research papers
A smooth and differentiable bulksolvent model for macromolecular diffraction
^{a}Department of Molecular and Cellular Physiology and Howard Hughes Medical Institute, Stanford, California, USA, ^{b}Department of Chemistry, Stanford, California, USA, and ^{c}Departments of Neurology and Neurological Sciences, Structural Biology and Photon Science, Stanford, California, USA
^{*}Correspondence email: brunger@stanford.edu
Inclusion of lowresolution data in macromolecular crystallography requires a model for the bulk solvent. Previous methods have used a binary mask to accomplish this, which has proven to be very effective, but the mask is discontinuous at the solute–solvent boundary (i.e. the mask value jumps from zero to one) and is not differentiable with respect to atomic parameters. Here, two algorithms are introduced for computing bulksolvent models using either a polynomial switch or a smoothly thresholded product of Gaussians, and both models are shown to be efficient and differentiable with respect to atomic coordinates. These alternative bulksolvent models offer algorithmic improvements, while showing similar agreement of the model with the observed amplitudes relative to the binary model as monitored using R, R_{free} and differences between experimental and model phases. As with the standard solvent models, the alternative models improve the agreement primarily with lower resolution (>6 Å) data versus no bulk solvent. The models are easily implemented into crystallographic software packages and can be used as a general method for bulksolvent correction in macromolecular crystallography.
Keywords: bulksolvent models.
1. Introduction
The size, shape and crystallographic packing of macromolecules leads to interstitial spaces that occupy a significant portion (typically >40%) of the crystal volume (Matthews, 1968). The solvent surrounding the protein is typically only visibly ordered within the first shell of hydration, while the scattering of the remainder can be approximated as arising from a continuum. In macromolecular crystallography, this effect is usually modeled by defining a region of zero density inside the solventaccessible surface, while the area outside is treated as a constant (i.e. flat) scattering volume (which we refer to as the `binary' model). The resulting binary mask is Fouriertransformed and added to the atomic structure factors, yielding a total scattering factor F_{t},
where F_{c} are the structure factors computed from the molecule, F_{s} are the structure factors from the Fouriertransformed binary mask, k_{s} is the electron density of the bulk solvent in units of electrons per Å^{3}, B_{s} is a B factor that represents the isotropic thermal disorder of the solvent and s is the reciprocallattice vector. The effect of exponential multiplication by B_{s} in is smoothing of the bulksolvent model in real space (Fokine & Urzhumtsev, 2002).
The binarymask model was initially proposed by Phillips (1980) and later adapted in XPLOR (Jiang & Brünger, 1994) using a version of the Lee and Richards solventaccessible surface model (Lee & Richards, 1971). The constant k_{s} and B factor B_{s} can be optimized against the diffraction data and recent efforts have improved the robustness of the solventparameter optimization in PHENIX and CNS (Fokine & Urzhumtsev, 2002; Afonine et al., 2005; Brunger, 2007; Adams et al., 2010). This approach represents the current standard and has been incorporated into most modern crystallographic software packages. However, by virtue of the binary nature of the mask used to calculate F_{s}, this bulksolvent model is `jumpdiscontinuous' at the solute–solvent boundary (i.e. the mask jumps from a value of zero to one) and therefore is not differentiable with respect to the atomic coordinates. As a result, chainrule terms arising from the binary mask cannot be included during optimization of the coordinates using positional minimization or simulated annealing (Brünger et al., 1987). Therefore, the bulksolvent model is kept fixed until an update is performed; thus, the overall target function is not continuous. A potential problem with the binary mask is made apparent by considering that values in the mask can be flipped upon infinitesimal atomic coordinate changes (see Fig. 1).
The application of Babinet's principle to macromolecular crystallography was originally proposed by Moews & Kretsinger (1975) and involves using the same Fourier coefficients as derived from the atoms (F_{c} in equation 1) but with opposite phases to describe the bulksolvent scattering. An overlooked aspect of this approach is that the bulk solvent is differentiable with respect to the individual atoms and relies on the same derivatives as computed for the atomic model structure factors. However, the use of Babinet's principle as a bulksolvent model is uncommon owing to the poor agreement with the diffraction data relative to the binary model. This is a consequence of the fact that the phaseinverted F_{c} is not an adequate description of the electron density in the bulksolvent region, as it is not a characteristic function that shows the relatively featureless electron density characteristic of bulk solvent. Here, we propose a modification of Babinet's principle that uses a characteristic function rather than the F_{c}s and retains differentiability with respect to the atomic coordinates.
To generate a differentiable characteristic function, we first use atomcentered Gaussians, which has been suggested and implemented in various contexts in the past. Phillips' initial description of a bulksolvent correction used Gaussians, but the resulting Gaussian density was isocontoured at a selected density level where all points outside the isocontour were set to a constant density and all points inside were set to zero, yielding a binary model (Phillips, 1980). Roversi and coworkers used Gaussian smoothing of the molecular surface to assist ab initio phasing methods (Roversi et al., 2000). Kostrewa suggested the use of exponential smoothing as an improvement over the binary model for crystallographic data sets (Kostrewa, 1997; Fokine & Urzhumtsev, 2002), and a Gaussian model for permittivity and is common in biomolecular Poisson–Boltzmann calculations (Grant et al., 2001).
As an alternative to atomcentered Gaussians, we use a polynomial switch at the solute–solvent boundary. The simplicity of loworder polynomials offers a potential speed benefit over the Gaussian treatment, which is an important consideration for macromolecules with many atoms. The utility of polynomials and their derivatives for describing solute–solvent boundaries has been duly noted (Im et al., 1998; Schnieders et al., 2007) and is a critical part of Poisson–Boltzmann calculations for large systems (Baker et al., 2001). Polynomials have also been noted to stabilize moleculardynamics simulations in which implicit bulksolvent models are used (Arnold & Ornstein, 1994).
We describe a simple replacement of the solventmodel structure factors F_{s} using either a polynomial switch (which we refer to as the `polynomial' model) or a smoothly thresholded version of atomcentered Gaussians (referred to as the `Gaussian' model). We show that the polynomial and Gaussian models result in continuous target values as a function of coordinates and similar agreement with the diffraction data as the binary model, as monitored using R, R_{free} and differences between model and experimental phases. Finally, the Gaussian and polynomial models are differentiable with respect to atomic coordinates such that chainrule terms arising from the bulksolvent model can be included during positional minimization and simulatedannealing protocols.
2. Methods
2.1. Babinet's principle
The total scattering of a macromolecule in bulk solution is depicted pictorially in Fig. 2. The scattering from the macromolecule alone (gray; F_{c}) is added to the constant scattering from a bulk scattering mass (blue; F_{b}) minus the bulk scattering effect that would arise from the macromolecular mask alone (F_{m}). For the sake of simplicity, the symmetry of the system is assumed to be P1. The situation simplifies in as the Fourier transform of the constant scattering volume is zero (except at zero frequency, which is ignored in this case). Therefore, (1) can be reformulated as
This is Babinet's principle as it is typically applied in macromolecular crystallography, with the exception of inverting the phase of F_{m} rather than F_{c}. This is therefore opposite to the binary model in that the realspace mask is one inside the protein mask and zero elsewhere. F_{m} can be any function that varies from zero in the bulk solvent to one in the solute region.
For both the polynomial and Gaussian models presented below, we assume n atoms at individual coordinates r_{i} and an arbitrary grid point at r_{g}. The distance between the two vectors is defined as r = r_{g} − r_{i}.
2.2. Gaussian model
The derivation of the Gaussian bulksolvent model follows that of Grant et al. (2001). Briefly, starting with a Gaussian with variance σ^{2},
a function for the bulk mask at grid point r_{g} can be defined as a product of the densities of individual atoms,
which, ignoring atomic overlaps, is approximately equivalent to
To generate a characteristic function that smoothly varies from zero to one, the solute mask (note that the general term `mask' can be any density function, not just limited to values of zero and one as for a binary mask) is
A is a constant that scales the Gaussians. For this work, we chose a value of 11.5 Å based on the results of Grant et al. (2001).
Computation of the solute mask requires two loops, the first being over the atoms (to generate ρ_{sum}) and the second over the map to carry out the exponentiation in the above equation. This is similar to the binary bulksolvent model, which requires an initial pass through the atoms to generate the mask and a second pass to shrink the mask based on the shrink radius (Jiang & Brünger, 1994).
This mask can be Fouriertransformed to yield F_{m} in (2). The solvent mask, which is necessary for the derivatives (see equation 9), is simply
The Gaussian functions provide an easily differentiable formalism with respect to the atomic coordinates,
where α ∈ {x, y, z}. This equation can be combined using the chain rule to yield the derivative of the bulk solvent at r_{g} for atom i,
which can be used with any targetfunction derivative with respect to the bulksolvent structure factors following the equations given in Brünger (1989).
It is worthwhile to point out that the atomic Gaussians used in the computation of F_{c} could be used for the purposes of generating the solute/solvent mask. However, this would come at a significant computational expense as the calculations for the Gaussian and polynomial model are performed in P1 (see Implementation section).
2.3. Polynomial switch model
For the polynomial model, we implemented a multiplicative cubic switch function with the endpoints fixed at zero and one (Im et al., 1998), although higher order functions are also possible (Schnieders et al., 2007). Given an atom radius a and a window size to compute the switch function w, the distance d between the grid point and atom is computed as r − a + w. The cubic polynomial function describing the solvent density is then
The switching function S is only computed within the window w,
The characteristic function to yield the solute mask is the product of the switch functions over atoms,
This has the benefit that only a single pass through the atoms is required, which provides a speed benefit over the Gaussian model (data not shown). As above, the solute mask can be used to compute F_{m} and the bulksolvent density is simply
The derivative of the switch with respect to atomic coordinates is only necessary inside the window region:
The derivative of the solute mask with respect to atomic coordinates can be obtained by
which, as with the Gaussian model, can be combined using the chain rule with the
target function of choice.2.4. Implementation
An important point regarding the implementation of the described models is the presence of the realspace solvent mask in the derivatives (equations 9 and 17). This requires that the solvent mask is stored in memory. Furthermore, as the solvent mask is a manybody equation (i.e. the solvent density at any given grid point may depend on several atoms, including those generated by crystallographic symmetry), the solvent mask must include contributions from the unique atom set and nearby symmetry atoms. To properly account for this, we first use the solute mask in computing F_{m} (equations 6 and 14) and then use a spatial decomposition routine to locate atoms within a 4.0 Å shell around the unique atom set, although the specific choice of this shell depends on the parameters of the bulksolvent model (e.g. w in the polynomial model). This calculation is performed in P1. The resulting mask is converted to a solvent mask (e.g. equation 7) and used for the derivative computation.
The Gaussian and polynomial models were computed with an experimental program system, Force Field X (FFX), which is a Java Virtual Machine (JVM) based framework aimed towards combining modules from several fields of molecular biophysics into an integrated platform (Schnieders & Fenn, in preparation). Bulksolvent structure factors from FFX were input to CNS v.1.3 (Schröder et al., 2010; Table 1). The CNS optimization of the solvent parameters used a grid search together with leastsquares optimization (Brunger, 2007). The leastsquares optimization employed a leastsquares target function [see equation 2 in Brünger (1989) and equation 11 of Jiang & Brünger (1994)]. The bulksolvent models were computed on a bounded grid that was calculated as onethird of the maximum resolution, but limited to 0.57 and 0.9 Å for high and lowresolution structures, respectively (Rees et al., 2005; Brunger, 2007). The default probe and shrink radius parameters (1.0 Å) were used for the binary model in CNS.

We also optimized the solvent models within the FFX framework (Supplementary Table 1^{1}). These optimizations started from a scale (k_{s}) that represents the electron density of bulk water at 267 K and 0.33 e Å^{−3} (Botti et al., 2002), and a B factor (B_{s}) of 50.0 Å^{2} (Fokine & Urzhumtsev, 2002). FFX employed the derivatives given in the appendix of Afonine et al. (2005) for the optimization of k_{s} and B_{s}. Analytic gradients were verified using finitedifference methods and optimizations were carried out using a limitedmemory BFGS minimizer until the r.m.s. gradient magnitude was reduced to less than 1.0 × 10^{−5}. In contrast to CNS, combined solventparameter grid searches/minimizations were not performed. Furthermore, the solventmodel grid size was simply computed as onethird of the maximum resolution (Bricogne, 2006), i.e. no grid bounding was performed.
3. Results
For the test cases used here, only the anisotropic scale and bulksolvent parameters were fitted to the diffraction data. Atomic coordinates were not altered from their deposited values and no B factors was performed (except for testing of the analytic derivatives; see below). All atoms, including ordered water molecules, were included as part of the molecular surface for use in computation of the solvent mask (e.g. equations 6 and 14). However, TLSbased (Schomaker & Trueblood, 1968) ANISOU records from the PDB files and ligands were ignored (MPD in the case of PDB entry 1n7s , zinc and phosphate in PDB entry 3dyc , GTP in PDB entry 3bbp and ATP in PDB entry 1nsf ) owing to a lack of support for these ligands in the software used.
of the positions or atomicWe initially sought to determine optimum values for the two variable parameters of the alternative bulksolvent models: σ for the Gaussian model and w for the polynomial model (the value of a in the polynomial model was set to the van der Waals radius). Using a grid search with the R and R_{free} values as a guide, a value of 0.55 times the van der Waals radius was determined for σ to consistently yield optimum R values (data not shown). In some cases, the optimum value for σ varied slightly; it is straightforward for these cases to implement gridsearch routines as part of the process (Brunger, 2007). For the polynomial model, we determined a value of 0.8 Å for w, which yielded similar R values to the Gaussian model. Furthermore, the profiles of the polynomial and Gaussian models with these values appeared to be similar (Fig. 3d and discussion below), suggesting that the two masks model the bulk solvent in a similar fashion.
A density slice of the binary and Gaussian bulksolvent models is presented in Fig. 3. Of note is the continuous from the protein region (red) to bulk solution (blue) in the Gaussian/polynomial mask (Fig. 3b) versus the sharper transitions in the binary mask (Fig. 3a). The other apparent feature of the density slices is that the alternative bulksolvent models do not use a probe radius to extend the solvent mask and therefore the resultant masks are more closely associated with the van der Waals surface definition of Richards (1977) and Connolly (1985) rather than the solventaccessible surface that defines the binary model (Lee & Richards, 1971). The shrink procedure of the binary mask reduces the solventaccessible surface to the molecular surface, with the intention that small internal cavities are excluded from the mask (compare Figs. 3a and 3b). This has the effect of preventing bulksolvent scattering in regions that should not scatter Xrays, such as hydrophobic cavities. However, it is not determined from the mask procedure what type of contacts are available (if any) in the excluded cavities to differentiate a cavity as hydrophobic or otherwise (Finney, 1975). In any case, significant experimental evidence suggests that internal cavities – even small hydrophobic cavities – can be partly occupied by water molecules that exhibit short bound lifetimes or are dynamically disordered such that the water molecules are not observed in crystallographic experiments (Richards, 1977; Tilton et al., 1986; Ernst et al., 1995; Buckle et al., 1996; Otting et al., 1997; Yu et al., 1999; Liu et al., 2008). Therefore, the use of the van der Waals surface in the alternative bulksolvent models has some physical footing, although it is not entirely correct since the average electron density in these small internal cavities is expected to be less than that of the bulk solvent. Nevertheless, we obtain similar R values with the binary, Gaussian and polynomial models (see discussion below) and the difference densities in small cavities appear to be similar (Supplementary Fig. 1^{1}). A pictorial representation of the mask is also shown in the onedimensional case for an atom located at the origin below each twodimensional density slice using either the binary model (Fig. 3c) or the Gaussian/polynomial model (Fig. 3d). The Gaussian and polynomial models generate a solvent distribution that asymptotes to bulk electron density approximately where the first shell of density in radial distributions of solvent about protein molecules appears (Pettitt et al., 1998; Makarov et al., 2002; Chen et al., 2008).
The similar R_{free} of the Gaussian and polynomial models compared with the binary model (Table 1 and Supplementary Table 1^{1}) suggests that the alternative bulksolvent models are similarly consistent with the diffraction data. This is also reflected in the agreement between the calculated and experimentally determined phases in the 1nsf test case. Fig. 4 shows that the bulksolvent models (magenta, yellow and green lines) primarily improve the R_{free} at a resolution lower than approximately 6 Å (0.03 Å^{−2}) compared with having no bulksolvent correction (blue lines), although the agreement with the highresolution data is also improved in most cases. The k_{s} values are higher on average for the polynomial/Gaussian models versus the binary model, perhaps compensating for an overall smaller solvent electrondensity volume owing to the soft nature of the transition at the solvent–solute boundary compared with the binary model.
To test the analytic gradients of the alternative bulksolvent models, a solvent water molecule in one of the tested structures (water 2126 in model 1exr ) was moved from its original position in 0.05 Å increments into the bulk solvent while monitoring the analytic atomic derivatives compared with finite differences of the log likelihood [LLK; for details on the computation of the loglikelihood target, see Cowtan (2005) and McCoy (2004)]. Finite differences were calculated using a double wide criterion
For this procedure, the scale and Bfactor values of the bulk solvent were held fixed as the water molecule was moved. The results are shown in Fig. 5. Using a Δx of 1.0 × 10^{−4} Å (Fig. 5a), the derivatives and finite differences match if derivatives based on either (9) or (17) are included (solid lines), but do not agree (dashed green and red lines in Fig. 5a) if the derivatives of the bulk solvent are not included.
The finite differences for the binary model with solventmodel updates performed at every step (blue dots in Fig. 5b) show larger fluctuations than the corresponding calculation without derivatives for the Gaussian and polynomial models (compare the dashed lines in Fig. 5a and the dotted lines in Fig. 5b). Δx was set to 0.01 Å in the binary case to avoid aliasing artifacts. Finer grid spacings could not improve this result. This example illustrates that the alternative bulksolvent models will be less prone to sawtoothlike (i.e. up and down) patterns during bulksolvent model updates (see, for example, Fig. 2 in Phillips, 1980). However, computation of the solvent model and its derivatives are required at every minimization or simulatedannealing step to achieve the improved stability.
4. Conclusions
Implicit continuum solvent models are important for crystallographic et al., 2000). The utility of a polynomial or Gaussian definition of the solvent density extends beyond crystallography, as continuum solvent electrostatics are a crucial component in analyses such as computations of binding and energies (for a review of this subject, see Kollman et al., 2000), as well as pK_{a} calculations, for which accurate continuum models and their derivatives are crucial in improving agreement with experiment (Simonson et al., 2004). It is also possible to combine implicit models based on reference interactionsite models (Lounnas et al., 1994) and explicit solvent, although both come with a greater time cost. These methods may be of interest in accounting for differences in solvation between multiple structures and to fully analyze the pattern of hydration around macromolecules (Makarov et al., 2002); studies of protein structures have suggested the need for such methods for quite some time (Savage & Wlodawer, 1986). Furthermore, crystal structures obtained with highly accurate experimental phases suggest that the outer shells of solvation about proteins may not be captured by a simple continuum model (Burling et al., 1996).
to improve the agreement between model and diffraction data at low resolution and such models also have potential for improving phasing methods (RoversiThe polynomial and Gaussian continuum solvent models offer a comparable agreement with the diffraction data versus the standard binary model as the R values and phase differences suggest. The continuous nature of the alternative models offer improved stability for atomic the latter of which acts as a `continuum boundary' on the atoms. These aspects of the polynomial and Gaussian models will be most powerful when the model is updated at each step during the process.
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