Size and Shape of Protein Molecules at the Nanometer Level Determined by Sedimentation, Gel Filtration, and Electron Microscopy相关交流-蚂蚁淘在线

Animportantpartofcharacterizinganyproteinmoleculeistodetermineitssizeandshape.Sedimentationandgelfiltrationarehydrodynamictechniquesthatcanbeusedforthismediumresolutionstructuralanalysis.Thisreviewcollectsanumberofsimplecalculationsthatareusefulforthinkingaboutproteinstructureatthenanometerlevel.ReadersareremindedthatthePerrinequationisgenerallynotavalidapproachtodeterminetheshapeofproteins.Instead,asimpleguidelineispresented,basedonthemeasuredsedimentationcoefficientandacalculatedmaximumS,toestimateifaproteinisglobularorelongated.ItisrecalledthatagelfiltrationcolumnfractionatesproteinsonthebasisoftheirStokesrADIus,notmolecularweight.Themolecularweightcanbedeterminedbycombininggradientsedimentationandgelfiltration,techniquesavailableinmostbiochemistrylaboratories,asoriginallyproposedbySiegelandMonte.Finally,rotaryshadowingandnegativestainelectronmicroscopyarepowerfultechniquesforresolvingthesizeandshapeofsingleproteinmoleculesandcomplexesatthenanometerlevel.Acombinationofhydrodynamicsandelectronmicroscopyisespeciallypowerful.

KeyWords:Proteinshape-hydrodynamics-gelfiltration-sedimentation-electronmicroscopy

Introduction

Mostproteinsfoldintoglobulardomains.Proteinfoldingisdrivenlargelybythehydrophobiceffect,whichseekstominimizecontactofthepolypeptidewithsolvent.Mostproteinsfoldintoglobulardomains,whichhaveaminimalsurfacearea.Peptidesfrom10to30kDatypicallyfoldintoasingledomain.Peptideslargerthan50kDatypicallyformtwoormoredomainsthatareindependentlyfolded.However,someproteinsarehighlyelongated,eitherasastringofsmallglobulardomainsorstABIlizedbyspecializedstructuressuchascoiledcoilsorthecollagentriplehelix.Theultimatestructuralunderstandingofaproteincomesfromanatomic-levelstructureobtainedbyX-raycrystallographyornuclearmagneticresonance.However,structuralinformationatthenanometerlevelisfrequentlyinvaluable.Hydrodynamics,inparticularsedimentationandgelfiltration,canprovidethisstructuralinformation,anditbecomesevenmorepowerfulwhencombinedwithelectronmicroscopy(EM).

Oneguidingprincipleenormouslysimplifiestheanalysisofproteinstructure.Theinteriorofproteinsubunitsanddomainsconsistsofcloselypackedatoms(1).Therearenosubstantialholesandalmostnowatermoleculesintheproteininterior.Asaconsequenceofthis,proteinsarerigidstructures,withaYoung’smodulussimilartothatofPlexiglas(2).EngineerssometimescategorizeBIOLOGyasthescienceof“softwetmaterials”.Thisistrueofsomehydratedgels,butproteinsarebetterthoughtofasharddryplastic.Thisisobviouslyimportantforallofbiology,tohavearigidmaterialwithwhichtoconstructthemachineryoflife.Asecondconsequenceoftheclosepackedinteriorofproteinsisthatallproteinshaveapproximatelythesamedensity,about1.37g/cm³.Formostofthefollowing,wewillusethepartialspecificvolume,v₂,whichisthereciprocalofthedensity.v₂variesfrom0.70to0.76fordifferentproteins,andthereisaliteratureoncalculatingordeterminingthevalueexperimentally.Forthepresentdiscussion,wewillignorethesevariationsandassumetheaveragev₂ = 0.73cm³/g.

HowBigIsaProteinMolecule?

Assumingthispartialspecificvolume(v₂ = 0.73cm³/g),wecancalculatethevolumeoccupiedbyaproteinofmassMinDaltonasfollows.

$V{left( {{ ext{nm}}^{3} } ight)} = frac{{{left( {{0.73;,{ ext{cm}}^{3} } mathord{left/ {vphantom {{0.73;,{ ext{cm}}^{3} } { ext{g}}}} ight. kern- ulldelimiterspace} { ext{g}}} ight)}; imes ;{left( {{{ ext{10}}^{{{ ext{21}}}} { ext{nm}}^{{ ext{3}}} } mathord{left/ {vphantom {{{ ext{10}}^{{{ ext{21}}}} { ext{nm}}^{{ ext{3}}} } {{ ext{cm}}^{{ ext{3}}} }}} ight. kern- ulldelimiterspace} {{ ext{cm}}^{{ ext{3}}} }} ight)}}}{{6.023; imes ;10^{{23}} {{ ext{Da}}} mathord{left/ {vphantom {{{ ext{Da}}} { ext{g}}}} ight. kern- ulldelimiterspace} { ext{g}}}}; imes ;M{left( {{ ext{Da}}} ight)} = 1.212; imes ;10^{{ - 3}} {left( {{{ ext{nm}}^{3} } mathord{left/ {vphantom {{{ ext{nm}}^{3} } {{ ext{Da}}}}} ight. kern- ulldelimiterspace} {{ ext{Da}}}} ight)}; imes ;M{left( {{ ext{Da}}} ight)}.$

(2.1)

Theinverserelationshipisalsofrequentlyuseful:M(Da) = 825V(nm³).

Whatwereallywantisaphysicallyintuitiveparameterforthesizeoftheprotein.Ifweassumetheproteinhasthesimplestshape,asphere,wecancalculateitsradius.WewillrefertothisasR_min,becauseitistheminimalradiusofaspherethatcouldcontainthegivenmassofprotein

$R_{{min }} = {left( {3V/4pi } ight)}^{{1/3}} = 0.066M^{{1/3}} { ext{ (for }}M{ ext{ in Dalton, }}R_{{{ ext{min}}}} { ext{ in nanometer)}}{ ext{.}}$

(2.2)

Someusefulexamplesforproteinsfrom5,000to500,000DaaregiveninTable1.

Table1R_minforproteinsofdifferentmass

Itisimportanttoemphasizethatthisistheminimumradiusofasmoothspherethatcouldcontainthegivenmassofprotein.Sinceproteinshaveanirregularsurface,evenonesthatareapproximatelysphericalwillhaveanaverageradiuslargerthantheminimum.

HowFarApartAreMoleculesinSolution?

Itisfrequentlyusefultoknowtheaveragevolumeofsolutionoccupiedbyeachmolecule,ormoredirectly,theaveragedistanceseparatingmoleculesinsolution.Thisisasimplecalculationbasedonlyonthemolarconcentration.

Ina1-Msolution,thereare6 × 10²³molecules/l,=0.6molecules/nm³,orinverting,thevolumepermoleculeisV = 1.66nm³/moleculeat1M.ForaconcentrationC,thevolumepermoleculeisV = 1.66/C.

Wewilltakethecuberootofthevolumepermoleculeasanindicationoftheaverageseparation.

ProteinM(kDa)	5	10	20	50	100	200	500
R_min(nm)	1.1	1.42	1.78	2.4	3.05	3.84	5.21

$d = V^{{1 mathord{left/ {vphantom {1 3}} ight. kern- ulldelimiterspace} 3}} = {1.18} mathord{left/ {vphantom {{1.18} C}} ight. kern- ulldelimiterspace} C^{{1 mathord{left/ {vphantom {1 3}} ight. kern- ulldelimiterspace} 3}} ,$

(3.1)

whereCisinmolaranddisinnanometer.Table2givessometypicalvalues.

Table2Distancebetweenmoleculesasfunctionofconcentration

Twointerestingexamplesarehemoglobinandfibrinogen.Hemoglobinis330mg/mlinerythrocytes,makingitsconcentration0.005M.Theaverageseparationofmolecules(centertocenter)is6.9nm.Thediameterofasinglehemoglobinmoleculeisabout5nm.Thesemoleculesareveryconcentrated,nearthehighestphysiologicalconcentrationofanyprotein(thecrystallinsinlenscellscanbeat>50%proteinbyweight).

Fibrinogenisalargerod-shapedmoleculethatformsafibrinbloodclotwhenactivated.Itcirculatesinplasmaataconcentrationofaround2.5g/l,about9μM.Thefibrinogenmoleculesarethereforeabout60nmapart,comparabletothe46-nmlengthoftherod-shapedmolecule.

TheSedimentationCoefficientandFrictionalRatio.IstheProteinGlobularorElongated?

Biochemistshavelongattemptedtodeducetheshapeofaproteinmoleculefromhydrodynamicparameters.Therearetwomajorhydrodynamicmethodsthatareusedtostudyproteinmolecules—sedimentationanddiffusion(orgelfiltration,whichistheequivalentofmeasuringthediffusioncoefficient).

Thesedimentationcoefficient,S,canbedeterminedinananalyticalultracentrifuge.Thiswasastandardpartofthecharacterizationofproteinsinthe1940sand1950s,andvaluesofS_20,w(sedimentationcoefficientstandardizedto20°Cinwater)arecollectedinreferencessuchastheChemicalRubberCo.(CRC)HandbookofBiochemistry(3).Today,Sismorefrequentlydeterminedbyzonesedimentationinasucroseorglycerolgradient,bycomparisontostandardproteinsofknownS.Fivetotwentypercentsucrosegradientshavebeenmostfrequentlyused,butweprefer15–40%glycerolgradientsin0.2Mammoniumbicarbonate,becausethisisthebufferusedforrotaryshadowingEM(Section6).Theproteinofinterestissedimentedinonebucketoftheswingingbucketrotor,andproteinstandardsofknownS(Table5)aresedimentedinaseparate(orsometimesthesame)gradient.Followingsedimentation,thegradientiselutedintofractionsandeachfractionisanalyzedbysodiumdodecylsulfatepolyacrylamidegelelectrophoresis(SDS-PAGE)tolocatethestandardsandthetestprotein.Figure1showsanexampledeterminingthesedimentationcoefficientofthestructuralmaintenanceofchromosome(SMC)proteinfromBacillussubtilis.

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Fig.1GlycerolgradientsedimentationanalysisofSMCproteinfromB.subtilis(BsSMC;upperpanel)andsedimentationstandardscatalaseandbovineserumalbumin(lowerpanel).A200-μlsamplewaslayeredona5.0-mlgradientof15–40%glycerolin0.2MammoniumbicarbonateandcentrifugedinaBeckmanSW55.1swingingbucketrotor,16h,38,000rpm,20°C.Twelvefractionsof400μleachwerecollectedfromaholeinthebottomofthetubeandeachfractionwasrunonSDS-PAGE.LaneSMshowsthestartingmaterial,andfraction1isthebottomofthegradient.Thebottompanelshowsthatthe11.3-Scatalaseelutedpreciselyinfraction4,whilethe4.6-SBSAelutedmostlyinfraction8,withsomeinfraction9.WeestimatedtheBSAtobecenteredonfraction8.2.Experimentswithadditionalstandardproteinshavedemonstratedthatthe15–40%glycerolgradientsarelinearovertherange3–20S,soalinearinterpolationisusedtodetermineSoftheunknownprotein.BsSMCisinfractions7and8,estimatedmorepreciselyatfraction7.3.Extrapolatingfromthestandards,wedetermineasedimentationcoefficientof6.0SforBsSMC.Otherexperimentsgaveanaveragevalueof6.3SforBsSMC(19).

Thesedimentationcoefficientofaproteinisameasureofhowfastitmovesthroughthegradient.Increasingthemassoftheproteinwillincreaseitssedimentation,whileincreasingitssizeorasymmetrywilldecreaseitssedimentation.TherelationshipofStosizeandshapeoftheproteinisgivenbytheSvedbergformula:

Concentration	1M	1mM	1μM	1nM
Distancebetweenmolecules(nm)	1.18	11.8	118	1,180

$S = {M{left( {1 - v_{2} ho } ight)}} mathord{left/ {vphantom {{M{left( {1 - v_{2} ho } ight)}} {N_{{ ext{o}}} f}}} ight. kern- ulldelimiterspace} {N_{{ ext{o}}} f} = {M{left( {1 - v_{2} ho } ight)}} mathord{left/ {vphantom {{M{left( {1 - v_{2} ho } ight)}} {,{left( {N_{{ ext{o}}} 6pi eta R_{{ ext{s}}} } ight)}}}} ight. kern- ulldelimiterspace} {,{left( {N_{{ ext{o}}} 6pi eta R_{{ ext{s}}} } ight)}}.$

(4.1)

MisthemassoftheproteinmoleculeinDalton;N_oisAvogadro’snumber,6.023 × 10²³;v₂isthepartialspecificvolumeoftheprotein;typicalvalueis0.73cm³/g;ρisthedensityofsolvent(1.0g/cm³forH₂O);ηistheviscosityofthesolvent(0.01g/cm^−sforH₂O).

Acriticalfactorintheequationisthefrictionalcoefficient,f(dimensionsgrampersecond)whichdependsonboththesizeandshapeoftheprotein.Foragivenmassofprotein(orgivenvolume),fwillincreaseastheproteinbecomeselongatedorasymmetrical(fcanbereplacedbyanequivalentexpressioncontainingR_s,theStokesradius,tobediscussedlater).Shasthedimensionsoftime(seconds).Fortypicalproteinmolecules,Sisintherangeof2–20 × 10⁻¹³s,andthevalue10⁻¹³sisdesignatedaSvedbergunit,S.Thus,typicalproteinshavesedimentationcoefficientsof2–20S.

Fromtheabovedefinitionofparameters,itisclearthatSdependsonthesolventandtemperature.Inclassicalstudies,thesolvent-dependentfactorswereeliminatedandthesedimentationcoefficientwasextrapolatedtothevalueitwouldhaveat20°Cinwater(forwhichρandηaregivenabove).ThisisreferredtoasS_20,w.Inthepresenttreatment,wewillbereferringmostlytostandardproteinsthathavealreadybeencharacterized,orunknownonesthatwillbereferencedtotheseingradientsedimentation,soouruseofSwillalwaysmeanS_20,w.

Ausefulconceptistheminimumvalueoff,whichwouldobtainifthegivenmassofproteinwerepackedintoasmoothunhydratedsphere.AswehavediscussedinSection1,theradiusofthisspherewillbeR_min = 0.066M^1/3(Eq.2.2).Inabout1850,G.G.Stokescalculatedtheoreticallythefrictionalcoefficientofasmoothsphere(notethattheequationissimilartothatfortheStokesradius,tobediscussedlater,buttheparametersherearedifferent):

$f_{{min }} = 6pi eta R_{{min }} .$

(4.2)

Wehavenowdesignatedf_minastheminimalfrictionalcoefficientforaproteinofagivenmass,whichwouldobtainiftheproteinwereasmoothsphereofradiusR_min.

Theactualfofaproteinwillalwaysbelargerthanf_minbecauseoftwothings.First,theshapeoftheproteinnormallydeviatesfromspherical,tobeellipsoidalorelongated;closelyrelatedtothisisthefactthatthesurfaceoftheproteinisnotsmoothbutratherroughonthescaleofthewatermoleculesitistravelingthrough.Second,allproteinsaresurroundedbyashellofboundwater,one–twomoleculesthick,whichispartiallyimmobilizedorfrozenbycontactwiththeprotein.Thiswaterofhydrationincreasestheeffectivesizeoftheproteinandthusincreasesf.

ThePerrinEquationDoesNotWorkforProteins

Ifonecoulddeterminetheamountofwaterofhydrationandfactorthisout,therewouldbehopethattheremainingexcessoffoverf_mincouldbeinterpretedintermsofshape.Algorithmshavebeendevisedforestimatingtheamountofboundwaterfromtheaminoacidsequence,butthesegenerallydonotdistinguishbetweenburiedresidues,whichhavenoboundwaterandsurfaceresidueswhichbindwater.Someattemptshavebeenmadetobasetheestimateofboundwaterbasedonpolarresidues,whicharemostlyexposedonthesurface.A0.3-gH₂O/gproteinisatypicalestimate,butinfact,thiskindofguessisalmostuselessforanalyzingf.

Intheolderdays,whentherewassomeconfidenceintheseestimatesofboundwater,physicalchemistscalculatedavaluecalledf_o,whichwasthefrictionalcoefficientforaspherethatwouldcontainthegivenprotein,butenlargedbytheestimatedshellofwater(otherauthorsusef_otodesignatewhatwetermf_min(3,4);werecommendusingf_mintoavoidambiguity).Themeasuredfforproteinswasalmostalwayslargerthanf_o,suggestingthattheproteinwasasymmetricalorelongated.Averypopularanalysiswastomodeltheproteinasanellipsoidofrevolutionandcalculatetheaxialratiofromf/f_o,usinganequationfirstdevelopedbyPerrin.Thisapproachisdetailedinmostclassicaltextsofphysicalbiochemistry.Infact,thePerrinanalysisalwaysoverestimatestheasymmetryoftheproteins,typicallybyafactoroftwotofive.Itshouldnotbeusedforproteins.

Theproblemisillustratedbyanearlycollaborativestudyofphosphofructokinase,inwhichthelaboratoryofJamesLeedidhydrodynamicsandourlaboratorydidEM(5).WefoundbyEMthatthetetramericparticleswereapproximatelycylinders,9nmindiameterand14nmlong.Theshapewasthereforelikearugbyball,withanaxialratioof1.5foraprolateellipsoidofrevolution.TheLeegroupmeasuredthemolecularweightandsedimentationcoefficient,determinedfandestimatedwaterofhydrationandf_o.TheythenusedthePerrinequationtocalculatetheaxialratio.Theratiowasfive,whichwouldsuggestthattheproteinhadtheshapeofahotdog.TheEMstructure(whichwaslaterconfirmedbyX-raycrystallography)showsthatthePerrinequationoverestimatedtheaxialratiobyafactorof3.

Telleretal.(6)summarizedthesituation:“Frequentlytheaxialratiosresultingfromsuchtreatmentareabsurdinlightofthepresentknowledgeofproteinstructure.”TheyexplainedthatthemajorproblemwiththePerrinequationisthatittreatstheproteinasasmoothellipsoid,wheninfactthesurfaceoftheproteinisquiterough.Telleretal.wentontoshowhowthefrictionalcoefficientcanactuallybederivedfromtheknownatomicstructureoftheprotein,bymodelingthesurfaceoftheproteinasashellofsmallbeadsofradius1.4Å.Theshellcoatedthesurfaceoftheprotein,modelingitsrugosity,andincreasingthesizeoftheproteinbytheequivalentofasinglelayerofboundwater.ThisanalysishasbeenextendedbyGarciaDeLaTorreandcolleagues(7).

InterpretingShapefromf/f_min = S_max/S

IfthePerrinequationisuseless,istheresomeotherwaythatshapecanbeinterpretedfromf?Theanswerisyes,atasemiquantitativelevel.Wehavediscoveredsimpleguidelineswheretheratiof/f_mincanprovideagoodindicationofwhetheraproteinisglobular,somewhatelongated,orveryelongated.

Insteadofproceedingwiththeclassicalratiof/f_min,wherefisinnonintuitiveunits,wewillreformulatetheanalysisdirectlyintermsofthesedimentationcoefficient,whichistheparameteractuallymeasured.WewilldefineavalueS_maxasthemaximumpossIBLesedimentationcoefficient,correspondingtof_min.S_maxistheSvaluethatwouldbeobtainediftheproteinwereasmoothspherewithnoboundwater.Thesetworatiosareequal:f/f_min = S_max/S.CombiningEqs.2.2,4.1,and4.2,wehave

$S_{{max }} = 10^{{13}} M{left( {1 - v_{2} ho } ight)}/N_{{ ext{o}}} {left( {6pi eta R_{{min }} } ight)} = M{left[ {2.378; imes ;10^{{ - 4}} } ight]}/R_{{min }}$

(4.3a)

$S_{{max }} = 0.00361M^{{2/3}} .$

(4.3b)

Theleadingfactorof10¹³inEq.4.3aconvertsS_maxtoSvedbergunits.ThenumbersinbracketsinEq.4.3aarecalculatedusingv₂ = 0.73cm³/g,ρ = 1.0g/cm³,η = 0.01gcm⁻¹s⁻¹ = 10⁻⁹gnm⁻¹s⁻¹.Thefinalexpression,Eq.4.3bexpressesS_maxinSvedbergsforaproteinofmassMinDaltons.SometypicalnumericalvaluesofS_maxforproteinsfrom10,000to1,000,000DaaregiveninTable3.

Table3S_maxcalculatedforproteinsofdifferentmass

WehavesurveyedvaluesofS_max/Sforavarietyofproteinsofknownstructure.Table4presentsS_max/Sforanumberofapproximatelyglobularproteinsandforarangeofelongatedproteins,allofknowndimensions.ItturnsoutthatS_max/Sisanexcellentpredictorofthedegreeofasymmetryofaprotein.Fromthissurveyofknownproteins,wecanproposethefollowinggeneralprincipals.

ProteinM_r(kDa)	10	25	50	100	200	500	1,000
S_maxSvedbergs	1.68	3.1	4.9	7.8	12.3	22.7	36.1

•	NoproteinhasS_max/S = f/f_minsmallerthan∼1.2.
•	Forapproximatelyglobularproteins: S_max/Sistypicallybetween1.2and1.3.
•	Formoderatelyelongatedproteins: S_max/Sisintherangeof1.5to1.9.
•	Forhighlyelongatedproteins(tropomyosin,fibrinogen,extendedfibronectin): S_max/Sisintherangeof2.0to3.0.
•	Forverylongthread-likemoleculeslikecollagen,orhugeextendedmoleculeslikethetenascinhexabrachion(notshown): S_max/Scanrangefrom3–4ormore.

Table4S_max/Svaluesforrepresentativeglobularandelongatedproteins

Apartfromindicatingtheshapeofaprotein,S_max/Scanoftengivevaluableinformationabouttheoligomericstate,ifonehassomeideaoftheshape.Forexample,ifoneknowsthattheproteinsubunitisapproximatelyglobular(fromEMforexample),butfindsS_max/S = 2.1,thiswouldsuggestthattheproteininsolutionisactuallyadimer.Ontheotherhand,ifonethinksaproteinisadimer,butfindsS_max/S < 1.0forthedimermass,theproteinisapparentlysedimentingasamonomer.

TheuseofS_max/Stoestimateproteinshapehasbeendescribedbrieflyin(8).

TheKirkwood/BloomfieldCalculation

TheunderstandingofhowproteinshapeaffectshydrodynamicsiselegantlyextendedbyananalysisoriginallydevelopedbyKirkwood(9)andlaterextendedbyBloomfieldandGarciaDeLaTorres(10–12).Initssimplestapplication,itcalculatesthesedimentationcoefficientofarigidoligomericproteincomposedofsubunitsofknownSandknownspacingrelativetoeachother.Inmorecomplexapplications,aproteinofanycomplexshapecanbemodeledasasetofnonoverlappingspheresorbeads.SeeByron(13)foracomprehensivereviewoftheprincipalsandapplicationsofhydrodynamicbeadmodelingofbiologicalmacromolecules.

ThebasisoftheKirkwood/Bloomfieldanalysisistoaccountforhoweachbeadshieldstheothersfromtheeffectofsolventflowandtherebydeterminethehydrodynamicsoftheensemblefromitscomponentbeads.Figure2showsasimpleexampleofthebeadmodelingapproachandprovidesaninstructivelookathowsizeandshapeaffectsedimentation.Thereareseveralimportantconclusions.

Protein	Dimensions(nm)	Mass	S_max	S	S_max/S
Globularproteinstandardsdimensionsarefrompdbfiles
Phosphofructokinase	14 × 9 × 9	345,400	17.77	12.2	1.46
Catalase	9.7 × 9.2 × 6.7	230,000	13.6	11.3	1.20
Serumalbumin	7.5 × 6.5 × 4.0	66,400	5.9	4.6	1.29
Hemoglobin	6 × 5 × 5	64,000	5.78	4.4	1.32
Ovalbumin	7.0 × 3.6 × 3.0	43,000	4.43	3.5	1.27
FtsZ	4.8 × 4 × 3	40,300	4.26	3.4	1.25
Elongatedproteinstandards—tenascinfragments(27,28);heatrepeat(29,30)
TNfn1–5	14.7 × 1.7 × 2.8	50,400	4.94	3.0	1.65
TNfn1–8	24.6 × 1.7 × 2.8	78,900	6.64	3.6	1.85
TNfnALL	47.9 × 1.7 × 2.8	148,000	10.1	4.3	2.36
PR65/AHEATrepeat	17.2 × 3.5 × 2.0	60,000	5.53	3.6	1.54
Fibrinogen	46 × 3 × 6	390,000	19.3	7.9	2.44

•	ArodofthreebeadshasaboutatwofoldhigherSthanasinglebead.
•	S_max/Sis1.18forthesinglebead(theeffectoftheassumedshellofwater),1.34forthethree-beadrod,and1.93forthestraight11-beadrod.ThisisconsistentwiththeprincipalsgiveninSection4forglobular,somewhatelongated,andveryelongatedparticles.
•	Bendingtherodat90°inthemiddlecausesonlyasmallincreaseinS.BendingitintoaU-shapewiththearmsaboutonebeaddiameterapartincreasesSabitmore.Bendingthissame11-beadstructuremoresharply,sothetwoarmsareincontact,causesasubstantialincreaseinS,from5.05to5.58.TheguidingprincipleisthatfoldingaffectsSwhenonepartofthemoleculeisbroughtcloseenoughtoanothertoshielditfromwaterflow.

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Fig.2Eachbeadmodelsa10-kDadomain,withanassumedsedimentationcoefficientof1.42S.Theradiusofthebeadis1.67nm,usingR_min = 1.42nm,andadding0.25nmforashellofwater.ThebeadsareanapproximationtoFN-IIIorIgdomains,whichare∼1.7 × 2.8 × 3.5nm.ThesedimentationcoefficientsofmultibeadstructureswerecalculatedbytheformulaofKirkwood/Bloomfield.

GelFiltrationChromatographyandtheStokesRadius

“Gelfiltrationchromatographyiswidelyusedfordeterminingproteinmolecularweight.”ThisquotefromSigma-Aldrichbulletin891Aisawidelyheldmisconception.Thefallacyisobscurelycorrectedbyalaternoteinthebulletinthat“Onceacalibrationcurveisprepared,theelutionvolumeforaproteinofsimilarshape,butunknownweight,canbeusedtodeterminetheMW.”Thekeyissueis“ofsimilarshape”.Generally,thecalibrationproteinsareallglobular,andiftheunknownproteinisalsoglobular,thecalibratedgelfiltrationcolumndoesgiveagoodapproximationofitsmolecularweight.Theproblemisthattheshapeofanunknownproteinisgenerallyunknown.Iftheunknownproteiniselongated,itcaneasilyeluteatapositiontwicethemolecularweightofaglobularprotein.

Thegelfiltrationcolumnactuallyseparatesproteinsnotontheirmolecularweightbutontheirfrictionalcoefficient.Sincethefrictionalcoefficient,f,isnotanintuitiveparameter,itisusuallyreplacedbytheStokesradiusR_s.R_sisdefinedastheradiusofasmoothspherethatwouldhavetheactualfoftheprotein.Thisismuchmoreintuitivesinceitallowsonetoimaginearealsphereapproximatelythesizeoftheprotein,orsomewhatlargeriftheproteiniselongatedandhasboundwater.

AsmentionedaboveforEq.4.2,Stokescalculatedtheoreticallythefrictionalcoefficientofasmoothspheretobe:

$f = 6pi eta R_{{ ext{s}}} .$

(6.1)

TheStokesradiusR_sislargerthanR_minbecauseitistheradiusofasmoothspherewhosefwouldmatchtheactualfoftheprotein.Itaccountsforboththeasymmetryoftheproteinandtheshellofboundwater.Morequantitatively,f/f_min = S_max/S = R_s/R_min.

SiegelandMonte(4)arguedconvincinglythattheelutionofproteinsfromagelfiltrationcolumncorrelatescloselywiththeStokesradius,R_s,presentingexperimentaldatafromawiderangeofglobularandelongatedproteins.TheStokesradiusisknownforlargenumberofproteins,includingonesconvenientforcalibratinggelfiltrationcolumns(Table5).Figure3showsanexamplewheretheR_softheunknownproteinSMCproteinfromB.subtiliswasdeterminedbygelfiltration.

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Fig.3DeterminationofR_sofBsSMCbygelfiltration.ThecolumnwascalibratedbyrunningstandardproteinsBSA,catalase,andthyroglobulinoverthecolumn,thenBsSMC.BsSMCelutedinfraction14.2,whichcorrespondstoanR_sof10nmontheextrapolatedcurve.Inrepeatedexperiments,theaverageR_swas10.3nm(19).

Table5Standardsforhydrodynamicanalysis

Gelfiltrationcalibrationkits,containingglobularproteinsofknownmolecularweightandR_s,arecommerciallyavailable(GEHealthcare,Sigma-Aldrich).Thesesameproteinscanbeusedforsedimentationstandards.Theproteinsinthesekitsareincludedinthetablealongwithsomeothersthatwehavefounduseful.ThevaluesforM_rgiveninthefirstcolumnarefromaminoacidsequencedata.ValuesforS_20,wandR_sarefromtheSiegel–Montepaper(indicatedS-Mundersource),ortheCRCHandbookofBiochemistry(3)(indicatedHBC).TheyagreewiththevalueslistedforR_sintheGEHealthcaregelfiltrationcalibrationkit,withtheexceptionofferritin.The“M_rcalc”inthelastcolumnwasobtainedbyoursimplificationoftheSiegel–Montecalculation(M = 4,205SR_s).Notethattheworstdisagreementwith“M_raaseq”isabout10%

^aSforribonucleaseAisquestionablebecauseofthelowS_max/S(1.05).Svaluesforbovineserumalbuminvaryintheliteraturefrom4.3to4.9.Manysourcesuse4.3,butwefindthat4.6givesabetterfitwithotherstandards(notethatthestandardcurveinFig.5used4.3,but4.6wouldhaveplaceditclosertotheline)

ThestandardproteinsshouldspanR_svaluesaboveandbelowthatoftheproteinofinterest(butinthecaseofSMCproteinfromB.subtilis,ashortextrapolationtoalargervaluewasused).TheliteraturegenerallyrecommendsdeterminingthevoidandincludedvolumesofthecolumnandplottingapartitioncoefficientK_AV(4).However,wehavefounditgenerallysatisfactorytosimplyplotelutionpositionvsR_sforthestandardproteins.Thisgenerallygivesanapproximatelylinearplot,butotherwise,itissatisfactorytodrawlinesbetweenthepointsandreadtheR_softheproteinofinterestfromitselutionpositiononthisstandardcurve.

AgelfiltrationcolumncandetermineR_srelativetotheR_softhestandardcalibrationproteins.TheR_softhesestandardswasgenerallydeterminedfromexperimentallymeasureddiffusioncoefficients.Sometabulationsofhydrodynamicdatalistthediffusioncoefficient,D,ratherthanR_s,soitisworthknowingtherelationship:

Protein	M_raaseq	S_20,w	S_max/S	R_s(nm)	Source	M_rS-M
RibonucleaseAbeefpancreas	14,044	2.0^a	1.05^a	1.64	HBC	13,791
ChymotrypsinogenAbeefpancreas	25,665	2.6	1.21	2.09	HBC	22,849
Ovalbuminhenegg	42,910s	3.5	1.27	3.05	HBC	44,888
Albuminbeefserum	69,322	4.6^a	1.33	3.55	S-M,HBC	68,667
Aldolaserabbitmuscle	157,368	7.3	1.45	4.81	HBC	147,650
Catalasebeefliver	239,656	11.3	1.21	5.2	S-M	247,085
Apo-ferritinhorsespleen	489,324	17.6	1.28	6.1	HBC	451,449
Thyroglobulinbovine	606,444	19	1.37	8.5	HBC	679,107
Fibrinogen,human	387,344	7.9	2.44	10.7	S-M	355,449

$D = {kT} mathord{left/ {vphantom {{kT} f}} ight. kern- ulldelimiterspace} f = {kT} mathord{left/ {vphantom {{kT} {{left( {6pi eta R_{{ ext{s}}} } ight)}}}} ight. kern- ulldelimiterspace} {{left( {6pi eta R_{{ ext{s}}} } ight)}}.$

(6.2)

wherek = 1.38 × 10⁻¹⁶gcm²s⁻²K⁻¹isBoltzman’sconstantandTistheabsolutetemperature.kisgivenhereincentimeter–gram–secondunitsbecauseDistypicallyexpressedincentimeter–gram–second;R_swillbeexpressedincentimeterinthisequation.TypicalproteinshaveDintherangeof10⁻⁶to10⁻⁷cm²s⁻¹.ConvertingtonanometerandforT = 300Kandη = 0.01:

$R_{{ ext{s}}} = {left( {1 mathord{left/ {vphantom {1 D}} ight. kern- ulldelimiterspace} D} ight)},2.2; imes ;10^{{ - 6}} ,$

(6.3)

whereR_sisinnanometerandDisincentimetersquaredpersecond.

Simplyknowing,R_sisnotveryvaluableinitself,exceptforestimatingthedegreeofasymmetry,butthiswouldbethesameanalysisdevelopedaboveforS_max/S.However,ifonedeterminesbothR_{sandS,thispermitsadirectdeterminationofmolecularweight,whichcannotbededucedfromeitheronealone.Thisisdescribedinthenextsection.}

DeterminingtheMolecularWeightofaProteinMolecule—CombiningSandR_sàlaSiegelandMonte

Withthecompletionofmultiplegenomesandincreasinglygoodannotation,theprimarysequenceofalmostanyproteincanbefoundinthedatabases.Themolecularweightofeveryproteinsubunitisthereforeknownfromitssequence.Butanexperimentalmeasureisstillneededtodetermineifthenativeproteininsolutionisamonomer,dimer,oroligomer,orifitformsacomplexwithotherproteins.Ifonehasapurifiedprotein,themolecularweightcanbedeterminedquiteaccuratelybysedimentationequilibriumintheanalyticalultracentrifuge.ThistechniquehasmadeastrongcomebackwiththeintroductionoftheBeckmanXL-Aanalyticalultracentrifuge.Thereareanumberofgoodreviews(14,15),andthedocumentationandprogramsthatcomewiththecentrifugeareveryinstructive.

WhatifonedoesnothaveanXL-Acentrifugeortheproteinofinterestisnotpurified?In1966,SiegelandMonte(4)proposedamethodthatachievestheresultsofsedimentationequilibrium,withtwoenormousadvantages.First,itrequiresonlyapreparativeultracentrifugeforsucroseorglycerolgradientsedimentationandagelfiltrationcolumn.Thisequipmentisavailableinmostbiochemistrylaboratories.Second,theproteinofinterestneednotbepurified;oneneedsonlyanactivityoranantibodytolocateitinthefractions.Thisisaverypowerfultechniqueandshouldbeintherepertoireofeveryproteinbiochemist.

Themethodologyisverysimple.TheproteinisrunoveracalibratedgelfiltrationcolumntodetermineR_sandhencef.Separately,theproteiniscentrifugedthroughaglycerolorsucrosegradienttodetermineS.OnethenusestheSvedbergequation(Eq.4.1)toobtainMasafunctionofR_sandS.

$M = SN_{{ ext{o}}} {{left( {6pi eta R_{{ ext{s}}} } ight)}} mathord{left/ {vphantom {{{left( {6pi eta R_{{ ext{s}}} } ight)}} {{left( {1 - v_{2} ho } ight)}}}} ight. kern- ulldelimiterspace} {{left( {1 - v_{2} ho } ight)}}$

(7.1a)

settingη = 0.01,v₂ρ = 0.73,convertingStoSvedbergunitsandR_stonanometer,wecansimplifyfurther:

$M = 4,205,{left( {SR_{s} } ight)}$

(7.1b)

whereSisinSvedbergunits,R_sisinnanometer,andMisinDaltons.

Thisisprettysimple!Importantly,intypicalapplications,thismethodgivestheproteinmasswithinabout±10%.Thisismorethanenoughprecisiontodistinguishbetweenmonomer,dimer,ortrimer.

ElectronMicroscopyofProteinMolecules

Sincetheearly1980s,electronmicroscopyhasbecomeapowerfultechniquefordeterminingthesizeandshapeofsingleproteinmolecules,especiallyoneslargerthan100kDa.TwotechniquesavailableinmostEMlaboratories,rotaryshadowingandnegativestain,canbeusedforimagingsinglemolecules.Cryo-EMisbecomingapowerfultoolforproteinstructuralanalysis,butitrequiresspecialequipmentandexpertise.Foralargenumberofapplications,rotaryshadowingandnegativestainprovidetheessentialstructuralinformation.

Forrotaryshadowing,adilutesolutionofproteinissprayedonmica,theliquidisevaporatedinahighvacuum,andplatinummetalisevaporatedontothemicaatashallowangle.Themicaisrotatedduringthisprocess,sotheplatinumbuildsuponallsidesoftheproteinmolecules.ThefirstEMimagesofsingleproteinmoleculeswereobtainedbyHallandSlayterusingrotaryshadowing(16).Theirimagesoffibrinogenshowedadistinctivetrinodularrod.However,rotaryshadowingfellintodisfavorbecausetheimagesweredifficulttoreproduce.Proteintendedtoaggregateandcollectsalt,ratherthanspreadassinglemolecules.In1976,JamesPullman,agraduatestudentattheUniversityofChicago,thendevisedaprotocolwithonesimplebutcrucialmodification—headded30%glyceroltotheproteinsolution.Forreasonsthatarestillnotunderstood,theglycerolgreatlyhelpsthespreadingoftheproteinassinglemolecules.

Pullmanneverpublishedhisprotocol,buttwolabssawhismimeographednotesandtestedouttheeffectofglycerol,asapartoftheirownattemptstoimproverotaryshadowing(17,18).Theyobtainedreproducibleandcompellingimagesoffibrinogen(thefirstsincetheoriginalHallandSlayterstudyandconfirmingthetrinodularrodstructure)andspectrin(thefirsteverimagesofthislargeprotein).Thetechniquehassincebeenusedincharacterizinghundredsofproteinmolecules.

Figure4showsrotaryshadowedSMCproteinfromB.subtilis,fibrinogen,andhexabrachion(tenascin).SMCproteinfromB.subtilisishighlyelongated,consistentwithitshighS_max/Sdiscussedabove(19).Thefibrinogenmoleculesshowthetrinodularrod,buttheseimagesalsoresolvedasmallfourthnodulenexttothecentralnodule(20),notseeninearlierstudies.Thecentralnoduleisabout50kDa,andthesmallerfourthnoduleisabout20kDa.The“hexabrachion”tenascinmolecule(21)illustratesthepowerofrotaryshadowingattwoextremes.First,themoleculeishuge.Eachofitssixarmsismadeupof∼30repeatingsmalldomains,totaling∼200,000Da.Atthelargerscale,theEMshowsthateacharmisanextendedstructure,matchingthelengthexpectediftherepeatingdomainsareanextendedstringofbeads.Atthefinerscale,theEMcandistinguishthedifferentsizeddomains.Theinnersegmentofeacharmisastringof3.5-kDaepidermalgrowthfactordomains,seenhereasathinnersegment.Astringof10-kDaFN-IIIdomainsisclearlydistinguishedasathickeroutersegment.Theterminalknobisasingle22-kDafibrinogendomain.TheR_minofthesedomainsare0.8,1.7,and2.8nm,andthesecanbedistinguishedbyrotaryshadowing.RotaryshadowingEMcanvisualizesingleglobulardomainsassmallas10kDa(3.5nmdiameter)andelongatedmoleculesasthinas1.5nm(collagen).

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Fig.4RotaryshadowingEMofthreehighlyelongatedproteinmolecules:theSMCproteinfromB.subtilis(19),fibrinogen(20),andthehexabrachionprotein,tenascin(21).

NegativestainisanotherEMtechniquecapableofimagingsingleproteinmolecules.Itisespeciallyusefulforimaginglargermoleculeswithacomplexinternalstructure,whichappearonlyasalargeblobinrotaryshadowing.Importantly,noncovalentprotein–proteinbondsaresometimesdisruptedintherotaryshadowingtechnique(8),buturanylacetate,inadditiontoprovidinghighresolutioncontrast,fixesoligomericproteinstructuresinafewmilliseconds(22).Anexcellentreviewofmoderntechniquesofnegativestaining,withcomparisontocryo-EM,isgivenin(23).

ThesimplepictureofthemoleculeproducedbyEMisfrequentlythemoststraightforwardandsatisfyingstructuralanalysisatthe1–2-nmresolution.Whenthestructureisconfirmedbyhydrodynamicanalysis,theinterpretationisevenmorecompelling.

HydrodynamicAnalysisandEMAppliedtoLargeMultisubunitComplexes

ThetextboxaboveshowedtheapplicationoftheSiegel–MonteanalysistoSMCproteinfromB.subtilis,whichhadonlyonetypesubunitandwasfoundtobeadimer.Similarhydrodynamicanalysiscanbeusedtoanalyzemultisubunitproteincomplexes.Therearemanyexamplesintheliterature;IwillshowhereanelegantapplicationtoDASH/Dam1.

TheproteincomplexcalledDASHorDam1isinvolvedinattachingchromosomalkinetochorestomicrotubulesinyeast.DASH/Dam1isacomplexoftenproteinsthatassembleintoaparticlecontainingonecopyofeachsubunit.Thesecomplexesfurtherassembleintoringsthatcanformaslidingwasheronthemicrotubule(24,25).Thebasicten-subunitcomplexhasbeenpurifiedfromyeastandhasalsobeenexpressedinEscherichiacoliandpurified(thisrequiredtheheroiceffortofexpressingalltenproteinssimultaneously(24)).Figure5showsthehydrodynamiccharacterizationofthepurifiedproteincomplexandillustratesseveralimportantfeatures.

ApplicationtoSMCproteinfromB.subtilis.Inthesectionsabove,weshowedhowSoftheSMCproteinfromB.subtiliswasdeterminedtobe6.3Sfromglycerolgradientsedimentation,andR_swas10.3nm,fromgelfiltration.PuttingthesevaluesinEq.7.1b,wefindthatthemolecularweightofSMCproteinfromB.subtilisis273,000Da.Fromtheaminoacidsequence,weknowthatthemolecularweightofoneSMCproteinfromB.subtilissubunitis135,000Da.TheSiegel–MonteanalysisfindsthattheSMCproteinfromB.subtilismoleculeisadimer.

KnowingthatSMCproteinfromB.subtilisisadimerwithmolecularweight270,000Da,wecannowdetermineitsS_max/S.S_maxis15.1(Eq.4.3b)soS_max/Sis2.4.TheSMCproteinfromB.subtilismoleculeisthusexpectedtobehighlyelongated.EM(seebelow)confirmedthisprediction.

•	Forboththegelfiltration(sizeexclusionchromatography,Fig.5a)andgradientsedimentation,Fig.5b,twocalibrationcurvesofknownproteinstandardsareshown,greenandblack.Theseareindependentcalibrationruns.Inthisstudy,thegelfiltrationcolumnwascalibratedintermsofthereciprocaldiffusioncoefficient,1/D,whichisproportionaltoR_s(Eq.6.2).
•	ThefractionswereanalyzedbyWesternblotforthelocationoftwoproteinsofthecomplex,Spc34pandHsk3p.Methodsnotesthat1mlfractionsfromgelfiltrationwereprecipitatedwithperchloricacidandrinsedwithacetonepriortoSDS-PAGE,anessentialamplificationforthedilutesamplesofyeastcytoplasmicextract.Thesetwoproteinselutedtogetherinbothgelfiltrationandsedimentation,consistentwiththeirbeingpartofthesamecomplex.
•	TheprofilesofthetwoproteinswereidenticalwhenanalyzedintheirnativeforminyeastcytoplasmicextractandasthepurifiedcomplexexpressedinE.coli.Thisisstrongevidencethattheexpressionproteiniscorrectlyfoldedandassembled.
•	Thereisminimaltrailingofanysubunits.Thismeansthatthereisnosignificantdissociationduringthetensofminutesforthegelfiltration,orthe12-hcentrifugation.Thecomplexisheldtogetherbyveryhighaffinitybonds,makingitessentiallyirreversible.
•	CombiningtheR_s = 7.6nm(from1/D = 0.35 × 10⁻⁷,andS = 7.4,Eq.7.1bgivesamassofM = 236kDa,closetothe204kDaobtainedfromaddingthemassofthetensubunits.S_maxis12.6givingS_max/S = 1.7,suggestingamoderatelyelongatedprotein.

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Fig.5HydrodynamicanalysisoftheDASH/Dam1complex.Gelfiltrationisshowninaandsucrosegradientsedimentationinb.Independentcalibrationcurvesusingstandardproteinsareshowninblackandgreen.DarkandlightblueshowSpc34pinyeastcytoplasmicextractandinthepurifiedrecombinantprotein.RedandpurpleshowHsk3p.TheproteinswereidentifiedandquantitatedbyWesternblotofthefractions,showninc.Thefourproteinbandselutedtogetherat1/D = 0.35 × 10⁷,correspondingtoR_s = 7.6nm,andat7.4S.ReproducedfromMirandaetal.(24)withpermissionoftheauthors.

Figure6showsEMimagesofDASH/DAM1byrotaryshadowing(a)andnegativestain(b).Rotaryshadowingshowedirregularparticlesabout13nmindiameter(24).Theparticleshadvariableandfrequentlyelongatedshapes,butinternalstructurecouldnotberesolved.Alaterstudyusedstateoftheartnegativestainingandsophisticatedcomputerprogramstosortimagesintoclassesandaveragethem(26).Theseimagesresolvedacomplexinternalstructure.Thenegativestainshowedmostoftheparticles(80%)tobedimers,with15%monomersand5%trimers.ThiscontradictsthehydrodynamicanalysisofMirandaetal.(24)showingthattheparticlesweremonomers.Thereasonforthisdiscrepancyisnotknown.

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Fig.6EMofDASH/Dam1.aRotaryshadowingshowsparticlesroughly13nminsize,withirregularshape.bState-of-the-artnegativestaincoupledwithsingleparticleaveragingshowsacomplexinternalstructureoftheelongatedparticles.Thescalebarindicates100nmfortheunprocessedimages.Theaveragedimagesontherightshowamonomer,dimer,andtrimer.Thesepanelsare14nmwide.Thedimerwasthepredominantspecies.Leftpanel(rotaryshadowing)reprintedwithpermissionofMirandaetal.(24).Rightpanels(negativestain)reprintedwithpermissionofWangetal.(26).

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