Matta, Boyd. The quantum theory of atoms in molecules

31211 Topological Analysis of Proteins as Derived from Medium and High-resolution Electron Density

11.5.3

Electrostatic Interaction Energies

Electrostatic interaction energy values were calculated between the adenine group of NADPþ and the adenine binding site. All three models were considered, at atomic resolution and using a smoothing degree of t ¼ 0:420 A˚ 2. The atomic charges of the adenine group were those of MacKerell et al. [55]. Interestingly, interaction energy values obtained within the framework of a simple point charge Coulombic interaction depict, in all models, stabilization of the adenine group within the protein cavity. Indeed, electrostatic interaction energies computed in

the atomic description using QTAIMEXP charges and multipolar database QVAL charges are 152.3 and 84.3 kJ mol 1, respectively, whereas the fragment description leads to 74.6, 49.8, and 46.5 kJ mol 1 with QTAIMEXP, QVAL and unit charges, respectively. This stabilization is emphasized in the framework of the QTAIMEXP charge model for both atomic and medium-resolution descriptions, because of to their greater magnitudes. Surprisingly, interaction energy obtained for the unit-charge model is very close to that obtained for QVAL fragment charges, despite discrepancies observed in electrostatic potential representations.

11.6

Conclusions and Perspectives

This chapter presents several concepts and techniques for conducting topological analysis of protein ED distributions at di erent levels of resolution. It particularly focuses on the transferability of electronic information from high to lowresolution representations.

Several descriptions of ED distributions were considered. First, experimental ED distributions were modeled at a subatomic resolution level (d < 0:9 A˚ ) by use of a multipolar description that enables determination of atomic charges in accordance with the Bader QTAIM approach [24, 36]. Second, a CP analysis technique developed by Johnson [26] was applied to both experimental and promolecular ED distributions at medium resolution (d A3 A˚ ) to locate their ED maxima. In this second approach, promolecular ED grids were obtained by Fourier transformation of calculated structure factors. Third, a smoothing algorithm was applied to a Gaussian promolecular description of the ED distribution to locate its maxima and to define their corresponding molecular fragments by a clustering procedure.

Application of these di erent approaches were proposed to study the electrostatic properties of the adenine binding site of the hAR structure bound with the cofactor NADPþ [6]. More precisely, the atomic charges that were obtained at a subatomic resolution level were used to calculate medium-resolution fragment charges. Two sets of atomic charges, characterized by overall charge neutrality, were used – database atomic charges directly obtained from the valence popula-

11.6 Conclusions and Perspectives 313

tions and topological QTAIM charges derived from the transferred ED multipolar database.

Topological analysis of ED distributions reconstructed from the multipolar database showed that atomic properties such as charge and volume enabled classification of the atoms according to their chemical function or their chemical environment. For example, atoms bonded to O and N, or to C, formed di erent clusters. This observation emphasized the e ect of the electronegativity of atoms involved in a chemical bond. This clustering of topological properties was described in more detail for the atoms belonging to the peptide group HNaHaCa aCbO, and to those forming the aromatic group of the amino acid tyrosine. The QTAIM charges seemed to be totally transferable and were thus regarded as simple point charges in electrostatic property calculations.

Molecular electrostatic potentials generated by the adenine binding site in the hAR structure and the electrostatic interaction energies of the adenine moiety with the protein binding site were calculated using several charge models. The two first models were built from the two sets of atomic charges. Each of these two sets was also used to calculate fragment charges. Finally, a fifth charge model was built by assigning formal unit charges to the ARG, LYS (þ1) and GLU, ASP ( 1) side-chains. Comparison of these five models led to several observations. First, although atomic database and QTAIM charges have similar signs, the QTAIM charges have more pronounced positive and negative values than the database charges. When applied to fragment representations, database and QTAIM fragment charge signs may di er, depending on fragment content. Second, all charge models, atomic or fragment-based, gave similar results, i.e. the adenine moiety is located in a slightly positive area and its aNH2 and H8A group/atom are pointing toward a negative area. It is however shifted away from the adenine group with the topological model when t increases. Fragment representations, up to medium smoothing (t ¼ 0:420 A˚ 2), are thus acceptable descriptions for modeling electrostatic properties of a protein site. Third, two positive regions occur on either side of the adenine plane, except with the formal unit charge model. In that last instance all four unit charges of the adenine binding site are located on one side only of the adenine rings and do not properly depict its 3D electrostatic properties. Finally, all electrostatic interaction energies adopt the same stabilization e ect toward the adenine group within its binding site, even with the formal unit charges. The topologically based electrostatic energy is, however, more negative, because of the more pronounced individual atomic charges.

This work will have a variety of applications. In this chapter, the clustering procedure used to partition a protein structure into fragments is based on progressive smoothing of the protein promolecular ED distribution, i.e. a change in the overall temperature factor. The decomposition of a protein structure as a function of the crystallographic resolution rather than the temperature factor could be considered. This would enable deeper comparison with the CPs obtained using Johnson’s approach [26]. For more general application purposes, a database of atomic charges could be combined with a database of protein fragment contents and used to automatically generate reduced protein steric and electrostatic representa-

314 11 Topological Analysis of Proteins as Derived from Medium and High-resolution Electron Density

tions. Such representations are expected to be useful in docking applications, for example, which are nowadays largely used to predict protein–protein, protein– DNA, and protein–ligand interaction patterns.

Acknowledgments

The Universite´ Henri Poincare´ – Nancy I, and CNRS are gratefully acknowledged for financial support. A.L. is indebted to the Ministe`re de l’Education Nationale, de l’Enseignement Supe´rieur et de la Recherche for a doctoral fellowship. C.L., B.G., V.P. and C.J. thank Dr A.D. Podjarny for providing us with the hAR crystallographic data. L.L. and D.P.V. thank L. Piela and R. Carbo`-Dorca for fruitful discussions. The FNRS-FRFC, the ‘‘Loterie Nationale’’ (convention no. 2.4578.02), and the Faculte´s Universitaires Notre-Dame de la Paix (FUNDP), are gratefully acknowledged for the use of the Interuniversity Scientific Computing Facility (ISCF) Center.

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317

12

Fragment Transferability Studied Theoretically and Experimentally with QTAIM – Implications for Electron Density and Invariom Modeling

Peter Luger and Birger Dittrich

12.1 Introduction

It is interesting to note that as early as 1915, three years after the discovery of X- ray di raction, Peter Debye made a note that the distribution of electron density should be obtainable from this new experimental method [1]. It took ninety years, however, until, in the autumn of 2005, Philip Coppens’ highlight article (entitled ‘‘Charge Densities Come of Age’’) appeared in ‘‘Angewandte Chemie’’ [2]. He pointed out it was a long way from Debye’s vision in 1915 to the current stage of electron-density work, that progress was slow, and that the major advances had all occurred in the last decade. The substantial progress made in electron density research in the last few years rests on three pillars:

1.advances in experimental techniques – high-brilliance X-ray sources (synchrotron beamlines), very low temperatures

(T ! 10 K), and CCD area detection;

2.theoretical developments, e.g. Bader’s QTAIM theory [3], enabling the derivation of quantitative topological data; and

3.computing – the development and distribution of specific computer program systems for all aspects of electronic density work, including refinement, analysis, and visualization of results.

As a result of these simultaneous developments, electron-density studies of entire classes of chemically related compounds or of larger molecules became feasible in a reasonable time. Routine application of electron-density work is in sight.

In this chapter we would like to start with a brief summary of recent experimental advances which enable verification of the transferability of atomic fragments from significant experimental data, a key feature of Bader’s QTAIM theory. Understanding of what constitutes chemical similarity led to the introduction of invarioms, pseudoatom fragments of electron density that are invariant in transfer from one molecule to another, and to their recent applications. Invarioms en-

318 12 Fragment Transferability Studied Theoretically and Experimentally with QTAIM

able replacement of the independent atom model (IAM), and their introduction will possibly add another pillar to high-resolution crystallographic work.

12.2

Experimental Electron-density Studies

12.2.1

Experimental Requirements

The electron density of a chemical structure consists of a large spherical contribution and a very small nonspherical contribution located mainly in the regions of the covalent bonds and some nonbonding (lone pair) regions. Because chemistry happens in the latter regions, one is more interested in the aspherical part of the electron density and observation of these small e ects requires very precise experiments. Atoms in crystals are not at rest, being described by well-known displacement terms which decrease as the temperature is reduced. Because thermal smearing should be as small as possible, data collection should be performed at the lowest temperature achievable. At low temperatures, moreover, high-order reflections are more likely to have significant intensities above the background. High-order data are needed not only to improve accuracy and resolution but also to provide su cient data for the refinement of the increased number of variables of the Hansen and Coppens multipole model [4]. To summarize, several experimental requirements must be met:

	excellent single-crystal quality
	precise intensity-data collection
	high resolution (sin y=l > 1:0 A˚ 1 or d < 0:5 A˚ )
	hard X-radiation, e.g. MoKa or shorter, synchrotron radiation
	high completeness and redundancy in reciprocal space,
	hence use of an area detector is preferable
	low temperature (T a100 K)

Whether or not a data set is suitable for electron-density determination must be carefully examined. Several criteria can be used:

conventional figures of merit, Rint, Rs, R(F) or R(F2), Rw(F), GoF

the Hirshfeld test [5] – a covalently bonded pair of atoms of comparable mass can be regarded as rigid with respect to thermal motion, so that the displacement parameter components of the atoms in question should be equal in bond direction

the quality of the residual electron density obtained by Fourier transformation of the di erence between observed and multipole-model structure factors – the residual density should be unstructured and low

Fig. 12.1 Electron density of the dipeptide l-phenylalanyl–l-proline.H2O, an example of a proper data set. Tetragonal, P43212, a; b ¼ 8:197, c ¼ 41:226 A˚ , V ¼ 2769 A˚ 3,

T ¼ 90 K, MoKa, (sin y=lÞmax ¼ 1:20 A˚ 1, Smart 1K CCD di ractometer, 112.138 measured reflections, 11.139 symmetryindependent, reflections with Fo > 2:5

12.2 Experimental Electron-density Studies 319

s(Fo):10.120 (91%!), completeness: 99.2%, redundancy: 9.92, Rint ¼ 2.8%. Multipole model: 653 parameter, R(F) ¼ 2.7%, Rw(F) ¼ 2.3%, GoF ¼ 1:82. Left: Molecular structure with hydrogen bonds to the water

molecule; middle/right: static/residual map in the plane of the phenyl ring, contour intervals 0.1/0.05 e A˚ 3.

Figure 12.1 illustrates an example of a proper electron-density data set measured for the dipeptide l-phenylalanyl–l-proline.H2O. All criteria mentioned above have been fulfilled. In particular, the residual density is featureless except for a small signal next to one of the hydrogen atoms. The Hirshfeld test was also satisfied in that the maximum di erence of displacement parameter components on a bond was a0.0005 A˚ 2.

12.2.2

Recent Experimental Advances

For several decades single-crystal X-ray data collection was performed with fourcircle di ractometers with point detectors operated in serial detection mode, so that usually no more than 500–1000 reflections could be collected per day. The appearance of CCD area-detection di ractometers in the mid nineties was a major breakthrough. It was shown that this type of detection led to data sets of su cient accuracy for electron-density work and that the data collection time could be reduced from weeks or months to one or a few days [6–8].

12.2.2.1 Synchrotron Radiation Compared with Laboratory Sources

Because optimum experimental conditions depend not only on detector type and quality but also on the choice of primary radiation and the data-collection temperature, we will discuss these aspects in some detail. On a laboratory scale most crystallographers have a sealed 2.4-kW Mo tube or sometimes a rotating anode available; synchrotron radiation should be considered as a favorable alternative. Synchrotron beamlines provide a bright source of X-radiation with outstanding

320 12 Fragment Transferability Studied Theoretically and Experimentally with QTAIM

Fig. 12.2 High-resolution synchrotron data set for the fullerene derivative dodecakisethoxycarbonyl-C60-fullerene. Left: summary of the crystallographic data. Right: static deformation density map in an equatorial plane of the C60 sphere [9] Copyright 2002 and Reproduction with Permission from American Chemistry Society.

properties. Most important for electron density experiments are the very high intensity and the tunable wavelength, enabling choice of l A0:5 A˚ or even shorter. This enables high-resolution data sets to be collected, if needed also for smaller crystals, for which absorption and extinction problems are minimized.

An example in which synchrotron radiation was essential for collection of a high-resolution data set is illustrated by Fig. 12.2 for a highly substituted C60 fullerene derivative.

Fullerenes are usually very unsuitable for experimental electron-density studies because of poor crystal quality and the high mobility of the molecules in the crystal lattice. This is why, for example, there is no electron-density study of an unsubstituted fullerene. After several unsuccessful attempts we were able to grow suitable crystals of the highly substituted Th-symmetrical derivative dodecakise- thoxycarbonyl-C60-fullerene, C102H60O24 [10] (Fig. 12.2), which co-crystallized with 1,2-difluorobenzene. On the basis of a data set of more than 350,000 reflections measured in five days at 100 K up to a resolution of sin y=l ¼ 1:26 A˚ 1 on beamline D3 of Hasylab/DESY, a properly resolved electron-density distribution and related bond-topological and atomic properties [9, 11] were derived.

If well-di racting crystals of su cient size can be grown, high-resolution data collection can be conducted with laboratory equipment. An example is a data set for vitamin B12 (Fig. 12.3). Although B12 crystals of desired size can be grown rather easily in a variety of solvents, they are extremely unstable when taken out of the mother liquor, being destroyed within seconds. This made careful, but also time-consuming crystal preparation necessary. To avoid the risk of wasting synchrotron beamtime, data were collected by use of a Bruker Smart 1K di ractometer with conventional MoKa radiation from a sealed 2.4-kW tube and were very suitable for further electron-density evaluation. Figure 12.3 shows the static defor-

12.2 Experimental Electron-density Studies 321

Fig. 12.3 Left: molecular structure of vitamin B12 in the crystalline modification cocrystallizing with 12 water and 3 propanol molecules. Summary of crystallographic data [12]: orthorhombic, P212121, cell volume 8952(3) A˚ 3, Z ¼ 4, MoKa, T ¼ 100 K, a total of 660,822 reflections collected,

(sin y=lÞmax ¼ 1:22 A˚ 1, 98,204 unique, 78,606 reflections with I > 2s(I) (80%), Rint ¼ 0:038, R(mul) ¼ 0:03. Right: experimental static deformation density in the plane of the corrin ring. Contour levels 0.1 e A˚ 3.

mation density distribution in the corrin plane; further derivation of topological properties is still in progress.

12.2.2.2 Data Collection at Ultra-low Temperatures (10–20 K)

Low-temperature di raction experiments are normally conducted at temperatures of approximately 100 K, making use of conventional nitrogen gas-stream devices. It has been shown [13], however, that substantial and, hence, favorable reduction of thermal motion occurs when the temperature is further reduced to 10–20 K. Because of reduction of the displacement parameters, the number of high-order reflections with intensities significantly above the background increases, and this is of utmost importance for electron-density work. These ultra-low temperatures can be achieved either by replacing nitrogen with helium in open-flow gas-stream devices or by using closed-cycle helium cryostats. Open-flow helium systems have the disadvantages of high cost of helium consumption and are, for this reason, rarely used in normal university laboratories.

Closed cycle helium cryostats which, in the double-stage version, reach a temperature of 10–20 K, are a low-running-cost alternative, because no gas is consumed. The cold head of the cryostat, which carries the crystalline sample, must, however, be enclosed in a cylindrical vacuum chamber. When point detectors were used, beryllium or carbon chambers were acceptable, but these result in an intolerable background pattern on CCD area detector frames. This technical