theoretical study of the interaction between sodium ion and a cyclopeptidic tubular structure

Theoretical Study of the Interaction Between Sodium

Ion and a Cyclopeptidic Tubular Structure

FRANCESCO FERRANTE, GIANFRANCO LA MANNA

Dipartimento di Chimica Fisica ‘‘F. Accascina’’, Universita degli Studi di Palermo,Viale delle Scienze, 90128 Palermo, Italy

Received 31 October 2005; Revised 16 March 2006; Accepted 17 March 2006DOI 10.1002/jcc.20677

Published online 20 April 2007 in Wiley InterScience (www.interscience.wiley.com).

Abstract: DFT calculations have been carried out to describe the pathway of a sodium ion along the stacking

direction of a tubular structure set up by five cyclopeptidic units, which can be considered a suitable model of a

hollow tubular structure of indefinite length. A lattice of points inside the tubular structure is defined and the DFT

interaction energy values with a sodium ion are obtained. The data allow predicting a zigzag path of the ion inside

the hosting structure.

q 2007 Wiley Periodicals, Inc. J Comput Chem 28: 2085–2090, 2007

Key words: tubular structure; cyclopeptide; nanotube; macrocyclic system

Introduction

Several classes of organic tubular structures are known, which

are made up of self-assembling of macrocyclic units;1,2 among

these, the hollow tubular systems set up by stacking of cyclopep-

tides derived from alternating D- and L-aminoacids, where the

self-complementary of the NH��CO group allows for intermo-

lecular hydrogen bonding, can be used as suitable devices for

building intermembrane ionic channels with performances com-

parable to those of the natural counterparts.3 The side groups of

the aminoacids can be chosen so that the resulting tubular struc-

ture can be inserted inside a lipidic double layer4 or inside a

monolayer adsorbed on a support.5 Moreover, a large variety of

hybrid cyclopeptides self-assembling into tubular structures

has been designed with potential applications in several fields of

physics and chemistry.6

We attempt here to study, using accurate quantum–mechanical

methods in the framework of the computational study of cyclo-

peptidic tubular systems undertaken by our research group,7 the

interaction between a cyclopeptidic tubular system and a sodium

ion, to give some insight in the transport process of a cation

through a tubular system.

The importance of such a study relies on the possible under-

standing of the basic elementary processes regulating the con-

ductivity properties inside artificial devices mimicking natural

systems. An example of this kind of study is reported in the

case of the insertion of lithium ion into carbon nanotubes.8

In the first part of the article, the choice of a suitable tubular

structure is dealt with, whereas in the second section the interac-

tion energy values between the chosen system and Naþ ion are

reported and discussed.

Computational Details

The full geometry optimization of the tubular aggregates was

carried out at the density functional theory level, using the gen-

eralized gradient BP86 functional9 and a split valence plus polari-

zation basis set (SVP) whose contraction scheme is (7s4p1d/

4s1p)/[3s2p1d/2s1p]. To reduce the computational time, the reso-

lution of identity (RI-DFT) approximation10 was used. In the

RI scheme, the electron density is expressed in terms of an

auxiliary basis set, which results in a better scaling behavior of

the Coulombian integrals relative to the number of basis set

functions, without consistent loss in accuracy. Its reliability was

already tested in the literature11 and also by us in the case of

the geometry optimization of dimeric and trimeric aggregates of

some cyclopeptides (Ferrante, F.; La Manna, G. unpublished

data). The auxiliary basis set corresponds to the SVP basis; its

contraction scheme is (8s3p3d1f/4s2p)/[6s3p3d1f/2s1p]. The

same model was used to calculate the energy values of all the

671 systems generated by placing, one at a time, a sodium

ion in a point of the lattice built up inside the cavity of the

aggregate, as described in The Interaction Between the Tubular

Structure and the Naþ Ion. The SVP and auxiliary basis set

Contract/grant sponsor: MIUR, Universita di Palermo

Correspondence to: G. La Manna; e-mail: [email protected]

q 2007 Wiley Periodicals, Inc.

contraction schemes for sodium atom are (10s5p1d)/[4s2p1d]

and (12s4p4d1f)/[5s2p2d1f], respectively. All calculations were

performed by using the package TURBOMOLE, Version 5.6.12

Results and Discussion

Definition of the Tubular Structure

The aim is to define the geometry of a tubular structure (TuS),

having the properties of an ideal system set up by an indefinite

number of monomeric units. So, once the monomeric unit was

selected, we optimized the geometries of the polymeric struc-

tures obtained, by stacking an increasing number of monomeric

units; when the geometry of the inner part of the polymeric

structure was independent on further addition of monomers, that

geometry was considered as representative of the whole TuS.

Cyclo[(Gly)8] was chosen as the monomeric unit because

glycine is the simplest aminoacidic structure and it is possible to

make computations on relatively large polymers. The presence

of side groups, in case different aminoacids were considered,

should not affect the conclusions, as the side groups are outside

the region under study, and the distortions on the structure of the

backbone, as well as the effects on the interaction of the TuS

with the ion, should be negligible.

The optimized geometrical parameters of the monomer are

given in Table1. The bond length values are similar to those

reported for cyclo[(Gly-D-Ala)4],13 with the largest deviation of

0.04 A in the case of the C��C bond. Conversely, dihedral

angles values are rather different, with the backbone torsional

angle ! less planar than in cyclo[(Gly-D-Ala)4].The dimer and the trimer were built up by bringing together

the monomers along the axis perpendicular to the plane of the

rings according to the most stable ‘‘antiparallel 1’’ geometry, as

Table 1. Relevant Geometrical Parameters of the Optimized Structure of cyclo [(Gly)8].

Bond lenghts (A) Bond angles (8) Dihedral angles (8)

C��O 1.233 C��C��N 115.4 C��N��C��C (�) 6128.4

C��N 1.364 C��C��O 120.4 N��C��C��N ( ) 6156.3

N��H 1.030 N��C��O 124.1 C��C��N��C� (!) 6170.8

N��C� 1.451 C��N��C 122.4

C��C 1.542 C��N��H 113.5

C��N��H 120.1

Figure 1. (a) Linearized schemes of the three stacking modes of a dimeric cyclopeptidic structure.

The antiparallel 1 stacking mode was shown to be the most stable structure. (b) Linearized scheme of

the trimer according to the antiparallel 1 stacking mode.

2086 Ferrante and La Manna • Vol. 28, No. 13 • Journal of Computational Chemistry

Journal of Computational Chemistry DOI 10.1002/jcc

shown in Figure 1. The minimum energy geometry of the dimer

corresponding to the ‘‘antiparallel 1’’ stacking mode is more

stable than the parallel mode by about 30 kJ mol�1 and by about

50 kJ mol�1 with respect to the ‘‘antiparallel 2’’ stacking mode,

which is in agreement with experimental data.14 In the trimer,

shown in Figure 1b, the same ‘‘antiparallel 1’’ arrangement was

adopted for both AB and BA0 couples of monomers.

The N��H��O¼¼C hydrogen bond lengths obtained for the

optimized dimer (AB) and the trimer (ABA0) of cyclo[(Gly)8]are reported in Table2. In the case of the dimer, the hydrogen

bond lengths are all approximately identical, whereas in the

trimer an alternation of shorter and longer hydrogen bonds is

observed. This can be rationalized in terms of the rotation of the

amidic groups of the central unit in the trimer that causes

the carbonylic oxygens to be closer to the amidic hydrogens of

the upper unit and farther from those of the lower unit. The val-

ues of the hydrogen bond lengths are in agreement with those

obtained in DFT calculations on aggregates of linear chains of

polyglycine.15

Afterwards, the tetramer and the pentamer were similarly built

up by adopting the same ‘‘antiparallel 1’’ stacking mode; the

hydrogen bond lengths obtained for the pentamer B0ABA0B@,whose optimized structure is shown in Figure 2a, are reported in

Table 2. We noted that the hydrogen bond lengths involving the

three central units were all in the range 1.870–1.877 A, whereas

the values concerning the terminal units showed noticeable

differences, as previously observed in the trimer. We conclude

that, since the geometry of the central monomer of the pentamer

is essentially replicated in the two adjacent monomers, the

pentamer can be considered as a representative structure of a

tubular {cyclo[(Gly)8]}n of indefinite length. Hence, the inner

trimer can be a good descriptor of the whole tubular structure

and our analysis will be limited to points lying inside it.

Table 2. N��H��O¼¼C Hydrogen Bond Lengths, in A, for the Optimized

Aggregates {cyclo[(Gly)8]}2 (AB), {cyclo[(Gly)8]}3 (ABA0), and

{cyclo[(Gly)8]}5 (B0ABA0B0 0).

Dimer Trimer Pentamer

AB AB BA0 B0A AB BA0 A0B@

1.908 1.921 1.889 1.912 1.877 1.876 1.902

1.907 1.890 1.918 1.878 1.870 1.875 1.876

1.907 1.915 1.889 1.905 1.872 1.875 1.901

1.908 1.885 1.919 1.883 1.875 1.873 1.887

1.908 1.917 1.890 1.907 1.874 1.874 1.918

1.907 1.889 1.923 1.883 1.876 1.8701 1.886

1.908 1.915 1.892 1.910 1.873 1.874 1.911

1.907 1.886 1.918 1.881 1.877 1.870 1.881

Figure 2. (a) Optimized structure of the pentamer tubular structure. (b) Scheme of the points inside

the tubular structure where the sodium ion was placed and interaction energy values were computed.

2087Interaction Between Sodium Ion and a Cyclopeptidic Tubular Structure


It is to be outlined that, given the antiparallel arrangement,

the central monomeric unit of the pentamer is not a symmetry

plane.

The resulting main features of {cyclo[(Gly)8]}5 are the fol-

lowing: height, 20.1 A; interring distance, 4.8 A, very close to

the experimental values found in several cyclopeptide nano-

tubes;1,3,13 and diameter of the cavity (central monomer), 9.1 A.

The Interaction Between the Tubular Structure

and the Na+ Ion

The transport of ions inside the tubular structure (TuS), driven

by some external force, is strongly dependent on the relative

values of the inner energetic barriers. The interaction energy

between the chosen ion, Naþ, and the pentamer was calculated

for a lattice of points that was built inside the cavity of the pen-

tamer, taking as reference element the longitudinal axis of the

tubular structure, which is defined as the straight line passing

through the two points at intermediate distance from two carbonylic

oxygens at opposite sides of the second and of the fourth unit.

The distance between the second and the fourth monomeric

unit of the pentamer, 9.6 A, was divided into 10 identical parts

and each one of the resulting 11 points was the center of a circle

perpendicular to the axis. Moreover, every circular contour was

divided into a number of parts, so that each resulting point was

at about the same distance, ranging from 0.8 to 1.0 A, from the

adjacent considered points. This allowed a homogeneous distri-

bution of the points, where the interaction with the sodium ion

was evaluated. The distance between two adjacent circular surfa-

ces along the axis is 0.96 A; the radius of the inner circle was

chosen as 0.83 A (r1) and the radii of the other three concentric

circles were 1.66 (r2), 2.49 (r3), and 3.32 A (r4), respectively.

The total number of the considered points, lying on 11 parallel

planes, resulted 671. The set of the points considered is shown

in Figure 2b.

An analysis of the interaction energy values is reported here

for the most representative regions, namely the circular contours

corresponding to the third plane (p3), crossing the H-bonds

between the second and the third monomeric unit, those in the

sixth plane (p6), crossing the central monomeric unit of the pen-

tamer and those in the ninth plane (p9), crossing the H-bonds

between the third and fourth unit of the pentamer.

The interaction energy values obtained for the points lying

on the outermost circular contour are reported in Figure 3. An

approximate periodicity of �/2 is apparent, and so we can draw

our attention on angular values lower than �/2.Some attractive energy minima are present in the cases of

the p3 and p9 planes, but the general aspect shows that the

region corresponding to this outer contour is repulsive every-

where, except for some small intervals of the angular coordinate.

This indicates that the presence of the sodium ion at distances

from the axis of the TuS larger than 3.3 A can be ruled out.

When going inside the TuS, the interaction energy values

show all negative (attractive) values, as shown in Figure 4. In

the case of the distance 2.49 A from the axis (Fig. 4a), the sec-

tions p3 and p9 show the deepest minima, corresponding to

Figure 3. Interaction energy between the tubular structure and a

Naþ ion placed at the outermost circular contour (r4) as a function

of its angular coordinate for three representative planes (see text).

Figure 4. Interaction energy between the tubular structure and a

Naþ ion placed at the circular contours r3 (a) and r2 (b) as a function

of its angular coordinate for three representative planes (see text).



interactions between the ion and the oxygen atoms of the TuS,

whereas the minima found in the section p6 can be referred to

interactions with the nitrogens; in all cases, the maxima have

been found in correspondence to interactions with the carbon

atoms of the carbonylic groups. For the distance of 1.66 A from

the axis, all sections show attractive values, with a less signifi-

cant angular modulation, especially in the case of the section p6,

and the energy values are all within the range of 25 kJ mol�1.

In Figure 5 an analysis of the interaction energies as a func-

tion of the distance from the central axis of the pentamer, for a

fixed value of the angular parameter and different sections p3,

p6 and p9, is reported. The minima of interaction energy are

located in the range of distances of 2.2–2.5 A, with lower

distance values in the case of the central section p6.

A further analysis of the interaction can be done by examin-

ing the energy values along the longitudinal direction (stacking

direction). Figure 6 shows the interaction energy values as a

function of the longitudinal coordinate z for different values of

the distance from the axis 0, r1, r2, and r3, at the fixed 0 value

of the angular coordinate. We observe the presence of two

minima and an energy barrier, approximately corresponding to

the p6 plane, which is less pronounced by decreasing the

distance of the ion from the longitudinal axis of the TuS.

Similar trends are observed in the case of different values of

the angular coordinate.

Conclusion

The overall information obtained from the analysis of the inter-

action energy values between the sodium ion and the cyclope-

ptidic tubular structure allows hypothesizing, in the presence of

a driving force along the stacking direction, going from p3

Figure 5. Interaction energy as a function of the ion’s radial coordi-

nate, at 0 radians, for three representative planes (see text).

Figure 6. Interaction energy as a function of the ion’s longitudinal coordinate, at 0 radians, for differ-

ent values of the radial coordinate [0 (a); r1 (b); r2 (c); r3 (d), see text].

2089Interaction Between Sodium Ion and a Cyclopeptidic Tubular Structure


towards p9, a sort of zigzag pathway. In fact, after an initial

optimal interaction at a distance around 2.3 A from the longitu-

dinal axis, corresponding to the energy minimum found in corre-

spondence with the p3 plane, it is necessary the ion gets closer

to the axis so as to overcome the lowest energetic barrier, and

then goes along, away from the axis, for experimenting the larg-

est attractive energy (minimum energy distance about 2.4 A at

the p9 plane).

Moreover, the ion’s movement shows an angular modulation;

according to the observed interaction energy values, it is likely

to expect a decreasing of the angular coordinate, from initial

values of 0.8–1.2 rad (plane p3) to about 0.2 rad (plane p9).

The apparent asymmetry shown from the initial to the final

values of the radial and angular coordinates is due to the intrin-

sic asymmetry of the pentamer because of the antiparallel

arrangement adopted in the building of the TuS.

This result can be considered as a first static computational

approach in the study of the ionic conductivity inside a tubular

structure and could be extended by a molecular dynamics

simulation.

Acknowledgment

We thank the Computing Center of the University of Palermo

(C.U.C.) for technical support.

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theoretical study of the interaction between sodium ion and a cyclopeptidic tubular structure

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