a cellular automata model of chromatography

A cellular automata model of chromatography

Lemont B. Kier1, Chao-Kun Cheng2 and H. Thomas Karnes3*1Department of Medicinal Chemistry, Virginia Commonwealth University, Richmond, VA 23298, USA2Department of Mathematical Sciences, Virginia Commonwealth University, Richmond, VA 23298, USA3Department of Pharmaceutics, Virginia Commonwealth University, Richmond, VA 23298, USA

Received 29 January 2000; accepted 16 February 2000

ABSTRACT: Dynamic models of the behavior of solvent and solute molecules can be made using cellular automata. Achromatographic column was represented by use of a cellular automata grid of 43� 200 spaces. Solvent (mobile phase), solute andstationary phase cells were designated to simulate the chromatographic situation. The movements of solute and solvent cells down thegrid were monitored for different numbers of iterations, different flow rates and different affinities of the solutes for the stationaryphase and the solvent for itself. The cellular automata dynamics were successfully able to model expected chromatographic behaviorexcept in a few cases where the number of cells was not large enough to provide an average value reflective of the molecular situation.Copyright# 2000 John Wiley & Sons, Ltd.

INTRODUCTION

Chromatographic processes involve various modes ofphysical separation of solutes based on their differinginteractions with a stationary packed bed of material anda mobile fluid or gas flowing through it. A number ofautomated systems have been developed and employed topredict chromatographic separation behavior based onempirical results (Snyderet al., 1986). A theoreticallybased computational model for chromatographic reten-tion and separation would allow the prediction ofchromatographic behavior for perturbations of the systemthat cannot easily be duplicated experimentally. Cellularautomata are proposed and evaluated in this work as sucha computational system to model chromatographicprocesses.

THE GENERAL MODEL FOR CELLULARAUTOMATA

Cellular automata are dynamic computational systemsthat are discrete in space, time and state and whosebehavior is specified completely by rules governing localrelationships. They are an attempt to simplify the oftennumerically intractable dynamic simulations into a set ofsimple rules that mirror intuition and that are easy tocompute. As an approach to the modeling of properties ofcomplex systems they have a great benefit in beingvisually informative of the progress of dynamic events.From their early development by von Neumann (1966), avariety of biological applications ranging from kinetics to

solution phenomena have been reported (Ermentrout, andEdelstein-Keshet, 1993, Chopard, and Droz, 1998; Kier,et al., 1999; and others).

Our model is composed of a grid of spaces called cellson the surface of a torus to remove boundary conditions.Each cell i has four tessellated neighbors,j, and fourextended neighbors,k, in what is called an extended vonNeumann neighborhood (Fig. 1). Each cell has a stategoverning whether it is empty or occupied by a solvent orsolute molecule. The contents of a cell move, join withanother occupied cell or break from a tessellatedrelationship according to probabilistic rules. These rulesare established at the beginning of each simulation. Therules are applied one after another to each occupied cell atrandom; the complete application of the rules to alloccupied cells constitutes one iteration. The rules areapplied uniformly to each cell type and are local, thusthere is no interaction at a distance. Our cellular automatamodel is kinematic, asynchronous and stochastic. Theinitial conditions are random hence they do not determinethe ultimate state of the cells, which is called theconfiguration. The same initial conditions do not there-fore yield the same configuration after a certain numberof iterations. The configuration achieved after manyiterations reach a collective organization that possessesrelative constancy in reportable counts of the distributionof cells. What we observe and record from the cellularautomata simulations are emergent cellular distributionsas a result of this complex system.

THE MOLECULAR SYSTEM

In order to understand how the model relates to thesystem under study, it must be made clear just what the

BIOMEDICAL CHROMATOGRAPHYBiomed. Chromatogr.14: 530–534 (2000)

*Correspondence to: H. T. Karnes, Department of Pharmaceutics,Virginia Commonwealth University, Richmond, VA 23298, USA.

Copyright 2000 John Wiley & Sons, Ltd.

ORIGINAL RESEARCH

cells, the configurationsgenerated,and the cellularautomatamodels represent.A cell with a state valueindicatingoccupationby somemoleculesis not a modelof a molecule with specified electronic and stericfeatures.Theseattributesareconsideredto besubsumedinto the rules. The molecular systemis intermediatebetweenthe molecularlevel andbulk phasemodelsofsystemsandmay be modeledwith moleculardynamicsor cellular automata.Molecular-level phenomenaaremodeled with molecular orbital theory, topologicalindicesor fragmentmethods.At the bulk level we usedescriptionsbased on statistical and thermodynamicmethods. Solution phenomenamight be effectivelystudied using molecular system models in order tounderstandthe processeswhereby single moleculesachieve configurationswhich can be related to bulkdistributions.Severalexamplesof this aredemonstratedin thisarticleincludingthedifferentialretentionof solutemolecules,the influence of mobile phasevelocity onretention,the effect of mobile phasepolarity, solvationof the stationaryphaseandpeakwidth asa function ofmigrationtime.

METHODS

The chromatography model

The simulationsof chromatographicphenomenain thisreportarederivedfrom cellularautomatadynamicson agrid of spaces43by 200,calledacolumn,seeFig. 2. The43� 200 grid was chosenbecauseit was the smallestgrid to provide resolutionof the two test solutes.Othergrid sizesthatweretestedandfailed to providebaselineseparationwere43� 43and43� 100grids.Thesolventcells representingthe mobile phase were randomlydistributedover this spaceat the initiation of eachrun.Thesecells,designatedW, constitute69%of thecells inthegrid beforetheintroductionof anyothercomponents.The stationaryphasecells, designatedB, weremodeledby thepresenceof 600cellsrandomlydistributedoverthe

grid, replacing600W cells.TheseimmobileB cellswerepositionedat least threecells from anotherB cell. Thesolutecells representingtwo different compoundsweremodeledby 10 cells each.Thesereplaceda correspond-ing numberof W cells and were positionedon the firstrow of thecolumnat thebeginningof eachrun.

The movementsof the solutesin the column weregovernedby rulesdenotingthe joining andbreakingoflike or unlike cells (Kier et al., 1995).The relationshipsamongthemobilephasecells,W, stationaryphasecells,B, andsolutemolecules,S1 andS2, aregovernedby thejoining andbreakingprobabilities,designatedPB andJrespectively.A standardsetof ruleswasdevelopedandemployed,andtheseareshownin Table1.Foreachstudythis standardset was used,modified in particularcaseswherea variablewasto beevaluatedby simulation.Thegravity term (Chengand Kier, 1995), representingtheforcepushingthemobilephase,wasuniformly appliedtoall componentsexcept the stationaryphasecells. Thegravity termis literally theprobabilityof acell movingtoa positionfurther downthecolumn.

Eachoccupiedcell, at random,interpretsandemploysits intrinsic stateandits probabilityof movement.Whenall occupied cells have computedand executedtheirtrajectory,this representsone iteration.The iteration inthiscaseis oneunit of ‘time’. Thepositionof eachsolutecell is recordedat a row from 1 to 200 after a certainnumberof iterations.Eachiterationis repeated100timesto achieveanaveragepositionon thecolumnequivalentto 1000solutecells.Thepositionof thepeakmaximumisdeterminedby summingthe numberof cells found ingroupsof 10 rows on the column then plotting theseaveragesagainstthe iteration time. Resultswere thencompared with expected chromatographic behavior(SnyderandKirkland, 1979).

Figure 1. Extendedvon Neumannneighborhood.

Figure 2. Schematicrepresentationof the cellular automata‘column’.

Copyright 2000JohnWiley & Sons,Ltd. Biomed.Chromatogr.14: 530–534(2000)

Cellular automatamodel ORIGINALRESEARCH 531

RESULTS AND DISCUSSION

Therateof migrationthroughthecolumnwascalculatedby recordingthe movementsof two solutes,S1 and S2,with different affinities for the stationaryphase,B. Theaffinities of S1 and S2 for B were defined by theprobabilistic rules governing the joining and breakingwith B. Therule setswerePB(S1B) = 0.90,J(S1B) = 0.10and PB(S2B) = 0.10, J(S2B) = 2.0. These rules werechosento modela solutewhich interactedstrongly(S2)and weakly (S1) with the stationary phase, B. Theprobability of breakingandjoining interactionswith thestationaryphasewould be expectedto vary from theserules.Themagnitudeof differencebetweenprobabilitiesfor S1 andS2 and the magnitudeof differencebetweenbreakingand joining probabilitiesfor eachsolutewerechosenarbitrarily and do not representany particularsolutepair.

Af®nity of solute for the stationary phase

Thisstudymodeledtheinfluenceof therelativeaffinitiesof solutesfor the stationaryphase,B. This affinity isencodedin the parameters,PB(SB) and J(SB). High

valuesof PB(SB) andlow valuesof J(SB) denotea weakaffinity, while reversalof theserelativevaluesdenotesastrongaffinity.

This is evident in the empirically basedchromato-graphicsenseas:

k0 � KAs

Am

wherek' is thechromatographiccapacityfactorandK isthe affinity constant for interaction between solutemoleculesin the stationaryphaserelative to those inthe mobile phase.As and Am representthe areasof thestationaryandmobilephasesrespectively.A singlesolutewas used in the study with five different sets ofparameters.Theresultsof thestudyareshownin Table2.

The solute affinity study revealsthat soluteswith agreateraffinity for the stationaryphasemigrated at aslower rate (higher capacity factor). Theseparameterscharacterizethestructuraldifferencesamongsolutesthatgive rise to retention characteristicsand serve as theprimary basisfor separationin chromatography.

Relative retention of two solutes

Thepositionsof thepeaksof two soluteswererecordedattheline of thegrid columnafterseveraliterationsof time.The line of thegrid columnrepresentsmigrationrateor,in chromatographicterms,theaveragevelocity of solutemolecules.Thenumberof iterationswouldcorrespondtoa ‘column length’ parameterin a chromatographicsenseandcanberepresentedfor chromatographicprocessesasfollows:

tR � I�x

wheretR is the retentiontime takenasthe inverseof the

Table 1. Trajectory parameters

Encounters PB(xy) J(xy) Interpretation

B–B 0.999 0.001 Stationary,at leastthreecells apart

B–S1 0.90 0.20 Affinity for BB–S2 0.10 2.00 Affinity for BB–W 0.90 0.10 Solvationof BS1–S1 0.90 0.10 Interrelationof S1S1–S2 0.90 0.10 Interrelationof S1 andS2S2–S2 0.90 0.10 Interrelationof S2S1–W 0.90 0.10 Solvationof S1S2–W 0.90 0.10 Solvationof S2W–W 0.90 0.10 Relativepolarity of solventG(S1) = 10 Flow rateof S1G(S2) = 10 Flow rateof S2G(W) = 10 Flow rateof S2

Columndimensions= 43� 200cells.Watercell content69.0%minus20 solutesand 600 B. S1 content= 10 cells. S2 content= 10 cells. B(stationary cells)= 600 cells. Run ‘time’ = 600 iterations. Usualnumber of runs= 100. System configuration is a cylinder withingredientsflowing backto the top of thesystem.

Table 2. Effect of soluteaffinity for the stationary phase

PB(SB) J(SB) Capacityfactor (k')

0.10 2.00 1.1930.30 1.20 0.2460.50 0.80 0.1580.70 0.60 0.0960.90 0.20 0.035

Table 3. Retention rates for two solutes

Peakretentiontime (tR, arbitraryunits) Peakwidth (time, arbitraryunits)at half-height

Iterationtime (I) S1 S2 S1 S2

300 0.0118 0.025 30 32350 0.0103 0.020 25 43400 0.0085 0.019 32 47450 0.0076 0.016 24 43600 0.0059 0.013 33 60


532 ORIGINALRESEARCH L. B. Kier et al.

peakposition on the grid, I is the numberof iterationsandmx representsthe averagevelocity of the solutemolecules.It canbeseenfrom theaboveequationthatthenumberof iterationsis expectedto be directly propor-tional to theretentiontimeif themodelbehavedsimilarlyto thoseexperimentallyobservedfor chromatography.Peakpositionshavebeenconvertedinto retentiontimesin Table 3 to facilitate comparisonto the chromato-graphic situation. Theseretention times are noted foreachtime interval,in Table3. Theretentiontime of eachsolutewas found to be linearly relatedto the iterationtime (I). The two relatingequationsare:

Migration line for S1 � 0:29ÿ 1:92� 10ÿ5 (iteration)

ÿ 0:017; r � 0:960

Migration line for S2 � ÿ3:70� 10ÿ5 (iteration)

� 0:034; r � 0:945:

In the caseof the peakwidth, it would be expectedthatthe peakwidth for the more strongly interactingsolute(S2) would begreaterthanthatof theweakerinteractingsolvent (S1) and that the peak width for both solutesshould increasewith larger iteration times. A generaltrendof increasingpeakwidth with iterationtime canbeobservedfrom thedata,but thedatadoesnotappearto bestronglyrelatedin thiswayasexpected.It is possiblethatthe numberof cells wasnot largeenoughto provideanaveragereflectiveof themolecularlevel.Thepeakwidthfor all iterationtimesis greaterfor S2 thanfor S1, whichisconsistentwith what is expectedfor chromatography.

Figure 3 showsthe positional distribution of solutemolecules(S1 and S2) along the grid in 10 line groups.Approximatebaselineseparationis achievedfollowing600 iterations.

The in¯uence of mobile phase ¯ow rate onmigration

The gravity parameterfor eachingredientin the studydefinesthemobilephaseflow rate.All ingredientsexceptthe stationaryphaseB were endowedwith the samegravity termwithin eachstudy.Threevalueswerechosento describethreedifferentflow rates.Thetimeneededforan unretained solute molecule to move through achromatographiccolumn (t0) is inverselyrelatedto themobile phaseflow rateasshown:

t0 � I�

where m is the velocity of the mobile phase.Solutemoleculesshould travel through the column at somefraction of the mobile phasevelocity andretentiontimeof the solute,hencetR shouldbe directly relatedto m.Theseresultsof the simulationof the effect of flow rateare shown in Table 4. As predicted,shorter retentiontimesoccurredwith thesolutesandwatergiventhefasterflow rates. In the case of the gravity term, it wasunexpectedthat the retentiontime for the solvent (W)was greaterthan that for the lower affinity solute(S2).This is likely againto be relatedto the relatively smallnumber of cells as comparedto the large number ofmoleculeswhich constituteaveragepeak retention inchromatography.

Effects of mobile phase solvent polarity onretention

The polarity of the mobile phasesolvent,W, is encodedin therelativeself-associationexperiencedamongmobilephasemolecules.This is governedby the rules,PB(W)andJ(W). High valuesof PB(W) andlow valuesof J(W)simulate a weak self-association,correspondingto arelatively non-polar solvent. The alternativevalues ofthese parameterssimulate a strong self-association,correspondingto a relativelypolarsolventsuchaswater.Threesetsof parameterswereselectedfor this studytoepitomizethe rangeof solventpolarity. The PB(W) andJ(W) values were varied in each set while all otherparameterswereheldconstantusingthestandardvalues.The migrationswere recordedafter 600 iterations,andthe resultsare shownin Table 5. Migration is faster innon-polarsolvents,which is consistentwith a reversed-phasechromatographicsystem.

Figure 3.Thepositionaldistributionof solutemoleculesS1 andS2 along the grid in 10-line groups.The standardparameterslisted in Table1 wereused.

Table 4. Effect of flow rate on the retention of solutes

Peakretentiontime (tR, arbitraryunits)

Gravity (flow rate) S1 S2 W

2 0.02 0.0077 0.00835 0.015 0.0063 0.0061

10 0.013 0.0059 0.0057


Cellular automatamodel ORIGINALRESEARCH 533

Effect of stationary phase solvation

Theeffectof solvatingthestationaryphaseis of interestas this property reflects the degreeto which solventmoleculescompetewith the solute for the stationaryphase.Thevariationin thispropertywasencodedinto thePB(WB) and J(WB) parameters.A low value for thePB(WB) and a high value for the J(WB) parameterdenotesa strong affinity of the solvent, W, for thestationaryphase,B. The oppositeconditionis found forthe reverseof thesevalues.Table6 showsthis influencefor three parametersets.The study revealsa modestinfluence on retention but a greater influence on therelativeresolutionamongthe two solutesin eachstudy.Thegreatertherelativesolvationof thestationaryphase,the poorer the resolution. The resolution, Rs, wascalculatedas:

Rs � tR1ÿ tR2

0:5 �peakwidth of S1� peakwidth of S2�

Theobservationregardingretentiontimesfor thethreeparameter sets was expected since the degree ofassociation of the solvent and stationary phase isindependentof solute–stationaryphaseinteractionsanda significanteffect on retentionwould not be expected.The greater effect on resolution was unexpectedandmight be due to the effect of greatersolvation of thestationaryphaseon peakwidth.

Effect of solute and stationary phase solvation

Thesolvationof boththesolutesandthestationary phaseswasstudiedasan indication of relative solventstrengthusingthevariationsin thePB andJ valuesto encodetheseconditions.Theresultsareshownin Table 7. Thedegreeof difficulty of the solvent–solute interaction greatlyreducesretentiontime, as expectedfor a strong mobile

phase solvent. Greater solute solvation retards themigrationanddecreasesthe resolution.

CONCLUSIONS

The cellular automata system was able to predictexpectedchromatographicbehaviorfor different soluteaffinities for the stationaryphase;the effect of iterationtime and flow rate on retention; the effect of solventpolarity andsoluteaffinity for the stationaryandmobilephases.Experimentscarried out at the systemslevelinvolveonly afew dozeninteractiveelementsandassuchmaynot havesufficientnumbersto accuratelypredictallphenomenawhich areobservedat thebulk level. This isprobablythe explanationfor the lack of a demonstrablerelationshipfor peakwidth anditerationtime. Thesameis true for theobservationthat thesolventwasshowntobemoreretainedthanthesolutefor oneparametersetintheflow-ratestudy.Perhapstheseinconsistencieswoulddisappearwith a greaternumberof resolutionelementsand/orrunsat theexpenseof computationtime.

The only trend clearly observedwith the cellularautomatamodelthat wasnot expectedwasthe apparenteffect that solvation of the stationary phasehad onresolution.The systemlevel observeslocal eventsthatmay be hardto duplicateat the bulk level becausetheyaremaskedby thelargenumberaveragebehavior.Theasyet unexplainedobservationinvolving stationaryphasesolvation could in fact be due to a local eventunobservableat the bulk level andthusdeservesfurtherstudy.

REFERENCES

ChengC.-K. andKier L. B. 1995.J. Chem.Inf. Comput. Sci.35, 1054.Chopard B. and Droz M. 1998. Cellular Automata Modeling of

PhysicalSystems,CambridgeUniversity Press,Cambridge.ErmentroutG. B. andEdelstein-KeshetL. 1993.J. Theor.Biol., 160,

97.Kier L. B., ChengC.-K. andTestaB. 1995.Pharm.Res.12, 615.Kier L. B., ChengC.-K. andTestaB. 1999.Fut. Gen.Comput.Sci.16,

273.SnyderL. R., Dolan J. W. andRigneyM. P. 1986.LC-GC 4,9,921.SnyderL. R. andKirkland J. J. 1979.Introductionto ModernLiquid

Chromatography,2ndEdn.J Wiley: New York.von Neumann,J. 1966,Theoryof Self-ReproducingAutomata,Burks

A. (Ed). University of Illinois Press,Champaigne,IL.

Table 5. The effect of solvent polarity on the migration ofsolutes

Retentiontime (tR, arbitraryunits)

PB(W) J(W) S1 S2

0.90 0.10 0.0125 0.00590.50 0.70 0.0143 0.00770.25 1.50 0.025 0.0167

Table 6. Effect of stationary phasesolvation

PB(WB) J(WB) tR tR Rs

0.90 0.10 0.013 0.0059 1.1250.50 0.80 0.0091 0.0059 0.6670.20 1.50 0.0091 0.0063 0.556

Table 7. Effect of soluteand stationary phasesolvation

Retentiontime(tR arbitraryunits)

PB(WB) = PB(WS) J(WB) = J(WS) S1 S2

0.90 0.20 0.04 0.00590.20 1.50 0.10 0.033


534 ORIGINALRESEARCH L. B. Kier et al.

a cellular automata model of chromatography

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