresponding to measured F_{obs} are calculated. The (F_{fill}, φ_{fill}) terms correspond to unmeasured (missing) reflections, while φ_{model} are either model phases (phases of F_{model}) or model phases combined with experimental phase information. The weights m and D are calculated as described in Read (1986) and Urzhumtsev et al. (1996).
A residual (difference) synthesis is computed similarly to (1),
2.2. Modeling missing reflections
Let d_{min} be the highest resolution limit of a set of F_{obs}. It has been demonstrated that if the terms corresponding to unmeasured reflections F_{obs} in the resolution range (d_{min}, ∞) are missing from the map calculation then the map quality is degraded (Lunin, 1988; Urzhumtsev et al., 1989; Lunin & Skovoroda, 1991; Tronrud, 1996; Cowtan, 1996; Lunina et al., 2002; Urzhumtseva & Urzhumtsev, 2011). To mitigate this negative effect, terms corresponding to missing F_{obs} are replaced (often referred to as `filled in') with values (F_{fill}, φ_{fill}) (Murshudov et al., 1997; Lunina et al., 2002; Altomare et al., 2008; Sheldrick, 2008). Ignoring missing reflections is essentially equivalent to postulating that their values are all zero (F_{fill} = 0). We have implemented two ways of approximating the terms corresponding to the missing F_{obs}.
The first method uses the densitymodification functionality of RESOLVE (Terwilliger, 2003). A feature of density modification is that it can be used to estimate the amplitudes as well as the phases of reflections that are missing. In the statistical densitymodification approaches implemented in RESOLVE, these amplitudes and phases are those that maximize the likelihood (plausibility) of the map based on such features as the flatness of the solvent, and density distributions (Terwilliger, 2003). The use of density modification to estimate missing amplitudes consists of carrying out several cycles of density modification and finally using the mapbased amplitudes for the missing reflections in the resolution range (d_{min}, ∞).
The second method for the estimation of missing amplitudes uses the available atomic model. A key element of this second approach is that atomic model truncation is carried out to eliminate unreliably placed atoms so that they do not feed back into the map through F_{fill}. Firstly, a Fourier map (1) is calculated with F_{fill} all zero and normalized based on the r.m.s. deviation of the density in the map. Each atom is then scored against this map and the average density at the atomic centers (ρ_{ave}) is noted. Atoms that have a map correlation worse than 0.7 or have a density value at the atom center of less than min{½ρ_{ave}, 1} are removed. Since we consider a situation close to the final structure, only a relatively small part of the atoms may be filtered out. This truncated model is then used to calculate F_{model} in the resolution range (d_{min}, ∞). A bulksolvent contribution to F_{model} is calculated as described in Afonine et al. (2005) and the mask is calculated from the truncated atomic model. This allows optimal k_{sol} and B_{sol} values to be obtained, as well as the overall anisotropic scale factor. The F_{model} values for the full resolution range are calculated and the missing reflections that are part of F_{model} are then assigned to F_{fill}. We note that using typical values of k_{sol} and B_{sol} of 0.35 e Å^{−3} and 46 Å^{2}, respectively (Fokine & Urzhumtsev, 2002), is suboptimal, occasionally resulting in maps with pronounced artifacts in the bulksolvent region.
We find that estimating values for missing reflections in this way generally reduces map noise. As an example, Fig. 2 shows the map calculated with (1) using F_{fill} = 0 as well as maps where missing terms (F_{fill}) were derived either using statistical density modification or based on a working model. PDB (Bernstein et al., 1977; Berman et al., 2000) entry 1nh2 was used in this example.
2.3. Removal of noise and model bias by map randomization and combination
The rationale behind map randomization followed by combination of the resulting perturbed maps is based on the hypothesis that map artifacts and noise are more affected by randomization than the signal in the map (Kirillova, 2008). In this way, the artifacts and noise might be reduced compared with the signal by map averaging or some other statistical treatment of the resulting maps. This approach is also the basis of kick maps (Gunčar et al., 2000; Turk, 2013).
Here, we took a different route to achieve a similar goal. Empirically, we found that the use of a weighting scheme in (1) such as
possesses the desired property of perturbing the map noise more than the signal. Here, α and β are constants individually defined for each reflection, I_{obs} and I_{model} are the observed and calculated structurefactor intensities, respectively, and are the experimental uncertainties of the measured I_{obs}. The sum spans over all reflections. If α and β are picked randomly for each reflection, then each new set of Fourier map coefficients (1) will have a unique set of weights w and therefore will generate a unique Fourier map. We found that a useful range of α and β values is between 0 and 5, allowing the generation of maps that are sufficiently different in noise but that have the signal perturbed only a little, as illustrated in Figs. 3(a), 3(b) and 3(c).
The idea of using the weights in (3) was inspired by the empirical weighting scheme used in SHELX (Sheldrick, 2008),
where the parameter δ is a small numerical value. By defining δ per reflection similarly to (3) one can also perturb the map. However, we noticed that small values of δ (0 < δ < 1) do not change the maps visibly compared with using unit weights w, while using δ > 1 results in severely modelbiased maps, as exemplified in Fig. 4.
In our approach, an ensemble of Fourier map coefficients is generated by repeated (N times) calculation of individual sets of map coefficients using equations (1) and (3) with F_{fill} obtained as described in §2.2.
Since an ensemble of Fourier maps (1) is generated, this provides an opportunity to randomize the generation of F_{fill} each time the new map is calculated. This is achieved by randomly removing an additional 10% of the atoms and randomly drawing k_{sol} and B_{sol} values from the distributions (0, 0.01, …, 0.4) and (20, 25, …, 80), respectively, for the purpose of F_{fill} calculation (§2.2).
A combined Fourier map is then obtained in the following way (step 3a in Fig. 1). Firstly, N calculated sets of Fourier map coefficients are split into n groups each containing N/n sets of map coefficients. The map coefficients in each group are then averaged, 5% of the map coefficients (reflections) are randomly removed from each averaged set of coefficients and a Fourier map is calculated with the remaining reflections. This is repeated for each group of map coefficients, resulting in n Fourier maps (each with 5% of the reflections randomly removed) that are then averaged in real space to yield a single map. Fig. 3 illustrates the use of the randomized weights (3) in the calculation of a map using (1).
The mapaveraging procedure as described above reduces or even completely eliminates a large number of the noise peaks, but not all. Some persistent noise may remain, and some very low map values (for example, peaks smaller than ρ_{cut} = 0.5σ) will also always be present in the averaged map. Since later steps of the FEM procedure include nonselective equalization of map values, where the noise may be enhanced as well as the signal, it is essential that these undesirable map values are largely eliminated. This is achieved by the following two steps. One step truncates low map values at a chosen cutoff. The other identifies connected regions of the electrondensity map (referred to as blobs) that have a smaller volume than a typical volume of a reliably placed atom at some threshold level t_{1} that can be thought of as noise and considered for elimination. Moreover, we would not only like to eliminate those blobs shown at a threshold t_{1}, but also we want to remove their `roots': these same blobs as they appear at a lower threshold t_{2} = t_{1} − δ. Lowering the threshold to t_{2} may result in some blobs that were originally identified for elimination merging with blobs that were not considered for elimination. Such blobs are retained. Schematically, this is illustrated in Fig. 5. Fig. 5(a) shows blobs numbered 0, 1, 2, 3 at a threshold level t_{1}, where blobs 1, 2 and 3 are selected for elimination based on their volume. Fig. 5(b) is the same as Fig. 5(a) but contoured at a threshold t_{2}. Here, we will remove blobs 2 and 3, but blob 1 will be kept since it merges with blob 0 that was not selected for elimination. The mapconnectivity analysis algorithm used for the identification of blobs has been described by Lunina et al. (2003).
2.4. Composite residual (difference) OMIT map
An important element in the FEM procedure is restricting the final map to regions where there is convincing evidence of density. This is achieved by using a composite OMIT map. In contrast to the classical approach for calculating a composite OMIT map (Bhat & Cohen, 1984; Bhat, 1988), we implemented a procedure that is similar to that described by Cowtan (2012). The main differences from the existing approaches are that instead of omitting actual atoms in each omit box we zero the F_{model} synthesis in each box (omitting both the atomic and bulksolvent contributions) and compute modified structure factors from it that are in turn used to restore the density in the box by calculating a residual synthesis (2). All maps are computed and manipulated in the (GrosseKunstleve et al., 2011).
We use the OMIT map as a filter: all grid nodes in the OMIT map with values below a certain threshold are set to zero and the remainder are set to one. Intermediate maps of the FEM procedure are multiplied by the binarized OMIT map (Fig. 1). The threshold used to binarize the OMIT map is a parameter of the procedure and by default is 1σ.
The procedure used to calculate the OMIT map consists of three principal stages (Fig. 6). The inputs to the procedure are the observed structure factors (F_{obs}) and the atomic model. Firstly, a sampled model density map and bulksolvent mask are calculated from the atomic model. At this stage an empty (initialized with zeros) map is defined. By the end of the calculations this map will be the resulting composite OMIT map. Next, the model density and bulksolvent mask are Fouriertransformed, yielding structure factors F_{calc} and F_{mask}, and scales necessary to calculate F_{model}, k_{total} and k_{mask} are then obtained as described in Afonine, GrosseKunstleve et al. (2013). The final stage consists of a loop over boxes that cover the entire For each box, a larger box (wide box) is defined by expanding the original box by one grid node in each direction (the `neutral' volume in Bhat, 1988); the corresponding volumes inside the wide box in the model map and bulksolvent mask are then set to zero. The modified model map and bulksolvent mask are used to calculate the corresponding structure factors F_{calc_omit} and F_{mask_omit}. In turn, these structure factors and the previously calculated scales k_{total} and k_{mask} are used to calculate F_{model_omit}, the coefficients m and D and the residual Fourier map coefficients F_{diff} (2), which are then Fouriertransformed into a residual synthesis. Finally, the smaller box corresponding to the wide box is extracted from this synthesis and placed into the initially preemptied OMIT map result. The final composite residual OMIT map is obtained upon the completion of iterations over all boxes.
To examine the efficacy of the procedure, we performed the following numerical experiment. Firstly, we created an example that generates a modelbiased map as detailed in Fig. 7. We then applied the OMIT procedure described above. The inputs to the procedure were the amplitudes of structure factors F_{B} and model 1 that contains both residues. The local map for residue 2 was 0.99 calculated using the (F_{B}, φ_{A}) synthesis and was smaller than 0.01 for the OMIT map. This indicates that information about residue 2 is not visibly present in the OMIT map.
2.5. Map combination
The goal of this step is to combine the values for a series of N maps obtained as variations around the initial conditions. In particular, the maps corresponding to the same structure and the same crystal are calculated on the same grid and the Fourier coefficients are different but not substantially different. Usually N is of the order of 8–16. The combination is performed independently for each grid node. Two problems need to be addressed: (i) how to store all N maps simultaneously in a memoryefficient way and (ii) what statistical procedure to use to extract a `signal' from N map values in a given grid node (having N maps implies that each grid node has N values associated with it).
Typically, map values are stored as a set of fourbyte or eightbyte real values (eight bytes in cctbx; GrosseKunstleve & Adams, 2002; GrosseKunstleve et al., 2002), which means that having more than one or two maps simultaneously in memory may be prohibitive. To address this issue, we propose a way to convert realvalue maps into onebyte integervalue maps (integer*1 in Fortran or uint8_t or unsigned char in C++). In turn, this opens the possibility of having eight integervalue maps that would occupy as much memory as one eightbyte realvalue map. Assuming that two realvalue maps are the maximum that could be held in memory simultaneously, we arrived at 16 as the maximum number of integervalue maps that would be used.
Firstly, we scale these maps in quantile ranks (the same procedure as used below in applying histogram equalization; see §2.7 for details). The resulting maps vary from 0 to 1. If the original map has been truncated to be flat (set to zero) below some threshold (σ_{0}), then the new map will be flat below some q_{0} < 1 value. We then convert these maps into integer numbers in the range (0, 255) as follows:
Note that multiplying by 256 instead of 255 is computationally faster.
Now for each grid node we have N integer values j_{1}, j_{2}, …, j_{N} in the range (0, 255), where N, as noted above, is 16. There are a number of options to analyze an array of N values. Since outliers may influence the average value, we decided to use the most frequent value (the mode), which is the most persistent value in a given node. Given the rather small set of data points, 16, spread across the range (0, 255), it is likely that none of the 16 numbers will coincide exactly, making it impossible to calculate the mode. To overcome this problem, working with one grid node at a time, we transform its 16integer values into a realvalue array of length 256,
Here, each of 16 values j_{n} is blurred by a Gaussian. By trial and error, we found that values of b between 2 and 5 are optimal for such blurring. We then search for the argument j_{max} of the global maximum of the function f(j); note that f(j) is an integerargument realvalue function. Finally, we construct a quadratic interpolation for f(j_{max} − 1), f(j_{max}), f(j_{max} + 1) and take the real argument x_{max} of the maximum of this interpolation as the final value which is assigned to this grid node in the output combined map. Obviously, the resulting map may be scaled again in quantile ranks or in r.m.s. deviations.
Two particular situations are considered separately. The first is when all values j_{1}, j_{2}, …, j_{N} are very different and the function f(j) has N peaks of equal height (equal to one or marginally higher). We consider that the synthesis does not have any information in this node and we assign it the minimal possible value, i.e. zero. This means that the map region composed from such points will show no structural features at any cutoff level.
Secondly, the function may have several (at least two) strong peaks of approximately equal height, i.e. the values j_{1}, j_{2}, …, j_{N} form a few groups (clusters). By default, we take the peak with the smallest argument, in other words we generate the least noisy synthesis at the risk of a possible loss or decrease of signal; we call such a synthesis `the minimum synthesis'. Alternatively, one can generate `the maximum synthesis', where the peak with the maximum argument is taken instead. This synthesis contains the maximum signal at the risk that some noise may be highlighted.
2.6. Enhancing the signal by map sharpening
Combination of several maps into one single map may result in smearing of the peaks corresponding to atoms in the structure. In addition, density may be weak owing to thermal or static disorder, as well as owing to low completeness or low resolution. For example, density corresponding to two macromolecular chains that are near to each other may merge into continuous density if the data resolution is relatively low or/and the corresponding atoms have large atomic displacement parameters. In the FEM procedure we use two methods of map sharpening to improve the map interpretability: exponential (Bfactor) sharpening and unsharp masking.
2.6.1. Global map sharpening: Bfactor (exponential) sharpening
Bfactor sharpening of the map consists of multiplying the Fourier map coefficients in (1) by an exponential function of resolution exp(−B_{sharp}s^{2}/4), where s is a reciprocalspace vector and B_{sharp} is a sharpening B value. This technique has been shown to be useful in improving the interpretability of lowresolution maps (see Brunger et al., 2009 and references therein). Obtaining an optimal B_{sharp} value is not straightforward because there is no mathematical criterion available that quantifies the map improvement as a function of the choice of B_{sharp} value. Typically, an optimal B_{sharp} is found by a trialanderror method consisting of systematically sampling a range of plausible B_{sharp} values. The best value provides a reasonable compromise between signal improvement and increased noise, which is typically judged by eye. Other empirical approaches exist (DeLaBarre & Brunger, 2006). Here, we use the kurtosis (Abramowitz & Stegun, 1972) of the map to quantify the sharpness of the map features (`peakedness') and thus determine an optimal B_{sharp},
where the sums span over all map grid points, N is the number of grid points and is the average density. Values of B_{sharp} are tested in a defined range and the value that maximizes the map kurtosis is selected. Others have also pointed out that kurtosis may be used to characterize crystallographic maps (see, for example, Lamzin & Wilson, 1993; Pai et al., 2006).
Below, we provide the basis for identifying map kurtosis as a metric for selecting B_{sharp}. We examined how data and model properties, e.g. resolution and B factors, are related to map kurtosis and why it may be a good measure to use in selecting B_{sharp}. For this, we used a test example of two atoms, Mg and O, placed inside a 10 × 5 × 5 Å box in P1 at approximately 3 Å distance from each other. Each atom has unit occupancy. Fig. 8(a) shows the distribution of 1.5 Å resolution Fourier syntheses values along the Mg—O vector corresponding to three cases with all atoms having B factors equal to 10, 30 and 50 Å^{2}, respectively. As expected, the larger the B factor the more smeared out the peaks are. Sampling a larger Bfactor range, we plotted map kurtosis as function of B (Fig. 8b). We observe that larger B factors smear peaks more and diminish the map kurtosis. In the next test, we varied the resolution of the synthesis, keeping the B factor fixed at 10 Å^{2}. Fig. 8(c) shows Fourier syntheses values along the Mg—O vector corresponding to 1, 2 and 3 Å resolution, respectively. We also plot map kurtosis as a function of resolution (Fig. 8d). We observe that lowering the resolution smears the peaks and also lowers the kurtosis of the map.
From the two numerical experiments presented above, we surmised that map kurtosis might be a good measure of not only the map sharpness but also of the heights of peaks in the map. Fig. 9 illustrates kurtosis using four scenarios: one sharp peak (Fig. 9a), the same sharp peak and some noise (Fig. 9b), a smeared peak (Fig. 9c) and a sharp peak and stronger noise (Fig. 9d). Clearly, broadening the peak lowers the kurtosis as well as adding noise. This example also illustrates the effect of Bfactor sharpening and its relationship to kurtosis. Indeed, Bfactor sharpening enhances both the signal and the noise.
To explore this further, we performed another test using the same construct of two atoms as above. Firstly, we set the B factors of both atoms to 25 Å^{2} and calculated a 1.5 Å resolution synthesis. We then sampled sharpening B factors, B_{sharp}, in the (−100, 100 Å^{2}) range, B_{sharp} scaling the map coefficients for each trial as described at the beginning of this section and computing the kurtosis of the corresponding synthesis. Fig. 10(a) shows the result of applying four selected B_{sharp} values and Fig. 10(b) shows the dependency of map kurtosis on B_{sharp}. We note that the sharpening B value that results in a maximal peak enhancement but also a minimal increase in noise is that which offsets the effective B factor of the two atoms to zero: a sharpening B value of 25 Å^{2}. This value also corresponds to the peak of map kurtosis shown in Fig. 10(b).
From this, we conclude that map kurtosis may be a good metric for map sharpening and can be used to obtain the optimal B_{sharp} value. Empirically, we note that it is most effective if map noise is reduced before Bfactor sharpening by density modification or by map combination as described above. §3 provides more examples using experimental data.
2.6.2. Local map sharpening: unsharp masking
Unsharp masking (see, for example, McHugh, 2005) is a digital imageprocessing technique that we use in the FEM procedure for the purpose of map sharpening. The calculation involves two steps. Firstly, a locally averaged map (M_{ave}) is calculated by replacing each grid point value in the original map (M_{orig}) with the average value calculated over all 27 nearestneighbor grid points. The sharpened map is then calculated as max(M_{orig} − M_{ave}, 0). This type of sharpening has the effect of amplifying highfrequency components in the map and is useful for improving the clarity of images, although the sharpened image can also have unwanted edge effects. We note that applying unsharp masking (UM) increases map kurtosis.
To illustrate the effect of the two mapsharpening options, exponential and unsharp masking, we consider a test example similar to that used in the previous section: C and N atoms are placed in the middle of a P1 box at about a distance of 1.45 Å apart. Each atom has unit occupancy and a fixed B factor of 15 Å^{2}. Fig. 11 shows five curves: blue is the exact electrondensity distribution calculated using the Gaussian approximation formula [see, for example, formula 12 in Afonine & Urzhumtsev (2004) implemented as described in GrosseKunstleve et al. (2004)] and red is for its Fourier image calculated at 1.5 Å resolution. Both exponential (green) and unsharp masking (purple) sharpen the map similarly, while applied together they act synergistically (black).
2.7. Equalizing the signal: histogram equalization (quantile rank scaling)
The goal of this step is to make the strong and weak signal in the map similar in strength. Indeed, when interpreting a map it is often less important how strong the strong signal is, as long as it is strong enough to be reliably distinguished from the noise and interpreted in terms of the atomic model. Therefore, assuming that the noise has been efficiently suppressed in a previous step, it is reasonable to expect that equalizing the map values across the unitcell volume will yield a more easily interpretable map.
The histogramequalization (HE) method (see, for example, Hummel, 1975, 1977) is used to transform map values over the entire unitcell volume to generate a uniform distribution of map values. This method nonlinearly (but monotonically) transforms the image values to enhance the contrast and is a routine tool in the digital imageprocessing field. Fig. 12 illustrates the effect of HE applied in two dimensions to a photograph. The method works similarly in three dimensions when applied to Fourier maps. After applying HE to a crystallographic map the strong and weak peaks become more equal (Fig. 13). Obviously, the noise is equalized as well (compare Figs. 14a and 14b), so it is essential that it is maximally removed before applying HE. Fig. 15 provides an additional (and somewhat simpler) illustration of the effect of HE on a 1 Å resolution Fourier map calculated using a test model of three atoms (Mg, O and H) placed in a P1 box. The three atoms were chosen specifically to yield map peaks of very different heights.
The histogramequalization procedure consists of two steps. Firstly, the cumulative histogram of the map is calculated. Each pixel of the map is then replaced with the corresponding value taken from the cumulative histogram. This is equivalent to rescaling the map in quantile ranks as described by Urzhumtsev et al. (2014). The cumulative histogram of the resulting map is a straight line, which is the diagonal of a (0, 1) box (Fig. 14d). Note that the HE map may be scaled in r.m.s. deviations: since this is a linear transformation, it does not modify the features of the HE map.
2.8. Other tools that are potentially useful in the FEM calculation procedure
As part of developing the FEM method, we also explored other alternatives for signal equalization or enhancement and noise reduction. They are not part of the current implementation of the FEM procedure and are listed below for the sake of completeness.
We developed and tried a procedure for the combination of two maps that enhances the features that are present in both maps. This procedure is related to the minimum function that is often used in the analysis of Patterson maps (Buerger, 1970; Terwilliger et al., 1987). Two maps are analyzed in this procedure, which are assumed to be on the same scale. For each threshold level ranging from s_{1} to s_{2} with a certain step, values in both maps are retained if they are simultaneously above or below the s_{1} + step level; otherwise, they are set to zero. The final map is the average of the two modified maps. The actual values of the thresholds s_{1} and s_{2} depend on how the maps are scaled. For example, if both maps are histogramequalized (see §2.7) then s_{1} and s_{2} are equal to 0 and 1, respectively, and the step can be 0.1, for instance. The problem that we encountered with this approach is that eventually it creates `holes' in the map set to zero if both values do not intersect while all surrounding grid nodes are different from zero. It may also result in the creation of sharp edges in the map, which may potentially result in strong Fourier ripples if such a map is subjected to a Fourier transform. Finally, combining maps considering two maps at a time is less powerful than the analysis of all available maps simultaneously.
We have tried other (than Bfactor or unsharp masking) sharpening techniques such as a Kuwahara filter (a type of median filter; Kuwahara et al., 1976), as has previously been reported (Diaconu et al., 2005) to be useful in interpretation of lowresolution electronmicroscopy maps. However, this method was not successful in our tests.
For map equalization an alternative is to partition the entire
into boxes, compute the r.m.s. deviation of the map in each box and then scale the map values in each box with it individually. This approach requires treating macromolecular and solvent regions separately, which is not always convenient or straightforward, and also requires knowledge of the solvent mask.An approach that may enhance weak signal in the map and suppress noise is to apply a polynomial density modification (Hoppe & Gassmann, 1968; Collins, 1975; Raghavan & Tulinsky, 1979). Our experience was that this works well in many instances, but there are caveats. Firstly, the density map has to be truncated and rescaled so that its values are between 0 and 1, which obviously requires the application of some cutoffs, the choice of which may not always be obvious (especially automatically in software). Secondly, the transformation function has an inflection point at 0.5, which means that values below 0.5 are suppressed and values above 0.5 are enhanced. Clearly, this implies an arbitrary assumption that all peaks below 0.5 are noise and are signal otherwise.
We therefore prefer the histogramequalization method as it efficiently equalizes the contrast while it does not require ad hoc parameters to be defined.
2.9. Note about the comparison of maps
In the examples below, we show maps obtained as the result of one manipulation or another. When several maps are shown it is often necessary to display them at equivalent contouring levels. Typically, this is performed by choosing a contouring threshold in r.m.s. deviations (σ) and using it for all maps that are subject to comparison. As pointed out by Urzhumtsev et al. (2014), this is at best suboptimal and at worst misleading, as even maps contoured at identical r.m.s. deviation thresholds may not be comparable. A better approach is to use thresholds that select equal volumes of the map, which may not correspond to identical thresholds as expressed in r.m.s. deviations (Urzhumtsev et al., 2014). In the cases in the present work, if a figure shows more than one map the contouring levels are chosen such that they encompass identical volumes of the map. In practice, obtaining equivalent contouring levels expressed in r.m.s. deviations is achieved in the following way. Firstly, two cumulative distribution functions (CDFs) are calculated from the two maps. Then, given the selected contouring level for the first map, the corresponding CDF value is obtained and this value is used to obtain the corresponding argument for the CDF of the second map, which is the desired threshold value (see Fig. 16). This can be performed using the phenix.map_comparison tool, which implements the methodologies described in Urzhumtsev et al. (2014).
3. Results
Unlike density modification (classical or statistical), the FEM is not an iterative phaseimprovement procedure and it is generally not expected to reveal features that are not already present in the original map from (1). Instead, it makes the available signal more apparent by suppressing the noise and by equalizing the magnitude of the weak signal with the strong signal.
The FEM procedure can be applied to Xray and nuclear density maps and has been tested at resolutions from high (1 Å) to low (about 4 Å). The FEM protocol was designed and optimized for maps corresponding to neartofinal models when the noise is usually smaller than the signal. For initial and intermediate cases the amount and magnitude of the noise may be comparable with the signal, making the noise very difficult to suppress efficiently. In such cases FEM maps should be used with care. However, FEM maps are expected to be less modelbiased than the starting maps as they are filtered by the composite residual OMIT map.
Keeping Fig. 15 in mind, we note that FEM maps remain similar over a broad range of contour levels. If the FEM is scaled by the r.m.s. deviation [which is likely to be the default in many moleculargraphics tools such as Coot (Emsley et al., 2010) and MAIN (Turk, 2013)], then thresholds between 0 and 1σ are generally desirable. Also, it is important to note that synthesis (1) and the FEM are likely to be less comparable at identical contouring levels (§2.9).
Below, we provide a number of examples of applying the FEM to selected structures that illustrate typical outcomes of the procedure. In the majority of cases the FEM map shows previously missing density for side chains, enhanced density for disordered loops and alternative conformations. FEM maps are typically cleaner overall and flatter in bulksolvent regions. They are also a better target for realspace refinement.
3.1. PDB entry 1f8t : better sidechain density
Fig. 17 shows no density (above the 1σ threshold) for the side chain of Arg74 (chain L) in the map from (1). At the same time, the composite residual OMIT map calculated with (2) as described in §2.4 shows sidechain density at a level of 1.5σ. FEM reveals density for the entire residue at a level equivalent to 1σ. For comparison, we calculated a RESOLVE densitymodified map, which did not show any density above the 0.8σ level.
3.2. PDB entry 1nh2 : cleaning an overall noisy map
The map from (1) is unusually noisy and notoriously patchy for this structure. Fig. 18 shows an example of typical map improvement throughout the entire model. Fig. 19 provides an overall view of the macromolecule–solvent interface and shows significant noise reduction in the bulksolvent region as a result of applying FEM.
3.3. PDB entry 2r24 : the structure of aldose reductase obtained using neutron data
We have examined the outcomes of FEM calculations for about a dozen neutron structures selected from the PDB. The general observations stated for Xray structures also hold for structures refined against neutron data. Fig. 20 illustrates the application of FEM to a neutron structure. In this example the somewhat broken ligand density map from (1) improves after calculating the FEM.
3.4. PDB entry 2g38 : a highly anisotropic data set
This is a classical example of data with severe anisotropy and has served as an example for the Anisotropy Correction Server (Strong et al., 2006). The original diffraction data were kindly provided by the author (Michael Sawaya), as the corresponding PDB entry contains only an anisotropycorrected version of the measured intensities. Fig. 21 shows three maps: the original map from (1), the FEM and a densitymodified RESOLVE map. The latter two maps show a notable reduction in anisotropy effects (elongation of map contours along the vertical direction).
3.5. PDB entry 1se6 : an incorrectly placed ligand
Pozharski et al. (2013) pointed out a misfitted ligand in this entry. While the Fourier map from (1) does not clearly identify the ligand at 1σ (Fig. 22a), featureenhanced maps calculated both with (Fig. 22b) and without (Fig. 22c) the ligand present in the input model file better identified the ligand. Also, composite residual OMIT maps calculated with (Fig. 22d) and without (Fig. 22e) the ligand both unambiguously confirm its identity. A RESOLVE densitymodified map barely shows the ligand density. Correcting the ligand fit is obvious based on the residual OMIT or FEM maps, and a subsequent round of yields a clear ligandomit Fourier map from (1).
3.6. PDB entries 2vui and 2ppi : the importance of map sharpening
The two examples in Fig. 23 illustrate the importance of using map sharpening. Typically, sharpening deblurs the image, making the features more distinctive, and often reveals features that are not otherwise visible, such as residue side chains. As expected, sharpening does increase the map noise; however, when applied as part of the FEM protocol the noise arising from sharpening is mostly eliminated.
3.7. PDB entry 3i9q : FEM as a better target for realspace refinement
To further illustrate the FEM procedure, we performed the following test. We selected PDB entry 3i9q (data resolution 1.45 Å), which we rerefined using phenix.refine (Afonine et al., 2012). Using this model, we calculated two maps: the FEM and the usual map from (1) (Figs. 24a and 24b). Since we wanted to compare the original σ_{A} map (Read, 1986) with the FEM, no modeling of missing reflections, anisotropy correction or sharpening was performed. What is remarkable about this structure is that the map from (1) is outstandingly poor, perhaps owing to poor completeness in the lowresolution zone (50% completeness in the 17.49–6.58 Å zone). We then removed everything but the macromolecule from the model and subjected this model to 1000 independent moleculardynamics (MD) simulation runs using phenix.dynamics, with each run continuing until the rootmeansquare deviation (r.m.s.d.) between the starting and current models exceeded 3 Å. This generated a diverse ensemble of 1000 structures as shown in Fig. 25. Next, each model from the ensemble was subjected to ten macrocycles of realspace using phenix.real_space_refine (Afonine, Headd et al., 2013) against the FEM and the map from (1) independently. Each macrocycle included model morphing (Terwilliger et al., 2012), simulated annealing with a slowcooling protocol starting from 5000 K and local and overall gradientdriven minimization (Afonine et al., 2012). The success was quantified by calculation of the r.m.s. deviation between the unperturbed model (before MD) that we consider to be the best available model and the model after realspace Fig. 26 shows a histogram of r.m.s. deviations for the outcomes against the FEM and the map from (1). Clearly, in a majority of cases against a featureenhanced map resulted in refined models that were significantly closer to the true structure than those refined against a standard map.
4. Conclusions
The Fourier maps routinely used in crystallographic structure solution are never perfect owing to errors in both the experimental data and the parameters of the structural model. Various kinds of map errors may impede structure solution and completion or result in erroneous map interpretation, which in turn leads to an incorrect atomic model.
Over the decades, great effort has been put into the development of methods to improve crystallographic maps. However, most existing methods only address one or a few map qualityrelated problems at a time. Also, some of the existing methods designed to address one problem may make other problems worse. More thorough and efficient methods are typically very computationally expensive (they may take hours or days to compute) and also may require casespecific parameterization as they use
or model building.In this manuscript, we have presented a novel method of crystallographic map modification that simultaneously combines several desirable maps, requires minimal inputs (i.e. the current atomic model and diffraction data), does not require timeconsuming calculations (such as or model building) and is relatively fast to calculate (from less than a minute to a few minutes). We call the map obtained as a result a featureenhanced map (FEM).
All of the key tools used in the FEM calculation (including the FEM protocol itself) are implemented as part of cctbx. The calculation of OMIT maps is available as a commandline tool called phenix.composite_omit_map (Echols & Afonine, 2014). The FEM calculation shown here is available in PHENIX (Adams et al., 2010) starting with version dev1832 from the command line (phenix.fem) and in the PHENIX GUI. The data and scripts used to obtain the figures in this manuscript are available at https://phenixonline.org/phenix_data/ .
Acknowledgements
This work was supported by the NIH (Project 1P01 GM063210) and the Phenix Industrial Consortium. This work was supported in part by the US Department of Energy under Contract No. DEAC0205CH11231. AU thanks the French Infrastructure for Integrated Structural Biology (FRISBI) ANR10INSB0501 and Instruct, which is part of the European Strategy Forum on Research Infrastructures (ESFRI). DT acknowledges support from Structural Biology Grant P0048 from the Slovenian Research Agency.
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