The focus of this paper is on practical applications, specifically on workflows and conventions that are implicit to Situs programs. Over the years, owing to the modular design, the number of possible combinations of programs has increased to a point where it has become difficult to document possible workflows in our online tutorials. Based on specific modeling tasks, some new usage examples are provided here to inspire users to experiment on their own. Also, in an effort to bridge Situs to other software, many of the implicit conventions are described for the first time.

Our online tutorials (http://situs.biomachina.org) now include Unix bash-shell scripts for the automatic generation of tutorial solutions. Much of the workflow complexity originates from the command-line-based scripting that allows programs to be combined in creative ways. Fig. 1 shows a typical example. The hypothetical problem is that one would like to bring two volumetric density maps into register. The maps can be format converted (§2) with the map2map tool or processed with ‘volume algebra’ tools (§8) such as voledit. For technical reasons, the rigid-body matching tools collage, colores and matchp(oin)t require an atomic PDB file for docking to a target map (Fig. 1 ). Therefore, the second map must be intermittently transformed to the atomic (PDB) format using vol2pdb so it is free from the cubic lattice (for the rotation and translation in the docking). After the matching of the pseudo-PDB map, it can be interpolated back into the volumetric format through projection onto the original lattice with pdb2vol.

Many of the Situs tools rely on implicit conventions for setting parameters. It is perhaps surprising to readers from the crystallographic community that important parameters of volumetric density maps such as resolution, density levels and even map formats are not strictly defined in the hybrid modeling community. For example, the surface isolevel used for visualizing a volume map is an intuitive concept that is surprisingly difficult to solve computationally. Although one can attempt to set the isolevel based on the enclosed volume (Harpaz et al., 1994 ), the resolution lowering leads to a shift in density, eroding convex features and filling up concave features of the atomic structure. To prevent convex features from protruding from the low-resolution surface, the isolevel of cryo-EM maps is empirically set to enclose 120–150% of the molecular volume depending on the overall shape of the system. This is just one example where empirical ‘fudge factors’ trump first principles. Over time, software developers have implemented conventions for a multitude of such quantities as they were breaking new ground. Although an effort is under way to standardize such conventions (Heymann et al., 2005 ), it is still often necessary to investigate the source code when sharing data between different software packages. In an effort to create more transparency, the most important Situs conventions are documented here.

Volumetric maps (Fig. 1 ) come in a variety of map formats, many of which were inspired by crystallographic formats. For example, the map2map format-conversion utility of Situs accepts cryo-EM and crystallographic density files in ASCII (text), CCP4, MRC, Situs, SPIDER and X-PLOR formats, automatically adjusts to the machine architecture (endianism) and supports permutation of axes as well as non-orthogonal unit cells (this is accomplished by trilinear interpolation to a cubic lattice, if necessary). In the late 1990s there were no universally accepted standards in the cryo-EM community for setting the origin of the map coordinate system, which is critical for the docking of atomic structures. The minimalist Situs format was conceived to keep track of the coordinate system, to enforce a cubic lattice and to be independent of the ever-changing map-format standards in the community. Although this ASCII-based (and thus easily readable/editable) format was initially meant to be of limited use only within the Situs package, it is now supported by the molecular-graphics programs VMD (Humphrey et al., 1996 ), Chimera (Pettersen et al., 2004 ) and Sculptor (Birmanns et al., 2011 ), by the EMAN2 reconstruction package (Ludtke et al., 1999 ) and also by the em2em format-conversion tool (http://www.imagescience.de/em2em). In the map format, a short header holds the voxel spacing WIDTH, the map origin as defined by the three-dimensional coordinates of the first voxel ORIGX, ORIGY, ORIGZ and the map dimensions (number of increments) NX, NY, NZ. This minimalist header is followed by the data fields such that x increments change fastest and z increments change slowest.

Situs also supports three-dimensional bead models from SAXS (Chacón et al., 2000 ). Fig. 2 shows an application of SAXS-related tools in the visualization and atomic interpretation of SAXS-derived shapes (Wriggers & Chacón, 2001b ). To test the docking accuracy, the pdb2saxs tool was created, which projects atomic structures onto a hexagonal close-packed lattice, with user-defined bead radii written to the PDB occupancy field. The resulting models (Fig. 2 b) served as ‘simulated’ low-resolution data in Wriggers & Chacón (2001b ).

The bead models in PDB format can be transformed into density maps for subsequent docking using a hard-sphere kernel in pdb2vol. The SAXS modeler then has access to all docking strategies supported by Situs, including correlation-based docking (collage or colores), feature-point based matching (matchpt) and even flexible real-space fitting (§8).

One problem in the interpretation of SAXS data is the visualization of the beads. It is useful to render not the densely packed beads themselves (Fig. 2 b), but a transparent wire mesh or envelope (Fig. 2 c) that shows the fitted atomic structure. This envelope was created by isocontouring an intermediate volumetric map, which was generated from the beads by convolution with a soft kernel such as a Gaussian (using pdb2vol).

Resolution R is a quantity in Fourier space and has dimension Å⁻¹, but the real-space quantity r = R ⁻¹ is also often termed ‘resolution’ in biophysical parlance. The symbols r and R are used here to differentiate between the spatial and frequency domain.

In crystallography, let R _c be the radius of a circular region within which Fourier terms contribute to the crystallographic synthesis; one can then say that r _c = R _c ⁻¹ is the crystallo­graphic resolution (Frank, 2006 ). In contrast, Rayleigh con­sidered the resolvability of two points in real space (Stenkamp & Jensen, 1984 ). The real-space image of one point, whose Fourier transform is limited by a disk of radius R _c, is the Airy pattern [J ₁(2πrR _c)/(2πrR _c)]², where J ₁ is the first-order Bessel function. The Rayleigh criterion is satisfied when the first minimum of one point’s Airy pattern coincides with the central maximum of the other. This critical point-to-point distance r _p turns out to be r _p = 0.610r _c (see Appendix A3 in Radermacher, 1988 ). The Airy pattern is dominated by a bell-shaped central Airy disk, which measures a ‘full-width at half-maximum’ FWHM = 0.514r _c. Ignoring the weak outer rings (resulting from the hard limit in Fourier space), the Airy pattern can be approximated by a Gaussian function that matches the Airy disk (http://en.wikipedia.org/wiki/Airy_disk).

In the development of the pdb2vol tool of Situs in the 1990s, the width of the Gaussian convolution kernel was set empirically to mimic cryo-EM maps at published resolution values (for resolution estimation in cryo-EM, see Frank, 2006 ). The empirical Situs resolution r _s was set to r _s = 2σ (or 1.471 FWHM), where σ is the three-dimensional standard deviation of the Gaussian kernel, exp(−3r ²/2σ²). One can relate the Situs resolution r _s to the crystallographic resolution r _c and to the point resolution r _p by matching the Gaussian to the Airy disk of the same FWHM. It follows that r _s = 1.239r _p = 0.756r _c. Note that the Sculptor molecular-graphics software (Birmanns et al., 2011 ) from our laboratory also follows this convention.

The pdb2mrc tool of the EMAN package (Ludtke et al., 1999 ; renamed e2pdb2mrc.py in EMAN2) is another popular resolution-lowering tool which established its own resolution measure, r _e. In EMAN, the functional form of the Gaussian real-space kernel is exp(−π² r ²/r _e ²). The Fourier transform exp(−r _e ²/R ²) is not strictly limited to a hard radius, as in the crystallographic case, but a reasonable radius of the soft Gaussian in Fourier space is set by EMAN at the 1/e cutoff where R = R _e. After matching the Gaussian real-space kernel to the Airy disk, it follows that r _e = 1.886 FWHM = 1.589r _p = 1.282r _s = 0.969r _c.

The idea of using coarse-grained models of feature points for the docking of multi-scale structures is a classic idea that predates Situs (Wriggers et al., 1998 ). We have shown in earlier work that this approach is advantageous at resolutions below 10 Å because the points provide ‘interior features’ (and an encoding of the molecular shape) even in the absence of interior density variations from the secondary structure. However, the earlier Situs tools had a limited number of feature points and required both point clouds to be of equal size. We have recently published a new anchor-point registration technique that overcomes these combinatorial limitations and is able to dock smaller point clouds to larger ones (Birmanns & Wriggers, 2007 ).

Fig. 3 shows a typical application of feature-point-based rigid-body matching. The matchpt utility is a command-line program for matching arbitrary-sized three-dimensional point sets (coarse-grained models), which can be generated on the fly or by using the output of the Situs programs qpdb and qvol. The utility can dock a subunit into a larger target map, i.e. find N feature points within another set with M points, N < M, and match them. To solve this problem, matchpt uses a heuristic and investigates only a subset of all possible permutations of feature points (Birmanns & Wriggers, 2007 ).

The idea of matching point sets was based on the observation that for many low-resolution maps numeric values of the cross-correlation (CC) are often in a narrow range and less discriminatory compared with the r.m.s.d. values of the feature points (Wriggers et al., 1999 ). This is a consequence of the fact that feature points can reliably and reproducibly encode the molecular shape, even in the absence of interior (secondary-structure) density features. Therefore, it makes sense for difficult low-resolution maps to use matchpt as an alternative to the CC-based tools colores and collage. In the default mode, a user would explore the quality of the match of the point clouds by minimizing their r.m.s.d. Alternatively, the minimum of the statistical variability (here the sum of average variabilities of both point sets) can be used to select an optimum N and M, since this variability was found to be a good estimator for the docking accuracy (Wriggers & Birmanns, 2001 ). Finally, a user may wish to explore the standard cross-correlation (CC), which is discretely sampled by the solutions of the point-cloud matching.

The Situs cross-correlation coefficient (CC) for volumetric maps ρ_vol(r) and ρ_calc(r) is defined aswhere ρ_vol(r) is a low-resolution map and ρ_calc(r) is an atomic structure subject to rigid-body movements, projection to the volume V and convolution with a Gaussian kernel (Wriggers & Chacón, 2001a ). The CC values are normalized, but unlike in the Pearson correlation coefficient the averages are not subtracted from the densities. Consequently, the calculation is slightly more efficient than that of the Pearson CC. The convention takes into account that ρ_calc(r) often has the physical meaning of a density with well defined positive amplitude that corresponds to the represented low-resolution structure. This way, CC [0, 1] for ρ_vol(r), ρ_calc(r) > 0. One can show that maximizing the Situs CC value also maximizes the Pearson correlation coefficient, so dropping the subtraction of averages in (1) yields no performance penalty. The Situs CC convention has also been adopted by the Sculptor visualization program (Birmanns et al., 2011 ).

Chacón & Wriggers (2002 ) introduced colores, a widely used CC-based fitting tool that takes advantage of Fourier correlation theory to rapidly scan the translational degrees of freedom of a probe molecule relative to a (fixed) target-density map (whereas the rotations are sampled exhaustively by the enumeration of a list of homogeneously distributed Euler angles). The performance of the standard CC (1) is limited to resolutions higher than 10 Å, where densities exhibit the internal (i.e. secondary) structure. The major advantage of colores is that it extends the viable resolution range to ~30 Å by means of an (optional) Laplacian operator [applied to both ρ_vol(r) and ρ_calc(r) in equation 1] that emphasizes shape-contour information in addition to the traditional volume correlation (also, a masking filter was implemented that suppresses singularities of the Laplacian of ρ_vol at density edges resulting from thresholding or segmentation).

Recently, the new refinement tool collage was introduced which performs a conjugate-gradient optimization of the same scoring functions known from colores. The main innovation is the simultaneous optimization of multiple rigid fragments that ‘see’ each other and avoid steric clashes by means of the normalization in (1). Birmanns et al. (2011 ) showed that this approach yields more accurate fits. Also, if all density is accounted for by the fragments it is no longer necessary to use the Laplacian filter option, even at low resolution.

The new collage tool and external symmetry-manipulation programs can be combined to impose symmetry constraints on the fragments during multi-fragment docking. Fig. 5 provides an overview of the workflow. Standard volumetric map formats are converted to cubic lattices in Situs format with the map2map utility. Subsequently, the volume data are inspected and, if necessary, prepared for the fitting using a variety of map tools (§8). A data-type conversion (§3) is optional. The fitting of multiple PDB input files to the target map is handled by collage as described in §6. Symmetry constraints are outsourced to a separate program in the Unix bash-shell script. The Situs-native pdbsymm tool allows the generation of multiple symmetry-related copies following symmetry-axis conventions based on the target map. C, D and H (helical) symmetry options are currently supported. For other specialized cases (e.g. crystallographic symmetry), an alternative program (such as the CCP4 pdbset tool; http://www.ccp4.ac.uk/html/pdbset.html) may be substituted for pdbsymm in the workflow (Fig. 5 ). The entire Unix bash-shell implementation then proceeds as follows.

The goal of the scripting approach is to keep Situs tools modular and to avoid having to write specialized tools for every possible symmetry scenario. This means that collage will technically still treat each fragment as independent. However, collage will take only a single conjugate-gradient step (step iv), after which the symmetry will again be enforced (in steps ii and vi). The net effect after several iterations of the loop is a symmetry-enforced conjugate-gradient optimization. An example of this approach is available online in the multi-fragment tutorial (file run_tutorial.bash at http://situs.biomachina.org/tutorial_multi.html).

One active research area that exceeds the scope of this article is flexible docking, which bring deviating features of multi-resolution structures into register (Wriggers et al., 2004 ; Wriggers, 2010 ). Fig. 6 shows the differences between flexed structures of RNA polymerase in the form of two-dimensional projections. Systematic tests and validations of flexible fitting with spatial interpolation have been carried out using the qplasty tool (Rusu et al., 2008 ) and experimental applications of flexible docking (using molecular-dynamics refinement) were performed in collaboration with experimental laboratories on systems such as RNA polymerase (Darst et al., 2002 ) and the thick filament of tarantula muscle (Alamo et al., 2008 ).

Another noteworthy application that exceeds the scope of this paper is ‘volume algebra’, in which map densities are modified according to simple algebraic operations (Wriggers et al., 2011 ). Volume-algebra operations are typically enabled by map-density registration (Fig. 1 ) and include map summation or averaging (volaver), difference mapping (voldiff), binary masking and map multiplication (voledit, volmult) and density matching using an affine transformation (volhist), as well as cropping, thresholding and segmentation (voledit). For application examples of volume algebra operations, see Fig. 1 in Wriggers et al. (2011 ) and the online tutorials.

The performance of multi-fragment-based refinement (Figs. 4 and 5 ) is the subject of ongoing research. Our empirical tests with experimental maps have shown that the refinement is very robust. The normalization of the cross-correlation coefficient (1) penalizes steric clashes between fragments, similar to features of a tabu search using genetic algorithms (Rusu & Birmanns, 2010 ). Also, the radius of convergence for the method appears to be rather large, reducing the need for exhaustive exploration. The performance will be evaluated further in future work.

Situs has been ported to multiple platforms and the source code is freely available at http://situs.biomachina.org.