Density functional theory fragment descriptors to

THE JOURNAL OF CHEMICAL PHYSICS 126, 145105 共2007兲

Density functional theory fragment descriptors to quantify the reactivity of a molecular family: Application to amino acids P. Seneta兲 and F. Aparicio Institut Carnot de Bourgogne, UMR 5209 CNRS, Université de Bourgogne, 9 Avenue Alain Savary, F-21078 Dijon Cedex, France

共Received 12 January 2007; accepted 16 February 2007; published online 12 April 2007兲 By using the exact density functional theory, one demonstrates that the value of the local electronic softness of a molecular fragment is directly related to the polarization charge 共Coulomb hole兲 induced by a test electron removed 共or added兲 from 共at兲 the fragment. Our finding generalizes to a chemical group a formal relation between these molecular descriptors recently obtained for an atom in a molecule using an approximate atomistic model 关P. Senet and M. Yang, J. Chem. Sci. 117, 411 共2005兲兴. In addition, a practical ab initio computational scheme of the Coulomb hole and related local descriptors of reactivity of a molecular family having in common a similar fragment is presented. As a blind test, the method is applied to the lateral chains of the 20 isolated amino acids. One demonstrates that the local softness of the lateral chain is a quantitative measure of the similarity of the amino acids. It predicts the separation of amino acids in different biochemical groups 共aliphatic, basic, acidic, sulfur contained, and aromatic兲. The present approach may find applications in quantitative structure activity relationship methodology. © 2007 American Institute of Physics. 关DOI: 10.1063/1.2715570兴 I. INTRODUCTION

There is a considerable interest in chemistry and pharmacology to quantify the similarities and differences of a set of molecules using descriptors of their chemical properties. One of the key concepts to quantify and rationalize the variations of chemical reactivity 共in particular, in acid-base reactions兲 consists to evaluate the responses of a molecule to a variation of its number of electrons. Former empirical concepts, as the Mulliken electronegativity1 and the chemical hardness/softness2 are of this kind, in density functional theory 共DFT兲,3–5 they are, respectively, formulated in terms of the chemical potential ␮v 共Ref. 6兲 and its derivatives relative to the number of electrons N.7,8 The energy ␮v is defined as in thermodynamics,6

␮v␦N = Ev共N + ␦N兲 − Ev共N兲,

共1兲

in which Ev共N兲 is the ground-state energy of a system and v is the electrostatic potential of the nuclei. The chemical hardness ␩v is a concept introduced because a finite system has a chemical potential varying with its number of electrons,7

␩v␦N = ␮v共N + ␦N兲 − ␮v共N兲.

共2兲

The chemical softness Sv is 1 / ␩v.8 Difference in hardnesses is fundamental to understand acid-base reactions as stated by the hard-soft-acid-base principle:9 hard acids prefer hard bases and soft acids prefer soft bases. This principle is a consequence of the maximum hardness principle10,11 which establishes that a molecule has a ground state which maximizes its chemical hardness.12 These global descriptors, Eqs. 共1兲 and 共2兲, can be applied to isolated functional groups to predict the reactivity of difa兲

Electronic mail: [email protected]

0021-9606/2007/126共14兲/145105/8/$23.00

ferent parts of organic molecules.13 On the other hand, since the seminal work of Fukui,14 the frontier highest occupied molecular orbital 共HOMO兲 and lowest occupied molecular orbital 共LUMO兲 have been widely used to describe the spatial variation of molecular reactivity to electrophilic and nucleophilic attacks. These orbitals represent strictly the variation of the electronic density when an electron is removed 共cation兲 or added 共anion兲 to the molecule for a frozen geometry and frozen self-consistent potential. The relaxation of the self-consistent potential due to a change in the total electron number is nicely encompassed in a DFT linear response function: the Fukui function f共r兲.5,15 The latter measures the variation of the electronic density ␦␳共r兲 to a shift in the overall number of electrons ␦N for a frozen nuclear configuration 共constant v兲,5,15 f共r兲␦N = ␳v共r;N + ␦N兲 − ␳v共r;N兲.

共3兲

The propensity of an infinitesimal volume dr around a point r in a molecule to accept 共␦N ⬎ 0兲 or donate 共␦N ⬍ 0兲 electrons in a chemical reaction is thus measured by f共r兲dr. The reactivity of a molecule is more easily analyzed by using a discrete index, the condensed Fukui function f k,16 which is obtained by partitioning the density between fragments, f k␦N = Nv共k;N + ␦N兲 − Nv共k;N兲,

共4兲

where Nv共k兲 is the number of electrons in fragment k of the molecule at constant total external potential v. The precise value of Nv共k兲 is obtained by direct integration of the total density and depends on the partitioning of the space.17–20 Another important DFT local descriptor,21 which we will study in more detail below, is the local softness s共r兲.8 In its condensed form, it is given by

126, 145105-1

© 2007 American Institute of Physics

Downloaded 19 Apr 2007 to 193.52.246.58. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp

145105-2

P. Senet and F. Aparicio

sk =

Nv共k;N + ␦N兲 − Nv共k;N兲 . ␮v共N + ␦N兲 − ␮v共N兲

J. Chem. Phys. 126, 145105 共2007兲

共5兲

For an isolated molecule, N is an integer and ␦N = ± 1 in Eqs. 共1兲–共5兲. The HOMO and LUMO correspond thus to the simplest approximation of the electrophilic 共␦N = 1兲 and nucleophilic 共␦N = −1兲 Fukui functions defined by Eqs. 共3兲 and 共4兲 and the hardness, Eq. 共2兲, measures the gap between single ionized states. Considering separate ionized states of a molecule is not always the best approach of the reactivity indices. Alternative differential forms of the reactivity indices for the ground-state molecule can be obtained either by considering an approximate energy model E共N , v兲 which extrapolates continuously between integer numbers N共␦N = dN兲 共Refs. 22–26兲 or by considering the variations of the chemical electronic potential to molecular geometry variations 共more generally the variations of the external potential 5 v兲. This permits to establish the relations between the polarizability kernel and frontier orbitals from the ground state without invoking the ionic states.5,27 In this differential approach, the local softness can be defined, for instance, by s共r兲 = S

冋 册 ␦␮ ␦v共r兲

.

共6兲

N

There is an enormous literature devoted to the Fukui function and other DFT related chemical reactivity descriptors which has been reviewed.28,29 One should emphasize that these descriptors permit in principle to describe the reactivity from the sole knowledge of the response of the molecular electronic density within a perturbation theory which do not invoke orbitals.5,23 Fukui function and nonlinear electronic responses, as the nonlinear Fukui function f ⬘共r兲23,27,30 recently renamed Fukui difference,31 contain indeed informations about orbital reorganizations 共relaxation effects兲 which are related to the selectivity of molecular interactions. The DFT descriptors have been largely applied in order to understand 共and predict兲 the site of in which the reaction takes place, probing the difference in reactivity as measured by the descriptors for atoms within the same molecule. On the opposite, application of these concepts to build score function measuring the chemical 共quantum兲 similarity32 within a molecular family is, in fact, in its infancy.33 The present work contributes to this last point. Being a perturbation theory, the DFT formulation of the chemical descriptors concerns separate fragments before reaction. When we compare molecules within a family, we are comparing covalently linked fragments. If the product AB is regarded as formed by two covalently linked fragments A and B, how the local softness of the product is related to the former interaction between A and B? We argue that the local softness of the whole AB system is somehow related to the ability of electron transfer between the different parts A and B of the product molecule. Indeed, in a recent paper,34 one of us develops a local point of view of the local softness showing that the softness condensed on an atom 共fragment A兲 is related to the polarization charge, we named Coulomb hole, induced by the local ionization of that atom on the rest of the molecule 共fragment B兲. This “gedanke” local ionization process demonstrated that the value of the condensed softness of

the atom is directly related to the 共integrated兲 positive screening charge accompanying a test electron in the molecule. Although related, this result is markedly different from the independent derivation of a formal relation between the asymptotic exchange-correlation hole and the Fukui function.35 In Ref. 34, it was shown that the Coulomb hole is also related to the change of softness of the system B when A is added. The relation between softnesses and the Coulomb hole derived previously for an atomic fragment34 is proven below in a more general case for any molecular fragment of a molecule. The usefulness of this relation is demonstrated by considering a family of molecules differing by a functional fragment. The present partitioning of molecules in constant and variable fragments and its relation to the Coulomb hole are explored numerically by a quantitative analysis of the biochemical properties of amino acids. To our knowledge, there are few previous studies addressing the chemical reactivity in gas phase of amino acids in the framework of DFT descriptors. In most cases, these works are devoted to determine the protonation site in the amino-acid region. Perez and Contreras36 studied the gas-phase protonation of hydroxylamine, its methylated derivatives, and the aliphatic amino acids glycine, alanine, and valine. They concluded that the condensed atomic Fukui function correctly discriminates the site of protonation in aliphatic amino acids. Melin et al.37 showed that the Fukui functions are poor descriptors of the preferred site of protonation for various hydroxylamine derivatives and for the aliphatic amino acids glycine, alanine, and valine. They proposed the charges as better descriptors in this type of reactions. The electron acceptor or donor character of amino acids 共histidine, phenylalanine, and tryptophan兲 upon interaction with dioxins has been probed by Arulmozhiraja et al. using Huhey’s formula.38 Finally, an interesting correlation between the global hardness of a subset of 16 amino acids and their experimental gas phase proton affinity has been found in an earlier study by Baeten et al.39 The hardness was computed using Eq. 共2兲 for a biradical ˙ HCHRC ˙ O 共R = lateral chain兲 which mimchemical group N ics the fragment obtained from the removal of a residue from a polypeptide chain. The present study differs from these previous works and aims to discriminate the global biochemical properties of amino acids by using a local 共fragment兲 reactivity descriptor characterizing their linked lateral chains. The paper is organized as follows. The relation between the Coulomb hole of a fragment, its Fukui function, and local softness is derived in the next section by using exact density functional relations. A practical computational scheme is presented in Sec. III and its application to amino acids at the Hartree-Fock 共HF兲 and second-order Moller-Plesset theory 共MP2兲 levels is described in Sec. IV. The paper ends with a discussion and summary of the main results. II. MODEL EQUATIONS AND COMPUTATIONAL SCHEME

For a molecule with a global softness S,5 the Fukui function obeys the following formal equation:40

Downloaded 19 Apr 2007 to 193.52.246.58. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp

145105-3

1 = S

J. Chem. Phys. 126, 145105 共2007兲

DFT fragment descriptors

冕

dr⬘h共r,r⬘兲f共r⬘兲,

共7兲

in which h共r , r⬘兲 is the so-called hardness kernel.41 One assumes next a partition of the space in two regions we named ⍀I and ⍀II, 1 = S

冕

⍀I

dr⬘h共r,r⬘兲f共r⬘兲 +

冕

⍀II

dr⬘h共r,r⬘兲f共r⬘兲.

共8兲

These partitions define subspaces in which the hardness kernel is bounded by the boundaries. We assume that this bounded hardness kernel to have an inverse. Although we do not provide an explicit proof of this assumption here, if the inverse would not exist the hardness kernel would have a zero eigenvalue corresponding to a fragment for which the electronic chemical potential cannot change: this case would correspond to an infinite reservoir.23 One defines the inverse kernel of the region I by

冕

⍀I

dr⬙h−1共r,r⬙兲h共r⬙,r⬘兲 = ␦共r − r⬘兲,

共9兲

where in Eq. 共9兲 the points r and r⬘ are in ⍀I. It follows immediately from Eqs. 共8兲 and 共9兲, sI共r兲 = f共r兲 − f II S

冕

⍀I

dr⬘␹I␮共r,r⬘兲vII共r⬘兲,

共10兲

in which we have introduced the following definitions: sI共r兲 ⬅

冕

dr⬘h−1共r,r⬘兲,

⍀I

共11兲

␹I␮共r,r⬘兲 ⬅ − h−1共r,r⬘兲, f II ⬅

冕

⍀II

vII共r⬘兲 ⬅

共12兲

dr⵮ f共r⵮兲,

冕

⍀II

共13兲

dr⬙h共r⬘,r⬙兲

f共r⬙兲 . f II

共14兲

The regional softness of the fragment ⍀I is defined here by Eq. 共11兲 whereas Eq. 共13兲 defines the global Fukui function of the fragment II. ␹I␮共r , r⬘兲 in Eq. 共12兲 represents the linear polarizability kernel at constant chemical potential23 of the region I and vII共r⬘兲 defines an electronic pseudopotential34 induced by the region II. Integrating both members of Eq. 共10兲 on the subspace ⍀I and using the following definition for the softness of a fragment: SI ⬅

冕

⍀I

dr⬘sI共r兲,

one finds SI = f I − f II S

共15兲

冕 ⬘冕 dr

⍀I

⍀I

dr␹I␮共r,r⬘兲vII共r⬘兲.

共16兲

Finally, using the normalization property of the Fukui function f I + f II = 1,5 one arrives to the final result,

FIG. 1. 共Color online兲 Schematic representation of the amino-acidic region 共fragment I兲 and the lateral-chain region 共fragment II兲 of amino acids. The lateral-chain formula 共R兲 and the three-letter symbol of each residue are shown.

f II =

1 − 共SI/S兲 1 − qhg共I兲

共17兲

,

in which the total screening charge induced on fragment I by the pseudopotential generated by the fragment II has been defined, qhg共I兲 ⬅

冕 ⬘冕 dr

⍀I

⍀I

dr␹I␮共r,r⬘兲vII共r⬘兲.

共18兲

Equation 共17兲 is formally similar to Eq. 16 in Ref. 34 derived in the approximate framework of electronegativityequalization method 共EEM兲.22 Its interpretation is, however, slightly different. In the present derivation, the fragments can be any part of the molecular system instead of being a single atom and the functional model implicit in Eq. 共7兲 has exchange-correlation contributions to the nondiagonal hardness matrix elements on the opposite of the electrostatic EEM approach.34 Equation 共17兲 tells us that the Fukui function of a fragment is related to the variation of the softness of the global system when the fragment is removed 关numerator of Eq. 共17兲兴 and to the total charge induced by the peudopotential vII of one electron removed from the fragment II with a charge distribution f共r兲 / f II. On the other hand, −qhg共I兲 can be interpreted as the polarization charge accompanying a test electron which would be “localized” on fragment II when the molecule is connected to an infinite reservoir 共constant chemical potential兲. It is interesting to remark that the Fukui function on fragment II will have low values when SI is nearly equal to S: a situation corresponding to a fragment II much harder than fragment I. On the opposite, one expects large values of f II in case of nearly perfect screening, i.e., for qhg共I兲 = 1 关for a detailed discussion of the limits of the two sites Eq. 共17兲 for an atom, see Ref. 34兴. The relation between the screening charge, Eq. 共18兲, and the reactivity descriptors can be useful in studying the reactivity of a series of molecules for which a practical implementation of Eq. 共17兲 can be derived as follows. Consider a series of molecules having a constant fragment I and a variable fragment II, as the amino acids studied below 共Fig. 1兲. We are interested to compare the Coulomb hole qhg共I兲 for a fragment II knowing its Fukui function f II by applying Eq. 共17兲. The latter function is easily evaluated

Downloaded 19 Apr 2007 to 193.52.246.58. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp

145105-4

J. Chem. Phys. 126, 145105 共2007兲

P. Senet and F. Aparicio

FIG. 2. 共Color兲 Global softness S as a function of the local softness of the lateral chain sII for all amino acids 共physiological form兲 at HF/ 6-311G共d , p兲. The full line represents the linear regression obtained with Eq. 共19兲. Dashed lines are linear interpolations 关Eq. 共19兲兴 for the different biochemical types 共see text for details兲.

from any electronic structure calculation using one of the methods available in the literature29 but application of Eq. 共17兲 requires also the softness of the constant fragment I. Whereas there are methods to evaluate the global softness of a molecule, there is currently no simple way to extract from an ab initio calculation the softness of a fragment of the molecule as defined by Eq. 共15兲. An approximation of SI can be, however, obtained by considering the following linear representation of Eq. 共17兲 for a molecular family: S = sII共1 − qhg共I兲兲 + SI ,

共19兲

where the condensed local softness of the fragment II is defined as usual,5 sII ⬅ Sf II .

共20兲

In Eq. 共19兲, S and sII are computed ab initio and a linear fit of their relation provides the average Coulomb hole qhg共I兲 and the average constant fragment softness SI of the molecular family. Equation 共19兲 is applied to the amino acids next. One concludes this section by emphasizing the different meanings of the regional, fragment, and condensed local softnesses of a molecular fragment defined by Eqs. 共11兲, 共15兲, and 共20兲, respectively. One has indeed the following unusual relation between the softness of a fragment and its condensed local softness:

SII − sII =

冕

⍀II

drsII共r兲 − S

冕

⍀II

drf共r兲,

共21兲

which by using Eq. 共11兲 and the relation between f共r兲 and the inverse kernel5 transforms to

SII − sII =

冕 冕 冕 冕 dr

⍀II

⍀II

dr⬘h−1共r,r⬘兲

dr

−

⍀II

⍀I+⍀II

dr⬘h−1共r,r⬘兲,

共22兲

which can be written in a closed form, SII − sII = −

冕

⍀II

drsI共r兲,

共23兲

showing that the global softness of the fragment and its condensed local softness do not coincide in general as it could be naively deduced from the usual relation5 兰drs共r兲 = S valid only for the whole system. III. COMPUTATIONAL DETAILS

In order to illustrate the relation between the softnesses and the Coulomb hole of a molecular family, we used the complete set of 20 amino acids in their standard physiological protonated forms 共Fig. 1兲. Using the 共L兲 form of amino acids as initial geometry, prebuilt in the MOLDEN program,42 the structure of each amino acid was first fully optimized at HF/ 6-311G共d , p兲 and MP2 / 6-311G共d , p兲 levels of theory using the GAUSSIAN03 package of programs.43 In a second step, we computed the mean geometry for the constant fragment of all amino acids by averaging the geometries of the amino-acidic region I of the optimized structures, we froze it, and we reoptimized the lateral chains for all residues. The reoptimization process induces small changes in the final conformation of the lateral chain of the amino acids, for the same level of theory; however, if we compare the HF and MP2 reoptimized structures, we found that the Arg, Lys, His, Phe, and Tyr residues show the most important changes. Basically, the changes are related to one rotation in the dihedral

Downloaded 19 Apr 2007 to 193.52.246.58. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp

145105-5

DFT fragment descriptors

J. Chem. Phys. 126, 145105 共2007兲

FIG. 3. 共Color兲 Global softness S as a function of the local softness of the lateral chain sII for all amino acids 共physiological form兲 at MP2 / 6-311G共d , p兲. The full line represents the linear regression obtained with Eq. 共19兲. Dashed lines are linear interpolations 关Eq. 共19兲兴 for the different biochemical types 共see text for details兲.

angle involving the C␥ – C␦ bond. The chemical reactivity descriptors were computed for these final structures with the same level of theory and basis sets used to generate these corresponding geometries. The chemical softness S was obtained from the expression S ⬇ 1 / 共⑀LUMO − ⑀HOMO兲, in terms of the one electron energies of the frontier HOMOs and LUMOs, ⑀HOMO and ⑀LUMO, respectively.5 The condensed Fukui function 共average of the electrophilic and nucleophilic兲 on an atom f a was evaluated from a single point calculation in terms of the molecular orbital coefficients and the overlap matrix using the method proposed by Contreras et al.44 The

condensed Fukui function of one fragment f k was computed by f k = 兺a苸M k f a, where M k is the number of atoms in the fragment k corresponding to Eq. 共17兲. The softness and the Coulomb hole of the constant fragment SI and qhg共I兲 were evaluated by applying a linear regression to the curve S共sII兲 关Eq. 共19兲兴 for the complete sets of amino acids. IV. COULOMB HOLE AND LOCAL SOFTNESS IN AMINO ACIDS

The linear regression to the curves S共sII兲 关Eq. 共19兲兴 computed at the HF and MP2 levels of the theory are shown,

FIG. 4. 共Color兲 Global softness S as a function of the local softness of the lateral chain sII for all amino acids 共neutral form兲 at HF/ 6-311G共d , p兲. The full line represents the linear regression obtained with Eq. 共19兲. Dashed lines are linear interpolations 关Eq. 共19兲兴 for the different biochemical types 共see text for details兲.

Downloaded 19 Apr 2007 to 193.52.246.58. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp

145105-6

J. Chem. Phys. 126, 145105 共2007兲

P. Senet and F. Aparicio

FIG. 5. 共Color兲 Global softness S as a function of the local softness of the lateral chain sII for all amino acids 共neutral form兲 at MP2 / 6-311G共d , p兲. The full line represents the linear regression obtained with Eq. 共19兲. Dashed lines are linear interpolations 关Eq. 共19兲兴 for the different biochemical types 共see text for details兲.

aromatic residues 共F, Y, and W兲. On the other hand, as nicely shown in Figs. 2 and 3, the amino acids can be partitioned formally in groups bounded by the range of values spanned by the local softness sII of their lateral chains. These groups overlap in a large extent the classification of the amino acids in biochemical types: aliphatic residues 关G, A, V, I, L, P 共imino acid兲兴, hydroxyl residues 共S, T兲, amide residues 共N, Q兲, acidic residues 共D, E兲, basic residues 共R, K兲, S-containing residue 共M and C兲, histidine 共H兲, and aromatic residues 共F, Y, W兲.45 The clustering of amino acids in different groups in Figs. 2 and 3 can be quantified by the local softness of the lateral chain and by the slope of the variation of this quantity 关Coulomb hole, Eq. 共19兲兴 computed within each biochemical type. The linear regression applied within each type is represented at Figs. 2 and 3 and the results are summarized in Table II. We emphasize that the global softnesses of the amino acids S 共vertical axis in Figs. 2 and 3兲 do not permit such a classification because amino acids of different types have similar global softnesses.

respectively, in Figs. 2 and 3 for all the amino acids in their physiological protonated forms 共the acidic, basic residues, and histidine are charged兲. For comparison, a similar regression of S共sII兲 has been also applied to the complete set of amino acids in their neutral forms in Figs. 4 and 5. The fitted values of the softness of the constant fragment SI and of the Coulomb hole qhg共I兲 are reported in Table I. The adjustment of the average parameters is slightly better for the set of neutral amino acids as the approximation of a constant fragment is better if all the lateral chains bear the same charge. However, in order to compare our results to the biochemical classification of the amino acids, we will consider only the set with charged amino acids represented at Figs. 2 and 3 in the rest of the discussion. It is, however, interesting to emphasize that the softnesses of acidic and basic residues depend strongly on their charges 共Tables II and III兲. On the other hand, for some amino acids, the inclusion of electronic correlation has a significant effect on their softnesses as it can be deduced by comparing Figs. 2 and 3. In both HF and MP2 methods, the value of SI is smaller than the global softness of any of the isolated amino acids 共Table II兲. The value of qhg共I兲 reflects the average slope of increase of the global softness of the amino acids with the local softness of its lateral chain but they are notable deviations from this mean slope as, for instance, in the case of

V. SUMMARY AND DISCUSSION

The main result of the present study is contained in a “softness-softness” equation. This Eq. 共19兲 relates the global

TABLE I. Global softness S共I兲 and Coulomb hole qhg共I兲 of the fragment I at the two levels of theory used in this work. The physiological and neutral forms are reported. All quantities are in a.u. R is the correlation coefficient of the linear approximation 关Eq. 共19兲兴. Physiological forms Method

S共I兲

qhg共I兲

HF MP2

1.7523 1.7401

0.7536 0.7331

Neutral forms R

S共I兲

qhg共I兲

R

0.8673 0.9012

1.7222 1.7196

0.7521 0.7476

0.9345 0.9259

Downloaded 19 Apr 2007 to 193.52.246.58. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp

145105-7

J. Chem. Phys. 126, 145105 共2007兲

DFT fragment descriptors

TABLE II. Local softness of the lateral chains sII and Coulomb hole qhg共I兲 of each biochemical type of amino acids 共physiological form兲 computed at HF/ 6-311G共d , p兲 and MP2 / 6-311G共d , p兲 levels of theory. All quantities are in a.u. R is the correlation coefficient of the linear approximation 关Eq. 共19兲兴 applied within each type. HF 共a.u.兲

MP2 共a.u兲 S

sII qhg = 0.7921

共R = 0.8070兲 1.8225 1.8437 2.0739 1.8458 1.8149 2.0718 1.8650 2.1420

Aliphatic and hydroxyl residues Gly 共G兲 Ala 共A兲 Ser 共S兲 共Hydroxyl兲 Val 共V兲 Ile 共I兲 Thr 共T兲 共Hydroxyl兲 Leu 共L兲 Pro 共P兲 共Imino acid兲

0.2302 0.2579 0.0997 0.4135 0.4228 0.1910 0.4936 0.4172

Amide residues Gln 共Q兲 Asn 共N兲

0.6335 1.0040

Acidic residues Asp 共D兲 Glu 共E兲

0.4327 0.3655

Basic residues Arg 共R兲 Lys 共K兲

qhg = −0.7930 1.1486 2.1171 1.2689 2.3326

Sulphur-containing residues Met 共M兲 Cys 共C兲

1.4974 1.5160

Histidine His 共H兲

1.2750

Aromatic residues Phe 共F兲 Tyr 共Y兲 Trp 共W兲

qhg = 0.2215 共R = 0.9971兲 2.0585 2.1872 2.1988 2.3144 2.2064 2.4628

sII

S

qhg = 0.9435 0.1785 0.2419 0.2911 0.3754 0.3939 0.4357 0.4603 0.4896

qhg = 1.1274

qhg = 0.8577 1.8742 1.8274

0.6765 0.8804

2.1012 2.0533

0.1379 1.1971

qhg = 0.8516

1.8741 1.8384 qhg = 1.5238 2.0279 2.0630

qhg = −0.8102 1.1329 2.1125 1.2530 2.3299

qhg = 5.0064

qhg = 4.5580 2.1091 2.0350

1.4950 1.6800

2.0189

1.7234

¯

softness of a molecule to the softness of a constant fragment and to the local softness and Coulomb hole of a variable fragment. Its application to a family of molecules is approximative, assuming that the constant fragment keeps its value TABLE III. Local softness of the lateral chains sII and Coulomb hole qhg共I兲 of acidic amino acids, basic amino acids, and histidine in neutral forms computed at HF/ 6-311G共d , p兲 and MP2 / 6-311G共d , p兲 levels of theory. All quantities are in a.u. HF 共a.u.兲

MP2 共a.u.兲 S

sII

sII

S

Acidic residues Asp 共D兲 Glu 共E兲

qhg = 2.2203 0.5777 1.7696 0.4877 1.8794

qhg = 1.5773 0.6550 1.7781 0.4835 1.8771

Basic residues Arg 共R兲 Lys 共K兲

qhg = −0.2989 1.3566 2.1338 1.1931 1.9214

qhg = −0.2972 1.3465 2.1240 1.1824 1.9112

Histidine His 共H兲

2.0392

¯

¯ 2.1583

2.1504

2.1925

共R = 0.9274兲 1.8427 1.8487 1.8223 1.8142 1.8172 1.8368 1.8479 1.8845

2.0989 2.0020 ¯ 2.1717

qhg = 0.1309 共R = 0.9994兲 2.1085 2.2175 2.2412 2.3428 2.4616 2.5259

independently of the chemical nature of the variable fragment to which it is linked. However, the numerical application to amino acids, using a frozen orbital approximation, demonstrates a very high correlation between the global softness of the residue and the local softness of its lateral chain. The lateral chain defines the biochemical properties as does the value of its local softness in the present approach. The numerical validity of the softness-softness relation would merit to be tested for other molecular families. The use of the local softness in scoring function has been recently discussed. The present work points that the local softness and the Coulomb hole could serve to define a distance of similarity between two chemical moieties. We hope that this contribution will stimulate other researches in this direction. ACKNOWLEDGMENTS

One of the authors 共F.A.兲 thanks the “Conseil Régional de Bourgogne” for financial support during his stay at the Université de Bourgogne. The use of the computer facilities of the CRI 共Dijon兲 and CINES 共Montpellier兲 is gratefully acknowledged.

Downloaded 19 Apr 2007 to 193.52.246.58. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp

145105-8

J. Chem. Phys. 126, 145105 共2007兲

P. Senet and F. Aparicio

R. S. Mulliken, J. Chem. Phys. 2, 782 共1934兲. R. G. Pearson, Science 151, 172 共1963兲. 3 P. Hohenberg and W. Kohn, Phys. Rev. 136, B864 共1964兲. 4 W. Kohn and L. J. Sham, Phys. Rev. 140, A1133 共1965兲. 5 R. G. Parr and W. Yang, Density-Functional Theory of Atoms and Molecules 共Oxford University Press, New York, 1989兲. 6 R. G. Parr, R. A. Donelly, M. Levy, and W. E. Palke, J. Chem. Phys. 68, 3801 共1978兲. 7 R. G. Parr and R. G. Pearson, J. Am. Chem. Soc. 105, 7512 共1983兲. 8 R. G. Parr and W. Yang, Proc. Natl. Acad. Sci. U.S.A. 82, 6723 共1985兲. 9 R. G. Pearson, Hard and Soft Acids and Bases 共Dowden, Hutchinson and Ross, Stroundenboury, PA, 1973兲. 10 R. G. Pearson, J. Chem. Educ. 45, 981 共1986兲. 11 P. K. Chattaraj and P. W. Ayers, J. Chem. Phys. 123, 068101 共2006兲. 12 R. G. Parr and P. K. Chattaraj, J. Am. Chem. Soc. 113, 1854 共1991兲. 13 F. De Proft, W. Langenaeker, and P. Geerlings, J. Phys. Chem. A 97, 1826 共1993兲. 14 K. Fukui, Science 218, 747 共1982兲. 15 R. G. Parr and W. Yang, J. Am. Chem. Soc. 106, 4049 共1984兲. 16 W. Yang and W. J. Mortier, J. Am. Chem. Soc. 108, 5708 共1986兲. 17 R. F. W. Bader, Chem. Rev. 共Washington, D.C.兲 91, 893 共1991兲. 18 F. L. Hirschfeld, Theoretical Chemistry Accounts: Theory, Computation and Modeling, Theor. Chim. Acta 44, 129 共1977兲. 19 P. W. Ayers, R. C. Morrison, and R. K. Roy, J. Chem. Phys. 116, 8731 共2002兲. 20 F. De Proft, R. Vivas-Reyes, A. Peeters, C. Van Alsenoy, and P. Geerlings, J. Comput. Chem. 24, 463 共2003兲. 21 See, e.g., S. Damoun, G. Van de Woude, F. Méndez, and P. Geerlings, J. Phys. Chem. A 101, 886 共1997兲. 22 See, e.g., W. J. Mortier, Structure and Bonding 共Springer, Berlin, 1993兲, Vol. 80, p. 126. 23 P. Senet, J. Chem. Phys. 105, 6471 共1996兲. 24 R. Balawender and P. Geerlings, J. Chem. Phys. 123, 124102 共2005兲.

R. Balawender and P. Geerlings, J. Chem. Phys. 123, 124103 共2005兲. J. P. Perdew, R. G. Parr, M. Levy, and J. L. Balduz, Phys. Rev. Lett. 49, 1691 共1982兲. 27 P. Senet, J. Chem. Phys. 107, 2516 共1997兲. 28 H. Chermette, J. Comput. Chem. 20, 129 共1999兲. 29 P. Geerlings, F. De Proft, and W. Langenaeker, Chem. Rev. 共Washington, D.C.兲 103, 1793 共2003兲. 30 P. Fuentealba and R. G. Parr, J. Chem. Phys. 94, 5559 共1991兲. 31 C. Morell, A. Grand, and A. Toro-Labbé, Chem. Phys. Lett. 425, 342 共2006兲. 32 M. Carbo, M. A. Arnau, and L. Leyda, Int. J. Quantum Chem. 17, 1185 共1980兲. 33 P. Geerlings, G. Boon, C. Van Alsenoy, and F. De Proft, Int. J. Quantum Chem. 101, 722 共2005兲. 34 P. Senet and M. Yang, J. Chem. Sci. 117, 411 共2005兲. 35 R. F. Nalewajski, Chem. Phys. Lett. 410, 335 共2005兲. 36 P. Perez and A. Contreras, Chem. Phys. Lett. 293, 239 共1998兲. 37 J. Melin, F. Aparicio, V. Subramanian, M. Galvan, and P. Chattaraj, J. Phys. Chem. A 108, 2487 共2004兲. 38 S. Arulmozhiraja, T. Fujii, and G. Sato, Mol. Phys. 100, 423 共2002兲. 39 A. Baeten, F. De Proft, and P. Geerlings, Int. J. Quantum Chem. 60, 931 共1996兲. 40 S. K. Ghosh, Chem. Phys. Lett. 172, 77 共1990兲. 41 M. Berkowitz and R. G. Parr, J. Chem. Phys. 88, 2554 共1988兲. 42 G. Schaftenaar and J. H. Noordik, J. Comput.-Aided Mol. Des. 14, 123 共2000兲. 43 M. J. Frisch, G. W. Trucks, H. B. Schlegel et al., GAUSSIAN 03, Revision B.05, Gaussian, Inc., Wallingford, CT, 2004. 44 R. Contreras, P. Fuentealba, M. Galván, and P. Pérez, Chem. Phys. Lett. 304, 405 共1999兲. 45 T. E. Creighton, Proteins, Structures and Molecular Properties, 2nd ed. 共Freeman, New York, 1994兲.

1

25

2

26

Downloaded 19 Apr 2007 to 193.52.246.58. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp