research papers
Polarizable atomic multipole Xray
application to peptide crystals^{a}Department of Chemistry, Stanford, CA 94305, USA, ^{b}Department of Molecular and Cellular Physiology, Stanford, CA 94305, USA, and ^{c}Howard Hughes Medical Institute, USA
^{*}Correspondence email: pande@stanford.edu, brunger@stanford.edu
Recent advances in computational chemistry have produced force fields based on a polarizable atomic multipole description of biomolecular electrostatics. In this work, the Atomic Multipole Optimized Energetics for Biomolecular Applications (AMOEBA) force field is applied to R_{free} by 20–40% relative to the original spherically symmetric scattering model.
of molecular models against Xray diffraction data from peptide crystals. A new formalism is also developed to compute anisotropic and aspherical structure factors using fast Fourier transformation (FFT) of Cartesian Gaussian multipoles. Relative to direct summation, the FFT approach can give a speedup of more than an order of magnitude for aspherical of ultrahighresolution data sets. Use of a formalism makes the method highly parallelizable. Application of the Cartesian Gaussian multipole scattering model to a series of four peptide crystals using multipole coefficients from the AMOEBA force field demonstrates that AMOEBA systematically underestimates electron density at bond centers. For the trigonal and tetrahedral bonding geometries common in organic chemistry, an atomic multipole expansion through hexadecapole order is required to explain bond electron density. Alternatively, the addition of interatomic scattering (IAS) sites to the AMOEBAbased density captured bonding effects with fewer parameters. For a series of four peptide crystals, the AMOEBA–IAS model loweredKeywords: scattering factors; aspherical; anisotropic; force fields; multipole; polarization; AMOEBA; bond density; direct summation; FFT; SGFFT; Ewald; PME.
1. Introduction
The number of Xray crystal structures in the Protein Data Bank (PDB) with a resolution of higher than 1.0 Å continues to increase rapidly (Berman et al., 2000). In late 2002, there were already over 100 structures available at subatomic resolution (Afonine & Urzhumtsev, 2004), while as of early 2009 the number had more than tripled to well over 300. Examples include the proteins lysozyme at 0.65 Å (Wang et al., 2007), aldose reductase at 0.66 Å (Howard et al., 2004) and serine protease at 0.78 Å (Kuhn et al., 1998), as well as nucleic acid structures such as BDNA at 0.74 Å (Kielkopf et al., 2000), ZDNA at 0.60 Å (Tereshko et al., 2001) and an RNA tetraplex at 0.61 Å (Deng et al., 2001). Crystals that diffract to high resolution are ideal for studying valenceelectron distributions (Jelsch et al., 2000; Muzet et al., 2003; Zarychta et al., 2007; Volkov et al., 2007; Coppens & Volkov, 2004) that dictate the electrostatic properties of macromolecules. Electrostatics, in turn, is one of the driving forces in protein and nucleic acid folding, which should be understood in detail in order to predict biomolecular thermodynamics and kinetics (Snow et al., 2002, 2005; Sorin & Pande, 2005; Pande et al., 2003). In this work, we contribute an improved theory and algorithm for computing the anisotropic and aspherical valenceelectron density of molecules for Xray refinement.
Calculation of structure factors is generally based on scattering factors defined by the isolatedatom model (IAM), which assumes that the electron density around each atom is spherically symmetric. However, subatomic resolution diffraction data capture aspherical features of the electron density that result from bonding and the local chemical environment. The difference between the IAM and the true electron density is defined as the deformation density. For example, aspherical electrondensity models of diamond, silicon and germanium developed by DeMarco and Weiss and later by Dawson explained the peaks of deformation density at bond midpoints observed in the experimental data (Dawson, 1967a,b,c; DeMarco & Weiss, 1965). In these works, the IAM was augmented by atomcentered spherical harmonic expansions, whose physical consequence was to redistribute electron density from nonbonding lobes into the tetragonal arrangement of bond centers.
A variety of radial functions have been used in combination with atomcentered spherical harmonic expansions. Modified Gaussians were promoted by Dawson (1967a), a set of harmonic oscillator wavefunctions by KurkiSuonio (1968) and more recently a formalism based on Slatertype orbitals (STO) was described by Stewart and coworkers (Epstein et al., 1977; Cromer et al., 1976; Stewart, 1979, 1977) and by Hansen & Coppens (1978), which represents the current standard (Jelsch et al., 2005; Zarychta et al., 2007; Volkov et al., 2007; Coppens, 2005). However, spherical harmonics are not the only basis set available to describe the angular dependence of the deformation density.
We first present a formulation of anisotropic and aspherical atomic densities based on Cartesian Gaussian multipoles, which leads to much simpler formulae for the calculation of structure factors via direct summation in than the STObased theory of Hansen & Coppens (1978). We also demonstrate that Cartesian Gaussian multipoles allow the computation of structure factors via fast Fourier transformation (FFT) of the realspace electron density (Cooley & Tukey, 1965). The latter approach, originally proposed by Ten Eyck (1973, 1977), is the basis of the efficient macromolecular algorithms (Brünger, 1989; Afonine & Urzhumtsev, 2004; Afonine et al., 2007; Agarwal, 1978) implemented in programs such as CNS (Brünger et al., 1998; Brunger, 2007) and PHENIX (Adams et al., 2002). The method implemented in CNS lends itself to efficient parallelization (Brünger, 1989).
Boys originally proposed Cartesian Gaussian functions as basis functions to solve the manyelectron Schrödinger equation (Boys, 1950). The advantage of Gaussians over STOs in this context is that twoelectron integrals have analytic forms, which has led to the adoption of Gaussian basis sets for many ab initio calculations (Hehre et al., 1969, 1970). We note that the equivalence of spherical harmonics and Cartesian tensors is well known, with key relationships having been presented by Stone (1996) and Applequist (1989, 2002).
We apply Cartesian Gaussian multipoles to restrained crystallographic refinements based on the Atomic Multipole Optimized Energetics for Biomolecular Applications (AMOEBA) forcefield electrostatic model (Ponder & Case, 2003; Ren & Ponder, 2002, 2003, 2004; Schnieders et al., 2007; Schnieders & Ponder, 2007). The AMOEBA electrostatic model is based on the superposition of permanent atomic multipoles truncated at quadrupoles and induced dipoles. Permanent electrostatics represents the electron density of a group of atoms in the absence of interactions with the environment, which may include other parts of the molecule or solvent. Groups are chosen to be relatively rigid in order to avoid conformational variability in the permanent multipole moments. Conversely, the induced dipoles of AMOEBA represent polarization, the response of the electron density to the local electric field.
Force fields are widely used to restrain macromolecular et al., 1987), with the latter used within simulatedannealing algorithms to promote global optimization (Brünger, 1988, 1991; Brünger et al., 1989, 1990, 1997; Kuriyan et al., 1989; Adams et al., 1997; Brünger & Rice, 1997). Up to now, force fields in crystallography have been largely limited to the geometric and repulsive terms and have had no influence on the atomic scattering factors. Therefore, using a scattering model based on AMOEBA electrostatics is novel and lends insight into the progress being made in the development of precise, transferable force fields. Another limitation of the use of force fields for restraining Xray has been the lack of proper treatment of longrange electrostatic interactions, which is overcome in this work via use of particlemesh Ewald summation (PME; Darden et al., 1993; Essmann et al., 1995; Sagui et al., 2004).
by contributing forces to local optimizations and (BrüngerIn addition to AMOEBA, polarizable force fields are being studied by a number of other groups. Maple and coworkers have pursued a model similar to AMOEBA, but with the permanent moments truncated at dipole order, which has shown promising results for protein–ligand complexes (Friesner et al., 2005; Maple et al., 2005). As an alternative to induced dipoles, Patel and Brooks employed a fluctuatingcharge model of polarization (Patel & Brooks, 2006), while Lamoureux and Roux have demonstrated success using classical Drude oscillators (Lamoureux et al., 2006; Lamoureux & Roux, 2003). In addition to polarization, Gresh and coworkers have developed a methodology to include nonclassical effects such as electrostatic penetration and charge transfer (Gresh, 2006; Gresh et al., 2007; Piquemal et al., 2006, 2007).
Although classical potentials can be validated against a range of experimental observables, for example smallmolecule solvation energies (Shirts et al., 2003; Shirts & Pande, 2005), highresolution diffraction data can pinpoint deficiencies in an electrostatics model with high precision. For example, we show that truncation of permanent atomic multipoles at quadrupole order limits the ability of the AMOEBA model to place charge density at bond midpoints. We use an efficient solution to this limitation by refining partial charges at bond centers as originally proposed by Afonine et al. (2007).
2. Theory
2.1. Subgrid fast Fourier transform
The starting point for this work is the subgrid fast Fourier transform algorithm (SGFFT), which will be briefly summarized (Brünger, 1989). In FFTbased methods, the electron density is computed over a lattice chosen to be fine enough to avoid aliasing effects at a given resolution. This computation can be made more efficient by an artificial increase in the atomic displacement parameters (ADPs) of all atoms. The optimum choice in CNS v.1.2 (Brunger, 2007) for the ADP offset and grid size follows the work of Bricogne (2001). An important point is that the electron density is only computed within a cutoff radius around each atom. As the resolution increases, the cutoff is increased based on an empirical scheme to maintain agreement between directsummation structure factors and derivatives and the SGFFT calculation (Brunger, 2007).
Structure factors are computed by FFT of the electron density of an ). The SGFFT is based on factorizing this computation into smaller FFTs that are computed separately on sublattices, which allows efficient parallelization since these tasks are independent (Brünger, 1989; Kay Diederichs, private communication). CNS v.1.21 has implemented this approach via an OpenMP environment (courtesy Kay Diederichs, University of Konstanz; available at https://cnsonline.org ). is then applied to the structure factors, and the target function and its derivatives with respect to structure factors are evaluated. Symmetry operators are applied to the derivatives of the target function with respect to the structure factors followed by inverse Fourier transform. Using the chain rule, derivatives of the target function with respect to atomic parameters are then computed by multiplication and summation over the local neighborhood around each atom of the derivatives of the electron density with respect to atomic parameters.
of atoms (Agarwal, 1978Although the original SGFFT method was developed with an isolatedatom description of electron density and isotropic ADPs, it is generalizable to aspherical Cartesian Gaussian multipoles and anisotropic ADPs. All that is needed are formulae for the electron density and the derivatives of the electron density with respect to atomic parameters, which then can be inserted into equations (29) and (40) of Brünger (1989). In the following sections, we develop these necessary formulae.
2.2. Isolatedatom Gaussian density
The key mathematical property of Gaussians with respect to efficient calculation of structure factors is that they are an eigenfunction of the Fourier transform (FT). In other words, a Gaussian in real space transforms to a Gaussian in vice versa. Consider the canonical spherically symmetric Gaussian (Agarwal, 1978),
andwhere a_{i} and b_{i} are constant parameters fitted to ab initio calculations on isolated atoms (this work is based on a sum of six Gaussians; n = 6; Su & Coppens, 1998), κ is an expansion/contraction parameter used to adjust the width of the density and r is a position vector relative to the center of the atom. Its FT is given by
where s is the reciprocallattice vector and we have used the FT definition given in Appendix A. The reciprocallattice vector is s = h^{t}A^{−1} = (A^{−1})^{t}h, where h is a column vector with the of a Bragg reflection and A is the fractionalization matrix that transforms coordinates r with respect to a Cartesian basis to fractional coordinates r_{frac} as defined in a set. The Debye–Waller factor (Waller, 1923) is given by
in U is defined via a Cartesian basis consistent with PDB ANISOU records (Trueblood et al., 1996; GrosseKunstleve & Adams, 2002). Multiplication of (3) by the atomic form factor from (2) gives the scattering factor
where each element of the symmetric positivedefinite matrixbased on U_{i} that are defined by
where U_{add} is the artificial isotropic increase or decrease in the ADP discussed above and I_{3} is a 3 × 3 identity matrix. Removal of U_{add} analytically from each after the FT is straightforward. The only difference, therefore, between each U_{i} is the isolatedatom scattering parameter b_{i}.
Application of the inverse FT to (4) gives the realspace anisotropic electron density
where U_{i} is the determinant of matrix U_{i} and U_{i}^{−1} is its inverse. This expression can also be viewed as the convolution of the Gaussian form factor of (1) with the inverse Fourier transform of the Debye–Waller factor of (3). Although the underlying isolatedatom scattering factor is spherically symmetric, convolution with anisotropic ADPs can lead to an angular dependence in ρ^{(n,κ)}(r). Using the relationship that B = 8π^{2}U, one can show that (6) reduces to the isotropic density expression reported by Brünger in equation (16) of Brünger (1989) if all diagonal elements of U_{i} are equal to U_{iso} + b_{i}/8π^{2} + U_{add} with zero offdiagonal components.
2.3. Polarizable atomic multipole electron density
For the derivation of an atomic multipole expansion from a collection of point charges we begin with the Taylor expansion of the V(r) at r arising from n partial point charges that represent the electron density of an atom,
where Δ_{i} is the position of partial charge c_{i}, ∇_{α} = ∂/∂r_{α} is one component of the del operator, α ∈ {x, y, z} and the Greek subscripts {α, β} represent the use of the Einstein summation convention for summing over tensor elements (Stone, 1996). We omit the constant factor of 1/4π∊_{0} throughout for compactness. Let the monopole, dipole and traceless quadrupole moments be defined as
where removal of the trace in the definition of the quadrupole moment is allowed because the potential satisfies the Laplace equation (i.e. ∇^{2}V = 0). Substitution of the relationships in (8) into the final expression of (7) gives the in terms of a Cartesian multipole expansion, which we truncate at quadrupole order
We now replace the Coulomb potential of (9) with the potential from the sum of Gaussians from (1), which is given by
and find
We now introduce unique superscripts on the charge, dipole and quadrupole Gaussian basis sets, denoted by {n_{q}, n_{d}, n_{Θ}} and {κ_{q}, κ_{d}, κ_{Θ}}, to allow them to differ in number and width.
The potential of the charge density of (12) quickly approaches the Coulomb potential as r increases since the error function goes to unity such that at large r this potential satisfies the Laplace equation and the use of a traceless quadrupole tensor is still justified. Application of the Laplace operator to both sides of (12) gives the negative of a continuous charge density based on Cartesian Gaussian multipoles,
In crystallography the convention is that electron density is positive, so we will keep the negative sign. Therefore, a negative partial charge equates to positive scattering density.
Inclusion of ADPs is described by convolution of (13) with the realspace temperature factor,
Based on the convolution differentiation rule
the solution to (14) is given by substituting for f(r) in (13) with the corresponding ρ(r) from (6) to give
However, since q only represents partial atomic charges, the contributions from valence and core electrons need to be added. Additionally, the AMOEBA force field divides each atomic into permanent (d) and induced (u) contributions to account for polarization. Therefore, we construct the total atomic electron density at a location r relative to the center of atom j by adding the contribution of core and valence electron density to (16) and splitting the dipole into permanent and induced components to give
where P_{j}^{(c)} is the integer number of core electrons (carbon has two) and P_{j}^{(v)} is the integer number of valence electrons (carbon has four). The superscripts on the anisotropic Gaussian form factors ρ_{j}^{(n,κ)}(r) have been made explicit for our model. We make the reasonable choice of using the isolatedatom scattering parameters for both core and valence electron densities. The width of the core electron density is frozen at the isolatedatom description (κ = 1) based on the observation that chemical bonding does not perturb it significantly (Hansen & Coppens, 1978). On the other hand, the width of the valence electron density expands or contracts relative to the isolatedatom model owing to a gain or reduction, respectively, of electron density from or to covalently bonded atoms. This effect is modeled by the width parameter of the valence density κ_{v}. In this work, the dipole and quadrupole densities are described by a single Gaussian (n_{d} = n_{Θ} = 1) based on a and b parameters set to unity. The widths of the dipole and quadrupole densities are controlled by the κ_{d} and κ_{Θ} parameters. In this work, the width parameters {κ_{v}, κ_{d}, κ_{Θ}} are optimized against the diffraction data for each AMOEBA multipole type. The multipole moments are fixed by the AMOEBA force field and are not refined against the data.
The partial derivatives through second order of the anisotropic and aspherical density defined in (6), which are required for the realspace multipolar density given in (17), are
where u_{α} is a unit vector in the α direction with α ∈ {x, y, z}. In addition, the third, fourth and fifthorder terms of the expansion are presented as supplementary information along with a Mathematica notebook.^{1}
To the best of our knowledge, (17) is the first expression reported in the literature for a realspace form factor that is the convolution of an atomic multipolar electron density with anisotropic ADPs. This equation opens the door to exploring precise polarizable atomic multipole refinements in tandem with efficient computation of structure factors via FFT.
Given a q, d, Θ) are converted via rotation from a local frame. For example, as shown in Fig. 1, the z axis of the local frame for the carbonyl O atom of the peptide bond is in the direction of the bond to the carbonyl C atom. Its positive x axis is located in the O=C—C^{α} plane in the direction of the C^{α} atom and the y axis is chosen to give a righthanded coordinate system (Ren & Ponder, 2002). The induced dipole (u) on each atom is determined via a selfconsistent field (SCF) calculation, where the field is a sum of contributions from the permanent atomic multipoles and induced dipoles. The AMOEBA polarization model is described in greater detail in work by Ren & Ponder (2002).
the AMOEBA permanent multipole moments for each atom in the global coordinate frame (2.4. Derivatives of the electron density
2.4.1. Atomic coordinates
As a simplification, the derivation up to this point has assumed that the atomic center was the origin of the coordinate system. However, for this section on the derivatives with respect to atomic coordinates we place atom j at r_{j} in the global frame. In order to keep the derivation manageable, we split the total electron density into that produced by permanent charges ρ_{perm} and that of induced charges ρ_{ind},
The derivative of the permanent multipole electron density of atom j with respect to the α coordinate of atom j is given by
where the derivative of the dipole and quadrupole densities are each composed of two terms owing to the chain rule. As described above, the dipole and quadrupole moments of each atom are implicitly a function of its coordinates and the coordinates of a few of its bonded neighbors (atoms k) that define the local frame of the multipole. Therefore, the derivative of the permanent multipole electron density of atom j with respect to the α coordinate of atoms k must also be considered,
where the derivatives of spherically symmetric terms are zero with respect to the coordinates of atom k because they have no dependence on the orientation of the local frame. Note that the partial derivative of an anisotropic and aspherical density tensor with respect to an atomic coordinate is the negative of the partial derivatives given in (18), simply due to the negative sign on r_{j}. The derivatives of the polarizable density with respect to atomic coordinates are very specific to the AMOEBA electrostatic model and are discussed in Appendix B. However, we note that computing the derivatives of a polarizable density with respect to atomic coordinates is O(n^{2}logn) using PME, which quickly becomes the most expensive part of the overall calculation.
2.4.2. ADPs
The derivative of the anisotropic electron density of atom j with respect to an anisotropic displacement parameter U_{j,τυ} is given by
and requires the partial derivatives of the Cartesian Gaussian tensors with respect to ADP components. Introducing a few relationships facilitates their presentation. Firstly, based on the equality
we have
where the Kronecker delta δ_{τυ} is unity for diagonal elements of U and zero otherwise. Differentiating an identity from matrix algebra U^{−1}U = I gives the following relationship
which makes it possible to differentiate U instead of its inverse. This is preferred since only one or two elements of ∂U/∂U_{τυ} are equal to unity and the rest are zero. Specifically, a single element is equal to unity if τ equals υ, while two elements are equal to unity otherwise, since U_{τυ} and U_{υτ} represent the same variable in this case. For convenience, we define a 3 × 3 matrix J^{(τυ)},
and based on the chain rule we have
Differentiating (6) with respect to U_{τυ} and using (24), (27) and the product rule gives
2.4.3. Gaussian width
The Gaussian width parameter κ controls radial expansion and contraction of the Cartesian Gaussian multipoles. Analogous parameters are used to optimize the STOs within the Hansen and Coppens scattering model (Hansen & Coppens, 1978). The derivative of the electron density with respect to this parameter is similar to the gradient for the ADP parameters. Two chainrule terms are necessary. Firstly, the gradient of the normalizing term
where
Secondly, the gradient of the inverse ADP matrix is most conveniently expressed using the gradient of the original ADP matrix,
where
For convenience the matrix J_{i}^{(κ)} is defined to more compactly represent this result,
Differentiating (6) with respect to κ and using (29), (33) and the product rule gives
together with the third and fourthorder terms available as supplementary information^{1}.
2.5. Fourier transform of the polarizable atomic multipole electron density
Remarkably, the FT of the anisotropic and aspherical density given in (17) is simply
where the dipole and quadrupole terms in (35) depend on the FT of the partial derivatives defined in (18). Through fifth order the reciprocalspace tensors are
and in compressed tensor notation the general expression for order u + v + w is
This expression is considerably more compact than any reported previously for an aspherical scattering factor in ). Notably, our formulation has no dependence on cumbersome Fourier Bessel transforms of Slatertype functions (Dawson, 1967a; Hansen & Coppens, 1978; Su & Coppens, 1990). Our equation (35) has been implemented by `direct summation' for comparison to the performance of the FFT algorithm.
particularly the formulation based on STOs and spherical harmonics (Hansen & Coppens, 19783. Scattering models
Four scattering models were implemented by modifying and combining the CNS (Brünger et al., 1998) and TINKER (Ponder, 2004) code bases. The scattering models were added to the CNS code base, while TINKER was used to compute AMOEBA chemical forces and to supply CNS with polarizable multipoles in the global frame.
3.1. Isolated atom
The first scattering model (`IAM') is the conventional IAM based on the relativistic ).
factors described by Su & Coppens (19983.2. Isolated atom with interatomic scattering
The second scattering model (`IAM–IAS') augments the IAM with interatomic scattering sites at bond centers (Afonine et al., 2007). Unlike the model of Afonine and coworkers, our implementation does not include IAS sites at lone pairs or at the center of aromatic rings. We have neglected these sites based on the rationale that the AMOEBA electrostatic model is sufficient to capture these details of the electron density, which we provide further evidence for below when discussing the of a TyrGlyGly tripeptide.
In our approach, chemically equivalent bonds are constrained to use the same IAS parameters. Charge density that is added to or removed from bond centers is exactly balanced by changing the net charge of the bonddefining atoms. For example, a bond charge of −0.2 e requires atomic charge increments that sum to 0.2 e. In this way, all molecules retain their original net charge. Each bond type requires three parameters: the charge increments of both atoms and the Gaussian width of the scattering site. Bond types are defined based on the concatenation of the AMOEBA forcefield atom types.
3.3. AMOEBA
The third scattering model (`AMOEBA') is based on the polarizable atomic multipoles of the AMOEBA force field. Each chemically unique multipole type requires three Gaussian width parameters as described in §2. The induced dipoles were iterated to selfconsistency using PME whenever any atomic coordinates were changed during (Darden et al., 1993; Sagui et al., 2004; Essmann et al., 1995).
3.4. AMOEBA with interatomic scattering
The final scattering model (`AMOEBA–IAS') augments AMOEBA electrostatics with interatomic scattering sites. It became clear during the course of this study that an atomic multipole expansion truncated at quadrupole order is insufficient to capture bond charge density for most molecular geometries. This is consistent with theoretical observations by Stone and coworkers that the convergence of a distributed multipole analysis (DMA) may be improved by using both atoms and bond centers as expansion sites (Stone & Alderton, 1985; Stone, 2005). Furthermore, experimental data from the Xray scattering of diamond and silicon, simple examples of tetrahedral bonding geometry, are explained by the superposition of one atomic octopole moment and one atomic hexadecapole moment (Dawson, 1967a,b). The characteristics of the four scattering models are further clarified below with respect to four peptide test cases.
The following computational details were constant across all of the refinements. The isotropic ADP offset U_{add} was set to 1/(4π^{2}), which is equivalent to B_{add} = 8π^{2}U_{add} = 2, the FFT grid factor to 0.33 (as appropriate for crystal structures at subatomic resolution), and the electrondensity cutoff around each atom was 18 (specified by the E_{lim} parameter in CNS). These conservative parameters led to close agreement between direct summation and FFT computation of structure factors. The CNS parameter w_{A} that controls the weighting of Xray target function relative to the forcefield energy was set to 1.0, although we also tested 0.2.
This raised R_{free} values by less than 0.1% and lowered the AMOEBA differences between refinements presented below, but did not alter any trends or our conclusions. It should be noted that forcefield restraints are not necessarily required for at subatomic high resolution. However, their use in this study gives an insight into the relative energetic cost of the structural changes arising from differences in the four scattering models. A modified version of the refine.inp CNS task file was used for all refinements using the MLI target function.
4. Applications
To demonstrate the behavior of Xray refinements based on Cartesian Gaussian multipoles, we present two sets of applications. The first set is simply to illustrate the performance of direct summation versus FFT and SGFFT computation of structure factors as a function of system size. The second set describes refinements on a series of four peptide crystals that diffract to 0.59 Å resolution or better. All examples use the AMOEBA force field for chemical forces, instead of the default CNS force field based on Engh & Huber parameters (Engh & Huber, 1991). Although the refinements were performed in the native of each crystal, AMOEBA energies and gradients as computed using the TINKER code base required expanding to P1. This did not increase the number of refined variables, but suggests the need for an AMOEBA code that takes advantage of crystal symmetry.
4.1. Runtime scaling on protein data sets
Evaluation of the target function and its derivatives by directsummation calculation of structure factors via (35) and (36) is O(N_{atoms} × N_{reflections} × N_{symm}). Alternatively, the FFT algorithm based on (17) and (18) is O(N_{grid} × logN_{grid}), where the number of grid points N_{grid} depends on the resolution of the diffraction data. Aspherical refinements based on the Hansen–Coppens formalism are currently limited to direct summation, since the realspace form of the electron density convolved with ADPs is unknown. Therefore, the performance of Xray refinements based on Cartesian Gaussian multipoles and FFT is of particular interest. The results are summarized in Table 1 and are plotted in Fig. 2. Although the performance difference is only about a factor of two for the small protein crambin, over an order of magnitude improvement is achieved for both ribonuclease A and aldose reductase. Parallelization with the SGFFT method results in a further significant speedup (a speedup of a factor of nearly four relative to a single processor on a fourprocessor machine).

4.2. of peptide crystals
In principle, a more precise scattering model based on Cartesian Gaussian multipoles with coefficients from the AMOEBA electrostatics model should improve the quality of refinements relative to the IAM as judged by both R_{free} and the of the Furthermore, the quality of the AMOEBA function can also be assayed, since it is reasonable to expect that and R_{free} should be correlated.
The peptide crystals studied include YG_{2} (PichonPesme et al., 2000), cyclic P_{2}A_{4} (Dittrich et al., 2002) and AYA with three waters or with an ethanol molecule (Chęcińska, Forster et al., 2006; Chęcińska, Mebs et al., 2006). Detailed descriptions of the unitcell parameters, number of atoms, resolution and measured reflections are given in Table 2. The results are summarized in Table 3 and compared with previous work below.


4.2.1. YG_{2}
The R_{free} values of the IAM and IAM–IAS refinements of YG_{2} (4.60 and 3.86%, respectively) are slightly lower than those reported by Afonine and coworkers (4.72 and 4.06%, respectively; Afonine et al., 2007). The R_{free} value of the AMOEBA–IAS (3.50%) is a significant improvement. The R_{work} value (3.17%) of the AMOEBA–IAS is also lower and is comparable to multipolar refinements reported by Volkov and coworkers using transferred or refined multipole coefficients (3.66% and 3.42%, respectively; Volkov et al., 2007). Crossvalidationbased comparisons are unavailable in this case. We note that the AMOEBA–IAS used a reflectionstoparameters ratio of 11.1, which is slightly higher than the value of 10.6 reported by Volkov and coworkers using refined multipole coefficients. This is computed based on the number of reflections reported in Table 2 and the number of parameters given in Table 3.
Electrondensity maps of the tyrosine ring for the four scattering models are shown in Fig. 3, which lend visual insight into their properties. The nonH atom positions are apparent in the 2F_{o} − F_{c} contours for each The standard IAM scattering model underestimates the electron density at bond centers and at the oxygen lonepair sites, as shown by the F_{o} − F_{c} contours. Our IAM–IAS scattering model explains the electron density at bond centers, but does not capture lonepair electron density. Conversely, the AMOEBA model places electron density approximately at the lonepair positions but not at bond centers. Finally, the AMOEBA–IAS model explains much of the lonepair and bonding electron densities.
4.2.2. P_{2}A_{4}
The R_{free} values of our IAM and IAM–IAS refinements of P_{2}A_{4} (3.73 and 3.01%, respectively) agree closely with the values of Afonine and coworkers (3.63 and 3.23%, respectively; Afonine et al., 2007). The R_{free} value of the AMOEBA–IAS (2.94%) is lower by 0.07%, which is the least amount of improvement seen for AMOEBA–IAS relative to IAM–IAS in this study. The R_{work} value (2.86%) of the AMOEBA–IAS is slightly higher, but comparable to those reported by Volkov and coworkers using transferred or refined multipole coefficients (2.60% and 2.53%), although this work uses a higher reflectionstoparameters ratio (50.3 compared with 43.6; Volkov et al., 2007). As for YG_{2}, crossvalidation was not performed. The similarity of the R values for YG_{2} and P_{2}A_{4} between the AMOEBA–IAS refinements and the multipolar refinements of Volkov and coworkers is consistent with the principle that bond scattering sites capture density that is represented by higher order atomic moments missing in the AMOEBA model (octopole and hexadecapole).
In Fig. 4 the precision of the R_{work} and R_{free} values computed using discrete FTs are compared with analytic direct summation for P_{2}A_{4} under the AMOEBA scattering model. Agreement to four decimal places is seen for B_{add} values between 0 and 3 Å^{2}, which serves as validation of the correctness of (17) and (35). These results support the conclusion that FFTbased computation of structure factors is appropriate for anisotropic and aspherical scattering models.
4.2.3. AYA
The AYA data sets were chosen because of the extremely low temperature achieved during the measurement of structure factors (9 K for AYA + three waters and 20 K for AYA + ethanol). For AYA + water, Chęcińska and coworkers (Chęcińska, Forster et al., 2006; Chęcińska, Mebs et al., 2006) originally reported an R value of 2.4%, which is in agreement with the R value of our IAM (2.67%). Addition of IAS lowered the R_{free} statistic from 2.71% to 2.39%, while addition of polarizable atomic multipole electron density showed a further improvement to an R_{free} of 1.95%. For AYA + ethanol the R_{work} value of the IAM (3.20%) is comparable to that reported originally by Chęcińska and coworkers (2.9%). IAM–IAS lowered R_{free} from 3.33 to 2.49%, while AMOEBA–IAS achieved 2.08%.
4.3. summary
The results for all four peptide refinements are summarized in Fig. 5. In every case, use of the AMOEBA–IAS scattering model relative to the IAM scattering model lowered both R_{free} and the of the crystal. When the IAM scattering model is used, molecular conformations are highly strained to compensate. For example, H—C atom bonds are too short because the IAM model centers electron density at the hydrogen nucleus. In the crystal structures, this electron density is shifted towards the C atom. As the description of the electron density is improved, the relaxes by approximately 16 kJ mol^{−1} per residue. The precise amount of relaxation depends on the weighting between the crystallographic target and the force field. Unrestrained refinements with an IAM scattering model could adopt even more unphysical conformations. This suggests that accurate chemical restraints are necessary even for ultrahighresolution refinements unless an anisotropic and aspherical scattering model is used.
In Fig. 6, we present plots of the IAS sites that were refined for each peptide system. Their Gaussian fullwidth at halfmaximum (FWHM) is plotted against charge magnitude for both the IAM–IAS and the AMOEBA–IAS models. The majority of the charges under the IAM–IAS model and all of the charges under the AMOEBA–IAS model refined to negative partial charge values (or positive scattering density), which is consistent with the physical concentration of charge density at chemical bonds. The similarity of the refined charges between the IAM–IAS and the AMOEBA–IAS models suggests that an atomic multipole description of electron density truncated at quadrupole order underestimates density at trigonal and tetrahedral bond centers.
5. Conclusions
Cartesian Gaussian multipoles offer an efficient alternative to the Hansen and Coppens formulation of aspherical scattering. They eliminate the use of Slatertype functions and allow structure factors to be computed by FFT. Numerical tests show that that FFT and directsummation implementations of Cartesian Gaussian multipoles agree to high precision. For subatomic resolution biomolecular data sets such as ribonuclease A and aldose reductase, parallelized computation of structure factors using the SGFFT method results in a speedup of one to two orders of magnitude compared with direct summation.
Ideally, a forcefield electrostatics model should be accurate enough to explain the electron density observed in Xray diffraction experiments. Although the AMOEBA polarizable multipole force field energetic model shows promise, truncation of the permanent moments at quadrupole order systematically underestimates electron density at bond centers. Our results suggest that the added computational expense of including hexadecapole moments in the
computation is justified. As supplementary information we have provided a Mathematica notebook and formulae that allow computation of Cartesian Gaussian multipoles up to the fourth order in anticipation of further improvements to force fields.In the near future, we will present the results of applying our polarizable atomic multipole et al., 2007), our scattering model significantly improves as it does for the simpler peptide cases presented here. Equally exciting will be the use of the AMOEBA force field and in particular the electrostatic forces to orient water molecules in the absence of clear Hatom electron density. We anticipate that of hydrogenbonding networks will enhance the usefulness of Xray crystallography experiments with respect to explaining pK_{a} shifts, ligandbinding affinities and enzymatic mechanisms.
methodology to macromolecules. For ultrahighresolution macromolecular data sets, such as HEWL at 0.65 Å (WangAPPENDIX A
Fourier transform definition
The definition and notation for the Fourier transform as used in this work is given by
and the corresponding inverse Fourier transform by
APPENDIX B
Derivative of the polarizable electron density with respect to atomic coordinates
The total polarizable electron density arising from the induced dipole of all atoms is given by
The gradient of this density with respect to the α component of atom j is
The second term is nonzero only for i = j and is simple to calculate. The first term, however, depends on ∂u_{i,β}/∂r_{j,α} which is the derivative of a component of the induced dipole of atom i with respect to the α component of atom j. In other words, perturbing the position of atom j affects not only its own scattering but that of all polarizable atoms. The induced dipole on atom i arises from the selfconsistent crystal field multiplied by the polarizability,
where α_{i} is the atomic polarizability of atom i, T_{ik}^{(1)} is a matrix of tensors that produces the field at site i
owing to the multipole M_{k} at site k
and T_{ik}^{(11)} is the matrix of tensors that produces the field at site i
owing to the induced dipole u_{k} at site k. For simplicity, we have not formulated (43) using PME electrostatics. Therefore, the sum over k includes all atoms in the crystal except atom i. The derivative of (43) with respect to coordinate r_{j,α} is given by
The first three terms on the righthand side are not difficult to compute. However, the fourth term shows that the gradients of the polarizable scattering are O(n^{3}) without use of PME. Specifically, there are 3n × 3n induced dipole density derivatives, each of which is the sum of 3n terms. In this work, we have computed these derivatives by finite differences using PME, which is O(n^{2}logn).
Supporting information
Higher order Cartesian Gaussian multipole tensors. DOI: 10.1107/S0907444909022707/dz5164sup1.pdf
Mathematica notebook. DOI: 10.1107/S0907444909022707/dz5164sup2.pdf
Acknowledgements
The authors wish to thank Jay W. Ponder and Chuanjie Wu for carefully editing the manuscript. We also thank Pengyu Ren, Paul D. Adams and Thomas A. Darden for helpful discussions. This work was supported by an award from the NSF to Vijay S. Pande, Jay W. Ponder, Teresa HeadGordon and Martin HeadGordon for `Collaborative Research: Cyberinfrastructure for Next Generation Biomolecular Modeling' (Award No. CHE0535675) and by the Howard Hughes Medical Institute.
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