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Article

Characterizing Cavities in Model Inclusion Fullerenes: A Comparative Study

Institut Universitari de Ciència Molecular, Universitat de València, Dr. Moliner 50, E-46100 Burjassot (València), Spain
Int. J. Mol. Sci. 2001, 2(2), 72-88; https://doi.org/10.3390/i2020072
Submission received: 16 November 2000 / Accepted: 3 April 2001 / Published: 2 June 2001

Abstract

:
The fullerene-82 cavity is selected as a model system in order to test several methods for characterizing inclusion molecules. The methods are based on different technical foundations such as a square and triangular tessellation of the molecular surface, spherical tessellation of the molecular surface, numerical integration of the atomic volumes and surfaces, triangular tessellation of the molecular surface, and cubic lattice approach to the molecular volume. Accurate measures of the molecular volume and surface area have been performed with the pseudorandom Monte Carlo (MCVS) and uniform Monte Carlo (UMCVS) methods. These calculations serve as a reference for the rest of the methods. The SURMO2 method does not recognize the cavity and may not be convenient for intercalation compounds. The programs that detect the cavities never exceed 1% deviation relative to the reference value for molecular volume and 5% for surface area. The GEPOL algorithm, alone or combined with TOPO, shows results in good agreement with those of the UMCVS reference. The uniform random number generator provides the fastest convergence for UMCVS and a correct estimate of the standard deviations. The effect of the internal cavity on the solvent-accessible surfaces has been calculated. Fullerene-82 is compared with fullerene-60 and -70.

Introduction

Analyses of molecular packing in liquids and crystals [1] have shown that cavities may occur as a result of local packing defects. In proteins, internal cavities large enough to accommodate even xenon atoms have been identified [2,3,4,5]. Although the existence of these internal cavities has been largely associated with conformational flexibility [6,7,8], their role remains poorly understood. Internal cavities affect protein stability [9], and play an important role in the close packing of amino acid side chains, particularly in the core of the protein [10,11]. The knowledge of the channels and cavity structures within crystals of globular proteins may be useful [12,13,14,15,16,17,18,19,20] in detecting the binding sites of specific ligands and ions on accessible surfaces of protein molecules within channels; in studying the structure and physical properties of the water near protein surfaces within the channels; for predicting the narrowest sites of the channels for diffusion of molecules, etc.
Molecular models are commonly represented as systems of fused hard spheres with unequal radii, and there has been an important body of work on the computation of surface area and volume of such systems [21] and on the topological description of their surfaces [22,23]. In large molecules such as proteins, where most of the van der Waals surface is buried in the interior, other, more suitable definitions such as the solvent-accessible surface [24,25] and solvent-excluding surface have been used. The corresponding surface areas can be computed from the atomic coordinates by approximate procedures or by using analytic algorithms such as those of Gibson and Scheraga [21], Connolly [26,27] or Richmond [28]. The analytic algorithms enable the calculation of first-order derivatives of the surface area, which is an important requirement for the use of surface area functions in optimization procedures [29]. Methods to compute atomic and molecular volumes of proteins from the atomic coordinates have also been derived by Richards [4] and Finney [30]. The most common shortcomings of the detection algorithms are (1) difficulties in detecting cavities smaller than a certain size, and (2) the tendency to detect spurious cavities. Alard and Wodak [31] have described a procedure that is able to detect cavities in a system of interpenetrating spheres, no matter what their size or shape.
In a previous article the (Ih)fullerene-60 (C60, buckminsterfullerene) and (D5h)fullerene-70 (C70) molecules were selected as model inclusion molecules [32]. In this work (C2)fullerene-82 (C82) has been selected as a model cavity. The following programs have been used for characterizing the molecular cavities of fullerenes: (1) SURMO2 (surface of molecule), (2) the MS (molecular surface) program by Connolly [26,27], (3) SCAP (solvent-dependent conformational analysis program), (4) the GEPOL (geometry of polyhedron) program, (5) a program for the computation of surface areas of molecules (CSAM), (6) the TOPO program, (7) pseudorandom Monte Carlo computation of the molecular volume and surface (MCVS), and (8) uniform Monte Carlo computation of the molecular volume and surface (UMCVS). In the next section, the different methods used in the preceding programs are briefly described, as well as the new features that have been included in them. Following that, results are presented and discussed. The last section summarizes my conclusions.

Methods

Program SURMO2-I represents the global shape of a molecule as an envelope of the van der Waals spheres of the external atoms [33]. This allows a numerical treatment of integrals defined on the molecular surface, as
I = g ( ρ , θ , ϕ ) d ω
where g is the position vector of a point in the surface and d ω = sin θ d θ d ϕ represents the elementary solid angle. The evaluation of the Equation (1)-type integrals is obtained from the finite sum I = g d ω i g i δ ω i . To achieve the summation one can take a mesh of points M i j ( θ i , ϕ j ) by cutting the surfaces on a single unitary sphere with a uniform distribution of 2N1 parallels and N2 meridians. If the molecule is cut by an axis passing by the origin, two intersecting points are obtained: g+ and g-. Denoting by M0 the upper pole, M1j the points placed on the first parallel, Mij those of the i-th parallel and M N 1 j those placed on the equator, the integral (1) can be measured as
I = N 2 3 ( g M 0 + + g M 0 ) + 7 6 j = 0 N 2 1 ( g M 1 j + + g M 1 j ) + 1 2 j = 0 N 2 1 ( g M N 1 j + + g M N 1 j ) + i = 2 N 1 1 j = 0 N 2 1 ( g M i j + + g M i j )
where j g M i j indicates a sum over all of the mesh points on the i-th parallel. The molecular volume is calculated as:
V = 1 3 ρ 3 d ω = 4 π 3 2 N 1 N 2 i = 0 N 1 r M i j = 0 N 2 1 [ ( g M i j + ) 3 + ( g M i j ) 3 ]
where ρ is the radial spherical coordinate, r = ( 1 / 3 ,   7 / 6 ,   1 ,   1 ,   , 1 / 2 ) and Equation (2) has been used. The calculation of the molecular surface area S has been estimated in SURMO-I as
S ρ 2 d ω = 4 π 2 N 1 N 2 i = 0 N 1 r M i j = 0 N 2 1 [ ( g M i j + ) 2 + ( g M i j ) 2 ]
and calculated in SURMO-II by dividing each g2 by the cosine of the angle formed by the semiaxis and the normal vector:
S = 4 π 2 N 1 N 2 i = 0 N 1 r M i j = 0 N 2 1 ( g M i j + ) 2 + ( g M i j ) 2 cos α M i j
Improving previous algorithms for calculating contact and reentrant surfaces by Greer and Bush [34], and for calculating the solvent-accessible surface area by Shrake and Rupley [25], Connolly [26,27] has developed program MS. MS places dots over the solvent-accessible surface of a molecule in order to calculate the solvent-excluding surface of a molecule. The analysis of a molecular surface by MS is carried out by rolling a probe sphere over the whole molecular surface. For each contact point with the van der Waals surface, there is a dot. The result is a smooth van der Waals surface, which represents the accessible zones for the solvent molecule, including the internal cavities.
The hypothesis of Hopfinger in program SCAP is that one can centre a solvation sphere on each group of the molecule [35,36]. The intersecting volume , between the solvation sphere and the van der Waals spheres of the remaining atoms, is calculated. SCAP manages up to four parameters for a given solvent: (1) n: maximum number of solvent molecules allowed for filling the solvation sphere; (2) Δgo: variation of Gibbs free energy associated with the extraction of one solvent molecule out of the solvation sphere [37,38]; (3) Rv: radius of the solvation sphere; and (4) Vf: free volume available for a solvent molecule in the solvation sphere [39,41,42,43,44,45,46]. In the solvation sphere, part of the volume excludes the solvent molecules. This volume consists of the van der Waals volume of the group at which the sphere is centred and of a volume representing the groups bonded to the central group. The latter volume is represented by a number of cylinders. The difference between the total volume of the solvation sphere and the volume excluded to the solvent molecules represents the volume V’ that is actually available for the n solvent molecules. Vf can be calculated as V f = V / n V s . The variation of Gibbs free energy associated with the extraction of all the solvent molecules out of the solvation sphere of a group, R, results Δ G R o = n   Δ g o ( 1 V o / V ) , and the solvation free energy of a molecule ensues Δ G solv o = R = 1 N Δ G R o . 1-Octanol—water partition coefficients P are calculated as
R T ln P = Δ G solv o ( water ) Δ G solv o ( 1-octanol )
at T = 298 K [47,48,49,50,51,52,53,54].
Program GEPOL sets a sphere on each atom and creates new spheres in the spaces inaccessible to the solvent [55]. Their spherical surfaces are divided into a set of triangular tessels by using a pentakis dodecahedron. The division can be increased by means of tessellation parameter NDIV. The molecular surface area is easily obtained by summing up the areas of all triangles: S = i S i where Si is the area of the i-th triangle. GEPOL defines a molecular origin to calculate the molecular volume. Let r ¯ i be the position vector of the i-th triangle centre and n ¯ i be the corresponding normal vector to the surface at this point. The volume is obtained by summing up all solid volumes made by the triangles vector surfaces and the origin of the molecule: V = ( 1 / 3 ) i S i n ¯ i r ¯ i . The calculation of the molecular globularity and rugosity indices as well as the fractal dimension of the solvent-accessible surface has been implemented in GEPOL.
The molecular volume can readily be computed by decomposing the space it occupies, using a cubic lattice [56,57,58,59]. In program CSAM, the problem of computing surfaces is reduced to the computation of volumes by converting the surface into a sheet of finite uniform thickness δh; by computing its volume the surface area can be estimated as A V / δ h [60]. If a plane cuts a randomly oriented cube, then the area of the cut will be Δ A 2 / 3 a 2 . If one corner of a cube is located on one side of the plane, then one has a triangular cut, and on average the areas of triangular cuts are smaller than ( 2 / 3 ) a 2 . The polygonal cuts are classified according to their number of sides and using different increments ΔAk for the five different types of cuts. The area is now A = k = 1 5 N k Δ A k , where Nk represents the number of cubes cut by the surface with the k-th type of cut for which the increment ΔAk is added to the surface area A. The limiting parameter a is the length of the edges of the cubes. The molecular volume could be estimated by adding the volumes of interior cubes and half of the volumes of surface cubes as V ( I + 0.5 P ) a 3 , where I is the number of cubes in the inside of the molecular surface and P is the number of cubes partially included in it. For a sphere of radius 3.0Å, the exact mean contribution results 0.482 P   a 3 . This value has been used now to calculate V as V = ( I + 0.482 P ) a 3 .
Our program TOPO [61] represents the surface of molecules by the external surface of a set of overlapping spheres with appropriate radii [62], centred on the nuclei of the atoms [63,64,65]. The molecule is treated as a solid in space, defined by tracing spheres about the atomic nuclei. It is enclosed in a box and the geometric descriptors evaluated by counting points within the solid or close to chosen surfaces. The molecular volume is calculated as V = P G R I D 3 , where P is the number of points within the molecular volume and GRID is the size of the mesh grid. The molecular bare surface area could be estimated as S Q G R I D 2 , where Q is the number of points close to the bare surface area (within a distance between RX and R X + G R I D of any atomic nucleus X). If the point falls exactly on the surface of one of the atomic spheres, it accounts indeed for GRID2 units of area on the molecular bare surface. This is because the total surface of atom X can accommodate 4 π R X 2 / G R I D 2 points. When a point falls beyond the surface, it represents GRID2 units of area on the surface of a sphere of radius R > R X , not on the surface of atom X. On the surface of X it accounts only for a fraction of this quantity: G R I D 2 ( R X / R ) 2 . The total bare surface area is calculated now as S = F G R I D 2 , where F is the sum of elements AF defined as A F = R X 2 / R 2 ( I ) for those points close enough to the surface of any atom X. R X 2 is the squared radius of atom X and R 2 ( I ) is the squared distance of point I from the atomic nucleus X. Two topological indices of molecular shape can also be calculated: G and G’. Consider Se as the surface area of a sphere whose volume is equal to the molecular volume V. The ratio G = S e / S is interpreted as a descriptor of molecular globularity. The ratio G = S / V is interpreted as a descriptor of molecular rugosity. The solvent-accessible surface (AS) is calculated in the same way as the molecular bare surface by mean of pseudoatoms, whose van der Waals radii have been increased by the radius R of the probe. The fractal dimension, D, of the molecules [66] may be obtained according to Lewis and Rees [67] as D = 2 d ( log A S ) / d ( log R ) , where R is the probe radius. A version of TOPO has been implemented in program AMYR for the theoretical simulation of molecular associations and chemical reactions [68,69,70,71] and in GEPOL.
Program MCVS performs the Monte Carlo (MC) measure of the molecular volume as follows [72]: (1) Build a routine able to detect whether a point is inside the region. (2) Find a rectangular box including the region. (3) Generate N points uniformly distributed in the box, and count the number Ni of points falling inside the region. Let p = N i / N and q = 1 p . The MC estimate of the volume is V = W p , W being the volume of the box, and the corresponding standard deviation is σ = W ( p q / N ) 1 / 2 . The MC measure of the surface area of the union of n spheres may be performed as follows [73]: (1) Let Si be the surface of the i-th sphere and T be the sum of all these surfaces. (2) Generate N points distributed uniformly on the total surface of all the spheres. (3) Count the number Nu of points lying on the surface of the union. Let p = N u / N and q = 1 p . The MC surface area is S = T p and the corresponding standard deviation is σ = T ( p q / N ) 1 / 2 .
To ensure an unbiased sampling of the whole space of parameters, the random number generator (RNG) of MCVS has been substituted in our program UMCVS, by a uniform random number generator (URNG) that yields uniformly distributed real numbers over a given finite range [ a , b ] [74,75]. URNG LOIUAB has been used.

Calculation results and discussion

Geometric descriptors

A summary of the basis of various programs and algorithms used in the present work to characterize the molecular cavity of fullerenes is presented in Table 1, along with usual values for the limiting parameters and the value used in this work [32]. Notice that some methods were designed for proteins and so the usual limiting parameters have been correspondingly increased whenever possible for the smaller molecules discussed in this work. For SURMO2 the limiting parameters N 1 = N 2 = 72 cut the unitary sphere in 2 N 1 N 2 = 10368 surface elements. For the MS algorithm the density of surface points, which has been taken as 100 points·Å-2, gives 26 251 total surface points for C82. No limiting parameter is given as input for the SCAP model. For the GEPOL program, variable NDIV controls the tessellation level and has been taken as its maximum value of 5; this gives 2 074 points for C82. The CSAM and TOPO methods have as limiting parameter the size of the mesh grid, which has been taken as 0.1Å. The molecular volume and surface area were measured accurately with the MCVS and UMCVS methods, using N = 10 7 observations for each measure. This represents 14 730 points·Å-3 for C82. The van der Waals radius for the carbon atom has been taken as 1.76Å for all the methods [32].
Table 1. Foundations of different programs used to characterize molecular cavities.
Table 1. Foundations of different programs used to characterize molecular cavities.
DescriptorBased on:Limiting ParameterUsualThis work
SURMO2Square/triangular tessellationN1 and N21272
MSSpherical tessellationDensity of surface pointsa10100
SCAPNumerical integrationbbb
GEPOLTriangular tessellationNDIV55
CSAMCubic latticeMesh gridc0.50.1
TOPOCubic latticeMesh gridc0.50.1
MCVSPseudorandom Monte CarloNumber of observations104107
UMCVSUniform Monte CarloNumber of observations104107
a Density of surface points (points·Å-2).b No limiting parameter is given as an input for this model.c Size of the mesh grid (Å).
The geometry of C82 has been optimized with the MM2-87 program [76]. The molecule is brought into its principal inertial coordinate system. The length x of the molecule is defined as the maximum length, the height z as its minimum thickness, and its width y is measured at right angles to the axes indicated by the length and height. The origin is the centre of mass for the molecule. The C82 structure is given in Figure 1. The molecular volumes of C82 are shown in Figure 2a. The total volume Vt is the sum of the molecular (Vm) and cavity (Vc) volumes: V t = V m + V c . The molecular surfaces of C82 are shown in Figure 2b. The molecular surface Sm, is the sum of the external (Se) and cavity (Sc) surfaces: S m = S e + S c .
Figure 1. Molecular image of fullerene-82.
Figure 1. Molecular image of fullerene-82.
Ijms 02 00072 g001
Figure 2. 2a. Molecular volumes and surfaces of fullerene-82: Molecular volume Vm, cavity volume Vc, and total volume Vt, 2b. Molecular volumes and surfaces of fullerene-82: External surface Se, cavity surface Sc, and molecular surface Sm.
Figure 2. 2a. Molecular volumes and surfaces of fullerene-82: Molecular volume Vm, cavity volume Vc, and total volume Vt, 2b. Molecular volumes and surfaces of fullerene-82: External surface Se, cavity surface Sc, and molecular surface Sm.
Ijms 02 00072 g002
The geometric descriptors and topological indices calculated for C82 are given in Table 2. The SURMO2 method is unable to recognize the fullerene cavity. Hence, the calculated volume V is a measure of the total volume Vt, and is 730.6Å3. The rest of procedures recognize the cavity, and the value of the molecular volume Vm is available and lies in the range 679-686Å3. These results compare well with the accurate UMCVS reference calculation, which gives a result of (678.9±0.5)Å3. The external surface area, S e = 391.3 2, is estimated by SURMO2-I, which has been modified for the calculation of molecular spherical surface areas [32]. This result is improved when the correction of the deviation from the spherical shape is taken into account with SURMO2-II ( S e = 429.4 Å2).
Geometric descriptors and topological indices for fullerene-82 calculated with different methods.a
Geometric descriptors and topological indices for fullerene-82 calculated with different methods.a
PropertyMolecule+cavityCavity free molecule
SURMO-IbSURMO-IIcSCAPMSGEPOLCSAMTOPOCombinedd
Ve730.6730.6678.5f686.4683.8678.8678.8
Sg391.3429.4503.6504.9505.9500.3477.6505.9
ASh598.0617.0633.8627.8628.8627.6616.5628.8
AS'i1061107310781072.71072107210631072
Gj1.00250.91350.7415f0.74380.75020.78200.7383
G'k0.53560.58780.7423f0.73710.73170.70360.7453
Dl1.441.461.491.481.481.481.471.48
PropertyCavity free molecule
MCVSUMCVS
Ve681.4 (≥0.5)678.9 (0.5)
Sg508.4 (≥0.7)505.5 (0.7)
ASh632.5 (≥1.5)629.7 (1.5)
AS'i1071 (≥3)1074 (3)
Gj0.7365 (≥0.0014)0.7390 (0.0014)
G'k0.7461 (≥0.0016)0.7445 (0.0016)
Dl1.49 (≥0.04)1.48 (0.04)
a Statistical errors for some methods are reported in parentheses.b Program SURMO2 modified for the calculation of molecular spherical surface areas.c Program SURMO2 after correcting the deviation from the spherical shape.d The combined method calculates the molecular volume with program TOPO and the molecular surface areas with program GEPOL.e Molecular volume (Å3).f This method does not allow the calculation of this property.g Molecular surface area (Å2).h Water-accessible surface area (Å2).i Side-chain accessible-surface area (Å2).j Molecular globularity.k Molecular rugosity (Å-1).l Fractal dimension of the solvent-accessible surface.
However, the actual (external plus internal) molecular surface area, Sm, lies in the range 478-508Å2 (programs from SCAP to UMCVS), which compares well with the UMCVS reference (505.5±0.7Å2). Not surprisingly, the globularity index G is rather greater as calculated by SURMO2 (close to unity, as expected for a sphere) compared with the rest of methods, and the rugosity index G’ is rather smaller. Notice that the internal cavity effect is difficult to appreciate in the context of the water-accessible surface area AS and of the side-chain accessible-surface area AS’ because of the small cavity contribution ( A S c a v 13 Å2 and A S c a v 1 Å2). Similarly, the cavity effect in the fractal dimension of the solvent-accessible surface is calculated as negligible in C82.
The comparison between the GEPOL and TOPO programs has a special interest because the former is an efficient algorithm but does not perform an atom-to-atom partition analysis of the geometric descriptors and topological indices of molecules [61]. It is interesting to propose a combined method that calculates the molecular volume with program TOPO and the molecular surface areas with program GEPOL [32]. In particular, the resulting G and G’ values show the best agreement with the UMCVS reference.
Both Monte Carlo (MC) methods provide statistical estimations of the standard deviation of the C82 molecular volume [ σ ( V ) = 0.26 Å3]. This quantity basically depends on the number of observations and is exact for UMCVS and approximate for MCVS. From the standard deviation σ it has been estimated the statistical absolute error E as E = t 95 % , N σ , where t95%,N is an integral of the Student distribution, which is easy to calculate by looking up the corresponding value of t in Fisher tables for a 95% confidence band and N degrees of freedom [77]. This statistical parameter is a function of N. It approaches an asymptotic limit of 1.96 when N approximates infinity. It is not very different from 1.96 for N 50 . Thus, with the number of observations given in Table 1 the statistical absolute error has been estimated as E ( V ) = 0.5 Å3 and the relative error as E r ( V ) = 0.07 % . The same procedure has been repeated for the molecular surface area [ σ ( S ) = 0.36 Å2] and a statistical absolute error of E ( S ) = 0.7 Å2 has been obtained. The relative error of the molecular surface area (0.14%) is doubled when compared with the molecular volume. This reflects the fact that molecular surface area is a more difficult magnitude to calculate than molecular volume [63,64]. Notice that this statistical error is not the only source of error in MCVS. When comparing the MCVS results with the UMCVS references a bias of 2.5Å3 can be seen for the molecular volume [5 times greater than E(V)] and 2.9Å2 for the molecular surface area [4 times greater than E(S)]. This feature is a consequence of the nonuniform random number generator (RNG) used in MCVS, which results in an underestimation of standard deviations and errors. This bias corroborates those observed for C60 and C70 [32].
The mean relative deviations of the geometric descriptors and topological indices calculated for C60, C70 and C82 with different cavity-sensitive methods (from SCAP to TOPO) relative to the UMCVS reference are reported in Table 3. Notice that only three sets of data have been averaged, so that when a calculated relative deviation is rather low [e.g., E r ( V ) = 0.01 % for TOPO] it could be underestimated. It can be seen from the data that differences in molecular volume V, relative to those obtained with the UMCVS reference, never exceed 1% in spite of the sometimes-relevant differences in method of calculation. The same observation can be made for the molecular surface area [ | E r ( S ) | 5 % ], water-accessible surface area [ | E r ( A S ) | 2 % ], side-chain accessible-surface area [ | E r ( A S ) | 1 % ], molecular globularity [ E r ( G ) 6 % ], molecular rugosity [ | E r ( G ) | 5 % ] and fractal dimension [ | E r ( D ) | 0.8 % ]. This is encouraging, since it means that all the algorithms, even starting from different points of view, are producing consistent results. Notice that TOPO gives the best molecular volume but the greatest errors for the surface-related properties. However, GEPOL shows, in general, the best results and the outcome is remarkably good for all the surface-related properties. In comparing CSAM and TOPO, the former, designed for dealing with the surface area, reveals a lower error for this property. However, TOPO, conceived for handling molecular volume, shows the least error of all methods for this property. It is thus appealing to combine GEPOL and TOPO. The combined (S from GEPOL/V from TOPO) method shows relative errors of less than 0.1% for all the descriptors in the three fullerenes and the smallest errors for indices G and G’.
Table 3. Mean relative deviations (percent) of the geometric descriptors and topological indices for fullerene-60, -70 and -82, calculated with different cavity-free molecule methods relative to the UMCVS reference calculations.
Table 3. Mean relative deviations (percent) of the geometric descriptors and topological indices for fullerene-60, -70 and -82, calculated with different cavity-free molecule methods relative to the UMCVS reference calculations.
PropertySCAPMSGEPOLCSAMTOPOaverageaCombinedbMCVS
Vc-0.06-10.7-0.010.4-0.010.4
Sd-0.20.007-0.04-1-51-0.040.01
ASe-0.8-0.6-0.008-0.2-20.7-0.0080.7
AS'f-0.7-0.4-0.1-0.1-10.5-0.10.6
Gg0.2-0.71620.040.3
G’h-0.1--1-2-52-0.03-0.4
Di-0.1-0.30.07-0.09-0.80.30.070.06
a Mean of the relative deviations (percent) in absolute value for the first four methods.b The combined method calculates the molecular volume with program TOPO and the molecular surface areas with program GEPOL.c Molecular volume (Å3).d Molecular surface area (Å2).e Water-accessible surface area (Å2).f Side-chain accessible-surface area (Å2).g Molecular globularity.h Molecular rugosity (Å-1).i Fractal dimension of the solvent-accessible surface.
From the calculation results in Table 2 and Reference 33, referring to the total (SURMO2) and cavity-sensitive (SCAP to UMCVS) molecular shapes, the geometric descriptors and topological indices for the cavities of the three fullerenes have been calculated. The data are given in Table 4 and show that for C60, the molecular volume V, surface area S and water-accessible surface area AS are smaller than for C70 and for C82 while the side-chain accessible-surface area AS’ remains almost constant (1Å) for all the fullerenes. Notice that, for the three fullerenes, S > A S > A S . This is because a water molecule with an effective radius of 1.41 Å and a volume of about 12Å3 has little space to move inside the cavity while a probe sphere representing a protein side chain with a radius of 3.5 Å and a volume of about 180 Å3 cannot be contained inside the C60, C70 or C82 cavity. In particular, for C82, the fractal dimension of the cavity solvent-accessible surface is strongly greater than for C70 and for C60, indicating that, for the former, this cavity surface is rather more irregular.
Table 4. Geometric descriptors and topological indices for the cavities of fullerene-60, -70 and –82.
Table 4. Geometric descriptors and topological indices for the cavities of fullerene-60, -70 and –82.
CavityVaSbAScAS'dGeG'fDg
Fullerene-6026.648.03.200.89781.80452.75
Fullerene-7039.563.88.420.87911.61522.89
Fullerene-8251.776.112.710.88191.47205.21
a Molecular volume (Å3).b Molecular surface area (Å2).c Water-accessible surface area (Å2).d Side-chain accessible surface area (Å2).e Molecular globularity.f Molecular rugosity (Å-1).g Fractal dimension of the solvent-accessible surface.

Solvation descriptors

The solvation descriptors calculated with program SCAP for the three fullerenes are listed in Table 5. The negative Gibbs free energy of solvation is slightly increased on going from C60 (15.60kJ·mol-1) to C82 (20.86kJ·mol-1). However, the negative Gibbs free energy of solvation in 1-octanol is rather increased from C60 (128.7kJ·mol-1) to C82 (172.4kJ·mol-1), so that the 1-octanol/water partition coefficient Po is smaller for C60 ( log P o = 19.9 ) than for C70 and for C82 (26.6).
Table 5. Solvation descriptors for fullerene-60, -70 and -82.
Table 5. Solvation descriptors for fullerene-60, -70 and -82.
FullereneΔGs,waΔGs,obΔGs,chcΔGs,cfdlog Po
(SCAP)e
log Po
(CDHI)f
log Pch
(SCAP)g
log Pchhlog Pcf
(SCAP)i
log Pcfh
fullerene-60-15.60-128.7-78.48-103.219.913.811.111.615.421.0
fullerene-70-18.09-148.4 -91.20-119.922.915.812.813.617.924.4
fullerene-82-20.86-172.4-106.2-139.226.617.615.016.120.828.6
a Gibbs free energy of solvation in water (kJ·mol-1).b Gibbs free energy of solvation in 1-octanol (kJ·mol-1).c Gibbs free energy of solvation in cyclohexane (kJ·mol-1).d Gibbs free energy of solvation in chloroform (kJ·mol-1).e Po is the 1-octanol/water partition coefficient.f CDHI: calculations carried out with a method developed by Kantola et al.g Pch is the cyclohexane/water partition coefficient.h Calculations carried out with a method developed by Leo et al.i Pcf is the chloroform/water partition coefficient.
The log P o results can be compared with data obtained with a method developed by Kantola et al. [78]. This method allows one to obtain conformationally dependent hydrophobic indices based on atomic contributions. Kantola et al. fitted the following expression for the 1-octanol/water partition coefficient:
log P = i α i ( N ) S i + β i ( N ) S i ( Δ q i ) 2 + γ i ( N ) Δ q i
where Si is the contribution of atom i to the molecular surface area; α, β and γ are fitting parameters dependent only on the atomic number of atom i; and Δq represents the atomic net charges [79], which has been computed with algorithm POLAR [80]. The comparison between SCAP and the method of Kantola et al. has special interest. The latter assigns a set of parameters for each atom, depending only on its atomic number and not on the surrounding atoms in the molecule. Instead, SCAP also takes into account the functional group to which each atom belongs in the molecule. A computer program, called CDHI, which uses the method of Kantola et al., has been written [81]. The contribution of each atom to the molecular surface area is calculated with TOPO. The calculated log P o values do not agree exactly with the CDHI values. However, both the trend (C82 > C70 > C60) and the basic information that one obtains from the results is the same. In fact, all the three fullerenes will remain in the organic solvent, as one would expect.
SCAP has allowed calculating other organic solvent/water partition coefficients [40,41,42,43,44,45] for cyclohehexane (ch) and chloroform (cf). L o g P c h and log P c f show the same trend and log P o . In particular, the results, especially for cyclohexane/water, compare better than log P o with calculations carried out with a method proposed by Leo et al. [82].

Conclusions

In this article, C82 has been selected as a model cavity in order to test eight different methods for characterizing inclusion molecules. From these results and previous work on C60 and C70, the following conclusions can be drawn.
  • The use of SURMO2, which do not recognize the cavities, may not be convenient for intercalation compounds, since this method gives an estimate of the global fraction of occupied space within the volume, not allowing distinction among different niches [83].
  • The programs that do recognize the cavities (SCAP, MS, GEPOL, CSAM and TOPO) never exceed 1% relative deviation in molecular volume and 5% in surface area, in spite of the sometimes relevant differences in method of calculation. This means that the choice should rely mainly on other possibilities offered by the different methods such as computational performance, possibility of fragment analysis, etc. GEPOL shows, in general, the best results and the outcome is remarkably good for the entire surface areas and surface-related properties. The combined (GEPOL/TOPO) method shows relative errors below 0.1% for all of the descriptors in C60, C70 and C82.
  • The molecular volume and surface area were measured accurately with MCVS and UMCVS. The UMCVS measures the molecular volume and surface areas with high precision, so that the standard deviation is divided by 10 each time the number of points (and the CPU time) is multiplied by 100. This may be compared with the CSAM and TOPO methods, using a three-dimensional grid, for which a multiplication of the CPU time by about 1 000 is required to divide the error by 10.
  • The URNG in UMCVS provides the fastest convergence for the algorithm and a better estimate of the standard deviations. The use of a pseudorandom generator in MCVS produces a bias in the calculated properties and underestimates their standard deviations and errors, so that it cannot be recommended for high-precision predictions or as a benchmark maker.
  • In C60, C70 and C82 the effect of including the internal cavity surface in the calculation of the solvent-accessible surfaces is small but, evidently, if the cavity were much bigger (e.g. fullerene-240, -540, and -960) this effect would be much greater because the solvent molecule would then have room inside; at present, this effect can be corrected with an additional calculation carried out by use of a method that does not recognize the cavity.

Acknowledgments

Acknowledgement: The author acknowledges financial support of the Dirección General de Enseñanza Superior of the Spanish MEC (Project PB97-1383).

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  • Sample availability: Not available.

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