Polarizable atomic multipole X-ray refinement: application to peptide crystals

Abstract

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 restrained refinement of molecular models against X-ray 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 refinement of ultrahigh-resolution data sets. Use of a sublattice 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 AMOEBA-based density captured bonding effects with fewer parameters. For a series of four peptide crystals, the AMOEBA-IAS model lowered R(free) by 20-40% relative to the original spherically symmetric scattering model.

Figure 1

The local multipole frame of the carbonyl O atom of the peptide backbone is shown. The positive z axis is along the C=O bond and the x axis is chosen in the O=C—C^α plane in the direction of the C^α atom. The y axis is directed into the page in order to achieve a right-handed coordinate system. Also shown are the nonzero multipole moments of the O atom and a qualitative representation of their shape. The d _z Cartesian Gaussian dipole (in Debye units) places electron density along the C=O bond, while the trace of the Cartesian Gaussian quadrupole (in Buckingham units) positions electron density approximately at lone-pair positions.

Figure 2

The scaling of the Cartesian Gaussian multipole model, truncated at quadrupole order, is plotted on a log–log scale for computation of the intensity-based maximum-likelihood target function (MLI) for direct summation, FFT and SGFFT. Direct summation scales linearly with the product of the number of atoms, the number of reflections and the number of symmetry operators. Computation of the crystallographic target function by FFT of the Cartesian Gaussian multipole electron density shows a speedup of a factor of between 1.8 and 14.5 compared with direct summation. A further speedup factor of nearly four is achieved using the SGFFT method on a four-processor machine.

Figure 3

(a) IAM, (b) IAM–IAS, (c) AMOEBA and (d) AMOEBA–IAM refinements, respectively, for GY₂. The F _o − F _c and 2F _o − F _c σ_A-weighted electron-density maps are contoured at 3.5σ and shown in green and gray, respectively. Both the IAM and AMOEBA models fail to explain the electron density at bond centers seen in the data. In addition, the IAM model does not account for lone-pair density on the O atom.

Figure 4

The precision of numerical computation of the R _work and R _free values via FFT is compared with analytic direct summation as a function of the isotropic increase B _add in ADP parameters for P₂A₄ under the AMOEBA scattering model. Note that B _add = 8π² U _add.

Figure 5

The improvement arising from the AMOEBA–IAS scattering model, relative to the IAM model, is plotted as a function of relative percentage improvement in R _free value and the relative AMOEBA potential energy per residue. For all data sets, the best R _free value and lowest potential energy per residue were achieved using the AMOEBA–IAS scattering model. 1 kcal mol⁻¹ = 4.186 kJ mol⁻¹.

Figure 6

For the inter-atomic scattering sites of the IAM–IAS (a) and AMOEBA–IAS (b) scattering models, the refined Gaussian full-width at half-maximum (FWHM) is plotted versus partial charge magnitude. The majority of charges for the IAM–IAS model and all charges for the AMOEBA–IAS are negative. The sub-angstrom FWHM values are consistent with very localized bond densities.