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Inclusion complexes of β-cyclodextrine with organicligands: molecular dynamics simulation of thethermodynamic stability in gas phase and in watersolutionA. V. Odinokov a , S. V. Titov b , V. A. Tikhomirov a , M. V. Basilevsky a & M. V. Alfimov aa The Photochemistry Center of the Russian Academy of Sciences, 7A, Bld. 1, NovatorovStreet, Moscow, 119421, Russiab Karpov Institute of Physical Chemistry, 3-1/12, Bld. 6, Obuha Side Street, Moscow, 05064,RussiaPublished online: 21 Jan 2013.

To cite this article: A. V. Odinokov , S. V. Titov , V. A. Tikhomirov , M. V. Basilevsky & M. V. Alfimov (2013): Inclusioncomplexes of β-cyclodextrine with organic ligands: molecular dynamics simulation of the thermodynamic stability in gasphase and in water solution, Molecular Simulation, 39:6, 442-452

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Inclusion complexes of b-cyclodextrine with organic ligands: molecular dynamics simulation ofthe thermodynamic stability in gas phase and in water solution

A.V. Odinokova*, S.V. Titovb, V.A. Tikhomirova, M.V. Basilevskya and M.V. Alfimova

aThe Photochemistry Center of the Russian Academy of Sciences, 7A, Bld. 1, Novatorov Street, Moscow 119421, Russia; bKarpov Instituteof Physical Chemistry, 3-1/12, Bld. 6, Obuha Side Street, Moscow 05064, Russia

(Received 26 March 2012; final version received 13 October 2012)

The double decoupling version of the thermodynamic integration procedure is applied to perform the molecular dynamicalmodelling of binding free energies DGbind of b-cyclodextrine (CD) with a number of organic ligands. Simulations for watersolutions show a satisfactory agreement (within 1–2 kcal/mol) with the experimentally measured equilibrium bindingconstants. The DGbind values are also reported for the gas phase complexation of the same ligands, although no experimentaldata are available for such systems. These gas phase computations have revealed the large stabilisation effect for the CDcomplexes of ionic ligands. Only in this special case the attempt of a qualitative rationalising the obtained simulation dataproved to be fairly successful. The problems specific for simulations for ionic ligands in water solution are discussed.

Keywords: cyclodextrine; inclusion complexes; binding affinity; molecular dynamics

1. Introduction

Molecular level computations of supramolecular complexes

are the developing branch of research in the recent

computational chemistry. They are capable to provide the

direct evaluation of the complex stability based on the

analysis of complicated mechanisms of intermolecular

interactions in flexible and versatile supramolecular struc-

tures. The binding equilibrium constantKbind, the measure of

the stability of a complex (often referred to as the ‘binding

affinity’ [1], commonly appears as a result of the molecular-

dynamical (MD) simulation for the equilibrium system

A þ B $ AB; ð1Þ

where AB means the associated complex, A and B being its

ingredients. In this study, we report the results of the MD

modelling for the Gibbs free energy change,

DGbind ¼ 2RT lnKbind; ð2Þ

(where R denotes the gas constant and T is the temperature)

derived by means of the thermodynamic integration (TI)

procedure. The complexes formed by b-cyclodextrine (CD)

with several organic ligands are considered in terms of

Scheme 1 where reactant A represents the CD and B is the

ligand. The methodology of such computations is described

[2,3] and its complete realisation is now available for a given

force field. Approximate computational algorithms have also

been devised and tested [4–7]. In particular, the stability of

several CD inclusion complexes in water solution was

simulated within the M2 algorithm [4,8], in which several

most stable configurations of the associate AB were

explicitly included in statistical averaging in terms of

Scheme 1, whereas the impact of the solvent (water) was

tested at the implicit level of a continuum solvation approach.

The computations reported below apply the full TI technique

including the explicit water solvent for the simulation of the

binding free energy 2 with several original details described

in Sections 2 and 3.

Many equilibrium systems 1 for CD in water have been

investigated experimentally [9], which provides the

necessary background for the validation of our simulation

results reported in Section 4. The concluding part of this

study (Section 5) is the attempt designed for the

interpretation of those numerical data and formulated in

terms of conventional models of intermolecular inter-

actions usually applied in theoretical modelling of

solvation phenomena.

An understandable motivation underlying such sort of

computational research is to cover the larger diversity of

binding effects. Unfortunately, the available list of

experimental binding affinities for CD inclusion complexes

discovers a surprisingly narrow range of variation [1,9]; it

does not even reveal any preference for several classes of

chemical structures. This is why our choice of guest ligands

looks rather arbitrarily. A number of aromatic ligands of

different size and several adamantane derivatives were

considered. The rigid structures of such molecules facili-

tated their investigation. The attempt of treating ionic

ligands provided a possibility to arrive beyond the limited

range of usual experimental material at the expense of

getting a risk to encounter uncommon methodological

q 2013 Taylor & Francis

*Corresponding author. Email: ale.odinokov@gmail.com

Molecular Simulation, 2013

Vol. 39, No. 6, 442–452, http://dx.doi.org/10.1080/08927022.2012.740636

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problems. Few other smaller ligand structures were also

considered for the sake of comparison.

The situation seems to be more favourable when gas

phase associates of CD are considered; here the variation of

computed binding effects becomes more remarkable and

systematic. However, there exists no experimental bench-

mark to validate this computational protocol. The strategy of

this study was to verify, as a first step, our computational

scheme based on a number of experimental examples which

are provided by measurements of association constants in

water solution for ligands of various structures. As the

simulation results looked rather optimistic, the computations

of the same gas phase associates within the same metho-

dology may be considered as reliable predictions. The

knowledge about the in vacuo stability of supramolecular

complexes is quite desirable for many applications, and, in

the absence of experimental evidence, the computer simula-

tions can serve as a valuable source of available information.

2. The algorithmic scheme

The value DGbind for the standard Gibbs free energy

change, accompanying the direct association process 1,

can be evaluated by means of the TI technique which

involves double decoupling of the ligand [2,11,12,18]. The

extra harmonic potential UðrÞ is imposed upon the

intermolecular distance r between the centres of mass of

the CD (the host) and the ligand (the guest) inserted in the

host’s cavity [2]. The present realisation of the TI included

the gradual variation of the parameters of the inter-

molecular potentials (both Coulomb and Lennard–Jones

(LJ) ones) between A and B in Equation (1) and also

between the ligand B and solvent particles. In addition, the

force constant lk, inherent to the restraining potential

UðrÞ ¼ 1=2lkr 2, was varied during the decoupling

process. Altogether, the value of the TI parameter l

changed between l ¼ 0 (when all intermolecular inter-

actions are working in full strength and UðrÞ ¼ 0) and

l ¼ 1 (when the ligand is withdrawn from the CD cavity,

being virtually transferred to the empty vacuum space, and

UðrÞ attains its maximum value).

The thermodynamic cycle performing the equilibration

process 1 can be formulated as the scheme displayed in

Figure 1. The subscript ‘sol’ indicates the presence of water

environment, while subscript ‘vac’ denotes the vacuum

state of the corresponding substrate. The total process of

complex formation is decomposed into several steps. The

free energy DG8(LW) (ligand/water) is the net free energy

outcome of the ligand extraction from water to vacuum,

when all interactions between ligand and water molecules

are gradually turning off. Actually, this quantity is opposite

to the hydration free energy: DG8ðLWÞ ¼ 2DGhydr. The

free energy difference DG8(LCW) (ligand/cyclodextrine/

water) corresponds to the annihilation of all intermolecular

interactions with ligand molecule, whereas restraining

potential UðrÞ is turning on. The last step describes the

change of the restrained state of the ligand to the ideal gas

state of standard concentration. The free energy difference

in this case can be calculated analytically [2]:

c ¼ 2RT ln C 0 2pRT

k

� �3=2" #

; ð3Þ

where the force constant k corresponds to l ¼ 1. The

standard concentration unit C 0 ¼ 1 mol/l is used in

Equation (3). For brevity, the quantity DG8ðLCWÞ2 c

will be denoted as DGðLCWÞ.

3. The computational procedure

The Optimised Potentials for Liquid Simulations, all-atom

version (OPLS AA) force field [13–15] was implemented,

combined with TIP4P water model [16], which is

compatible with the OPLS AA. The force constant of the

restraining potential UðrÞ was chosen as k ¼ 200 kJ/

(mol nm2). The pertaining correction value 3 equals to

c ¼ 2:566 kcal/mol. Computations at 298 K were carried

out with the isobaric–isothermal ensemble and periodic

Asol+Bsol

Asol+Bvac

Avac+Bvac (AB)vac

Asol+Bvac+U(r)

Avac+Bvac+U(r)

(AB)sol

∆Gbind

∆Gbind

∆G° (L

W)

∆G

° (LCW

)

∆G° (L

CW

)

C

C

Figure 1. Realisation of reaction 1 as a thermodynamic cyclefor aqueous media (top) and vacuum (bottom). The standardbinding free energy can be expressed as DGbind ¼ DG8ðLWÞ2DG8ðLCWÞ2 c for water and DGbind ¼ 2DG8ðLCWÞ2 c forvacuum.

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boundary conditions. The GROMACS MD package [17]

was implemented throughout. The MD cubic cell

contained a single particle of the CD–ligand complex

and 873 water molecules. The double decoupling version

of the TI eliminated gradually the intermolecular Coulomb

and Lennard–Jones interactions between the ligand and its

environment. The integration was performed using 100

points of the TI parameter l with the constant step 0.01. In

routine computations, the stabilisation period during 50 ps

was accepted for every value of l and the modelling was

continued for the next 200 ps. To avoid singularities in the

non-bonded interactions, a soft-core modification of the

Lennard–Jones potentials was used.

The above-listed duration of the MD modelling within a

single TI window (200 ps) was established as the

appropriate trajectory length in special tests with several

ligands. The further increase of such time period did not

affect the ultimate TI results. This piece of trajectory serves

as a measure for the array of spatial configurations which

surround the stable conformation of the CD–ligand

complex, and it can be considered as valuable alternative

to a straightforward configurational constraint convention-

ally imposed according to the recommendations in the

literature [2,3]. At the initial TI step, the size and the shape

of this space region can be outlined based on a trajectory

sampled at l ¼ 0. In test computations for several ligands

with l ¼ 0, the trajectory length was deliberately increased

up to 1 – 2 ns, and even in these tests separated

configurations of complex fragments were never observed.

The smooth profile of the mean force kdH=dll was always

obtained with short steps 1022 along the total TI interval.

We observed no significant hysteresis (the area of the loop

was less than 0.5 kcal/mol) when the direction of the TI

variable l was inverted. Special care was taken for the

convergence of ion computations. Here, two different

procedures for annihilating the interactions were tried: in

the first version, the sequential elimination of the

electrostatic potential (decharging) and next of the LJ

interaction was made. The second version made these two

steps simultaneously as described above. The complex

formation for 1-adamantyl carboxylate ion (see below) and

the hydration of tret-butyl ammonium cation were tested.

The free energy misfit between these two procedures did not

exceed 1 kcal/mol.

These results provided another evidence for the

consistency of the described procedure and assured its

convergence. The general arguments given above are valid

for the case of ‘strong binding’ complexes and it seems,

based on our test computations, that the systems considered

(2DG(LCW) . 3 RT, see Table 1) indeed fall into this

category.

The restraining potential was conformed to the spatial

distributions of complex configurations extracted from the

l ¼ 0 trajectories, as recommended in the literature [10].

The distribution of distances u between centres of mass of

the CD and a ligand was taken as a benchmark. For several

test ligands, this distribution comprised narrow peaks with

their maxima positioned close to u ¼ 0 and the widths of

similar magnitude. Thereby, the minimum of the

restraining potential was set at u ¼ 0, whereas the force

strength, listed above as k ¼ 200 kJ/(mol nm2), corre-

sponded to the average width of the u-distributions.

Table 1. Standard free energies (kcal/mol) of dehydration and complex formation with CD for several ligands.

DG8ðLWÞ DGbind DGbindðLCÞ

Ligand Calc. Exp.a Calc. Exp.b Calc.c

Benzene 0.1 0.9 21.6 22.8, 23.0 27.8Naphthalene 1.2 2.4 23.1 23.9 211.6Anthracene 2.2 4.2 23.8 24.5 214.6Adamantane 21.0 – 24.8 – 29.4Amantadine 4.7 4.7 26.4 – 210.8Methylamine 3.3 4.6 21.3 – 23.2Pyrrole 3.5 4.8 20.8 – 27.2Pyrene 3.2 – 22.9 22.2, 23.7 215.1Resorcinol 8.9 – 22.1 22.8e 214.61-Adamantyl ammonium 65.0 – 24.1 25.3 238.4Tert-butyl ammonium 63.2 65–67 – – 243.01-Adamantyl carboxylate 85.3 – 21.6 25.0 252.61-Adamantyl carboxylated 80.7 – 26.1 25.0 252.6Acetate 85.4 77–80 – – 249.5Acetated 80.1 77–80 – – 249.5

a Experimental values of dehydration free energies [24–27].b Experimental values of free energies for the complex formation in water for uncharged ligands [9,28–30] and for ions [24,25,53].c Calculated free energies of the complex formation in the gas phase.d Computations with the modified LJ diameter s0

OO ¼ 3.11 A (see the text).e T ¼ 303 K.

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Special attention was paid for treating electrostatic

interactions in the case of charged ionic ligands. The

conventional reaction field (RF) procedure [17,19] was used

at the stage of the trajectory computation. According to the

RF, the continuum solvent approach was applied beyond the

cut-off radius Rcut ¼ 14 A. The application of this algorithm

instead of the alternative periodic particle mesh Ewald

technique [17] saves a lot of computational effort. Such

simplified treatment of the electrostatic forces was further

refined in the course of the TI and at the final stage of the

Coulomb free energy computation. Each solute atom was

surrounded by a Born sphere with the radius R ¼ Rcut, and

the unified solute cavity was constructed as a collection of

overlapping atomic spheres according to the polarisable

continuum model (PCM, [20]) prescription. Inside this

cavity, all solute–solvent interactions were explicitly

treated, whereas the impact of the external solvent was

worked up at the continuum level, following the PCM. This

methodology has been elaborated earlier (MD/PCM [21]).

Its application for ionic ligands improved significantly the

straightforward RF results. No advantage was revealed with

uncharged ligands, as compared to the RF treatment.

The following modifications in the standard compu-

tational scheme for electrostatic interactions thereby

appeared. After computing the trajectory in a conventional

RF way, the modified calculation of the long-range part of

the electrostatic energy was carried out. The single non-

spherical cut-off cavity was assembled around the solute,

based on atomic Born-like spheres with the common radius

Rcut ¼ 14 A. The free energy fraction for the interaction of

solute atomic charges with the external solvent polarisation

was calculated by the PCM technique [20]. The term ‘Born

contribution’ will be applied henceforth for this piece of the

electrostatic solvation energy.

The enthalpy part (DH) of free energy differences (see

Appendix) was calculated as a difference of the internal

energy between the two states [7,22,23]. In the process of

complex formation, the number of particles varies and the

correction term 2pv must be added to obtain true enthalpy

value, where p is pressure and v is molar volume. All

computations were carried out at standard conditions, so

this term was supposed to be 20.59 kcal/mol. After the

enthalpy calculation, the entropic part (TDS) of the free

energy was obtained as TDS ¼ DH 2 DG.

4. The results

4.1 The binding affinities

The contents of Table 1 are exposed with the notation

introduced in Section 2. Calculated and experimental

values of hydration energies (with relevant signs) are listed

in columns 2 and 3. Rigid ligands were only selected in

order to avoid the multiplicity of conformations. The

binding energies, both calculated and experimental

energies, for CD þ ligand complexes in water solution

appear in columns 4 and 5. Column 6 contains binding

energies for the gas phase. They are actually treated as the

vacuum ones and denoted as DG8ðLCÞ (ligand/cyclodex-

trine). The comparison with experimental data is

unavailable for this case.

Such a comparison for water solution provides the basic

test for the validation of the computational algorithm. The

agreement within 1–2 kcal/mol, which is observed, we

consider as satisfactory. The similar level of accuracy has

been reported in the recent systematic studies on hydration

energies for uncharged organic ligands, which were based

on the thermodynamic perturbation (TP) techniques

[27,31,32]. The misfit, not exceeding 2 kcal/mol, appears

to impose the natural bound for both TI and TP metho-

dologies. Closer agreement with the experiment can be

gained at the expense of a special force field calibration for a

limited family of ligands.

In our computations, a single ligand, the 1-adamantyl

carboxylate anion (structure I), shows a significant deviation

beyond this boundary and thus deserves a special discussion.

For making a definite diagnosis, the consideration of its

hydration step could be helpful; unfortunately, there is no

hydration experiment for this ligand. Only implicit

comparison can be made based on the hydration behaviour

of the acetate ion, which reveals indeed the big disagreement

(7 kcal/mol) between the theory and the experiment. The

direction of the spurious hydration energy shift correlates

with the discrepancy found in the binding affinity.

Similar anomalies are known about the hydration of

other anions [33–37]. (The comprehensive reference list,

covering the interpretation of hydration free energies for

cations and anions in terms of their Born radii, can be found

in Refs [35,36].) The necessary correction can be made by a

minor modification of the force field. Indeed, after changing

a single parameter, we were able to get a satisfactory

agreement between the calculations and experimental data

for both the binding free energy of I and the hydration free

energy of the acetate anion, imitating the structure I in the

hydration experiment. The LJ cross diameter

sOO1¼ 3:06 A (in the OPLS AA prescription) for the

interaction between carboxyl oxygen O of I and the water

oxygen O1 was substituted by s0OO1

¼ 3:11 A, specially

fitted in order to improve the calculation of the hydration

energy for the acetate ion. Simultaneously, this modifi-

cation improved the binding affinity result for the CD þ I

associate.

4.2 Observed structures of CD complexes

The spatial configurations of CD complexes with electri-

cally neutral ligands are similar both in the gas phase and in

the water solution. The ligand is positioned close to the

centre of the CD cavity. In the gas phase, the advantage of

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the central position results from the maximisation of the

total amount of pairwise interactions between the ligand

atoms and their environment in the cavity. The same

arrangement is also natural in the presence of bulk water

because of the hydrophobic properties of the CD cavity.

The ligand orientation is a secondary effect determined

by steric restrictions; so there is no definitely preferable

orientation for symmetric ligands like benzene and

adamantane, whereas the spatially extended molecular

structures of naphthalene and anthracene are arranged

along the central axis of the CD cavity.

The combination of hydrophobic and hydrophilic

interactions determines the configurations of ionic

complexes in water. The adamantyl residues of structures

I and II remain immersed in the CD cavity, but their

charged fragments are directed outside, in order to

facilitate the formation of hydrogen bonds with water

molecules. In the gas phase, it is seen in Figure 3 that the

ligand structures are shifted towards the exit of the CD

cavity. Such conformations, with ionic groups appearing

outside, are stabilised owing to their interaction with the

external hydroxyles and the nucleophilic oxygens which

belong to the CD glucose residues and serve as stabilising

centres. So, in Figure 3(a) the carboxylate group of the

anion I forms H-bonds with the nearest external CD

hydroxyles, whereas the ammonium centre of the cation II

in Figure 3(b) is involved in specific binding with the

nearest external oxygen-containing groups of the CD

structure.

Strong reorganisation of the CD basket skeleton

accompanies its association with ionic ligands in the gas

phase. With anion I inside, its conic structure is distorted

as a result of H-bonding. In the case of cation II, the

rotation of monomeric glucose fragments of the CD

provides the necessary contacts of their oxygen atoms with

the charged ammonium group of the ligand. In such a way,

the equilibrated low energy structures of the complex are

organised.

4.3 Free energy computations for ionic ligands: themethodological comments

The consideration of ionic ligands was deliberately

included in this study as an attempt to bring more

diversity in the list of binding affinities found for the CD

complexes with uncharged ligand components. This step,

associated with the appearance of strong electrostatic

effects, is also accompanied with methodological pro-

blems which remain not fully resolved until now and are

briefly discussed below.

The absolute free energy scale for the hydration of

electrically charged molecular objects cannot be estab-

lished based on the direct experimental evidence. As a

result, a significant scattering of experimental hydration

free energies for ions is observed in the recent literature

(the surveys in key Refs [24,25,53,54] can be rec-

ommended as an introduction to the problem). A purely

theoretical solution of this problem at a molecular level is

hindered by several drawbacks inherent to the existing

simulation techniques. They are mainly associated with

the widely applied and commonly accepted treatment of

long-range Coulomb interactions in terms of the lattice

summation algorithm (the Ewald method). There are two

significant corrections required when the periodic

boundary conditions are introduced in a simulation

combined with the Ewald algorithm [38–40]. First of

them accounts for the removal of the ion–ion self-

potential arising due to Coulomb interactions between the

original ion and its periodic replicas. This well-known

correction is readily evaluated [40–42]. Another one

appears due to the possible presence of the interface

dividing the explicit bulk solvent (i.e. water) and the gas

phase volume. It has been revealed in numerical

simulations as the stepwise change of the potential and

its electrostatic component arising when an ionic solute

penetrates through the interface surface [41,55–57]. Such

effect is missing in the Ewald schemes [38–40] and it,

therefore, must be added as a free energy correction (’the

surface correction’), which amounts as much as 10–

15 kcal/mol [38,39,56] and is proportional to the ion

charge (i.e. it has opposite signs for cations and anions

keeping unchanged the absolute value).

The large corrections described above are specific for

the Ewald algorithm but do not appear in the alternative

simulation schemes which invoke true Coulomb inter-

actions complemented by a consistent truncation

approaches for treating long-range effects [38–40].

Being just of this sort, our present methodology avoids a

necessity of introducing any extra corrections (see Section

3). The only problem still remaining is the lack of carefully

elaborated force field adapted for simulations of

polyatomic organic ions in solution.

In the forthcoming analysis, we use a tentative

standard OPLS AA parameterisation [13] applied for the

computations reported in Table 1. As a supporting notion

for this we can invoke the general argument [38–42] that

the non-trivial electrostatic stepwise effects arising at the

bulk water–gas phase boundary depend linearly on the ion

charge and hence they are the same for all cations and all

anions. The possible errors, which could arise in their

estimates due to the improper force field parameters, are

therefore cancelled provided the net free energy change

represents the complexation process under study which

proceeds entirely in the bulk water phase. For every

particular ligand, either a cation or an anion, the errors

appearing at the stage of treating hydration (LW) free

energies are expected to disappear in the ultimate binding

affinity value.

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4.4 Free energy partitioning for ionic ligands

The inspection of hydration energies (the LW step) of small

organic ions, which are included in Table 1, reveals a

significant discrepancy between the computed and

experimental data. These ions were considered as structural

analogues of adamantane ions (Figure 2) for which experi-

mental data on hydration are available. Such observation

reflects and illustrates the general problem of estimating the

absolute free energy scale for ionic hydration energies as

discussed in Section 3. It is not directly connected to the

factors considered usually in solvation theories. The

inferiority of the standard OPLS AA parameterisation

fails to ensure the transferability of parameters for different

kinds of solutes in the water solution.

Fortunately, this flaw is specific for water solutions.

Being additive, it makes little effect on the processes

proceeding entirely in the water phase. This is why we

expect that it is cancelled in our computations of CD

binding energies considered as differences of LW and LCW

free energies.

Based on the above arguments, the properties of ionic

ligands I and II were investigated in more detail. It is

expedient to slightly rearrange the schemes from Figure 1 in

order to reduce the binding energy data to a single bench-

mark system, in which the ligand is transferred from a given

bound state to the vacuum (the gas phase). Three types of

bound states then appear in our computations, namely, the

water solution (LW), the CD complex in water solution

(LCW) and the CD complex in vacuum (LC). The corres-

ponding transfer free energies are listed in three respective

columns of Table 2. For ligands I and II, the electrostatic

interactions dominate in their energetic trends, which opens

the possibility for a sensible qualitative analysis.

For three types of systems (LW, LCW and LC) the

recalculated, in this way, relevant values of free energy

changes (DG) are listed together with their electrostatic

(DGel) and non-electrostatic (DGnel) components.

Obviously, DG ¼ DGel þ DGnel. In LW (hydration)

systems, the non-electrostatic fraction is additionally

subdivided into Lennard-Jones energy (DGLJ) and the

cavitation energy (DGcav):

DGnelðLWÞ ¼ DGLJ þ DGcav: ð4Þ

The first term is the collection of pairwise van der Waals

interactions of the ligand (it is negative), the second term

represents the free energy excess required for a preparation

of the excluded volume cavity in the water solvent (it is

positive). Its estimates in Table 2 follow from the relation

DGcav ¼ jV þ hxns [43], where the non-spherical index

xns ¼ 1 2 RV=RS is defined via the solute radii determined

in terms of the volume V and surface area S of the solute

cavity of the excluded volume, i.e. RV ¼ ð3V=4pÞ1=3 and

RS ¼ ðS=4pÞ1=2. The parameters j and h were fitted

empirically in order to reproduce the cavitation energies in

water computed by the Monte Carlo technique for a

diverse number of organic molecules with different sizes

and shapes [43]. For LW and LCW systems, we also

separate the Born fraction of the electrostatic free energy

(see Section 3), which originates from the polarisation of

water bulk outside the PCM cavity and constitutes

approximately 210 kcal/mol. In Table 2, the pertaining

electrostatic free energies are doubled as ( . . . )/( . . . ) where

the left-hand part counts the explicit electrostatic

interactions within the PCM cavity surrounding the solute

particle as described in Section 3. The Born contribution is

added to it in the right-hand part, denoting the total

electrostatic free energy. The magnitude of the cut-off

radius Rcut ¼ 14 A warrants that the CD particle is always

confined inside the cut-off cavity. As a result, the Born

energy fraction is the same for LW and LCW systems (the

external bulk region is filled by water) but it does not exist

in the LC system (the external region is empty).

Table 2. Free energies (kcal/mol) of hydration and complexformation with the CD and their components for 1-adamantylcarboxylate and 1-adamantyl ammonium ions.a

DGhydr DGðLCWÞ DG8ðLCÞ

1-Adamantylammonium (II)DGel 256.3/266.4 254.0/264.1 226.4DGnel 1.4 25.0 212.0DGLJ 217.6b

DGcav 19.0b

Total 265.0 269.1 238.41-Adamantyl carboxylate (I)DGel 274.9/285.3 272.3/282.6 242.7DGnel 0 23.6 210.0DGLJ 221b

DGcav 21b

Total 285.3 286.2 252.7

a See the notation for free energies in Figure 1.b Cavitation energies based on the methodology using molecular volumeswith a non-spherical correction, as defined in Section 6 of Ref. [43]; theLJ contributions were found as differences according to Equation 4.

Figure 2. Structures of ionic ligands.

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Figure 4 illustrates the TI process for three electrostatic

free energies DGelðLWÞ, DGelðLCWÞ and DGelðLCÞ. The

variable TI parameter l (0 # l # 1) coincides in the

present case with the absolute magnitude q of the ionic

electric charge (performing the charging process of a

ligand). The LJ parameters are kept fixed in this procedure,

and there is no need of invoking the auxiliary restraining

potential UðrÞ with the pertaining correction 3. The mean

force 2k›H=›ll; where H denotes the system interaction

Hamiltonian, is plotted as a function of l ¼ q along the

ordinate axis. For water solutions of a ligand L the

observed plots are close to linearity. The strong nonlinear

effects promoted by the presence of the CD appear in

LCW and LC systems for the anion I (Figure 4(a)). This

nonlinearity points to a significant distortion of the flexible

CD structure during the electrostatic charging process. It

becomes clearly visible by the inspection of computer

models for the gas phase structures shown in Figure 3. The

nonlinear effect seems to be weakened for the cation II, as

can be inferred from Figure 4(b) Its average impact upon

the ultimate free energy value may be not large; in this

case the linearised plots of k›H=›ll with the effective

average slope can be considered. Such approach reveals

very similar effective slopes for LW and LCW systems

with the same ligands, thus explaining the similarity of the

corresponding free energy changes DGelðLWÞ and

DGelðLCWÞ. The small electrostatic contributions to

binding energies DGbindðLCWÞ of water complexes appear

as the ultimate result. Contrary to that, the effective slopes

reduce significantly for the vacuum LC systems.

5. Discussion

It is hardly expedient to look for a regularity in binding

energies DGbindðLCWÞ (the water solution). They all vary

within a narrow range (1–5 kcal/mol). Here the contri-

butions of different sorts of interaction are similar in

absolute magnitudes but alternate in signs, which results in

a mutual compensation of the individual free energy trends

and smearing the observed effects. The only definite

observation worth of notion is the correlation of

DGbindðLCWÞ values with the size of a ligand particle

provided ionic ligands are excluded. This trend is

conformed by computations of the small ligands (resorci-

nol, benzene, pyrrole and methylamine, see Table 1).

For ionic ligands I and II, some qualitative

consequences of the energy partitioning (Section 4.4)

can be considered. They follow from the linearised TI

treatment of the electrostatic free energies (see Figure 4).

Figure 3. (Colour online) Inclusion complexes with ionic ligands I (a) and II (b) in the gas phase.

0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.00

200

400

600

800

LC

LCW

LW

0

100

200

–·∂

H/∂

lÒ, k

cal/m

ol

–·∂

H/∂

lÒ, k

cal/m

ol

300

400

LC

LCW

LW

Figure 4. The illustration of the TI procedure in the LW, LCW and LC systems for the anion I (a) and cation II (b).

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For the electrostatic contribution, the linear approxi-

mation generates the effective Born free energy DGel ¼

21=2Bq2 with the effective slope

B ¼1

a1 2

1

1

� �; ð5Þ

where a and 1 denote the effective ion radius and the

effective dielectric permittivity, respectively. It is the

increase of B which promotes the seemingly anomalous

magnitude of the electrostatic DGelðLWÞ and DGelðLCWÞ

free energy changes for the anion I relative to the cation II.

The fact that the ionic hydration energies for anions are

systematically larger in magnitude, relative to their

cationic counterparts of the similar structure, is known

and repeatedly discussed [33–37]. The formal interpret-

ation, in terms of Equation (5), as a variation of the

effective Born radius a, conforms to the discussion in the

literature. This notion underlined our attempt of the

parameter modification when fitting in Section 4, the TI

computation for the anion I. More recent argumentation

[38–42] relates this discrepancy to the step of the

electrostatic mean force potential on the bulk water–air

interface which contributes to the solvation free energy

with opposite sign for cations and anions (see Section 3).

In Table 2, more definite conclusions can be extracted

by a comparison of total free energies of hydration

(DGðLWÞ) and formation of the vacuum complex

(DGðLCÞ). Their difference amounts to approximately

30 kcal/mol (32 and 27 kcal/mol for the anion and cation,

respectively). The similar electrostatic difference

DGelðLCÞ2 DGelðLWÞ increases up to 40 kcal/mol (42

and 40 kcal/mol, respectively). It could be concluded that

the disagreement between the total and electrostatic free

energies arises from the electrostatic Born contribution

(210 kcal/mol). The changes in the non-electrostatic free

energies DGnelðLCÞ2 DGnelðLWÞ are 10 and 13 kcal/mol,

respectively. This amount compensates, in a first

approximation, the loss of the Born energy in the vacuum

complex. The qualitative conjecture, which follows,

claims that the external electrostatic interactions of the

ligand, which are responsible for the large difference

between the total LC and LW formation free energies, are

confined within a limited space region. This space includes

the volume of the CD particle with its close external

neighbourhood and is filled by water in a LW system,

which is substituted by the material of the CD particle in

the vacuum complex. The corresponding large free energy

misfit, owing to its electrostatic origin, can be attributed to

the discrepancy of the effective Born factors (Equation

(5)) and, ultimately, to the different screening by the

effective dielectric constant 1, which is different (water is

substituted by the CD) in the two systems under

consideration.

We consider finally the non-electrostatic free energy

contributions, as defined by Equation (4). In the water

solution, the cavitation and the attractive LJ terms cancel

one another in a large extent [22,44–46]. The data of

Table 2 are in accord with this observation. By noting that

the cavitation term is absent in the vacuum LC system (the

cavity does already exist in the initial CD structure), one

could expect the energy gain of order of 20 kcal/mol for

both ions. The observed effect proves to be significantly

lower (10 and 12 kcal/mol). This can be understood from

the examination of the structure of ionic CD complexes

reported in Section 4.2. It has revealed that the charged

centres of the ligand molecule tend to escape the host’s

cavity preferring to form H-bonds with the external

hydroxyl groups belonging to the CD. By this means,

when the case of the vacuum complex is considered, the

number of van der Waals contacts formed by the ligand

with its environment is reduced, being followed by the

decrease in the attractive LJ interactions.

For ionic ligands, the comparison of LW and LC

systems has discovered the origin of the out of ordinary

stabilisation for their gas phase complexes with the CD.

The situation becomes different when LW and LCW

systems are compared. It is seen from Table 2 that the

LW ! LCW transition changes quite weakly the electro-

static free energy, as explained above based on the analysis

of the TI charging plots. The variation of the non-

electrostatic free energy component proves to be small as

well. The CD cavity is filled by water molecules when the

ligand is absent, so there is no background for changing

markedly the cavitation energy owing to the ligand

transfer from the water bulk. Also, the average number of

attractive van der Waals contacts shows no trend to

change, keeping almost constant the LJ component of the

DGnel. Moreover, as observed from Table 2, the small

variations of electrostatic and non-electrostatic ingredients

of the total DG have opposite signs and mutually

annihilate, leaving remarkably small the net effect in the

binding affinity.

6. Conclusion

The attempts to reveal at a physico-chemical level those

factors which are responsible for binding affinities of

organic ligands in the particular case of their complexation

with CD are seriously encumbered by the fact that the

corresponding free energy changes are small and vary

within a narrow range of magnitude [1–9]. Our computa-

tions agree with this notion for a number of typical

uncharged ligands. For ionic ligands a special problem

appears, because no satisfactory force fields do exist at

present for molecular simulations of ions in the water

solution [38–42]. It is not a simple task to establish the

origin of the free energy scale for ionic hydration. Our

investigation of the pertaining effects in this study was

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based on the OPLS AA parameterisation [13–15], which

cannot predict absolute values of ionic hydration energies.

This is not, however, necessary, provided only the binding

affinities in the water solution are considered. Because the

complexation process does not involve the transfer of ions

through the gas phase–bulk water interface, the true (gas

phase) origin of free energy changes does not contribute to

the ultimate binding effect.

At such level of accuracy, the numerical treatment and

analysis of electrostatic free energy components for ionic

ligands affords a guideline for understanding at a

qualitative level the trends and mechanisms controlling

the formation of their supramolecular complexes. We have

observed the strong stabilisation effect which

accompanied the formation of ionic inclusion complexes

of CD in the gas (vacuum) phase. Electrostatic

interactions, typically following the linear response

behaviour, were revealed as its main source. The similar

trend can be expected to retain for equilibrated solutions of

strongly charged ligands in non-polar or weakly polar

solvents. However, the majority of present experimental

measurements of supramolecular association phenomena

are carried out in the water solution. In a solvent of high

polarity, the large amount of desolvation energy is

required for an ionic guest-ligand to bring it inside the

body of the host component of the inclusion complex. This

penalty suppresses the binding affinity, as already noted by

several authors [50,51]. In the absence of strong

electrostatic interactions, the remaining combination of

interfering nonlinear effects of different nature produces

weak binding, with the results which are hardly

predictable without a rigorous computation at a molecular

level. Such a phenomenon is usually referred to as ‘the

compensation effect’ [1,4,8,22]. This term implies the

compensation of enthalpic and entropic contributions

having opposite signs, which accompanies the equilibrium

formation of the association complexes AB in Equation (1)

from the ingredients A and B. The total effect occurs as a

loss of motional freedom in the reaction system followed

by its conversion in the binding volume [1,2]. The specific

mechanisms of such stabilising compensation may be

quite diverse; the present discussion in Section 5 can serve

as an illustration. For the case of uncharged ligands in

water solution, the direct MD simulation, which supports

this conjecture, is reported in Appendix. Most remarkable

is that the compensation effect is mostly visible at the

preliminary (LW and LCW) stages when either a ligand or

ligand–CD complex is transferred in the water solution

from the gas phase. The minor free energy residues cancel

one another once again at the ultimate stage resulting in

the formation of the inclusion complex. In such a

complicated situation with a number of interfering factors,

invoking the argumentation of formal thermodynamics

[22,47–49] remains as a sole approach for the inter-

pretation of the observed results of experimental

measurements. The present argument serves as a

rationalising factor possibly explaining the extremely

low diversity of the combined stability effects observed in

the water solution of CD inclusion complexes. It follows

that only a reliable numerical investigation is apt for the

differentiation of stabilisation effects in the supramole-

cular compounds under consideration. The present

computational scheme seems to be properly adapted for

this purpose, provided an adequate force field is available.

This conjecture, which is rather pessimistic, seems to be

shared in the recent literature devoted to binding affinities

of complicated supramolecular aggregates [51,52].

Acknowledgements

The authors thank the Ministry of Education and Science ofRussian Federation (the state contract 02.523.11.3014) for thefinancial support and Russian Foundation of Basic Research(Project No. 08-03-00993) for partial financial support and accessto computational resources of Nvidia CUDA (Tesla C870 GPU)workstation of Karpov Institute of Physical Chemistry.

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Appendix: The entropy/enthalpy analysis of binding

effects

Some computations illustrating the partitioning of free energychanges into their enthalpic and entropic components are listed inTable A1 for several uncharged ligands. These data support theguess about mutual compensation of DH and TDS contributionsin the free energy change. Indeed, the absolute magnitudes oftheir individual values exceed significantly the total absolutevalue of DG. This effect is observed at both LCW and LW stages,

as combined in Figure 1 in the total complexation thermodyn-

amic cycle. The resulting DG values become quite small already

at these preliminary stages, and their further compensation

appears at the final stage of combination in the binding free

energy.On the other hand, the problem of a consistent molecular

simulation of entropies/enthalpies for the ion hydration case

seems to remain open nowadays [39,41,57,58]. We did not

include the attempts of such calculations in Table A1.

Table A1. The entropy/enthalpy partitioning of free energy changes involved in the complex formation between the CD and unchargedorganic ligands (all energy changes in kcal/mol).

Hydration Binding

DG DH TDS DG DH TDS

Benzene 20.1 26.5 26.4 21.6 0.6 2.2Naphthalene 21.2 25.4 24.2 23.1 28.1 25.0Anthracene 22.2 27.4 25.2 23.8 26.3 22.5Pyrrole 23.5 27.6 24.1 20.8 219.8 218.1Pyrene 23.2 211.0 27.8 22.9 216.6 212.5Resorcinol 28.9 216.9 28.0 22.1 0.0 2.1

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