Thermodynamics of biochemical reactions xxAlberty生物化学生物技术教案 2

Thermodynamics of biochemical reactions xx Alberty生物化学生物技术教案 2 130Chapter7Thermodynamics of the Bindingof Ligandsby Proteinsquadratic formulashows that the equilibriumconcentration of TotD is given by-2+(4+16K”heme)”28K“TotD=The fractionof hemein the dimer is given by12To tD-=2TotD+4TotT-1+2K”TotDSubstituting equation7.4-4into equation7.4-5yields2.fD=1+(1+4K”heme)i71Substituting thisequation into equation7.4-2yields2(YD-Y,)Y=Y,+1+(I+4K”heme)(7.4-4)(7.4-5)(7.4-6)(7.4-7)At specifiedO,Y is a functiononly ofhemein away that is describedby threeparameters,Y,Y,and K“.This equationis thesame asthat derivedby Ackersand Halvorson (1974),although ithas arather differentform.If itwere possible to titratehemoglobin withoxygen atsufficiently highheme,Y,could beobtained directly.However,for the values of the sevenequilibrium constantsobtained byMills.Johnson,and Ackers (1976),the tet-ramer ispartially dissociatedat thehighest practicalheme concentrationsof about5mM.Equation7.4-7indicates thatif Ycan bedetermined atseveral highheme,a linearextrapolation is possible(see equation7.4-12).As isevident fromequation7.4-7,the questionas towhetherhemeis highor lowdepends onwhetherhemebK”orheme1,equation7.4-7reduces to-T)Y=Y,+(K”)12heme1(7.4-8)Thus a plot of Y versusheme-”2at aspecifiedO,must approachlinearity ashemeis increased.The interceptof the limiting slope of aplot of Y versusathemel-I2=0is Y,and the limiting slopeis(Y,-Y,)/(K”)1i2.This slopeis determined by twofactors,(Y,-YT)and KI.The slopewill below at highO,because Y,-YT is small.The slopewill below atlowO,because K“is large.Once Y,has been determined at a series ofO,by use of extrapolationsof thistype,Kk,K&,Kk3,and Kk4can becalculated by the method of nonlinearleast squares.The values of Y,at variousO,can bedetermined byextrapolations atlowheme.If4K“heme1,the squareroot termin equation7.4-7can berewritten usingx(1+x)lr2%1+7(x1)(7.4-9)to obtain(D-T)Y=Y,+1+K”hemeSince4K”heme1is satisfied,we canuse(7.4-10)(7.4-11)7.4Experimental Determinationof SevenApparent EquilibriumConstants131I.111111.I I.I I IIheme500100015002000Figure7.3Calculated plotsofYversushemel-at fiveO,expressed asmolar concentrations.Starting at the topthe oxygenconentrations are2x lo-,lo-,5x and10-M.The interceptsgive values ofY,.Reprinted from R.A.Alberty,Biophys.Chem.63,119132 (1997),with permissionfrom ElsevierScience.to obtainY=Y,-(Y,-Y,)Kheme(7.4-12)Thus asheme+0,Y beesa linearfunction ofheme,and Yapproaches Y,in thelimit ofheme=0.The conditionthat4Kheme1is hardto satisfyexperimentally becauselow concentrationsof hemehave to be used.There is a steepslope atlowO,because(Y,-Y,)and Kare bothlarge.Note thatwhen the slopeof aplotis large,the determination of theintercept ismore uncertain.At highO,the slopewill besmaller because(YD-Y,)and Kare smaller.The determination of YDat aseries ofO,yields K;,and K;,.Once YTand Y,have been determinedbyextrapolation,theslopeof eachplot at specifiedO,yields K.It isalso possibleto CalculateKfrom anymeasured Yvalue byuseofequation7.4-7written in the form=K2(YD-Y,)/Y-YT2-14heme(7.4-13)The determinationof Kat aseriesofO,and knowledgeof YTand YDas a function ofO,makes itpossibleto calculate OK;(see equation7.3-9),the apparentassociation constantfor the reaction2D=T at the specified7;P,pH,Cl-1,ionic strength,etc.To showhow thclimiting formsofYas a function ofhemecan be used todetermine all the equilibrium constants for the binding of oxygen by hemoglobinthat ispartially dissociatedinto dimers,values ofY werecalculated with the parametersat21.4C,pH7.4,Cl-=0.2M,and0.2M ionic strength.These calculatedvalues ofY werethen plotted versusheme-2.In Fig.7.3,the interceptsat theY axiscorrrespond tobindings atvery highheme concentrationswhere the dissociation intodimers isnegligible.Thus theintercepts can be used to calculate the fourequilibrium constants for thetetramer.These plotsshow that the extrapolationbees linearasheme/,is reduced.Since thelimiting slopeis(Y,-Y,)/(K)12,the valueof Kat aparticularO,can becalculated fromthelimiting slope afterYT has beendetermined.To determihe bindingconstants for the dimer,Y is plottedversushemeatthelowest possibleheme concentrations,as shown in Fig.7.4.The intercepts132Chapter7Thermodynamics of the Bindingof Ligandsby Proteins-aheme2x10-R4x106x108x10-R1x10Figure7.4Calculated values ofYversushemeat hemeconcentrations startingatthetop of2x5x and10-7M.The interceptsgive thevaluesofY,.(see Problem7.5).Reprinted from R.A.Alberty,Biopkys.Chem.63,119-132 (1997),with permissionfrom ElsevierScience.attheY-axis correspondto the bindings by the dimer.Thus theintercepts can beused tocalculatethe two equilibrium constantsfor the binding ofoxygenby thedimer.The plotsshow that the extrapolationbees linearashemeis reducedto low values,but thesehave to be very lowvalues,especially atO,that halfsaturate thedimer.The directdeterminationof KL1and K;,from oxygenbinding experimentswill requirevery lowheme,which hasnot yetbeen achievedin oxygenbinding experiments.This maybe achievableusing longabsorption cells,multipath cells,or aFourier transformspectrometer.Since thelimitingslopeis(Y,-YT)/8.2426(K)Ii2,thevalueof Kat aparticularO,can bedetermined in this way.A checkon thevaluesof the sevenapparent equilibrium constants is that theycan beused tocalculatetheshapes ofboth of these plots,including thenonlinear regions.7.5DISSOCIATION OFA DIPROTICACID Beforediscussing theeffect of pH onprotein-ligand equilibria,it isnecessary todiscuss anaspect of acid dissociationsthat wastoo advancedfor Chapter1.Consider a protein Athat hastwo acid groups.The aciddissociation constantsare definedby HA-=H+A-K,(7.5-1)H,A=Hf+HA-K2(7.5-2)The binding polynomial for this systemis The average binding of hydrogen ions is given byH+dP-H+dln P.-PdH+-dHf(7.5-3)(7.5-4)7.5Dissociation of a Diprotic Acid133The plotof NHversus pHis the titration curvefor H,A.Note thatIn Pcan becalculated byintegrating(mH/Hf)dHf.Applying equation7.5-4toequation7.5-3yields(7.5-5)At veryhigh pH,the binding of Hiapproaches zero,and atvery lowpH itapproaches2.There isanother wayto lookat thisbinding,and that is to assume that the two groups are independent.In thiscase the dissociation reactions are writtenHA-=H+A2-K l(7.5-6)HAH=Hf+HA-K2(7.5-7)In otherwords,Kis the dissociation constantfor theleft hydrogenatom and K,is thedissociation constantfor theright hydrogenatom.In thiscase the binding issimply the sum of the bindingsatthetwo sites:(7.5-8)This equationcan berearranged toThus(7.5-10)K,=rcl+K,(7.5-11)If IC+10-2pH+hw+Phpi(7.6-9)K=Kref1+10-pH+phrll+-P H+PP LSI+P KP LIThe pKsoftheacidgroupsin thecatalytic site of fumarase at298K and an ionicstrength of0.01M aregiven inTable7.4along withthe equilibriumconstantsfor the referencereactions.More informationon theexperimental determinationoftheseparameters isavailable inWigler andAlberty (1960).The pKsfor thetwo acidgroupsin the unoupied catalytic siteof fumaraseare pKPsl=6.9and pK,=6.3.These twoacidgroupscan beconsidered to be identical and independentbecause theirdifference is0.6=log4.The pKsfor site-L-tartrate arepK,=7.5and pK,=7.4,indicating there isacooperative effectbecause they are closerthan0.6pH units(see Section1.2).The aciddissociationsofthe ligandsare ignoredin thesecalculations becausewe areprimarily concernedwith whathappens in the neighborhoodof pH7.With thesevaluesofthepKs,the pHdependence ofthe apparentequilibriumconstantfor Table7.4Fumarase at298.15K and an Ionic Strength of0.01M pKsof AcidGroups inthe CatalyticSite inLigand PK,PK,Kref Unoupied6.96.3suinate7.56.51.2x10-3D-tartrate7.86.92.5x10-L-tartrate7.57.44.110-3ineso-tartrate7.15.74.6Source:With permissionfrom P.W.Wigler andR.A.Alberty,J.Am.Chem.SOC.82,54825488 (1960).Copyright1960American ChemicalSociety.136Chapter7Thermodynamics ofthe Bindingof Ligandsby Proteins-0.25-0.5_-_-1!Figure7.5Plot oflog Kfor thereaction site-L-tartrate=site+L-tartrate overa rangeofpHat25C andan ionicstrength of0.01M.With permissionfrom R.A.Alberty,J.Phys.Chem.B104,9929-9934 (2000).Copyright2000American ChemicalSociety.-0.75-1-1.25-1.5-1.75thedissociation of L-tartrate fromthe plexis givenby7;-!-.-71+10-pH+6.9+10-2pH+13.2K=ref1+10-pH+7.5+10-2pH+14.9(7.6-10)-0.5-0.75-1-1.25-1.5-1.75The base10logarithm ofKcalculated usingequation7.6-10is givenas a functionofpH in Fig.7.5.The constantKref=4.1x lop3has beenomitted inmaking Fig.7.5because itis notinvolved inthe pHdependence.The change in bindingof hydrogen ions inthedissociationofthesite-L-tartrate plexcan becalculated bytaking thederivative oflog Kwith respectto pH(equation4.7-5).The pHdependence of ArNH isshown in Fig.7.6.Since theproducts(unoupied siteplus L-tartrate)bind hydrogen ions lessstrongly thanthe plex,A,NH isnegative atall pH values.Another wayto expressthis isthat hydrogen ionsareproduced inthedissociation,except inthelimit of veryhigh andverylowpH valueswhere ArNH=0.The precedingdiscussion hasbeen concernedwiththe apparentdissociationconstant ofaprotein-ligand plexand thechange inbindingofH+inthedissociationoftheligand.Now wereturn to the discussionofthehydrogenionbinding(Chapter1)ofthe unoupied siteof fumarase and especiallythesiteoupied byL-tartrate.Theaveragenumber of hydrogenions bound bythe.=./-.i t1;1#Figure7.6Plot of ArNH for thereactionsite-L-tartrate=site+L-tartrate overa rangeofpHat25C andan ionicstrength of0.01M.With permissionfrom R.A.Alberty,J.Phys.Chem.B104,99299934 (2000).Copyright2000American ChemicalSociety.catalytic siteisgivenby R,=-(l/ln10)d InP(P,)/pH.This quantityfor theunoupied siteis plottedin Fig.7.7.This plothas thesame shapeas thetitration curve ofa single sitebecauseK,=4K,but withthe ordinatemultiplied by2.The bindingcapacity forhydrogenionsis definedby1d21nP dRHdpH ln (10)dpH2-(7.6-11)Di Cera,Gill,and Wyman (1988)adopted thisname becausethis quantityis analogousto theheat capacity,which isgivenbythe secondderivative ofthe Gibbs energy Gwith respectto temperature(equation2.5-25).They pointedout thatthebindingcapacity isa measureof cooperativity.The bindingcapacity for theunoupiedsite,which iscalculated usingequation7.6-11,is plottedversus pHin Fig.7.8.The numberofhydrogenionsboundbythecatalytic siteinthefumarase-r.-tartrate plexisplottedinFig.7.9.This is steeper thanthetitrationcurveofa diproticacid withidenticaland independent groups.The bindingcapacity for thesiteoupied bymeso-tartrate isshown inFig.7.10.The slopeofthebinding curveissteeperthan for theunoupiedsite showninFig.7.6,as expectedsince thebinding iscooperative.The precedingexample ofthedeterminationofthepKsofacidgroupsinthebindingsiteofaprotein and thebindingcapacity is based on the studyofthe-1.25-1.5-1.756789Figure7.8Plot ofthebindingcapacity(see equation7.6-11)for an unoupiedcatalytic siteof fumaraseat25C andan ionicstrength of0.01M.With permissionfrom R.A.Alberty,J.Phys.Chem.B104,9929-9934 (2000).Copyright2000American ChemicalSociety.138Chapter7Thermodynamics ofthe Bindingof Ligandsby Proteins21.5I-05-1Figure7.9Plot ofthe averagenumberofhydrogenionsbound nk,bythecatalytic siteoffumaraseoupicd byL-tartrate at25C andan ionicstrength of0.01M.With permissionfrom R.A.Alberty,J.Phys.Chcm.B104,9929-9934 (2000).Copyright2000American ChemicalSociety.petitive inhibitionof anenzyme,but themethodofcurve fittingthepHdependenceofK(see equation7.6-9)can beused whentheapparentequilibriumconstantcan bemeasured spectrophotometricallyor byequilibrium dialysis(Klotz,1997).rn7.7CALCULATION OFSTANDARD TRANSFORMEDGIBBS ENERGIESOF FORMATIONOF THECATALYTIC SITEOF FUMARASEThe apparentequilibriumconstantfor abiochemical reactionataspecified pHcan becalculated fromthe standard transformed Gibbs energies of formation ofthe reactants,and thestandard transformedGibbs energy of formationofthereactants arecalculated usingisomer groupthermodynamics(see Section4.5-1).Alberty(1999a)has shownthat AfG:for abiochemical reactantisgivenby A,G;O=A,G,-RTln P(7.7-1)where AfG;is thestandardtransformedGibbs energy of formationfor thespecies withthe fewesthydrogen atomsand Pis thebindingpolynomialfor thereactant.This equationcan beapplied to the enzymaticsite forfumaraseand to theplexes formedwith petitive.inhibitors(Alberty,2000d).dNii(PLtotYdpH-1.751f._I_._I._I.,H6789Figure7.10Plot ofthebindingcapacity(see equation7.5-11)forthecatalyticsiteoffumaraseoupied byL-tartrate at25C andan ionicstrength of0.01M.With permissionfromR.A.Alberty,J.P hjChern.B104,9929-9934 (2000).Copyright2000American ChemicalSociety.7.7Calculation ofStandard Transformed Gibbs Energies of Formation139Table7.5of Fumarasein kJmo1-l at25C andIonicStrength0.01M Standard TransformedGibbsEnergiesofFormation forthe CatalyticSite PH5PH6PH7PH8PH9site suinatesite-suinate D-tartrate site-D-tartrate L-tartrate site-L-tartrate meso-tart ratesite-meso-tartrate-18.39-576.28-615.86114.1672.45114.1672.56114.16101.51-7.96-553.45-582.23136.99106.43136.99106.71136.99133.36-1.66-530.62-551.35159.83138.75159.83140.02159.83161.52-0.19-507.78-525.15182.66166.52182.66168.21182.66186.15-0.02-484.95-501.70205.49190.48205.49191.78205.49209.24Source:With permissionfromR.A.Albcrty.J.PhJs.Chem.B104,9929-9934 (2000).Copyright2000American ChemicalSociety.*This tableisbased onthe convention thatA=0at25C andzero ionicstrength forthe doublycharged ionsof D-tartrate.L-tartarate,and meso-lartratc.In addition,the conventionisthatA,G”=0forthebindingsiteathigh pH.The apparentdissociationconstantofthefumarase site-suinate plexto yieldunoupiedsiteand suinateis representedbythefollowing functionofpH:(7.7-2)Aording toequation7.1-19,this Kisgivenby-RTln K=A,G”(site)+A,GO(SUCC)A,G”(site-su)(7.7-3)The valueof A,Go(su2-)at25C andzero ionicstrength is-690.44kJ mol-,and thepHdependenceofA,Go(su)isgivenby-690.44-4RTln(see equation4.4-lo),neglecting theeffect ofthebindinghydrogenionsat lowerpH values.This valueis independentoftheionicstrengthbecause2;=NH(i).A,G”(site)is takenas zerointhelimitofhighpHby conventionso thatA,G”(site)=-RTln(l+106.y-pH+1013.3-2pH1(7.7-4)Equation7.7-3can bewritten as)K+1013.3-2pH RTln(1+i06.9-pH=-RTln-960.44-4RTln(lOPPH)-A,G”(site-su)(7.7-5)Substituting equation7.6-2yields A,G”(site-su)=-707.11-4RTlr1(10-)l(l+107.5-p+1014.0-PH)(7.7-6)The valuesofA,G”forthecatalyticsite,suinate,and site-suinate calculatedinthisway areshowninTable7.5.Similar calculationshave beenmade forD-tartrate,L-tartrate,and meso-tartrate usingdata fromTable7.4.Since the AfGo valuesfor thesethreereactantsare notknown,theconventionhasbeenadopted thattheyareequal tozero.Table7.5shows thatstandardtransformedGibbsenergiesofformationat specifiedpHvaluescan becalculated foranunoupiedbindingsiteand thebindingsiteoupiedbyaligand.8.18.28.38.48.58.68.78.8Two-Phase Systemswithout Chemical Reactions Two-Phase Systemwith aChemical ReactionandaSemipermeable MembraneTwo-Phase Systemwith aChemical Reactionand Membrane Permeable bya Single Ion Two-Phase Systemwith aChemical Reactionand MembranePermeable bya SingleIon TransformedGibbs EnergyofaTwo-Phase Systemwith aChemicalReactionandaMembranePermeablebyaSingleIonEffects ofElectric Potentialson MolarProperties ofIons EquilibriumDistribution ofCarbon Dioxidebetween theGas Phaseand AqueousSolution PhaseSeparation in Aqueous SystemsContaining HighPolymers Whena systeminvolves twoor morephases,there isasinglefundamental equation forthe Gibbsenergythatis thesum ofthefundamental equationsfortheseparate phases:dG=dG,+dG,+.The fundamental equation forthe systemprovides the criterion for spontaneous change and equilibrium.However,thereisa separateGibbs-Duhem equation for each phase becauseany intensiveproperty ofa phase is relatedtothe other intensiveproperties ofthat phase.In thetreatments herethe amount of material intheinterface isignored onthe assumptionthatthe amounts there are negligiblepared withthe amounts inthebulk phases.The effectsof smallpressure differences between thephases arealso ignored.New phasesmay formspontaneously undercertain circumstances,but phasescanalsobe separated by membraneswith specifiedpermeabilities.Phase equilibriumacross semipermeablemembranes isof specialinterest inbiological applications.First,we willconsider two-phase aqueoussystems with-out chemicalreactions,then introducereactions,and finallyelectric potentialdifferencesbetweenphases.The numbersof intensivedegrees offreedom Fand141Thernwdyanamics of Biochemical Reactions.Robert A.Alberty Copyright0xxJohn Wiley&Sons,Inc.ISBN0-471-22851-6142Chapter8Phase Equilibriumin AqueousSystems extensivedegrees offreedom Dhave been discussed in Chapter3,and Fand DatspecifiedpH havebeendiscussedinChapter4.That discussionis continuedhere.The distributionof carbondioxide betweenthe gasphase andaqueous solutionis discussedas afunctionofpH andionicstrength.8.1TWO-PHASE SYSTEMSWITHOUT CHEMICAL REACTIONS OneSpecies,Two PhasesThis systemisnotuseful forrepresenting abiochemical systembut isneeded asa foundation.The fundamental equation for G fora systemcontaining alphaand betaphases dG=-SdT+VdP+pAzd/TA1+pAgdMA,j(8.1-1)This shows thatthe natural variables for Gforthis system beforephase equilibrium is establishedare7;P,nAa,andt7!,.When Ais transferredfrom onephase totheother.di?,=-dnA8.Substituting thisconservation relationinto equation8.1-1yields(8.1-2)which showsthat/iAS=pAO=p,at phase equilibrium.Substituting thisequilib-rium conditioninequation8.1-1yields d