Global optimization of cholic acid aggregatesBalázs Jójárt, Béla Viskolcz, Mihalj Poša, and Szilard N. Fejer

Citation: The Journal of Chemical Physics 140, 144302 (2014); doi: 10.1063/1.4869832 View online: http://dx.doi.org/10.1063/1.4869832 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/140/14?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Hydrogen-bonded aggregates in precise acid copolymers J. Chem. Phys. 140, 054902 (2014); 10.1063/1.4863326 Atomistic simulations of pH-dependent self-assembly of micelle and bilayer from fatty acids J. Chem. Phys. 137, 194902 (2012); 10.1063/1.4766313 Modeling of hyaluronic acid containing anti-cancer drugs-loaded polylactic-co-glycolic acid bioconjugates fortargeted delivery to cancer cells AIP Conf. Proc. 1482, 75 (2012); 10.1063/1.4757441 Aggregation in dilute aqueous tert-butyl alcohol solutions: Insights from large-scale simulations J. Chem. Phys. 137, 034509 (2012); 10.1063/1.4731248 A computational study of ultrafast acid dissociation and acid–base neutralization reactions. II. The relationshipbetween the coordination state of solvent molecules and concerted versus sequential acid dissociation J. Chem. Phys. 134, 094505 (2011); 10.1063/1.3554654

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THE JOURNAL OF CHEMICAL PHYSICS 140, 144302 (2014)

Global optimization of cholic acid aggregatesBalázs Jójárt,1 Béla Viskolcz,1 Mihalj Poša,2 and Szilard N. Fejer1,3,a)

1Department of Chemical Informatics, Faculty of Education, University of Szeged,Boldogasszony sgt. 6, H-6725, Szeged, Hungary2University of Novi Sad, Faculty of Medicine, Department of Pharmacy, Novi Sad, Serbia3Pro-Vitam Ltd., str. Muncitorilor nr. 16, 520032 Sfantu Gheorghe, Romania

(Received 17 December 2013; accepted 14 March 2014; published online 8 April 2014)

In spite of recent investigations into the potential pharmaceutical importance of bile acids as drugcarriers, the structure of bile acid aggregates is largely unknown. Here, we used global optimizationtechniques to find the lowest energy configurations for clusters composed between 2 and 10 cholatemolecules, and evaluated the relative stabilities of the global minima. We found that the energeti-cally most preferred geometries for small aggregates are in fact reverse micellar arrangements, andthe classical micellar behaviour (efficient burial of hydrophobic parts) is achieved only in systemscontaining more than five cholate units. Hydrogen bonding plays a very important part in keepingtogether the monomers, and among the size range considered, the most stable structure was foundto be the decamer, having 17 hydrogen bonds. Molecular dynamics simulations showed that the de-camer has the lowest dissociation propensity among the studied aggregation numbers. © 2014 AIPPublishing LLC. [http://dx.doi.org/10.1063/1.4869832]

I. INTRODUCTION

Bile acids (BAs), the major constituents of bile, aremetabolites of cholesterol, and play a significant role in theabsorption of cholesterol, lipids, and fat-soluble nutrientsfrom the intestines. BAs belong to the class of facial am-phiphilic molecules, and behave differently from classical“head-tail” surfactants. As a general feature, the convex sideof BAs is hydrophobic, and the concave side is hydrophilic,respectively, due to the arrangement of different functionalgroups (OH and CH3) attached to the sterane core. Besidesacting like detergent molecules, recent research shows thatbile acids have other important physiological functions aswell, e.g., hormonal functions, cellular pathway activation,regulation of lipid and glucose metabolism.1, 2

Cholic acid (CA) is one of the two primary bile acidssynthesized in the liver of most mammals, and contains threeOH groups responsible for the hydrophilic character of theconcave side (Figure 1). The other primary bile acid, chen-odeoxycholic acid is less hydrophilic in character, having onlytwo OH groups.

BAs readily form mixed micelles with other surfactants,such as phosphatidylcholines (PCs) and sodium-dodecyl-sulphate (SDS), and are capable of self-aggregation as well.The size of pure BA micelles is considerably smaller, con-taining usually less than 10 molecules. The fine structure ofsuch BA micelles is not well understood. BAs also form sec-ondary aggregates above a certain concentration in solution.The aggregation properties of BAs can only be studied in-directly with experimental techniques.3, 4 Therefore, in orderto understand the structure and behaviour of such micelles insolution, it is necessary to use computational modelling ap-proaches. Several atomistic and coarse-grained models have

a)Electronic mail: szilard.fejer@cantab.net

been employed so far in modelling BA aggregation,5–9 and theresults of different modelling approaches can only be com-pared qualitatively. However, to the best of our knowledge,no global optimization studies have been carried out so far,using fully atomistic models of bile acids.

In solution, BA monomers are in a dynamic equilib-rium with their aggregates, and the dynamic nature of ag-gregation means that BA micelles are polydisperse. The av-erage aggregation number of various bile acids depends onthe local concentration and “availability” of monomers, andon various other factors such as hydrophilicity, hydrogenbonding propensity, and solution parameters (ionic strength,pH, etc.).

The aggregation behaviour of BAs can be readily studiedwith extensive molecular dynamics simulations. After reach-ing equilibrium, the various micelles can be characterized.However, on the vast energy landscape associated with BA ag-gregation, certain structures might exist that are not encoun-tered during an equilibrium MD simulation, which might belower in energy than all other identified micelles.

The aim of the current study was to find the lowestenergy configurations of small CA clusters in implicit sol-vent, evaluate the stability of these structures, and to ana-lyze apparent discrepancies in literature data about the aver-age aggregation number of cholate, the stable form of cholicacid under physiological pH. Putative global energy min-ima were identified for cluster sizes 2–10, and short MDruns were started from these configurations, in order to as-sess their stability at 300 K, and implicitly the role of en-tropy in micelle formation. Clearly, association of CA isstrongly concentration- and temperature-dependent.10 Low-energy CA structures are likely not the dominant form of as-sociation at physiological temperatures, but we can deduceimportant clues from the way they associate preferentially atzero temperature, which can be relevant for designing BAs as

0021-9606/2014/140(14)/144302/7/$30.00 © 2014 AIP Publishing LLC140, 144302-1

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144302-2 Jójárt et al. J. Chem. Phys. 140, 144302 (2014)

top view from the hydrophilic side side view

FIG. 1. Three-dimensional structure of cholate, the deprotonated form of cholic acid.

drug carriers.11 By adapting global optimization methods toclusters of biomolecules, we can evaluate which interactionsstabilize a particular structure. Based on the obtained struc-tural knowledge, functional groups on the molecules canbe modified to enhance aggregation into a target structure.Molecular dynamics simulations carried out at room tempera-ture have a very low chance of finding structures related to theglobal energy minima for different CA cluster sizes, due to thecomplex energy landscape, therefore the use of global opti-mization techniques is necessary to efficiently find the lowestenergy structures.

II. METHODS

A. Force field

The parameter set for CA molecules was built with theGeneralized Amber Force Field (GAFF).12 A widely usedvariant of the Generalized Born (GB) implicit solvent model(igb = 2)13 was employed to model the aqueous environment,with an ionic strength of 0.1 M, and mbondi2 GB solvationradii. No cut-off values were used for the potential terms. Thepairwise summation for calculating the Born radii was doneon atoms within 25 Å of each other.

B. Global optimization

In order to find global minima for CA clusters efficiently,we used the basin-hopping parallel tempering approach byWales et al., as implemented in GMIN.14, 15 The compu-tation of energies and gradients was carried out by theAMBER9 code, interfaced to the main GMIN source. Globaloptimization runs were started from randomly distributed CAmolecules, on multiple processors, with each thread havingits own Monte-Carlo (MC) temperature. Starting structureswere minimized using the L-BFGS method,16 and the foundminima were used as new starting structures in the Markovchain, if their energy was lower than that of the previous min-imum. If the energy of the new minimum was higher thanthe previous one, the step was accepted or rejected basedon an acceptance criterion, defined by the MC temperature.In order to perturb the structure of the accepted minimum,each CA molecule in the cluster was translated and rotatedby random amounts, as a rigid body. The molecules were al-lowed to explore their conformational space through relax-ation from highly strained starting structures, which were en-

countered during the random rigid body rotational and trans-lational moves.17 If the cluster contains too many atomicclashes and overlaps after the MC step, it is possible thatsome carbon atoms undergo a chirality inversion during en-ergy minimization, therefore the chirality of each carbon atomwas checked in all newly obtained minima, and structurescontaining inverted carbons were discarded. CA monomerswere prevented from dissociating using a system based onmaintaining a percolated graph of all atoms,15 with a distancethreshold of 5 Å, above which two molecules are considereddissociated. High temperature and low temperature replicasin parallel runs could be exchanged similarly to the paralleltempering method, in order to ensure a better coverage of con-figuration space. Global optimization was stopped when 80%of the threads found the same lowest energy minimum. Allstructures were optimized to an RMS force of 10−6. In orderto reach convergence to the global minima, between 2 × 106

and 2.4 × 107 basin-hopping steps (local minimisations) hadto be taken, depending on the cluster size, totalling 8.56 × 107

steps. We have used 32 replicas, and the temperature rangewas chosen to be wide enough to allow higher temperaturereplicas to get an acceptance ratio close to 1 for higher energyminima encountered during basin-hopping (0.1 to 10 kBT).

C. Molecular dynamics

The stability of clusters of different sizes was assessedby starting 100 short molecular dynamics runs (5 ns) fromthe global minimum structures (N = 2 to 10) at 300 K, withrandom starting velocities, and determining the percentage ofdissociated clusters by the end of the run. The global mini-mum structures were heated from 10 K to 300 K in 6 steps.The duration of each step was 5 ps and the time step wasset to 0.5 fs. Temperature was controlled with the Berendsenthermostat18 using τT = 1.0 ps. After the heating process, thesystem was subjected to 5 ns long constant temperature sim-ulation using the Langevin thermostat,19, 20 with a timestep of2 fs, and the damping coefficient set to 1 ps−1. This protocolwas repeated 100 times (using different, random initial veloc-ities) in order to obtain statistically relevant results. The sameimplicit solvent model and non-bonded interactions were usedfor the molecular dynamics runs, as for the global optimiza-tion. The calculations were performed with the sander moduleof the AMBER12 program package.21 The structures savedduring each run have been analyzed for signs of dissociation.

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144302-3 Jójárt et al. J. Chem. Phys. 140, 144302 (2014)

Eto

t/N

(kca

l/m

ol)

N

−24

−26

−28

−30

−32

−34

−36

−38

−40

−421 2 3 4 5 6 7 8 9 10

CAfitted function

FIG. 2. Energy/molecule of the lowest energy CA clusters as a function of cluster size.

III. RESULTS AND DISCUSSION

A. Global optimization

Figure 2 shows the total energy of the global minimadivided with the number of molecules in the cluster, as afunction of cluster size, and a decay function fitted to thefirst seven data points. The fitting of an analytical functionto the data points can be done using the assumption that ina general case every CA monomer contributes to the totalenergy with its own energy, E0, plus a stabilization energyEstab, which arises due to new contacts being formed betweenthe molecules. We assume the stabilization energy to be in-dependent of the number of possible nearest neighbours, be-cause of the facial amphiphilicity of the molecule and the factthat it can only make a fixed number of hydrogen bonds withneighbours. In fact, the average number of hydrogen bondsper molecule in the global minima varies between 1.1 and1.8. Also, the charges on the carboxylate groups further re-strict the favoured orientation between two molecules, evenwhen screened by the implicit solvent. The total energy willbe therefore

Etot = N · E0 + (N − 1) · Estab, (1)

where N is the number of molecules in the cluster. The en-ergy/molecule quantity, as a function of N, will be therefore

Etot

N= E0 + Estab − Estab

N. (2)

The only unknown in the above equation is the stabiliza-tion energy gained during aggregation, which we assume tobe constant, and it will be the only parameter in the curvefitting. As seen in Figure 2, Etot/N for N ≥ 8 is signifi-cantly lower than indicated by the trend for N = 1 to 7,therefore curve fitting was done only for data points upto N = 7. The fitted parameter, the stabilization energy is

Estab = −15.8479 ± 0.03903 kcal/mol (about the same as theenergy of 3 hydrogen bonds). Above N = 8, extra stabiliza-tion is happening in the lowest energy structures, which canbe explained by structural features.

The global minimum for the CA dimer is an inverted mi-celle, and is stabilized by hydrogen bonding. The structure ofthe dimer is peculiar, belonging to the C2 symmetry group,if only the heavy atoms are taken into account (Figure 3(a)),and contains three hydrogen bonds. The trimer has a similarstructure to that of the most favoured arrangement of threechiral rigid building blocks, described previously,22, 23 havinga twisted parallel relative orientation (Figure 3(b)). This ori-entation is kept in the tetramer as well, but no such orientationcan be seen for larger aggregates. In case of N = 5, four CAunits are arranged so that each two are aligned, and the pairsare stacked on top of each other and rotated by 90◦.

For aggregation numbers N = 2 to 5, there is a steady in-crease in hydrogen bonding propensity among the molecules(Table I). These structures are inverted micelles, having theirhydrophilic part pointing inside. However, at N = 5, the trendstarts changing, with two CAs orienting their hydrophilicsides towards the solvent. For the global minimum of 6 CAmolecules, an asymmetric distribution of hydrogen bonds canbe observed, the aggregate being similar to a secondary mi-celle. In this cluster, three CAs form a normal micelle, withtheir hydrophilic parts pointing towards the solvent, and therest have their hydrophilic part pointing inside. The sameasymmetry of hydrogen bonding distribution can be seen forN = 7, which also contains a 3-molecule inverted micelle.

The character of the aggregate changes significantly atN = 8, since no more inverted micelle arrangements can befound in the lowest energy structure. The distribution of hy-drogen bonds is more uniform, and many H-bonds are beingformed on the surface of the aggregate. However, at N = 9,two CAs form an inverted micelle, with hydrogen bondingsimilar to that in N = 2. Although this is the lowest energy

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144302-4 Jójárt et al. J. Chem. Phys. 140, 144302 (2014)

(a) (b) (c) (d)

(e) (f)

FIG. 3. Structures of selected global minima. (a) N = 2, C2 symmetry for the heavy atoms; (b) N = 3; (c) N = 4; (d) N = 5; (e) N = 9, close-up view on thechain of four hydrogen bonds; (f). N = 10, a structure in which the central CA unit is completely covered.

minimum for this micelle, inverted micelles necessarily meanmore exposed hydrophobic parts, unless these parts can becovered. This is not possible in the present structure, thereforethe energy/molecule of this structure is only slightly lowerthan that of N = 8. A chain of four consecutive hydrogenbonds, in which three hydroxyl groups are both donors andacceptors, is formed between the inverted micelle part andanother CA unit in the aggregate, probably contributing to itsstabilization (Figure 3(e)).

Among all cluster sizes considered (N = 2 to 10), themost stable structure was found for N = 10, having a total en-ergy lower by about 9 kcal/mol than expected from the ener-getic trend shown in Figure 2. This is the only structure whichcontains a CA molecule having its sterane core completelysurrounded by other CAs. The small increase of the solventaccessible surface area from N = 9 to 10, relative to the trendset between N = 2 and 9 is due to the efficient packing of one

TABLE I. Total energy of global minima, solvent accessible surface areasand number of hydrogen bonds as a function of N.

N E (kcal/mol) E/N (kcal/mol) SASA (Å2) SASAOH (%) H-bonds

1 − 25.799 − 25.799 616.1 19.1 02 − 67.530 − 33.765 940.8 7.7 33 − 108.885 − 36.295 1207.8 6.5 54 − 150.441 − 37.610 1528.7 5.3 75 − 192.098 − 38.420 1734.0 11.0 96 − 234.223 − 39.037 1950.5 15.7 87 − 276.507 − 39.501 2159.7 18.9 88 − 322.430 − 40.304 2327.0 16.9 129 − 363.943 − 40.438 2608.9 19.9 1110 − 409.809 − 40.981 2657.3 15.9 17

CA inside the micelle, shielding it completely from the sol-vent (Figure 4). Similarly to the aggregates larger than 5, thismicelle is also mainly stabilized by hydrogen bonding on itssurface.

Figure 5 shows the distribution of hydrogen atoms partic-ipating in H-bonds from the centre of mass of each aggregate,relative to the distance of the atom furthest away from thecentre. This rough measure gives us an indication about thedegree by which hydrogen bonding occurs inside the micelle.It can be seen that for small aggregates, most of the hydrogenbonding occurs closer to the centre of mass, in contrast to thetrend observed for large aggregates. There are no hydrogenbonds around the central region of N ≥ 5 micelles, indicat-ing that above this size, the cluster starts to behave more like

SASA

(A2)

N

3000

2500

2000

1500

1000

500

01 2 3 4 5 6 7 8 9 10 11

totalOH groups

FIG. 4. Solvent accessible surface area of global minima, compared to thatcalculated for the OH groups.

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144302-5 Jójárt et al. J. Chem. Phys. 140, 144302 (2014)

0 − 0.2 0.2 − 0.4 0.4 − 0.6 0.6 − 0.8 0.8 − 1.00

2

4

6

8

1023456789

10

FIG. 5. Distribution of hydrogen bond—centre of mass distances withineach global minimum (calculated between H atoms taking part in the hy-drogen bond and the centre of mass), relative to the atom furthest away fromthe centre of mass of the aggregate.

an ordinary micelle. Below N = 5, the aggregates have an in-verted micellar character. Unfortunately, it is not feasible atthe moment to find the lowest energy configurations for thesecluster sizes using explicit water, due to the large system sizeand many degrees of freedom, hence we cannot directly studywith global optimization techniques the competition betweenCA-CA hydrogen bonds and those formed between CA andwater molecules.

CA has the most OH groups among the family of bileacids, and therefore it has the highest propensity of forminghydrogen bonds. Recent experimental results24 show that theaverage aggregation number is 13 for CA, at concentrationshigher than the critical micelle concentration (cmc), this num-ber being well above the average aggregation number deter-mined for oxoderivatives of CA. All determined aggregationnumbers (Fig. 7 of Ref. 24) show some linear concentrationdependence, with that of CA depending the most on monomerconcentration. When extrapolated to zero concentration (in-finitely dilute solution), the average aggregation number ofCA drops to 9.67 ± 1.08. The slope of the aggregation num-ber versus concentration graph is the largest for CA amongthe bile acids considered in that study.24

Previous MD simulations8, 9 have determined differentaverage association numbers for cholate aggregates. The mostobvious discrepancy is between results reported for high CAmonomer concentrations in Refs. 8 and 9, the average micellesizes being 19.7 (300 mM)8 and less than 6 (270 mM).9 How-ever, the employed force fields were different in the two stud-ies. These discrepancies suggest that force field parametersshould be improved to properly capture the weak intermolec-ular interactions between bile acid molecules. The relativelylow association numbers reported for a high CA concentra-tion in Ref. 9 can also be due to an underestimation of theenergetic contributions of hydrogen bonds by the CHARMMGeneral Force Field, the authors concluding that “hydrogenbonding among CA primary micelles is weak and unstable.”However, in experimental setups, the monotonic correspon-

dence between the number of hydroxyl groups in the bile acidmonomer and their aggregation numbers suggests that hydro-gen bonding has a significant role in stabilizing aggregateslarger than 8.24 The experimental observations fit well to ourresults, since we see a strong extra stabilization effect occur-ring at precisely N = 8, due to the global minimum containing3-4 hydrogen bonds more than the smaller aggregates. The to-tal number of hydrogen bonds in the global minima increasesmonotonically until N = 5 (9 H-bonds), decreases to 8 forN = 6 and 7, and increases to 12 for N = 8.

B. Dissociation propensity of micelles

The potential energy calculated using force fields withan implicit solvent contains some estimated entropy changesof the solvent during the solvation process. Bile acid ag-gregation is a process driven by the entropy gain of boundwater molecules that are released during association, simi-larly to the aggregation of surfactants and most virus cap-sids. The calculated energies for the most stable minima con-tain therefore implicitly an estimated entropy change duringsolvation, compared to that of the monomers. However, oneshould keep in mind that these structures are the most stable at0 K. The correlation between the global potential energy andfree energy minima is not guaranteed at room temperature,where the role of entropy is significant. In the current contri-bution, we have used the global minima as starting structuresin short MD runs, to evaluate the likelihood of dissoci-ation, once structures are formed that are similar to thedetermined global minima. The energy landscape of such ag-gregates is very complex, given the dynamic nature of BA ag-gregation, therefore we consider this hybrid global optimiza-tion/MD approach to be the most accessible, in order to learnabout the structural features of the energetically most stableaggregates.

For each cluster size, starting from the respective globalminimum, we ran 100 short MD simulations (5 ns) at300 K, according to the protocol described in Sec. II. After5 ns, the number of undissociated structures was determined,and their distribution is plotted in Figure 6(a). However, fromthis plot alone it is not possible to determine the propensityof dissociation, since in some cases dissociation and reasso-ciation events can occur. We did not use any periodic bound-ary conditions, therefore the system cannot reach an equilib-rium state until all molecules are dissociated, but nevertheless,dissociation and reassociation events, right after the dissoci-ation step, are possible. During the 5 ns MD runs, reassocia-tion occurs in about 16% of the cases after dissociation. Thelikelihood of reassociation events decreases rapidly, once thedissociated CA monomer gets further away from the cluster.Therefore, by evaluating the time needed for the first disso-ciation step, we can have a better indication about the rela-tive stabilities of the global minima (N = 2 to 10) at roomtemperature.

Figure 6(d) shows the characteristic decay time τ of thefirst-order dissociation, for each cluster size, obtained by lin-ear fitting on the ln nc vs time data (see Figure 1 of the supple-mentary material17), where nc is the number of undissociated

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(a) (b)

(c) (d)

coun

ts

coun

ts

coun

ts

NN

NN

50

150

200

250

300

350

400

1

1

1

1

2

2

2

2

22

2

3

3

3

33

4

4

4

44

4

5

5

5

55

6

6

6

66

6

7

7

7

77

8

8

8

88

8

9

9

9

99

12

14

16

18

20

40

60

80

100

100

10

10

10

1010

10

00

00

0.5

1.5

2.5

11

τ(n

s)

FIG. 6. (a) Number of undissociated micelles at the end of 100 short MD runs (5 ns); (b) combined distribution of micelle sizes after 100 short MD runs;(c) same distribution as in (b), but showing the contribution of each set of starting structures (N = 2 to 10); (d) characteristic decay time for each cluster, and itsstandard deviation, as a function of cluster size.

clusters out of the 100 parallel MD runs at a certain time step.For cluster sizes N = 2 and 3, dissociation occurs rapidly.There is a monotonic increase in τ , until N = 5, followedby a decrease until N = 8. τ increases again significantly forN = 9 and 10. This trend suggests that, out of the size rangeconsidered, the most stable micelle size is 10, followed by9, and the stability of smaller micelles in the range of 6–8is similar. Interestingly, the pentamer is also very resistant tofragmentation, probably due to the large number of hydrogenbonds per particle (1.8). Micelles containing less than 5 CAunits are likely to dissociate rapidly in dilute solutions.

The extra stability of N = 9 and 10 suggests that escapingthe basin of attraction25 of the global minimum is somewhatharder at room temperature for these sizes, than for smalleraggregates. Since the global minima after N = 8 do have extrastabilization due to increased hydrogen bonding, we speculatethat at room temperature, for these sizes, the role of entropiccontribution is comparable to that of the energetics, and theglobal free energy minimum of such clusters will be similarto the global minima depicted in Figure 3. The average aggre-gation number for CA in a dilute solution (30 mM) was deter-mined as 4.47 in Ref. 8. However, during those simulations itwas not possible to identify aggregates larger than 10, giventhat the simulation box contained only 10 CA molecules. Itwould be instructive to carry out long MD simulations inlarger boxes, containing 20–30 CA monomers, and analyzetheir equilibrium behaviour for large dilutions.

IV. CONCLUSIONS

The aim of this study was to find the lowest energy struc-tures for different aggregation numbers of cholate, and eval-uate the potential of cholate micelles to be used as a drugcarrier. We have identified the global minima for aggrega-tion numbers N = 2 to 10, and showed that extra stabiliza-tion effects occur after N = 8, which can be explained by in-creased hydrogen bonding, and overall better packing of thehydrophobic side of cholate. Moreover, the global minimumfor N = 10 contains a completely buried CA molecule, havingonly its -COO− group in contact with the surface. Therefore,drugs having a similar size or anisotropy to CA can be ef-ficiently packed with at least 9 CA units. It is also possibleto refine the structure of CA in order to further stabilize the“cage” surrounding the packed molecule, by introducing newfunctional groups and extra hydrogen bonding. In this way,global optimization can guide the search for bile acid deriva-tives allowing optimized packing of certain drugs.

Global optimization shows that among the size rangeconsidered, the most stable structure is that of N = 10, whichis in a good agreement with recent experimental results,24

suggesting that the average aggregation number for dilute so-lutions is between 8.59 and 10.75. However, since we did notconsider larger clusters, there is a possibility that larger clus-ters are even more stable than the decamer. This issue will beinvestigated in future work, but we expect N = 10 to be a true

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144302-7 Jójárt et al. J. Chem. Phys. 140, 144302 (2014)

“magic number,” because the main stabilizing factor in thatstructure is the complete encapsulation of a monomer. MDruns at 300 K confirmed the increased stability of the lowestenergy structures for N = 9 and 10, since CA monomers dis-sociated significantly later from these larger aggregates thanfrom smaller micelles.

ACKNOWLEDGMENTS

The authors thank the support of the Hungary-Serbia IPA Cross-border Co-operation programmeHUSRB/1002/214/193, Bile Acid Nanosystems as MoleculeCarriers in Pharmaceutical Applications (BANAMOCA).Infrastructural support was provided by the TÁMOP-4.2.2.A-11/1/KONV-2012-0047, “New Functional Materials andTheir Biological and Environmental Answers.” S.N.F. isa John von Neumann Fellow supported by the EuropeanUnion and the State of Hungary, co-financed by the Euro-pean Social Fund in the framework of TÁMOP-4.2.4.A/2-11/1-2012-0001 “National Excellence Program.”

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