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Point Scattering: A New Geometric Invariant with

Applications From (Nano)Clusters to Biomolecules

ERNESTO ESTRADA

Complex Systems Research Group, X-ray Unit, RIAIDT, Edificio CACTUS, University of Santiagode Compostela, 15782 Santiago de Compostela, Spain

Received 2 May 2006; Revised 10 August 2006; Accepted 16 August 2006DOI 10.1002/jcc.20541

Published online 16 January 2007 in Wiley InterScience (www.interscience.wiley.com).

Abstract: A new geometric invariant is defined from ‘‘first principles’’ for a point ensemble, which can represent

clusters, molecules, crystals, and biomolecules. The scattering of a point ensemble is defined in terms of the Euclidean

distance matrix and a vector measuring the weighted departure of the points from the cluster centre. Using the Rayleigh–

Ritz theorem this function is maximized obtaining the point scattering of the ensemble. The point scattering shows

several properties which are useful for studying clusters, molecules, crystals, and biomolecules. We examined different

natural clusters of hard spheres such as colloidal particles and fullerenes, as well as protein–peptide complexes and the

effect of temperature on protein structure. In all cases point scattering differentiates point ensembles with different

structures, which are not distinguished by other geometric invariants, such as the second moment of mass distribution,

surface areas, and volumes. Point scattering also shows better correlation with thermodynamic parameters of binding

and describes the interior cavities of hollowed ensembles better than the other geometric measures.

q 2007 Wiley Periodicals, Inc. J Comput Chem 28: 767–777, 2007

Key words: Euclidean distance; discrete geometry; molecular geometry; protein packing; fullerenes; clusters

Introduction

A geometric invariant is a quantity which remains unchanged

under certain classes of transformation, such as the group of

translations or rotations.1 They are useful for comparing objects

because they usually reflect intrinsic properties of objects. Among

the most well-known geometrical invariants we can mention the

perimeter, second moment of mass distribution, surface area, and

volume.1 The last two are particularly of great interest in chemis-

try because of their relationship with the concept of packing,

which is a fundamental and essential characteristic of natural sys-

tems.2,3 In particular these concepts are necessary in understand-

ing protein structure and for uncovering the relationship between

packing and stability as well as for the study of solute hydropho-

bicity.4 A very well known example is provided by the average

packing density inside proteins,5–8 which is as high as in crystal-

line solids. It is known that a number of oligomeric proteins,

including, acetylcholine receptors, aquaporins, tight junction

occluding, and claudins, and gap junction channels arrange into

densely packed clusters, arrays or strands in the plasma mem-

brane.9–13 The optimal packing of tubes has also received atten-

tion as a model for understanding the way in which a DNA mole-

cule could be packed within a small virus.14–17 Packing is also an

important characteristic of clusters, such as colloidal materials

like photonic crystals and macroporous media.18,19

In macromolecules, such as proteins and nucleic acids, the sur-

face area and volume are commonly used as geometric invari-

ants,6,20,21 which are related to various molecular properties, such

as stability, solubility, crystal packing or molecular recognition.

Some of these measures, such as the hydrophobic surface area,

excluded volume, and radius of gyration, have been incorporated

into energy minimization algorithms used in studies of the three-

dimensional (3D) structure of proteins.22–24 The development of

global optimization procedures for clusters, crystals, and biomole-

cules is of great importance in fields ranging from protein struc-

ture prediction to the design of microprocessor circuitry.25 In

computational geometry, the second moment of mass distribution

is often used as a geometric invariant which is related to packing

density26–30 and has been found to be of great utility in the study

of colloidal clusters obtained from densely packed micro-

spheres.18,19

In this article, we introduce a new measure of packing that

accounts for the scattering of points in a discrete ensemble. This

measure solves some of the problems found with other packing

measures and reveals several other useful properties with which

to study clusters, molecules, crystals, and biomolecules.

Contract/grant sponsor: Ramon y Cajal, Spain.

Correspondence to: E. Estrada; e-mail: [email protected]

q 2007 Wiley Periodicals, Inc.

Theoretical Approach

Some ‘‘Classical’’ Geometric Invariants

Some of the most widely used geometric invariants are related to

the area and volume of surfaces. There are three definitions of

‘‘surface’’ which are widely used as geometric invariants of

points, particularly molecules, as well as for the definition of

packing measures.31–33 They are the van der Waals (vdW) sur-

face, the solvent accessible surface (SA), and the molecular sur-

face (MS). The first is defined as the surface of what is covered

by the points, i.e., atoms, which are represented by spherical balls

with radii equal to their vdW radii. The second surface is gener-

ated by the center of the solvent, which is modeled by a rigid

sphere, when rolling over the vdW surface of the cluster or mole-

cule. The third surface is generated by the front of the same sol-

vent sphere. The area and volume of such surfaces can then be

determined by approximation techniques or by analytic methods.

One of these methods, which treats volume and area overlaps

fully and accurately, is the ‘‘alpha shape method.’’20

The second moment of mass distribution is defined as: M2 ¼Si¼1n kri � r0k2, where ri is the centre coordinate of the ith vertex,

e.g., centre of cluster or atom, and r0 ¼ n�1 Si¼1n ri is the centre

of mass of the object, e.g., cluster or molecule, subject to the con-

straint kri � rjk � 2 for i = j. (Here k. . .k denotes the usual Eu-

clidean distance). M2 is the sum, divided by n, of the squared dis-

tances between all pairs of points. That is, M2 ¼ 1n

Pi>jðrijÞ2,

which geometrically represents the sum of the areas of the squares

formed by the edges of length rij.26 M2 has also been interpreted as

an energy function in determining clusters of hard spheres26 as

well as in communications, where the points represent a constella-

tion of n signals with total energyM2.34,35 It is known that the clus-

ters obtained using this energy function are quite different from

those using other potentials, such as Lennard–Jones for n � 8. The

second moment is intimately related to the radius of gyration, Rgyr,

another parameter which provides information regarding the global

conformation of a system, which is widely used in polymer statis-

tics.36 It can be calculated from the second moment M2 as follows:

Rgyr ¼ (M2/n)1/2.

Definition of Point Scattering

In this work we will use the term ‘‘point’’ to designate a body

whose spatial extent and internal motion and structure, if any, are

irrelevant to the specific problem under study, which is the pack-

ing of points in the discrete object. Consequently, a point here

can be a sphere, colloidal particle or an atom which is part of a

cluster or a molecule, which represents the discrete object.

We start by considering an object O formed by n points. Let us

represent the object by means of a column vector x, whose kthentry captures the relative departure of point k from the geometri-

cal centre, o, of the object. The entries of the vector x, xi, take val-ues between 0 and 1. They represent a sort of weighted distance

from the center of the object in which the outlying points receives

more weights than the points closest to the center. We impose the

restriction that the norm of this weight vector x be one or xxT ¼ 1.

Then, based on the Euclidean distances between the pairs of

points in the object, rij, we can define a measure for the spreading

of the points in O in a similar way as used in spectral clustering

techniques37–39:

SðOÞ ¼Xni¼1

Xnj¼1

rijxixj ¼ xTDx (1)

where D is the Euclidean distance matrix of the points in the

object. The function S(O) increases with the increase in the sepa-

ration between points as well as with the departure of the points

from the centre of the object. Consequently, we consider S(O) asa measure of the scattering of the points in the discrete object,

which will take minimum values for the least scattered objects.

We are interested in finding the maximum value of the scatter-

ing function, which can be obtained by maximizing the expres-

sion (2). Let {�1, �2, . . . , �n} be the nondecreasing order of the

eigenvalues of D and let xi be the orthonormal eigenvector corre-

sponding to the ith eigenvalue:kxk2 ¼ xxT ¼ 1.40 Then, according

to the Rayleigh–Ritz theorem,41 we have:

SðOÞ ¼ maxx

xTDxjxTx ¼ 1� � ¼ �1ðDÞ (2)

where �1(D) is the spectral radius or largest eigenvalue of D. Thismaximum is attained when x ¼ x1, where x1 is the principal eigen-

vector of D. It is clear that �1(D), which is our measure of point

scattering, remains unchanged under translation and rotation. This

geometric measure can also be invariant to scaling by normalizing

the distances by using a canonical representation of the object.

However, we will not consider this item in the current work.

The interpretation of the principal eigenvector of D as a rela-

tive weighted measure of the departure of a point from the centre

of the object can be understood by means of the following analy-

sis. It is known that the principal eigenvector is proportional to

the row sum of a matrix M formed by summing all powers of the

distance matrix, weighted by corresponding powers of the recip-

rocal of the principal eigenvalue:

M ¼ limn!1

1

nDþ ��1

1 D2 þ ��21 D3 þ � � � þ ��n

1 Dnþ1� ��

: (3)

Then, we can consider an object formed by three points placed

on a straight line, in such a way that point i is equidistant from

points j and k, i.e., i is placed at the centre of the object. Let

�a ¼ Pb rab be the sum of distances for the point a in the discrete

object, i.e., the sum of the ath row or column of D. It is obvious

that the minimal value of �a will be obtained for the point which is

at the centre of the object, which has the lowest entries in D: min

�a ¼ �i. The same is also true for the different powers of D, which

makes that the lowest value of the row sum for the matrixM corre-

sponds to the point at the centre of the object. As the ith row sum

of this matrix is proportional to the ith entry of the principal eigen-

vector of D we have that the lowest entry of xl corresponds to the

point at the center of the object. As far as we add more points, for

instance in different orbits from the center, the points more distant

from the center will have larger values of their sum rows in M and

of its components in xl. Then, it is clear that xl is the relative depar-

ture of the points from the centre of the cluster weighted in a way

that the most distant points are weighted more strongly than the

768 Estrada • Vol. 28, No. 4 • Journal of Computational Chemistry

Journal of Computational Chemistry DOI 10.1002/jcc

points closest to the centre. This measure is not appropriate for

comparing points in different clusters because points which are

equidistant from the centre in two different clusters will give the

same values of xl. For instance, the value of xl for the arrangement

of unit spheres in a square and in a tetrahedron are identical, e.g.,

x1 ¼ (0.5 0.5 0.5 0.5) for both objects. In these objects the points

are equidistant from the respective centers, but the distances from

the center to the points in both objects are different, which makes

xl inappropriate for comparing points in the square to points in the

tetrahedron. Consequently, the use of the following measure is

more appropriated for comparing points in different objects. Since

xl is a �1-eigenvector, one has x1(i) ¼ Sjrijx1(j)/�1, so that

SðiÞ � x1 ið Þ�1ðDÞ (4)

can be considered as a measure of local geometric invariant for

the points of the cluster.

Another consequence of the current representation of discrete

objects is that M2 can also be calculated from the spectrum of D. It

can be shown thatM2 is equal to the half sum of the diagonal entries

of D2, i.e., M2 ¼ 12nTrðD2Þ ¼ 1

2n

Pni¼1ðD2Þii, where Tr stands for

the trace of the matrix. It is well-known that Tr Dk� � ¼ Pn

j¼1 �j

� �k,

from which it follows that M2 ¼Pn

j¼1 �j

� �2. The radius of gyra-

tion, Rgyr is then easily expressed in terms of the spectrum of D

as:Rgyr ¼ ½1n

Pnj¼1 �j

� �2�1=2.Computational Methods

The calculations of the point scattering measure for the different

objects described in this work were carried out using a Matlab1 pro-

gram developed in-house. The input for the program is the distance

matrix, which is previously obtained from the Cartesian coordinates of

the points using an implementation in MODESLAB (www.mode-

slab.com). The output of the Matlab1 program are the principal eigen-

value and eigenvector of the distance matrix, which correspond to the

geometric invariants introduced here. Other geometric invariants were

also calculated in this work for the sake of comparison. The first of

them is the second moment of mass distribution, which was calculated

using the squared eigenvalues of the distance matrix obtained from the

Matlab1 program according to: M2 ¼P

nj¼1 �j

� �2. Surface areas and

volumes (SA or vdW) were calculated from the grid method

according to the implementation of the Bodor et al.42 approach

used in HyperchemTM. This method uses the atomic radii of Gave-

zotti.43 In SA calculations the solvent probe radius used was 1.4 A

and we always used 50 points on cube side. The calculations of SA

surface area and volumes of the square grid with holes were car-

ried out using the same grid approach but using hydrogen atoms

with radius equal to 1 A (unit radius spheres) instead of using the

Gavezotti radius for hydrogen, which is 1.17 A (this value can be

modified in the file VDWGRID.TXT of HyperchemTM).

Results and Discussion

Point Scattering in Clusters and Crystals

Clusters of hard spheres can represent a variety of structures

found in nature ranging from pollen grains to crystals and virus

capsids. When applicable, the examples of clusters studied here

can also be considered as crystals in which the spheres are occu-

pying the lattice points of space lattice.44 When considering clus-

ters of hard spheres we will take the centers of the spheres as the

points forming the object. Here we will consider hard spheres,

i.e., noninterpenetrating spheres, of unit radius. Thus, rij � 2 for

every pair of nodes in G.

Scattering in Clusters with Degenerated Second Moments

As a first example we will consider the optimal clusters in two-

dimensions (2D). Graham and Sloane conjectured 15 years ago

that for n = 4 every optimal packing is (up to limits imposed by

symmetry) a subset of the hexagonal lattice A2, which is gener-

ated by (1,0) and �1=2;ffiffiffi3

p=2

� �.29

The exclusion of the four point cluster appears to be due to the

well known fact that the second moment of mass distribution is

unable to differentiate a square and a rhombic arrangements of

points as they both have: M2 ¼ 8a2, where a is the radius of the

circles. This result contradicts the intuitive idea that the rhombus,

which is a subset of A2, is more tightly packed than the square,

which is not a subset of the hexagonal lattice. This situation is found

not only for this pair of clusters but for several other pairs of clus-

ters. In Figure 1 we illustrate some of these examples. On the third

and fourth lines of Figure 1 we show two examples that extend this

observation to 3D clusters. They correspond to a pair of clusters of

six spheres forming a rhombic bipyramid (rhombic octahedron) and

a triangular prism. The last example is provided by the pair of clus-

ters with eight spheres forming a cube and a triangular biprism.

The consideration of S(O) clearly differentiates all these pairs

of different clusters according to their packings. First, S(O) indi-cates that the rhombus has lower point scattering than the square

as expected from the fact that the first is a subset of the hexagonal

lattice. It also indicates that the prism and the biprism are less

scattered than the rhombic pyramid and the cube, respectively

(Fig. 1). In all cases the clusters with optimal packing (according

to second moment of mass distribution)26 having 4 (tetrahedron),

5 (triangular bipyramid), 6 (octahedron or square bipyramid), and

8 (dodecadeltahedron or snub disphenoid) nodes show the lowest

values of S(O) (see last column of Fig. 1).

Square Lattices with Holes

As a second example we build a toy model consisting of square

lattice of 16 spheres in which we remove two spheres at random.

In total there are 21 different configurations of 14 spheres plus

two holes in this square lattice. In Figure 2 we illustrate the

square lattice with sphere numbering as used here and three con-

figurations of 14 spheres and two holes. In Table 1 we give the

values of the several geometric invariants: second moment, SA sur-

face areas and volumes and the values of point scattering, for these

21 configurations of 14 spheres. The vdW surface area and the vol-

umes are the same for all these objects: 175.80 A2 and 58.91 A3,

respectively. As can be seen, there are four pairs and a triple of

clusters having identical values of SA surface areas, one pair, two

triples, and one quadruple of clusters with identical values of the

SA volumes and three pairs of clusters with degenerate values of

M2. However, there is not a single pair of clusters with identical

values of S(O). The geometric invariants show good linear correla-

769New Geometric Invariant with Applications From Clusters to Biomolecules


Figure 1. Clusters of hard spheres with identical values of the second moment of mass distribution (first

two columns), which are differentiated by the point scattering. The last column shows the optimal pack-

ing for the corresponding number of spheres according to Sloane et al.30



tions between each other. The best linear correlation is observed for

M2 and S(O) with a correlation coefficient > 0.99. In general, S(O)shows excellent correlations with the other geometric invariants

(Table2). For instance, S(O) has larger correlation coefficients with

the SA surface areas and the volumes thanM2.

Hollowed Clusters: Fullerenes and Virus Capsids

The third example of clusters to be studied here is that of carbon

clusters or fullerenes. A fullerene is a closed cage formed by ncarbon atoms distributed in a Degree-3 network of pentagons and

hexagons on the surface of a spheroid.45 Consequently, fullerenes

are good representatives of hollow clusters in which the carbon

atoms on the surface create nonpolar cavities of different sizes in

the interior of the cluster.46 In fact, these cavities have been used

to model the penetration of water molecules to nonpolar interior

of proteins. Vaitheeswaran et al. have shown that up to nine water

molecules can be easily accommodated in the internal cavity of

C180.47 They have shown that the stability of encapsulated water

clusters depends critically on cavity size. The existence of other

endohedral complexes with fullerenes has been observed experi-

mentally. Among the most significant, one can cite the existence

of ‘‘bucky-onions,’’ in which one fullerene is encapsulated inside

another one, such as C60 @ C240 @ C540 . . . ,48 or the existence

of endohedral metal encapsulated fullerenes, such as Sc2 @ C84

or Sc3 @ C82.49 Thus, the study of the geometric invariants of

these carbon clusters is of great importance for understanding

these inclusion phenomena, which critically depend on the rela-

tion between the size of the cluster and the size of the interior cavity.

Fullerenes also represent interesting models of viral protective pro-

Figure 2. Square grids of spheres of unit radius numbered from 1 to 16 (A). Three clusters of 14

spheres, which are subsets of the square grid, having the lowest (B) (1,4-cluster), intermediate (C) (1,7-

cluster), and the largest scattering (D) (6,11-cluster) among the 21 clusters. The numbers indicate the

spheres which were removed according to the numbering given in A.



tein shells (capsids), as it is well known that there are numerous

roughly spherical viruses whose capsids display perfect icosahedral

symmetry.50 For instance, the tobacco ringspot virus capsid is com-

posed of 60 copies of a 513-amino-acid capsid protein, each of which

corresponds to one of the atoms in C60 ‘‘buckminsterfullerene.’’51

Here we study the areas and the volumes (vdW and surface

accessible) as well as the second moment of mass distribution

and point scattering of 19 fullerenes ranging from C20 to C540. In

Table3 we give the values of the geometric invariants for all these

carbon clusters.

In general there is very good linear correlation between the

different pairs of measures. However, plots of the geometric

invariants versus the size of the carbon clusters reveal interesting

characteristics of the different invariants. In Figure 3 we illustrate

these plots for S(O) and the vdW area and volume.

The behavior of M2 is similar to that of S(O). The surface acces-sible area and the volume plots are similar to those of the vdW ana-

logues. As can be seen in Figure 3, surface area and volume increase

linearly with the size of the clusters (these values are plotted on an

inverse scale in Fig. 3). This means that the size of the interior cav-

ities of these fullerenes increase linearly with the size of the cluster.

However, a different picture is provided by S(O) (as well as by M2)

which shows a nonlinear increment of the scattering as a function of

the cluster sizes. This indicates that according to S(O) the size of theinterior nonpolar cavity of a fullerene increases nonlinearly with the

increase of the number of carbon atoms—an observation with im-

portant consequences for the study of inclusion complexes in fuller-

ene cavities. For instance, while the optimal number of water mole-

cules that can be encapsulated in C140 is four, giving (H2O)4 @

C140, this number increases up to eight for C180, which leads to

(H2O)8 @ C180.47 Torrens has calculated the volume and area of the

internal cavities of smaller fullerenes, C60, C70, and C82, in which it

is observed that the volume of the internal cavity increases nonli-

nearly with the size of the carbon cluster in agreement with our cur-

rent findings even for these small fullerenes.52 These results illus-

trate the utility of the point scattering as a geometric invariant which

contains information not duplicated by other invariants.

Point Scattering in Proteins

The previous analysis of S(O) in clusters of hard spheres provides

several suggestions as to how this measure quantifies the scatter-

ing of a set of points. Models based on sphere packing have been

used for studying optimal properties of proteins.53 In this sense,

the study of the toy model of spheres in a square lattice with two

holes clearly indicates that the least scattered structures are those

in which the holes are as far as possible from the centre of the

cluster. On the contrary, the structure with the largest scattering

(see D in Fig. 2) corresponds to the one in which the two holes

are adjacent to the middle point of the cluster. This is an impor-

tant characteristic for the study of proteins, which are believed to

have efficiently packed interiors. The existence of internal cav-

ities has also been considered to be an important characteristic of

these macromolecules, which in general are well accounted for

by the S(O) index (see structure B in Fig. 2). Consequently, we

will study here the relationships between S(O), as well as other

geometric invariants, and the thermodynamic parameters of bind-

ing between peptides and proteins in the complex RNase-S. In

this case, point scattering can be referred to as the atomic scatter-

ing as it is calculated for all atoms in the proteins.

The protein–peptide complex RNase-S is obtained by cleavage

of bovine pancreatic ribonuclease A (RNase A) with subtilisin to

give an ‘‘S protein’’ and an ‘‘S peptide.’’ These two fragments can

be reconstituted to give rise to RNase-S, which is catalytically

active with a structure very similar to that of ribonuclease-A

(RNase-A).54,55 The S peptide consists of the first 20 amino acids

of RNase-A, but it has been shown that a truncated version formed

by Residues 1–15 forms a complex with protein S which is struc-

turally identical with RNase-S. There are two hydrophobic resi-

dues in the S peptide that contribute significantly to the stability of

RNase-S. These residues, methionine 13 (M13) and phenylalanine

8 (F8), are buried inside the RNase-S core.55,56 Mutation experi-

ments have led to their replacement by several other smaller

hydrophobic amino acids of different sizes resulting in mutated S

peptides of the type F8X and M13X, where X represents alanine

(A), methionine (M), norleucine (Nle), �-aminobutyric acid

(ANB), valine (V), leucine (L) or isoleucine (I). The structures of

Table 1. Values of Geometric Invariants for the 21 Clusters of

Spheres of Unit Radius Which are Subsets of the Square Grid

as Explained in Figure 2.

Cluster S(O) M2 SASA (A2) VSA (A3)

1,4 53.239 121.43 301.68 386.13

1,16 53.633 123.99 306.13 388.62

1,2 54.587 128.28 301.69 386.13

1,3 54.927 129.14 316.03 399.08

1,8 55.254 130.85 315.52 399.08

1,12 55.362 131.71 315.52 399.08

2,3 56.620 137.43 323.42 402.26

1,6 56.648 137.71 316.38 404.09

1,7 56.704 138.57 316.40 404.09

2,5 56.738 137.71 329.87 412.01

1,11 56.823 139.43 316.40 404.09

2,8 56.937 138.57 330.40 412.05

2,12 57.063 139.43 330.40 412.05

2,14 57.098 139.71 329.85 412.05

2,15 57.129 139.99 329.85 412.05

2,6 58.339 146.57 332.28 415.51

2,7 58.422 146.85 330.73 417.06

2,10 58.529 147.43 330.73 417.06

2,11 58.568 147.71 330.73 417.06

6,7 59.980 155.71 332.19 420.54

6,11 60.042 155.99 331.06 422.08

Table 2. Correlation Coefficients for the Pairs of Geometric Invariants

Analyzed for the 21 Clusters of 14 Spheres of Unit Radius Which are

Subsets of the Square Grid.

M2 SASA VSA

S(O) 0.998 0.867 0.952

M2 0.838 0.936

SASA 0.963

SASA, solvent accessible solvent area; VSA, solvent accessible volume.



the RNase-S complexes with S peptide mutants were determined

by X-ray crystallography and the free energies (DG8) and enthal-

pies (DH8) of the S-peptide-protein binding were determined using

titration calorimetry.57 In Table 4 we show the values of the differ-

ences in the thermodynamic parameters upon mutation: DDG8 andDDH8, e.g., DDG8 ¼ DG8 (mutant) � DG8 (wild type).

We calculated the values of S(O) as well as areas and vdW

and SA volumes for eight protein–peptide complexes. The values

are given in Table 4 where we also provide the correlation coeffi-

cients of the linear fits between the thermodynamic parameters

and the geometric measures. As can be seen S(O) gives the best

linear correlations for both thermodynamic properties, which has

Table 3. Values of Geometric Invariants for Several Fullerenes Studied Here.

Fullerene S M2 SAvdW (A2) SASA (A2) VvdW (A3) VSA (A3)

C20 (Ih) 53.68 82.22 170.27 318.28 186.06 525.78

C24 (D6) 71.51 121.32 195.21 353.05 221.84 598.12

C26 (D3h) 80.61 142.18 205.91 361.59 238.09 630.73

C28 (Td) 89.81 163.53 216.32 376.62 254.00 660.92

C30 (D5h) 100.68 192.98 227.75 391.77 270.24 696.19

C32 (D3) 109.86 213.92 239.76 398.30 287.05 726.42

C36 (D6h) 131.20 271.05 262.47 425.65 320.10 791.65

C50 (D5h) 214.86 521.84 342.04 500.42 436.83 1,016.75

C60 (Ih) 280.98 742.29 397.31 552.89 516.99 1,169.50

C76 (D2) 405.93 1,223.88 499.18 637.41 662.44 1,453.72

C78 (D3h) 420.25 1,276.38 507.84 651.25 676.12 1,478.92

C78 (D3h) 421.07 1,283.57 508.42 648.06 676.15 1,477.48

C78 (D3) 421.14 1,283.94 508.27 648.89 675.92 1,478.12

C78 (C2v) 420.40 1,277.84 508.27 648.89 675.92 1,478.12

C80 (Ih) 438.35 1,353.59 522.74 668.90 697.89 1,522.54

C180 (Ih) 1,470.1 6,756.38 1,097.36 1,200.40 1536.2 3,139.81

C240 (Ih) 2,277.8 12,161.70 1,472.53 1,542.21 2,070.28 4,177.71

C320 (Ih) 3,476.6 21,244.27 1,917.71 1,952.61 2,722.84 5,430.73

C540 (Ih) 7,596.0 60,084.84 3,196.6 3,131.93 4,571.25 9,004.91

Figure 3. Plot of geometric measures (in reverse scale) versus number of carbon atoms in fullerenes.

[Color figure can be viewed in the online issue, which is available at www.interscience.wiley.com.]



correlation coefficients identical to those of VSA. The best linear

correlation previously obtained for DDH8 was by using the

occluded surface area, which measures the internal packing of a

protein, and shows a correlation coefficient identical to those

obtained here with S(O) and VSA, i.e., 0.98. Occluded surface

area shows a correlation coefficient of 0.95 for DDG8, which is

slightly lower than the one obtained here by using S(O) and VSA.

In contrast, other geometric invariants, such as cavity volumes

and SA area, show very poor correlations, while the packing mea-

sure called ‘‘depth’’ shows correlation coefficients of approxi-

mately 0.90 for both thermodynamic parameters of binding.

As previously indicated, the values of S(i) can be used as a

local scattering parameter for the points of a cluster. In the partic-

ular case of proteins, S(i)corresponds to atomic contributions to

protein scattering. It is also possible to consider the average val-

ues of S(i) for the atoms in an amino acid as the scattering of

amino acid residues in the protein: S(res). Using this approach we

have calculated the values of S(res) for all the amino acids in the

wild type RNAse A solved at 1.6 A (2rns). In Figure 4 we plot

the values of S(res) for all residues in 2rns, where the lowest val-

ues of S(res) correspond to the least scattered residues and the

higher values to the most scattered ones. In the same figure we

illustrate the ten least scattered residues as well as the ten most

scattered ones. As can be seen, the least scattered residues corre-

spond to those which are located in the interior of the protein in a

region close to the centre of the structure, which is commonly

identified as the most packed region. The most scattered residues

are located far away from the middle of the protein, in the periph-

eral regions of the protein where residues 1, 68, 88–94, and 113

are found.

In a similar way S(res) can be used to analyze the changes on

residue scattering produced by external factors such as temperature.

As an example we calculate the values of S(res) for all the amino

acids in the RNAse-A, which is a kidney-shaped monomeric

enzyme of 124 residues. The structure of this protein has been

determined by X-ray diffraction at a resolution of 1.5 A at nine dif-

ferent temperatures ranging from 98 to 320 K.53 In previous studies

it has been shown that the protein molecule expands slightly with

increasing temperature which affects principally the degree of fold-

ing of the main backbone of the protein.6,58–60 It was also previ-

ously observed that the 3D structure of RNAse-A undergoes no

dramatic change over the range of temperatures analyzed. Here we

analyze the changes in scattering of individual amino acids with

changing temperature. In Figure 5 we plot the change in the values

of S(res) during step-by-step ‘‘heating’’ of RNAse-A. The first plot

shows the difference in S(res) for the amino acids in the structures

determined at 98 and 130 K, i.e., DS(res) ¼ S130 K(res) �S98 K(res). As can be seen in this figure, most of the amino acids

suffer small changes in their scattering when analyzed in this step-

by-step method. However, a dramatic change in scattering is

observed for residue Gln101, which appears as an intense peak in

most of the plots. For instance, in the 130–98 K plot this residue

appears with an intense positive peak, which indicates that Gln101

is least scattered at 98 K than at 130 K. However, this situation is

inverted in the next step, which indicates that Gln101 returns to a

least scattered conformation when passing from 130 to 160 K. In

Figure 6 we illustrate these two conformations for Gln101. This

alternating change of conformation of Gln101 is observed for all

RNAse-A structures below 220 K and it still appears for the changes

at the highest temperatures, i.e., 120–260 K and 320–280 K. This

residue is located in one of the extended loops of the protein, which

have been previously observed to undergo the most intense move-

ments. In general, the most packed regions at the interior of the pro-

tein structure do not suffer large variations in the scattering com-

pared to the atoms which are in the protruding loops. An exception

is the 180–220 K transition where the scattering of most of the resi-

dues is altered. This change coincides with the known fact that in

the neighborhood of 200 K changes occur in the dynamic properties

of many proteins in solutions and in the crystal state.58 These

dynamic changes are believed to be produced primarily in the coor-

dination shells of water that are bound to the surface of the protein,

which in this case is expressed as an increment in the packing

changes of amino acid residues in the whole protein.

These results clearly indicate that the point scattering appears

to be a convenient geometric invariant, which can account for the

vdW interactions between peptides and proteins as well for the

effects of temperature on protein structure and dynamics. In con-

sequence, this measure can be useful as a geometric parameter

for empirical potentials of energy minimization algorithms for the

study of the 3D structure of proteins.

Table 4. Values of Geometric Invariants as well as Thermodynamic Binding Parameters for the Protein–Peptide Complexes in RNAse-S.

PDB S SAvdW (A2) SASA (A2) VvdW (A3) VSA (A3) DDH8 (258 C) DDG8 (kcal mol�1)

2rln 18,137 11,753.2 6,653.4 10,343.8 22,508.9 �7.9 �0.8

1rbg 18,029 11,758.4 6,586.4 10,324.9 22,391.9 �2.5 �0.7

1rbh 18,030 11,755.7 6,596.6 10,324.9 22,380.3 �2.2 �0.5

1rbi 18,031 11,745.4 6,577.7 10,311.2 22,307.4 �1.9 �0.2

1rbd 18,019 11,719.5 6,539.8 10,292.7 22,294.0 1.3 0.7

1d5e 17,618 11,438.7 6,653.3 10,113.1 22,117.1 10.0 2.9

1d5d 17,554 11,384.2 6,674.8 10,083.6 22,073.1 14.7 3.6

1d5h 17,543 11,447.9 6,593.7 10,117.4 22,037.2 17.7 5.1

r(DDH8) �0.98 �0.94 �0.28 �0.96 �0.98

r(DDG8) �0.96 �0.94 �0.31 �0.95 �0.96

The correlation coefficients between the geometric invariants and thermodynamic parameters are given in the last two rows.



Conclusions

We have analyzed the main deficiencies of some of the most

widely used ‘‘classical’’ geometric invariants for the analysis of

clusters, molecules, and macromolecular systems, such as pro-

teins. This analysis has led to the introduction of a new measure,

which is based on first principles, and resolves the deficiencies

previously observed associated with other geometric measures,

such as second moment of mass distribution, surface areas, and

volumes. This new geometric invariant accounts for the scattering

of points from the center of the object, e.g., cluster, molecule,

protein, etc. We have proved that the principal eigenvalue of the

Figure 4. Values of residues scattering in ribonuclease-S (PDB 2rns). The largest values of the scatter-

ing correspond to most packed residues, e.g., 88–94 and the lowest values of scattering correspond to

the least scattered residues, e.g., 44–47.



Euclidean distance matrix represents the point scattering of the

object studied. The Euclidean distance matrix is built by consider-

ing the distances between all pairs of points in the discrete object,

such as the center of spheres in clusters or interatomic distances

in molecules. This matrix can be obtained experimentally, for

example by X-ray diffraction, or by empirical or ab initio optimi-

Figure 5. Change of residues scattering in RNAse-A as an effect of temperature.

Figure 6. Change in conformation of residue Gln101 at two different temperatures which produces the

change in scattering observed in Figure 5. At 98 K this residue is least scattered than at 130 K, which

produces a large positive peak in the first plot of Figure 5. [Color figure can be viewed in the online

issue, which is available at www.interscience.wiley.com.]



zation procedures, such as molecular mechanics or quantum

chemical calculations.

In this work we have illustrated some of the advantages of the

point scattering for studying hypothetical and real clusters, such as

those that can arise from dense packings of colloidal microspheres

as well as carbon clusters or fullerenes. We have also studied

‘‘atomic’’ scattering in proteins by illustrating the relationship

between this measure and the binding energetics of peptide–pro-

tein interaction as well as the effects of temperature on protein

structure and dynamics. In all cases point scattering appears to be

a convenient geometric invariant, which is easily and exactly com-

puted for any discrete configuration of points, accounting for im-

portant structural characteristics of the objects studied, which are

invariants to the group of rotation and translation of the object.

One of the most significant applications of geometric invari-

ants in chemistry is that they are used in measuring packing.61

For instance, packing efficiency of a given atom is simply defined

as the ratio of the space it could minimally occupy to the space

that it actually does occupy.61 In a similar way as volume and

surface area are applied to define packing efficiency in different

ways, the new geometric invariant we have introduced here can

be used for similar purposes giving it a wide spectrum of applica-

tions in different areas of research.

Acknowledgments

The author thanks Ms. Y. Gutierrez for assistance in developing a

computer program to calculate the point scattering. Prof. D. J.

Klein and Dr. J. A. Rodrıguez-Velazquez are also acknowledged

for useful comments and clarifications.

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