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<record><TEI><teiHeader><fileDesc><titleStmt><title xml:lang="en">Polarizable atomic multipole X-ray refinement: application to peptide crystals</title><author><name sortKey="Schnieders, Michael J" sort="Schnieders, Michael J" uniqKey="Schnieders M" first="Michael J." last="Schnieders">Michael J. Schnieders</name><affiliation><nlm:aff id="a">Department of Chemistry, Stanford, CA 94305,<country>USA</country></nlm:aff></affiliation></author><author><name sortKey="Fenn, Timothy D" sort="Fenn, Timothy D" uniqKey="Fenn T" first="Timothy D." last="Fenn">Timothy D. Fenn</name><affiliation><nlm:aff id="b">Department of Molecular and Cellular Physiology, Stanford, CA 94305,<country>USA</country></nlm:aff></affiliation><affiliation><nlm:aff id="c">Howard Hughes Medical Institute,<country>USA</country></nlm:aff></affiliation></author><author><name sortKey="Pande, Vijay S" sort="Pande, Vijay S" uniqKey="Pande V" first="Vijay S." last="Pande">Vijay S. Pande</name><affiliation><nlm:aff id="a">Department of Chemistry, Stanford, CA 94305,<country>USA</country></nlm:aff></affiliation></author><author><name sortKey="Brunger, Axel T" sort="Brunger, Axel T" uniqKey="Brunger A" first="Axel T." last="Brunger">Axel T. Brunger</name><affiliation><nlm:aff id="b">Department of Molecular and Cellular Physiology, Stanford, CA 94305,<country>USA</country></nlm:aff></affiliation><affiliation><nlm:aff id="c">Howard Hughes Medical Institute,<country>USA</country></nlm:aff></affiliation></author></titleStmt><publicationStmt><idno type="wicri:source">PMC</idno><idno type="pmid">19690373</idno><idno type="pmc">2733883</idno><idno type="url">http://www.ncbi.nlm.nih.gov/pmc/articles/PMC2733883</idno><idno type="RBID">PMC:2733883</idno><idno type="doi">10.1107/S0907444909022707</idno><date when="2009">2009</date><idno type="wicri:Area/Pmc/Corpus">000473</idno></publicationStmt><sourceDesc><biblStruct><analytic><title xml:lang="en" level="a" type="main">Polarizable atomic multipole X-ray refinement: application to peptide crystals</title><author><name sortKey="Schnieders, Michael J" sort="Schnieders, Michael J" uniqKey="Schnieders M" first="Michael J." last="Schnieders">Michael J. Schnieders</name><affiliation><nlm:aff id="a">Department of Chemistry, Stanford, CA 94305,<country>USA</country></nlm:aff></affiliation></author><author><name sortKey="Fenn, Timothy D" sort="Fenn, Timothy D" uniqKey="Fenn T" first="Timothy D." last="Fenn">Timothy D. Fenn</name><affiliation><nlm:aff id="b">Department of Molecular and Cellular Physiology, Stanford, CA 94305,<country>USA</country></nlm:aff></affiliation><affiliation><nlm:aff id="c">Howard Hughes Medical Institute,<country>USA</country></nlm:aff></affiliation></author><author><name sortKey="Pande, Vijay S" sort="Pande, Vijay S" uniqKey="Pande V" first="Vijay S." last="Pande">Vijay S. Pande</name><affiliation><nlm:aff id="a">Department of Chemistry, Stanford, CA 94305,<country>USA</country></nlm:aff></affiliation></author><author><name sortKey="Brunger, Axel T" sort="Brunger, Axel T" uniqKey="Brunger A" first="Axel T." last="Brunger">Axel T. Brunger</name><affiliation><nlm:aff id="b">Department of Molecular and Cellular Physiology, Stanford, CA 94305,<country>USA</country></nlm:aff></affiliation><affiliation><nlm:aff id="c">Howard Hughes Medical Institute,<country>USA</country></nlm:aff></affiliation></author></analytic><series><title level="j">Acta Crystallographica Section D: Biological Crystallography</title><idno type="ISSN">0907-4449</idno><idno type="eISSN">1399-0047</idno><imprint><date when="2009">2009</date></imprint></series></biblStruct></sourceDesc></fileDesc><profileDesc><textClass></textClass></profileDesc></teiHeader><front><div type="abstract" xml:lang="en"><p>A method to accelerate the computation of structure factors from an electron density described by anisotropic and aspherical atomic form factors <italic>via</italic> fast Fourier transformation is described for the first time.</p></div></front><back><div1 type="bibliography"><listBibl><biblStruct></biblStruct><biblStruct></biblStruct><biblStruct></biblStruct><biblStruct></biblStruct><biblStruct></biblStruct><biblStruct></biblStruct><biblStruct></biblStruct><biblStruct></biblStruct><biblStruct></biblStruct><biblStruct></biblStruct><biblStruct></biblStruct><biblStruct></biblStruct><biblStruct></biblStruct><biblStruct></biblStruct><biblStruct></biblStruct><biblStruct></biblStruct><biblStruct></biblStruct><biblStruct></biblStruct><biblStruct></biblStruct><biblStruct></biblStruct><biblStruct></biblStruct><biblStruct></biblStruct><biblStruct></biblStruct><biblStruct></biblStruct><biblStruct></biblStruct><biblStruct></biblStruct><biblStruct></biblStruct><biblStruct></biblStruct><biblStruct></biblStruct><biblStruct></biblStruct><biblStruct></biblStruct><biblStruct></biblStruct><biblStruct></biblStruct><biblStruct></biblStruct><biblStruct></biblStruct><biblStruct></biblStruct><biblStruct></biblStruct><biblStruct></biblStruct><biblStruct></biblStruct><biblStruct></biblStruct><biblStruct></biblStruct><biblStruct></biblStruct><biblStruct></biblStruct><biblStruct></biblStruct><biblStruct></biblStruct><biblStruct></biblStruct><biblStruct></biblStruct><biblStruct></biblStruct><biblStruct></biblStruct><biblStruct></biblStruct><biblStruct></biblStruct><biblStruct></biblStruct><biblStruct></biblStruct><biblStruct></biblStruct><biblStruct></biblStruct><biblStruct></biblStruct><biblStruct></biblStruct><biblStruct></biblStruct><biblStruct></biblStruct><biblStruct></biblStruct><biblStruct></biblStruct><biblStruct></biblStruct><biblStruct></biblStruct><biblStruct></biblStruct><biblStruct></biblStruct><biblStruct></biblStruct><biblStruct></biblStruct><biblStruct></biblStruct><biblStruct></biblStruct><biblStruct></biblStruct><biblStruct></biblStruct><biblStruct></biblStruct><biblStruct></biblStruct><biblStruct></biblStruct><biblStruct></biblStruct><biblStruct></biblStruct><biblStruct></biblStruct><biblStruct></biblStruct><biblStruct></biblStruct><biblStruct></biblStruct><biblStruct></biblStruct><biblStruct></biblStruct><biblStruct></biblStruct><biblStruct></biblStruct><biblStruct></biblStruct><biblStruct></biblStruct><biblStruct></biblStruct></listBibl></div1></back></TEI><pmc article-type="research-article"><pmc-dir>properties open_access</pmc-dir>
<front><journal-meta><journal-id journal-id-type="nlm-ta">Acta Crystallogr D Biol Crystallogr</journal-id><journal-id journal-id-type="publisher-id">Acta Cryst. D</journal-id><journal-title-group><journal-title>Acta Crystallographica Section D: Biological Crystallography</journal-title></journal-title-group><issn pub-type="ppub">0907-4449</issn><issn pub-type="epub">1399-0047</issn><publisher><publisher-name>International Union of Crystallography</publisher-name></publisher></journal-meta><article-meta><article-id pub-id-type="pmid">19690373</article-id><article-id pub-id-type="pmc">2733883</article-id><article-id pub-id-type="publisher-id">dz5164</article-id><article-id pub-id-type="doi">10.1107/S0907444909022707</article-id><article-id pub-id-type="coden">ABCRE6</article-id><article-id pub-id-type="pii">S0907444909022707</article-id><article-categories><subj-group subj-group-type="heading"><subject>Research Papers</subject></subj-group></article-categories><title-group><article-title>Polarizable atomic multipole X-ray refinement: application to peptide crystals</article-title><alt-title>Polarizable atomic multipole X-ray refinement</alt-title></title-group><contrib-group><contrib contrib-type="author"><name><surname>Schnieders</surname><given-names>Michael J.</given-names></name><xref ref-type="aff" rid="a">a</xref></contrib><contrib contrib-type="author"><name><surname>Fenn</surname><given-names>Timothy D.</given-names></name><xref ref-type="aff" rid="b">b</xref><xref ref-type="aff" rid="c">c</xref></contrib><contrib contrib-type="author"><name><surname>Pande</surname><given-names>Vijay S.</given-names></name><xref ref-type="aff" rid="a">a</xref><xref ref-type="corresp" rid="cor">*</xref></contrib><contrib contrib-type="author"><name><surname>Brunger</surname><given-names>Axel T.</given-names></name><xref ref-type="aff" rid="b">b</xref><xref ref-type="aff" rid="c">c</xref><xref ref-type="corresp" rid="cor">*</xref></contrib><aff id="a"><label>a</label>Department of Chemistry, Stanford, CA 94305,<country>USA</country></aff><aff id="b"><label>b</label>Department of Molecular and Cellular Physiology, Stanford, CA 94305,<country>USA</country></aff><aff id="c"><label>c</label>Howard Hughes Medical Institute,<country>USA</country></aff></contrib-group><author-notes><corresp id="cor">Correspondence e-mail: <email>pande@stanford.edu</email>, <email>brunger@stanford.edu</email></corresp></author-notes><pub-date pub-type="ppub"><day>01</day><month>9</month><year>2009</year></pub-date><pub-date pub-type="epub"><day>14</day><month>8</month><year>2009</year></pub-date><pub-date pub-type="pmc-release"><day>14</day><month>8</month><year>2009</year></pub-date><pmc-comment> PMC Release delay is 0 months and 0 days and was based on the
<volume>65</volume><issue>Pt 9</issue><issue-id pub-id-type="publisher-id">d090900</issue-id><fpage>952</fpage><lpage>965</lpage><supplementary-material content-type="local-data"><media xlink:href="d-65-00952-sup1.pdf" mimetype="application" mime-subtype="pdf"><caption><p>Click here for additional data file.</p></caption></media><p>Supplementary material file. DOI: <ext-link ext-link-type="uri" xlink:href="http://dx.doi.org/10.1107/S0907444909022707/dz5164sup1.pdf">10.1107/S0907444909022707/dz5164sup1.pdf</ext-link></p><media xlink:href="d-65-00952-sup1.pdf" xlink:type="simple" id="d32e135" position="anchor" mimetype="application" mime-subtype="pdf"></media></supplementary-material><supplementary-material content-type="local-data"><media xlink:href="d-65-00952-sup2.pdf" mimetype="application" mime-subtype="pdf"><caption><p>Click here for additional data file.</p></caption></media><p>Supplementary material file. DOI: <ext-link ext-link-type="uri" xlink:href="http://dx.doi.org/10.1107/S0907444909022707/dz5164sup2.pdf">10.1107/S0907444909022707/dz5164sup2.pdf</ext-link></p><media xlink:href="d-65-00952-sup2.pdf" xlink:type="simple" id="d32e142" position="anchor" mimetype="application" mime-subtype="pdf"></media></supplementary-material><history><date date-type="received"><day>24</day><month>4</month><year>2009</year></date><date date-type="accepted"><day>13</day><month>6</month><year>2009</year></date></history><permissions><copyright-statement>© Schnieders et al. 2009</copyright-statement><copyright-year>2009</copyright-year><license license-type="open-access" xlink:href="http://creativecommons.org/licenses/by/2.0/uk/"><license-p>This is an open-access article distributed under the terms of the Creative Commons Attribution Licence, which permits unrestricted use, distribution, and reproduction in any medium, provided the original authors and source are cited.</license-p></license></permissions><self-uri xlink:type="simple" xlink:href="http://dx.doi.org/10.1107/S0907444909022707">A full version of this article is available from Crystallography Journals Online.</self-uri><abstract abstract-type="toc"><p>A method to accelerate the computation of structure factors from an electron density described by anisotropic and aspherical atomic form factors <italic>via</italic> fast Fourier transformation is described for the first time.</p></abstract><abstract><p>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 <italic>R</italic><sub>free</sub> by 20–40% relative to the original spherically symmetric scattering model.</p></abstract><kwd-group><kwd>scattering factors</kwd><kwd>aspherical</kwd><kwd>anisotropic</kwd><kwd>force fields</kwd><kwd>multipole</kwd><kwd>polarization</kwd><kwd>AMOEBA</kwd><kwd>bond density</kwd><kwd>direct summation</kwd><kwd>FFT</kwd><kwd>SGFFT</kwd><kwd>Ewald</kwd><kwd>PME</kwd></kwd-group></article-meta></front><body><sec id="sec1" sec-type="introduction"><label>1.</label><title>Introduction</title><p>The number of X-ray crystal structures in the Protein Data Bank (PDB) with a resolution of higher than 1.0 Å continues to increase rapidly (Berman <italic>et al.</italic>, 2000<xref ref-type="bibr" rid="bb8"> ▶</xref>). In late 2002, there were already over 100 structures available at subatomic resolution (Afonine & Urzhumtsev, 2004<xref ref-type="bibr" rid="bb4"> ▶</xref>), 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 <italic>et al.</italic>, 2007<xref ref-type="bibr" rid="bb86"> ▶</xref>), aldose reductase at 0.66 Å (Howard <italic>et al.</italic>, 2004<xref ref-type="bibr" rid="bb44"> ▶</xref>) and serine protease at 0.78 Å (Kuhn <italic>et al.</italic>, 1998<xref ref-type="bibr" rid="bb48"> ▶</xref>), as well as nucleic acid structures such as B-DNA at 0.74 Å (Kielkopf <italic>et al.</italic>, 2000<xref ref-type="bibr" rid="bb47"> ▶</xref>), Z-DNA at 0.60 Å (Tereshko <italic>et al.</italic>, 2001<xref ref-type="bibr" rid="bb82"> ▶</xref>) and an RNA tetraplex at 0.61 Å (Deng <italic>et al.</italic>, 2001<xref ref-type="bibr" rid="bb32"> ▶</xref>). Crystals that diffract to high resolution are ideal for studying valence-electron distributions (Jelsch <italic>et al.</italic>, 2000<xref ref-type="bibr" rid="bb46"> ▶</xref>; Muzet <italic>et al.</italic>, 2003<xref ref-type="bibr" rid="bb54"> ▶</xref>; Zarychta <italic>et al.</italic>, 2007<xref ref-type="bibr" rid="bb87"> ▶</xref>; Volkov <italic>et al.</italic>, 2007<xref ref-type="bibr" rid="bb84"> ▶</xref>; Coppens & Volkov, 2004<xref ref-type="bibr" rid="bb25"> ▶</xref>) 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 <italic>et al.</italic>, 2002<xref ref-type="bibr" rid="bb70"> ▶</xref>, 2005<xref ref-type="bibr" rid="bb71"> ▶</xref>; Sorin & Pande, 2005<xref ref-type="bibr" rid="bb72"> ▶</xref>; Pande <italic>et al.</italic>, 2003<xref ref-type="bibr" rid="bb55"> ▶</xref>). In this work, we contribute an improved theory and algorithm for computing the anisotropic and aspherical valence-electron density of molecules for X-ray crystal structure refinement.</p><p>Calculation of structure factors is generally based on scattering factors defined by the isolated-atom 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 electron-density 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, 1967<italic>a</italic><xref ref-type="bibr" rid="bb28"> ▶</xref>,<italic>b</italic><xref ref-type="bibr" rid="bb29"> ▶</xref>,<italic>c</italic><xref ref-type="bibr" rid="bb30"> ▶</xref>; DeMarco & Weiss, 1965<xref ref-type="bibr" rid="bb31"> ▶</xref>). In these works, the IAM was augmented by atom-centered spherical harmonic expansions, whose physical consequence was to redistribute electron density from nonbonding lobes into the tetragonal arrangement of bond centers.</p><p>A variety of radial functions have been used in combination with atom-centered spherical harmonic expansions. Modified Gaussians were promoted by Dawson (1967<italic>a</italic><xref ref-type="bibr" rid="bb28"> ▶</xref>), a set of harmonic oscillator wavefunctions by Kurki-Suonio (1968<xref ref-type="bibr" rid="bb50"> ▶</xref>) and more recently a formalism based on Slater-type orbitals (STO) was described by Stewart and coworkers (Epstein <italic>et al.</italic>, 1977<xref ref-type="bibr" rid="bb35"> ▶</xref>; Cromer <italic>et al.</italic>, 1976<xref ref-type="bibr" rid="bb26"> ▶</xref>; Stewart, 1979<xref ref-type="bibr" rid="bb74"> ▶</xref>, 1977<xref ref-type="bibr" rid="bb73"> ▶</xref>) and by Hansen & Coppens (1978<xref ref-type="bibr" rid="bb41"> ▶</xref>), which represents the current standard (Jelsch <italic>et al.</italic>, 2005<xref ref-type="bibr" rid="bb45"> ▶</xref>; Zarychta <italic>et al.</italic>, 2007<xref ref-type="bibr" rid="bb87"> ▶</xref>; Volkov <italic>et al.</italic>, 2007<xref ref-type="bibr" rid="bb84"> ▶</xref>; Coppens, 2005<xref ref-type="bibr" rid="bb24"> ▶</xref>). However, spherical harmonics are not the only basis set available to describe the angular dependence of the deformation density.</p><p>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 <italic>via</italic> direct summation in reciprocal space than the STO-based theory of Hansen & Coppens (1978<xref ref-type="bibr" rid="bb41"> ▶</xref>). We also demonstrate that Cartesian Gaussian multipoles allow the computation of structure factors <italic>via</italic> fast Fourier transformation (FFT) of the real-space electron density (Cooley & Tukey, 1965<xref ref-type="bibr" rid="bb23"> ▶</xref>). The latter approach, originally proposed by Ten Eyck (1973<xref ref-type="bibr" rid="bb80"> ▶</xref>, 1977<xref ref-type="bibr" rid="bb81"> ▶</xref>), is the basis of the efficient macromolecular refinement algorithms (Brünger, 1989<xref ref-type="bibr" rid="bb12"> ▶</xref>; Afonine & Urzhumtsev, 2004<xref ref-type="bibr" rid="bb4"> ▶</xref>; Afonine <italic>et al.</italic>, 2007<xref ref-type="bibr" rid="bb3"> ▶</xref>; Agarwal, 1978<xref ref-type="bibr" rid="bb5"> ▶</xref>) implemented in programs such as <italic>CNS</italic> (Brünger <italic>et al.</italic>, 1998<xref ref-type="bibr" rid="bb15"> ▶</xref>; Brunger, 2007<xref ref-type="bibr" rid="bb14"> ▶</xref>) and <italic>PHENIX</italic> (Adams <italic>et al.</italic>, 2002<xref ref-type="bibr" rid="bb1"> ▶</xref>). The sublattice method implemented in <italic>CNS</italic> lends itself to efficient parallelization (Brünger, 1989<xref ref-type="bibr" rid="bb12"> ▶</xref>).</p><p>Boys originally proposed Cartesian Gaussian functions as basis functions to solve the many-electron Schrödinger equation (Boys, 1950<xref ref-type="bibr" rid="bb9"> ▶</xref>). The advantage of Gaussians over STOs in this context is that two-electron integrals have analytic forms, which has led to the adoption of Gaussian basis sets for many <italic>ab initio</italic> calculations (Hehre <italic>et al.</italic>, 1969<xref ref-type="bibr" rid="bb43"> ▶</xref>, 1970<xref ref-type="bibr" rid="bb42"> ▶</xref>). We note that the equivalence of spherical harmonics and Cartesian tensors is well known, with key relationships having been presented by Stone (1996<xref ref-type="bibr" rid="bb75"> ▶</xref>) and Applequist (1989<xref ref-type="bibr" rid="bb6"> ▶</xref>, 2002<xref ref-type="bibr" rid="bb7"> ▶</xref>).</p><p>We apply Cartesian Gaussian multipoles to restrained crystallographic refinements based on the Atomic Multipole Optimized Energetics for Biomolecular Applications (AMOEBA) force-field electrostatic model (Ponder & Case, 2003<xref ref-type="bibr" rid="bb61"> ▶</xref>; Ren & Ponder, 2002<xref ref-type="bibr" rid="bb62"> ▶</xref>, 2003<xref ref-type="bibr" rid="bb63"> ▶</xref>, 2004<xref ref-type="bibr" rid="bb64"> ▶</xref>; Schnieders <italic>et al.</italic>, 2007<xref ref-type="bibr" rid="bb66"> ▶</xref>; Schnieders & Ponder, 2007<xref ref-type="bibr" rid="bb67"> ▶</xref>). 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.</p><p>Force fields are widely used to restrain macromolecular refinement by contributing forces to local optimizations and molecular dynamics (Brünger <italic>et al.</italic>, 1987<xref ref-type="bibr" rid="bb19"> ▶</xref>), with the latter used within simulated-annealing algorithms to promote global optimization (Brünger, 1988<xref ref-type="bibr" rid="bb11"> ▶</xref>, 1991<xref ref-type="bibr" rid="bb13"> ▶</xref>; Brünger <italic>et al.</italic>, 1989<xref ref-type="bibr" rid="bb17"> ▶</xref>, 1990<xref ref-type="bibr" rid="bb18"> ▶</xref>, 1997<xref ref-type="bibr" rid="bb16"> ▶</xref>; Kuriyan <italic>et al.</italic>, 1989<xref ref-type="bibr" rid="bb49"> ▶</xref>; Adams <italic>et al.</italic>, 1997<xref ref-type="bibr" rid="bb2"> ▶</xref>; Brünger & Rice, 1997<xref ref-type="bibr" rid="bb20"> ▶</xref>). 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, refinement 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 X-ray refinement has been the lack of proper treatment of long-range electrostatic interactions, which is overcome in this work <italic>via</italic> use of particle-mesh Ewald summation (PME; Darden <italic>et al.</italic>, 1993<xref ref-type="bibr" rid="bb27"> ▶</xref>; Essmann <italic>et al.</italic>, 1995<xref ref-type="bibr" rid="bb36"> ▶</xref>; Sagui <italic>et al.</italic>, 2004<xref ref-type="bibr" rid="bb65"> ▶</xref>).</p><p>In 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 <italic>et al.</italic>, 2005<xref ref-type="bibr" rid="bb37"> ▶</xref>; Maple <italic>et al.</italic>, 2005<xref ref-type="bibr" rid="bb53"> ▶</xref>). As an alternative to induced dipoles, Patel and Brooks employed a fluctuating-charge model of polarization (Patel & Brooks, 2006<xref ref-type="bibr" rid="bb56"> ▶</xref>), while Lamoureux and Roux have demonstrated success using classical Drude oscillators (Lamoureux <italic>et al.</italic>, 2006<xref ref-type="bibr" rid="bb51"> ▶</xref>; Lamoureux & Roux, 2003<xref ref-type="bibr" rid="bb52"> ▶</xref>). In addition to polarization, Gresh and coworkers have developed a methodology to include nonclassical effects such as electrostatic penetration and charge transfer (Gresh, 2006<xref ref-type="bibr" rid="bb38"> ▶</xref>; Gresh <italic>et al.</italic>, 2007<xref ref-type="bibr" rid="bb39"> ▶</xref>; Piquemal <italic>et al.</italic>, 2006<xref ref-type="bibr" rid="bb59"> ▶</xref>, 2007<xref ref-type="bibr" rid="bb58"> ▶</xref>).</p><p>Although classical potentials can be validated against a range of experimental observables, for example small-molecule solvation energies (Shirts <italic>et al.</italic>, 2003<xref ref-type="bibr" rid="bb69"> ▶</xref>; Shirts & Pande, 2005<xref ref-type="bibr" rid="bb68"> ▶</xref>), high-resolution 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 <italic>et al.</italic> (2007<xref ref-type="bibr" rid="bb3"> ▶</xref>).</p></sec><sec id="sec2"><label>2.</label><title>Theory</title><sec id="sec2.1"><label>2.1.</label><title>Subgrid fast Fourier transform</title><p>The starting point for this work is the subgrid fast Fourier transform algorithm (SGFFT), which will be briefly summarized (Brünger, 1989<xref ref-type="bibr" rid="bb12"> ▶</xref>). In FFT-based 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 <italic>CNS</italic> v.1.2 (Brunger, 2007<xref ref-type="bibr" rid="bb14"> ▶</xref>) for the ADP offset and grid size follows the work of Bricogne (2001<xref ref-type="bibr" rid="bb10"> ▶</xref>). 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 direct-summation structure factors and derivatives and the SGFFT calculation (Brunger, 2007<xref ref-type="bibr" rid="bb14"> ▶</xref>).</p><p>Structure factors are computed by FFT of the electron density of an asymmetric unit of atoms (Agarwal, 1978<xref ref-type="bibr" rid="bb5"> ▶</xref>). 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<xref ref-type="bibr" rid="bb12"> ▶</xref>; Kay Diederichs, private communication). <italic>CNS</italic> v.1.21 has implemented this approach <italic>via</italic> an OpenMP environment (courtesy Kay Diederichs, University of Konstanz; available at <ext-link ext-link-type="uri" xlink:href="http://cns-online.org">http://cns-online.org</ext-link>). Crystallographic symmetry 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.</p><p>Although the original SGFFT method was developed with an isolated-atom 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<xref ref-type="bibr" rid="bb12"> ▶</xref>). In the following sections, we develop these necessary formulae.</p></sec><sec id="sec2.2"><label>2.2.</label><title>Isolated-atom Gaussian density</title><p>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 reciprocal space and <italic>vice versa</italic>. Consider the canonical spherically symmetric Gaussian atomic scattering factor (Agarwal, 1978<xref ref-type="bibr" rid="bb5"> ▶</xref>),<disp-formula id="fd1"><graphic xlink:href="d-65-00952-efd1"></graphic></disp-formula>where <italic>a</italic><sub><italic>i</italic></sub> and <italic>b</italic><sub><italic>i</italic></sub> are constant parameters fitted to <italic>ab initio</italic> calculations on isolated atoms (this work is based on a sum of six Gaussians; <italic>n</italic> = 6; Su & Coppens, 1998<xref ref-type="bibr" rid="bb79"> ▶</xref>), κ is an expansion/contraction parameter used to adjust the width of the density and <bold>r</bold> is a position vector relative to the center of the atom. Its FT is given by<disp-formula id="fd2"><graphic xlink:href="d-65-00952-efd2"></graphic></disp-formula>where <bold>s</bold> is the reciprocal-lattice vector and we have used the FT definition given in Appendix <italic>A</italic><xref ref-type="app" rid="appa"></xref>. The reciprocal-lattice vector is <bold>s</bold> = <bold>h</bold><sup><italic>t</italic></sup><bold>A</bold><sup>−1</sup> = (<bold>A</bold><sup>−1</sup>)<sup><italic>t</italic></sup><bold>h</bold>, where <bold>h</bold> is a column vector with the Miller indices of a Bragg reflection and <bold>A</bold> is the fractionalization matrix that transforms coordinates <bold>r</bold> with respect to a Cartesian basis to fractional coordinates <bold>r</bold><sub>frac</sub> as defined in a crystallographic basis set. The Debye–Waller factor (Waller, 1923<xref ref-type="bibr" rid="bb85"> ▶</xref>) is given by<disp-formula id="fd3"><graphic xlink:href="d-65-00952-efd3"></graphic></disp-formula>in reciprocal space, where each element of the symmetric positive-definite matrix <bold>U</bold> is defined <italic>via</italic> a Cartesian basis consistent with PDB ANISOU records (Trueblood <italic>et al.</italic>, 1996<xref ref-type="bibr" rid="bb83"> ▶</xref>; Grosse-Kunstleve & Adams, 2002<xref ref-type="bibr" rid="bb40"> ▶</xref>). Multiplication of (3)<xref ref-type="disp-formula" rid="fd3"></xref> by the atomic form factor from (2)<xref ref-type="disp-formula" rid="fd2"></xref> gives the scattering factor<disp-formula id="fd4"><graphic xlink:href="d-65-00952-efd4"></graphic></disp-formula>based on <bold>U</bold><sub><italic>i</italic></sub> that are defined by<disp-formula id="fd5"><graphic xlink:href="d-65-00952-efd5"></graphic></disp-formula>where <italic>U</italic><sub>add</sub> is the artificial isotropic increase or decrease in the ADP discussed above and <italic>I</italic><sub>3</sub> is a 3 × 3 identity matrix. Removal of <italic>U</italic><sub>add</sub> analytically from each structure factor after the FT is straightforward. The only difference, therefore, between each <bold>U</bold><sub><italic>i</italic></sub> is the isolated-atom scattering parameter <italic>b</italic><sub><italic>i</italic></sub>.</p><p>Application of the inverse FT to (4)<xref ref-type="disp-formula" rid="fd4"></xref> gives the real-space anisotropic electron density<disp-formula id="fd6"><graphic xlink:href="d-65-00952-efd6"></graphic></disp-formula>where |<bold>U</bold><italic><sub>i</sub></italic>| is the determinant of matrix <bold>U</bold><italic><sub>i</sub></italic> and <bold>U</bold><italic><sub>i</sub></italic><sup>−1</sup> is its inverse. This expression can also be viewed as the convolution of the Gaussian form factor of (1)<xref ref-type="disp-formula" rid="fd1"></xref> with the inverse Fourier transform of the Debye–Waller factor of (3)<xref ref-type="disp-formula" rid="fd3"></xref>. Although the underlying isolated-atom scattering factor is spherically symmetric, convolution with anisotropic ADPs can lead to an angular dependence in ρ<sup>(<italic>n</italic>,κ)</sup>(<bold>r</bold>). Using the relationship that <italic>B</italic> = 8π<sup>2</sup><italic>U</italic>, one can show that (6)<xref ref-type="disp-formula" rid="fd6"></xref> reduces to the isotropic density expression reported by Brünger in equation (16) of Brünger (1989<xref ref-type="bibr" rid="bb12"> ▶</xref>) if all diagonal elements of <bold>U</bold><sub><italic>i</italic></sub> are equal to <italic>U</italic><sub>iso</sub> + <italic>b<sub>i</sub></italic>/8π<sup>2</sup> + <italic>U</italic><sub>add</sub> with zero off-diagonal components.</p></sec><sec id="sec2.3"><label>2.3.</label><title>Polarizable atomic multipole electron density</title><p>For the derivation of an atomic multipole expansion from a collection of point charges we begin with the Taylor expansion of the electric potential <italic>V</italic>(<bold>r</bold>) at <bold>r</bold> arising from <italic>n</italic> partial point charges that represent the electron density of an atom,<disp-formula id="fd7"><graphic xlink:href="d-65-00952-efd7"></graphic></disp-formula>where Δ<sub><italic>i</italic></sub> is the position of partial charge <italic>c</italic><sub><italic>i</italic></sub>, ∇<sub>α</sub> = ∂/∂<italic>r</italic><sub>α</sub> is one component of the del operator, α ∈ {<italic>x</italic>, <italic>y</italic>, <italic>z</italic>} and the Greek subscripts {α, β} represent the use of the Einstein summation convention for summing over tensor elements (Stone, 1996<xref ref-type="bibr" rid="bb75"> ▶</xref>). We omit the constant factor of 1/4π∊<sub>0</sub> throughout for compactness. Let the monopole, dipole and traceless quadrupole moments be defined as<disp-formula id="fd8"><graphic xlink:href="d-65-00952-efd8"></graphic></disp-formula>where removal of the trace in the definition of the quadrupole moment is allowed because the potential satisfies the Laplace equation (<italic>i.e.</italic> ∇<sup>2</sup><italic>V</italic> = 0). Substitution of the relationships in (8)<xref ref-type="disp-formula" rid="fd8"></xref> into the final expression of (7)<xref ref-type="disp-formula" rid="fd7"></xref> gives the electric potential in terms of a Cartesian multipole expansion, which we truncate at quadrupole order<disp-formula id="fd9"><graphic xlink:href="d-65-00952-efd9"></graphic></disp-formula>We now replace the Coulomb potential of (9)<xref ref-type="disp-formula" rid="fd9"></xref> with the potential from the sum of Gaussians from (1)<xref ref-type="disp-formula" rid="fd1"></xref>, which is given by<disp-formula id="fd10"><graphic xlink:href="d-65-00952-efd10"></graphic></disp-formula>and find<disp-formula id="fd11"><graphic xlink:href="d-65-00952-efd11"></graphic></disp-formula>We now introduce unique superscripts on the charge, dipole and quadrupole Gaussian basis sets, denoted by {<italic>n<sub>q</sub></italic>, <italic>n<sub>d</sub></italic>, <italic>n</italic><sub>Θ</sub>} and {κ<italic><sub>q</sub></italic>, κ<italic><sub>d</sub></italic>, κ<sub>Θ</sub>}, to allow them to differ in number and width.<disp-formula id="fd12"><graphic xlink:href="d-65-00952-efd12"></graphic></disp-formula>The potential of the charge density of (12)<xref ref-type="disp-formula" rid="fd12"></xref> quickly approaches the Coulomb potential as <italic>r</italic> increases since the error function goes to unity such that at large <italic>r</italic> 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)<xref ref-type="disp-formula" rid="fd12"></xref> gives the negative of a continuous charge density based on Cartesian Gaussian multipoles, <disp-formula id="fd13"><graphic xlink:href="d-65-00952-efd13"></graphic></disp-formula>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.</p><p>Inclusion of ADPs is described by convolution of (13)<xref ref-type="disp-formula" rid="fd13"></xref> with the real-space temperature factor,<disp-formula id="fd14"><graphic xlink:href="d-65-00952-efd14"></graphic></disp-formula>Based on the convolution differentiation rule<disp-formula id="fd15"><graphic xlink:href="d-65-00952-efd15"></graphic></disp-formula>the solution to (14)<xref ref-type="disp-formula" rid="fd14"></xref> is given by substituting for <italic>f</italic>(<bold>r</bold>) in (13)<xref ref-type="disp-formula" rid="fd13"></xref> with the corresponding ρ(<bold>r</bold>) from (6)<xref ref-type="disp-formula" rid="fd6"></xref> to give<disp-formula id="fd16"><graphic xlink:href="d-65-00952-efd16"></graphic></disp-formula>However, since <italic>q</italic> only represents partial atomic charges, the contributions from valence and core electrons need to be added. Additionally, the AMOEBA force field divides each atomic dipole moment into permanent (<bold>d</bold>) and induced (<bold>u</bold>) contributions to account for polarization. Therefore, we construct the total atomic electron density at a location <bold>r</bold> relative to the center of atom <italic>j</italic> by adding the contribution of core and valence electron density to (16)<xref ref-type="disp-formula" rid="fd16"></xref> and splitting the dipole into permanent and induced components to give<disp-formula id="fd17"><graphic xlink:href="d-65-00952-efd17"></graphic></disp-formula>where <italic>P<sub>j</sub></italic><sup>(<italic>c</italic>)</sup> is the integer number of core electrons (carbon has two) and <italic>P<sub>j</sub></italic><sup>(<italic>v</italic>)</sup> is the integer number of valence electrons (carbon has four). The superscripts on the anisotropic Gaussian form factors <italic>ρ<sub>j</sub></italic><sup>(<italic>n</italic>,κ)</sup>(<bold>r</bold>) have been made explicit for our model. We make the reasonable choice of using the isolated-atom scattering parameters for both core and valence electron densities. The width of the core electron density is frozen at the isolated-atom description (κ = 1) based on the observation that chemical bonding does not perturb it significantly (Hansen & Coppens, 1978<xref ref-type="bibr" rid="bb41"> ▶</xref>). On the other hand, the width of the valence electron density expands or contracts relative to the isolated-atom 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<sub><italic>v</italic></sub>. In this work, the dipole and quadrupole densities are described by a single Gaussian (<italic>n</italic><sub><italic>d</italic></sub> = <italic>n</italic><sub>Θ</sub> = 1) based on <italic>a</italic> and <italic>b</italic> parameters set to unity. The widths of the dipole and quadrupole densities are controlled by the κ<sub><italic>d</italic></sub> and κ<sub>Θ</sub> parameters. In this work, the width parameters {κ<sub><italic>v</italic></sub>, κ<sub><italic>d</italic></sub>, κ<sub><italic>Θ</italic></sub>} 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.</p><p>The partial derivatives through second order of the anisotropic and aspherical density defined in (6)<xref ref-type="disp-formula" rid="fd6"></xref>, which are required for the real-space multipolar density given in (17)<xref ref-type="disp-formula" rid="fd17"></xref>, are<disp-formula id="fd18"><graphic xlink:href="d-65-00952-efd18"></graphic></disp-formula>where <bold>u</bold><sub>α</sub> is a unit vector in the α direction with α ∈ {<italic>x</italic>, <italic>y</italic>, <italic>z</italic>}. In addition, the third-, fourth- and fifth-order terms of the expansion are presented as supplementary information along with a Mathematica notebook.<xref ref-type="fn" rid="fn1">1</xref></p><p>To the best of our knowledge, (17)<xref ref-type="disp-formula" rid="fd17"></xref> is the first expression reported in the literature for a real-space 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 <italic>via</italic> FFT.</p><p>Given a molecular conformation, the AMOEBA permanent multipole moments for each atom in the global coordinate frame (<italic>q</italic>, <bold>d</bold>, Θ) are converted <italic>via</italic> rotation from a local frame. For example, as shown in Fig. 1,<xref ref-type="fig" rid="fig1"> ▶</xref> the <italic>z</italic> 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 <italic>x</italic> axis is located in the O=C—C<sup>α</sup> plane in the direction of the C<sup>α</sup> atom and the <italic>y</italic> axis is chosen to give a right-handed coordinate system (Ren & Ponder, 2002<xref ref-type="bibr" rid="bb62"> ▶</xref>). The induced dipole (<bold>u</bold>) on each atom is determined <italic>via</italic> a self-consistent 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<xref ref-type="bibr" rid="bb62"> ▶</xref>).</p></sec><sec id="sec2.4"><label>2.4.</label><title>Derivatives of the electron density</title><sec id="sec2.4.1"><label>2.4.1.</label><title>Atomic coordinates</title><p>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 <italic>j</italic> at <bold>r</bold><sub><italic>j</italic></sub> in the global frame. In order to keep the derivation manageable, we split the total electron density into that produced by permanent charges ρ<sub>perm</sub> and that of induced charges ρ<sub>ind</sub>,<disp-formula id="fd19"><graphic xlink:href="d-65-00952-efd19"></graphic></disp-formula>The derivative of the permanent multipole electron density of atom <italic>j</italic> with respect to the α coordinate of atom <italic>j</italic> is given by<disp-formula id="fd20"><graphic xlink:href="d-65-00952-efd20"></graphic></disp-formula>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 <italic>k</italic>) that define the local frame of the multipole. Therefore, the derivative of the permanent multipole electron density of atom <italic>j</italic> with respect to the α coordinate of atoms <italic>k</italic> must also be considered,<disp-formula id="fd21"><graphic xlink:href="d-65-00952-efd21"></graphic></disp-formula>where the derivatives of spherically symmetric terms are zero with respect to the coordinates of atom <italic>k</italic> 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)<xref ref-type="disp-formula" rid="fd18"></xref>, simply due to the negative sign on <bold>r</bold><italic><sub>j</sub></italic>. The derivatives of the polarizable density with respect to atomic coordinates are very specific to the AMOEBA electrostatic model and are discussed in Appendix <italic>B</italic><xref ref-type="app" rid="appb"></xref>. However, we note that computing the derivatives of a polarizable density with respect to atomic coordinates is <italic>O</italic>(<italic>n</italic><sup>2</sup>log<italic>n</italic>) using PME, which quickly becomes the most expensive part of the overall calculation.</p></sec><sec id="sec2.4.2"><label>2.4.2.</label><title>ADPs</title><p>The derivative of the anisotropic electron density of atom <italic>j</italic> with respect to an anisotropic displacement parameter <italic>U</italic><sub><italic>j</italic>,τυ</sub> is given by<disp-formula id="fd22"><graphic xlink:href="d-65-00952-efd22"></graphic></disp-formula>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<disp-formula id="fd23"><graphic xlink:href="d-65-00952-efd23"></graphic></disp-formula>we have<disp-formula id="fd24"><graphic xlink:href="d-65-00952-efd24"></graphic></disp-formula>where the Kronecker delta δ<sub>τυ</sub> is unity for diagonal elements of <bold>U</bold> and zero otherwise. Differentiating an identity from matrix algebra <bold>U</bold><sup>−1</sup><bold>U</bold> = <bold>I</bold> gives the following relationship<disp-formula id="fd25"><graphic xlink:href="d-65-00952-efd25"></graphic></disp-formula>which makes it possible to differentiate <bold>U</bold> instead of its inverse. This is preferred since only one or two elements of ∂<bold>U</bold>/∂<italic>U</italic><sub>τυ</sub> 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 <italic>U</italic><sub>τυ</sub> and <italic>U</italic><sub>υτ</sub> represent the same variable in this case. For convenience, we define a 3 × 3 matrix <bold>J</bold><sup>(τυ)</sup>,<disp-formula id="fd26"><graphic xlink:href="d-65-00952-efd26"></graphic></disp-formula>and based on the chain rule we have<disp-formula id="fd27"><graphic xlink:href="d-65-00952-efd27"></graphic></disp-formula>Differentiating (6)<xref ref-type="disp-formula" rid="fd6"></xref> with respect to <bold>U</bold><sub>τυ</sub> and using (24)<xref ref-type="disp-formula" rid="fd24"></xref>, (27)<xref ref-type="disp-formula" rid="fd27"></xref> and the product rule gives<disp-formula id="fd28"><graphic xlink:href="d-65-00952-efd28"></graphic></disp-formula></p></sec><sec id="sec2.4.3"><label>2.4.3.</label><title>Gaussian width</title><p>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<xref ref-type="bibr" rid="bb41"> ▶</xref>). The derivative of the electron density with respect to this parameter is similar to the gradient for the ADP parameters. Two chain-rule terms are necessary. Firstly, the gradient of the normalizing term<disp-formula id="fd29"><graphic xlink:href="d-65-00952-efd29"></graphic></disp-formula>where<disp-formula id="fd30"><graphic xlink:href="d-65-00952-efd30"></graphic></disp-formula>Secondly, the gradient of the inverse ADP matrix is most conveniently expressed using the gradient of the original ADP matrix,<disp-formula id="fd31"><graphic xlink:href="d-65-00952-efd31"></graphic></disp-formula>where<disp-formula id="fd32"><graphic xlink:href="d-65-00952-efd32"></graphic></disp-formula>For convenience the matrix <bold>J</bold><sub><italic>i</italic></sub><sup>(κ)</sup> is defined to more compactly represent this result, <disp-formula id="fd33"><graphic xlink:href="d-65-00952-efd33"></graphic></disp-formula>Differentiating (6)<xref ref-type="disp-formula" rid="fd6"></xref> with respect to κ and using (29)<xref ref-type="disp-formula" rid="fd29"></xref>, (33)<xref ref-type="disp-formula" rid="fd33"></xref> and the product rule gives<disp-formula id="fd34"><graphic xlink:href="d-65-00952-efd34"></graphic></disp-formula>together with the third- and fourth-order terms available as supplementary information<xref ref-type="fn" rid="fn1">1</xref>.</p></sec></sec><sec id="sec2.5"><label>2.5.</label><title>Fourier transform of the polarizable atomic multipole electron density</title><p>Remarkably, the FT of the anisotropic and aspherical density given in (17)<xref ref-type="disp-formula" rid="fd17"></xref> is simply<disp-formula id="fd35"><graphic xlink:href="d-65-00952-efd35"></graphic></disp-formula>where the dipole and quadrupole terms in (35)<xref ref-type="disp-formula" rid="fd35"></xref> depend on the FT of the partial derivatives defined in (18)<xref ref-type="disp-formula" rid="fd18"></xref>. Through fifth order the reciprocal-space tensors are<disp-formula id="fd36"><graphic xlink:href="d-65-00952-efd36"></graphic></disp-formula>and in compressed tensor notation the general expression for order <italic>u</italic> + <italic>v</italic> + <italic>w</italic> is<disp-formula id="fd37"><graphic xlink:href="d-65-00952-efd37"></graphic></disp-formula>This expression is considerably more compact than any reported previously for an aspherical scattering factor in reciprocal space, particularly the formulation based on STOs and spherical harmonics (Hansen & Coppens, 1978<xref ref-type="bibr" rid="bb41"> ▶</xref>). Notably, our formulation has no dependence on cumbersome Fourier Bessel transforms of Slater-type functions (Dawson, 1967<italic>a</italic><xref ref-type="bibr" rid="bb28"> ▶</xref>; Hansen & Coppens, 1978<xref ref-type="bibr" rid="bb41"> ▶</xref>; Su & Coppens, 1990<xref ref-type="bibr" rid="bb78"> ▶</xref>). Our equation (35)<xref ref-type="disp-formula" rid="fd35"></xref> has been implemented by ‘direct summation’ for comparison to the performance of the FFT algorithm.</p></sec></sec><sec id="sec3"><label>3.</label><title>Scattering models</title><p>Four scattering models were implemented by modifying and combining the <italic>CNS</italic> (Brünger <italic>et al.</italic>, 1998<xref ref-type="bibr" rid="bb15"> ▶</xref>) and <italic>TINKER</italic> (Ponder, 2004<xref ref-type="bibr" rid="bb60"> ▶</xref>) code bases. The scattering models were added to the <italic>CNS</italic> code base, while <italic>TINKER</italic> was used to compute AMOEBA chemical forces and to supply <italic>CNS</italic> with polarizable multipoles in the global frame.</p><sec id="sec3.1"><label>3.1.</label><title>Isolated atom</title><p>The first scattering model (‘IAM’) is the conventional IAM based on the relativistic elastic scattering factors described by Su & Coppens (1998<xref ref-type="bibr" rid="bb79"> ▶</xref>).</p></sec><sec id="sec3.2"><label>3.2.</label><title>Isolated atom with inter-atomic scattering</title><p>The second scattering model (‘IAM–IAS’) augments the IAM with inter-atomic scattering sites at bond centers (Afonine <italic>et al.</italic>, 2007<xref ref-type="bibr" rid="bb3"> ▶</xref>). 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 refinement of a Tyr-Gly-Gly tripeptide.</p><p>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 bond-defining 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 force-field atom types.</p></sec><sec id="sec3.3"><label>3.3.</label><title>AMOEBA</title><p>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 §<xref ref-type="sec" rid="sec2"></xref>2. The induced dipoles were iterated to self-consistency using PME whenever any atomic coordinates were changed during refinement (Darden <italic>et al.</italic>, 1993<xref ref-type="bibr" rid="bb27"> ▶</xref>; Sagui <italic>et al.</italic>, 2004<xref ref-type="bibr" rid="bb65"> ▶</xref>; Essmann <italic>et al.</italic>, 1995<xref ref-type="bibr" rid="bb36"> ▶</xref>).</p></sec><sec id="sec3.4"><label>3.4.</label><title>AMOEBA with inter-atomic scattering</title><p>The final scattering model (‘AMOEBA–IAS’) augments AMOEBA electrostatics with inter-atomic 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<xref ref-type="bibr" rid="bb77"> ▶</xref>; Stone, 2005<xref ref-type="bibr" rid="bb76"> ▶</xref>). Furthermore, experimental data from the X-ray 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, 1967<italic>a</italic><xref ref-type="bibr" rid="bb28"> ▶</xref>,<italic>b</italic><xref ref-type="bibr" rid="bb29"> ▶</xref>). The characteristics of the four scattering models are further clarified below with respect to four peptide test cases.</p><p>The following computational details were constant across all of the refinements. The isotropic ADP offset <italic>U</italic><sub>add</sub> was set to 1/(4π<sup>2</sup>), which is equivalent to <italic>B</italic><sub>add</sub> = 8π<sup>2</sup><italic>U</italic><sub>add</sub> = 2, the FFT grid factor to 0.33 (as appropriate for crystal structures at subatomic resolution), and the electron-density cutoff around each atom was 18 (specified by the <italic>E</italic><sub>lim</sub> parameter in <italic>CNS</italic>). These conservative parameters led to close agreement between direct summation and FFT computation of structure factors. The <italic>CNS</italic> parameter <italic>w</italic><sub>A</sub> that controls the weighting of X-ray target function relative to the force-field energy was set to 1.0, although we also tested 0.2.<disp-formula id="fd38"><graphic xlink:href="d-65-00952-efd38"></graphic></disp-formula>This raised <italic>R</italic><sub>free</sub> values by less than 0.1% and lowered the AMOEBA potential energy differences between refinements presented below, but did not alter any trends or our conclusions. It should be noted that force-field restraints are not necessarily required for refinement 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 <monospace>refine.inp</monospace>
<italic>CNS</italic> task file was used for all refinements using the MLI target function.</p></sec></sec><sec id="sec4"><label>4.</label><title>Applications</title><p>To demonstrate the behavior of X-ray refinements based on Cartesian Gaussian multipoles, we present two sets of applications. The first set is simply to illustrate the performance of direct summation <italic>versus</italic> 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 <italic>CNS</italic> force field based on Engh & Huber parameters (Engh & Huber, 1991<xref ref-type="bibr" rid="bb34"> ▶</xref>). Although the refinements were performed in the native space group of each crystal, AMOEBA energies and gradients as computed using the <italic>TINKER</italic> code base required expanding to <italic>P</italic>1. This did not increase the number of refined variables, but suggests the need for an AMOEBA code that takes advantage of crystal symmetry.</p><sec id="sec4.1"><label>4.1.</label><title>Runtime scaling on protein data sets</title><p>Evaluation of the target function and its derivatives by direct-summation calculation of structure factors <italic>via</italic> (35)<xref ref-type="disp-formula" rid="fd35"></xref> and (36)<xref ref-type="disp-formula" rid="fd36"></xref> is <italic>O</italic>(<italic>N</italic><sub>atoms</sub> × <italic>N</italic><sub>reflections</sub> × <italic>N</italic><sub>symm</sub>). Alternatively, the FFT algorithm based on (17)<xref ref-type="disp-formula" rid="fd17"></xref> and (18)<xref ref-type="disp-formula" rid="fd18"></xref> is <italic>O</italic>(<italic>N</italic><sub>grid</sub> × log<italic>N</italic><sub>grid</sub>), where the number of grid points <italic>N</italic><sub>grid</sub> depends on the resolution of the diffraction data. Aspherical refinements based on the Hansen–Coppens formalism are currently limited to direct summation, since the real-space form of the electron density convolved with ADPs is unknown. Therefore, the performance of X-ray refinements based on Cartesian Gaussian multipoles and FFT is of particular interest. The results are summarized in Table 1<xref ref-type="table" rid="table1"> ▶</xref> and are plotted in Fig. 2<xref ref-type="fig" rid="fig2"> ▶</xref>. 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 four-processor machine).</p></sec><sec id="sec4.2"><label>4.2.</label><title>Refinement of peptide crystals</title><p>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 <italic>R</italic><sub>free</sub> and the potential energy of the asymmetric unit. Furthermore, the quality of the AMOEBA potential energy function can also be assayed, since it is reasonable to expect that potential energy and <italic>R</italic><sub>free</sub> should be correlated.</p><p>The peptide crystals studied include YG<sub>2</sub> (Pichon-Pesme <italic>et al.</italic>, 2000<xref ref-type="bibr" rid="bb57"> ▶</xref>), cyclic P<sub>2</sub>A<sub>4</sub> (Dittrich <italic>et al.</italic>, 2002<xref ref-type="bibr" rid="bb33"> ▶</xref>) and AYA with three waters or with an ethanol molecule (Chęcińska, Forster <italic>et al.</italic>, 2006<xref ref-type="bibr" rid="bb21"> ▶</xref>; Chęcińska, Mebs <italic>et al.</italic>, 2006<xref ref-type="bibr" rid="bb22"> ▶</xref>). Detailed descriptions of the unit-cell parameters, number of atoms, resolution and measured reflections are given in Table 2<xref ref-type="table" rid="table2"> ▶</xref>. The refinement results are summarized in Table 3<xref ref-type="table" rid="table3"> ▶</xref> and compared with previous work below.</p><sec id="sec4.2.1"><label>4.2.1.</label><title>YG<sub>2</sub></title><p>The <italic>R</italic><sub>free</sub> values of the IAM and IAM–IAS refinements of YG<sub>2</sub> (4.60 and 3.86%, respectively) are slightly lower than those reported by Afonine and coworkers (4.72 and 4.06%, respectively; Afonine <italic>et al.</italic>, 2007<xref ref-type="bibr" rid="bb3"> ▶</xref>). The <italic>R</italic><sub>free</sub> value of the AMOEBA–IAS refinement (3.50%) is a significant improvement. The <italic>R</italic><sub>work</sub> value (3.17%) of the AMOEBA–IAS refinement 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 <italic>et al.</italic>, 2007<xref ref-type="bibr" rid="bb84"> ▶</xref>). Cross-validation-based comparisons are unavailable in this case. We note that the AMOEBA–IAS refinement used a reflections-to-parameters 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<xref ref-type="table" rid="table2"> ▶</xref> and the number of parameters given in Table 3<xref ref-type="table" rid="table3"> ▶</xref>.</p><p>Electron-density maps of the tyrosine ring for the four scattering models are shown in Fig. 3<xref ref-type="fig" rid="fig3"> ▶</xref>, which lend visual insight into their properties. The non-H atom positions are apparent in the 2<italic>F</italic><sub>o</sub> − <italic>F</italic><sub>c</sub> contours for each refinement. The standard IAM scattering model underestimates the electron density at bond centers and at the oxygen lone-pair sites, as shown by the <italic>F</italic><sub>o</sub> − <italic>F</italic><sub>c</sub> contours. Our IAM–IAS scattering model explains the electron density at bond centers, but does not capture lone-pair electron density. Conversely, the AMOEBA model places electron density approximately at the lone-pair positions but not at bond centers. Finally, the AMOEBA–IAS model explains much of the lone-pair and bonding electron densities.</p></sec><sec id="sec4.2.2"><label>4.2.2.</label><title>P<sub>2</sub>A<sub>4</sub></title><p>The <italic>R</italic><sub>free</sub> values of our IAM and IAM–IAS refinements of P<sub>2</sub>A<sub>4</sub> (3.73 and 3.01%, respectively) agree closely with the values of Afonine and coworkers (3.63 and 3.23%, respectively; Afonine <italic>et al.</italic>, 2007<xref ref-type="bibr" rid="bb3"> ▶</xref>). The <italic>R</italic><sub>free</sub> value of the AMOEBA–IAS refinement (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 <italic>R</italic><sub>work</sub> value (2.86%) of the AMOEBA–IAS refinement 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 reflections-to-parameters ratio (50.3 compared with 43.6; Volkov <italic>et al.</italic>, 2007<xref ref-type="bibr" rid="bb84"> ▶</xref>). As for YG<sub>2</sub>, cross-validation was not performed. The similarity of the <italic>R</italic> values for YG<sub>2</sub> and P<sub>2</sub>A<sub>4</sub> 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).</p><p>In Fig. 4<xref ref-type="fig" rid="fig4"> ▶</xref> the precision of the <italic>R</italic><sub>work</sub> and <italic>R</italic><sub>free</sub> values computed using discrete FTs are compared with analytic direct summation for P<sub>2</sub>A<sub>4</sub> under the AMOEBA scattering model. Agreement to four decimal places is seen for <italic>B</italic><sub>add</sub> values between 0 and 3 Å<sup>2</sup>, which serves as validation of the correctness of (17)<xref ref-type="disp-formula" rid="fd17"></xref> and (35)<xref ref-type="disp-formula" rid="fd35"></xref>. These results support the conclusion that FFT-based computation of structure factors is appropriate for anisotropic and aspherical scattering models.</p></sec><sec id="sec4.2.3"><label>4.2.3.</label><title>AYA</title><p>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 <italic>et al.</italic>, 2006<xref ref-type="bibr" rid="bb21"> ▶</xref>; Chęcińska, Mebs <italic>et al.</italic>, 2006<xref ref-type="bibr" rid="bb22"> ▶</xref>) originally reported an <italic>R</italic> value of 2.4%, which is in agreement with the <italic>R</italic> value of our IAM refinement (2.67%). Addition of IAS lowered the <italic>R</italic><sub>free</sub> statistic from 2.71% to 2.39%, while addition of polarizable atomic multipole electron density showed a further improvement to an <italic>R</italic><sub>free</sub> of 1.95%. For AYA + ethanol the <italic>R</italic><sub>work</sub> value of the IAM (3.20%) is comparable to that reported originally by Chęcińska and coworkers (2.9%). IAM–IAS lowered <italic>R</italic><sub>free</sub> from 3.33 to 2.49%, while AMOEBA–IAS achieved 2.08%.</p></sec></sec><sec id="sec4.3"><label>4.3.</label><title>Refinement summary</title><p>The results for all four peptide refinements are summarized in Fig. 5<xref ref-type="fig" rid="fig5"> ▶</xref>. In every case, use of the AMOEBA–IAS scattering model relative to the IAM scattering model lowered both <italic>R</italic><sub>free</sub> and the potential energy 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 molecular conformation relaxes by approximately 16 kJ mol<sup>−1</sup> 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 ultrahigh-resolution refinements unless an anisotropic and aspherical scattering model is used.</p><p>In Fig. 6<xref ref-type="fig" rid="fig6"> ▶</xref>, we present plots of the IAS sites that were refined for each peptide system. Their Gaussian full-width at half-maximum (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.</p></sec></sec><sec id="sec5" sec-type="conclusions"><label>5.</label><title>Conclusions</title><p>Cartesian Gaussian multipoles offer an efficient alternative to the Hansen and Coppens formulation of aspherical scattering. They eliminate the use of Slater-type functions and allow structure factors to be computed by FFT. Numerical tests show that that FFT and direct-summation 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.</p><p>Ideally, a force-field electrostatics model should be accurate enough to explain the electron density observed in X-ray 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 atomic scattering factor 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.</p><p>In the near future, we will present the results of applying our polarizable atomic multipole refinement methodology to macromolecules. For ultrahigh-resolution macromolecular data sets, such as HEWL at 0.65 Å (Wang <italic>et al.</italic>, 2007<xref ref-type="bibr" rid="bb86"> ▶</xref>), our scattering model significantly improves refinement statistics, 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 H-atom electron density. We anticipate that refinement of hydrogen-bonding networks will enhance the usefulness of X-ray crystallography experiments with respect to explaining p<italic>K</italic><sub>a</sub> shifts, ligand-binding affinities and enzymatic mechanisms.</p></sec><sec sec-type="supplementary-material"><title>Supplementary Material</title><supplementary-material content-type="local-data" xlink:href="d-65-00952-sup1.pdf" position="float" xlink:type="simple"><p>Supplementary material file. DOI: <ext-link ext-link-type="uri" xlink:type="simple" xlink:href="http://dx.doi.org/10.1107/S0907444909022707/dz5164sup1.pdf">10.1107/S0907444909022707/dz5164sup1.pdf</ext-link></p><media mimetype="application" mime-subtype="pdf" xlink:href="d-65-00952-sup1.pdf" position="float" xlink:type="simple"></media></supplementary-material><supplementary-material content-type="local-data" xlink:href="d-65-00952-sup2.pdf" position="float" xlink:type="simple"><p>Supplementary material file. DOI: <ext-link ext-link-type="uri" xlink:type="simple" xlink:href="http://dx.doi.org/10.1107/S0907444909022707/dz5164sup2.pdf">10.1107/S0907444909022707/dz5164sup2.pdf</ext-link></p><media mimetype="application" mime-subtype="pdf" xlink:href="d-65-00952-sup2.pdf" position="float" xlink:type="simple"></media></supplementary-material></sec></body><back><app-group><app id="appa"><label>Appendix A </label><title>Fourier transform definition</title><p>The definition and notation for the Fourier transform as used in this work is given by<disp-formula id="fd39"><graphic xlink:href="d-65-00952-efd39"></graphic></disp-formula>and the corresponding inverse Fourier transform by<disp-formula id="fd40"><graphic xlink:href="d-65-00952-efd40"></graphic></disp-formula></p></app><app id="appb"><label>Appendix B </label><title>Derivative of the polarizable electron density with respect to atomic coordinates</title><p>The total polarizable electron density arising from the induced dipole of all atoms is given by<disp-formula id="fd41"><graphic xlink:href="d-65-00952-efd41"></graphic></disp-formula>The gradient of this density with respect to the α component of atom <italic>j</italic> is<disp-formula id="fd42"><graphic xlink:href="d-65-00952-efd42"></graphic></disp-formula>The second term is nonzero only for <italic>i</italic> = <italic>j</italic> and is simple to calculate. The first term, however, depends on ∂<italic>u</italic><sub><italic>i</italic>,β</sub>/∂<italic>r</italic><sub><italic>j</italic>,α</sub> which is the derivative of a component of the induced dipole of atom <italic>i</italic> with respect to the α component of atom <italic>j</italic>. In other words, perturbing the position of atom <italic>j</italic> affects not only its own scattering but that of all polarizable atoms. The induced dipole on atom <italic>i</italic> arises from the self-consistent crystal field multiplied by the polarizability,<disp-formula id="fd43"><graphic xlink:href="d-65-00952-efd43"></graphic></disp-formula>where α<sub><italic>i</italic></sub> is the atomic polarizability of atom <italic>i</italic>, <bold>T</bold><sub><italic>ik</italic></sub><sup>(1)</sup> is a matrix of tensors that produces the field at site <italic>i</italic><disp-formula id="fd44"><graphic xlink:href="d-65-00952-efd44"></graphic></disp-formula>owing to the multipole <bold>M</bold><sub><italic>k</italic></sub> at site <italic>k</italic><disp-formula id="fd45"><graphic xlink:href="d-65-00952-efd45"></graphic></disp-formula>and <bold>T</bold><sub><italic>ik</italic></sub><sup>(11)</sup> is the matrix of tensors that produces the field at site <italic>i</italic><disp-formula id="fd46"><graphic xlink:href="d-65-00952-efd46"></graphic></disp-formula>owing to the induced dipole <bold><italic>u</italic></bold><sub><italic>k</italic></sub> at site <italic>k</italic>. For simplicity, we have not formulated (43)<xref ref-type="disp-formula" rid="fd43"></xref> using PME electrostatics. Therefore, the sum over <italic>k</italic> includes all atoms in the crystal except atom <italic>i</italic>. The derivative of (43)<xref ref-type="disp-formula" rid="fd43"></xref> with respect to coordinate <italic>r</italic><sub><italic>j</italic>,α</sub> is given by<disp-formula id="fd47"><graphic xlink:href="d-65-00952-efd47"></graphic></disp-formula>The first three terms on the right-hand side are not difficult to compute. However, the fourth term shows that the gradients of the polarizable scattering are <italic>O</italic>(<italic>n</italic><sup>3</sup>) without use of PME. Specifically, there are 3<italic>n</italic> × 3<italic>n</italic> induced dipole density derivatives, each of which is the sum of 3<italic>n</italic> terms. In this work, we have computed these derivatives by finite differences using PME, which is <italic>O</italic>(<italic>n</italic><sup>2</sup>log<italic>n</italic>).</p></app></app-group><fn-group><fn id="fn1"><label>1</label><p>Supplementary material has been deposited in the IUCr electronic archive (Reference: <ext-link ext-link-type="uri" xlink:href="http://scripts.iucr.org/cgi-bin/paper?dz5164">DZ5164</ext-link>). Services for accessing this material are described at the back of the journal.</p></fn></fn-group><ack><p>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. 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The positive <italic>z</italic> axis is along the C=O bond and the <italic>x</italic> axis is chosen in the O=C—C<sup>α</sup> plane in the direction of the C<sup>α</sup> atom. The <italic>y</italic> 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 <italic>d</italic><sub><italic>z</italic></sub> 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.</p></caption><graphic xlink:href="d-65-00952-fig1"></graphic></fig><fig id="fig2" position="float"><label>Figure 2</label><caption><p>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.</p></caption><graphic xlink:href="d-65-00952-fig2"></graphic></fig><fig id="fig3" position="float"><label>Figure 3</label><caption><p>(<italic>a</italic>) IAM, (<italic>b</italic>) IAM–IAS, (<italic>c</italic>) AMOEBA and (<italic>d</italic>) AMOEBA–IAM refinements, respectively, for GY<sub>2</sub>. The <italic>F</italic><sub>o</sub> − <italic>F</italic><sub>c</sub> and 2<italic>F</italic><sub>o</sub> − <italic>F</italic><sub>c</sub> σ<sub>A</sub>-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.</p></caption><graphic xlink:href="d-65-00952-fig3"></graphic></fig><fig id="fig4" position="float"><label>Figure 4</label><caption><p>The precision of numerical computation of the <italic>R</italic><sub>work</sub> and <italic>R</italic><sub>free</sub> values <italic>via</italic> FFT is compared with analytic direct summation as a function of the isotropic increase <italic>B</italic><sub>add</sub> in ADP parameters for P<sub>2</sub>A<sub>4</sub> under the AMOEBA scattering model. Note that <italic>B</italic><sub>add</sub> = 8π<sup>2</sup><italic>U</italic><sub>add</sub>. </p></caption><graphic xlink:href="d-65-00952-fig4"></graphic></fig><fig id="fig5" position="float"><label>Figure 5</label><caption><p>The improvement arising from the AMOEBA–IAS scattering model, relative to the IAM model, is plotted as a function of relative percentage improvement in <italic>R</italic><sub>free</sub> value and the relative AMOEBA potential energy per residue. For all data sets, the best <italic>R</italic><sub>free</sub> value and lowest potential energy per residue were achieved using the AMOEBA–IAS scattering model. 1 kcal mol<sup>−1</sup> = 4.186 kJ mol<sup>−1</sup>.</p></caption><graphic xlink:href="d-65-00952-fig5"></graphic></fig><fig id="fig6" position="float"><label>Figure 6</label><caption><p>For the inter-atomic scattering sites of the IAM–IAS (<italic>a</italic>) and AMOEBA–IAS (<italic>b</italic>) scattering models, the refined Gaussian full-width at half-maximum (FWHM) is plotted <italic>versus</italic> 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.</p></caption><graphic xlink:href="d-65-00952-fig6"></graphic></fig><table-wrap id="table1" position="float"><label>Table 1</label><caption><title>Comparison of computational efficiency of direct-summation, FFT and SGFFT methods for the computation of the Cartesian Gaussian multipole scattering factors</title><p>The permanent multipole expansion was truncated at atomic quadrupoles and polarization was included <italic>via</italic> induced dipoles. The FFT method shows a speedup factor of 1.8–14.5 relative to direct summation. Parallelization by SGFFT provided an additional factor of 3.7–3.9 using four processors. All calculations were performed on a MacPro workstation with 2 × 2.66 GHz Dual Core Intel Xeon processors.</p></caption><table frame="hsides" rules="groups"><thead valign="bottom"><tr><th style="border-bottom:1px solid black;" rowspan="1" colspan="1" align="left" valign="bottom">PDB code</th><th style="border-bottom:1px solid black;" rowspan="1" colspan="1" align="left" char="." charoff="50" valign="bottom">Atoms</th><th style="border-bottom:1px solid black;" rowspan="1" colspan="1" align="left" char="." charoff="50" valign="bottom">Reflections</th><th style="border-bottom:1px solid black;" rowspan="1" colspan="1" align="left" char="." charoff="50" valign="bottom"><italic>N</italic><sub>symm</sub></th><th style="border-bottom:1px solid black;" rowspan="1" colspan="1" align="left" char="." charoff="50" valign="bottom">Atoms × reflections × <italic>N</italic><sub>symm</sub> × 10<sup>−6</sup></th><th style="border-bottom:1px solid black;" rowspan="1" colspan="1" align="left" char="." charoff="50" valign="bottom">Direct (s)</th><th style="border-bottom:1px solid black;" rowspan="1" colspan="1" align="left" char="." charoff="50" valign="bottom">FFT (s)</th><th style="border-bottom:1px solid black;" rowspan="1" colspan="1" align="left" char="." charoff="50" valign="bottom">Direct/FFT</th><th style="border-bottom:1px solid black;" rowspan="1" colspan="1" align="left" char="." charoff="50" valign="bottom">SGFFT (s)</th><th style="border-bottom:1px solid black;" rowspan="1" colspan="1" align="left" char="." charoff="50" valign="bottom">Direct/SGFFT</th></tr></thead><tbody valign="top"><tr><td style="" rowspan="1" colspan="1" align="left" valign="top"><ext-link ext-link-type="uri" xlink:href="http://scripts.iucr.org/cgi-bin/cr.cgi?rm=pdb&pdbId=1ejg">1ejg</ext-link></td><td style="" rowspan="1" colspan="1" align="char" char="." charoff="50" valign="top">642</td><td style="" rowspan="1" colspan="1" align="char" char="." charoff="50" valign="top">112209</td><td style="" rowspan="1" colspan="1" align="char" char="." charoff="50" valign="top">2</td><td style="" rowspan="1" colspan="1" align="char" char="." charoff="50" valign="top">144.1</td><td style="" rowspan="1" colspan="1" align="char" char="." charoff="50" valign="top">49.9</td><td style="" rowspan="1" colspan="1" align="char" char="." charoff="50" valign="top">28.1</td><td style="" rowspan="1" colspan="1" align="char" char="." charoff="50" valign="top">1.8</td><td style="" rowspan="1" colspan="1" align="char" char="." charoff="50" valign="top">7.3</td><td style="" rowspan="1" colspan="1" align="char" char="." charoff="50" valign="top">6.8</td></tr><tr><td style="" rowspan="1" colspan="1" align="left" valign="top"><ext-link ext-link-type="uri" xlink:href="http://scripts.iucr.org/cgi-bin/cr.cgi?rm=pdb&pdbId=2vb1">2vb1</ext-link></td><td style="" rowspan="1" colspan="1" align="char" char="." charoff="50" valign="top">2544</td><td style="" rowspan="1" colspan="1" align="char" char="." charoff="50" valign="top">187165</td><td style="" rowspan="1" colspan="1" align="char" char="." charoff="50" valign="top">1</td><td style="" rowspan="1" colspan="1" align="char" char="." charoff="50" valign="top">476.1</td><td style="" rowspan="1" colspan="1" align="char" char="." charoff="50" valign="top">301.8</td><td style="" rowspan="1" colspan="1" align="char" char="." charoff="50" valign="top">91.5</td><td style="" rowspan="1" colspan="1" align="char" char="." charoff="50" valign="top">3.3</td><td style="" rowspan="1" colspan="1" align="char" char="." charoff="50" valign="top">23.6</td><td style="" rowspan="1" colspan="1" align="char" char="." charoff="50" valign="top">12.8</td></tr><tr><td style="" rowspan="1" colspan="1" align="left" valign="top"><ext-link ext-link-type="uri" xlink:href="http://scripts.iucr.org/cgi-bin/cr.cgi?rm=pdb&pdbId=1fn8">1fn8</ext-link></td><td style="" rowspan="1" colspan="1" align="char" char="." charoff="50" valign="top">4294</td><td style="" rowspan="1" colspan="1" align="char" char="." charoff="50" valign="top">158550</td><td style="" rowspan="1" colspan="1" align="char" char="." charoff="50" valign="top">1</td><td style="" rowspan="1" colspan="1" align="char" char="." charoff="50" valign="top">680.8</td><td style="" rowspan="1" colspan="1" align="char" char="." charoff="50" valign="top">245.1</td><td style="" rowspan="1" colspan="1" align="char" char="." charoff="50" valign="top">45.8</td><td style="" rowspan="1" colspan="1" align="char" char="." charoff="50" valign="top">5.4</td><td style="" rowspan="1" colspan="1" align="char" char="." charoff="50" valign="top">12.4</td><td style="" rowspan="1" colspan="1" align="char" char="." charoff="50" valign="top">19.8</td></tr><tr><td style="" rowspan="1" colspan="1" align="left" valign="top"><ext-link ext-link-type="uri" xlink:href="http://scripts.iucr.org/cgi-bin/cr.cgi?rm=pdb&pdbId=1dy5">1dy5</ext-link></td><td style="" rowspan="1" colspan="1" align="char" char="." charoff="50" valign="top">4835</td><td style="" rowspan="1" colspan="1" align="char" char="." charoff="50" valign="top">159422</td><td style="" rowspan="1" colspan="1" align="char" char="." charoff="50" valign="top">2</td><td style="" rowspan="1" colspan="1" align="char" char="." charoff="50" valign="top">1541.6</td><td style="" rowspan="1" colspan="1" align="char" char="." charoff="50" valign="top">505.6</td><td style="" rowspan="1" colspan="1" align="char" char="." charoff="50" valign="top">37.0</td><td style="" rowspan="1" colspan="1" align="char" char="." charoff="50" valign="top">13.7</td><td style="" rowspan="1" colspan="1" align="char" char="." charoff="50" valign="top">9.7</td><td style="" rowspan="1" colspan="1" align="char" char="." charoff="50" valign="top">52.1</td></tr><tr><td style="" rowspan="1" colspan="1" align="left" valign="top"><ext-link ext-link-type="uri" xlink:href="http://scripts.iucr.org/cgi-bin/cr.cgi?rm=pdb&pdbId=1us0">1us0</ext-link></td><td style="" rowspan="1" colspan="1" align="char" char="." charoff="50" valign="top">6865</td><td style="" rowspan="1" colspan="1" align="char" char="." charoff="50" valign="top">511265</td><td style="" rowspan="1" colspan="1" align="char" char="." charoff="50" valign="top">2</td><td style="" rowspan="1" colspan="1" align="char" char="." charoff="50" valign="top">7019.7</td><td style="" rowspan="1" colspan="1" align="char" char="." charoff="50" valign="top">2346.2</td><td style="" rowspan="1" colspan="1" align="char" char="." charoff="50" valign="top">162.3</td><td style="" rowspan="1" colspan="1" align="char" char="." charoff="50" valign="top">14.5</td><td style="" rowspan="1" colspan="1" align="char" char="." charoff="50" valign="top">42.3</td><td style="" rowspan="1" colspan="1" align="char" char="." charoff="50" valign="top">55.5</td></tr></tbody></table></table-wrap><table-wrap id="table2" position="float"><label>Table 2</label><caption><title>Refinement systems</title></caption><table frame="hsides" rules="groups"><thead valign="bottom"><tr><th style="border-bottom:1px solid black;" rowspan="1" colspan="1" align="left" valign="bottom">Molecule</th><th style="border-bottom:1px solid black;" rowspan="1" colspan="1" align="left" valign="bottom">Space group and unit-cell parameters (Å, °)</th><th style="border-bottom:1px solid black;" rowspan="1" colspan="1" align="left" valign="bottom">Non-H atoms</th><th style="border-bottom:1px solid black;" rowspan="1" colspan="1" align="left" valign="bottom">H atoms</th><th style="border-bottom:1px solid black;" rowspan="1" colspan="1" align="left" valign="bottom">Bonds</th><th style="border-bottom:1px solid black;" rowspan="1" colspan="1" align="left" valign="bottom"><italic>d</italic><sub>min</sub> (Å)</th><th style="border-bottom:1px solid black;" rowspan="1" colspan="1" align="left" valign="bottom">Reflections</th></tr></thead><tbody valign="top"><tr><td style="" rowspan="1" colspan="1" align="left" valign="top">YG<sub>2</sub></td><td style="" rowspan="1" colspan="1" align="left" valign="top"><italic>P</italic>2<sub>1</sub>2<sub>1</sub>2<sub>1</sub>, <italic>a</italic> = 7.98, <italic>b</italic> = 9.54, <italic>c</italic> = 18.32</td><td style="" rowspan="1" colspan="1" align="char" char="." charoff="50" valign="top">22</td><td style="" rowspan="1" colspan="1" align="char" char="." charoff="50" valign="top">19</td><td style="" rowspan="1" colspan="1" align="char" char="." charoff="50" valign="top">40</td><td style="" rowspan="1" colspan="1" align="char" char="." charoff="50" valign="top">0.43</td><td style="" rowspan="1" colspan="1" align="char" char="." charoff="50" valign="top">4766</td></tr><tr><td style="" rowspan="1" colspan="1" align="left" valign="top">P<sub>2</sub>A<sub>4</sub></td><td style="" rowspan="1" colspan="1" align="left" valign="top"><italic>P</italic>2<sub>1</sub>2<sub>1</sub>2<sub>1</sub>, <italic>a</italic> = 10.13, <italic>b</italic> = 12.50, <italic>c</italic> = 19.50</td><td style="" rowspan="1" colspan="1" align="char" char="." charoff="50" valign="top">35</td><td style="" rowspan="1" colspan="1" align="char" char="." charoff="50" valign="top">36</td><td style="" rowspan="1" colspan="1" align="char" char="." charoff="50" valign="top">72</td><td style="" rowspan="1" colspan="1" align="char" char="." charoff="50" valign="top">0.37</td><td style="" rowspan="1" colspan="1" align="char" char="." charoff="50" valign="top">24878</td></tr><tr><td style="" rowspan="1" colspan="1" align="left" valign="top">AYA + 3 waters</td><td style="" rowspan="1" colspan="1" align="left" valign="top"><italic>P</italic>2<sub>1</sub>, <italic>a</italic> = 8.12, <italic>b</italic> = 9.30, <italic>c</italic> = 12.53, β = 91.21</td><td style="" rowspan="1" colspan="1" align="char" char="." charoff="50" valign="top">26</td><td style="" rowspan="1" colspan="1" align="char" char="." charoff="50" valign="top">27</td><td style="" rowspan="1" colspan="1" align="char" char="." charoff="50" valign="top">50</td><td style="" rowspan="1" colspan="1" align="char" char="." charoff="50" valign="top">0.59</td><td style="" rowspan="1" colspan="1" align="char" char="." charoff="50" valign="top">5019</td></tr><tr><td style="" rowspan="1" colspan="1" align="left" valign="top">AYA + ethanol</td><td style="" rowspan="1" colspan="1" align="left" valign="top"><italic>P</italic>2<sub>1</sub>, <italic>a</italic> = 8.85, <italic>b</italic> = 9.06, <italic>c</italic> = 12.36, β = 94.56</td><td style="" rowspan="1" colspan="1" align="char" char="." charoff="50" valign="top">26</td><td style="" rowspan="1" colspan="1" align="char" char="." charoff="50" valign="top">27</td><td style="" rowspan="1" colspan="1" align="char" char="." charoff="50" valign="top">52</td><td style="" rowspan="1" colspan="1" align="char" char="." charoff="50" valign="top">0.59</td><td style="" rowspan="1" colspan="1" align="char" char="." charoff="50" valign="top">5258</td></tr></tbody></table></table-wrap><table-wrap id="table3" position="float"><label>Table 3</label><caption><title>Refinement statistics and the relative AMOEBA potential energy per asymmetric unit are given for four small peptide crystals using the IAM, IAM–IAS, AMOEBA and AMOEBA–IAS scattering models</title><p>In all cases, the lowest <italic>R</italic><sub>free</sub> was found using the AMOEBA–IAS scattering model. Furthermore, the structure with the lowest AMOEBA potential energy per asymmetric unit also corresponded to AMOEBA–IAS refinement.</p></caption><table frame="hsides" rules="groups"><thead valign="bottom"><tr><th style="" rowspan="1" colspan="1" align="left" valign="bottom"> </th><th style="" rowspan="1" colspan="1" align="left" valign="bottom"> </th><th style="" rowspan="1" colspan="1" align="left" valign="bottom"> </th><th style="" rowspan="1" colspan="1" align="left" valign="bottom"> </th><th style="border-bottom:1px solid black;" rowspan="1" colspan="2" align="left" valign="bottom"><italic>R</italic><sub>work</sub>/<italic>R</italic><sub>free</sub> (%)</th><th style="" rowspan="1" colspan="1"> </th></tr><tr><th style="border-bottom:1px solid black;" rowspan="1" colspan="1" align="left" valign="bottom">Molecule</th><th style="border-bottom:1px solid black;" rowspan="1" colspan="1" align="left" valign="bottom">Scattering model</th><th style="border-bottom:1px solid black;" rowspan="1" colspan="1" align="left" valign="bottom"><italic>N</italic><sub>param</sub></th><th style="border-bottom:1px solid black;" rowspan="1" colspan="1" align="left" valign="bottom"><italic>N</italic><sub>data</sub>/<italic>N</italic><sub>param</sub></th><th style="border-bottom:1px solid black;" rowspan="1" colspan="1" align="left" valign="bottom"><italic>I</italic><sub>obs</sub>/σ(<italic>I</italic><sub>obs</sub>) > 0</th><th style="border-bottom:1px solid black;" rowspan="1" colspan="1" align="left" valign="bottom"><italic>I</italic><sub>obs</sub>/σ(<italic>I</italic><sub>obs</sub>) > 3</th><th style="border-bottom:1px solid black;" rowspan="1" colspan="1" align="left" valign="bottom">Energy<xref ref-type="table-fn" rid="tfn1">†</xref> (kcal mol<sup>−1</sup>)</th></tr></thead><tbody valign="top"><tr><td style="" rowspan="1" colspan="1" align="left" valign="top">YGG</td><td style="" rowspan="1" colspan="1" align="left" valign="top">IAM</td><td style="" rowspan="1" colspan="1" align="left" valign="top">274</td><td style="" rowspan="1" colspan="1" align="left" valign="top">17.4</td><td style="" rowspan="1" colspan="1" align="left" valign="top">4.73/4.74</td><td style="" rowspan="1" colspan="1" align="left" valign="top">4.41/4.60</td><td style="" rowspan="1" colspan="1" align="char" char="." charoff="50" valign="top">36.5</td></tr><tr><td style="" rowspan="1" colspan="1" align="left" valign="top"> </td><td style="" rowspan="1" colspan="1" align="left" valign="top">IAM–IAS</td><td style="" rowspan="1" colspan="1" align="left" valign="top">349</td><td style="" rowspan="1" colspan="1" align="left" valign="top">13.7</td><td style="" rowspan="1" colspan="1" align="left" valign="top">3.93/4.01</td><td style="" rowspan="1" colspan="1" align="left" valign="top">3.59/3.86</td><td style="" rowspan="1" colspan="1" align="char" char="." charoff="50" valign="top">7.2</td></tr><tr><td style="" rowspan="1" colspan="1" align="left" valign="top"> </td><td style="" rowspan="1" colspan="1" align="left" valign="top">AMOEBA</td><td style="" rowspan="1" colspan="1" align="left" valign="top">355</td><td style="" rowspan="1" colspan="1" align="left" valign="top">13.4</td><td style="" rowspan="1" colspan="1" align="left" valign="top">4.50/4.56</td><td style="" rowspan="1" colspan="1" align="left" valign="top">4.16/4.37</td><td style="" rowspan="1" colspan="1" align="char" char="." charoff="50" valign="top">6.8</td></tr><tr><td style="border-bottom:1px solid black;" rowspan="1" colspan="1" align="left" valign="top"> </td><td style="border-bottom:1px solid black;" rowspan="1" colspan="1" align="left" valign="top">AMOEBA–IAS</td><td style="border-bottom:1px solid black;" rowspan="1" colspan="1" align="left" valign="top">430</td><td style="border-bottom:1px solid black;" rowspan="1" colspan="1" align="left" valign="top">11.1</td><td style="border-bottom:1px solid black;" rowspan="1" colspan="1" align="left" valign="top">3.54/3.72</td><td style="border-bottom:1px solid black;" rowspan="1" colspan="1" align="left" valign="top">3.17/3.50</td><td style="border-bottom:1px solid black;" rowspan="1" colspan="1" align="char" char="." charoff="50" valign="top">0.0</td></tr><tr><td style="" rowspan="1" colspan="1" align="left" valign="top">PPAAAA</td><td style="" rowspan="1" colspan="1" align="left" valign="top">IAM</td><td style="" rowspan="1" colspan="1" align="left" valign="top">339</td><td style="" rowspan="1" colspan="1" align="left" valign="top">73.4</td><td style="" rowspan="1" colspan="1" align="left" valign="top">4.25/4.22</td><td style="" rowspan="1" colspan="1" align="left" valign="top">3.65/3.73</td><td style="" rowspan="1" colspan="1" align="char" char="." charoff="50" valign="top">32.2</td></tr><tr><td style="" rowspan="1" colspan="1" align="left" valign="top"> </td><td style="" rowspan="1" colspan="1" align="left" valign="top">IAM–IAS</td><td style="" rowspan="1" colspan="1" align="left" valign="top">417</td><td style="" rowspan="1" colspan="1" align="left" valign="top">59.7</td><td style="" rowspan="1" colspan="1" align="left" valign="top">3.56/3.48</td><td style="" rowspan="1" colspan="1" align="left" valign="top">3.00/3.01</td><td style="" rowspan="1" colspan="1" align="char" char="." charoff="50" valign="top">18.3</td></tr><tr><td style="" rowspan="1" colspan="1" align="left" valign="top"> </td><td style="" rowspan="1" colspan="1" align="left" valign="top">AMOEBA</td><td style="" rowspan="1" colspan="1" align="left" valign="top">417</td><td style="" rowspan="1" colspan="1" align="left" valign="top">59.7</td><td style="" rowspan="1" colspan="1" align="left" valign="top">4.24/4.23</td><td style="" rowspan="1" colspan="1" align="left" valign="top">3.69/3.77</td><td style="" rowspan="1" colspan="1" align="char" char="." charoff="50" valign="top">12.9</td></tr><tr><td style="border-bottom:1px solid black;" rowspan="1" colspan="1" align="left" valign="top"> </td><td style="border-bottom:1px solid black;" rowspan="1" colspan="1" align="left" valign="top">AMOEBA–IAS</td><td style="border-bottom:1px solid black;" rowspan="1" colspan="1" align="left" valign="top">495</td><td style="border-bottom:1px solid black;" rowspan="1" colspan="1" align="left" valign="top">50.3</td><td style="border-bottom:1px solid black;" rowspan="1" colspan="1" align="left" valign="top">3.42/3.42</td><td style="border-bottom:1px solid black;" rowspan="1" colspan="1" align="left" valign="top">2.86/2.94</td><td style="border-bottom:1px solid black;" rowspan="1" colspan="1" align="char" char="." charoff="50" valign="top">0.0</td></tr><tr><td style="" rowspan="1" colspan="1" align="left" valign="top">AYA + 3 waters</td><td style="" rowspan="1" colspan="1" align="left" valign="top">IAM</td><td style="" rowspan="1" colspan="1" align="left" valign="top">342</td><td style="" rowspan="1" colspan="1" align="left" valign="top">14.7</td><td style="" rowspan="1" colspan="1" align="left" valign="top">2.75/2.79</td><td style="" rowspan="1" colspan="1" align="left" valign="top">2.67/2.71</td><td style="" rowspan="1" colspan="1" align="char" char="." charoff="50" valign="top">17.5</td></tr><tr><td style="" rowspan="1" colspan="1" align="left" valign="top"> </td><td style="" rowspan="1" colspan="1" align="left" valign="top">IAM–IAS</td><td style="" rowspan="1" colspan="1" align="left" valign="top">411</td><td style="" rowspan="1" colspan="1" align="left" valign="top">12.2</td><td style="" rowspan="1" colspan="1" align="left" valign="top">2.24/2.47</td><td style="" rowspan="1" colspan="1" align="left" valign="top">2.16/2.39</td><td style="" rowspan="1" colspan="1" align="char" char="." charoff="50" valign="top">4.1</td></tr><tr><td style="" rowspan="1" colspan="1" align="left" valign="top"> </td><td style="" rowspan="1" colspan="1" align="left" valign="top">AMOEBA</td><td style="" rowspan="1" colspan="1" align="left" valign="top">423</td><td style="" rowspan="1" colspan="1" align="left" valign="top">11.9</td><td style="" rowspan="1" colspan="1" align="left" valign="top">2.40/2.55</td><td style="" rowspan="1" colspan="1" align="left" valign="top">2.31/2.47</td><td style="" rowspan="1" colspan="1" align="char" char="." charoff="50" valign="top">4.7</td></tr><tr><td style="border-bottom:1px solid black;" rowspan="1" colspan="1" align="left" valign="top"> </td><td style="border-bottom:1px solid black;" rowspan="1" colspan="1" align="left" valign="top">AMOEBA–IAS</td><td style="border-bottom:1px solid black;" rowspan="1" colspan="1" align="left" valign="top">492</td><td style="border-bottom:1px solid black;" rowspan="1" colspan="1" align="left" valign="top">10.2</td><td style="border-bottom:1px solid black;" rowspan="1" colspan="1" align="left" valign="top">1.72/2.03</td><td style="border-bottom:1px solid black;" rowspan="1" colspan="1" align="left" valign="top">1.64/1.95</td><td style="border-bottom:1px solid black;" rowspan="1" colspan="1" align="char" char="." charoff="50" valign="top">0.0</td></tr><tr><td style="" rowspan="1" colspan="1" align="left" valign="top">AYA + ethanol</td><td style="" rowspan="1" colspan="1" align="left" valign="top">IAM</td><td style="" rowspan="1" colspan="1" align="left" valign="top">342</td><td style="" rowspan="1" colspan="1" align="left" valign="top">15.4</td><td style="" rowspan="1" colspan="1" align="left" valign="top">3.30/3.50</td><td style="" rowspan="1" colspan="1" align="left" valign="top">3.20/3.33</td><td style="" rowspan="1" colspan="1" align="char" char="." charoff="50" valign="top">23.1</td></tr><tr><td style="" rowspan="1" colspan="1" align="left" valign="top"> </td><td style="" rowspan="1" colspan="1" align="left" valign="top">IAM–IAS</td><td style="" rowspan="1" colspan="1" align="left" valign="top">423</td><td style="" rowspan="1" colspan="1" align="left" valign="top">12.4</td><td style="" rowspan="1" colspan="1" align="left" valign="top">2.32/2.66</td><td style="" rowspan="1" colspan="1" align="left" valign="top">2.21/2.49</td><td style="" rowspan="1" colspan="1" align="char" char="." charoff="50" valign="top">14.8</td></tr><tr><td style="" rowspan="1" colspan="1" align="left" valign="top"> </td><td style="" rowspan="1" colspan="1" align="left" valign="top">AMOEBA</td><td style="" rowspan="1" colspan="1" align="left" valign="top">435</td><td style="" rowspan="1" colspan="1" align="left" valign="top">12.1</td><td style="" rowspan="1" colspan="1" align="left" valign="top">3.42/3.75</td><td style="" rowspan="1" colspan="1" align="left" valign="top">3.32/3.58</td><td style="" rowspan="1" colspan="1" align="char" char="." charoff="50" valign="top">3.7</td></tr><tr><td style="" rowspan="1" colspan="1" align="left" valign="top"> </td><td style="" rowspan="1" colspan="1" align="left" valign="top">AMOEBA–IAS</td><td style="" rowspan="1" colspan="1" align="left" valign="top">516</td><td style="" rowspan="1" colspan="1" align="left" valign="top">10.2</td><td style="" rowspan="1" colspan="1" align="left" valign="top">1.90/2.25</td><td style="" rowspan="1" colspan="1" align="left" valign="top">1.79/2.08</td><td style="" rowspan="1" colspan="1" align="char" char="." charoff="50" valign="top">0.0</td></tr></tbody></table><table-wrap-foot><fn id="tfn1"><label>†</label><p>1 kcal mol<sup>−1</sup> = 4.186 kJ mol<sup>−1</sup>.</p></fn></table-wrap-foot></table-wrap></floats-group></pmc></record>Pour manipuler ce document sous Unix (Dilib)
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