Isotopes2 Isotopes4 TheaverageatomicmassalsocalledtheaverageatomicweightorjustatomicweightofanelementisdenedastheweightedaverageofthemassesofallitsnaturallyoccurringstableisotopesForexampleth ID: 365657
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Isotopes(2) Notethatthelightestisotopeisalsothemostabundantonefortheseelements.Hereisalistoftheheavyisotopes,sortedbyabundance:IsotopeMass[Da]%Abundance34S33.9678684.2113C13.0033551.1033S32.9714590.7515N15.0001090.3718O17.9991590.2017O16.9991310.0382H2.0141020.015Weseethatsulfurhasabigimpactontheisotopedistribution.Butitisnotalwayspresentinapeptide(onlytheaminoacidsCysteinorMethionincontainsulfur).Apartfromthat,13Cismostabundant,followedby15N.Theseisotopesleadto“+1”peaks.Theheavyisotopes18Oand34Sleadto“+2”peaks.Notethat17Oand2Hareveryrare.10002 Isotopes(4) Theaverageatomicmass(alsocalledtheaverageatomicweightorjustatomicweight)ofanelementisdenedastheweightedaverageofthemassesofallitsnaturallyoccurringstableisotopes.Forexample,theaverageatomicmassofcarboniscalculatedas(98.9%12.0+1.1%13.003355) 100%.=12.011Formostpurposessuchasweighingoutbulkchemicalsonlytheaveragemolecularmassisrelevantsincewhatoneisweighingisastatisticaldistributionofvaryingisotopiccompositions.Themonoisotopicmassisthesumofthemassesoftheatomsinamoleculeusingtheprincipleisotopemassofeachatominsteadoftheisotopeaveragedatomicmassandismostoftenusedinmassspectrometry.Themonoisotopicmassofcarbonis12.10004 Isotopicdistributions Themassspectralpeakrepresentingthemonoisotopicmassisnotalwaysthemostabundantisotopicpeakinaspectrum–althoughitstemsfromthemostabundantisotopeofeachatomtype.Thisisduetothefactthatasthenumberofatomsinamoleculeincreasestheprobabilityoftheentiremoleculecontainingatleastoneheavyisotopeincreases.Forexample,ifthereare100carbonatomsinamolecule,eachofwhichhasanapproximately1%chanceofbeingaheavyisotope,thenthewholemoleculeisnotunlikelytocontainatleastoneheavyisotope.Themonoisotopicpeakissometimesnotobservableduetotwoprimaryreasons. Themonoisotopicpeakmaynotberesolvedfromtheotherisotopicpeaks.Inthiscaseonlytheaveragemolecularmassmaybeobserved. Eveniftheisotopicpeaksareresolved,themonoisotopicpeakmaybebelowthenoiselevelandheavyisotopomersmaydominatecompletely.10006 Isotopicdistributions(3) Example: 10008 Isotopicdistributions(5) Example:Isotopicdistributionofhumanmyoglobin Screenshot:http://education.expasy.org/student_projects/isotopident/10010 Isotopicdistributions(6) Abasiccomputationaltaskis: Givenanionwhoseatomiccompositionisknown,howcanwecomputeitsisotopicdistribution?Wewillignorethemassdefectsforamoment.Itisconvenienttonumberthepeaksbytheirnumberofadditionalmassunits,startingwithzeroforthelowestisotopicpeak.Wecancallthistheisotopicrank.LetEbeachemicalelement.LetE[i]denotetheprobabilityoftheisotopeofEhavingiadditionalmassunits.ThustherelativeintensitiesoftheisotopicpeaksforasingleatomofelementEare(E[0],E[1],E[2],...,E[kE]).HerekEdenotestheisotopicrankoftheheaviestisotopeoccurringinnature.WehaveE[`]=0for`�k.ForexamplecarbonhasC[0]=98.9%=0.989(isotope12C)andC[1]=1.1%=0.011(isotope13C).10011 Isotopicdistributions(8) Clearlythesametypeofformulaappliesifwecomposealargermoleculeoutofsmallermoleculesorsingleatoms.Moleculeshaveisotopicdistributionsjustlikeelements.Forsimplicity,letusdeneconvolutionpowers.Let1:=andn:=n�1,foranyisotopicdistribution.Moreover,itisnaturaltodene0by0[0]=1,0[`]=0for`�0.Thisway,0willactasneutralelementwithrespecttotheconvolutionoperator,asexpected.ThentheisotopicdistributionofamoleculewiththechemicalformulaE1n1E`n`,composedoutoftheelementsE1,...,E`,canbecalculatedasE1n1E`n`=n1E1...n`E`.10013 Isotopicdistributions(10) BoundingtherangeofisotopesForpracticalcases,itispossibletorestrictthesummationrangeintheconvolutionoperation.Inprincipleitispossibletoformamoleculesolelyoutoftheheaviestisotopes,andthisdeterminesthesummationrangeneededintheconvolutioncalculations.However,theabundanceofsuchanisotopomerisvanishinglysmall.Infact,itwillsoonfallbelowtheinverseoftheAvogadronumber(6.02214151023),sowewillhardlyeverseeasinglemoleculeofthistype.Forsimplicity,weletusrstconsiderasingleelementEandassumethatkE=1.(Forexample,Ecouldbecarbonornitrogen.)Inthiscasetheisotopicdistributionisabinomialwithparameterp:=E[1],En[k]=nkpk(1�p)n�k.Themeanofthisbinomialdistributionispn.Largedeviationscanbeboundedasfollows.10015 Isotopicdistributions(12) Moregenerally,apeptideiscomposedoutoftheelementsC,H,N,O,S.Foreachoftheseelementsthelightestisotopehasanaturalabundanceabove95%andthehighestisotopicrankisatmost2.Againwecanboundthesumoftheabundancesofheavyisotopicvariantsbyabinomialdistribution:XjiE1n1E`n`[j]Xji=2nj0.05j0.95n�j.(Inordertogetiadditionalmassunits,atleasti=2oftheatomsmustbe`heavy'.)10017 Isotopicdistributions(14) Tosummarize:Therstk+1abundancesEn1En``[i],i=0,...,k,oftheisotopicdistributionofamoleculeEn11En``canbecomputedinO(`klogn)timeandO(`k)space,wheren=n1+...+n`.(Exercise:Totestyourunderstanding,checkhowtheseresourceboundsfollowfromwhathasbeensaidabove.)10019 Massdecomposition(2) Thisisformalizedbytheconceptofacompomer[BL05].Wearegivenanalphabetofsizejj=k,whereeachletterhasamassai,i=1,...,k.Theseletterscanrepresentatomtypes,isotopes,oraminoacids,ornucleotides.Weassumethatallmassesaredifferent,becauseotherwisewecouldneverdistinguishthemanyway.Thuswecanidentifyeachletterwithitsmass,i.e.,=fa1,...,akgN.Thisissometimescalledaweightedalphabet.Themassofastrings=s1...sn2isdenedasthesumofthemassesofitsletters,i.e.,mass(s)=Pjsji=1si.Formally,acompomerisanintegervectorc=(c1,...,ck)2(N0)k.Eachcirepresentsthenumberofoccurrencesofletterai.Themassofacompomerismass(c):=Pki=1ciai,asopposedtoitslength,jcj:=Pki=1ci.Inshort:Acompomertellsushowmanyinstancesofanatomicspeciesarepresentinamolecule.Wewanttondallcompomerswhosemassisequaltotheobservedmass.10021 Massdecomposition(4) Usingdynamicprogramming,wecansolvethefollowingproblemsefciently(:weightedalphabet,M:mass):1.Existenceproblem:Decidewhetheracompomerscwithmass(c)=Mexists.2.OneWitnessproblem:Outputacompomercwithmass(c)=M,ifoneexists.3.Allwitnessesproblem:Computeallcompomerscwithmass(c)=M.10023 Massdecomposition(5) Thedynamicprogrammingalgorithmisavariationoftheclassicalalgorithmorigi-nallyintroducedforthe`CoinChangeProblem',originallyduetoGilmoreandGo-mory.GivenaquerymassM,atwo-dimensionalBooleantableBofsizekMisconstructedsuchthatB[i,m]=1()misdecomposableoverfa1,...,aig.Thetablecanbecomputedwiththefollowingrecursion:B[1,m]=1()mmoda1=0andfori�0,B[i,m]=8:B[i�1,m]mai,B[i�1,m]_B[i,m�ai]otherwise.ThetableisconstructeduptomassM,andthenastraight-forwardbacktrackingalgorithmcomputesallwitnessesofM.10024 Massdecomposition(7) ThenumberofcompomersisO(M).(Exercise:why?)Dependingonthemassresolution,theresultscanbeusefulforMupto,say,1000Da,butingeneralonehastotakefurthercriteriaintoaccount.(Figurefrom[BL].) 10026 Massdecomposition(8) Exampleoutputfromhttp://bibiserv.techfak.uni-bielefeld.de/decomp/#imsdecomp1.3#Copyright2007,2008InformaticsforMassSpectrometrygroup#atBielefeldUniversity##http://BiBiServ.TechFak.Uni-Bielefeld.DE/decomp/##precision:4e-05#allowederror:0.1Da#massmode:mono#modifiers:none#fixedmodifications:none#variablemodifications:none#alphabet(character,mass,integermass):#H1.00782525196#C12300000#N14.003074350077#O15.994915399873#P30.973761774344#S31.972071799302#constraints:none#chemicalplausibilitycheck:off##Showninparenthesesaftereachdecomposition:#-actualmass#-deviationfromactualmass10027 ##mass218.03has1626decompositions(showingthebest100):H2C6N8O2(218.03007;+7.1384e-05)H8C7N1O7(218.03008;+7.6606e-05)H13C2N5O1P1S2(218.02991;-8.7014e-05)H139O1P2(218.03012;+0.000117068)H19C1N1O3P3S1(218.02985;-0.000151322)H16N3O4S3(218.03029;+0.000293122)H5C2N9O2P1(218.03038;+0.00038199)H11C3N2O7P1(218.03039;+0.000387212)H10C6N4O1S2(218.0296;-0.00039762)H16C5O3P2S1(218.02954;-0.000461928)H16C3N3P4(218.02947;-0.000531458)H21C2N1P1S4(218.02944;-0.000556018)H8N7O5S1(218.03076;+0.000762126)H14C1O10S1(218.03077;+0.000767348)H18C5O1P4(218.03081;+0.000811196)H13C7N2P3(218.02916;-0.000842064)H18C6S4(218.02913;-0.000866624)H12C8N1O2S2(218.03095;+0.000945034)H9C1N5O6P1(218.02904;-0.000955442)H3N12O1P1(218.02904;-0.000960664)H21C1N1O1P5(218.03112;+0.001121802)H10C11N1P2(218.02885;-0.00115267)H137O3S1(218.02884;-0.001156056)H15C4N2O2P1S2(218.03126;+0.00125564)H6C5N4O6(218.02873;-0.001266048)C4N11O1(218.02873;-0.00127127)H4C8N5O3(218.03141;+0.001414038)H17C1N1O5P1S2(218.02858;-0.001424446) H12C3N3O4S2(218.02692;-0.003077706)H12C1N6O1P2S1(218.02685;-0.003147236)H18N2O3P4(218.02679;-0.003211544)H12C2N4O4P2(218.03338;+0.003377904)H6C3N8O2S1(218.03344;+0.003442212)H12C4N1O7S1(218.03345;+0.003447434)H9C5N5O1P1S1(218.02654;-0.003457842)H15C4N1O3P3(218.02648;-0.00352215)H20C1N2P2S3(218.02638;-0.00361624)H15N2O7P1S1(218.03376;+0.00375804)H6C9N4O1S1(218.02623;-0.003768448)H12C8O3P2(218.02617;-0.003832756)H17C5N1P1S3(218.02607;-0.003926846)H8C2N3O9(218.02605;-0.003946134)H2C1N10O4(218.02605;-0.003951356)H2C11N6(218.03409;+0.004094124)H22C2O1P4S1(218.03418;+0.004182024)H14C9S3(218.02576;-0.004237452)H16C5N1O2S3(218.03432;+0.004315862)H5C7N7P1(218.0344;+0.00440473)H11C8O5P1(218.03441;+0.004409952)H10C1N6O3S2(218.02558;-0.00442036)H16N2O5P2S1(218.02552;-0.004484668)H133C3O3(218.02547;-0.004526884)H19C1N2O2P1S3(218.03463;+0.004626468)H8C3N8P2(218.03472;+0.004715336)H14C4N1O5P2(218.03472;+0.004720558)H8C5N5O3S1(218.03478;+0.004784866)H13C4N1O5P1S1(218.0252;-0.004795274)H7C3N8P1S1(218.0252;-0.004800496) Massdefect Thedifferencebetweentheactualatomicmassofanisotopeandthenearestinte-gralmassiscalledthemassdefect.ThesizeofthemassdefectvariesoverthePeriodicTable.Themassdefectisduetothebindingenergyofthenucleus: http://de.wikipedia.org/w/index.php?title=Datei:Bindungsenergie_massenzahl.jpg10028 Massdefect(3) HighresolutionandlowresolutionisotopicspectraforC15N20S4O8.hires.datlores.dat715.909120100.0716100.0716.9061597.271726.8716.9085093.2716.91247916.1716.9133290.2717.9031890.271822.6717.90491917.6717.9055490.2717.9095201.1717.9118700.5717.9133601.6717.9158301.1718.9019601.37195.0718.9043100.4718.9082792.8718.9103990.1718.9167200.2719.9007191.17201.8719.9053190.2719.9091600.2719.9116290.2720.9040800.27210.2Isotopicpattern: Zoomon”+2”masspeak(718): 10030