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Revealing Noncovalent Interactions

Revealing Noncovalent Interactions
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  See discussions, stats, and author profiles for this publication at: Revealing Noncovalent Interactions  Article   in  Journal of the American Chemical Society · May 2010 DOI: 10.1021/ja100936w · Source: PubMed CITATIONS 741 READS 65 6 authors , including:Shahar KeinanCloud Pharmaceuticals 47   PUBLICATIONS   2,097   CITATIONS   SEE PROFILE Julia Contreras-GarcíaPierre and Marie Curie University - Paris 6 63   PUBLICATIONS   1,856   CITATIONS   SEE PROFILE Weitao YangZhejiang University 284   PUBLICATIONS   63,019   CITATIONS   SEE PROFILE All content following this page was uploaded by Julia Contreras-García on 30 June 2015. The user has requested enhancement of the downloaded file. All in-text references underlined in blue are added to the srcinal documentand are linked to publications on ResearchGate, letting you access and read them immediately.  Revealing Non-Covalent Interactions Erin R. Johnson 1, Shahar Keinan 1, Paula Mori-Sánchez 1, Julia Contreras-García 1,  Aron J.Cohen 2, and Weitao Yang 1 1  Department of Chemistry, Duke University, Durham, North Carolina, 27708 2  Department of Chemistry, University of Cambridge, Lensfield Road, Cambridge, CB2 1EW, UnitedKingdom  Abstract Molecular structure does not easily identify the intricate non-covalent interactions that govern manyareas of biology and chemistry, including design of new materials and drugs. We develop an approachto detect non-covalent interactions in real space, based on the electron density and its derivatives.Our approach reveals underlying chemistry that compliments the covalent structure. It provides arich representation of van der Waals interactions, hydrogen bonds, and steric repulsion in smallmolecules, molecular complexes, and solids. Most importantly, the method, requiring onlyknowledge of the atomic coordinates, is efficient and applicable to large systems, such as proteinsor DNA. Across these applications, a view of non-bonded interactions emerges as continuoussurfaces rather than close contacts between atom pairs, offering rich insight into the design of newand improved ligands. 1 Introduction Chemical interactions between a protein and a drug, or a catalyst and its substrate, self-assemblyof nanomaterials, 1 , 2  and even some chemical reactions, 3 , 4  are dominated by non-covalentinteractions. This class of interactions spans a wide range of binding energies, and encompasseshydrogen bonding, dipole-dipole interactions, steric repulsion, and London dispersion. 5 Molecular structure is governed by covalent, non-covalent, and electrostatic interactions, thelatter two of which are the driving force in most biochemical processes. The three-dimensionalmolecular structure defines covalent bonds, however, non-covalent interactions are hiddenwithin voids in the bonding network. Although there are several ways to view and analyzecovalent and electrostatic interactions, an analogous method for non-covalent interactions isconspicuously missing. Such a method would aid understanding of the complex interactions between biomolecules, and the design of self-assembled materials and drugs, among others. 6 In this work, we present an approach to map and analyze non-covalent interactions, requiringonly molecular geometry information, which compliments existing methods for covalent andelectrostatic interactions. Covalent bonds are intuitively represented using conventional Lewisstructures.7 They can be visualized from properties of the electron density with modernquantum-mechanical models of bonding, such as the electron localization function (ELF)8,9and atoms-in-molecules (AIM) theory.10 – 12 Also, purely electrostatic interactions can be Corresponding author: Information Available. DFT results for the s22 set of bi-molecular complexes and carbon-carbon covalent bonds, MP2 and promolecular results for seclected small molecules and complexes. Additional methodology details include comparison of other density-gradient ratios, comparison with AIM and ELF quantum-chemical bonding theories, promolecular atomic-density parameters, completereference 29, and molecular geometries for selected species. This material is available free of charge via the Internet at  NIH Public Access Author Manuscript  J Am Chem Soc . Author manuscript; available in PMC 2011 May 12. Published in final edited form as:  J Am C hem Soc . 2010 May 12; 132(18): 6498–6506. doi:10.1021/ja100936w. NI  H-P A A  u t  h  or M an u s  c r i   p t  NI  H-P A A  u t  h  or M an u s  c r i   p t  NI  H-P A A  u t  h  or M an u s  c r i   p t    analyzed using electrostatic potential maps.13 Non-covalent interactions are frequentlyvisualized using distance-dependent contacts, generally without consideration of hydrogenatoms.14 –  16  Hydrogen-bonds can be identified from the molecular geometry 17  and from ELF, 18  while grid-based calculations based on classical force fields are used to model other van der Waals interactions. 19  Our approach outlined below, using the density and its derivatives, allowssimultaneous analysis and visualization of a wide range of non-covalent interactions types asreal-space surfaces and adds an important tool to a chemist's arsenal. 2 Theory 2.1 Background The quantum-mechanical electron density,  ρ , from which all chemical properties can, in principle, be obtained 20  is the key quantity in density functional theory (DFT). The reduceddensity gradient, coming from the density and its first derivative, ( ) is afundamental dimensionless quantity in DFT used to describe the deviation from ahomogeneous electron distribution.20 – 22 Properties of the reduced gradient have beeninvestigated in depth in the process of developing increasingly accurate functionals. 23  Indensity tails (i.e. regions far from the molecule, in which the density is decaying to zeroexponentially) the reduced gradient will have very large positive values. Conversely, thereduced gradient will assume very small values, approaching zero, for regions of both covalent bonding and non-covalent interactions. 2.2 Identifying Non-Covalent Interactions To explore the features associated with small reduced gradients, we first examine plots of s versus  ρ  (Figure 1). These plots were generated by evaluating the B3LYP 24 , 25  density andreduced gradients on cuboid grids, with a 0.1 au step size, for each molecule or dimer. To provide even more sampling of the small low-density, low-gradient regions in hydrogen- bonded complexes, additional calculations were performed for water and formic acid dimerswith a much denser 0.025 au grid.Plotting s  versus  ρ , as in Figure 1, reveals the basic pattern of intramolecular interactions.Methane (Fig. 1a) illustrates the typical covalent bond pattern. The top left-side points (smalldensity and large reduced gradient) correspond to the exponentially-decaying tail regions of the density, far from the nuclei. The points on the bottom right side (density values of ca. 0.25au and low reduced gradient) correspond to the C-H covalent bonds. Covalent bonds have acharacteristic saddle point in the electron density (bond critical points10 – 12), correspondingto s  = 0. Regions near the nuclei have larger density values and appear beyond the right edgeof the plot. The plot has an overall shape of the form a  ρ − 1/3  because atomic and molecular densities are piecewise exponential. The results for water are very similar, the only difference being that the covalent bonds lie at higher density values, past the edge of the plot. In Figure1b-d, we consider six examples of chemical systems displaying various types of non-covalentinteractions. Plots of s  versus  ρ  for these systems all exhibit a new feature: one or more spikesin the low-density, low-gradient region, a signature of non-covalent interactions. The srcin of this feature is made apparent by considering the formation of an intermolecular complex. The predominant change in the density-gradient profile occurs for the low-density region betweenthe two monomers. The reduced gradient changes from very large values in the monomers tonear zero upon dimer formation. This is the basis of our approach.We also explored other ratios of density and gradient values. Indeed, the reduced gradient(regardless of the constant), the fundamental dimensionless variable in DFT, is also found to be the ratio of density and gradient values that most clearly isolates non-covalent interactionsin real space (see supplementary information). In order for some ratio of the density and Johnson et al.Page 2  J Am Chem Soc . Author manuscript; available in PMC 2011 May 12. NI  H-P A A  u t  h  or M an u s  c r i   p t  NI  H-P A A  u t  h  or M an u s  c r i   p t  NI  H-P A A  u t  h  or M an u s  c r i   p t    gradient of the form | ∇  ρ |/  ρ n  to be successful, it must distinguish between non-covalentinteractions and the exponentially-decaying tails of the density that occur far from themolecular system. Both types of regions are characterized by low densities. In density tails, both the density and gradient approach zero exponentially. In regions of non-covalentinteractions, the gradient will again approach zero and will be identically zero at the critical point.To this point, we have found that non-covalent interactions can be isolated as regions with lowdensity and low reduced gradient. The density values of the low-gradient spikes also appear to be an indicator of the interaction strength. However, very different types of interactions (i.e.hydrogen-bonding and steric crowding) appear in the same region of density/reduced-gradientspace. To distinguish between these interactions, we consider second derivatives of the density. 2.3 Identifying Interaction Types Although localizing low-density, low-gradient regions enables identification of weak interactions in a molecular system, more specific interaction types cannot be determined fromthe density values alone. Low density regions are obviously related to the weakest interactions,such as van der Waals, whereas those with higher densities will be related to stronger (either stabilizing or de-stabilizing) interactions. 26  Density derivatives can be used to this end.The sign of the Laplacian of the density, ∇ 2  ρ , is a widely used tool to distinguish betweendifferent types of strong interactions. 27  To understand bonding in more detail, the Laplacianis often decomposed into a sum of contributions along the three principal axes of maximalvariation. These components are the three eigenvalues  λ i  of the electron-density Hessian(second derivative) matrix, such that ∇ 2  ρ  =  λ 1  +  λ 2  +  λ 3 , (  λ 1   ≤    λ 2   ≤    λ 3 ). Analysis of thesecomponents has been widely applied to chemical bonding. 11 , 27 , 28 At nuclei (or non-nuclear attractors), the density reaches a local maxima and all three eigen-values are negative. Interatomic regions between bonded atoms are characterized by the presence of one positive and two negative eigenvalues (  λ 1   <  0 ;  λ 2   <  0 ;  λ 3   >  0). In the case of covalent interactions, the negative contributions are dominant and the resultant Laplacian isnegative. For weaker, non-covalent interactions, the Laplacian in the interatomic region isdominated by the positive contribution, 27  independently of whether they are bonding or non- bonding. Bonding interactions can be identified by the negative sign of  λ 2 , as for the hydrogen- bond in the water dimer example. Conversely, if atoms are in non-bonded contact,  λ 2   >  0 inthe interatomic region (  λ 3   >  0 and  λ 1  can be either positive or negative). An example of thissituation occurs in the center of the bicyclo[2,2,2]octene cage. Cases where several atomsinteract, but are not bonded, correspond to steric crowding in the context of classical chemistry.Therefore, we can utilize the sign of  λ 2  to distinguish bonded (  λ 2   <  0) from non-bonded (  λ 2 >  0) interactions. Analysis of the sign of  λ 2  thus helps to discern between different types of non-covalent interactions, whereas the density itself provides information about their strength.This is illustrated in Figure 2, which shows a modification of our earlier reduced gradient anddensity plots, such that the ordinate is now sign(  λ 2 )  ρ . The figure shows data for the hydrogen- bonded water dimer, with the low-density, low-gradient spike now lying at negative valuesindicative of stabilizing interactions. Conversely, the low-density, low-gradient spike for thesterically-crowded bicyclo[2,2,2]octene molecule remains at positive values indicating the lack of bonding in the central area of this moleule. Finally, the low-density, low-gradient spike for the dispersion-bound methane dimer is very near zero, with slightly negative values, indicativeof weak attraction.There have been many previous studies of  λ 2  in different bonding situations and attempts torationalise it in terms of movements (accumulation/depletion) of the density, that is often Johnson et al.Page 3  J Am Chem Soc . Author manuscript; available in PMC 2011 May 12. NI  H-P A A  u t  h  or M an u s  c r i   p t  NI  H-P A A  u t  h  or M an u s  c r i   p t  NI  H-P A A  u t  h  or M an u s  c r i   p t    understood as due to attractive/repulsive interactions.11,27,28 For non-covalent interactions,the main features of the electron density and its derivatives appear clearly if the density isconstructed from something as simple as a sum of atomic densities (see Section 4.3). Indeed,for all cases considered, results at the self-consistent and promolecular level are qualtiativelyequivalent, which rules out any simple connection to accumulation or depletion of the density.However, when the effect of self-consistent calculations on the density and eigenvalues areanalyzed, we find that some quantitative differences are introduced by density relaxation. Asexpected, the density-gradient peaks are shifted to more bonding regimes when comparing promolecular and self-consistent densities. Specifically, a large shift toward smaller densityvalues is observed in the peak corresponding to non-bonded overlap, introducing less repulsionand greater stability.To summarize, since non-covalent interactions are characterized by low density and reducedgradient values, they can be located by generating gradient isosurfaces enclosing thecorresponding regions of real space. The interaction types can be further understood by thevalues of sign(  λ 2 )  ρ  in these regions. These isosurfaces are the basis of our non-covalentinteraction method. 3 Computational Details To obtain plots of the electron density (  ρ ) and reduced density gradient ( ), density-functional theory calculations were performed for a selected set of small molecules and dimers.Calculations on methane, water, branched octane, bicyclo[2,2,2]octene, and the homo-molecular dimers of methane, benzene, water, and formic acid, were performed with theB3LYP functional24,25 and the 6-31G* basis set, using the Gaussian 03 program.29 Molecular geometries of methane, water, branched octane, and bicyclo[2,2,2]octene are the same as thoseused in the G3X procedure.30 The geometries of the methane, water, and formic acid dimerswere obtained from Ref.31 and the benzene dimer geometry was obtained from Ref.32Additional B3LYP/6-31G* calculations were performed on Hobza's set of 22 bi-molecular complexes.33 Diamond and graphite calculations were performed using the CRYSTAL program, 34  with the BPW91 functional 35 , 36  and modified 6-21G* basis sets optimized for diamond or graphite. 34 To generate approximate promolecular densities, fully-numerical, LSDA,37 free-atomicdensities were generated for the atoms H-Ar using the NUMOL program.38, 39  These densitieswere spherically averaged over space and summed over spins. Because atomic densities are piecewise exponential for each shell of electrons, they were then fit to one (H, He), two (Li- Ne), or three (Na-Ar) Slater-type functions of the form  ρ at   = Σ  j  c  j e ( − r / ζ   j ) , with the c  and ζ   parameter values given in the supporting information. The use of simple exponential functionsto construct the promolecular density also allows first and second derivatives to be obtainedanalytically.To demonstrate our method for the interaction between a ligand and a protein active site, weapply it to a complex of a bacterial regulatory protein of the tetR family with a tetracyclineinhibitor.40 The geometry was obtained from the protein data bank  41  (pdb file 2UXO, onlychain B). All crystal waters were included in the calculations. Protonation states weredetermined by WhatIf. 42  Hydrogens were added using the hbuild function in CHARMM.43The positions of the protein and water hydrogen atoms were optimized, followed by the positions of the ligand hydrogen atoms. In both these geometry optimizations, the positions of all other atoms were frozen. The protein was described with the CHARMM27 force field, thewater molecules were described with TIP3 force field, and the ligand was described with thePM3 semi-empirical Hamiltonian. Since the exponential atomic densities die off fairly quickly,only atoms lying within 6 Å of the tetracycline inhibitor were used in calculation of the Johnson et al.Page 4  J Am Chem Soc . Author manuscript; available in PMC 2011 May 12. NI  H-P A A  u t  h  or M an u s  c r i   p t  NI  H-P A A  u t  h  or M an u s  c r i   p t  NI  H-P A A  u t  h  or M an u s  c r i   p t  
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