Volume 69 Received 13 December 2012  Bulksolvent and overall scaling revisited: faster calculations, improved results^{a}Lawrence Berkeley National Laboratory, One Cyclotron Road, MS64R0121, Berkeley, CA 94720, USA,^{b}Department of Bioengineering, University of California, Berkeley, Berkeley, CA 94720, USA,^{c}IGBMC, CNRSINSERMUdS, 1 Rue Laurent Fries, BP 10142, 67404 Illkirch, France, and ^{d}Université de Lorraine: Département de Physique  Nancy 1, BP 239, Faculté des Sciences et des Technologies, 54506 VandoeuvrelèsNancy, France A fast and robust method for determining the parameters for a flat (maskbased) bulksolvent model and overall scaling in macromolecular crystallographic structure refinement and other related calculations is described. This method uses analytical expressions for the determination of optimal values for various scale factors. The new approach was tested using nearly all entries in the PDB for which experimental structure factors are available. In general, the resulting R factors are improved compared with previously implemented approaches. In addition, the new procedure is two orders of magnitude faster, which has a significant impact on the overall runtime of refinement and other applications. An alternative function is also proposed for scaling the bulksolvent model and it is shown that it outperforms the conventional exponential function. Similarly, alternative methods are presented for anisotropic scaling and their performance is analyzed. All methods are implemented in the Computational Crystallography Toolbox (cctbx) and are used in PHENIX programs. Keywords: bulk solvent; scaling; anisotropy; structure refinement; PHENIX. 
Macromolecular crystals typically contain a substantial amount of disordered solvent, ranging from approximately 20 to 90% of the crystal volume, with a mean of 55%, in the Protein Data Bank (PDB; Bernstein et al., 1977; Berman et al., 2000). Anisotropy in the diffracted intensities is another common feature of macromolecular crystals that arises from various sources including crystal lattice vibrations (Shakked, 1983; Sheriff & Hendrickson, 1987). When modelling diffracted intensities, for example in structure refinement or automated model building, it is therefore critical to account for these two phenomena (see, for example, Jiang & Brünger, 1994; Urzhumtsev & Podjarny, 1995; Kostrewa, 1997; Badger, 1997; Urzhumtsev, 2000; Fokine & Urzhumtsev, 2002a; Fenn et al., 2010). The flat bulksolvent model (Phillips, 1980; Jiang & Brünger, 1994) combined with overall anisotropic scaling in either exponential (Sheriff & Hendrickson, 1987) or polynomial (Usón et al., 1999) forms is a well established and computationally efficient approach. Alternatives have been proposed (Tronrud, 1997; Vassylyev et al., 2007), but are not currently in wide use.
In the commonly used approach, the total structure factor is defined as
where k_{total} is the overall Millerindexdependent scale factor, F_{calc} and F_{mask} are the structure factors computed from the atomic model and the bulksolvent mask, respectively, and k_{mask} is a bulksolvent scale factor. The mask can be computed efficiently using exact asymmetric units as described in GrosseKunstleve et al. (2011).
The overall scale factor k_{total} can be thought of as the product
where k_{overall} is the overall scale factor and k_{isotropic} and k_{anisotropic} are the isotropic and anisotropic scale factors, respectively.
k_{overall} is a scalar number that can be obtained by minimizing the leastsquares residual
where F_{obs} are the observed structure factors and
The sum is over all reflections. Solving LS/k_{overall} = 0 leads to
In the exponential model the anisotropic scale factor is defined as
where U_{cryst} is the overall anisotropic scale matrix equivalent to U^{*} defined in GrosseKunstleve & Adams (2002); s^{t} = (h, k, l) is the transpose of the Millerindex column vector s.
Usón et al. (1999) define a polynomial anisotropic scaling function that can be rewritten in matrix notation as follows:
where V_{0} and V_{1} are symmetric 3 × 3 matrices, s^{2} = s^{t}G^{*}s and G^{*} is the reciprocalspace metric tensor. Expression (7) is equivalent to the first terms in the Taylor series expansion of the exponential function (6),
with the constant term omitted. The omission of the constant 1 means that k_{anisotropic} is equal to zero for the reflection F_{000}, as follows from (7). Therefore, in this work we modify (7) by adding the constant
The bulksolvent scale factor is traditionally defined as
where k_{sol} and B_{sol} are the flat bulksolvent model parameters (Phillips, 1980; Jiang & Brünger, 1994; Fokine & Urzhumtsev, 2002b).
Depending on the calculation protocol, k_{isotropic} may be assumed to be a part of k_{anisotropic} or it can be assumed to be exponential: k_{isotropic} = exp(Bs^{2}/4), where B is a scalar parameter. Alternatively, it may be determined as described in §2.3 below.
The determination of the anisotropic scaling parameters (U_{cryst} or V_{0} and V_{1}) and the bulksolvent parameters k_{sol} and B_{sol} requires the minimization of the target function (3) with respect to these parameters. Despite the apparent simplicity, this task is quite involved owing to a number of numerical issues (Fokine & Urzhumtsev, 2002b; Afonine et al., 2005a). Previously, we have developed a robust and thorough procedure (Afonine et al., 2005a) to address these issues. This procedure is used routinely in PHENIX (Adams et al., 2010). However, owing to its thoroughness the procedure is relatively slow and may account for a significant fraction of the execution time of certain PHENIX applications (for example, phenix.refine).
In this paper, we describe a new procedure which is approximately two orders of magnitude faster than the approach described in Afonine et al. (2005a) and often leads to a better fit of the experimental data. The speed gain is the result of an analytical determination of the optimal bulksolvent and scaling parameters. The better fit to the experimental data is partially the result of employing a more detailed model for k_{mask} compared with the exponential model in equation (10) and is partially a consequence of the new analytical optimization method. Analytical optimization eliminates the possibility of becoming trapped in local minima, which exists in all iterative local optimization methods, including the procedure used previously.
To obtain the elements of the anisotropic scaling matrix (6), the minimization of (3) is replaced by the minimization of
For this, we assume that F_{obs} and F_{model} are positive. We also assume that the minima of (3) and (11) are at similar locations. This assumption is not obvious and, as discussed below, may not always hold (see §3.3 and Table 2). Expression (11) can be rewritten as
Here, Z = [1/(2^{2})]ln[F_{obs}(k_{overall}k_{isotropic}F_{calc} + k_{mask}F_{mask})^{1}]. Defining
and using
the target function determining the optimal U_{cryst} is
The U_{cryst} values that minimize (15) are determined from the condition = 0, which gives a system of six linear equations
where M = , V = (h^{2}, k^{2}, l^{2}, 2hk, 2hl, 2kl)^{t}, denotes the outer product and b = .
The desired U_{cryst} matrix is determined by solving the system (16):
Crystalsystemspecific symmetry constraints can be incorporated via a constraint matrix (C), which we derive from first principles by solving the system of linear equations R^{t}UR = U for all rotation matrices R of the crystalsystem point group. Alternatively, symmetry constraints are often derived manually and tabulated (Nye, 1957; Giacovazzo, 1992). For example, the constraint matrix for the tetragonal crystal system is
The number of rows in C determines the number of independent coefficients of U_{cryst}. Let U_{ind} be the column vector of independent coefficients; the (redundant) set of six coefficients U_{cryst} is then obtained via
The constraint matrix C is introduced into equations (16) and (17) above as follows:
with M_{C} = , V_{C} = CV, b_{C} = and
The polynomial model (Usón et al., 1999) for anisotropic scaling allows the direct use of the residual (3) to find the optimal coefficients for V_{0} and V_{1} in equation (9). An advantage of this model is that no assumptions about the similarity of the location of the minima of targets (3) and (11) are required. Conceptually, a disadvantage of equation (9) is that it is only an approximation of equation (6), as was shown above. However, the number of parameters is doubled in equation (9) compared with equation (6), since V_{0} and V_{1} are treated independently. The increased number of degrees of freedom may therefore compensate for approximation inaccuracies.
Similarly to §2.1, the optimal coefficients for V_{0} and V_{1} are determined by the condition _{V}LS = 0 and can be obtained by solving a system of 12 linear equations. We follow the arguments of Usón et al. (1999) for not using symmetry constraints in this case.
Defining K = k^{2}_{total} = (k_{overall}k_{isotropic}k_{anisotropic})^{2}, the determination of the desired scaling parameters k_{isotropic} and k_{mask} is reduced to minimizing
in resolution bins, where k_{overall} and k_{anisotropic} are fixed. This minimization problem is generally highly overdetermined because the number of reflections per bin is usually much larger than two.
Introducing w = F_{mask}^{2}, v = (F_{calc}, F_{mask}) and u = F_{calc}^{2} and substitution into (22) leads to
Minimizing (23) with respect to K and k_{mask} leads to a system of two equations:
Developing these equations with respect to k_{mask},
and introducing new notations for the coefficients, we obtain
Multiplying the second equation by Y_{2} and substituting KY_{2} from the first equation into the new second equation, we obtain a cubic equation
The senior coefficient in (27) satisfies the CauchySchwarz inequality:
Therefore, equation (27) can be rewritten as
and solved using a standard procedure.
The corresponding values of K are obtained by substituting the roots of equation (29) into the first equation in (26):
If no positive root exists k_{mask} is assigned a zero value, which implies the absence of a bulksolvent contribution. If several roots with k_{mask} 0 exist then the one that gives the smallest value of LS_{s}(K, k_{mask}) is selected.
If desired, one can fit the righthand side of expression (10) to the array of k_{mask} values by minimizing the residual
for all k_{mask} > 0. This can be achieved analytically as described in Appendix A. Similarly, one can fit k_{overall}exp(B_{overall}s^{2}/4) to the array of K values.
In case of twinning with N twinrelated domains, the total model intensity is
where _{j} is the twin fraction of the jth domain, T_{j} is the corresponding twin operator (a 3 × 3 rotation matrix) and
k_{total} includes all scale factors (overall, isotropic and anisotropic). We make the reasonable assumption that k_{total} and k_{mask} are identical for all twin domains.
Finding the twin fractions _{j} can be achieved by solving the minimization problem
with the constraint condition
where I(s) = F^{2}_{obs} and s_{j} = T_{j}s. This constrained minimization problem can be reformulated as an unconstrained minimization problem by the standard technique of introducing a Lagrange multiplier:
The values {_{1}, ..., _{N}, } that minimize (36) are the solution of the system of N + 1 linear equations with N + 1 variables:
or
The solution of this system is
with the (N + 1) × (N + 1) matrix
and
Here, 1 is a row or column containing N unit elements to complete the matrix M and
The values of are expected to be between 0 and 1, and is proportional to the sum of squared intensities. Therefore, it is numerically beneficial to multiply the C(_{1}, ..., _{N}) term in (36) by a constant in order to make the value for numerically similar to the values for the twin fractions .
Once the twin fractions have been found, the procedure described in §2.3 can be used to obtain the overall and bulksolvent scale factors. Similarly to (23), we can write
where _{j} are known twin fractions and K and k_{mask} are the scale factors to be determined. Similarly to §2.3, we obtain
Introducing new variables as before for equation (23) leads to
The determination of the twin fractions and scales k_{total} and k_{mask} are iterated several times until convergence. The determination of does not guarantee that the individual twin fractions _{j} are in the range 01. For any _{j} outside this range the corresponding twin operation is ignored for the current iteration and the new smaller set of twin fractions and scales are redetermined. However, in the next iteration the full set of is tried again.
The scale factors involved in the calculation of F_{model} according to equation (1) are highly correlated. Therefore, the order of their determination is important. Empirically, we found that the determination of k_{isotropic} and k_{mask} followed by the determination of k_{anisotropic} works optimally in most cases. The determination of (k_{mask}, k_{isotropic}) and k_{anisotropic} is repeated several times until the R factor decreases by less than 0.01% between cycles. The number of cycles required to reach convergence is typically between 1 and 5.
To determine k_{anisotropic}, our protocol can make use of three available scaling methods: polynomial (poly; §2.2), exponential with analytical calculation of the optimal parameters (exp_{anal}; §2.1) and exponential with the optimal parameters obtained via LBFGS (Liu & Nocedal, 1989) minimization (exp_{min}; Afonine et al., 2005a). The three methods can be tested independently, in which case the result with the lowest R factor is accepted. However, because exp_{min} is up to an order of magnitude slower than the other two methods it is not expected to be used routinely.
The calculation of k_{isotropic} and k_{mask} requires dividing the data into resolution bins (§3.2). If oscillation of k_{mask} between bins occurs, smoothening (Savitzky & Golay, 1964) is applied to the binwise determined values of k_{mask} such that it reduces the oscillations without altering the monotonic behavior of k_{mask} as a function of resolution (see Fig. 1). Finally, the smoothed values are assigned to individual reflections using linear interpolation. The k_{isotropic} scales are updated using equation (5) in order to account for the changed k_{mask}.
 Figure 1 Examples of smoothening of k_{mask}. The original k_{mask} (blue; obtained as the solution of equation 29) and that after smoothening (red) are shown for three PDB entries with the PDB codes shown on the plots. 
As illustrated in §3.2, the minimum of the Rfactor function
and the minimum of the leastsquares function (22) can be at significantly different locations in the (k_{mask}, k_{isotropic}) parameter space. To assure that the final (k_{mask}, k_{isotropic}) values correspond to the lowest R factor, a fast grid search is performed around the optimal values of the leastsquares function.
The goal of binning is to group data by common features to characterize each group by a set of common parameters. Here, the key parameter is the resolution d of reflections. Binning schemes with bins containing an approximately equal number of reflections (i.e. the resolution range is uniformly sampled in d^{3}) or a predefined number of bins are typically used. Since the lowresolution region of the data is sparse, such binning schemes tend to produce only one or very few lowresolution bins, which is insufficient to best model the bulksolvent contribution. Unfortunately, decreasing the number of reflections per bin will disproportionally increase the number of bins (N_{bins}) at higher resolution and may still provide insufficient detail for the lowresolution data (Table 1).

An alternative approach which divides the resolution range uniformly on a logarithmic scale ln(d) (Urzhumtsev et al., 2009) efficiently solves this problem. The flowchart of the algorithm is shown in Fig. 2. This scheme allows the higher resolution bins to contain more reflections than the lower resolution bins and more detailed binning at low resolution without increasing the total number of bins. An additional reason for using logarithmic binning is that the dependence of the scales on resolution is approximately exponential (see previous sections), which makes the variation of scale factors more uniform between bins when a logarithmic binning algorithm is used. Table 1 compares binning performed uniformly in d^{3} and in ln(d) spacing for three data sets (PDB entries 3hay , 1kwn and 3gk8 ). Note the data completeness of the lowresolution bins.
 Figure 2 Flowchart of the logarithmic resolutionbinning algorithm. 
We evaluated the performance of the new scaling protocol by applying it to approximately 40 000 data sets selected from the PDB. The structures were selected by evaluating all PDB entries using phenix.model_vs_data (Afonine et al., 2010) and excluding all entries for which the recalculated R_{work} was greater than the published value by five percentage points.
To score the test results three crystallographic R factors (46) were computed using all reflections, using only lowresolution reflections and using only highresolution reflections. Lowresolution reflections were selected using the condition d_{min} > 8 Å but selecting at least the 500 lowest resolution reflections. Highresolution reflections were taken from the highest resolution bin. Each of the three anisotropic scaling methods (poly, exp_{anal} and exp_{min}) was tested independently within each run. Additionally, two other tests were performed: one combining poly and exp_{anal} as described in §3.1 (referred to as poly+exp_{anal}) and the other using the protocol of Afonine et al. (2005a) (referred to as old).
Fig. 3 shows a comparison of the alternative methods for determining k_{anisotropic} (see §3.1). Comparing the polynomial model (poly) versus the analytical exponential model (exp_{anal}), with a few minor exceptions poly results in slightly lower R factors overall and for the lowresolution reflections, while exp_{anal} results in lower R factors for the highresolution reflections. Comparing poly versus the original exponential model using minimization (exp_{min}), the R factors are very similar overall and for the highresolution reflections, while poly often results in lower R factors for the lowresolution reflections. Comparing the two different exponential models, exp_{min} results in lower R factors overall and nearly identical results for lowresolution reflections, but exp_{anal} results in lower R factors for the highresolution reflections. Fig. 4 compares the new protocol combining poly and exp_{anal} with the old protocol. With very few exceptions, the new protocol performs better for all three resolution groups.
 Figure 3 A comparison of the new scaling protocol using different models for the anisotropic scale factor. R versus R factor scatter plots for (a) poly versus exp_{anal}, (b) poly versus exp_{min} and (c) exp_{anal} versus exp_{min} R factors were computed using all reflections (left), lowresolution reflections only (middle) and highresolution reflections only (right). See §3.3 for details. 
 Figure 4 R versus R factor scatter plots comparing the new scaling protocol using poly+exp_{anal} for the anisotropic scale factor with the old protocol. For each structure the full set of structure factors available from the PDB was used to calculate scale factors and to calculate R factors (left). Using the same scalefactor values the R factors were calculated separately for the lowresolution reflections (middle) and highresolution reflections (right). A large spread of points in the vertical direction above the diagonal (red line) in these latter plots indicates that in many cases the scale factors produced by the old protocol resulted in a poorer fit to the data at low and high resolutions, while the new protocol generates scale factors with a good fit across all resolution ranges. See §3.3 for details. 
As described above, occasionally the minima of the Rfactor function (46) and the LS function (22) are at significantly different locations in the (k_{mask}, k_{isotropic}) parameter space (see Fig. 5). For example, considering k_{isotropic} to be a singlevalue scalar the pair (k_{mask}, k_{isotropic}) that minimizes the R factor in the lowresolution range of PDB data set 1kwn is (0.2913, 0.0961), while the pair (0.3218, 0.0863) minimizes the LS function. The corresponding R factors are 0.3073 and 0.3372, respectively. The data for PDB entry 1hqw lead to an even more dramatic difference, in which the pairs (k_{mask}, k_{isotropic}) that minimize the R factor and the LS function are (0.25, 0.0131) and (0.6166, 0.0151), respectively, and the corresponding R factors are 0.2924 and 0.5046. We made a similar observation for the overall anisotropic scale k_{anisotropic}, as illustrated in Table 2. For this, the best values for U_{cryst} were determined via a systematic search for the minima of the functions (3), (11) and (46) for three combinations of structures and highresolution cutoffs. Note the difference in the optimal U_{cryst} values and the corresponding R factors.

 Figure 5 Plots of R factors (with k_{isotropic} = 0.0961) and the LS function (with k_{isotropic} = 0.0863) for PDB entry 1kwn (left) and R factors (with k_{isotropic} = 0.0131) and the LS function (with k_{isotropic} = 0.0151) for PDB entry 1hqw (right), illustrating that the minima of the Rfactor function (46) and the LS function (22) can be at significantly different locations in parameter space. In such cases, a line search around the value of k_{mask} obtained by minimization of the LS function is necessary in order to obtain a value that minimizes the R factor. For plotting purposes, the values of the LS function were scaled to be similar to the R factors. 
The parameterization of the total model structure factor (1) does not make any assumption about the shape of k_{mask}; for example, it does not assume it to be exponential (10). This provides an opportunity to explore the behavior of k_{mask} as a function of resolution and compare it with k_{mask} obtained via (10). Fig. 6 illustrates the differences between the two methods of determining k_{mask} for six representative PDB entries selected from approximately 40 000 entries after inspection of the k_{mask} values. We observe that the plots of the values obtained using our new approach are in general significantly different from the exponential function. This observation is in line with Fig. 1 of Urzhumtsev & Podjarny (1995).
 Figure 6 Plots of k_{mask} as a function of resolution (s^{2}) for six selected PDB entries. The blue lines show k_{mask} as determined using the new method. The red lines show k_{mask} based on the exponential function (10) using optimized k_{sol} and B_{sol} parameters. 
At very low resolution the structure factors computed from the atomic model are approximately anticorrelated to the structure factors computed from the bulksolvent mask:
Here, p is a scale factor (Urzhumtsev & Podjarny, 1995). Relation (47) is the basis for alternative bulksolvent scaling methods that employ the Babinet principle (Moews & Kretsinger, 1975; Tronrud, 1997). Substitution of relation (47) into equation (1) yields
Obviously, F_{model} is invariant for any combination of scale factors k_{total} and k_{mask} satisfying the condition
Since our new scaling procedure determines k_{mask} and k_{isotropic} (which are part of k_{total}) simultaneously, without imposing constraints on their values, these scale factors may assume unusual values in the lowresolution range. However, we observe that in practice this only happens for a very small number of the test cases.
A new method for overall anisotropic and bulksolvent scaling of macromolecular crystallographic diffraction data has been developed which is an improvement over the existing algorithm of flat (maskbased) bulksolvent modeling and overall anisotropic scaling, versions of which are routinely used in various refinement packages such as CNS (Brunger, 2007), REFMAC (Murshudov et al., 2011) and phenix.refine (Afonine et al., 2012). In the process of developing this method, we concluded that the bulksolvent scale factor k_{mask} deviates quite significantly from the exponential model that has traditionally been used. This new method is approximately two orders of magnitude faster than the previous implementation and yields similar or often better R factors. Table 3 compares runtimes for a number of selected cases covering a broad range of resolutions and atomic model sizes. Therefore, the computational speed of the new method makes it possible to robustly compute bulksolvent and anisotropic scaling parameters even as part of semiinteractive procedures.

An inherent feature of the maskbased bulksolvent model is that it relies on the existing atomic model to compute the mask. This in turn implies that any unmodeled (as atoms) parts of the unit cell are considered to belong to the bulksolvent region. This may obscure weakly pronounced features in residual maps such as partially occupied solvent or ligands. This is common to all maskbased bulksolvent modeling methods, leading to the development of algorithms to account for missing atoms (Roversi et al., 2000). In the future, improved maps may be obtained by combining this latter approach with the new fast overall anisotropic and bulksolvent scaling method that we have presented.
The new method is implemented in the cctbx project (GrosseKunstleve et al., 2002) and is used in a number of PHENIX applications since v.1.8 of the software, most notably phenix.refine (Afonine et al., 2005b, 2012), phenix.maps and phenix.model_vs_data (Afonine et al., 2010). The cctbx project is available at http://cctbx.sourceforge.net under an opensource license. The PHENIX software is available at http://www.phenixonline.org .
All results presented are based on PHENIX v.1.8.1.
Our goal is to approximate a set of data points {Y(x)}^{N}_{j} = 1 with a Gaussian function,
For this, we use the standard approach of minimizing a leastsquares (LS) function,
If Y(x_{j}) 0 x_{j}, j = 1, N, the minimization of LS can be replaced by the minimization of
The minimum of this LSL function can be determined analytically,
Defining u = ln(a), v_{j} = x_{j}^{2}, d_{j} = ln[Y(x_{j})], we obtain
The variables {a, b} minimizing the LSL function are determined by the condition
This leads to
and
Defining p = , q = , r = and s = , we obtain
and
From this, we obtain
and finally
The authors thank the NIH (grant GM063210) and the PHENIX Industrial Consortium for support of the PHENIX project. This work was supported in part by the US Department of Energy under Contract No. DEAC0205CH11231.
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