A new approach for the thermodynamic modeling of the solubility of amino acids and β-lactam compounds as a function of pH

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Fluid

Phase

Equilibria

jo u rn al h om ep ag e :w w w . e l s e v i e r . c o m / l o c a t e / f l u i d

A

new

approach

for

the

thermodynamic

modeling

of

the

solubility

of

amino

acids

and

␤-lactam

compounds

as

a

function

of

pH

Luís

Fernando

Mercier

Franco

a

_,

_Silvana

_Mattedi

b

_,

_Pedro

_de

_Alcântara

_Pessôa

_Filho

a,∗

a_{BioprocessEngineeringGroup,ChemicalEngineeringDepartment,SchoolofEngineering,UniversidadedeSãoPaulo,CaixaPostal61548,05424-970São}

Paulo,SP,Brazil

b_{ChemicalEngineeringGraduateProgram,PolytechnicSchool,UniversidadeFederaldaBahia,Brazil}

a

r

t

i

c

l

e

i

n

f

o

Articlehistory:

Received24January2013

Receivedinrevisedform5June2013 Accepted6June2013

Available online 15 June 2013 Keywords: Solid–liquidequilibrium Modeling Aminoacids ␤-Lactamcompounds

a

b

s

t

r

a

c

t

Anewmodeltodescribethesolubilityofaminoacidsand␤-lactamcompoundsasafunctionofpH isproposed.Thedescriptionofthesolid–liquidequilibriumentailssimultaneouscalculationofphase andchemicalequilibria.Theliquid-phasenon-idealitywasaccountedforthroughanextendedPitzer modelforpolyelectrolytesolutions.AstatisticalthermodynamicinterpretationofPitzer’sbinary inter-actionparametersisalsopresented.Themodelwassuccessfullyappliedtodifferentsystemscontaining amino-acids,aswellasglycineanditsoligopeptidesand␤-lactamcompoundssuchasampicillinand 6-aminopenicillanicacid.

© 2013 Elsevier B.V. All rights reserved.

1. Introduction

Whenmodelingthesolubilityofaminoacidsor_␤-lactam antibi-oticsasafunctionofpH,themostimportantaspectistheliquid phasemodeling,sincesolidphasecanbeassumedtobeunique andconstitutedonlybysolutemolecules.Aspecificissue concern-ingthesolubilityofaminoacidsisthefactthatinaqueoussolutions theydissociate,sothatasingleaminoacidcanbefoundinthree differentstates:asacation,asananionandasazwitterion–the neutralnonpolarspeciesisvirtuallynon-existentinthesolution. Manystrategieshavebeenproposedtocalculateaminoacid activ-itycoefficients inliquid phase:throughthePoisson–Boltzmann equation[1],throughexcessGibbsenergymodels[2–8],through group-contributionmodels[9,10],throughstatistical thermody-namicapproaches[11]andthroughvolumetricequationsofstate [12–14].

Tothebestofourknowledge,thefirstmodeltodescribeamino acidsolubilitycurveasafunctionofpHwasbasedentirelyupon considerationsaboutchemicaldissociation[15]without consider-ingliquid-phasenon-ideality.Thesolubilitycurveoftyrosinein aqueoussolutionasa functionofpHwascalculated withgood agreementwithexperimentaldata.

TheworkbyKirkwood[1]isamongthefirstonestodescribe solutions containing zwitterions. It is based on the analytic

∗ Correspondingauthor.Tel.:+551130911106;fax:+551130912284. E-mailaddress:[email protected](P.d.A.PessôaFilho).

solutionofthelinearizedPoisson–Boltzmannequation, consider-ingthethreedirectionsontheLaplacianoperatorforaspherical zwitterion. The author obtained an expression that relatesthe logarithmofanionactivitycoefficienttoaseriesinvolving mul-tipolemoments.Thefirsttermofthisseriesisanalogoustothe Debye–Hückelterm,beingnullforzwitterions, astheyhave no netcharge.Withsomesimplifications,theauthoralsoshowedthe applicationofthisapproachtoanellipsoidzwitterion[16].

Nass[2]modeledthesolubilityofaminoacids asafunction ofpHassumingthattheexcessGibbsenergyisgivenbythesum ofchemicalandphysicalcontributions.Thechemicalcontribution wascalculatedconsideringaminoaciddissociation,massbalance equationsand phaseequilibrium conditions,whereas the phys-icalcontributionwascalculated throughWilson’sequation[17] forthebinarymixtureofwaterandaminoacid.The experimen-taldatawerewellcorrelated,attheexpenseofahighnumberof adjustableparameters.Chenetal.[3]consideredthattheexcess Gibbsenergycouldbesplitinthreeterms, twoofthemrelated tolong-rangeinteractions,viz.,aPitzer–Debye–HückelandaBorn terms,andanotheronetoaccountforshort-rangeinteractions,the NRTLequation[18].Fortheactivitycoefficientofthezwitterions, thePitzer–Debye–Hückelcontributionbecomesnull.Theauthors showedthattheexperimentalsolubilitydataweresatisfactorily correlatedwithonlyoneortwoadjustableparameters foreach compound.

TheworkbyGuptaandHeidemann[9]laidthefoundationfor mostensuingworks.Theirapproachrelateschemicaldissociation of amino acids and the solid–liquidequilibrium condition.The 0378-3812/$–seefrontmatter © 2013 Elsevier B.V. All rights reserved.

Listofsymbols Latinletters

a universal parameter of the Pitzer’s model (kg1/2_mol−1/2₎

ai activityofcomponenti

b universal parameter of the Pitzer’s model (kg1/2_mol−1/2₎

Bij osmotic second virial coefficient for interaction

betweencomponentsiandj(m3₎

Cij(1) adjustable parameter for the Pitzer’s model

(kgmol−1)

Cij(2) adjustable parameter for the Pitzer’s model

(kgmol−1K−1)

ds solventdensity(kgm−3)

e electriccharge(As) I ionicstrength(molkg−1)

kB Boltzmann’sconstant(m2kgs−2K−1)

KA1 equilibriumconstantofacidprotonation

KA2 equilibriumconstantofamineprotonation

KD equilibriumconstantofneutralmolecules

m0 molality of electrically neutral amino acids

(molkg−1)

mi molalityofcomponenti(molkg−1)

m* molality of component i in the reference state (molkg−1)

M numberofexperimentaldatasets Ms solventmolecularmass(kgmol−1)

N numberofexperimentaldatapoints NA Avogadro’snumber(mol−1)

r radialcoordinate(m)

R gasconstant(kgm2_mol−1_K−1_s−2₎

S solubility(molkg−1)

Sexp_i experimentalsolubility(molkg−1) Scalc

i calculatedsolubility(molkg−1)

T temperature(K)

Wij potential of mean force between i and j

(kgm2_mol−1_s−2₎

zi electricchargeofcomponenti

zj electricchargeofcomponentj

Greekletters

˛AA− ionizedfractionofaminoacidinanionicform

˛AA+ ionizedfractionofaminoacidincationicform

0 electricallyneutralaminoacidactivitycoefficient

AA− zwitterionicactivitycoefficient

i activitycoefficientofcomponenti

ε van der Waals attractive interaction parameter (m2_kgs−2₎

ε0 vacuumpermittivity(A2s4kg−1m−3)

εr dielectricconstant

 inverseofDebye’slength

ij Pitzer’sbinaryinteractionparameter(kgmol−1)

ijk Pitzer’sternaryinteractionparameter(kg2mol−2)

L

i chemical potential of component i in the liquid

phase(kgm2_mol−1_s−2₎

S

i chemicalpotentialofcomponentiinthesolidphase

(kgm2_mol−1_s−2₎

∗_i chemicalpotentialofcomponentiinthereference state(kgm2_mol−1_s−2₎

i numberdensityofcomponenti(m−3)

j numberdensityofcomponentj(m−3)

vanderWaalsdiameter(m)

0 fractionofelectricallyneutralaminoacids

AA fractionofnonpolaraminoacidmolecules

_AA− fractionofanionicaminoacidmolecules AA+ fractionofcationicaminoacidmolecules

AA− fractionofzwitterionicaminoacidmolecules

non-idealityofaqueoussolutions ofnativeand modifiedamino acidswascalculatedthroughthemodifiedUNIFACmodelofLarsen etal.[19]withnewgroupssuchasglycineandproline.Following theirwork,Pinhoetal.[10]proposedaUNIFACmodelcombined withaDebye–Hückeltermtocalculatetheactivitycoefficientsof aminoacids;thesolubilityofglycinewasmodeledasafunctionof pHwithverygoodagreementwithexperimentaldata. Neverthe-less,someauthorsquestiontheintroductionoftheDebye–Hückel terminthiscase[5,11].Rodríguez-Raposoetal.[4]fittedosmotic coefficientsofglycineinaqueoussolutionusingthePitzer equa-tionandcalculatedtheactivitycoefficientofglycineinaqueous solution,withverygoodagreementtotheexperimentaldata.

KhoshkbarchiandVera[5]splittheexcessGibbsenergyin long-rangeandshort-rangeinteractioncontributions.Theytestedthe modelsbyBromley[20]andtheirown[21]forthelong-range inter-actioncontributionand,fortheshort-rangeterm,theNRTLand Wilsonmodels[17].Whicheverthemodelchosen,theapproach requiredonlytwoparameters,andbinarywater–aminoacid sys-temsandternarywater–electrolyte–aminoacidsystemswerewell correlated.

Eventhoughthebasicequationtopredicttheglycinesolubility asafunctionofpHusedbyKhoshkbarchiandVera[11]follows theapproach by Guptaand Heidemann[9], theyintroduce the useoftheperturbationtheorytocalculatethenon-idealityofthe amino acidin theaqueoussolution.Theymodeled aminoacids asLennard–Jonessphereswithdipole–dipoleinteractions.Santana etal.[22]showedthatthismodelfailstoquantitativelypredictthe solubilityofampicillin,d-phenylglycineand6-aminopenicillanic acidasafunctionofpHevenadjustingfiveparameters.Pradhan andVera[6],usinganequationsimilartothatproposedbyGupta and Heidemann[9] withtheNRTLequation,correlated the dl-alaninesolubilityasafunctionofpH,withgoodagreementwith theexperimentaldatainthevicinityoftheisoelectricpoint.

Parketal.[12]correlatedtheactivitycoefficientofvaline, ␣-aminobutyricacid,alanine,glycineandglycylglycineinaqueous solutionbyapplyingthehydrogen-bondinglatticefluidequation ofstate,withnoconsiderationaboutaminoaciddissociation.

Xuetal.[7]proposeda modificationoftheWilsonmodelto calculatetheexcessGibbsenergyof aminoacidsolutions. They assumedthatallaminoacidmoleculeswereinzwitterionicform inpurewater,whichsomehowrestrictstheapplicationoftheir modelforcalculatingaminoacidsolubilityatpHinthevicinityof isoelectricpoint.

UsingthePC-SAFTequationofstateandaccountingforamino aciddissociation insolution,Fuchsetal.[13]describedthe sol-ubility curve as a function of pH for glycine, dl-alanine and dl-methionine in aqueous solution withgood agreement with experimentaldata.Similarly,Seyfietal.[14],followingGuptaand Heidemann[9]andapplyingSAFTequationofstate,correlatedthe dl-methioninesolubilitycurveasafunctionofpHandtemperature withhighaccuracy.

Tseng et al. [8] investigated the solubility of dl-alanine, l-leucine,l-isoleucine,l-serineanddl-phenylalanineasafunction ofpHinaqueoussolution.Consideringaminoaciddissociationin aqueoussolutionsandusingtheNRTLmodel,theyobtained satis-factoryresultswithtwoadjustableparameters.Dalrupetal.[23] modeledsystemscontainingtwoaminoacidsinsolutionusing PC-SAFTequationofstate.

Thus,whilemodelsdifferconsiderablyonwhichcontributions areconsideredintheactivitycoefficientmodel,thecalculationof acid–baseequilibriumremainsasthefundamentalstepin describ-ingthesolubilitycurve.

2. Theoreticalframework

Thebasisofthemodelisarecentworkonmodelingprotein solubilitycurveasafunctionofpH[24].Inthiscase,theequation foraminoacids(regardlessoftheresidue)isjustaparticularcase ofthesameequationwithtwoorthreeionizablegroups.

Theconditionofsolid–liquidequilibriumforagivensystem,in whichsolidphaseisconsideredtobeuniqueandtocontainonly moleculesofsolute,is:

S_i =L_i (1)

whereinS

i isthechemicalpotentialofsoluteiinsolidphaseand

L

i isthechemicalpotentialofsoluteiinliquidphase.

Neglect-ingtheeffectofpressureonchemicalpotentialandconsideringan isothermalprocess,thechemicalpotentialofsolidphasewillnot change,asitscompositionisfixed.Thus,thechemicalpotentialof soluteinliquidphasemustremainconstant,andthepHvaluedoes notalterthesolutechemicalpotentialinliquidphase.

Usingmolalityscale[25]: L_i =∗_i+RT lnai=∗i+RTln



_ imi m∗



(2) wherein ∗_i is the standard state chemical potential of solute i in a hypothetical ideal solution with unit concentration (m*=1.0molkg−1),iistheactivitycoefficientofsoluteiandmiis

themolalityofiinliquidphase.

Aminoacidor␤-lactamantibioticsmayhavefourdifferent ion-izationstatesinaqueoussolutions.Eq.(1)isvalidonlyforthose whichareelectricallyneutralspecies.Thisrestrictionisduetothe factthatsolidphaseiselectricallyneutral,andhencethephase equilibriumisestablishedonlybyelectricallyneutralspecies of aminoacids. Thus, theactivityof electricallyneutral molecules remains constantwithany variation ofpH.The experimentally determinedsolubility,however,isobtainedconsideringthe con-centrationof allkindsof amino acidsmoleculesirrespectiveto theirionizationstate.Thus,themolalityoftheelectricallyneutral moleculesisafractionofthesolubility:

m0(pH)=0(pH)S(pH) (3)

whereinm0isthemolalityoftheelectricallyneutralmoleculesof

aminoacids,0isthefractionofelectricallyneutralmoleculesof

aminoacidsandSistheaminoacidsolubility.Astheactivityof electricallyneutralmoleculesofaminoacidsisconstant,itsvalue atanypHisequaltoitsvalueatisoelectricpoint(pI).Thus,onecan write:

S(pH)= 0(pI)0(pI)S(pI)

0(pH)0(pH)

(4) Iftheaminoacidconcentrationinliquidphaseissmall,Henry’s lawappliesandEq.(4)canberewrittensimplyas[24]:

S(pH)= 0(pI)S(pI)

0(pH)

(5)

2.1. Chemicalequilibrium

Let us firstly consider amino acids with only two ionizable groupsinitschain–extensionforaminoacidswhichhavethree ionizablegroupsisstraightforward.

Inaqueoussolutions,thefollowingequilibriatakeplace:

NH2RCOOH NH3+RCOO− (6)

NH2RCOO−NH2RCOO−+H+ (7)

NH3+RCOOH NH3+RCOO−+H+ (8)

Therearefourdifferentspeciesofaminoacidmolecules:the neutral non-ionized one (AA), thezwitterion (AA±),the cation (AA+)andtheanion(AA−).Theequilibriumrelationshipassociated toeachreactionis:

KD= [AA±] [AA] (9) KA1= [AA−][H+] [AA±] (10) KA2= [AA±][H+] [AA+] (11)

ThevalueofKDissohightheneutralnon-ionizedformis

vir-tuallyinexistent,soitwillnolongerbeconsidered.Thefractionof ionizedspeciesofeachreaction,excludingthefirstone,isgivenby theHenderson–Hasselbalchequations:

˛_AA−(pH)= 10

pH−pKA1

1+10pH−pKA1 (12)

˛_AA+(pH)= 1

1+10pH−pKA2 (13)

Thus,thefractionofeachspeciesofaminoacidissimply: AA (pH)=[1−˛AA−(pH)][1−˛AA+(pH)] (14)

AA±(pH)=˛AA−(pH)˛AA+(pH) (15)

_AA−(pH)=˛_AA−(pH)[1−˛AA+(pH)] (16)

_AA+(pH)=[1−˛_AA−(pH)]˛_AA+(pH) (17) Eqs.(14)or(15)canbeusedin Eqs.(4) and (5).Thechoice betweenthemiscompletelyarbitrary[24],andsothezwitterionic speciesischosenfornumericalreasons,becauseabsolutevaluesof thefractioncalculatedthroughEq.(14)isvanishingsmall,several ordersofmagnitudesmallerthanabsolutevaluesofthefraction evaluatedthroughEq.(15).

2.2. ExtendedPitzermodel

Toaccountforliquidphasenon-ideality,theextendedPitzer modeldevelopedbyPessôaFilhoandMaurer[26]wasapplied.This formulationismoreadequatetoapplicationtoaminoacid solu-tions,whereintheelectricallyneutralformisazwitterionicone. Forasoluteiinanelectrolyteaqueoussolution,thenatural loga-rithmoftheactivitycoefficientisgivenbythesumofatermdue tolongrangeinteractionsandoneduetoshortrangeinteractions: ln i=lniLR+ln

SR

i (18)

Thelongrangetermisgivenby: ln _iLR=−Azi2



2 bln



1+b√I



+ √ I 1+b√I



(19) wherein: A= 1 3



e2 ε0εrkBT

3/2 (2dsNA)1/2 8 (20)

wherein ε0 is the vacuum permittivity

(ε0=8.85419×10−12A2s4kg−1m−3), εr is the

dielec-tric constant, ds is the solvent density, NA is Avogadro’s

number (NA=6.0221415×1023), e is the electronic

charge (e=1.60218×10−19As), kB is Boltzmann constant

andbisauniversalparameterequalto1.2kg1/2_mol−1/2_[25].The

shortrangetermisgivenby: ln_iSR=2

j/=s ij(I)mj+3

j/=s

k/=s ijkmjmk−zi2Ms

j/=s

k/=s (1)_ij 1 a2_I2 ×



1−



1+a√I+a₂2I

exp(−a√I)



mjmk (21)

whereinMsisthemolarmassofthesolventinkgmol−1,aisanother

universalparameterequal to2.0kg1/2_mol−1/2_,

ijk isthe

three-bodyinteractioncoefficient,and ij isthetwo bodyinteraction

coefficient,whichdependsuponionicstrength: ij(I)=(0)ij + (1) ij 2 a2_I2[1−(1+a √

I)exp(−a√I)] (22) Theionicstrengthisdefinedas:

I=1 2

i

mizi2 (23)

Forthezwitterionicform,thelongrangetermisnullsince zwit-terionhasnonetcharge.Twoothersimplificationscanbemade: thefirstoneconsistsinneglectingthree-bodyinteractions(ijk=0,

foranyi,jandk)andthesecondoneinneglectingthedependence uponionicstrengthintwo-bodyinteractions(ij(1)=0,foranyiand

j).Thedependenceoftheparameterduetotwo-bodyinteractions upontemperaturehasbeencorrelatedbytheproposedequationby PitzerandPeiper[27]withtheminimumofadjustableparameters (andforwhichatheoreticaljustificationcanbeprovided):

ij(T )=C_ij(1)+

C_ij(2)

T (24)

Thereby,onecanrewriteEq.(4)usingtheextendedPitzermodel toaccountfor liquidphase non-ideality duetothepresence of electricchargesinthesolutionas:

lnS(pH) S(pI) =ln AA±(pI) AA±(pH)+2AA−[AA−(pI)S(pI)−AA−(pH)S(pH)] +2_AA+[_AA+(pI)S(pI)−_AA+(pH)S(pH)] +2AA[AA(pI)S(pI)−AA(pH)S(pH)]

+2AA±[AA±(pI)S(pI)−AA±(pH)S(pH)] (25)

Theempirical observationofamino acidsolubilitycurves as afunctionofpHrevealscertainsymmetryaroundtheisoelectric point.Thisallowsinferringthatinteractionsamongtheelectrically neutralmoleculesofaminoacidandelectricallychargedmolecules areindistinguishable,i.e.,anelectricallyneutralmoleculeinteracts inthesamewaywithapositiveornegativechargedmolecule.This hypothesisimpliesthefollowingidentityapriori:

_AA−=_AA+=AA=AA±= (26)

Evidently,ifsuchsymmetryisnotobserved,thishypothesiswill nothold.ConsideringthesimplificationintroducedbyEq.(26),Eq. (25)canberewrittenas:

lnS(pH) S(pI) =ln

AA±(pI)

AA±(pH)+2[S(pI)−S(pH)] (27)

ByreplacingEqs.(12)–(17)inEq.(27),onegetsanexplicit rela-tionbetweenaminoacidssolubilityandpH:

logS(pH) S(pI) =pI−pH+log



1+10pH−pKA1 1+10pI−pKA1



+log



1+10pH−pKA2 1+10pI−pKA2



+_ln2₁₀ [S(pI)−S(pH)] (28)

2.3. Astatisticalthermodynamicinterpretationofinteraction parameters

ConsideringMcMillan–Mayer’stheory,fromthestatistical ther-modynamicpointof view,it ispossibletoemulatetheosmotic pressuregeneratedbyasoluteinacontinuumliquidsolventusing thesameexpressionsfornon-idealityofgases[28].Therefore,one canwritethenaturallogarithmoftheactivitycoefficientofthe zwitterionicmoleculeas:

lnAA±=2

j

jBj,AA± (29)

whereinB_j,AA±istheosmoticsecondvirialcoefficientandjisthe

numberofmoleculesofcomponentjpervolumeunity.

Consideringthat theosmoticsecondvirial coefficientcanbe relatedtothepotentialofmeanforceofinteractionamongsolutes intoagivensolventinthelimitoflowsoluteconcentrations,one gets[29]: Bij =− 1 162



2 0



 0



2 0



2 0



 0



+∞ 0



exp



−W_kij(r) BT

−1



r2sin ϕsinˇdrdϕd d˛dˇd (30)

Consideringthemeanforcepotentialasgiven bythesumof threecontributions:hardsphere(WHS_{),attractivepotentialdueto}

vanderWaalsforces(WvdW_{)andelectrostaticinteractions(W}elet₎

assuggestedbytheDLVOtheory:

Wij(r)=

⎧

⎨

⎩

+∞, ifr≤ zizje2exp[−(r− )] ∈r(1+ ) −ε



r



6 , ifr> (31) whereinziistheelectricchargeofmoleculei,zjistheelectriccharge

ofmoleculej,eistheelectroniccharge, isthemeanvander Waalsdiameterofmoleculesiandj,istheinteractionparameter betweenmoleculesiandj.

One can assume that any configuration of an amino acid moleculehasthesamevanderWaalsdiameter, –areasonable hypothesissincetheonlydifferencebetweenthemisthecharge. WealsoassumedthatthemagnitudeofvanderWaalsinteraction parameterbetweenzwitterionicformandtheothersisthesame, whichalsoseemsreasonableforsuchsimilarchemicalstructures astheionizedformsofanaminoacid.

Beingmoleculeithezwitterion,electrostaticcontributionisnull sinceziequalszero.Thus,theosmoticsecondvirialcoefficientfor

asystemwithisotropicinteractionscanbewrittenas: Bij= 2 3 3



1−_kε BT



(32) The natural logarithm of zwitterionic activity coefficient becomes: ln AA±=4 3 3



1− ε kBT



j j (33)

Table1

Valuesoftheinteractionparameter()andRMSD–Eq.(37)–foraminoacids.

Aminoacids (kgmol−1) RMSD(%) Experimentaldata

dl-Alanine 0.0267 2.0 [8] dl-Methionine 0.0935 6.8 [13] l-Isoleucine 0.234 5.2 [8] l-Leucine 0.273 9.4 [8] dl-Phenylalanine 3.95 2.9 [8] l-Serine 0.0165 4.6 [8]

ReplacingEq.(33)inEq.(4),onehas: ln S(pH) S(pI) =ln AA±(pI) AA±(pH)+ 4 3_N A 3



1− ε kBT



[S(pI)−S(pH)] (34) FromthecomparisonbetweenEqs.(27)and(34),onehasthat thephysicalmeaningofparametersestablishedinEq.(24)should begivenby: C_ij(1)=2 3NA 3 (35) C_ij(2)=−C(1) ij ε kB (36)

3. Resultsanddiscussions

3.1. SolubilityofaminoacidsasafunctionofpH

The roots of Eq. (28) were obtained through the Newton–Raphsonmethod.ParametersC_ij(1)andC_ij(2)wereobtained usingthesimplexmethodofNelderandMead[30].Therelative meansquaredeviation(RMSD)wasusedasanobjectivefunction toadjustinteractionparametersalongwithEq.(28):

RMSD=









1 N N

i=1



Sexp_i −Scalc i Sexp_i

2 (37)

whereinS_iexp is the ith experimentalsolubility and Scalc i is the

relatedcalculatedsolubilitybyEq.(28).

Table1presentstheresultsforinteractionparameterfor sys-temscontainingaminoacids.ThevaluesofpKAforsuchsystemsare

presentedinTable2.Onecanobservethatthemaximumvalueof RMSDforsuchadjustmentsislessthan10.0%andthatthe interac-tionparametermagnitudeincreaseswiththeaminoacidmolecular weight.Thisfindingshowsthatnon-idealityismainlygivenbyan entropic(combinatorial)effect,sinceitisdependentonthesizeof themolecules.TheresultsarepresentedinFigs.1–6.Fromthem,it ispossibletostatethattheapplicationofthemodelisaproficuous approach.

Eq.(37)canbeusedasacriterionforaquantitativecomparison amongdifferentmodels.Wecomparedtheresultsofthemodel hereinpresentedwiththeidealsolutionmodelandwiththeNRTL modelappliedbyTsengetal.[8].Thereisanimportantphysical differencebetweentheapplicationofthismodelandthatofNRTLas appliedbyTsengetal.[8].WhilethismodelisinMcMillan–Mayer ensemble,aswellasPitzer’smodel,consideringthatnon-idealityis establishedbyinteractionamongdifferentaminoacidsmolecules, theNRTLmodelconsidersthatnon-idealityisduetotheinteraction betweentheaminoacidandthesolventmolecule.Table3shows thecomparisonamongthesemodels.

Whilethemodelhereinpresentedrequiresonlyoneadjustable parameter,theNRTLequationrequiresatleasttwo.Evenso,asone canseeinTable3,thismodelcorrelatesexperimentaldatawith relativelymoreaccuracyforallthesystemsstudied.Forsystems

0.0 2.0 4.0 6.0 8.0 10.0 12.0 1.5 2.0 2.5 3.0 3.5 So lubility / mol.kg -1

pH

Fig.1.Solubilityofdl-alanineasafunctionofpHat298.15Kinaqueoussolution. Experimentaldata()[8];NaClorHClsolution()[6];KOHorHNO3solution()

[6].Modeling:idealsolution(dottedline),thiswork(continuousline).

suchassolutionscontainingdl-alanineandl-serine,despitethe vastliteratureconcerningthenon-idealitymodelingofthose sys-tems,thesimplesthypothesisofidealsolutionisreasonablewhen theRMSDiscomparedamongdifferentmodels.

Table2

ValuesofpKAforaminoacids,peptidesand␤-lactamcompounds.

Compound T(K) pKA1 pKA2 Reference dl-Alanine 298.15 2.35 9.87 [8] dl-Methionine 298.15 2.12 9.28 [13] l-Isoleucine 298.15 2.32 9.76 [8] L-Leucine 298.15 2.33 9.74 [8] dl-Phenylalanine 298.15 2.58 9.24 [8] L-Serine 298.15 2.21 9.15 [8] Tyrosinea _{298.15 2.20 9.11 [32]} Glycine 298.15 2.34 9.60 [32] Dyglycine 298.15 3.12 8.17 [32] Triglycine 298.15 3.26 7.91 [32] Tetraglycine 298.15 3.05 7.75 [32] Pentaglycine 298.15 3.05 7.70 [32] Hexaglycine 298.15 3.05 7.60 [32] Ampicillin 283.1 2.27 7.66 [22] 288.1 2.24 7.61 [22] 293.1 2.14 7.45 [22] 298.1 2.14 7.31 [22] 6-Aminopenicillanic acid 283.1 1.98 5.34 [22] 288.1 2.04 5.25 [22] 293.1 2.1 5.18 [22] 298.1 2.20 4.83 [22] a_ThepK

A3valueoftyrosineis10.07accordingtoGreensteinandWinitz[32].

Table3

ValuesofRMSD–Eq.(37)–foraminoacidsapplyingthemodels:idealsolution,this workandNRTL.ExperimentalsolubilitydataobtainedbyTsengetal.[8].

Aminoacid Model

Idealsolution(%) Thiswork(%) NRTLa_(%)

dl-Alanine 2.4 2.0 2.2

l-Isoleucine 24.9 5.2 6.3

l-Leucine 32.3 9.4 13.3

dl-Phenylalanine 62.5 2.9 3.0

l-Serine 6.0 4.6 5.1

0.0 2.0 4.0 6.0 8.0 10.0 12.0 0.0 0.2 0.4 0.6 0.8 1.0

So

lubility / mol.kg

-1

pH

Fig.2.Solubilityofdl-methionineasafunctionofpHat303.0Kinaqueous solu-tion.Experimentaldata()[13].Modeling:idealsolution(dottedline),thiswork (continuousline).

Themodelmustbeslightlymodifiedtoaccountforthepresence ofaminoacidswithionizableresidues.Withonemoreionizable group, there will beeight different configurations that can be assumedbyanaminoacidmolecule.Therefore,Eq.(25)wouldhave eight,insteadoffour,interactionparameters:oneparameterforthe interactionofeachaminoacidconfigurationwiththezwitterionic form.IfweadoptasimilarhypothesisasthatadoptedinEq.(26), Eq.(28)canthusberewrittenas:

log S(pH) S(pI) =pI−pH+log



1+10pH−pKA1 1+10pI−pKA1



+log



1+10pH−pKA2 1+10pI−pKA2



+log



1+10pH−pKA3 1+10pI−pKA3



+_ln2₁₀[S(pI)−S(pH)] (38) 0.0 2.0 4.0 6.0 8.0 10.0 12.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0

So

lubility / mol.kg

-1

pH

Fig.3. Solubilityofl-isoleucineasafunctionofpHat298.15Kinaqueous solu-tion.Experimentaldata()[8].Modeling:idealsolution(dottedline),thiswork (continuousline). 0.0 2.0 4.0 6.0 8.0 10.0 12.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0

So

lubility / mol.kg

-1

pH

Fig.4. Solubilityofl-leucineasafunctionofpHat298.15Kinaqueoussolution. Experimentaldata()[8].Modeling:idealsolution(dottedline),thiswork (con-tinuousline).

Therearefewexperimentaldataavailableintheopen litera-tureofsystemscontainingaminoacidswiththreeionizablegroups inchain.Here,weusedthetyrosinesolubilitydatapublishedby Hitchcock[15].Tothebestofourknowledge,inthatarticle, Hitch-cockpresentedtheveryfirstmodelingforanaminoacidsolubility curve asa function of pH,considering idealsolution. Although tyrosinehasanionizableR-group,itcommonlyappearsinthe bio-chemical specializedliteratureas nonpolaraminoacidsinceits pKA3istoohigh.InFig.7,weshowhowimportantistoconsider

thetyrosineR-groupasionizableevenwhenconsideringonlyideal solution(ij=0).Anditisinterestingtonotethatconsideringthe

tyrosineR-groupasan ionizableonedoesimprovesolid–liquid equilibriumdescription.

3.2. SolubilityofglycineanditsoligopeptidesasafunctionofpH Thesolubilityofsmallpeptideswithonlyterminalgroups,such asionizableoneswasalsoinvestigated.Table4presentsthevalues

0.0 2.0 4.0 6.0 8.0 10.0 12.0 0.0 0.1 0.2 0.3 0.4

So

lubility / mol.kg

-1

pH

Fig.5.Solubilityofdl-phenylalanineasafunctionofpHat298.15Kinaqueous solution.Experimentaldata()[8].Modeling:idealsolution(dottedline),thiswork (continuousline).

0.0 2.0 4.0 6.0 8.0 10.0 12.0 3.0 4.5 6.0 7.5 9.0 10.5 12.0 So lubility / mol.kg -1

pH

Fig.6. Solubilityofl-serineasafunctionofpHat298.15Kinaqueoussolution. Experimentaldata()[8].Modeling:idealsolution(dottedline),thiswork (con-tinuousline). 0.0 2.0 4.0 6.0 8.0 10.0 12.0 0.0 10.0 20.0 30.0 40.0

So

lubility / mol.kg

-1

pH

Fig.7. SolubilityoftyrosineasafunctionofpHat298.15Kinaqueoussolution. Experimentaldata()[15].Modeling:consideringR-groupasnon-ionizable(dotted line),consideringR-groupasionizable(continuousline).

Table4

Valuesofinteractionparameter()andRMSD–Eq.(37)–forglycineandits oligopeptidesasafunctionofpH.ExperimentalsolubilitydataobtainedbyLuetal. [31]. Compound (kgmol−1_{) RMSD(%)} Glycine 0.200 11.0 Dyglycine 3.26 6.9 Triglycine 13.8 9.8 Tetraglycine 93.6 1.7 Pentaglycine 238.8 4.7 Hexaglycine 258.2 7.9

ofinteractionparameterijandofRMSD–definedinEq.(37)–for

aqueoussolutionscontainingglycineoritsoligopeptides(upto hexaglycine).ThevaluesofpKAarealsopresentedinTable2.Fig.8

allowsobservingthatthemodelcanaccuratelydescribesolubility curves.Itisinterestingthat,assimilarasforsimpleaminoacids, the magnitude of interaction parameter ij increases withthe

incrementinthepeptidechainsize.Consideringtheinterpretation oftheparameterpresentedearlier,onecanwrite:

=2 3 3 NA



1− ε kBT



(39)

Assumingthatεshouldincreaseasthepeptidenonpolarchain increases,wewouldexpecttodecreasewiththeincrementin peptidechainsize,sincetherewouldbeanincrementin hydropho-bicinteractionrepresentedbytheattractiveparameterε.However, thevalueofincreases,whichmeansthattheeffectofvander Waalsdiameter ispredominant.

3.3. Solubilityofˇ-lactamcompoundsasafunctionofpHand temperature

Similarly to amino acids, ␤-lactam compounds present one amine and one carboxylic acid that participate in protonation equilibria.Thiscommonfeatureturnsthedevelopedmodelingfor aminoacidsapplicableto␤-lactamcompounds.Fig.9showsthe chemical structuresoftwo ␤-lactamcompounds: theantibiotic ampicillinanditsprecursor6-aminopenicillanicacid,forwhich sol-ubilitydataasafunctionofpHandtemperatureareavailable[22]. Inthiscase, onecanapplyEq.(28)bysimultaneouslyadjusting parametersC_ij(1)andC_ij(2),whicharerelatedto andεaccordingto Eqs.(35)and(36).TheregressionwasperformedusingNelderand

0.0 2.0 4.0 6.0 8.0 10.0 12.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 So lubility / mol.kg -1 A 2.0 3.0 4.0 5.0 6.0 7.0 0.00 0.01 0.02 0.03 0.04 0.05 B So lubility / mol.kg -1

Fig.8. SolubilityofglycineanditsoligopeptidesasafunctionofpH.Experimentaldata[31]:()glycine(A),tetraglycine(B);()dyglycine(A),pentaglycine(B);and() triglycine(A),hexaglycine(B).Ourmodel:(continuousline)glycine(A),tetraglycine(B);(dashedline)dyglycine(A),pentaglycine(B);and(dashed-dottedline)triglycine (A),hexaglycine(B).

Fig.9. Chemicalstructureofampicillin(A)and6-aminopenicillanicacid(B).

Mead[30]simplexmethodwiththeobjectivefunctionofglobal relativemeansquaredeviation(GRMSD):

GRMSD=









1 M×N M

j=1 N

i=1



S_i,jexp−Scalc i,j

Sexp_i,j



2

(40)

whereinNis thenumber ofexperimentaldatapointata given temperatureandMisthenumberofsetsofexperimentaldataat differenttemperatures.ThevaluesofpKAusedforampicillinand

6-aminopenicillanicacidarelistedinTable2.

Table5presentsvaluesofparameter andεadjustedaswell asthevalueofGRMSDdefinedinEq.(40).Theadjustedvaluesare coherent,asthevaluesof areinternallyconsistent,considering thesizeofampicillinand6-aminopenicillanicacidmolecules.The valuesof␧/kBshowananalogouscoherence.

Accountingfornon-idealityfor␤-lactamsolutionsisnecessary todescribethesolubilityexperimentaldata,ascanbeviewedin Figs.10and11.Inbothcases,thedependenceofsolubilityon tem-peratureisweak,butnon-negligible.Thevaluesofε/kBarerelated

tothedeviation(positiveornegative)fromideality:ifε/kBislarger

thanthetemperatureoftheexperimentaldataset,theactivity coef-ficientofzwitterionicspeciesislowerthantheunity,resultingina negativedeviationfromideality.

Table6 presentsthecomparisonamong three models:ideal solution,KhoshkbarchiandVera[11]modelandthemodelherein presented.ThevalueofGRMSDforthemodelhereinpresentedcan beseentobelowerthanthatobtainedbyapplyingtheideal solu-tionmodel,andevenlowerthantheKhoshkbarchiandVeramodel [11]–whichrequiresfiveadjustableparameters.

Table5

Valuesofparameters andε,andvaluesofGRMSD–Eq.(40)–forsystems contain-ingampicillinand6-aminopenicillanicacid.Experimentalsolubilitydataobtained bySantanaetal.[22]. Compound ( ˚A) ε/kB(K) GRMSD(%) Ampicillin 5.25 299.8 8.2 6-Aminopenicillanicacid 2.68 275.6 12.0 5.0 5.5 6.0 6.5 7.0 7.5 8.0 0.0 10.0 20.0 30.0 40.0 50.0 60.0 Sol ub ility / m mo l.L -1 pH

A

5.0 5.5 6.0 6.5 7.0 7.5 8.0 0.0 10.0 20.0 30.0 40.0 50.0 60.0 Sol ub ility / m mo l.L -1 pH

B

Fig.10.SolubilityofampicillinasafunctionofpHinaqueoussolution.Experimental data[22]:()283.06K,()288.01K,()292.95Kand()298.03K.Modeling: dot-tedline–283.06K,dot-dashedline–288.01K,dashedline–292.95K,continuous line–298.03K.(A)Thiswork;(B)idealsolution.

Table6

ValuesofGRMSD–Eq.(40)–fordifferentmodelsappliedtosystemscontaining ampicillinand6-aminopenicillanicacid.Experimentalsolubilitydataobtainedby Santanaetal.[22].

Model Compound

Ampicillin(%) 6-Aminopenicillanic acid(%)

Idealsolution 16.6 148.4

KhoshkbarchiandVera[11]a _{8.6 55.6}

Thiswork 8.2 12.0

a_{ModelbyKhoshkbarchiandVera[11]appliedwithparametersestimatedby}

5.0 5.5 6.0 6.5 7.0 7.5 8.0 0.0 100.0 200.0 300.0 400.0 500.0 600.0 Sol ub ility / m mo l.L -1 pH

A

5.0 5.5 6.0 6.5 7.0 7.5 8.0 0.0 100.0 200.0 300.0 400.0 500.0 600.0 Sol ub ility / m mo l.L -1

B

Fig.11.Solubilityof6-aminopenicillanicacidasafunctionofpHinaqueous solu-tion.Experimentaldata[22]:()283.06K,()288.01K,()292.95Kand() 298.03K[22].Modeling:dottedline–283.06K,dot-dashedline–288.01K,dashed line–292.95K,continuousline,298.03K.(A)Thiswork;(B)idealsolution.

4. Conclusions

Anewmodelfordescribingthesolubilitycurveofaminoacids and␤-lactamcompoundsasafunctionofpHispresented.This

modelrequireslessadjustableparametersandallowsabetter cor-relationoftheexperimentaldatathantraditionalapproaches.The applicabilityofthismodelinthecorrelationofseveralsetsof sol-ubility experimentaldatais stressed.Animportant observation concernsthecombinatorialentropiceffect,whichmightexplain theincreasingnon-idealitywithincrementofmolecularsize.This factalliedtotheliteraturereviewshowsthattheuseofvery com-plicatedmodelstodescribenon-idealityinsystemssuchastheones studiedheremaynotbethebestwaytodevelopthermodynamic modelingforengineeringpurposes.

Acknowledgments

The authors gratefully acknowledge the financial support fromtheBrazilianagenciesFAPESP(processes2008/11232-1and 2011/22070-5),CNPqandCAPES.

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