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Journal of Chromatography A, 1291 (2013) 73– 83

Contents lists available at SciVerse ScienceDirect

Journal of Chromatography A

jou rn al hom epage: www.elsev ier .com/ locate /chroma

n expanded cellular automata model for enantiomer separationssing a �-cyclodextrin stationary phase

arren DeSoi ∗, Lemont B. Kier, Chao-Kun Cheng, H. Thomas Karnesirginia Commonwealth University, School of Pharmacy, Department of Pharmaceutics, 410 North 12th Street, P.O. Box 980533, Richmond,A 23298-0533, USA

a r t i c l e i n f o

rticle history:eceived 4 January 2013eceived in revised form 15 March 2013ccepted 21 March 2013vailable online 26 March 2013

eywords:yclodextrin

a b s t r a c t

Chromatographic scale enantiomer separation has not been modeled using cellular automata (CA). CAuses easy to adjust equations to different enantiomers under various chromatographic conditions. Previ-ous work has demonstrated that CA modeling can accurately predict the strength of one-to-one bindinginteractions between enantiomers and �-cyclodextrin (CD) [1]. In this work, the model is expanded to achromatographic scale grid environment in order to transform model output into HPLC chromatograms.The model accurately predicted the lack of chromatographic selectivity of mandelic enantiomers (1.05published, 1.01 modeled) and the separation of brompheniramine enantiomers (1.13 published, 1.12

ellular automatahiral separationsodeling

modeled) previously modeled in one-to-one interactions. By examining cyclohexylphenylglycolic acid(CHPGA) enantiomers, the model accurately predicted both the selectivity and resolution of the enan-tiomer peaks at varying chromatographic temperatures. Modeled changes in mobile phase pH agreewith laboratory outcomes when examining peak resolution and selectivity. Changes in injection volumeresulted in an increase in retention time of the modeled enantiomers as was observed in the publishedlaboratory results.

© 2013 Elsevier B.V. All rights reserved.

. Introduction

Contrasting models that simulate enantiomeric interactions at one-to-one scale to study binding energies, the expansion toodel a chromatographic environment using cellular automata

CA) allows for the generation of model chromatograms. CA allowsor the easy determination of equations based on fundamentalinding energies (van der Waals, hydrogen bonding, hydropho-icity) to calculate probabilities and factors that guide ingredient

nteractions. Once chromatograms are predicted, then the degreef separation of enantiomers can be more accurately compared toaboratory results. Additionally, chromatographic conditions thatre traditionally altered in the laboratory to improve peak sep-ration can be modeled to see their impact on the degree ofhromatographic separation: mobile phase temperature and pH,nd injection volume.

Chromatographic retention of analytes in a �-cyclodextrin (CD)PLC column involves multiple interactions with the CD station-ry phase. This is incorporated into the model environment along

∗ Corresponding author at: Virginia Commonwealth University, School of Phar-acy, Department of Pharmaceutics, 410 North 12th Street, Richmond, VA

3298-0581, USA. Tel.: +1 804 852 6033; fax: +1 804 335 2087.E-mail address: desoidj@mymail.vcu.edu (D. DeSoi).

021-9673/$ – see front matter © 2013 Elsevier B.V. All rights reserved.ttp://dx.doi.org/10.1016/j.chroma.2013.03.060

with the addition of a dual solvent mobile phase. Along with thenew ingredient cells (enantiomers, mobile phase, and multiple CDsites), new interactions had to be accounted for. In addition to theenantiomer-CD interactions, mobile phase interactions with theenantiomers, CD stationary phase, and enantiomer are incorpo-rated into the model.

The chromatographic model is tested to verify that chro-matograms of mandelic acid and brompheniramine enantiomersagree with selectivity from the first one-to-one model [1]. Modelrules are then adjusted to examine the enantiomeric separationof CHPGA. Changes to modeled HPLC column temperature, mobilephase pH, and sample injection volume are compared to publishedlaboratory results.

2. Experimental

2.1. Software

The cellular automata model is written in JavaTM and executed

using Eclipse (version 3.1.2, The Eclipse Foundation) as an inte-grated development environment. All calculations and plots areperformed using Microsoft Office Excel 2003 (Microsoft Corpora-tion).

7 atogr. A 1291 (2013) 73– 83

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4 D. DeSoi et al. / J. Chrom

.2. Computer

Model runs are performed on a Toshiba SatelliteTM A305 laptop,ith Intel CoreTM Duo CPU 1.83 GHz, 3.00 GB RAM, on a Windows

32-bit operating system.

.3. Cellular automata model environment design

Cellular automata models are designed around a grid of cells,ngredient(s) location and amount, rules governing the interactionsnd movement of ingredients, and a specified runtime (itera-ions). Thousands of interactions occur between many analyte andtationary phase molecules depending on the amount of sam-le injected onto the chromatographic column and the column’sesign. Thousands of interactions are impractical to model, as thisould vastly slow down the model and require substantial compu-

ational power. There may be no benefit in expanding the model tohis degree, since the model is designed to predict chiral separationsn an efficient manner and not replicate the physical environmentf a chromatographic column. The number of analytes and �-CDtationary sites needs to be increased in a manner that allows mul-iple site interactions as the analytes move through the column.-CD stationary sites are evenly spaced in an alternating manner

o avoid possible solvent channels so that the analytes will havetationary phase interaction and not move through the columnithout interaction.

Analytes are placed randomly at the beginning of the modeledolumn, prior to the stationary phase, to represent the beginning ofn injection of sample onto a HPLC column. Two different types ofellular automata cells are used representing enantiomer pairs. Athe beginning of each run the analyte cells are randomly intermixednd have the ability to rotate as they move, which plays a part inheir separation behavior since they are variegated cells.

Mobile phase cells are also added to the model. Two differ-nt types of cells are used so that dual solvents may be modeled.heir polarity and densities are incorporated into the model toest represent their chemical and physical properties under lab-ratory conditions. The mobile phase cells interact with analytesnd stationary phase in various ways according to their chemicalature.

Flow (gravity factor in the model) is incorporated on analytesnd mobile phase cells at equal values. The gravity parameter inhe cellular automata model represents the tendency to move in aertain direction.

As in the one-to-one model, a �-CD is represented using severalells (see Fig. 1). The �-CD is now made of variegated cells thatre divided into two types of sites, each having their own set ofnteraction rules. B0 sides represent the secondary hydroxyl groupsocated at carbon two of the �-CD in addition to the hydrophobicnterior of the �-CD depending on what side of the analyte cell it isnteracting with. B1 sides represent the hydrophobic interior of the-CD, while B2 represent the primary hydroxyl groups of the �-CDt carbon six. The outside or exterior of the �-CD denote sites thatave minimal interaction with analytes and are in yellow using theariable C.

There are two types of analyte cells in the model, A and D, toepresent an enantiomer set. Analyte cells are variegated with 4ides to represent a chiral molecule. Each side has its own set ofnteraction rules with stationary phase sites (B0–B2), mobile phaseells W1 or W2 (water and acetonitrile), and each other. Mobilehase cells are non-variegated and have the same set of interactionules on each side.

The cellular automata model environment consists of a grid ofells 40 columns wide and 800 rows long for a total of 32,000 cellso represent a chromatographic column. This grid design evolvedhrough several steps of runs and observations. The grid is designed

Fig. 1. The two dimensional, cellular automata grid representing a cyclodextrin ringwith a variegated analyte cell for chromatographic scale.

as a torus with no edges so that ingredients may pass from one sideof the grid to the other. The first 10 rows of the grid does not containany CD cells so that analyte cells may start there at the beginningof each run to represent a sample injection. One hundred of eachanalyte are randomly placed within this space so that analytes areintermixed with each other and mobile phase cells.

�-CD cells begin at row 11 in the orientation as in Fig. 1. Ori-ented on its side the �-CD sites have analytes enter due to theirattraction or lack thereof. The side direction that the �-CD facesdoes not matter since all moving cells may exit one side of the gridand reappear on the opposite side, eliminating analyte and mobilephase cell movement boundaries. �-CD sites were placed in a man-ner so that they are spaced five cells apart on any side. This typeof placement continues until a total of 100 �-CD sites exist, so thatstationary phase is present from rows 10 to 204. The remaining595 rows are left empty for mobile phase at the start of each run.These rows provide an area for the analytes to move into after theirinteraction with the stationary phase sites. It is in these rows thatthe analytes will be examined, as in high performance liquid chro-matography when analytes leave the column and continue onward

to the detector.

Placement of mobile phase cells is random; however, severalfactors need to be considered to determine their concentration:empty space available in the grid after placement of stationary

D. DeSoi et al. / J. Chromatogr. A 1291 (2013) 73– 83 75

Table 1Cell population of water and acetonitrile at varying temperatures.

Temperature Density Number of cells

Water ACN Water ACN

W1 W2 W1 W2

24 0.9973 0.7793 13,376 627131 0.9954 0.7716 13,350 620937 0.9934 0.7650 13,323 615644 0.9907 0.7573 13,287 6094

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phenyl group [6]. For (R)-brompheniramine then, Eq. (8) is used

50 0.9881 0.7507 13,252 604157 0.9848 0.7430 13,208 5979

hase and analytes, empty space left to allow for ingredient cellovement, concentration of solvent in mobile phase, and density

f solvent at varying temperatures. Of the 32,000 available cells,nly 31,100 are available once analytes (A and D) and CD sitesre subtracted out. It has been previously demonstrated in aque-us systems that 69% occupancy of the cell grid allows for waterells to behave chemically similar to actual conditions [2]. In thehromatographic system modeled [3], the mobile phase consists of2.5% water, 31.5% acetonitrile, and 6.0% methanol. Since acetoni-rile and methanol both have the potential for hydrogen bonding,lbeit weaker than water [4] their model probabilities would be theame. For simplicity, the model will consist of a two solvent mobilehase, 62.5% water and 37.5% acetonitrile. Additionally, taking intoccount the density (W, number of cells to populate in grid) of theolvents involved [5] results in Eq. (1):

= 31, 100 × 0.69 × C × � (1)

here 31,100 is the number of empty cells in grid, 0.69 is the per-ent of occupied space, C is the concentration of solvent in mobilehase and � is the density of solvent.

Using Eq. (1), the cell concentration of water and acetonitrileas determined for each chromatographic temperature modeled

see Table 1). The calculated number of cells for each solvent werelaced randomly throughout the grid. See Fig. 2 for an example of

model generated environment (note: only the upper portion isisplayed due to space constraints).

.4. Development of model rule equations and PB values forandelic acid and brompheniramine

In the one-to-one model, several interactions were addressedor mandelic acid and brompheniramine. Hydrogen bindingetween the enantiomers at the secondary hydroxyl groups of the-CD, along with van der Waals, hydrophobicity, hydrogen bind-

ng, and steric interactions at the interior of the �-CD [1]. Eq. (3)1] is used to determine breaking probability of A0 (S-enantiomer)nd D0 (R-enantiomer) with cell B0 of the CD. Eq. (5) [1] is used toetermine breaking probability of A1–3 and D1–3 with B0 and B1.

With a system that now includes additional forces that resultn further movement, such as attractive and repulsive forces onhe analytes from mobile phase interactions and flow, interactionf the analytes with each other, and with the primary hydroxylroups of the CD should be incorporated. It has been previouslyhown that for mandelic acid the main interaction is in the interiorf the �-CD with the phenyl group [6]. Since the phenyl group isydrophobic, there should be a repulsion force with the primaryolar hydroxyl groups of the �-CD, hence an increase in the break-

ng probability between the two. In Eq. (5) [1], the hydrophobic

nergy of a molecule was determined by the sum of each group ortom of the molecule. Using this same approach, and that decahy-ronaphthalene is again used as the molecule to have a maximum

Fig. 2. Model generated chromatographic scale environment with analyte,cyclodextrin, and mobile phase cells.

hydrophobic energy at 10.0 kcal/mol [7], the following equation isdeveloped:

PB(B2) = 0.5 +(

W

10.0 kcal/mol× 0.45

)(2)

where W is the hydrophobic energy of enantiomer in CDcavity, 10.0 kcal/mol is the modeled maximum hydrophobicenergy potential and 0.45 is the maximum inhibition ofPB(B2).

A starting breaking probability value of 0.5 means that there isequal attraction and repulsion force between the analyte and theprimary hydroxyl groups of �-CD, therefore repulsion due to thehydrophobicity of the phenyl group will increase the value of PB(B2)with sides A1–3 and D1–3 of the mandelic acid enantiomers. Allother mandelic acid interactions with �-CD were either consideredimprobable or insignificant and assigned a breaking probability of0.5.

It has been shown that the brompheniramine (S) enantiomerprefers the insertion of the pyridine ring into the �-CD whilethe (R) enantiomer prefers the insertion of the para-substituted

to determine the hydrophobic repulsive force between D1–3 andB2. (S)-brompheniramine though has the potential for hydrogenbonding from the pyridine group with the primary hydroxyl groups

76 D. DeSoi et al. / J. Chromatogr. A 1291 (2013) 73– 83

Table 2Breaking probability and joining factors for mandelic acid and brompheniramine with �-cyclodextrin cells.

Interaction Interaction

Mandelic acid PB J Brompheniramine PB J

A0 B0 0.2656 1.591 A0 B0 0.3922 1.027A0 B1 0.5000 0.7079 A0 B1 0.5000 0.7079A0 B2 0.5000 0.7079 A0 B2 0.5000 0.7079A1 B0 0.3035 1.396 A1 B0 0.2858 1.484A1 B1 0.3035 1.396 A1 B1 0.2858 1.484A1 B2 0.7070 0.3463 A1 B2 0.3043 1.392A2 B0 0.3035 1.396 A2 B0 0.2858 1.484A2 B1 0.3035 1.396 A2 B1 0.2858 1.484A2 B2 0.7070 0.3463 A2 B2 0.3043 1.392A3 B0 0.3035 1.396 A3 B0 0.2858 1.484A3 B1 0.3035 1.396 A3 B1 0.2858 1.484A3 B2 0.7070 0.3463 A3 B2 0.3043 1.392

D0 B0 0.4219 0.9271 D0 B0 0.3453 1.208D0 B1 0.5000 0.7079 D0 B1 0.5000 0.7079D0 B2 0.5000 0.7079 D0 B2 0.5000 0.7079D1 B0 0.3035 1.396 D1 B0 0.3568 1.161D1 B1 0.3035 1.396 D1 B1 0.3568 1.161D1 B2 0.7070 0.3463 D1 B2 0.6980 0.3573D2 B0 0.3035 1.396 D2 B0 0.3568 1.161D2 B1 0.3035 1.396 D2 B1 0.3568 1.161D2 B2 0.7070 0.3463 D2 B2 0.6980 0.3573D3 B0 0.3035 1.396 D3 B0 0.3568 1.161D3 B1 0.3035 1.396 D3 B1 0.3568 1.161D3 B2 0.7070 0.3463 D3 B2 0.6980 0.3573

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f the CD [8]. Eq. (2) should not be used to predict the interac-ion between A1–3 and B2, rather an equation for attractive forcesnvolving hydrogen bonding like Eq. (3) [1]. (S)-brompheniramineas the potential for the nitrogen from the pyridine to bond with therimary hydroxyl groups of the �-CD. With O H· · ·N having a bondtrength of ∼6.9 kcal/mole [9], and pyridine undergoing hydrogenonding with a strength of ∼3 kcal/mol [8] Eq. (3) results:

B(B2) = 0.50 −(

Hprim.

6.9 kcal/mol× 0.45

)(3)

here 0.50 is the equal attraction and repulsion value, Hprim is theotential hydrogen bonding for enantiomer from N, 6.9 kcal/mol ishe maximum hydrogen bonding potential and 0.45 is the maxi-

um decrease in breaking probability.Again, a starting breaking probability value of 0.5 means that

here is equal attraction and repulsion force between the analytend the primary hydroxyl groups of �-CD, therefore attraction dueo the hydrogen bonding of the pyridine group will reduce the valuef PB(B2) with sides A1–3 of the (S)-brompheniramine.

Breaking probabilities and their corresponding joining fac-ors (using equation log J = −1.5PB + 0.6 [10], are calculated for thenantiomers of mandelic acid and brompheniramine using the pre-iously mentioned equations (see Table 2). However there are othernteractions that require breaking probabilities and joining factors:

Enantiomer to mobile phase.Enantiomer to enantiomer.Mobile phase to �-CD.Mobile phase to mobile phase.

Enantiomer interaction with mobile phase (water and acetoni-

rile) should involve no overall significant attraction or repulsion.lthough competing interactions of repulsion (water with ben-ene) and attraction (hydrogen bonding) are acknowledged,hey should work against each other. Therefore, all enantiomer

interactions with mobile phase were assigned the breaking proba-bility of 0.5 (no attraction or repulsion).

Enantiomer to enantiomer interaction also has competinginteractions. Hydrophobic attraction may occur between benzeneportions of the enantiomers and hydrogen bonding attractionbetween the alpha hydroxy acids may occur. Nevertheless repuls-ing interactions will occur when the opposite portions of theenantiomers interact, therefore enantiomer interactions wereassigned the breaking probability of 0.5.

Mobile phase interactions with �-CD are a little more complexsince water (W1) and acetonitrile (W2) will behave differently inthe hydrophobic interior and hydroxyl groups of the CD. At the B0cells of the CD, which represent both the secondary hydroxyl sitesand the upper hydrophobic interior of the CD, water is attracteddue to hydrogen binding but repelled due to the hydrophobicity.Since these forces are opposite, a breaking probability of 0.5 isassigned. B1 represents the stronger hydrophobic interior of the CD,an environment where water would be repelled. Therefore a break-ing probability of 1.0 is assigned between water and B1. B2 cellsdenote the primary hydroxyl groups that water will be attracted todue to hydrogen binding. The breaking probability between waterand the B2 cells follows the same probability of water–water inter-actions, varying with temperature described later. The exterior ofthe �-CD (C0) is considered to have no attractive or repulsive forceswith either mobile phase solvent and has a breaking probability of0.5 with each.

Like water, acetonitrile has the same competing interactionsas water at B0 and is assigned a breaking 0.5. In the hydropho-bic interior of the CD acetonitrile is not repelled like water and hasa breaking probability of 0.5. At the primary hydroxyl groups, ace-tonitrile may have some attraction and repulsion and is thereforegiven a value of 0.5.

Solvents that make up the mobile phase interact but vary with

temperature. Since temperature variation in the chromatographicsystem will be studied, then these interaction changes need to beaccounted for. Water breaking probabilities vary with temperature[10]. Acetonitrile breaking probabilities vary with temperature but

D. DeSoi et al. / J. Chromatogr

Table 3Breaking probability and joining factors for water (W1) and acetonitrile (W2) witheach other.

Temperature, ◦C

24 31 37 44 50 57

W1–W1PB 0.240 0.310 0.370 0.440 0.500 0.570J 1.74 1.36 1.11 0.871 0.708 0.556

W1–W2

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W2–W2PB 0.544 0.599 0.647 0.702 0.749 0.805J 0.608 0.503 0.426 0.352 0.300 0.247

re different than water since its liquid range is −45.7 ◦C to 81.6 ◦C5] (see Table 3). Breaking probabilities of 0.00–1.00 are used toover the liquid range of acetonitrile temperature.

.5. Development of model rule equations and PB values forHPGA

In the development of model equations for mandelic acid andrompheniramine, several types of interactions were considered.o study these interactions further another enantiomer set washosen, CHPGA. It is similar to mandelic acid except that a hydrogenas been replaced with cyclohexane (see Fig. 3). Where mandeliccid has minimal chiral separations since the only chiral discrim-nating characteristic is the different interactions of the alphaydroxy acid, CHPGA has the potential for different interactionsetween the benzene and cyclohexane groups in the interior of the-CD. Eq. (3) [1], Eq. (5) [1], and Eq. (2) should be used to calculate

he appropriate breaking probabilities, but they need to be adaptedo the new molecule being analyzed.

With O H· · ·O having a bond strength of ∼5.0 kcal/mol [9],HPGA has a possible total hydrogen bonding potential of5.0 kcal/mol. This represents the highest hydrogen bonding poten-ial due from the alpha hydroxy acid with the secondary hydroxylroups of �-CD. Taking this into account, a maximum PB of 0.05,nd a starting value of 0.50, results in the equation for the breakingrobability of A0 and D0 with cell B0:

B(B0) = 0.50 − Hsec.

15.0 kcal/mol× 0.45 (4)

here 0.50 is the equal attraction and repulsion value, Hsec. is theotential hydrogen bonding for enantiomer (additive of O atoms),5.0 kcal/mol is the maximum hydrogen bonding potential and 0.45

s the maximum decrease in breaking probability.

Fig. 3. Cyclohexylphenylglycolic acid (CHPGA).

. A 1291 (2013) 73– 83 77

A breaking probability for the interior cavity of �-CD takes intoaccount, van der Waals forces, hydrophobicity for the portion ofanalyte in �-CD, and steric hindrance inside �-CD. Hydrogen bond-ing in the interior of the �-CD is no longer incorporated sinceneither benzene or cyclohexane are likely to interact in this manner.

CHPGA contains both a benzene and cyclohexane that may enterthe �-CD hydrophobic interior. It has been shown with brompheni-ramine that different groups attached to the chiral carbon can havepreferential interaction with the interior of the �-CD [6]. How-ever, benzene and cyclohexane do not differ enough in their vander Walls volume (79.85 A3 and 98.66 A3) [11] or hydrophobicbinding potential (4.6 A and 5.8 A) [7] to explain significant chro-matographic separation. It has been shown with mandelic acid thatthe phenyl group preferably enters the �-CD cavity, but there is alsoenough volume for cyclohexane to enter either interchangeablyor simultaneously. Other studies have demonstrated this type ofinteraction, as in fenoprofen [12] where one enantiomer prefers theinsertion on one benzene ring into the �-CD, while the other prefersboth benzene rings (161 A3 van der Waals volume) to form interac-tions within the �-CD. Combined, benzene and cyclohexane havea van der Waals volume 178 A3 compared to the internal volumeof �-CD 262 A3 [13]. Due to CD’s ability to undergo conformationalchanges and that they have two open ends, it is possible for evenlarger molecules to enter �-CD partially [14]. Since cyclohexaneand �-CD may change their spatial conformation, one enantiomerof CHPGA may have this type of interaction while the other enan-tiomer has only the original benzene interaction. Chromatographicretention and laboratory selectivity have been demonstrated toimprove as the hydrogen on the chiral center of mandelic acidis replaced with larger substituted groups [3], evidence that thesecond group plays an important role in increasing the labora-tory selectivity of the retention interaction. Therefore the proposalthat cyclohexane plays a role in the interaction of retention willbe incorporated into the model. d-CHPGA (D cell) is retained theleast in the system studied [3] and will be modeled with the phenylgroup entering the �-CD interior, while l-CHPGA (A cell) will bemodeled with the phenyl and/or cyclohexane interacting with the�-CD interior.

One of the strongest bonding forces occurring between enan-tiomers and the �-CD cavity is hydrophobicity. Benzene andcyclohexane combined have a maximum hydrophobic bondingpotential (W) at 10.4 kcal/mol. Since neither the phenyl or cyclo-hexane group have additional groups attached, steric hindrancewill not be included.

Once more, a breaking probability of 0.5 means that there isequal bonding attraction and repulsion force between the ana-lyte and �-CD, which is the starting value for PB. van der Waals(8.8% contribution) and hydrophobic bonding (91.2% contribution)combine to have a total potential bonding energy of 11.4 kcal/mol.

Taking into account the attracting interactions involved, theresulting equation for the breaking probability of A1–3 and D1–3with B0 and B1 is shown in Eq. (5):

PB(B0,1) = 0.5 −(

V

178.5 Å3

× 0.040 + W

10.4 kcal/mol× 0.410

)

(5)

where V is the van der Waals volume of portion of enantiomerin CD cavity, 178.5 A3 is the benzene and cyclohexane van derWaals volume, 0.040 is the van der Waals contribution to 0.45 ofPB(B0,1), W is the hydrophobic bonding energy of enantiomer in CDcavity, 10.4 kcal/mol is the modeled maximum hydrophobic bond-

ing potential and 0.410 is the hydrophobic contribution to 0.45 ofPB(B0,1) (see Table 4).

As in Eq. (2), the phenyl or cyclohexane groups will not have anattraction to the primary hydroxyl groups due to their hydrophobic

78 D. DeSoi et al. / J. Chromatogr. A 1291 (2013) 73– 83

Table 4Bonding energy contributions for PB(B0,1) with A and D1–3.

Max kcal/mol % contribution Contribution to0.45 of PB(B0,1)

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0.0

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Cell

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2 per. Mov. Avg. (Bro mpheni ram ine (S))

2 per. Mov. Avg. (Brompheniramine (R))

van der Waals 1.0 8.8 0.040Hydrophobic 10.4 91.2 0.410

ature. Therefore, there should be a repulsion between the two. Eq.6) is identical to Eq. (2) except the maximum hydrophobic energys now increased:

B(B2) = 0.5 +(

W

10.4 kcal/mol× 0.45

)(6)

here W is the hydrophobic energy of enantiomer in CD cavity,0.4 kcal/mol is the modeled maximum hydrophobic energy poten-ial and 0.45 is the maximum inhibition of PB(B2).

Repulsion due to the hydrophobicity of the phenyl cyclohex-ne groups will increase the value of PB(B2) with sides A1–3 and1–3 of the CHPGA enantiomers. Using the above described equa-

ions, breaking probabilities with corresponding joining factors arealculated for the CHPGA enantiomers (see Table 5).

. Results and discussion

.1. Correlation of model results for chromatographic scaleeparations for mandelic acid and brompheniramine

.1.1. Conversion of CA output in Excel graphsBefore the model data output can be compared to the selectivity

f laboratory generated chromatograms, it must be transformedrom columns of numbers in a Windows Notepad file into an ExcelX Y (Scatter)” graph (© Windows Corporation). In order to do thiseveral steps are involved.

The cellular automata model output is a data file that reportsell population, location of D (R-enantiomer) and A (S-enantiomer)

f mandelic acid, and brompheniramine in the following format:

Run iterationRow

able 5reaking probability and joining factors for l-CHPGA and d-CHPGA with �-CD cells.

PB J

l-Cyclohexylphenylglycolic acidA0 B0 0.0500 3.350A0 B1 1.000 0.1260A0 B2 1.000 0.1260A1 B0 0.0500 3.350A1 B1 0.0500 3.350A1 B2 0.9500 0.1500A2 B0 0.0500 3.350A2 B1 0.0500 3.350A2 B2 0.9500 0.1500A3 B0 0.0500 3.350A3 B1 0.0500 3.350A3 B2 0.9500 0.1500

d-Cyclohexylphenylglycolic acidD0 B0 0.0500 3.350D0 B1 1.000 0.1260D0 B2 1.000 0.1260D1 B0 0.3008 1.409D1 B1 0.3008 1.409D1 B2 0.6990 0.3560D2 B0 0.3008 1.409D2 B1 0.3008 1.409D2 B2 0.6990 0.3560D3 B0 0.3008 1.409D3 B1 0.3008 1.409D3 B2 0.6990 0.3560

Fig. 4. Excel chromatogram of brompheniramine enantiomers.

• A average population at the above iteration• A standard deviation• D average population at the above iteration• D standard deviation

The run iteration is the iteration in the run that is being observed.Since runs go into thousands of iterations, it was decided to recordevery 10th iteration. The row represents how far through the col-umn a cell has moved, with row 204 being the end of the modeledcolumn. The cell average population is the average population ofcells A or D at that row at a specific point in time (e.g. iteration).For the analysis of mandelic acid and brompheniramine, each runreported was the average of 60 runs.

Enantiomers cell location of A and D were examined when 90%(or 90 of the 100 cells) were past the last �-CD cells at row 204.With 90% of the enantiomer cells eluted from the stationary phase,the cell population was sufficient to generate Excel simulated chro-matograms. At lower percentages of elution, peaks were distortedwith a tailing shoulder that represented enantiomer cells still in thestationary phase. This model measurement is taking a snapshot ofcell location of every enantiomer cell at a particular iteration. Notethat the number of iterations for A and D will be different depend-ing on their retention by the �-CD cells resulting a in degree ofseparation. Once an iteration is determined for the average num-ber of cells A where 90 have moved beyond row 204, all row datafor that iteration is imported into Excel. This is repeated for D cells.Since this results in a large number of rows (i.e. 800 rows), the sumof every 10 rows is determined.

3.1.2. Comparison of model chromatograms to model outputExcel simulated chromatograms are generated for brompheni-

ramine (see Fig. 4) and mandelic acid enantiomers (see Fig. 5). Thesimulated chromatograms were generated using a “X Y (Scatter)”chart with straight lines and a moving average of 2. It shouldbe noted that peaks are in order of row location (their spatialdistribution), therefore they are in the opposite order of typicalchromatograms. Additionally the peaks themselves are reversedin shape with tailing being on the left side of the peak. For thisportion only selectivity is being evaluated and compared to pub-lished results, therefore peak shape and elution order will not bealtered since this will not affect selectivity calculations.

In high performance chromatography, the selectivity betweenpeaks on a chromatogram is calculated by:

= kB

kA(7)

D. DeSoi et al. / J. Chromatogr

0.0

2.0

4.0

6.0

8.0

10.0

12.0

14.0

16.0

18.0

130 180 230 280 330

Cell

Popu

la�o

n

2 per. Mov. Avg. (Brompheniramine (S))

2 per. Mov. Avg. (Bro mpheni ram ine (R))

wkt

aemetR

hbtrwwacgrae

datrsr

3s

3

teiFTts

tsrm

Correcte d Row

Fig. 5. Excel simulated chromatogram of mandelic acid enantiomers.

here kB (=(tR − t0)/t0) is the retention factor of analyte B,A(=(tR − t0)/t0) is the retention factor of analyte A, tR is the reten-ion time of analyte and t0 is the void time.

Since the Excel simulated chromatograms are not traditionalnd do not have a void time, selectivity was evaluated by two differ-nt techniques. First by finding the row location of the populationaximum for the enantiomer. The ratio of the row location for each

nantiomer set was determined. Another approach was to takehe ratio of the number of iterations to elute for the enantiomers.esults were the compared to published results.

Mandelic acid had a laboratory selectivity of 1.05, resulting inardly any separation [6]. Evaluating the model results, selectivityy maximum row population gave a value of 1.01 and by itera-ion a value of 1.01. While these values are lower than laboratoryesults, the model accurately predicts that insufficient separationill occur. Brompheniramine has a greater laboratory separationith a selectivity of 1.13, while the model predicts a row and iter-

tion selectivity of 1.12 and 1.12, respectively. Here, the modelorrectly predicts brompheniramine enantiomer selectivity withreater chromatographic separation. Therefore, the model accu-ately predicts the lack of chromatographic selectivity in mandeliccid while predicting greater selectivity with brompheniraminenantiomers.

Predicting chromatographic selectivity is useful; however, itoes not consider all of the measurements on how well the peaksre separated. Brompheniramine enantiomers have greater selec-ivity but are still not baseline resolved (see Fig. 4). Peak tailing andesolution provide even more information on prediction of peakeparation. To accomplish this, more information from the modelesults is needed.

.2. Correlation of model results for chromatographic scaleeparations for CHPGA

.2.1. Expansion modeled Excel simulated chromatograms dataIn predicting the chromatographic separation of CHPGA enan-

iomers, more than selectivity is needed. Peak shape needs to bexamined. To accomplish this more information from the models extracted. Along with flow rate, void time will be determined.rom this, capacity factors will be compared to published values.ailing factors and resolution will be determined manually fromhe simulated chromatograms. With this information, enantiomereparation is examined more closely.

It was previously demonstrated that a run with 60 repeti-

ions produced simulated chromatograms with appropriate peakhape. To run with a number of repetitions that produces reliableesults, but is also efficient, several runs were made and the chro-atograms were evaluated. Runs with 10, 50, and 100 repetitions

. A 1291 (2013) 73– 83 79

were performed. Ten repetitions resulted in a peak shape that wasdistorted and not ideal. Although for examining selectivity and arough estimation for resolution, 10 repetitions may be sufficient.Fifty repetitions had appropriate peak shape with a single maxi-mum and peak tailing. One hundred repetitions did not improvethe precision of selectivity and only a slight improvement on peakshape. Therefore 50 repetitions was chosen, and all runs preformedgoing forward were run this way.

Void retention (I0 in iterations) was determined by settingbreaking probabilities of the analytes equivalent to the mobilephase and determining how many iterations it takes for unretainedcells to move through the stationary phase cells beyond row 205.Flow rate (rows/iteration) was then determined by taking the 205rows the unretained analytes traveled and dividing by the voidretention. Retention factors may now be determined:

k = (IR − I0)/I0T

(8)

where IR is the retention iterations of analyte, I0 is the void retentioniterations and T is the tailing factor of analyte.

The number of retention iterations represents when 99% (or 99of the 100 cells) are past the last �-CD cells at row 204. This isan increase in the percentage of 90% from the previous analysisof mandelic acid and brompheniramine. Measuring at 99% elutionfrom the �-CD stationary phase gave greater repeatability of thepeak shape; however, the retention iteration is the iteration forelution at the tail end of the peak. To account for peak shape, theretention factor is divided by the tailing factor. Selectivity is nowcalculated directly from Eq. (9).

= kA

kD(9)

where kD is the retention factor of analyte D and kA is the retentionfactor of analyte A.

As before with mandelic acid and brompheniramine, once aniteration is determined for the average number of cells A, where99 have moved beyond row 204, all row data for that iteration isimported into Excel. This is repeated for D cells. Every 10 rows aresummed.

Enantiomer population row position is now determined byknowing the enantiomer iteration, row position, and void iteration:

RowA = Row × (IR − I0)10, 000

(10)

where RowA is the row relative position adjusted for iteration, Rowis the row position unadjusted, IR is the retention iterations of ana-lyte, I0 is the void retention iterations and 10,000 is the factor tonormalize RowA to a manageable number.

With the row position now representing a relative position tothe time the enantiomers were retained by the stationary phase toeach other and the void retention, simulated chromatograms aregenerated for CHPGA using a “X Y (Scatter)” chart with straight linesand a moving average of 6. Using a straight line plot gave point topoint lines, making it difficult to determine the peak tangent linesfor calculation of resolution. A moving average of 6 was used inthe Excel plots to better define the peaks curvature that eliminatedthis problem. Note that the peaks are reversed in shape with tailingbeing on the left side of the peak. To correct this, the row popula-tions of each enantiomer are reversed and re-plotted resulting in amore conventional chromatogram (see Fig. 6).

3.2.2. Comparison of model chromatograms to model output3.2.2.1. Temperature impact on chromatography. When complexesform between the enantiomers of CHPGA with �-CD, the strengthof those complexes can be affected by temperature [15]. Because

80 D. DeSoi et al. / J. Chromatogr. A 1291 (2013) 73– 83

-1.0

0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

8.0

0 20 0 40 0 60 0

Cell

Popu

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n

Row po si�on adj uste d for iter a�on

6 per. Mov. Avg. (l-cyclohexylphenylglycolic acid)

6 per. Mov. Avg. (d -cyclohexylphe nylgl ycolic acid)

Ft

itbrtid[4ar

pett0aitciG

R

wiah

dho

Fv

y = -0.005x + 1.688

y = -0.005x + 1.699

1.35

1.40

1.45

1.50

1.55

1.60

20 25 30 35 40 45 50 55 60

Sele

c�vi

ty

Temperature, °C

Lab Selec�vity

Model Selec�vity

ig. 6. Excel simulated chromatogram of cyclohexylphenylglycolic acid enan-iomers at 24 ◦C.

nteractions of analytes with the interior of �-CD are slow relativeo other chromatographic interactions [3], slower mass transferetween the enantiomers and the interior of the �-CD can affect theesolution of the enantiomers. Higher temperatures can increasehe speed of this mass transfer, decreasing retention time whilencreasing peak resolution as in the modeled chromatographic con-itions of 24 ◦C and 57 ◦C. This was demonstrated in the laboratory3]. The chromatographic system in Feitsma’s was run at 24, 31, 37,4, 50, and 57 ◦C. Using the solvent cell populations from Table 1nd model probabilities and factors from Table 3, the model wasun at these temperatures.

Model results and published laboratory results [3] are com-ared in Fig. 7. It was first observed that the model predictednantiomer resolutions did increase with an increase in modeledemperature, as in the laboratory results. However, the slopes ofhe model resolutions differed from laboratory results at 0.012 vs..006, respectively. As can be seen in Fig. 7, the plots of laboratorynd model resolutions do not overlap. In fact the greatest differences at the lowest temperature of 24 ◦C. This difference decreases ashe temperature is increased, with the laboratory and model resultsonverging at 50 ◦C. Upon examination of the laboratory results,t was discovered that peak resolution was calculated assuming aaussian shaped peak [3] using the equation:

= (tR,l − tR,d)2(�l + �d)

(11)

here R is the resolution, tR,l is the retention time of l-CHPGA, tR,ds the retention time of d-CHPGA, � l is the peak width of l-CHPGAt 1/2 peak height and �d is the peak width of d-CHPGA at 1/2 peakeight.

Of the three chromatograms provided in the publication atifferent temperatures (24, 31, and 57 ◦C), they were found toave tailing factors that ranged from 3.9 to 2.3. The assumptionf a Gaussian shaped peaked therefore appeared non-ideal. The

y = 0. 006x + 1. 267

y = 0. 012x + 0. 938

y = 0.011x + 0.894

1.00

1.10

1.20

1.30

1.40

1.50

1.60

1.70

1.80

20 30 40 50 60

Reso

lu�o

n

Temperature, °C

Lab Reso lu�on

Model Reso lu�on

Lab Resolu�on Recalculated

ig. 7. Chromatographic resolution of cyclohexylphenylglycolic acid enantiomerss. temperature.

Fig. 8. Chromatographic selectivity of cyclohexylphenylglycolic acid enantiomersvs. temperature.

chromatograms from the publication were enlarged on a copymachine and the resolutions were calculated manually. It wasfound that the laboratory resolutions where now lower than thepreviously reported laboratory resolutions, with a slope of 0.011.The change in model resolution was now more parallel to labora-tory results. Model results also fall in between the two differenttechniques of interpreting reported laboratory resolutions.

There was still the question as to why model results andreported laboratory results converge as temperatures increase,agreeing at 50 and 57 ◦C. Laboratory chromatograms show that d-CHPGA tailing factor decreases from 3.88 at 31 ◦C to 2.25 at 57 ◦C.While l-CHPGA decreased slightly from 2.67 to 2.50. Therefore thepeaks are becoming more Gaussian at higher temperatures and Eq.(11) is more appropriate to use as temperatures increase. As thepeaks become more Gaussian, resolution values from Eq. (11) willapproach values calculated from baselines peak widths as in themodel.

Since higher temperatures decrease retention time of the enan-tiomers by increasing the speed of mass transfer, selectivitybetween the peaks may decrease as temperature increases. Thiswas demonstrated in the laboratory [3]. Model results and pub-lished laboratory results are compared in (see Fig. 8). Selectivityvalues of the model are nearly equivalent to the laboratory values.When plotted vs. temperature, the decreasing slope of selectivitieswith increasing temperature of both laboratory and model resultsnearly overlap with identical slopes of −0.005.

Model resolution values fall between two different techniquesof interpreting laboratory peak resolution and the differencesbetween model and laboratory results are explainable due to peakshape. Therefore, it is correct in concluding that the model accu-rately predicts the resolution of CHPGA enantiomers under varyingtemperature conditions. Additionally, the model accurately pre-dicts the selectivity of the laboratory chromatographic results.By this technique, the separation of CHPGA enantiomers can bemodeled for chromatographic temperature optimization.

3.2.2.2. pH impact on chromatography. Changes in pH have beenshown to have an effect on the chromatographic separation ofCHPGA [3]. Mandelic acid, which is structurally similar, waspreviously modeled where selectivity was shown to be mainlydependent on the interactions of the alpha hydroxyl acid with thesecondary hydroxyl groups of the �-CD. It is not unexpected thenthat the chromatographic separation of mandelic acid, with a pKa

of 3.85 [5], is influenced by pH. Using the same buffered mobilephase (pH of 4.2) from the separation of CHPGA enantiomers, man-delic acid enantiomers separated very slightly. When the pH wasincreased to 6.5, the mandelic acid enantiomers were no longer

separated [3].

CHPGA only differs from mandelic acid by the substitution of thehydrogen on the chiral carbon of mandelic acid with cyclohexane.At a pH of 4.2 the enantiomers of CHPGA separate under laboratory

D. DeSoi et al. / J. Chromatogr. A 1291 (2013) 73– 83 81

-1.0

1.0

3.0

5.0

7.0

9.0

0 20 40 60 80 100 120

Cell

Popu

la�o

n

Row po si�on

6 per. Mov. Avg. (l -cyclohexylphe nylgl ycolic acid)

6 per. Mov. Avg. (d-cyclohexylphenylglycolic acid)

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c

0.95

1.05

1.15

1.25

1.35

1.45

1.55

1.65

0.0 20.0 40.0 60.0 80.0 100.0

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Rete

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Rela�ve Concentra�on

d-CHPGA lab reten�on

l-CHPGA lab reten�on

d-CHPGA model reten�on

l-CHPGA model reten�on

ig. 9. Excel simulated chromatogram of cyclohexylphenylglycolic acid enan-iomers at pH 6.5.

onditions [3] with a resolution of 1.14. When the pH of the phos-hate buffer in the mobile phase is raised to 6.5, the resolution ofhe peaks decreases to 0.67 under laboratory conditions. Althoughot fully understood, it was hypothesized by the authors that this

s the result from differences in dissociations of the alpha hydroxycids. This results in a change of hydrogen bonding with the sec-ndary hydroxyl groups of the �-CD, stronger for mandelic acid andHPGA at a pH of 4.2 and weaker at 6.5. This hypothesis was inves-igated by modeling changes in the breaking probability of A0 and0 (alpha hydroxy acid portion of CHPGA) with cell B0 (secondaryydroxyl groups of the �-CD) from 0.05 for strong interaction to.50, a neutral interaction. Results of this model can be found inig. 9.

Both the laboratory and model’s resolution decreased signif-cantly from a mobile phase pH of 4.2 to 6.5. The resolutionifferences between the laboratory and model results are small±0.1) at both pHs. As can be seen visually in the simulated chro-

atograms (Fig. 6 for pH 4.2 and Fig. 9 for pH 6.5), at a pH of.2 the CHPGA enantiomer peaks are clearly retained longer basedn row position and peak overlap versus a pH 6.5. Peak retentionhange is also consistent with laboratory results [3], retention timesecreasing with a pH of 6.5.

Peak selectivity was likewise compared. At a pH of 4.2, bothhe laboratory and model results had a selectivity of 1.6. When theH of the mobile phase was changed to 6.5, laboratory selectiv-

ty decreased to 1.3, while the model selectivity decreased to 1.1.lthough slightly different, the selectivity results both decreasedignificantly.

Based on the model’s changes in peak resolution and selectiv-ty at different pH, results agree with laboratory outcomes. The

odel’s results support the hypothesis that a change in hydrogenonding between the alpha hydroxy acid group of CHPGA with theecondary hydroxyl groups of the �-CD plays a role in the enan-iomeric separation. To verify this experimentally, CHPGA coulde structurally altered to replace the carboxylic acid and hydroxylroup with structures incapable of hydrogen binding while main-aining the chiral center and not causing steric interference that

ight change the interaction with �-CD. Run under the same chro-atographic system, the separation should be similar to that of

HPGA at a pH of 6.5.

.2.2.3. Injection volume impact on chromatography. In high per-ormance liquid chromatography, retention of analytes on thetationary phase is affected by many factors. Selection of the propertationary phase and mobile phase are critical; however, there areome factors that effect analyte retention that are not as easily to

nderstand. In a linear chromatographic system, the distributionf analyte in the column is represented by [16]:

S = KC × cM (12)

Fig. 10. Relative concentration of enantiomers versus their relative retention underlaboratory and model conditions.

where cs is the concentration of analyte in stationary phase, cM is theconcentration of analyte in mobile phase and KC is the distributionconstant.

As can be seen from Eq. (12), analyte movement through thecolumn should not be impacted by overall analyte concentrationresulting in a linear isotherm. Under ideal conditions, changes insample concentration should not change the analytes retentiontime.

In chromatographic systems where there is peak tailing, convexadsorption isotherms typically predominate resulting in increasedretention time with decreased sample amounts [17]. However,when sample amounts are low, all distribution isotherms are likelyto be linear [16].

It has been observed with CHPGA enantiomers that as theamount of sample injected onto the column is reduced, the reten-tion time of the analytes increases [3]. At the lowest concentrationsthe increase in retention time is greatest and still increasingin a non-linear manner. This phenomenon is not explained bytraditional convex isotherms, since at lower concentrations theadsorption isotherm should become linear [16].

Laboratory conditions involved injecting approximately 11 �gof each CHPGA enantiomer onto the chromatographic system andsequentially reducing the amount in half, for concentration valuesof 100, 50, 25, 12.5, 6.3, 3.1, and 1.6% of the original concentration.To model this, enantiomer concentrations were changed similarlyin concentration. The number of cells for each enantiomer has been100 for each. Enantiomer cell concentrations were reduced from100 to 50, 25, 13, 6, 3, and 2 cells.

Determining retention time of enantiomers in the model couldno longer be determined by 99% elution and peak shape as hasbeen done to this point, since at lower cell concentration (i.e. 2 and3 cells) there is no tradition peak shape. At these low levels of enan-tiomer cells, there is no cell population distribution to plot in Excel.Instead, retention time will be relatively compared by determiningthe iteration when the average cell concentration of one cell is pastthe stationary phase or beyond row 205. Using this approach, allcell concentrations can be compared in the same manner.’

Model results at the different concentrations were then com-pared to published laboratory results.

Sample amounts were relative to the 100% level (100 cellsfor each enantiomer) and rounded to eliminate decimals, sincecell population must be modeled in integer values. For laboratoryresults, retention times were normalized against the retention at100% sample concentration. Model retention iterations were nor-malized against the retention at 100 cell population. In both thelaboratory and model results, the retention of the enantiomersincreased as sample amount decreased. As can be seen by theplots through the modeled and laboratory data points, Fig. 10, the

model’s retention was nearly identical to laboratory results untilrelative concentration levels reached 6% and below. Unlike labo-ratory conditions where dilution levels of 2% still result in many

82 D. DeSoi et al. / J. Chromatogr

0.951.001.051.101.151.201.251.301.351.40

0 20 40 60 80 100

Rela

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Rete

n�on

Rela�ve Concentra�on

d-CHP GA mode l ret en�on , PB = 0. 20

d-CHPGA model reten�on, PB = 0.35

d-CHPGA model reten�on, PB = 0.50

d-CHPGA model reten�on, PB = 0.80

d-CHP GA lab ret en�on

Fdm

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ig. 11. Relative concentration of d-cyclohexylphenylglycolic acid enantiomer atifferent breaking probability factors versus relative retention under laboratory andodel conditions.

nalyte molecules interacting, the model is limited to just 2 analyteells. The model does predict the increase in retention with changesn analyte concentration; however, it’s accuracy of the degree ofhange is limited when less than 13 cells were modeled.

There was peak tailing under all conditions [3] in the chromato-raphic system studied, therefore, convex adsorption isothermsppear to drive the retention. Yet, even at the most diluted sam-les the isotherms do not become linear as would be expected [16].his observation was not explainable by the authors. With all otherarameters being held constant, it was only the sample concen-ration that was changing and affecting retentions. The question issked, since analytes have the ability to interact with each other, ifhe tendency towards analyte-to-analyte interaction changes, doeshis impact the retention of analytes and how would this affect aonvex isotherm?

Previous model runs had a breaking probability between enan-iomers of 0.5, meaning there is equal attraction and repulsionorce between them. These breaking probabilities between enan-iomer cells A0–3 and B0–3 will be changed to have less interactionetween themselves with a value of 0.80. Analytes will be modeledor greater interaction between themselves with breaking proba-ilities of 0.20 and 0.35. Model runs and their interpretation will behe same as before.

Changing the strength of the interactions between enantiomersad an impact on the phenomenon of increasing retention timeith decreased sample amounts. This can best be examined by

ooking at how the individual enantiomers changed versus labo-atory results graphically (see Fig. 11).

In Fig. 11, the initial breaking probability conditions for d-CHPGAnantiomer-to-enantiomer interactions are displayed versus theaboratory results. As before, the model values for relative reten-ion versus the 100% original amount of the enantiomer agrees withaboratory conditions at samples amounts greater 6%. When thereaking probability of enantiomer interactions is increased to 0.80,odeling less enantiomer interactions, the relative retention val-

es do not change significantly from a breaking probability of 0.50.t may be that once repulsion between analytes reaches a certaintrength and rapidly become dispersed in mobile phase, retentionecomes solely driven by interactions with the stationary phaseince analyte cells are no longer near each other. Therefore con-inuing to increase the analyte-to-analyte repulsion has minimalmpact.

At sample amounts greater 6% the values remain similar to lab-ratory results, and at 6% and less retention increases significantly.hen the analyte-to-analyte breaking probability is lowered to

.20 and 0.35 the results change.A breaking probability of 0.20 represents a stronger attraction

etween CHPGA cells. Model results show that this type of attrac-

ion shows no change in relative retention with dilution of theample down to 25%. In this range there appears to be a linearsotherm. At dilution values of 13 and 6%, relative retention beginso increase but is still below laboratory results. Not until dilution

. A 1291 (2013) 73– 83

values of 3 and 2% do model values exceed experimental resultsand retain similarly as they did at higher breaking probabilities. Abreaking probability of 0.35 was modeled for the enantiomer withrelative retention results falling between the breaking probabilitiesof 0.20 and 0.50. Depicting a movement from a linear to a convexadsorption isotherm. Similar results were found for l-CHPGA.

This phenomenon is not explained by just a convex isotherm.Although a convex isotherm is present, it is proposed that analyte-to-analyte interaction is playing a role in retention in additionto a convex isotherm adsorption of analyte-to-stationary phase.As the attraction between analytes increases there is less of anincrease in overall relative retention as sample amounts decreased.Additionally, as the sample concentration becomes so low thatanalyte-to-analyte interactions become less likely to occur, the lesssignificant analyte attraction has on retention. Since changing theinteractive forces between enantiomers would involve changingthe mobile phase composition to make the environment more orless likely for solvation of the enantiomers, hence affecting analyteretention, this phenomenon would be difficult to analyze underlaboratory conditions.

4. Conclusions

A cellular automata model was developed to analyze and pre-dict the retention and chromatographic separation of enantiomerson �-CD stationary phases. One-to-one interactions do not takeinto account many factors that affect chromatographic separations.Therefore the model was expanded to represent a chromatographicscale. From the one-to-one modeled interactions, mandelic acidand brompheniramine were run under chromatographic modelconditions to compare results. The model accurately predicts thelack of chromatographic selectivity in mandelic acid while alsopredicting greater selectivity with brompheniramine enantiomers.

To further examine the chromatographic predictability of themethod, CHPGA enantiomers were modeled for separation undervarying chromatographic conditions. The model accurately pre-dicted both the resolution and selectivity of the enantiomer peaksat varying temperatures, while modeled changes in mobile phasepH of 4.2–6.5 agree with laboratory outcomes when examiningthe loss of peak resolution and selectivity at pH 6.5. Changes insample amount on the column resulted in an increase in retentiontime of the modeled enantiomers as was observed in the pub-lished laboratory results. As a model for one-to-one enantiomerbinding interactions with �-CD, it has proven to be accurate in theprediction of binding strengths. In its expansion to the chromato-graphic scale, the model has been proven rugged under the varyingchromatographic conditions studied that affect peak separation.To develop new equations for other enantiomers only requires abasic understanding of their chemical structure to determine theirpotential interactions occurring in CD separation, thus only takinghours to develop and run simulations under varying chromato-graphic conditions. Used as a tool for method development, themodel has potential for reducing the time and cost in enantiomerseparations using CDs.

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59.[4] A. Chaudhari, S. Lee, Int. J. Quantum Chem. 102 (2004) 106.

[5] R.C. Weast, CRC Handbook of Chemistry and Physics, 69th ed., CRC Press Inc.,

Boca Raton, FL, 1988, pp. D-162, F-10.[6] D. Durham, Chirality 8 (1996) 58.[7] T. Kakitani, H. Kakitani, S. Yomosa, Biophys. Struct. Mech. 7 (1980) 101.[8] D. Dimitrova, S. Ilieva, B. Galabov, SAR QSAR Environ. Res. 15 (2004) 311.

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[

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[9] S. Ege, Organic Chemistry: Structure and Reactivity, 5th ed., Houghton Mifflin

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10] L.B. Kier, P.G. Seybold, C. Cheng, Modeling Chemical Systems using CellularAutomata, 1st ed., Springer, Netherlands, 2005.

11] Y. Zhoa, M. Abraham, A. Zissimos, J. Org. Chem. 68 (2003) 7368.12] Y.H. Choi, C. Yang, H. Kim, S. Jung, Carbohydr. Res. 3 (2000) 393.

[[[[

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13] W. Saenger, J. Jacob, K. Gessler, T. Steiner, D. Hoffmann, H. Sanbe, K. Koizumi,

S.M. Smith, T. Takaha, Chem. Rev. 98 (1998) 1787.

14] W. Chen, C. Chang, M. Gilson, Biophys. J. 87 (2004) 3035.15] W.L. Hinze, Sep. Purif. Methods 10 (2) (1981) 159.16] J.A. Jonsson, Pure Appl. Chem. 68 (8) (1996) 1591.17] E. Heftmann, J. Chrom. Library 22A (1983) A32.