Solutions Manual For Biomolecular Thermodynamics From Theory to Application 1st Edition

Solutions Manual for
M M M




Biomolecular Therm M




odynamics, From Th M M




eory to Application, 1
M M M




e Douglas Barrick (Al
M M M




l Chapters)
M

,Solution Manual M




CHAPTERM 1
1.1 UsingMtheMsameMVennMdiagramMforMillustration,MweMwantMtheMprobabilit
yMofMoutcomesMfromMtheMtwoMeventsMthatMleadMtoMtheMcross-
hatchedMareaMshownMbelow:




A1 A1M n MB2 B2


ThisMrepresentsMgettingMAMinMeventM1MandMnotMBMinMeventM2,MplusMnotMgettingMA
inMeventM1MbutMgettingMBMinMeventM2M(theseMtwoMareMtheMcommonM“orMbut
MnotMboth”McombinationMcalculatedMinMProblemM1.2)MplusMgettingMAMinMeven
tM1MandMBMinMeventM2.

1.2 FirstMtheMformulaMwillMbeMderivedMusingMequations,MandMthenMVennMdiagra
msM willMbeMcomparedMwithMtheMstepsMinMtheMequation.MInMtermsMofMform
ulasMandMprobabilities,MthereMareMtwoMwaysMthatMtheMdesiredMpairMofMoutc
omesMcanMcomeMabout.MOneMwayMisMthatMweMcouldMgetMAMonMtheMfirstMev
entMandMnotMBMonMthe
secondM(MA1M∩M(∼B2M)).MTheMprobabilityMofMthisMisMtakenMasMtheMsimpleMprodu
ct,MsinceMeventsM1MandM2MareMindependent:

pA1M∩M(∼B2M)M =M pAM×
Mp∼B (A.1.1)
=M pAM×(1−M
pBM)
=M pAM−MpA
pB


TheMsecondMwayMisMthatMweMcouldMnotMgetMAMonMtheMfirstMeventMandMweMcouldMget
BMonMtheMsecondM((∼MA1)M∩MB2M)M,MwithMprobability

p(∼A1)M∩M B2M =M p∼AM×
MpB (A.1.2)
=M(1−MpAM)
×MpB
=M pBM−MpA
pB

,2 SOLUTIONM MANUAL


SinceMeitherMoneMwillMwork,MweMwantMtheMorMcombination.MBecauseMtheMtw
oMwaysMareMmutuallyMexclusiveM(havingMbothMwouldMmeanMbothMAMandM∼AMi
nMtheMfirstMoutcome,MandMwithMequalMimpossibility,MbothMBMandM∼B),MthisMor
McombinationMisMequalMtoMtheMunionM{MA1M∩M(∼B2M)}M∪M{(∼MA1)M∩MB2},MandMitsM
probabilityMisMsimplyMtheMsumMofMtheMprobabilityMofMtheMtwoMseparateMwaysM
aboveM(EquationsMA.1.1MandMA.1.2):

p{A1M∩M(∼B2M)}M∪M{(~A1)M∩MB2}M =M pA1M∩M(∼B2M)M +Mp(∼A1)M∩MB2
=M pAM−MpApBM+MpBM−MpApB
=MpAM+MpBM−M2pApB

TheMconnectionMtoMVennMdiagramsMisMshownMbelow.MInMthisMexerciseMweMwill
MworkMbackwardMfromMtheMcombinationMofMoutcomesMweMseekMtoMtheMindividu
alMoutcomes.MTheMprobabilityMweMareMafterMisMforMtheMcross-
hatchedMareaMbelow.
{MA1M∩M(∼B2M)}M∪M{(∼MA1)M∩MB2M}




A1 B2


AsMindicated,MtheMcirclesMcorrespondMtoMgettingMtheMoutcomeMAMinMeventM1
M(left)MandMoutcomeMBMinMeventM2.MEvenMthoughMtheMeventsMareMidentical,M
theMVennMdiagramMisMconstructedMsoMthatMthereMisMsomeMoverlapMbetweenMt
heseMtwoM(whichMweMdon’tMwantMtoMincludeMinMourM“orMbutMnotMboth”Mcom
bination.MAsMdescribedMabove,MtheMtwoMcross-
hatchedMareasMaboveMdon’tMoverlap,MthusMtheMprobabilityMofMtheirMunionMisMt
heMsimpleMsumMofMtheMtwoMseparateMareasMgivenMbelow.


A1MnM~B2
~MA1M nMB2


pAM×Mp~B
p ~A ×MpB
=MpAM(1M–MpB)
=M(1M–M
AM MpBM )p
A1MnM~B2 ~MA1M nMB2



AddingMtheseMtwoMprobabilitiesMgivesMtheMfullM“orMbutMnotMboth”Mexpress
ionMabove.MTheMonlyMthingMremainingMisMtoMshowMthatMtheMprobabilityMof
MeachMofMtheMcrescentsMisMequalMtoMtheMproductMofMtheMprobabilitiesMasM
shownMinMtheMtopMdiagram.MThisMwillMonlyMbeMdoneMforMoneMofMtheMtwoM
crescents,MsinceMtheMotherMfollowsMinManMexactlyManalogousMway.MFocusin
gMonMtheMgrayMcrescentMabove,Mit
representsMtheMAMoutcomesMofMeventM1MandMnotMtheMBMoutcomesMinMeventM
2.MEachMofMtheseMoutcomes MisMshownMbelow:

EventM 1 EventM 2


A1 ~B

p~BM=M1M–MpB
pA

, SOLUTIONM MANUAL 3
A1 ~B2