research article optimization of ic separation based on...

Hindawi Publishing CorporationJournal of Analytical Methods in ChemistryVolume 2013 Article ID 549729 11 pageshttpdxdoiorg1011552013549729

Research ArticleOptimization of IC Separation Based onIsocratic-to-Gradient Retention Modeling in Combination withSequential Searching or Evolutionary Algorithm

Šime UkiT Marko RogošiT Mirjana Novak Ena ŠimoviTVesna Tišler and Tomislav BolanIa

Faculty of Chemical Engineering and Technology University of Zagreb Marulicev trg 19 10000 Zagreb Croatia

Correspondence should be addressed to Sime Ukic sukicfkithr

Received 11 June 2013 Revised 3 October 2013 Accepted 17 October 2013

Academic Editor Miguel de la Guardia

Copyright copy 2013 Sime Ukic et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Gradient ion chromatography was used for the separation of eight sugars arabitol cellobiose fructose fucose lactulose melibioseN-acetyl-D-glucosamine and raffinoseThe separationmethodwas optimized using a combination of simplex or genetic algorithmwith the isocratic-to-gradient retentionmodeling Both the simplex and genetic algorithms providedwell separated chromatogramsin a similar analysis time However the simplex methodology showed severe drawbacks when dealing with local minimaThus thegenetic algorithm methodology proved as a method of choice for gradient optimization in this case All the calculatedpredictedchromatograms were compared with the real sample data showing more than a satisfactory agreement

1 Introduction

Isocratic elution exhibits some advantages over the gradientone such as greater simplicity lower cost simpler instrumen-tation and no need of column reequilibration between con-secutive injections [1] However gradient elution is becomingalmost unavoidable in conventional liquid chromatographyincluding ion chromatography (IC) In gradient mode theelution strength usually increases during analysis thus pro-viding much narrower chromatographic peaks and signifi-cantly shorter analysis time Such characteristics are favoredin cases with multicomponent samples which span a wideretention range [2]

The implementation of gradient elution implies findingthe suitable gradient program The major criterion for opti-mal elution is a good resolution between analytes accompa-nied with an acceptable analysis time Existence of a goodmodel to predict the column output is a crucial item inoptimization of chromatographic elution IC development inthe last several decades was accompanied with the increasingnumber of models either theoretical or empirical which canbe used to predict or to explain experimental chromatograms

[3ndash18] Among numerous approaches the retention isocratic-to-gradient (iso-to-grad) model [12 16] appears to be par-ticularly interesting This model is primarily developed forsystems with a single-competing eluent it is based on thetransfer of retention information from isocratic to gradientelution mode

The linear solvent strength model is the first and prac-tically the most important model that describes the rela-tion between component retention and concentration ofcompeting ion for isocratic elution This theoretical modeldeveloped by Snyder et al in 1979 [3] in its origin considersonly electrostatic effects leading to ion exchange retentionTherefore the presence of othermechanisms or occurrence offactors influencing the retentionwill result in deviations fromthe linear model To overcome this problem some authorsinclude an additional factor as a correction for nonlinearity[12 19 20] in IC and other LC techniques Such empiricallyobtained polynomial models were shown to fit isocraticexperimental data significantly better than the linear oneThe selection of polynomial instead of linear model doesnot have significant influence on the calculation time withcontemporary computers Therefore the authors decided to

2 Journal of Analytical Methods in Chemistry

describe the isocratic behavior with the better fitting modelthat is the polynomial dependence between the logarithm ofretention coefficient 119896 and concentration of competing ionin eluent 119888

log 119896 = 1198860+ 1198861log 119888 + 119886

2log2119888 (1)

In principle only three isocratic experiments are sufficientfor the determination of regression coefficients 119886

0minus2 of the

model described by (1)The retention coefficient is calculated as

119896 =119905119877minus 119905119872

119905119872

(2)

where 119905119877is a component retention time and 119905

119872is the columnrsquos

holdup time For the ion chromatography case the columnrsquoshold-up time is equal to the retention time of unretainedcompounds that is column void time 119905

0[21]

On the other hand gradient elution can be describedgenerally by the integral elution equation

1199050= int

119905119877minus1199050

0

119889119905

119896 [119888 (119905)] (3)

The integral from (3) can be approximated with a sum ofintegrals over small time intervals Inside those intervals thecoefficient 119896 can be assumed constant and can be calculatedas an average of 119896 values at the interval boundaries obtainedaccording to (1) In this way isocratic retention informationis transferred into gradient environment

The iso-to-grad approach allows for the prediction ofretention time for practically any gradient using only sev-eral isocratic experiments Nevertheless the optimizationof gradient elution remains a severe and complex problemespecially knowing that the number of different gradientprograms in any gradient domain is practically unlimitedThe problem can be simplified by setting different constraintson the domain (ie defining the sets of finite intervals inwhich the gradient curve may change finite sets of gradientcurve shapes or finite sets of allowed gradient slopes [22ndash25]) Unfortunately the usage of any constraint involves thepossibility of missing the true optimum An increase ofthe number of allowable options inside each finite set willproduce a finer description of the real experimental domainHowever it will at the same time increase significantly thenumber of possible gradient profiles This leads to anotherproblem a domain of profiles that is too large substantiallyincreases both the experimental effort and requiredmodelingtime which may even exceed the performance of moderncomputers

A mathematical model that is able to describe the opti-mization surface for gradient IC is generally a nonlinear andvery complex function Therefore it is not feasible to solve it(to find optimum) analytically The fact that there are oftenmultiple solutions (local extremes) complicates the situationeven more There are several classes of approaches availablefor solving this kind of problems among which are sequentialsearching procedures and evolutionary algorithms [26]

One of the best known sequential methods is the Nelder-Mead simplex method [27] The schematic block diagram of

simplex methodology is presented in Figure 1(a) The userdefines a very restricted number of simplex points that isvertexes usually one more than the number of factors tobe varied 119873

119865 The function to be optimized is evaluated at

the simplex vertexes either experimentally or by calculationusing a previously constructed model Then a decision ismade about the worst vertex which is replaced by a vertexthat is reflected through the 119873

119865-dimensional surface (the

reflection being defined by the rest119873119865vertexes) According to

the situation the reflected vertex can be further expanded orcontractedThus the next119873

119865+1 simplex is created Valuation

and new simplex creation processes repeat until a predefinedcriterion is fulfilled

Evolutionary algorithms mimic different biological pro-cesses in an attempt to optimize highly complex functions[28] Genetic algorithms [29 30] (Figure 1(b)) as a rep-resentative of the evolutionary algorithms are based ongenetic inheritance and Darwinian strive to survival [31]In other words they simulate the biological evolution Theyallow a population composed of many individuals to evolveunder specified selection rules to a state that optimizes thepredefined objective function value [28]

This work is focused on the application of simplex meth-odology and genetic algorithm in the optimization of gra-dient elution in IC The conventional application of thesetwo methodologies in gradient IC implies the search forthe optimum in the real experiment domain A limited setof experiments is performed initially and subsequently theexperiments with the worst output are replaced by new betterones This might take a long time and a lot of experimentaleffort In the particular approach described by this paper thenumber of experiments was practically minimized the novelldquoexperimentsrdquo to replace the old ones are not performed inthe real domain but by applying the initially constructed iso-to-grad model that is on a computer

2 Materials and Methods

21 Instrumentation The experiments were performed ona Dionex ICS-5000 (Thermo Fisher Scientific) ion chro-matographic system equipped with a dual pump (DP-5)eluent generatormodule (EG-5)with EGC IIIKOHcartridgedegas unit on eluent generator continuously regeneratedanion trap column (CR-ATC) thermostatically controlleddetection module (DC-5) and an autosampler (AS-AP)For performing the separation the system was equippedwith a Dionex strong anion exchange column CarboPacPA20 (3 times 150mm) and a respective guard column (3 times

30mm) The detection mode was pulse amperometry withreference electrode AgAgCl and working gold electrodeTheseparation was performed at constant temperature of 30∘Cwhile the detection temperature was 20∘C The eluent flowrate was 05mLmin the sample loop volume was 10 120583Land the data collection rate was 1Hz The whole system wascomputer controlled by the Chromeleon 7 software

The simplex and genetic algorithm modeling requiredusing a computer We wrote the codes for both algorithms inthe Matlab 2010b environment

Journal of Analytical Methods in Chemistry 3

Generation of initial


simplex point

simplex matrix

Calculation of

vertex

Evaluation Optimal point

Detection of vertex

Calculation of CP fromthe other vertexes

Creation of a newvertex by reflection

expansion orcompression of CP

Replacement of theworst vertex with

the new one

CF-value for each

with worst CF

(a)

Generation ofinitial population

Calculation of

Evaluation

Unit ranking according

Optimal unit

Selection of units

Unitsrsquo crossover

Mutation of genes

Generation of thenew population

CF-value for each unit

to CF-value

(b)Figure 1 Block diagrams of (a) simplex and (b) genetic algorithm methodology CF represents criterion function and CP is the centroidpoint of the hyperspace excluding the worst vertex

22 Reagents and Solutions Standard solutions of 8 sug-ars arabitol (150 ppm) cellobiose (600 ppm) fructose (1000ppm) fucose (100 ppm) N-acetyl-D-glucosamine (500 ppm)lactulose (3000 ppm) melibiose (600 ppm) and raffinose(1000 ppm) were prepared by diluting appropriate amountsof solid compounds in deionized water (the first six sugarswere from Sigma-Aldrich USA and the last two from DrEhrenstorfer Germany) The solid compounds were 97 orhigher purity Working mix solutions were prepared fromstandard solutions in concentrations 100 times lower than thestandard ones All solutions were preserved at 4∘C Workingeluent solutions were hydroxide solutions prepared online inthe eluent generator module by pumping water through theeluent generator chamber

In all cases 18MΩcmminus1 water (Millipore USA) was used

3 Modeling

31 Retention Modeling and Calculation of Resolution Inorder to determine parameters of isocratic retention model(1) for each sugar a set of 5 isocratic experiments wasperformed the experiments were equidistantly distributedwithin the range from 2 to 98mM KOH Peak maximumretention was modeled as common however isocratic mod-els were also defined for the 50 peak height point at thefronting side as well as for the 50 peak height point at thetailing side These three retention times associated with eachpeakwere used to predict the resolution between the analyzedsugars according to [32 33]

119877119878= 118

119905119877(2)

minus 119905119877(1)

11990805(1)

+ 11990805(2)

(4)

where 119905119877(1)

and 119905119877(2)

are retention times of two adjacent sugarsand 119908

05(1)and 119908

05(2)are corresponding peak widths at half

peak heightsThe void time needed for prediction of component reten-

tion under gradient elution was the same as in the isocraticcase

32 Criterion Function Simplex vertexes or units from thegenetic population were valued by the criterion function CF

CF = (1

sum119877119878

)

120572

sdot 119905119860

120573 (5)

Two optimization goals were implemented into the criterionthe maximal sum of resolutions between eluted componentssum119877119878[brackets in (5)] and the shortest analysis time 119905

119860 The

desired balance between the goals was adjusted by the properchoice of criterion function weights 120572 and 120573 A possiblemisbalance would produce an elution that is too long at oneextreme or a peak overlapping at another Therefore it wasnecessary to select the appropriate weight values For theselection purpose both weights were varied from 1 to 5 usingthe step of 1

33 Gradient Domain Scanning The same gradient domainwas searched for the optimal separation conditions with boththe simplex and genetic algorithmsThe eluent concentrationwas ranging from 5 to 95mM KOH only the gradientsproducing elution times shorter than 30min were taken asthe acceptable ones

In the case of simplex approach the acceptable timerange of 30min was split into 119873

119879equidistant intervals (with


varying 119873119879) This provided 119873

119879+ 1 simplex factors (time

points) in which the change of eluent concentration waspossible In the genetic algorithm case the equivalents to thesimplex factors were termed genes In the applied geneticalgorithm the 30-minute time range was split into 3minintervals which provided 11 genes in which the concentrationchange was possible

A linear gradient inside each of the intervals was pre-sumed The gradients were defined by eluent concentrations(the values of factors or genes) at start and end of eachinterval For both applied approaches the elution continuedisocratically after the first 30 minutes taking the last factor orgene value

The optimization procedure implied searching for theoptimal eluent concentration for each factor or gene There-fore an appropriate design in creation of initial matrix ofpoints or units was needed

34 Simplex Optimization In the simplex optimization ap-proach the initial matrix of points was created using theDoehlert design [34 35] which provides the mesh of latticepoints uniformly and equidistantly distributed in the spacearound some central point [36] If there are multiple optimain the searched domain local optima (instead of the globaloptimum) will probably be reached using different initialmatrices To test this we varied the parameters of theDoehlert design These were (1) the number of simplex fac-tors (2) characteristic distance (concentration interval) and(3) central point position (central concentration) The num-ber of simplex factors was varied from 1 to 5 using step 1Theconcentration intervals were selected equidistantly within thepredefined gradient domain range of 90mM KOH to takevalues of 5 10 15 30 45 and 90mMKOH Central point wasselected as an isocratic elution at a midpoint concentrationwithin the selected concentration interval For the threecharacteristic Nelder-Mead parameters that is coefficient ofreflection expansion and contraction standard values of 12 and 05 were taken respectively [37ndash39]

The optimization was set to terminate when the sum ofabsolute differences between the factor values of two consec-utive iterations became lower than the predefined objectivevalue of 10minus15

35 Genetic Algorithm Optimization In the genetic algo-rithm approach the first 30 minutes of elution were dividedinto 10 equally sized time intervals providing the set of 11genes The set of all genes is commonly termed the chromo-some it completely describes a unit with its peculiar elutioncharacteristics

The genetic populations were encoded by integer valuesThe initial population of 100 units was created by randomlyassigning the concentration values from the predefined gra-dient domain range to all genes in the population A ratherlow degree of elitism within the population units was setThis means that every unit from the population was valuedaccording to (5) afterwards theworst 70were removedTherest of units were used as parents for the crossover procedureThe crossover was performed according to the rule two

parents-two descendants We applied the uniform crossoverprocedure [28] the genes forwarded from the first parent tothe offspring were chosen randomly and the rest of the genescame from the other parent The new-born population wasjoined with the parents afterwards such a created populationwas valued again The worst 50 of units was removed andthe rest became the new parent population

Themutation of genes is reality in every living populationCrossover without mutations restricts the characteristics ofthe new-born population from the characteristics of theirparents mutations bring new characteristics to the popula-tion The percentage of mutations in new-born populationmay be considered an adjustable parameter The number ofmutations in new-born population was varied from 5 to 110(using step 5) of overall 330 genes The CF value (5) of thebest unit was recorded after each crossover + mutation cycleThe optimization stopped after a predefined number of 200cycles

Since the random setting of initial population and appli-cation ofmutations generally produces diverse results (differ-ent optima) a set of 10 consecutive runs was performed in allcases of genetic algorithm calculations

36 Peak Shape Description The generalized logistic distri-bution function was used for the description of peak shape ofindividual components in calculated chromatograms

119891 (119905) =119862

119861sdot

e(119860minus119905)119861

(1 + e(119860minus119905)119861)119862+1 (6)

The median of distribution is associated with parameter119860 parameter 119861 characterizes the distribution width andparameter 119862 bears information about the distribution skew-ness [40 41] Peak description according to generalizedlogistic distribution function requires only four experimental(or calculated) data These are the retention time of peakmaximum retention times of peak half-heights at frontingand tailing peak sides and the stretching factor 119878

119865

ℎ (119905) = 119878119865sdot 119891 (119905) (7)

The stretching factor in fact equals the real chromatographicpeak area [16]

4 Results and Discussion

Simplex and genetic algorithms were applied for the opti-mization of IC separation of the solution of 8 sugars Theconventional approach of simplex and genetic algorithmswasimproved by incorporating the iso-to-grad model [15] Thusthe simplex and genetic algorithmswere allowed to search forthe optimum in the virtual experiment domain instead ofthe real experimental space which produced the significantreduction of time and costs

41 Isocratic Model Parameters In principle only three iso-cratic runs are needed to obtain coefficients of the quadraticdependence described by (1) Nevertheless the higher num-ber of isocratic experiments generally provides a more


Table 1 Parameters of isocratic retention models described by (1) The eluent concentration is expressed as mM

50 peak height at fronting side Peak maximum 50 peak height at tailing side1198860

1198861

1198862

1198772

1198860

1198861

1198862

1198772

1198860

1198861

1198862

1198772

Arabitol minus0201 minus0009 minus0031 09580 minus0188 minus0007 minus0030 09723 minus0153 minus0020 minus0027 09689Cellobiose 1808 0127 minus0286 09999 1814 0125 minus0285 09999 1820 0126 minus0285 09999Fructose 1224 0128 minus0261 09999 1230 0128 minus0261 09999 1237 0124 minus0259 10000Fucose 0353 0282 0234 10000 0363 0278 minus0231 10000 0374 0274 minus0228 10000Lactulose 1552 0152 minus0291 10000 1558 0152 minus0291 10000 1566 0149 minus0289 10000Melibiose 1298 0205 minus0288 10000 1304 0204 minus0288 10000 1311 0201 minus0286 10000N-Acetyl-D-glucosamine 1168 0036 minus0243 09998 1177 0028 minus0239 09998 1185 0028 minus0238 09998Raffinose 1327 0346 minus0262 09989 1335 0345 minus0261 09989 1343 0341 minus0260 09989

reliable isocratic model In addition it cannot be knowna priori how the competing anion in eluent would affectthe peak area and this is the feature essential for the peakshape predictionTherefore we decided to perform 5 isocraticexperiments Where overlapping of the components wasobserved working solutions of pure analytes were eluted aswell Thus a set of isocratic retention data was obtained toallow for the calculation of model parameters of (1) Thecalculated parameters are presented in Table 1

Apparently the chosen polynomial model fits the reten-tion behavior well for 7 of the 8 investigated sugars (1198772 ge

09989) In the case of arabitol there are some discrepanciesThe observed error may be explained by comparing theretention times of all the 8 sugars at the lowest eluentconcentration (2mM KOH) The elution order observed wasarabitol 153min fucose 385 N-acetyl-D-glucosamine 1583fructose 1887melibiose 2398 raffinose 2792 lactulose 4012and cellobiose 6128 and void time was 093 In comparison toother sugars the difference between arabitol elution time andvoid time is particularly small only 060min It is importantto recur that the data collection rate of the detector wasonly 1Hz Such a small rate could probably incorporate someerror in the retention prediction which would be especiallynoticeable for extremely fast eluting components like arabitol

42 Investigated Gradient Domain Some constrains were seton the investigated gradient domain in both applied opti-mization approaches In the case of simplex optimization wehad to limit the number of factors as well as the concentrationinterval investigated These in turn determined the size ofthe simplex Constrains of the genetic algorithm were posedby the number of genes (11) in population units as well asby the allowed concentration values for each gene (integerencoding of the population) Although the acceptable timeof IC analysis was set to 30min for both approaches itshould be pointed out that the procedure was allowed topredict the optimal separation with elution longer than30min However such events were then simply excludedfrom further considerations As for the eluent concentrationrange concentrations higher than 95mM produced poorseparation of the several least retained sugars concentrationsbelow 5mM led to very long and economically unjustifiedelutions

Although the negative gradients are quite uncommon inIC since they deteriorate the baseline behavior the authorsdid not exclude them from the simulation In some casesafter reaching the satisfactory separation of a few first elutingcomponents the negative gradient may in principle slowdown the elution of the remaining components that wouldotherwise lead to overlapping of their peaks

43 Criterion FunctionWeights Asmentioned before we hadto assign the appropriate weights to the two contributionsincorporated in the criterion function CF (5) that wereselected for the valuation of simplex vertexes or genetic unitsAn appropriate balance between the weights should diminishthe possibility of obtaining elution times that are too longor elutions with overlapped peaks In addition the simplexmethodology always produces the same output for the samestarting conditions which is not the case in genetic approachdue to mutations Therefore the selection of CF weightswas performed according to the results obtained by simplexcalculations 25 simplex runs with different weight valueswere performed The number of simplex factors was heldconstant at the value of 5 and the concentration interval was15mMKOHThe calculated optimal elutions were comparedaccording to the existence of peak overlapping or analysistime that is too long The peak overlapping threshold wasset to 119877

119878lt 3 which is somewhat higher than the common

values 15 or 2 [42] We decided to set a more rigorousoverlapping criterion sincewewere dealingwith the retentionmodels associated with possible errors The obtained resultsas shown in Figure 2 indicate that contribution of analysistime must be favored over the contribution of overlappingfor the acceptable separation that is 120573 must be higherthan 120572 The unacceptable weight combinations are markedwith hollow circles in Figure 2 We decided to select thecombination of weight factors 120572 = 1 and 120573 = 2 for furthercalculations

44 Simplex Optimization The size of simplex was var-ied in the simplex optimization 6 different concentrationintervals and 5 different factor numbers were used Thecharacteristic results are shown in Figure 3 For the 2- and 3-factor cases the optimizations eventually reached the isocraticchromatograms regardless of the size of the concentration


1

2

3

4

5

1 2 3 4 5

120573

120572

Figure 2 Selection of CF-function weights Dark circles represent elutions with well separated peaks (all 119877119878ge 3) in acceptable analysis time

(maximal 30min) and the white ones are those with at least one overlapped peak pair (119877119878lt 3) or with exceeded analysis time

interval applied (an example is presented in Figure 3(a))In the case of 4-factor optimization four different chro-matograms were obtained depending on the starting point(Figures 3(a)ndash3(d)) The acceptable separation was obtainedfor the 5mM concentration interval case although the lastsugar eluted at the edge of the tolerated period of 30 minutes(Figure 3(b)) With further increase of the concentrationinterval the analysis time decreased at the same time thepeak overlapping was observed (Figures 3(a) 3(c) and 3(d))The results for the 5-factor optimization eventually producedfive different chromatograms (Figures 3(a) and 3(e)ndash3(h))The analysis time again decreased with the increase of theconcentration interval The smallest concentration intervalprovided the peak overlapping and analysis that is too long atthe same time (Figure 3(e)) Good separations were predictedfor the 10 and 15mM concentration intervals (Figures 3(f)and 3(g)) with a distinction of significantly shorter analysistime for the 15mMcase Further increase of the concentrationinterval produced peak overlapping (Figures 3(a) and 3(h))Therefore the best optimization result occurred when gradi-ent domain was scanned by the algorithm characterized with5 simplex factors and the concentration interval of 15mMKOH as the starting point It is important to notice that somedifferent starting conditions produced significantly differentldquooptimalrdquo separationsThis indicates that the applied simplexalgorithm did not reach the global optimum in every casethat is it had serious problems when dealing with localminima

It can be noticed that gradients shown in Figure 3 con-tinue even after the last component eluted from the columnTherefore it is necessary to clarify such observation to avoidany possible misunderstanding The concentration and timedomains were defined prior to the search for optimal sep-aration conditions Therefore even for those cases wherecomponents eluted very fast the remaining part of eluentconcentration profile must exist (regardless of the fact thatit has no influence on separation) but only as an artifactof the applied calculation procedure The specific eluentconcentration profiles shown in Figure 3 after the last elutedcomponent are simply the first ones that the algorithm foundThe selection of any other concentration profile continuing

the last eluted component will have no effect on separationsince it is already achieved

45 Genetic Optimization In the case of genetic algorithmwe were dealing with the percentage of mutations in new-born population and the number of crossover cycles as theimportant issues Figure 4 represents the results obtained fordifferent number of mutations

Although the final number of crossover cycles was set to200 the improvement of the best unitrsquos CF value vanishednormally after a much lower number of cycles We termedit as the threshold number of cycles and we recorded it whenthe CF value dropped below 10minus12The gray circles in Figure 4represent the median of the threshold number of cycles for10 consecutive runs and the bars represent the correspond-ing span of values According to the results plotted themedians seem to be relatively independent of the numberof mutations However considerably more scattering in thethreshold number of cycles was observed whenmore than 90genes were allowed to mutate In conclusion the crossoverwith 60 mutated genes (ie 182 of mutations) producedthe smallest median of the threshold number of cycles (theshortest calculation time) At the same time it produced theminimum scattering of results Therefore this percentageof mutations was taken as the best one Among the resultscalculated with this percentage of mutation the gradientelution profile with the shortest elution time (1889min) wasselected as the optimal one It is important to point out that all220 calculations using the genetic algorithm regardless of thenumber of mutations applied or diverse initial populationsrandomly selected produced extremely similar resolutions ofadjacent components (119877

119878values Table 2) The elution order

of the analyzed sugars was always the same and none ofthe peaks overlapped The standard deviation of predictedresolutions for all sugars and all 220 calculations was nothigher than 128 Also the standard deviation of maximalelution time was only 039

46 Comparison of Calculated Optimal and ExperimentalChromatograms In order to test the results obtained by


100

20

40

60

80

0 5 15 2085

90

95

100 100Re

spon

se (n

C)

Time (min)

c(K

OH

) (m

M)

(a)

0 5 10 15 20 25 30 35 400

10

20

30

40

50

60

15

20

25

30

Resp

onse

(nC)

Time (min)

c(K

OH

) (m

M)

(b)

5 10 15 20 25 3030

35

40

45

00

10

20

30

40

50

60

Resp

onse

(nC)

Time (min)

c(K

OH

) (m

M)

(c)

5 10

70

0 15 2055

60

65

70

0102030405060

Resp

onse

(nC)

Time (min)

c(K

OH

) (m

M)

(d)

0 10 20 30 40 5010

15

20

25

0

10

20

30

40

50

60

Resp

onse

(nC)

Time (min)

c(K

OH

) (m

M)

(e)

0 10 20 30 4020

25

30

35

0

10

20

30

40

50

60

Resp

onse

(nC)

Time (min)

c(K

OH

) (m

M)

(f)

25

30

35

40

4570

0102030405060

5 10 15 20 25 300

Resp

onse

(nC)

Time (min)

c(K

OH

) (m

M)

(g)

70

75

80

85

90

5 100 15 200

20

40

60

80

100

Resp

onse

(nC)

Time (min)

c(K

OH

) (m

M)

(h)

Figure 3 Searching for the appropriate number of simplex factors (119873119865) and concentration interval (Δ119888) The above chromatograms were

obtained for (a) 119873119865

= 2 and 3 for all Δ119888 119873119865

= 4 and Δ119888 = 30 45 and 90mM KOH 119873119865

= 5 and Δ119888 = 45 and 90mM KOH (b) 119873119865

= 4 andΔ119888 = 5mM KOH (c) 119873

119865= 4 and Δ119888 = 10mM KOH (d) 119873

119865= 4 and Δ119888 = 15mM KOH (e) 119873

119865= 5 and Δ119888 = 5mM KOH (f) 119873

119865= 5 and

Δ119888 = 10mMKOH (g) 119873119865

= 5 and Δ119888 = 15mMKOH and (h) 119873119865

= 5 and Δ119888 = 30mMKOH

the simplex and genetic algorithm optimization the real ICanalyses were performed under the calculated optimal KOHgradients (Figure 5) The experimental chromatograms arecompared with the predicted ones However for the full

description of predicted chromatogram one has to choose anappropriate distribution function to model the peak shapeThe generalized logistic function proved a good choice forthe peak shape prediction of inorganic anions in IC [16 25]


60

50

40

30

20

10

0

0 10 20 30 40 50 60 70 80 90 100 110

Number of mutations

Num

ber o

f cro

ssin

g cy

cles

Figure 4 Relation between the threshold number of crossover cycles and number of mutated genes in offspring population each populationhas a total number of 330 genes Grey circles represent the average threshold number of crossover cycles among the 10 repeated calculationsblack bars denote the scattering of 10 repeated calculations

0

20

40

60

80

0 10 2025

30

30

35

40

45

Resp

onse

(nC)

Time (min)

c(K

OH

) (m

M)

(a)

020406080

100120140

0

20

40

60

80

100

0 10 20 30

Resp

onse

(nC)

Time (min)

c(K

OH

) (m

M)

(b)

0

20

40

60

80

0 10 20 3025

30

35

40

45

1

2

3

4 5

6

78

Resp

onse

(nC)

Time (min)

c(K

OH

) (m

M)

(c)

020406080

100120140

0 10 20 300

20

40

60

80

100

12

34 5

6

7

8

Resp

onse

(nC)

Time (min)c(K

OH

) (m

M)

(d)

Figure 5 Comparison of experimental and predicted chromatograms obtainedwith optimal gradient profile (a) predicted optimal separationby simplex algorithm (b) predicted optimal separation by genetic algorithm (c) real chromatogram corresponding to simplex optimizationand (d) real chromatogram corresponding to genetic algorithm Dashed lines indicate the calculated optimal gradients Analyzed sugars aremarked according to elution order (1) arabitol (2) fucose (3)N-acetyl-D-glucosamine (4) fructose (5)melibiose (6) lactulose (7) raffinoseand (8) cellobiose

due to its simplicity and fair representation of experimentalchromatograms Therefore in order to create a better pre-sentation of the separated sugars it was applied in this caseas well All the data needed for calculation of 119860 119861 and 119862

parameters of the generalized logistic distribution functionare presented in Table 1 and the calculation procedure isdescribed elsewhere [16 25] Therefore only the stretchingfactor of the generalized logistic distribution function (7) willbe discussed here As mentioned before the stretching factorin this case equals the peak area of a real chromatogramAs said before in this research the amperometric detector

was applied Since this detector is a concentration selectivedetector an influence of competing anion concentration onthe peak area is observed (Figure 6) For all the analyzedsugars a decrease of peak area with the increase of eluentconcentration is observed Isocratic experimental data wereused to estimate the stretching factor that is peak areaSince the stretching factor depends on the competing anionconcentration the appropriate values for every peak werecalculated in the following manner First peak retentiontime is identified Second competing anion concentrationat that retention time is detected from the gradient profile


Table 2 Comparison of the calculated resolutions of 220 calculations using genetic algorithm

Minimal value Maximal value Standard deviationResolution

Arabitol and fucose 1198 1575 061Fucose and N-acetyl-D-glucosamine 1830 2493 128N-Acetyl-D-glucosamine and fructose 464 583 025Fructose and melibiose 495 581 015Melibiose and lactulose 579 1131 081Lactulose and raffinose 471 977 062Raffinose and cellobiose 300 368 013

Maximal elutionmin 1898 2076 039

0

5

10

15

20

25

30

35

2 26 50 74 98c(KOH) (mM)

Are

a (nC

middotmin

)

Figure 6 Peak areas for five isocratic experimentsThe componentsare marked as follows arabitol (X) cellobiose (998771) fructose (◻)fucose () lactulose (I) melibiose (times) N-acetyl-D-glucosamine(∙) and raffinose (◼)

calculated Third peak area is estimated at that competinganion concentration using the nearby experimental valuesfrom Figure 6 and linear interpolation

The predicted and real chromatograms are compared inFigure 5 The predicted chromatograms have much broaderbandwidths in comparison to the experimental ones offeringslight apparent overlapping of the two last eluted compo-nents (Figures 5(a) and 5(b)) Such overlapping was not aconsequence of invalid optimization (all the calculated 119877

119878

values were higher than 3) but simply the product of thechosen peak-shape function which seems to be less adequatefor sugars than it was for inorganic anions Despite thatthe chromatograms generally match each other confirmingthe applicability of both methodologies in the optimizationof IC separation Although the genetic algorithm may beconsidered somewhat better with respect to the analysis time(the predicted retention of the last eluting sugar of 1898minand 1919min for genetic and simplex optimization resp)it is obvious that both approaches provide similar analysistime In both cases the last two eluted sugars (raffinose andcellobiose) have very close retention times Their predictedresolutions are significantly smaller than for the other adja-cent sugar pairs and almost equal to the predefined thresholdvalue of119877

119878= 3The required separation of these two sugars is

therefore the most probable reason of such a long calculatedanalysis time

5 Conclusions

This work describes the application of the two method-ologies that is simplex and genetic algorithms combinedwith the iso-to-grad retention model in the optimizationof gradient IC separation This combination allowed thefinding of the optimumgradient profile in the virtual domainthus minimizing the experimental effort and costs The CFfunction weight factors were selected to favor the analysistime contribution (120572 = 1 for the resolution and 120573 = 2 for theanalysis time) The optimal gradient profile for the simplexapproach was obtained using the 5 simplex factors andthe concentration interval of 15mM KOH in initiating thesearch In the case of genetic algorithm 182 of mutationsin each offspring population were found to be the optimalpercentage in fast and reliable finding of the optimum profileThe simplex methodology exhibited problems when dealingwith local minima that is different starting conditionsresulted in significantly different component elution timesAt the same time the genetic algorithm did not suffer fromthe local minima problem 220 calculations provided 220almost identical separations The real sample experimentalchromatograms obtained by applying the calculated optimalgradient profiles were compared with the predicted onesand good agreement was found Both simplex and geneticalgorithms in combination with iso-to-grad model proved apotential to be applied in gradient IC optimization Howeverthe genetic algorithmwas coping betterwith the localminimaproblem The developed approach offers a significant reduc-tion of the experimental effort (only 5 isocratic experimentsneeded + few more for the overlapping peaks) In additionthere is no practical limitation on the gradient profile to betested Based on the results of this study the genetic algorithmin combination with the iso-to-grad retention modeling maybe recommended as themethod of choice for optimization ingradient ion chromatography

Conflict of Interests

Theauthors have declared no financial or commercial conflictof interests


Acknowledgments

This study was a part of the research projects Ion-ExchangeProcesses in Industrial Water-Quality System 125-1253092-3004 and Bioceramic Polymer and Composite Nanostruc-turedMaterials 125-1252970-3005 supported by theMinistryof Science Education and Sports of the Republic of CroatiaThe authors are grateful for this support The authors thanktheThermoFisher Scientific for technological support duringexperimentation and particularly Dr Nebojsa Avdalovic forhis valuable recommendations and suggestions

References

[1] M C Garcıa-Alvarez-Coque J R Torres-Lapasio and J JBaeza-Baeza ldquoModels and objective functions for the optimi-sation of selectivity in reversed-phase liquid chromatographyrdquoAnalytica Chimica Acta vol 579 no 2 pp 125ndash145 2006

[2] A P Schellinger and P W Carr ldquoIsocratic and gradient elutionchromatography a comparison in terms of speed retentionreproducibility and quantitationrdquo Journal of Chromatography Avol 1109 no 2 pp 253ndash266 2006

[3] L R Snyder J W Dolan and J R Gant ldquoGradient elution inhigh-performance liquid chromatography I Theoretical basisfor reversed-phase systemsrdquo Journal of Chromatography A vol165 no 1 pp 3ndash30 1979

[4] T B Hoover ldquoMultiple eluent and pH effects on ion chro-matography of phosphate and arsenaterdquo Separation Science andTechnology vol 17 no 2 pp 295ndash305 1982

[5] D R Jenke and G K Pagenkopf ldquoOptimization of anionseparation by nonsuppressed ion chromatographyrdquo AnalyticalChemistry vol 56 no 1 pp 85ndash88 1984

[6] D R Jenke and G K Pagenkopf ldquoModels for prediction ofretention in nonsuppressed ion chromatographyrdquo AnalyticalChemistry vol 56 no 1 pp 88ndash91 1984

[7] D R Jenke ldquoModeling of analyte behavior in indirect photo-metric chromatographyrdquo Analytical Chemistry vol 56 no 14pp 2674ndash2681 1984

[8] P Hajos O Horvath and V Denke ldquoPrediction of retention forhalide anions and oxoanions in suppressed ion chromatographyusing multiple species eluentrdquo Analytical Chemistry vol 67 no2 pp 434ndash441 1995

[9] J E Madden and P R Haddad ldquoCritical comparison ofretentionmodels for optimisation of the separation of anions inion chromatographyI Non-suppressed anion chromatographyusing phthalate eluents and three different stationary phasesrdquoJournal of Chromatography A vol 829 no 1-2 pp 65ndash80 1998

[10] J E Madden and P R Haddad ldquoCritical comparison of reten-tion models for the optimisation of the separation of anionsin ion chromatography II Suppressed anion chromatographyusing carbonate eluentsrdquo Journal of Chromatography A vol 850no 1-2 pp 29ndash41 1999

[11] J E Madden N Avdalovic P E Jackson and P R HaddadldquoCritical comparison of retention models for optimisation ofthe separation of anions in ion chromatography III Anionchromatography using hydroxide eluents on a Dionex AS11stationary phaserdquo Journal of Chromatography A vol 837 no 1-2pp 65ndash74 1999

[12] T Bolanca S Cerjan-Stefanovic M Lusa M Rogosic andS Ukic ldquoDevelopment of an ion chromatographic gradientretention model from isocratic elution experimentsrdquo Journal ofChromatography A vol 1121 no 2 pp 228ndash235 2006

[13] T Bolanca S Cerjan-Stefanovic S Ukic M Rogosic and MLusa ldquoApplication of different training methodologies for thedevelopment of a back propagation artificial neural networkretention model in ion chromatographyrdquo Journal of Chemomet-rics vol 22 no 2 pp 106ndash113 2008

[14] T Bolanca S Cerjan-Stefanovic M Lusa S Ukic and MRogosic ldquoEvaluation of separation in gradient elution ionchromatography by combining several retention models andobjective functionsrdquo Journal of Separation Science vol 31 no4 pp 705ndash713 2008

[15] T Bolanca S Cerjan-Stefanovic S Ukic M Lusa and MRogosic ldquoFrom isocratic data to a gradient elution retentionmodel in IC an artificial neural network approachrdquo Chro-matographia vol 70 no 1-2 pp 15ndash20 2009

[16] T Bolanca S Cerjan-Stefanovic S Ukic M Rogosic and MLusa ldquoApplication of a gradient retention model developed byusing isocratic data for the prediction of retention resolutionand peak asymmetry in ion chromatographyrdquo Journal of LiquidChromatography and Related Technologies vol 32 no 10 pp1373ndash1391 2009

[17] M Novic J Zupan and M Novic ldquoComputer simulation ofion chromatography separation an algorithm enabling con-tinuous monitoring of anion distribution on an ion-exchangechromatography columnrdquo Journal of Chromatography A vol922 no 1-2 pp 1ndash11 2001

[18] V Drgan M Novic andM Novic ldquoComputational method formodeling of gradient separation in ion-exchange chromatogra-phyrdquo Journal of Chromatography A vol 1216 no 37 pp 6502ndash6510 2009

[19] P J Schoenmakers H A H Billiet R Tussen and L de GalanldquoGradient selection in reversed-phase liquid chromatographyrdquoJournal of Chromatography A vol 149 pp 519ndash537 1978

[20] J R Torres-Lapasio M Roses E Bosch and M C Garcıa-Alvarez-Coque ldquoInterpretive optimisation strategy applied tothe isocratic separation of phenols by reversed-phase liquidchromatography with acetonitrile-water and methanol-watermobile phasesrdquo Journal of Chromatography A vol 886 no 1-2pp 31ndash46 2000

[21] L S Ettre ldquoNomenclature for chromatographyrdquo Pure amp AppliedChemistry vol 65 pp 819ndash872 1993

[22] Y Shan and A Seidel-Morgenstern ldquoOptimization of gradientelution conditions in multicomponent preparative liquid chro-matographyrdquo Journal of Chromatography A vol 1093 no 1-2pp 47ndash58 2005

[23] S Yamamoto and A Kita ldquoTheoretical background of shortchromatographic layers optimization of gradient elution inshort columnsrdquo Journal of Chromatography A vol 1065 no 1pp 45ndash50 2005

[24] T Ishihara and S Yamamoto ldquoOptimization of monoclonalantibody purification by ion-exchange chromatography appli-cation of simple methods with linear gradient elution experi-mental datardquo Journal of Chromatography A vol 1069 no 1 pp99ndash106 2005

[25] S Ukic T Bolanca and M Rogosic ldquoNovel criteria for fastsearching for optimal method in gradient ion chromatographyan integrated approachrdquo Journal of Separation Science vol 34no 7 pp 780ndash788 2011

[26] D L Massart B G M Vandeginste L M C Buydens Sde Jong P J Lewi and J Smeyers-Verbeke Handbook ofChemometrics and Qualimetrics Part A Elsevier 1998


[27] J A Nelder and R Mead ldquoA simplex method for functionminimizationrdquoThe Computer Journal vol 7 no 4 pp 308ndash3131965

[28] R L Haupt and S E Haupt Practical Genetic Algorithms JohnWiley amp Sons 1998

[29] J H Holland Adaptation in Natural and Artificial Systems TheUniversity of Michigan Press 1975

[30] D E GoldbergGenetic Algorithms in Search Optimization andMachine Learning Addison-Wesley 1989

[31] W G Quirino K C Teixeira C Legnani et al ldquoImprovedmultilayer OLED architecture using evolutionary genetic algo-rithmrdquoThin Solid Films vol 518 no 5 pp 1382ndash1385 2009

[32] J M Miller Chromatography Concepts and Contrasts JohnWiley amp Sons 2nd edition 2005

[33] J W Dolan ldquoPeak tailing and resolutionrdquo LC-GC North Amer-ica vol 20 no 5 pp 430ndash436 2002

[34] R E Bruns I S Scarminio and B de Barros Neto StatisticalDesignmdashChemometrics Elsevier 2006

[35] D H Doehlert ldquoUniform shell designsrdquo Applied Statistics vol19 no 3 pp 231ndash239 1970

[36] H Fauduet C Porte J-L Havet and D Daguet ldquoModellingof influential parameters on a continuous evaporation processby Doehlert shellsrdquo Journal of AutomatedMethods andManage-ment in Chemistry vol 25 no 1 pp 21ndash30 2003

[37] S-K S Fan andE Zahara ldquoAhybrid simplex search and particleswarm optimization for unconstrained optimizationrdquo EuropeanJournal ofOperational Research vol 181 no 2 pp 527ndash548 2007

[38] P C Wang and T E Shoup ldquoParameter sensitivity study ofthe Nelder-Mead Simplex Methodrdquo Advances in EngineeringSoftware vol 42 no 7 pp 529ndash533 2011

[39] A Karimi and P Siarry ldquoGlobal Simplex Optimizationmdashasimple and efficientmetaheuristic for continuous optimizationrdquoEngineering Applications of Artificial Intelligence vol 25 no 1pp 48ndash55 2012

[40] T Bolanca S Cerjan-Stefanovic S Ukic M Rogosic and MLusa ldquoPrediction of the chromatographic signal in gradientelution ion chromatographyrdquo Journal of Separation Science vol32 no 17 pp 2877ndash2884 2009

[41] T Bolanca S Ukic and M Rogosic ldquoPrediction of nonlineargradient signal in ion chromatography based on ldquoexperiment-freerdquo methodologyrdquo Journal of Liquid Chromatography andRelated Technologies vol 33 no 20 pp 1831ndash1841 2010

[42] J Weiss Handbook of Ion Chromatography Wiley-VCH 3rdedition 2004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Inorganic ChemistryInternational Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

International Journal ofPhotoenergy


Carbohydrate Chemistry

International Journal of


Journal of

Chemistry


Advances in

Physical Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom

Analytical Methods in Chemistry

Journal of

Volume 2014

Bioinorganic Chemistry and ApplicationsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

SpectroscopyInternational Journal of


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Medicinal ChemistryInternational Journal of


Chromatography Research International


Applied ChemistryJournal of



Theoretical ChemistryJournal of


Journal of

Spectroscopy

Analytical ChemistryInternational Journal of


Journal of


Quantum Chemistry


Organic Chemistry International

ElectrochemistryInternational Journal of



CatalystsJournal of


describe the isocratic behavior with the better fitting modelthat is the polynomial dependence between the logarithm ofretention coefficient 119896 and concentration of competing ionin eluent 119888

log 119896 = 1198860+ 1198861log 119888 + 119886

2log2119888 (1)

In principle only three isocratic experiments are sufficientfor the determination of regression coefficients 119886

0minus2 of the

model described by (1)The retention coefficient is calculated as

119896 =119905119877minus 119905119872

119905119872

(2)

where 119905119877is a component retention time and 119905

119872is the columnrsquos

holdup time For the ion chromatography case the columnrsquoshold-up time is equal to the retention time of unretainedcompounds that is column void time 119905

0[21]

On the other hand gradient elution can be describedgenerally by the integral elution equation

1199050= int

119905119877minus1199050

0

119889119905

119896 [119888 (119905)] (3)

The integral from (3) can be approximated with a sum ofintegrals over small time intervals Inside those intervals thecoefficient 119896 can be assumed constant and can be calculatedas an average of 119896 values at the interval boundaries obtainedaccording to (1) In this way isocratic retention informationis transferred into gradient environment

The iso-to-grad approach allows for the prediction ofretention time for practically any gradient using only sev-eral isocratic experiments Nevertheless the optimizationof gradient elution remains a severe and complex problemespecially knowing that the number of different gradientprograms in any gradient domain is practically unlimitedThe problem can be simplified by setting different constraintson the domain (ie defining the sets of finite intervals inwhich the gradient curve may change finite sets of gradientcurve shapes or finite sets of allowed gradient slopes [22ndash25]) Unfortunately the usage of any constraint involves thepossibility of missing the true optimum An increase ofthe number of allowable options inside each finite set willproduce a finer description of the real experimental domainHowever it will at the same time increase significantly thenumber of possible gradient profiles This leads to anotherproblem a domain of profiles that is too large substantiallyincreases both the experimental effort and requiredmodelingtime which may even exceed the performance of moderncomputers

A mathematical model that is able to describe the opti-mization surface for gradient IC is generally a nonlinear andvery complex function Therefore it is not feasible to solve it(to find optimum) analytically The fact that there are oftenmultiple solutions (local extremes) complicates the situationeven more There are several classes of approaches availablefor solving this kind of problems among which are sequentialsearching procedures and evolutionary algorithms [26]

One of the best known sequential methods is the Nelder-Mead simplex method [27] The schematic block diagram of

simplex methodology is presented in Figure 1(a) The userdefines a very restricted number of simplex points that isvertexes usually one more than the number of factors tobe varied 119873

119865 The function to be optimized is evaluated at

the simplex vertexes either experimentally or by calculationusing a previously constructed model Then a decision ismade about the worst vertex which is replaced by a vertexthat is reflected through the 119873

119865-dimensional surface (the

reflection being defined by the rest119873119865vertexes) According to

the situation the reflected vertex can be further expanded orcontractedThus the next119873

119865+1 simplex is created Valuation

and new simplex creation processes repeat until a predefinedcriterion is fulfilled

Evolutionary algorithms mimic different biological pro-cesses in an attempt to optimize highly complex functions[28] Genetic algorithms [29 30] (Figure 1(b)) as a rep-resentative of the evolutionary algorithms are based ongenetic inheritance and Darwinian strive to survival [31]In other words they simulate the biological evolution Theyallow a population composed of many individuals to evolveunder specified selection rules to a state that optimizes thepredefined objective function value [28]

This work is focused on the application of simplex meth-odology and genetic algorithm in the optimization of gra-dient elution in IC The conventional application of thesetwo methodologies in gradient IC implies the search forthe optimum in the real experiment domain A limited setof experiments is performed initially and subsequently theexperiments with the worst output are replaced by new betterones This might take a long time and a lot of experimentaleffort In the particular approach described by this paper thenumber of experiments was practically minimized the novelldquoexperimentsrdquo to replace the old ones are not performed inthe real domain but by applying the initially constructed iso-to-grad model that is on a computer

2 Materials and Methods

21 Instrumentation The experiments were performed ona Dionex ICS-5000 (Thermo Fisher Scientific) ion chro-matographic system equipped with a dual pump (DP-5)eluent generatormodule (EG-5)with EGC IIIKOHcartridgedegas unit on eluent generator continuously regeneratedanion trap column (CR-ATC) thermostatically controlleddetection module (DC-5) and an autosampler (AS-AP)For performing the separation the system was equippedwith a Dionex strong anion exchange column CarboPacPA20 (3 times 150mm) and a respective guard column (3 times

30mm) The detection mode was pulse amperometry withreference electrode AgAgCl and working gold electrodeTheseparation was performed at constant temperature of 30∘Cwhile the detection temperature was 20∘C The eluent flowrate was 05mLmin the sample loop volume was 10 120583Land the data collection rate was 1Hz The whole system wascomputer controlled by the Chromeleon 7 software

The simplex and genetic algorithm modeling requiredusing a computer We wrote the codes for both algorithms inthe Matlab 2010b environment




simplex point

simplex matrix

Calculation of

vertex


Detection of vertex





the new one

CF-value for each

with worst CF

(a)


Calculation of

Evaluation


Optimal unit

Selection of units


Mutation of genes



to CF-value




3 Modeling


119877119878= 118

119905119877(2)

minus 119905119877(1)

11990805(1)

+ 11990805(2)

(4)

where 119905119877(1)

and 119905119877(2)


05(1)and 119908





CF = (1

sum119877119878

)

120572

sdot 119905119860

120573 (5)


119860 The



















119891 (119905) =119862

119861sdot

e(119860minus119905)119861

(1 + e(119860minus119905)119861)119862+1 (6)


119865

ℎ (119905) = 119878119865sdot 119891 (119905) (7)








1198861

1198862

1198772

1198860

1198861

1198862

1198772

1198860

1198861

1198862

1198772












1

2

3

4

5

1 2 3 4 5

120573

120572












100

20

40

60

80

0 5 15 2085

90

95

100 100Re

spon

se (n

C)

Time (min)

c(K

OH

) (m

M)

(a)

0 5 10 15 20 25 30 35 400

10

20

30

40

50

60

15

20

25

30

Resp

onse

(nC)

Time (min)

c(K

OH

) (m

M)

(b)

5 10 15 20 25 3030

35

40

45

00

10

20

30

40

50

60

Resp

onse

(nC)

Time (min)

c(K

OH

) (m

M)

(c)

5 10

70

0 15 2055

60

65

70

0102030405060

Resp

onse

(nC)

Time (min)

c(K

OH

) (m

M)

(d)

0 10 20 30 40 5010

15

20

25

0

10

20

30

40

50

60

Resp

onse

(nC)

Time (min)

c(K

OH

) (m

M)

(e)

0 10 20 30 4020

25

30

35

0

10

20

30

40

50

60

Resp

onse

(nC)

Time (min)

c(K

OH

) (m

M)

(f)

25

30

35

40

4570

0102030405060

5 10 15 20 25 300

Resp

onse

(nC)

Time (min)

c(K

OH

) (m

M)

(g)

70

75

80

85

90

5 100 15 200

20

40

60

80

100

Resp

onse

(nC)

Time (min)

c(K

OH

) (m

M)

(h)



= 2 and 3 for all Δ119888 119873119865

= 4 and Δ119888 = 30 45 and 90mM KOH 119873119865

= 5 and Δ119888 = 45 and 90mM KOH (b) 119873119865

= 4 andΔ119888 = 5mM KOH (c) 119873

119865= 4 and Δ119888 = 10mM KOH (d) 119873

119865= 4 and Δ119888 = 15mM KOH (e) 119873

119865= 5 and Δ119888 = 5mM KOH (f) 119873

119865= 5 and

Δ119888 = 10mMKOH (g) 119873119865

= 5 and Δ119888 = 15mMKOH and (h) 119873119865

= 5 and Δ119888 = 30mMKOH




60

50

40

30

20

10

0

0 10 20 30 40 50 60 70 80 90 100 110

Number of mutations

Num

ber o

f cro

ssin

g cy

cles


0

20

40

60

80

0 10 2025

30

30

35

40

45

Resp

onse

(nC)

Time (min)

c(K

OH

) (m

M)

(a)

020406080

100120140

0

20

40

60

80

100

0 10 20 30

Resp

onse

(nC)

Time (min)

c(K

OH

) (m

M)

(b)

0

20

40

60

80

0 10 20 3025

30

35

40

45

1

2

3

4 5

6

78

Resp

onse

(nC)

Time (min)

c(K

OH

) (m

M)

(c)

020406080

100120140

0 10 20 300

20

40

60

80

100

12

34 5

6

7

8

Resp

onse

(nC)

Time (min)c(K

OH

) (m

M)

(d)










0

5

10

15

20

25

30

35

2 26 50 74 98c(KOH) (mM)

Are

a (nC

middotmin

)




119878




5 Conclusions





Acknowledgments


References





















































Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of




simplex point

simplex matrix

Calculation of

vertex


Detection of vertex





the new one

CF-value for each

with worst CF

(a)


Calculation of

Evaluation


Optimal unit

Selection of units


Mutation of genes



to CF-value




3 Modeling


119877119878= 118

119905119877(2)

minus 119905119877(1)

11990805(1)

+ 11990805(2)

(4)

where 119905119877(1)

and 119905119877(2)


05(1)and 119908





CF = (1

sum119877119878

)

120572

sdot 119905119860

120573 (5)


119860 The



















119891 (119905) =119862

119861sdot

e(119860minus119905)119861

(1 + e(119860minus119905)119861)119862+1 (6)


119865

ℎ (119905) = 119878119865sdot 119891 (119905) (7)








1198861

1198862

1198772

1198860

1198861

1198862

1198772

1198860

1198861

1198862

1198772












1

2

3

4

5

1 2 3 4 5

120573

120572












100

20

40

60

80

0 5 15 2085

90

95

100 100Re

spon

se (n

C)

Time (min)

c(K

OH

) (m

M)

(a)

0 5 10 15 20 25 30 35 400

10

20

30

40

50

60

15

20

25

30

Resp

onse

(nC)

Time (min)

c(K

OH

) (m

M)

(b)

5 10 15 20 25 3030

35

40

45

00

10

20

30

40

50

60

Resp

onse

(nC)

Time (min)

c(K

OH

) (m

M)

(c)

5 10

70

0 15 2055

60

65

70

0102030405060

Resp

onse

(nC)

Time (min)

c(K

OH

) (m

M)

(d)

0 10 20 30 40 5010

15

20

25

0

10

20

30

40

50

60

Resp

onse

(nC)

Time (min)

c(K

OH

) (m

M)

(e)

0 10 20 30 4020

25

30

35

0

10

20

30

40

50

60

Resp

onse

(nC)

Time (min)

c(K

OH

) (m

M)

(f)

25

30

35

40

4570

0102030405060

5 10 15 20 25 300

Resp

onse

(nC)

Time (min)

c(K

OH

) (m

M)

(g)

70

75

80

85

90

5 100 15 200

20

40

60

80

100

Resp

onse

(nC)

Time (min)

c(K

OH

) (m

M)

(h)



= 2 and 3 for all Δ119888 119873119865

= 4 and Δ119888 = 30 45 and 90mM KOH 119873119865

= 5 and Δ119888 = 45 and 90mM KOH (b) 119873119865

= 4 andΔ119888 = 5mM KOH (c) 119873

119865= 4 and Δ119888 = 10mM KOH (d) 119873

119865= 4 and Δ119888 = 15mM KOH (e) 119873

119865= 5 and Δ119888 = 5mM KOH (f) 119873

119865= 5 and

Δ119888 = 10mMKOH (g) 119873119865

= 5 and Δ119888 = 15mMKOH and (h) 119873119865

= 5 and Δ119888 = 30mMKOH




60

50

40

30

20

10

0

0 10 20 30 40 50 60 70 80 90 100 110

Number of mutations

Num

ber o

f cro

ssin

g cy

cles


0

20

40

60

80

0 10 2025

30

30

35

40

45

Resp

onse

(nC)

Time (min)

c(K

OH

) (m

M)

(a)

020406080

100120140

0

20

40

60

80

100

0 10 20 30

Resp

onse

(nC)

Time (min)

c(K

OH

) (m

M)

(b)

0

20

40

60

80

0 10 20 3025

30

35

40

45

1

2

3

4 5

6

78

Resp

onse

(nC)

Time (min)

c(K

OH

) (m

M)

(c)

020406080

100120140

0 10 20 300

20

40

60

80

100

12

34 5

6

7

8

Resp

onse

(nC)

Time (min)c(K

OH

) (m

M)

(d)










0

5

10

15

20

25

30

35

2 26 50 74 98c(KOH) (mM)

Are

a (nC

middotmin

)




119878




5 Conclusions





Acknowledgments


References





















































Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of















119891 (119905) =119862

119861sdot

e(119860minus119905)119861

(1 + e(119860minus119905)119861)119862+1 (6)


119865

ℎ (119905) = 119878119865sdot 119891 (119905) (7)








1198861

1198862

1198772

1198860

1198861

1198862

1198772

1198860

1198861

1198862

1198772












1

2

3

4

5

1 2 3 4 5

120573

120572












100

20

40

60

80

0 5 15 2085

90

95

100 100Re

spon

se (n

C)

Time (min)

c(K

OH

) (m

M)

(a)

0 5 10 15 20 25 30 35 400

10

20

30

40

50

60

15

20

25

30

Resp

onse

(nC)

Time (min)

c(K

OH

) (m

M)

(b)

5 10 15 20 25 3030

35

40

45

00

10

20

30

40

50

60

Resp

onse

(nC)

Time (min)

c(K

OH

) (m

M)

(c)

5 10

70

0 15 2055

60

65

70

0102030405060

Resp

onse

(nC)

Time (min)

c(K

OH

) (m

M)

(d)

0 10 20 30 40 5010

15

20

25

0

10

20

30

40

50

60

Resp

onse

(nC)

Time (min)

c(K

OH

) (m

M)

(e)

0 10 20 30 4020

25

30

35

0

10

20

30

40

50

60

Resp

onse

(nC)

Time (min)

c(K

OH

) (m

M)

(f)

25

30

35

40

4570

0102030405060

5 10 15 20 25 300

Resp

onse

(nC)

Time (min)

c(K

OH

) (m

M)

(g)

70

75

80

85

90

5 100 15 200

20

40

60

80

100

Resp

onse

(nC)

Time (min)

c(K

OH

) (m

M)

(h)



= 2 and 3 for all Δ119888 119873119865

= 4 and Δ119888 = 30 45 and 90mM KOH 119873119865

= 5 and Δ119888 = 45 and 90mM KOH (b) 119873119865

= 4 andΔ119888 = 5mM KOH (c) 119873

119865= 4 and Δ119888 = 10mM KOH (d) 119873

119865= 4 and Δ119888 = 15mM KOH (e) 119873

119865= 5 and Δ119888 = 5mM KOH (f) 119873

119865= 5 and

Δ119888 = 10mMKOH (g) 119873119865

= 5 and Δ119888 = 15mMKOH and (h) 119873119865

= 5 and Δ119888 = 30mMKOH




60

50

40

30

20

10

0

0 10 20 30 40 50 60 70 80 90 100 110

Number of mutations

Num

ber o

f cro

ssin

g cy

cles


0

20

40

60

80

0 10 2025

30

30

35

40

45

Resp

onse

(nC)

Time (min)

c(K

OH

) (m

M)

(a)

020406080

100120140

0

20

40

60

80

100

0 10 20 30

Resp

onse

(nC)

Time (min)

c(K

OH

) (m

M)

(b)

0

20

40

60

80

0 10 20 3025

30

35

40

45

1

2

3

4 5

6

78

Resp

onse

(nC)

Time (min)

c(K

OH

) (m

M)

(c)

020406080

100120140

0 10 20 300

20

40

60

80

100

12

34 5

6

7

8

Resp

onse

(nC)

Time (min)c(K

OH

) (m

M)

(d)










0

5

10

15

20

25

30

35

2 26 50 74 98c(KOH) (mM)

Are

a (nC

middotmin

)




119878




5 Conclusions





Acknowledgments


References





















































Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of




1198861

1198862

1198772

1198860

1198861

1198862

1198772

1198860

1198861

1198862

1198772












1

2

3

4

5

1 2 3 4 5

120573

120572












100

20

40

60

80

0 5 15 2085

90

95

100 100Re

spon

se (n

C)

Time (min)

c(K

OH

) (m

M)

(a)

0 5 10 15 20 25 30 35 400

10

20

30

40

50

60

15

20

25

30

Resp

onse

(nC)

Time (min)

c(K

OH

) (m

M)

(b)

5 10 15 20 25 3030

35

40

45

00

10

20

30

40

50

60

Resp

onse

(nC)

Time (min)

c(K

OH

) (m

M)

(c)

5 10

70

0 15 2055

60

65

70

0102030405060

Resp

onse

(nC)

Time (min)

c(K

OH

) (m

M)

(d)

0 10 20 30 40 5010

15

20

25

0

10

20

30

40

50

60

Resp

onse

(nC)

Time (min)

c(K

OH

) (m

M)

(e)

0 10 20 30 4020

25

30

35

0

10

20

30

40

50

60

Resp

onse

(nC)

Time (min)

c(K

OH

) (m

M)

(f)

25

30

35

40

4570

0102030405060

5 10 15 20 25 300

Resp

onse

(nC)

Time (min)

c(K

OH

) (m

M)

(g)

70

75

80

85

90

5 100 15 200

20

40

60

80

100

Resp

onse

(nC)

Time (min)

c(K

OH

) (m

M)

(h)



= 2 and 3 for all Δ119888 119873119865

= 4 and Δ119888 = 30 45 and 90mM KOH 119873119865

= 5 and Δ119888 = 45 and 90mM KOH (b) 119873119865

= 4 andΔ119888 = 5mM KOH (c) 119873

119865= 4 and Δ119888 = 10mM KOH (d) 119873

119865= 4 and Δ119888 = 15mM KOH (e) 119873

119865= 5 and Δ119888 = 5mM KOH (f) 119873

119865= 5 and

Δ119888 = 10mMKOH (g) 119873119865

= 5 and Δ119888 = 15mMKOH and (h) 119873119865

= 5 and Δ119888 = 30mMKOH




60

50

40

30

20

10

0

0 10 20 30 40 50 60 70 80 90 100 110

Number of mutations

Num

ber o

f cro

ssin

g cy

cles


0

20

40

60

80

0 10 2025

30

30

35

40

45

Resp

onse

(nC)

Time (min)

c(K

OH

) (m

M)

(a)

020406080

100120140

0

20

40

60

80

100

0 10 20 30

Resp

onse

(nC)

Time (min)

c(K

OH

) (m

M)

(b)

0

20

40

60

80

0 10 20 3025

30

35

40

45

1

2

3

4 5

6

78

Resp

onse

(nC)

Time (min)

c(K

OH

) (m

M)

(c)

020406080

100120140

0 10 20 300

20

40

60

80

100

12

34 5

6

7

8

Resp

onse

(nC)

Time (min)c(K

OH

) (m

M)

(d)










0

5

10

15

20

25

30

35

2 26 50 74 98c(KOH) (mM)

Are

a (nC

middotmin

)




119878




5 Conclusions





Acknowledgments


References





















































Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of


1

2

3

4

5

1 2 3 4 5

120573

120572












100

20

40

60

80

0 5 15 2085

90

95

100 100Re

spon

se (n

C)

Time (min)

c(K

OH

) (m

M)

(a)

0 5 10 15 20 25 30 35 400

10

20

30

40

50

60

15

20

25

30

Resp

onse

(nC)

Time (min)

c(K

OH

) (m

M)

(b)

5 10 15 20 25 3030

35

40

45

00

10

20

30

40

50

60

Resp

onse

(nC)

Time (min)

c(K

OH

) (m

M)

(c)

5 10

70

0 15 2055

60

65

70

0102030405060

Resp

onse

(nC)

Time (min)

c(K

OH

) (m

M)

(d)

0 10 20 30 40 5010

15

20

25

0

10

20

30

40

50

60

Resp

onse

(nC)

Time (min)

c(K

OH

) (m

M)

(e)

0 10 20 30 4020

25

30

35

0

10

20

30

40

50

60

Resp

onse

(nC)

Time (min)

c(K

OH

) (m

M)

(f)

25

30

35

40

4570

0102030405060

5 10 15 20 25 300

Resp

onse

(nC)

Time (min)

c(K

OH

) (m

M)

(g)

70

75

80

85

90

5 100 15 200

20

40

60

80

100

Resp

onse

(nC)

Time (min)

c(K

OH

) (m

M)

(h)



= 2 and 3 for all Δ119888 119873119865

= 4 and Δ119888 = 30 45 and 90mM KOH 119873119865

= 5 and Δ119888 = 45 and 90mM KOH (b) 119873119865

= 4 andΔ119888 = 5mM KOH (c) 119873

119865= 4 and Δ119888 = 10mM KOH (d) 119873

119865= 4 and Δ119888 = 15mM KOH (e) 119873

119865= 5 and Δ119888 = 5mM KOH (f) 119873

119865= 5 and

Δ119888 = 10mMKOH (g) 119873119865

= 5 and Δ119888 = 15mMKOH and (h) 119873119865

= 5 and Δ119888 = 30mMKOH




60

50

40

30

20

10

0

0 10 20 30 40 50 60 70 80 90 100 110

Number of mutations

Num

ber o

f cro

ssin

g cy

cles


0

20

40

60

80

0 10 2025

30

30

35

40

45

Resp

onse

(nC)

Time (min)

c(K

OH

) (m

M)

(a)

020406080

100120140

0

20

40

60

80

100

0 10 20 30

Resp

onse

(nC)

Time (min)

c(K

OH

) (m

M)

(b)

0

20

40

60

80

0 10 20 3025

30

35

40

45

1

2

3

4 5

6

78

Resp

onse

(nC)

Time (min)

c(K

OH

) (m

M)

(c)

020406080

100120140

0 10 20 300

20

40

60

80

100

12

34 5

6

7

8

Resp

onse

(nC)

Time (min)c(K

OH

) (m

M)

(d)










0

5

10

15

20

25

30

35

2 26 50 74 98c(KOH) (mM)

Are

a (nC

middotmin

)




119878




5 Conclusions





Acknowledgments


References





















































Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of


100

20

40

60

80

0 5 15 2085

90

95

100 100Re

spon

se (n

C)

Time (min)

c(K

OH

) (m

M)

(a)

0 5 10 15 20 25 30 35 400

10

20

30

40

50

60

15

20

25

30

Resp

onse

(nC)

Time (min)

c(K

OH

) (m

M)

(b)

5 10 15 20 25 3030

35

40

45

00

10

20

30

40

50

60

Resp

onse

(nC)

Time (min)

c(K

OH

) (m

M)

(c)

5 10

70

0 15 2055

60

65

70

0102030405060

Resp

onse

(nC)

Time (min)

c(K

OH

) (m

M)

(d)

0 10 20 30 40 5010

15

20

25

0

10

20

30

40

50

60

Resp

onse

(nC)

Time (min)

c(K

OH

) (m

M)

(e)

0 10 20 30 4020

25

30

35

0

10

20

30

40

50

60

Resp

onse

(nC)

Time (min)

c(K

OH

) (m

M)

(f)

25

30

35

40

4570

0102030405060

5 10 15 20 25 300

Resp

onse

(nC)

Time (min)

c(K

OH

) (m

M)

(g)

70

75

80

85

90

5 100 15 200

20

40

60

80

100

Resp

onse

(nC)

Time (min)

c(K

OH

) (m

M)

(h)



= 2 and 3 for all Δ119888 119873119865

= 4 and Δ119888 = 30 45 and 90mM KOH 119873119865

= 5 and Δ119888 = 45 and 90mM KOH (b) 119873119865

= 4 andΔ119888 = 5mM KOH (c) 119873

119865= 4 and Δ119888 = 10mM KOH (d) 119873

119865= 4 and Δ119888 = 15mM KOH (e) 119873

119865= 5 and Δ119888 = 5mM KOH (f) 119873

119865= 5 and

Δ119888 = 10mMKOH (g) 119873119865

= 5 and Δ119888 = 15mMKOH and (h) 119873119865

= 5 and Δ119888 = 30mMKOH




60

50

40

30

20

10

0

0 10 20 30 40 50 60 70 80 90 100 110

Number of mutations

Num

ber o

f cro

ssin

g cy

cles


0

20

40

60

80

0 10 2025

30

30

35

40

45

Resp

onse

(nC)

Time (min)

c(K

OH

) (m

M)

(a)

020406080

100120140

0

20

40

60

80

100

0 10 20 30

Resp

onse

(nC)

Time (min)

c(K

OH

) (m

M)

(b)

0

20

40

60

80

0 10 20 3025

30

35

40

45

1

2

3

4 5

6

78

Resp

onse

(nC)

Time (min)

c(K

OH

) (m

M)

(c)

020406080

100120140

0 10 20 300

20

40

60

80

100

12

34 5

6

7

8

Resp

onse

(nC)

Time (min)c(K

OH

) (m

M)

(d)










0

5

10

15

20

25

30

35

2 26 50 74 98c(KOH) (mM)

Are

a (nC

middotmin

)




119878




5 Conclusions





Acknowledgments


References





















































Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of


60

50

40

30

20

10

0

0 10 20 30 40 50 60 70 80 90 100 110

Number of mutations

Num

ber o

f cro

ssin

g cy

cles


0

20

40

60

80

0 10 2025

30

30

35

40

45

Resp

onse

(nC)

Time (min)

c(K

OH

) (m

M)

(a)

020406080

100120140

0

20

40

60

80

100

0 10 20 30

Resp

onse

(nC)

Time (min)

c(K

OH

) (m

M)

(b)

0

20

40

60

80

0 10 20 3025

30

35

40

45

1

2

3

4 5

6

78

Resp

onse

(nC)

Time (min)

c(K

OH

) (m

M)

(c)

020406080

100120140

0 10 20 300

20

40

60

80

100

12

34 5

6

7

8

Resp

onse

(nC)

Time (min)c(K

OH

) (m

M)

(d)










0

5

10

15

20

25

30

35

2 26 50 74 98c(KOH) (mM)

Are

a (nC

middotmin

)




119878




5 Conclusions





Acknowledgments


References





















































Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of






0

5

10

15

20

25

30

35

2 26 50 74 98c(KOH) (mM)

Are

a (nC

middotmin

)




119878




5 Conclusions





Acknowledgments


References





















































Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of


Acknowledgments


References





















































Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of



























Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of

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