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Improvement of cryo-EM maps by density modification Thomas C. Terwilliger 1,2 , Steven J. Ludtke 3 , Randy J. Read 4 , Paul D. Adams 5,6 , Pavel V. Afonine 5 1 Los Alamos National Laboratory, Los Alamos NM 87545 USA 2 New Mexico Consortium, Los Alamos NM 87544 USA 3 Baylor College of Medicine, Houston, TX 77030 USA 4 Cambridge Institute for Medical Research, Cambridge, CB2 0XY, UK 5 Molecular Biophysics & Integrated Bioimaging Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720-8235, USA 6 Department of Bioengineering, University of California Berkeley, Berkeley, CA, USA Abstract A density modification procedure for improving maps produced by single-particle electron cryo- microscopy is presented. The theoretical basis of the method is identical to that of maximum- likelihood density modification, previously used to improve maps from macromolecular X-ray crystallography. Two key differences from applications in crystallography are that the errors in Fourier coefficients are largely in the phases in crystallography but in both phases and amplitudes in electron cryo-microscopy, and that half-maps with independent errors are available in electron cryo-microscopy . These differences lead to a distinct approach for combination of information from starting maps with information obtained in the density modification process. The applicability of density modification theory to electron cryo- microscopy was evaluated using half-maps for apoferritin at a resolution of 3.1 Å and a matched 1.8 Å reference map. Error estimates for the map obtained by density modification were found to closely agree with true errors as estimated by comparison with the reference map. The density modification procedure was applied to a set of 104 datasets where half-maps, a full map and a model all had been deposited. The procedure improved map-model correlation . CC-BY 4.0 International license a certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under The copyright holder for this preprint (which was not this version posted March 6, 2020. ; https://doi.org/10.1101/845032 doi: bioRxiv preprint

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Page 1: Improvement of cryo-EM maps by density modificationIn macromolecular crystallography, the amplitudes of Fourier coefficients are generally measured accurately and the phases are poorly

Improvementofcryo-EMmapsbydensitymodification

ThomasC.Terwilliger1,2,StevenJ.Ludtke3,RandyJ.Read4,PaulD.Adams5,6,PavelV.Afonine5

1LosAlamosNationalLaboratory,LosAlamosNM87545USA2NewMexicoConsortium,LosAlamosNM87544USA3BaylorCollegeofMedicine,Houston,TX77030USA4CambridgeInstituteforMedicalResearch,Cambridge,CB20XY,UK5MolecularBiophysics&IntegratedBioimagingDivision,LawrenceBerkeleyNational

Laboratory,Berkeley,CA94720-8235,USA6DepartmentofBioengineering,UniversityofCaliforniaBerkeley,Berkeley,CA,USA

Abstract

Adensitymodificationprocedureforimprovingmapsproducedbysingle-particleelectroncryo-

microscopyispresented.Thetheoreticalbasisofthemethodisidenticaltothatofmaximum-

likelihooddensitymodification,previouslyusedtoimprovemapsfrommacromolecularX-ray

crystallography.Twokeydifferencesfromapplicationsincrystallographyarethattheerrorsin

Fouriercoefficientsarelargelyinthephasesincrystallographybutinbothphasesand

amplitudesinelectroncryo-microscopy,andthathalf-mapswithindependenterrorsare

availableinelectroncryo-microscopy.Thesedifferencesleadtoadistinctapproachfor

combinationofinformationfromstartingmapswithinformationobtainedinthedensity

modificationprocess.Theapplicabilityofdensitymodificationtheorytoelectroncryo-

microscopywasevaluatedusinghalf-mapsforapoferritinataresolutionof3.1Åanda

matched1.8Åreferencemap.Errorestimatesforthemapobtainedbydensitymodification

werefoundtocloselyagreewithtrueerrorsasestimatedbycomparisonwiththereference

map.Thedensitymodificationprocedurewasappliedtoasetof104datasetswherehalf-maps,

afullmapandamodelallhadbeendeposited.Theprocedureimprovedmap-modelcorrelation

.CC-BY 4.0 International licenseacertified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under

The copyright holder for this preprint (which was notthis version posted March 6, 2020. ; https://doi.org/10.1101/845032doi: bioRxiv preprint

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Page 2: Improvement of cryo-EM maps by density modificationIn macromolecular crystallography, the amplitudes of Fourier coefficients are generally measured accurately and the phases are poorly

andincreasedthevisibilityofdetailsinthemaps.Theprocedurerequirestwounmaskedhalf-

mapsandasequencefileorothersourceofinformationonthevolumeofthemacromolecule

thathasbeenimaged.

.CC-BY 4.0 International licenseacertified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under

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Page 3: Improvement of cryo-EM maps by density modificationIn macromolecular crystallography, the amplitudes of Fourier coefficients are generally measured accurately and the phases are poorly

Single-particleelectroncryo-microscopy(cryo-EM)israpidlybecomingthedominanttechnique

fordeterminationoflargethree-dimensionalstructuresofmacromoleculesandtheir

complexes1.Theresultofacryo-EManalysisisathree-dimensionalmapreflectingtheelectric

potentialofthemacromolecule2andwhichhasmapvaluesandanappearancecloselyrelated

tomapsobtainedfromX-raycrystallography3.Inbothcryo-EMandinmacromolecular

crystallography,theaccuracyofthemapisanimportantcharacteristic.Inmacromolecular

crystallography,theamplitudesofFouriercoefficientsaregenerallymeasuredaccuratelyand

thephasesarepoorlyestimated.Itiscommonpracticeinthatfieldtocarryoutaprocedure

knownasdensitymodificationtoreducetheerrorsinthephasesandtherebyimprovethe

resultingmap4-7.Thesourceofnewinformationincrystallographicdensitymodificationisprior

knowledgeaboutexpectedvaluesinamap.Forexample,theprobabilitydistributionofmap

valuesmaybeknown,ortheremaybeknowledgeaboutspecificfeaturesinthemapsuchasa

flatsolventregion.Informationaboutthetruedensityinpartorallofthemapcanbeusedto

obtainimprovedestimatesofthephases,andtheseimprovedphasesleadtoanimprovedmap

everywhere,notjustwheretheinformationwasapplied6.

Incryo-EMaformofdensitymodificationmaybeappliedduringtheprocessofimage

reconstruction.Themacromoleculetypicallyoccupiesonlyasmallpartofthevolumeofthe

reconstruction,andduringreconstructionnoiseisremovedfromthepartofthemapthatis

outsidethemacromolecule8,9.Thiscanimprovethemapintheregionofthemacromolecule

andisrelatedtothe“solventflattening”aspectofcrystallographicdensitymodification10.Local

denoising11orfiltering12,13proceduresareoftenappliedtoimprovetheinterpretabilityofcryo-

EMmaps.Aprocedureforhistogram-matchingandresolutionfilteringhasalsobeen

developed14.Thoughtheoverallprocessofdensitymodificationasimplementedina

crystallographicsettingisthoughttobeinappropriateforcryo-EM15,ithasbeensuggestedthat

thegeneralconceptcouldbeadaptedandapplied8,15.Hereweshowthataversionofdensity

modificationwiththesametheoreticalbasisascrystallographicdensitymodificationbutwith

keydifferencesreflectingthedifferencesbetweencrystallographicandcryo-EMmapscanbe

usedtoimprovecryo-EMmaps.

.CC-BY 4.0 International licenseacertified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under

The copyright holder for this preprint (which was notthis version posted March 6, 2020. ; https://doi.org/10.1101/845032doi: bioRxiv preprint

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Page 4: Improvement of cryo-EM maps by density modificationIn macromolecular crystallography, the amplitudes of Fourier coefficients are generally measured accurately and the phases are poorly

Thereareseveralframeworksfordensitymodificationthatcouldbeappliedtocryo-EMmaps4.

Hereweusemaximum-likelihooddensitymodification,asitmakesacleardistinctionbetween

informationcomingfromtheoriginaldataandinformationthatcomesfromexpectationsabout

thefeaturesinthemap5.Theprocessformapimprovementbymaximum-likelihooddensity

modificationinvolvesidentifyinghowthecurrentFouriercoefficientscanbechangedsoasto

increasetheplausibilityofthemap(expressedasalikelihood),whileretainingcompatibility

withtheoriginalexperimentalmap(seeMethods).

Cryo-EMmapsdifferinfundamentalwaysfromcrystallographicmaps,andtheactualprocess

ofdensitymodificationcannotbeappliedinthesamewayinthetwosituations.Asdetailedin

Methods,onekeydifferenceisthatincrystallographyonlytheamplitudesofFourier

coefficientsaremeasureddirectly(phasesaregenerallyestimatedindirectlybycomparisonof

amplitudesmeasuredatdifferentwavelengthsorfromslightlydifferingcrystals),whilein

electroncryo-microscopybothphaseandamplitudesaredirectlyavailablefromexperiment.

Anotheristhathalf-mapswithrelativelyindependenterrorsareavailableinelectroncryo-

microscopy16butnotincrystallography.

Wetestedtheapplicabilityofdensitymodificationtheorytocryo-EMbyapplyingittoamapof

apoferritinatareportedresolutionof3.1Å(EMDataBankentryEMD-2002817).Weuseda

matched1.8Åreferencemap(EMD-20026)toevaluatetheerrorestimatesthatmakeupakey

partofthedensitymodificationprocessandtotesttheeffectsofdensitymodificationonmap

quality.

Thedensitymodificationprocedureforthe3.1ÅmapwastobecarriedoutusingFourier

coefficientstoaresolutionof2.5Å,sowefirstcheckedtheaccuracyofthe1.8Åmapuptothis

resolutionbycalculatingtheFouriershellcorrelation(FSC)ofindependenthalf-maps16,18.Fig.

1Ashowsthatthetwohalf-mapscorrespondingtothefull1.8ÅmaphaveanFSCvalueabove

0.97atallinverseresolutionsofupto0.4Å-1(correspondingtoaresolutionof2.5Å).Half-

.CC-BY 4.0 International licenseacertified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under

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Page 5: Improvement of cryo-EM maps by density modificationIn macromolecular crystallography, the amplitudes of Fourier coefficients are generally measured accurately and the phases are poorly

datasetFSCvaluescanbeusedtoestimatetheexpectedcorrelationofFouriercoefficientsfora

maptoFouriercoefficientsrepresentingthetruemapusingtheformula18,

FSCref=[2FSC/(1+FSC)]½(1)

HereweusethenotationFSCreftoemphasizethatthisisaFSC-basedresolution-dependent

estimateofmapsimilaritytoareference(ideal)map.Thisusagecorrespondstotheprevious

use18ofCrefandissimilartotheuseofthenotationCC*incrystallography19).AccordingtoEq.

(1),anFSCvalueof0.97correspondstoavalueofFSCref=0.99,indicating(asidefrom

systematicerrorsaffectingbothhalf-maps15)thatuptoaresolutionof2.5Å,the1.8Åmap

closelymatchesaperfectmapofthisstructure.

Weappliedourdensitymodificationproceduretothetwohalf-mapsfromthe3.1Ådataset,

yieldingtwointermediatemap-phasinghalf-maps,twodensity-modifiedhalf-maps,andafinal

densitymodifiedfullmap.AsdescribedinMethods,eachFouriercoefficientineachmap-

phasinghalf-mapisobtainedindividuallybyadjustingittomaximizethelikelihood

(believability)ofamapcalculatedfromthiscoefficientplusallother(constant)Fourier

coefficientsinthecorrespondingoriginalhalf-map.Thesemap-phasinghalf-mapsarethen

recombinedwiththeoriginalhalf-mapsusingaresolution-dependentweightingapproachto

yielddensity-modifiedhalf-maps.Finallythedensity-modifiedhalf-mapsareaveragedto

produceasingledensity-modifiedmap.Asapartofthedensitymodificationprocess,estimates

areobtainedoftheFouriershellcorrelationsFSCrefbetweentheinitialfullmapandatruemap,

andalsoestimatedcorrelationsFSCrefbetweenthedensity-modifiedmapandatruemap.The

estimatedFouriershellcorrelationbetweentheinitialfullmapandatruemapcomesfrom

applyingEq.(1)toahalf-datasetFSC(suchastheoneshowninFig.1Aforthe3.1Åmap).Fig.

1Billustratestheseestimatedresolution-dependentmapaccuracyvalues(FSCref)forthe3.1Å

map(orangedots)andshowsthattheyareverysimilartoactualmapaccuracy(theFourier

correlationbetweenthe3.1Åandreference1.8Åmaps,blueline).

Forthedensity-modifiedmap,valuesofestimatedFouriershellcorrelationtoatruemap,

FSCref,areestimatedfromresolution-dependenterrorestimates.Theseerrorestimatesarein

.CC-BY 4.0 International licenseacertified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under

The copyright holder for this preprint (which was notthis version posted March 6, 2020. ; https://doi.org/10.1101/845032doi: bioRxiv preprint

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Page 6: Improvement of cryo-EM maps by density modificationIn macromolecular crystallography, the amplitudes of Fourier coefficients are generally measured accurately and the phases are poorly

turnbasedoncorrelationsamongthetwooriginalhalfmapsandthetwomap-phasinghalf

maps(seeMethods).Fig.1Bdisplaysestimatesofresolution-dependentmapqualityFSCreffor

thedensity-modifiedmap(purpletriangles)andshowsthattheyareverysimilartotheactual

Fouriershellcorrelationvaluesbetweenthedensity-modifiedmapandthe1.8Åreferencemap

(greenline).Fig.1BalsoshowsthatthevaluesofbothestimatedFouriercorrelationFSCrefand

actualFouriercorrelationtothe1.8Åreferencemapareconsistentlyimprovedoverthe

inverseresolutionrangefromabout0.3to0.4Å-1.Thismeansthatintheresolutionrange

betweenabout2.5Åand3.3Åthedensitymodifiedmapismoreaccuratethantheoriginal

map.

Figure1.Densitymodificationofapoferritin3.1Åmapandevaluationusingapoferritin1.8Å

map.A.Fouriershellcorrelation(FSC)curvesforEMD-20028(3.1Å)andEMD-20026(1.8Å).B.

Orangecirclesareestimatedresolution-dependentmapquality(FSCref)oforiginal3.1Åmap

(seetext).BluelineistheFSCoftheoriginal3.1Åmaptothe1.8Åmap.Purpletrianglesare

.CC-BY 4.0 International licenseacertified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under

The copyright holder for this preprint (which was notthis version posted March 6, 2020. ; https://doi.org/10.1101/845032doi: bioRxiv preprint

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Page 7: Improvement of cryo-EM maps by density modificationIn macromolecular crystallography, the amplitudes of Fourier coefficients are generally measured accurately and the phases are poorly

FSCrefofdensity-modified3.1Åmap,andthegreenlineistheFSCofthedensity-modified3.1Å

mapto1.8Åmap.HorizontallinesaredrawnatFSCvaluesofzeroand½.Verticaldottedlines

aredrawnatinverseresolutionsof1/2.78Å-1and1/2.97Å-1.C,D.Densitymodificationof

apoferritin1.8Åmap,carryingoutdensitymodificationwithoutauto-sharpeningthestarting

half-mapsandusingadensity-modificationresolutionof1.8Å.C.Depositedmapwith

resolution-dependencematchedtothatofthedensity-modifiedmapusingthePhenix20tool

auto_sharpenwiththeexternal_map_sharpeningmethod.D.Density-modifiedmap.Contours

inCandDaredrawntoencloseequalvolumes21.TheX-raystructureofhumanapoferritin(PDB

entry3ajo22)isshownafterdockinginthedepositedmapandre-refinementagainstthedensity

–modifiedmap.ArrowsindicatethelocationsofF81(multipleconformationsin3ajo)andY27

wherethearomaticringsareclearafterdensitymodification.

Inarealcase,therewillnormallybenoreferencemapforcomparison.TheanalysisinFig.1B

indicatesthatitisreasonabletousevaluesofestimatedFouriercorrelationFSCref,obtained

fromthecorrelationsbetweenoriginalanddensity-modifiedhalf-datasets,asroughestimates

ofFouriercorrelationtothetruemap.

Figs.1CandDillustratethevisualeffectsofdensitymodificationofahigh-resolutionmap,that

ofhumanapoferritinat1.8Å.Thedensity-modifiedmapinDshowshigh-resolutiondetailsthat

arenotpresentintheresolution-matchedmapinC,includingcleardensityforthearomatic

ringsofF81andY27.Takenasawhole,Fig.1indicatesthatapplicationofdensitymodification

toapoferritinmapsat3.1Åor1.8Åimprovestheminasignificantwaythatisingeneral

agreementwiththatexpectedfromtheerrormodelfordensitymodificationdescribedin

Methods.

Wenexttestedthegeneralityofmaximum-likelihooddensitymodificationofcryo-EMmapsby

applyingitto104setsofhalf-mapsandfullmapsavailablefromtheEMDatabank(EMDB17).

Themainfocusofthistestwasonmapsintheresolutionrangeof2Åto4.5Åwherewe

developedtheparametersandexpectedtheproceduretowork,butweincludedmapsatlower

.CC-BY 4.0 International licenseacertified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under

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Page 8: Improvement of cryo-EM maps by density modificationIn macromolecular crystallography, the amplitudes of Fourier coefficients are generally measured accurately and the phases are poorly

resolution(upto8Å)aswell.Toevaluatetheeffectsofdensitymodification,wecalculatedthe

Fouriershellcorrelation16ofeachmap(beforeandafterdensitymodification)tothedeposited

atomicmodelfromtheProteinDataBank(PDB23),afterre-refiningthemodeltothemaptobe

evaluated(seeMethods).Thepurposeofthere-refinementwastocreatesimilarbiasesforthe

analysisoforiginalanddensitymodifiedmapsandyieldacomparisonthatwasfair.Usingthe

map-modelFSCvaluesforanoriginalanddensity-modifiedmap,wegeneratedtwometrics

thatreflectdifferentaspectsofrelativeaccuraciesofthetwomaps.Thefirstmetricwasthe

resolutionatwhichthemap-modelFSCfallstoapproximately½,anindicationofaresolution

wherethereissubstantialinformationpresent,andthesecondwastheaverageFSC(inthe

sameresolutionrangeforthetwomaps),ameasureofoverallqualityofeachmap.Such

metricsofmapqualityhavemanylimitations18,24-26andarethereforefarlessusefulthana

directcomparisonwithanessentiallyidealmapasinFig.1B.However,iftheyarefairtheycan

atleastgiveageneralideaastowhetherthemethodisuseful.

Fig.2Aillustratestheresolutionatwhichthemap-modelFSCfallstoapproximately½for

originalmaps(bluedots)anddensity-modifiedmaps(orangecircles),plottedasafunctionof

thereportedresolutionoftheoriginalmaps.Theresolutionatwhichthemap-modelFSCfalls

toapproximately½isgenerallyimproved(resultinginasmallervalue)bydensitymodification

overtheentirerangeofreportedresolutions,thoughformapswithreportedresolutionworse

than4.5Åthevariabilityinthismetricisquitelarge.

Fig2Bshowstheaveragemap-modelFSCforeachoriginalmap(bluedots)anddensity-

modifiedmap(orangecircles)asreflectedinthemeanmap-modelFouriershellcorrelation,

againplottedasafunctionofreportedresolution.Thismetricalsogenerallyimproves(is

increased)afterdensitymodificationbutthereissubstantialvariabilityintheamountof

improvement(averageimprovementwithdensitymodificationwas0.07,standarddeviationof

0.08).Overall,boththeresolutionatwhichthemap-modelFSCfallstoapproximately½and

averageFSCgenerallyimprovewithdensitymodificationbutthereisconsiderablevariability,

particularlyatlowerresolution.

.CC-BY 4.0 International licenseacertified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under

The copyright holder for this preprint (which was notthis version posted March 6, 2020. ; https://doi.org/10.1101/845032doi: bioRxiv preprint

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Page 9: Improvement of cryo-EM maps by density modificationIn macromolecular crystallography, the amplitudes of Fourier coefficients are generally measured accurately and the phases are poorly

Toexaminethevisualeffectsofdensitymodificationinspecificcases,Fig.2panelsC-Fshow

twomatchedpairsofdepositedmapsandcorrespondingdensitymodifiedmaps.Tomakethe

visualcomparisonasfairaspossible,eachdepositedmapwassharpenedautomaticallyto

matchtheresolution-dependenceofthedensity-modifiedmap,andcontoursformatched

mapswerechosentoencloseequalvolumes21asinFig1.Figs.2Cand2Dshowaloopregion

(residuesA793-A804)thatispoorlyresolvedintheb-galactosidasemapataresolutionof2.2Å

(EMDBentry298427;PDBentry5a1a;Fig.2CandSupplementaryFig.1A),butisclearinthe

density-modifiedmap(Fig.2D).Figs.2Eand2Fshowa5.8Åcryo-EMmapofaguanylate

cyclase28(EMDBentry20282,PDBentry6pas).Intheoriginalmap(Fig.2E,resolutionmatched

tothedensity-modifiedmap)helicesareessentiallyfeaturelesstubesofdensity,whileinthe

density-modifiedmap(Fig.2F)theheliceshaveclearperiodicitycorrespondingtothehelical

repeat.Sharpeningoftheoriginalmapdoesnotyieldamoreinterpretablemap

(SupplementaryFig.1B).Ineachcasethedensitymodifiedmapshowssubstantiallymore

detailthantheoriginalmap.

.CC-BY 4.0 International licenseacertified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under

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Page 10: Improvement of cryo-EM maps by density modificationIn macromolecular crystallography, the amplitudes of Fourier coefficients are generally measured accurately and the phases are poorly

Figure2.ApplicationofdensitymodificationtomapsfromtheEMDataBank.A.Estimated

resolutionatwhichmap-modelFSCfallstoapproximately½fororiginalmaps(bluedots)and

density-modifiedmaps(orangedots)basedonmap-modelFSCusingmodelsrefinedagainstthe

mapsbeingexamined.B.Meanmap-modelFSCfororiginalmaps(bluedots)anddensity-

modifiedmaps(orangedots),calculateduptoinverseresolutionofcorrespondingto5/6the

statedresolutionofthemap(thevalueof5/6themapresolutionisatypicalvalueofthe

resolutionusedindensitymodification).C,D.Depositedanddensity-modifiedmapsforb-

galactosidase(2.2Å,EMDB2984).Theresolution-dependenceofthedepositedmapismatched

tothatofthedensity-modifiedmap,andcontoursincludeequalvolumes.SupplementaryFig.

S1AshowsthesameregionforthedepositedmapforEMDB2984.E,F,Depositedanddensity-

modifiedmapsofaguanylatecyclase28at5.8Å(EMDBentry20282,PDBentry6pas).

SupplementaryFig.S1Bshowsthesameregionforasharpenedversionofthedepositedmap.

.CC-BY 4.0 International licenseacertified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under

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Page 11: Improvement of cryo-EM maps by density modificationIn macromolecular crystallography, the amplitudes of Fourier coefficients are generally measured accurately and the phases are poorly

Alimitationinourcurrentimplementationofdensitymodificationforcryo-EMisthe

assumptionofrelativelyuniformnoiselevelsthroughouttheregionoutsidethe

macromolecule,whileactualmapstypicallyhavenoiselevelsthatvarywithdistancefromthe

macromolecule.AsdescribedinMethods,usingasub-volume(box)containingthe

macromoleculeandasmallregionarounditindensitymodificationreducesthenon-uniformity

innoiselevelsbutdoesnoteliminateit.Weinvestigatedwhetherareconstructionmethodthat

producedmoreuniformnoisewouldimprovedensitymodification.Weprocessedasubsetof

imagesavailableforb-galactosidase(EMPIAR-1006127,29)withtwodifferentproceduresusing

EMAN230,yieldingmapswithresolutionsofabout3.9Å.Thefirstprocedurewasastandard

reconstructionwithdefaultparametersincludingaGaussiankernelwitharesolution-

dependentwidthexceptthatnofinalmaskingwasappliedtothehalf-maps.Afullmapwith

maskingwasalsogeneratedfromthesehalfmapsandusedforcomparison.Thesecond

procedurewasareconstructionwithafixed-widthGaussiankernel,expectedtoyieldamore

uniformnoisedistributioninreal-space,totestwhetherthenon-uniformityinnoiseinacryo-

EMmapisalimitingfactor.Themapsobtainedfromeachprocedurewerethendensity

modified.EachmapwasthenevaluatedbasedonFouriershellcorrelationtothedeposited2.2

Åmapforb-galactosidase(EMD298427),superimposedonthereconstructions.Theresolution

atwhichthisFouriershellcorrelationfallstoapproximately½wasusedasaqualitymeasureof

thecorrespondingmap.

Simpleaveragingofthehalf-mapsfromthestandardreconstructiongaveamapwherethe

Fouriershellcorrelationfallsto½ataresolutionof4.0Åbasedonacomparisontothe

deposited2.2Åmap.Densitymodificationofthesemapsgaveamapwhereitwas3.9Å.

Simpleaveragingofthehalf-mapswithmoreuniformnoiseyieldedamapforwhichthe

resolutionwheretheFouriershellcorrelationfallsto½was3.9Å.Thiswasimprovedbydensity

modificationtoavalueof3.7Å.

.CC-BY 4.0 International licenseacertified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under

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Page 12: Improvement of cryo-EM maps by density modificationIn macromolecular crystallography, the amplitudes of Fourier coefficients are generally measured accurately and the phases are poorly

Figure3.Effectofareconstructionprocedureyieldingmoreuniformnoiseontheoutcomeof

densitymodification.A.Standardreconstructionprocedureappliedtosubsetofimagesoffor

b-galactosidase(EMPIAR-10061).B.Density-modifiedversionofthemapinA.C.Density

modifiedversionofreconstructiondesignedtoimprovetheuniformityofnoiseinthemap.D.

Deposited2.2Åmapofb-galactosidase,superimposedonthemapinAandlow-passfilteredat

aresolutionof3.5Å.Eachmapwasautomaticallysharpenedusingmodel-basedsharpening

withthedepositedmodelforthe2.2Åmapofb-galactosidase(5a1a)superimposedonthe

mapinA.Contourswerechosentoencloseequalvolumesinallfourmaps21.

AportionoftheinitialfullmapisshowninFig.3A.Forcomparison,Fig.3Dshowsthedeposited

high-resolution(2.2Å)map,low-passfilteredataresolutionof3.5Å.Thedensity-modified

.CC-BY 4.0 International licenseacertified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under

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Page 13: Improvement of cryo-EM maps by density modificationIn macromolecular crystallography, the amplitudes of Fourier coefficients are generally measured accurately and the phases are poorly

maps(Fig.3Band3C)botharemoresimilartothehigh-resolution,low-passfilteredmapthan

theinitialfullmap(Fig.3A),withthemapobtainedbydensitymodificationofthehalf-maps

withmoreuniformnoisebeingtheclearerofthetwomapsandthemostsimilartothehigh-

resolution,low-passfilteredmap(Fig3C).Theobservationthatthereconstructionprotocol

withrelativelyuniformnoiseproducedtheclearestmapindicatesthatsuchaprotocolmaybe

particularlywell-suitedfordensitymodification.Ingeneral,densitymodificationismostlikely

toimprovemapsthathavenotbeenmaskedandinwhichthemacromoleculeissurroundedby

asolventregionthatretainstheoriginalnoisefromthereconstruction.Thedensity

modificationprocesscanthenusethatnoiseinthesolventtoidentifyerrorsinFourier

coefficientsandtherebytoreducenoiseintheregionofthemacromolecule.

Thedensitymodificationproceduredescribedhereisfullyautomaticandrequiresonlytwo

half-maps,anoptionalnominalresolution,andinformationaboutthemolecularvolumeofthe

macromolecule(suchasasequencefileormolecularmass).Eachofthe104datasetsanalyzed

inFig.2tookfrom1to100cpu-minutes(averageof12minutes)on2.3GHzAMDprocessors.

Alloftheseanalyseswerecarriedoutwithdefaultparametersin“quick”mode,supplyinga

sequencefileandthereportedresolution,andcarryingoutonecycleofdensitymodification.

Inmanyofthesecases,improvedmapsareobtainedwithadditionalcyclesofdensity

modification.Inadditiontoafinaldensity-modifiedmap,theprocedureyieldstwodensity-

modifiedhalf-maps.AsnotedinMethods,thesemapsmayhavesomecorrelationsintroduced

bythemaskingeffectsofdensitymodification,butwiththiscaveattheycaninprinciplebe

furtherprocessedwithlocalsharpening31,weightedcombinationofhalf-mapsandother

methodsforoptimizingthefinalmap.

Therearenumerousextensionstothemethodsthataredescribedherethatcouldimprovethe

outcomeofdensitymodification.Inparticular,theanalysiscouldincludeinformationaboutthe

macromoleculefromothersourcessuchasmodelsbuiltusingthemapsorfittedintothem32,33.

Densitymodificationincludingsymmetrynotusedinthereconstructionprocesscouldbe

carriedoutaswell34.Theprocedurecouldallowstartinghalf-mapsthathaveerrorsthatare

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Page 14: Improvement of cryo-EM maps by density modificationIn macromolecular crystallography, the amplitudes of Fourier coefficients are generally measured accurately and the phases are poorly

correlatedwitheachotherorthathavedifferentexpectederrors,anderrorsthatdonotfollow

Gaussiandistributions.Itcouldbecarriedoutusingjustonemapormorethantwo‘half-maps”.

Mapscouldbedensitymodifiedwithoutboxingbyintroducingalocation-dependent

expectationformapvaluesoutsidethemacromolecule4.Errorscouldbeestimatedinregionsof

reciprocalspaceoranisotropicallyratherthaninshellsofresolution.Thedensitymodification

stepcouldalsobecarriedoutbyothermethods,forexamplesolventflipping35.Themolecular

compositioncouldbecalculatedbyanalysisofthemap,suchasusinglocalhistogramsofmap

values,allowingtheidentificationofunexpectedcomponents.Moregenerally,theentire

analysisdescribedherecouldbeextendedtoanysituationwhereamapofanydimensionality

haserrorsthatareatleastpartiallyindependentintheFourierdomainandinwhichsome

informationaboutexpectedvaluesinthemapisavailable.Assuggestedsometimeago8,a

densitymodificationproceduresuchastheonedescribedherecouldbeincorporatedintothe

overallprocessofiterativecryo-EMmaprefinementaswell.

Methods

Mapsandmodels

ThemapsusedtogenerateFig.1areapoferritinmapsEMD20026and20028andtheir

associatedhalf-maps.Thesemapshavereportedresolutionsof1.75Åand3.08Å,respectively

andwerefertothemasthe“1.8Å”and“3.1Å”apoferritinmaps.Theestimatesofresolution

forthesemapsbasedoncomparisonofmaskedhalf-mapsobtainedinthisworkareslightly

different,presumablyduetodifferentmaskingprocedures,withvaluesof1.93Åand2.97Å,

respectively.ThemodelshowninFig1isderivedfromPDBentry3ajo22andhasbeen

superimposedonthe1.8ÅmapandrefinedwiththePhenixtoolreal_spacerefine36againstthe

density-modified1.8Åmap.

The104setsofdatausedinFig.2werechosenfromtheEMDBusingdataatresolutionsfrom2

Å–8Åthathadassociatedhalf-datasetsandmatchingmodelsinthePDB.

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Page 15: Improvement of cryo-EM maps by density modificationIn macromolecular crystallography, the amplitudes of Fourier coefficients are generally measured accurately and the phases are poorly

Calculationsofmap-modelFouriershellcorrelationsweredonewithasoftmaskaroundthe

atomsinthemodel.Thesoftmaskwascalculatedasamaskaroundeachatomwitharadius

givenbytheatomicradiusoftheatomplustwicetheresolutionofthemap,followedby

smoothingofthemapwithaGaussiansmoothingfunctionwithafalloffto1/eoftwicethe

resolutionofthecorrespondingmap.AnalysesinFig.2werecarriedoutusingthePhenix20tool

resolve_cryo_em,providingtwohalf-maps,thereportedresolutionofthedataset,andthe

sequenceofthemacromoleculeobtainedfromthedepositedcorrespondingmodel.All104

analysesweredoneusingthesameversion(3689)andthesameparametersexceptfor

resolutionandsequence.

ThedataforFigures1-3areavailableasanExcelworksheetinSupplementaryDataI,andthe

(sharpened)originalanddensity-modifiedmapsshowninFigs.1-2alongwithChimerascripts

todisplaythemareavailableonlineat:http://phenix-

online.org/phenix_data/terwilliger/denmod_2020/.

Procedureforevaluationofmapquality

Weusedanautomatedproceduretoevaluatemapaccuracyandtochoosematchingmap

contoursfordisplaysothatmapcomparisonswouldbeasfairaspossible.Forevaluationof

mapaccuracywecalculatedFouriershellcorrelationsbetweenamapandanatomicmodel

refinedagainstthatmap.Therationaleforthisprocedureisthattheatomicmodelsavailable

fromthePDBarenormallyalreadyrefinedagainstthedepositedmap.Thisnecessarilybiases

themap-modelFSCcalculation.Tomakeacomparisonwithanewmap,themodelisfirstre-

refinedagainsttheoriginalmap.Thenthere-refinedmodelrefinedagainstthenewmapbefore

FSCcalculation.Thisisintendedtoleadtosimilarbiasesformodelsrefinedagainstoriginaland

densitymodifiedmaps,leadingtoarelativelyfaircomparisonbetweenmaps.

Ourlargelyautomatedprocedureforevaluationanddisplayofonemapwasthen(1)

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Page 16: Improvement of cryo-EM maps by density modificationIn macromolecular crystallography, the amplitudes of Fourier coefficients are generally measured accurately and the phases are poorly

refinementofthecorrespondingmodelfromthePDBusingthatmap,(2)boxingthemapwitha

rectangularboxaroundthemodelwithsoftedges,(3)calculationofmap-modelFSC,(4)

sharpeningthemapbasedonthemap-modelFSC18,37,(5)calculationoftheresolutionatwhich

themap-modelFSCfallstoapproximately½,and(6)calculationofaveragemap-modelFSC38up

toaresolutionof5/6thestatedresolutionofthemap(i.e.,0.83dmin).Thentocomparea

density-modifiedandoriginalmapvisually,themapswerevisuallyexaminedandaregionof

themapandcontourlevelforthedensity-modifiedmapwerechosenwheredifferencesfrom

theoriginalmapwereclear.Thecontourlevelfortheoriginalmapwasthenchosen

automaticallytoyieldthesameenclosedvolumesinthetwomaps21.Thiscontourlevelforthe

originalmapwasalwaysclosetothatobtainedbysimplyadjustingittomakeabouthalfthe

surfacethecoloroftheoriginalmapandhalfthecolorofthedensity-modifiedmapwhenthe

twomapsaredisplayedatthesametimeinChimera39.Finally,keepingthesamecontourlevels,

themapsinFig.3weremasked3Åaroundtheatomsintheregiontobedisplayedtomakeit

easiertoseetheregionofinterest.ForthemapsinFigs.1and2,anadditionalstepwasadded

inwhichtheresolution-dependenceoftheoriginalmapwasmatchedtothatofthedensity-

modifiedmap(usingthePhenixtoolauto_sharpenwiththeexternal_map_sharpeningmethod)

tomakethecomparisonofmapsasfairaspossible.

ErrorsinFouriercoefficientsrepresentingcryo-EMmaps

WeassumethatthedistributionoferrorsinFouriercoefficientsrepresentingcryo-EMmaps

canberepresentedbyatwo-dimensionalGaussianinthecomplexplane8.Thisassumptionis

evaluatedinFig.4whichcomparesFouriercoefficientsforapoferritinfromthe3.1Åand1.8Å

mapsanalyzedinFig.1.Fouriercoefficientsfortheshellofresolutionfrom3.0Åto3.1Åwere

calculatedforeachmapafterboxingthemapsaroundthefittedmodelusedinFig.1.The

Fouriercoefficientsforthe1.8Åmapweretreatedasperfectvalues.Thesevalueswere

multipliedbythecorrelationcoefficientbetweenthetwosetsofFouriercoefficientsand

subtractedfromtheFouriercoefficientsfromthe3.1Åmaptoyieldestimatesoftheerrorsin

the3.1Åmap.Fig.4showshistogramsoftheseerrorsalongdirectionsparalleland

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Page 17: Improvement of cryo-EM maps by density modificationIn macromolecular crystallography, the amplitudes of Fourier coefficients are generally measured accurately and the phases are poorly

perpendiculartotheFouriercoefficientsfromthe3.1Åmap.IneachcaseaGaussian

distributionisfittedtothesehistogramsandisshownaswell.Itcanbeseenthattheerrorsare

notquiteGaussianandarenotquitethesameinthetwodirections,butthataGaussianisa

goodfirstapproximation.Inthisexample,thenormalizederrorsperpendiculartotheFourier

coefficientsfromthe3.1Åmaphaveameanofzeroandastandarddeviationof0.63,while

thoseparallelhaveameanof0.1andastandarddeviationof0.70.

Fig.4.AnalysisofdistributionoferrorsinFouriercoefficientsforapoferritin3.1Åmap.A.

EstimatederrorsinFouriercoefficientsparalleltocoefficientsfor3.1Åmap.B.Errors

perpendiculartocoefficientsfor3.1Åmap.SolidlinesineachcasecorrespondtoaGaussian

fittedtothevaluesshown(seetext).

Procedurefordensitymodificationofcryo-EMmaps

Densitymodificationofacryo-EMmapisbasedonthemaximum-likelihoodformalismthatwe

developedpreviouslyforcrystallographicdensitymodification5.Therearetwoimportant

differences,however.OneisthatthestartingprobabilitydistributionsforFouriercoefficients

(thoseavailablebeforedensitymodification)areverydifferentinthetwocases.Theotheris

thattypicallyinacryo-EMexperiment,twoindependenthalf-mapsareavailable(twomaps

witherrorsthatareuncorrelated).

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Page 18: Improvement of cryo-EM maps by density modificationIn macromolecular crystallography, the amplitudes of Fourier coefficients are generally measured accurately and the phases are poorly

Maximum-likelihooddensitymodificationhastwooverallsteps.Inthefirststepaprobability

distribution(calledthe"map-phasing"probabilitydistribution)isobtainedforeachFourier

coefficient.Thismap-phasingdistributionhaserrorswhich,inanoptimalsituation,are

independentoftheerrorsinthecorrespondingFouriercoefficientinthestartingmap.Inthe

secondstepthemap-phasingprobabilitydistributionforeachFouriercoefficientisrecombined

withthestartinginformationaboutthatFouriercoefficienttoyieldanew"densitymodified"

estimateofthatFouriercoefficient.

Thefirststageisthesameincrystallographicandcryo-EMcases.Itstartswithamap

representedbyFouriercoefficients.Itrequiresafunctionthatdescribeshowthelikelihood

(believability)ofthatmapwouldchangeifthevaluesinthemapchange5.Thislikelihood

functionmightforexamplesaythatifthevaluesinthemapintheregionoutsidethe

macromoleculeallmovetowardsacommonvalue,thebelievabilityincreases.Itmightalsosay

thatifthedistributionofvaluesintheregionofthemacromoleculebecomesclosertoan

idealizeddistribution,thatmap'sbelievabilityimproves.Aspecificexampleofalikelihood

functionthathasboththesepropertieshasbeendescribed5(Eq.17inthisreference).Given

suchamapandlikelihoodfunction,itispossibletocalculatea"map-phasing"probability

distributionanditsmaximumorweightedmeanforeachFouriercoefficient40.Thisyieldsa

"map-phasing"map.

Themap-phasingmaphastheimportantpropertythatthenewestimateofaFourier

coefficientdoesnotdependatallonthevalueofthatFouriercoefficientinthestartingmap40.

Thisrathernon-intuitivesituationispossiblebecausethemap-phasingprobabilitydistribution

foraparticularFouriercoefficientcomesonlyfromalltheotherFouriercoefficientsandthe

characteristicsofthemapasreflectedinthelikelihoodfunction.Inothermethodsofdensity

modificationsuchassolventflippingasimilareffectisobtainedbyspecificallyremovingthe

informationcorrespondingtotheoriginalmap4,35.Itshouldbenotedhoweverthatiftheother

FouriercoefficientshaveinformationabouttheerrorsintheFouriercoefficientinquestion(for

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Page 19: Improvement of cryo-EM maps by density modificationIn macromolecular crystallography, the amplitudes of Fourier coefficients are generally measured accurately and the phases are poorly

examplethroughpreviousdensitymodificationorbymaskingofthemaparoundthe

molecule),thatinformationcanindeedaffectthemap-phasingestimateoftheFourier

coefficientofinterest.

Thekeydifferencesinimplementationbetweencrystallographicandcryo-EMcasesariseinthe

secondstep.First,theinformationabouttheFouriercoefficientthatisavailablebeforedensity

modificationisverydifferentinthetwocases.Inthecrystallographiccase,theamplitudeof

eachFouriercoefficientistypicallyknownquiteaccurately(oftenintherangeof5-30%

uncertainty)andtheremaybesomeinformationaboutthephase(thismightrangefromno

informationtophaseuncertaintiesintherangeof45degrees).Theresultingdistributionof

likelyvaluesforaparticularFouriercoefficientmightessentiallybearingofrelativelyconstant

amplitudeora"boomerang"withpartially-definedphaseandrelativelyconstantamplitude.In

contrast,inthecryo-EMcase,phaseandamplitudearebothuncertain,andthedistributionof

likelyvaluesbeforedensitymodificationcanberepresentedbyatwo-dimensionalGaussianin

thecomplexplane8.

Thisqualitativeandverysubstantialdifferenceintheformofthedistributionoflikelyvaluesfor

aFouriercoefficientpriortodensitymodificationmeansthatwhenrecombininginformation

betweenthestartingmapandmap-phasingdistributions,differentapproachesarebestsuited

tothetwosituations.Forcrystallographicapplications,recombinationessentiallyamountsto

testingpossiblevaluesofthephaseofaFouriercoefficientatconstantamplitudefor

consistencywithpriorandmap-phasingdistributions.Incontrast,forcryo-EMapplicationsas

describedhere,recombinationconsistsofcalculatingtheproductoftwo2-dimensional

distributionsandfindingthemaximumoftheresultingdistribution.Ifthedistributionsare

Gaussian,thisamountstoasimpleweightedaverageofthepriorandmap-phasingFourier

coefficients.

Thesecondkeydifferenceisthatcryo-EManalysesaretypicallycarriedoutinawaythatyields

twohalf-mapswithlargelyindependenterrors.Thismeansthatoverallmean-squarevaluesof

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Page 20: Improvement of cryo-EM maps by density modificationIn macromolecular crystallography, the amplitudes of Fourier coefficients are generally measured accurately and the phases are poorly

errorscanbeestimatedinastraightforwardwayinbinsorshellsofresolutionbycomparisonof

Fouriercoefficientsfromthetwohalf-mapsandfromthetwomap-phasinghalfmaps(see

below).

Theoverallprocedurefordensitymodificationoftwohalf-mapsisthen:(1)averagethetwo

halfmapsandsharpen/blurtheresultingmapbasedonresolution-dependenthalf-map

correlation18,37toobtainanoptimizedstartingfullmap,(2)calculatetargethistograms(see

below)forthemacromoleculeandnon-macromoleculeregionsofthisfullmap,(3)usethe

histogramsandFouriercoefficientsrepresentingeachhalf-mapinthefirststepofdensity

modificationtoobtainamap-phasingprobabilitydistributionforeachFouriercoefficientfor

thathalf-map,and(4)calculateaweightedaverageofvaluesofeachFouriercoefficient

obtainedfromthetwostartinghalf-mapsandthemap-phasingmapsobtainedfromthemto

yielda"density-modified"mapalongwithcorrespondingweightedhalf-mapsandresolution-

dependentestimatesoftheaccuraciesandresolutionofeachmap.Theoptimalweightingis

discussedbelowintermsofasimpleerrormodel.Finally(5)theentireprocesscanbe

repeated,usingthedensitymodifiedhalfmapsfromonecycleinstep(3)ofthenextcycle.Itis

alsopossible(butnotdonebydefaultinourcurrentprocedure)tousethefulldensitymodified

mapfromonecycletoobtainhistogramsforthenextcycle.Theprocessisconcludedaftera

specifiednumberofcycles.Fortheexamplesinthisworkonlyonecyclewascarriedout

(additionalcyclesdidimprovedensitymodificationformanyofthe104casesinFig.2,

however).

Onevariationontheoverallprocedurecansometimesimprovedensitymodification.Thisisto

carryoutdensitymodificationstartingdirectlywiththedepositedmapsinsteadofauto-

sharpeningthem37instep(1)above.Thisiscontrolledbythekeyword

density_modify_unsharpened_maps=TrueinthePhenixtoolresolve_cryo_em.

Targethistogramsofdensitydistributions

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Page 21: Improvement of cryo-EM maps by density modificationIn macromolecular crystallography, the amplitudes of Fourier coefficients are generally measured accurately and the phases are poorly

Akeyelementofthemaximum-likelihooddensitymodificationprocedureistheuseoftarget

histogramsrepresentingtheexpecteddistributionofmapvaluesforthe“true”(desired)mapin

theregioncontainingthemacromoleculeandintheregionoutsideit5,14.Thesehistogramscan

beobtainedinanyofseveralways.Oneistouseamapormapscorrespondingtohigh-quality

structuresthatarealreadydetermined.Anotheristousehistogramsusedpreviouslyfor

crystallographicanalyses.Thedefaultmethodusedhereistousehistogramsbasedoncrystal

structures,withanoptiontousehistogramsderivedfromthefullmapobtainedbyaveraging

thetwocurrenthalf-maps.Theseaveragedhistogramshavetheadvantagethattheyare

automaticallyatthecorrectresolutionandrepresentmacromoleculeandsurroundingregionin

justthesamewayasthehalfmaptobeanalyzed,butinourteststhehistogramsfrom

crystallographicanalysesresultedinthelargestimprovementssotheseareusedbydefault.

ErrormodelforanalysisofFSCcurvesanditsuseinoptimizingweightsandestimating

correlationstotruemaps.

Weuseasimpleerrormodelwiththefollowingassumptions:

A. Startinghalf-mapshaveerrorsthatareuncorrelatedbetweenhalf-maps.This

assumptionisbasedontheconstructionofhalf-maps,inwhichtheyarederivedfrom

independentsubsetsofthedata.Therearehoweversomeaspectsofhalf-map

constructionthatcouldleadtocorrelationoferrors,includingtheuseofthesame

referenceinsomestagesofanalysisandmaskingofthemaps18.

B. Density-modifiedhalf-mapshaveerrorsthatareuncorrelatedbetweenhalf-mapsand

theymayalsohaveerrorsthatarecorrelatedwiththecorrespondingstartinghalf-maps

anderrorsthatarecorrelatedwiththeeachother.Errorscorrelatedwiththe

correspondingstartinghalf-mapscouldcomefromthedensitymodificationprocedure

notyieldingfullyindependentinformation.Errorscorrelatedwiththeotherdensity-

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Page 22: Improvement of cryo-EM maps by density modificationIn macromolecular crystallography, the amplitudes of Fourier coefficients are generally measured accurately and the phases are poorly

modifiedhalf-mapcouldcomefrommaskingeffectsintroducedfromsolventflattening

proceduresindensitymodification.

C. Meansquarevaluesoferrorsareresolution-dependent.Thisassumptionsimplifiesthe

analysisoferrorsbydescribingtheerrorsintermsofresolutionandallowingthemtobe

estimatedinshellsofresolution.

D. Meansquareerrorsforeachmemberofapairofhalfmapsarethesame.This

assumptioncomesfromtheconstructionofhalf-maps,wheretheytypicallycomefrom

equalnumbersofimages.

E. Errorshavetwo-dimensionalGaussiandistributionswithmeanexpectedvaluesofzero.

Thissimplifiestheanalysis.

F. Fouriercoefficientsrepresentingindividualhalf-mapsareequaltothetrueFourier

coefficientsplusuncorrelatederrorsuniquetothatmapandcorrelatederrorsshared

amongtwoormoremaps.ThisyieldsasimpleformfortheFouriercoefficientsthatis

amenabletoestimationoferrorsfromcorrelationsofFouriercoefficients.

TheseassumptionsyieldasimpleerrormodelwhereFouriercoefficientsfortheoriginaland

density-modifiedhalf-mapscanberepresentedas:

Originalhalf-maps:

F1a=F+ga+sa (2a)

F1b=F+gb+sb (2b)

Density-modifiedhalf-maps:

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Page 23: Improvement of cryo-EM maps by density modificationIn macromolecular crystallography, the amplitudes of Fourier coefficients are generally measured accurately and the phases are poorly

F2a=F+ga+a+ba (3a)

F2b=F+gb+a+bb (3b)

Inthisdescription,FrepresentsthetruevalueofoneFouriercoefficient.Thereareestimatesof

Fthatcomefromeachhalf-mapandeachdensity-modifiedhalf-map.F1aandF1brepresent

Fouriercoefficientsforhalf-mapsaandb,andF2aandF2brepresentFouriercoefficientsfor

density-modifiedhalf-mapsaandb.Thetermssaandsbrepresentuncorrelatederrorsinhalf

mapsF1aandF1b.ThemeansquarevaluesofeachareS:<sa2>=<sb

2>=S,andthemeanvalues

ofallerrorsinthisanalysisarezero.Thetermgarepresentserrorsthatarecorrelatedbetween

halfmapF1aanditscorrespondingdensity-modifiedhalfmapF2a(presentinhalfmapF1aand

notcorrectedbydensitymodification),andthetermgbrepresentserrorscorrelatedbetween

F1bandF2b.Themeansquarevaluesofgaandgbare<ga2>=<gb2>=C.Thetermsbaandbb

representuncorrelatederrorsinhalfmapsF2aandF2b,where<ba2>=<ba2>=B.Theterma

representserrorscorrelatedbetweenhalfmapsF2aandF2b,where<a2>=A.Asitisassumed

thaterrorsareresolution-dependent,theestimatesofmeansquareerrors(A,B,C,S)arein

turnassumedtoberesolution-dependentandinourproceduretheyareestimatedinshellsof

resolution.

Forsimplicityinnotation,weassumebelowthattheFouriercoefficientsforeachhalf-mapare

normalizedinsuchawaythatthemeansquarevalueofFisunity.Asthefollowingcalculations

onlyinvolvecorrelationcoefficients,theoverallscaleonFouriercoefficientshasnoeffecton

thevaluesobtained,sothissimplificationdoesnotaffecttheoutcomeoftheanalysis.

Usingthiserrormodelandnormalization,theexpectedvaluesofcorrelationsbetweenhalf-

mapscanbecalculated.Theseareasfollows,wherethebracketsrepresentexpectedvalues,

andthenotationCC(x,y)representsthecorrelationcoefficientrelatingvaluesofxandy.

Theexpectedcorrelationbetweenhalf-mapsaandbisgivenby,

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Page 24: Improvement of cryo-EM maps by density modificationIn macromolecular crystallography, the amplitudes of Fourier coefficients are generally measured accurately and the phases are poorly

<CC(F1a,F1b)>=[<(F+ga+sa)(F+gb+sb)>/(<(F+ga+sa)2><(F+gb+sb)2>]½ (4)

whichreducesto,

<CC(F1a,F1b)>=1/(1+C+S) (5a)

Similarly,correlationbetweendensity-modifiedhalf-mapsaandbisgivenby:

<CC(F2a,F2b)>=(1+A)/(1+A+B+C) (5b)

Cross-correlationbetweenhalfmapaanddensity-modifiedhalfmapa(andalsobetween

correspondingmapsb)isgivenby:

<CC(F1a,F2a)>=(1+C)/[(1+C+S)(1+C+A+B)]½ (5c)

Cross-correlationbetweenhalfmapaanddensity-modifiedhalfmapb(andalsobetweenhalf

mapbanddensity-modifiedhalfmapa)isgivenby:

<CC(F1a,F2b)>=1/[(1+C+S)(1+C+A+B)]½ (5d)

Astherearefourrelationshipsandfourparametersdescribingerrors,therelationshipscanbe

usedtoestimatethevaluesoftheerrorsA,B,C,andS,leadingtotheformulas:

A=[CC(F1a,F1b)CC(F2a,F2b)/CC(F1a,F2b)2]-1 (6a)

B=[CC(F1a,F1b)-CC(F1a,F1b)CC(F2a,F2b)-CC(F1a,F2a)CC(F1a,F2b)+CC(F1a,F2b)2]/

CC(F1a,F2b)2 (6b)

C=[CC(F1a,F2a)/CC(F1a,F2b)]–1 (6c)

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Page 25: Improvement of cryo-EM maps by density modificationIn macromolecular crystallography, the amplitudes of Fourier coefficients are generally measured accurately and the phases are poorly

D=[1/CC(F1a,F1b)]–[CC(F1a,F2a)/CC(F1a,F2b)] (6d)

Asnotedinmoredetailbelow,forshellsathighresolutiontheuncertaintiesinthecorrelations

suchasCC(F1a,F1b)canbelargecomparedtothecorrelationsthemselves.Inthesesituationsthe

valuesofcorrelationsaresmoothedandadditionalassumptionsaremadeaboutthe

relationshipsamongtheerrorestimatesinorderreducethenumberofparametersthatneed

tobeobtainedfromthedataatthatresolution.

Afterestimationoferrors,allestimatesofFcanbeaveragedwithresolution-dependent

weightingfactorsw.Basedontheassumptionofequalmeansquareerrorsinmembersofa

pairofhalf-maps,theweightsoneachhalfmapinapairarealwaysequal.Therecombined

(density-modified)estimate(G)ofFisthengivenby,

G=wF1+(1-w)F2 (7a)

Wherewistheweightontheoriginalaveragedhalfmaps(F1)and(1-w)istheweighton

averageddensity-modifiedaveragedhalf-maps(F2)andtheaveragedmapsaregivenby:

F1=½(F1a+F1b) (7b)

F2=½(F2a+F2b) (7c)

TheweightwthatmaximizestheexpectedcorrelationoftheestimatedFouriercoefficient,G

withthetrueone,F,is:

w=(2A+B)/(2A+B+S) (8)

andtheestimatedcorrelationofGwithF,thecorrelationofthefinalestimateoftheFourier

coefficientwiththetrueone,represented18byFSCrefis:

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Page 26: Improvement of cryo-EM maps by density modificationIn macromolecular crystallography, the amplitudes of Fourier coefficients are generally measured accurately and the phases are poorly

FSCref=[2(A+½B+½S)/[A(C+S+2)+B(½C+½S+1)+(½C+1)S]]½ (9)

Asmentionedabove,assumptionsaremadetoallowestimationofcorrelatedanduncorrelated

errorsfromFSCplotsforresolutionshellswhereuncertaintiesincorrelationestimatesare

large.Theseadditionalassumptionsare:

A. Inthehighestresolutionshellconsideredthereisnoremainingsignalandall

correlationsofFouriercoefficientsareduetocorrelatederrors.Thelimitingresolutions

ofFSCplotsinthisanalysisaresetinsuchawaythatthereislittlesignalatthose

resolutionsandcorrelationsarelargelyduetonoiseandcorrelatederrors.FSCplotsare

calculatedinshellsofresolution(d).Thehighestresolutioninerroranalysesconsidered

(dmin)istheresolutionusedfordensitymodificationmultipliedbyafixedratio(typically

5/6).Astheresolutionusedfordensitymodificationisnormallyabout0.5-1Åfinerthan

thenominalresolutionoftheoverallmap,fora4Åmapthishigh-resolutiondminwould

typicallybeintherangeof2.5Åto3Å.

B. Inhigh-resolutionshellswherethereissubstantialuncertaintyintheestimatesoferrors

(typicallywherehalf-mapcorrelationsarelessthanabout0.05),ratiosofcorrelatedto

uncorrelatederrorsareassumedtobethesameasthoseestimatedinlower-resolution

shells.

C. Forshellsofresolution(d)wherethevaluesofFSCarebelowafixedminimumFSC,

(typicallyFSC_min=0.2),smoothedFSCvaluesarecalculatedbyfittingtheobservedvalues

toasimpleexponentialfunctionwithonefreeparameter.Thefunctionusedis

FSC=(FSC_d1-FSC_d_min)exp(-H/d2)+FSC_d_min.ThefreeparameterisH,thefall-offwith

1/d2).FSC_d1isthevalueofFSCatresolutiond1,thehighestresolutionwhereFSCis

higherthanafixedminimumvalue(typically0.2).FSC_d_ministheestimatedFSCatthe

highestresolutionintheanalysis.Asnotedaboveitisassumedthatanynon-zeroFSC

foundatthisresolutionisduetocorrelatederrorsintheanalysis.

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Page 27: Improvement of cryo-EM maps by density modificationIn macromolecular crystallography, the amplitudes of Fourier coefficients are generally measured accurately and the phases are poorly

Real-spaceweightingandweightingofindividualFouriercoefficientsincalculationofthefinal

map

Anoptionavailableattheendofacycleofdensitymodificationistoapplyalocalweighting

schemetothefinalcombinationoforiginalanddensity-modifiedmaps.Theideaistoidentify

localaccuracyintheoriginalmapfromlocalsimilaritybetweenoriginalhalf-maps,andalso

localaccuracyinthedensitymodifiedmapsfromlocalsimilaritybetweendensity-modifiedhalf

maps.Theprocedureforonepairofhalf-mapsistosubtractthemaps,squaretheresulting

map,andsmooththesquaredmapwithasmoothingradiustypicallygivenbytwicethe

resolutionofthemaptogivealocalvarianceforthosehalf-maps.Thenalocalweightforeach

setofhalf-mapsobtainedastheinverseofthelocalvarianceofthosehalf-maps.Theselocal

weightsarethenscaledtoyieldanaveragelocalweightofunityandthenareappliedtothe

individualhalf-mapsbeforetheyareaveraged.Asecondoptionforrecombinationofmapsis

toweightindividualFouriercoefficientsbasedontheestimatedvarianceforthesecoefficients.

ThevarianceforanindividualcoefficientisestimatedfromthefourFouriercoefficients

representingthefourhalf-mapsavailableattheendoftheprocedure(thetwooriginalhalf-

mapsandthetwodensity-modifiedhalf-maps).Thesetwoproceduresarenotappliedby

defaultbutcanimprovemapsinsomecases.

Overallspectralscaling

Theprocedureatthispointyieldsweights(w,1-w)forthetwopairsofhalf-mapsthatare

combinedtoyieldadensity-modifiedmap,andanestimateofthequalityofthemap

coefficientsinthisresolutionshell,(FSCref).Thiscalculationiscarriedoutinshellsofresolution

(typically100shells)andallowscalculationofFouriercoefficientsineachshellofresolution.A

finalresolution-dependentweighting(spectralscaling)isthenappliedtotheFourier

coefficients.Thereareseveraloptionsforthisfinalscaling.

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Page 28: Improvement of cryo-EM maps by density modificationIn macromolecular crystallography, the amplitudes of Fourier coefficients are generally measured accurately and the phases are poorly

Oneoption(thedefault,controlledbythekeywordfinal_scale_with_fsc_ref=True)beginsby

applyingascalefactor(Aj)foreachshelljtoyieldaconstantvalueofaverageamplitudesof

Fouriercoefficientsineachshell(essentiallycreatinganE-mapincrystallographicterminology;

notethatalthoughscalefactorsarecalculatedinshellsofresolution,theyareappliedasa

smoothfunctionofinverseresolution).Wenotethatthisprocedurehasasignificant

disadvantageinthattheFouriercoefficientsinthelowest-resolutionshellstypicallyhavevery

highvariationinmagnitudecomparedtothoseinhigher-resolutionshells,sothatnosingle

scalefactorreallycanbesuitable.Afterthisnormalizationstep,Fouriercoefficientsare

multipliedbythevalueofFSCrefinthatshell,correspondingtotheapproachoftenusedto

sharpenamapbasedonitscorrespondinghalf-maps18.

Asecondoptionforfinalscalingissimilartothefirst,exceptthatinthefirststepthescale

factorappliedisthesquarerootofthescalefactor(Aj)describedabove.Thisoptionisan

attempttoreducetheeffectofscalingonthelow-resolutioninformationandcanimprovethe

low-resolutionmap-modelcorrelationfoundattheconclusionofdensitymodificationinsome

cases.Thisprocedureiscontrolledbythekeywordgeometric_scale=Trueandisthedefaultif

final_scale_with_fsc_ref=False.

Athirdoption,onethatcanbecarriedoutaftereitherofthefirsttwo,istoapplyaresolution-

dependentscalefactortoallFouriercoefficientsthatcorrespondstotheresolution-

dependenceofacalculated“typical”proteinstructure(weusedthe2.2ÅmapEMD-2984ofβ-

galactosidasetocalculatethisresolutiondependence).Thisiscontrolledbythekeyword,

spectral_scaling=Trueandisnotcarriedoutbydefault.

Boxingofcryo-EMmaps

Inourprocedure,arectangularsolidportionofacryo-EMmapthatcontainsthe

macromoleculeofinterestiscutoutfromthemapandisusedintheanalysis.This“boxing”of

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Page 29: Improvement of cryo-EM maps by density modificationIn macromolecular crystallography, the amplitudes of Fourier coefficients are generally measured accurately and the phases are poorly

themapiscarriedoutwitha“soft”(Gaussian)maskwithasmoothingradiustypicallyequalto

theresolutionofthemaptoreducetheintroductionofcorrelationsinFouriercoefficients

betweendifferentmapsboxedinthesameway18.Theedgesoftheboxarechosenusing

boundsineachdirectionidentifiedusingalow-resolution(typically20Å)maskcalculatedfrom

thefullmapwithavolumebasedontheexpectedmolecularvolumeofthemacromolecule.

Typically,abufferof5Åisaddedtotheboundsineachdirectiontoyieldaboxthathas

dimensions10Åbiggerthanthesizeineachdirectionofthemacromolecule.

Therearetwoimportanteffectsofboxing.Oneistoreducethevariationofnoiseinthemapin

theregionoutsidethemacromolecule.Inatypicalcryo-EMmapthereissubstantialnoise

(fluctuationinmapvaluesnotrepresentingthemacromolecule)nearthemacromolecule,and

progressivelylessfurtherfromthemacromolecule(thevariationinnoiselevelsmayalsobe

morecomplicated).Thisvariationinnoiselevelsislargelyduetotheuseofproceduresthat

smoothFouriercoefficientsinreciprocalspacewiththeeffectofmaskingaroundthe

macromolecule30.Inourprocedureitisassumedthatthedistributionofmapvaluesinthe

regionoutsidethemacromoleculecanberepresentedbyasimplehistogram.Asthereisa

distance-dependentvariationinthelevelofnoiseinunboxedcryo-EMmaps,ourprocedurecan

bemademoreapplicablebyboxingthemaps.

ThesecondeffectofboxingthemapistoreducethecorrelationofFouriercoefficientsinthe

map.IfasmallobjectisplacedinalargeboxandFouriercoefficientsarecalculated

representingtheobjectinthebox,coefficientswithsimilarindices(neighboringFourier

coefficients)willbehighlycorrelated41,42.ThesignificanceofcorrelationsbetweenFourier

coefficientsisthaterrorsmaybecorrelatedaswell,resultinginmap-phasingFourier

coefficientsthatarenotfullyindependentfromtheoriginalFouriercoefficients.Boxingreduces

theemptyvolumeofthemapandreducesthiscorrelation.

Effectofmaskingindensitymodificationduetosolventflattening

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Page 30: Improvement of cryo-EM maps by density modificationIn macromolecular crystallography, the amplitudes of Fourier coefficients are generally measured accurately and the phases are poorly

Anindirecteffectofdensitymodificationismaskingofthemap.Densitymodificationincludes

astepinwhichnoiseinthesolventregionofthemapisreduced,andthisprocesscanleadtoa

mapthatispartiallymasked.Aconsequenceoffullmasking(settingvaluestoaconstantvalue

outsidethemask)isthatcorrelationswithaperfectmap,withamodel-basedmap,orbetween

half-maps16areincreasedrelativetoanunmaskedmap.Wecarriedoutatesttoevaluate

whetherthiseffectcontributesinasubstantialwaytotheimprovementinresolutionby

densitymodificationshowninFig.1B.ThetestconsistedofrepeatingtheanalysisinFig.1B

exceptthatinsteadofactuallycarryingoutdensitymodification,eachinitialhalf-mapwas

simplymultipliedbytheprobabilisticmaskthatwastobeusedinidentifyingtheproteinregion

fordensitymodification,andonlyonecyclewascarriedout(thiscanbedoneusingthe

keywordcontrol_no_denmodinthePhenixtoolresolve_cryo_em).InFig.1Btheresolutionat

whichthemap-modelFSCfallstoapproximately½changedwithdensitymodificationfrom2.98

Åto2.77Å.Inthetestanalysisusingjustmasking,thisvaluechangedbyonly0.01Å(from2.98

Åto2.97Å),indicatingthatthemaskingeffectisverysmallinthiscase.

Resolutioncutoffusedfordensitymodification.

Inordertoincludeinformationathighresolution,theresolution(d_dm)ofFouriercoefficients

usedinthedensitymodificationprocedureistypicallyfinerthantheresolutionofthefullmap.

Therelationshipbetweentheresolutiondofamapandtheoptimalresolutiond_dmfordensity

modificationisnotobvious,soweusedananalysisof51half-mapsfromtheEMDBand

associatedmodelsfromthePDBtodevelopanempiricalrelationship.Theempiricalfunction

wasobtainedbycarryingouttheentiredensitymodificationprocedureforeachdataset,each

witharangeofvaluesofd_dm.ThentheaverageFSCbetweenmapandmodel-basedmapwas

calculatedforeachanalysisandasimplefunctionwasdevelopedforchoosingtheresolutions

wheredensitymodificationwasoptimal.Thisfunction,developedforresolutionsbetween2.4

Åand5Å,was:

d_dm=2.4+0.99(d-3)-0.2(d-3)2 (10)

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Page 31: Improvement of cryo-EM maps by density modificationIn macromolecular crystallography, the amplitudes of Fourier coefficients are generally measured accurately and the phases are poorly

Ataresolutiond=2.4Å,thisyieldsd_dm=1.9.Formapswithresolutionfinerthan2.4Å,we

simplysubtract0.5Åfromtheresolutiontoyieldd_dm,exceptthatd_dmisneverallowedtobe

lessthan½d.Forresolutionsgreaterthan5Åtheoptimalresolutionfordensitymodification

hasnotbeenidentified.InFig.2Eq.(10)isused.

Threeoptionsforchoiceofresolutioncutofffordensitymodificationd_dmareavailableinthe

currentimplementationofdensitymodification.Oneisdirectlyspecifyingd_dm,oneisusingEq.

10toestimated_dm,andthelastistotryvariousvaluesofd_dmandchoosetheonethatleadsto

themostfavorableestimatedimprovementintheresolutionwhereFSCrefis½basedonEq.9.

Adjustableparameters

Therearemanyadjustableparametersinourprocedurebutallhavedefaultvalues(default

valuesforversion3689ofPhenixwereusedinall104testsinFig.2).Someoftheparameters

thatcansubstantiallyaffecttheresultsandthatausermightvaryiftheinitialresultsarenot

optimalarelistedhere.Theresolutionusedfordensitymodificationisnotfullyoptimizedand

canaffecttheoutcomesubstantially.Additionalcyclesofdensitymodificationcanimprovethe

outcomeinmanycases.Thechoiceoffinalscalingprocedure(final_scale_with_fsc_ref)can

affectthedensity-modifiedmap,ascanthechoiceofhalf-mapsharpeningatthestartof

densitymodification(density_modify_unsharpened_maps).Thenumberofshellsofresolution

usedinthecalculationofcorrelationsbetweenFouriercoefficientshasadefaultof100;more

shellscanpotentiallyimprovetheaccuracybynotgroupingcoefficientsthathaveverydifferent

valuessimplyduetoresolution-dependentvariationbutcouldreduceitduetofewer

coefficientsinacalculation.Theoptionaluseofspectralscaling,real-spaceweighting,and

individualweightingofFouriercoefficientsattheendoftheprocedurecansometimesaffect

theresultingmap.

Histogram-matching

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Page 32: Improvement of cryo-EM maps by density modificationIn macromolecular crystallography, the amplitudes of Fourier coefficients are generally measured accurately and the phases are poorly

Weexaminedwhethermapimprovementcanalsobeobtainedbyusinghistogram-matching43

withoutdensitymodification.Withhistogram-matchingalone,anewmapvalueateachgrid

pointisobtainedbasedonthedistributionsofexpectedandobservedvaluesinthemapand

thevalueatthatgridpoint.Withhistogram-matchingasapartofdensitymodification,this

histogram-matchedmapisnotuseddirectlyasthenewmap,butratheritisusedasa

likelihoodtargetthatindicatestheplausibilityofacandidatemap.Thisallowsinformation

fromallpartsofthemaptobecombinedtoyieldnewestimatesofeachFouriercoefficient.

Totestwhetherthemapimprovementobtainedwithdensitymodificationcouldbeobtained

usingreal-spacemethodsalone,weappliedhistogram-matching43tothe1.8Åapoferritinmap

showninFig.1C.SupplementaryFig.S2illustratesthatourhistogram-matchingapproach

improvesthismap,butdensitymodificationimprovesthismapconsiderablymore.PanelA

showstheaverageofdepositedhalf-maps,sharpenedtomatchdensity-modifiedmapinpanel

C.PanelBshowsthehistogram-matchedversionofthemapinA,andpanelCshowsthe

density-modifiedmap.NotetheconsiderableimprovementinclarityintheringofF81inthe

density-modifiedmap.

Effectsofdensitymodificationonindividualhalf-maps

SupplementaryFig.S3showstheindividualoriginalhalf-mapsandthetwodensity-modified

halfmapsproducedbythe1.8Åapoferritindensitymodificationexampleshownin

SupplementaryFig.S2.Itcanbeseenthattheoriginalhalfmapshavesimilarclaritybutare

different,andthatbothhalfmapsareimprovedbydensitymodificationandareagain

somewhatdifferent.

Sensitivityofdensitymodificationtoparameters

Wetestedthesensitivityofdensitymodificationtotwoparametersthatseemedlikelytohave

substantialeffectsontheprocedure,theboxsize,andtheresolutionusedfordensity

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Page 33: Improvement of cryo-EM maps by density modificationIn macromolecular crystallography, the amplitudes of Fourier coefficients are generally measured accurately and the phases are poorly

modification.Ineachcaseweusedthe3.1ÅapoferritinmapanalyzedinFig.1.Supplementary

Fig.S4showstheeffectofvaryingtheresolutionusedfordensitymodificationonthequalityof

densitymodification,asmeasuredbytheestimatedimprovementinresolutionandactual

improvement(changeintheresolutionwherethemap-modelFSCwasapproximately½).Itcan

beseenthatthedensitymodificationresolutionhasasmalleffectontheactualimprovement

overtherangeof2Å–2.6Åbut,asexpected,theimprovementbecomessmallerasthe

densitymodificationresolutionapproachesthenominalresolutionofthemap.Theestimated

resolutionhasmorevariabilitybutasimilarrelationship.

Weinvestigatedtheeffectofboxsizebycarryingoutdensitymodificationusingdifferentsize

boxestoextractthemolecule,rangingfromaboxjustthesizeofthemolecule(definedasthe

regionwithexpectedvolumewheredensityishighest),toaboxwithedges30Åbiggerineach

direction(thedefaultisabox10Åbiggerthanthemolecule).Overthisrangeofboxsizesthe

resultingresolutionwheretheFSCtothemodelequaled½variedbyasmallamount(ranging

from2.76Åto2.83Åforallboxesexceptthesmallestonewhichhadavalueof2.92Å),and

withthebestresolutionof2.76Åwithabox4Åbiggerthanthesmallestone.Thisindicates

thatatleastinthiscasetheexactboxsizeisnotveryimportantbutthatasmallboxmaybe

slightlybetterthanabiggerone.

Changesinmap-modelmetricsafterdensitymodification

Weexaminedwhethercommonmodelandmap-modelmetricsweresubstantiallydifferent

usingdensity-modifiedmapscomparedtotheoriginalmaps.SupplementaryFig.5showsthat

map-modelcorrelation(calculatedataresolutiongivenby0.83timesthenominalresolutionof

thestartingmaptoemphasizehigh-resolutioninformation)generallyimproves,butthatthe

othermetricsexamined(rotameroutliers,ClashScore44,Ramachandran%favored,EMRinger45

scores)didnotchangeinasystematicwayovertheresolutionrangeof2Å–4.5Åwhereour

methodisdesignedtoapply.

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Page 34: Improvement of cryo-EM maps by density modificationIn macromolecular crystallography, the amplitudes of Fourier coefficients are generally measured accurately and the phases are poorly

SupplementaryFig.S6showsthelocalmap-modelcorrelationforEMD-7544andassociated

PDBentry6coy46beforeandafterdensitymodification.Thedensitymodifiedmaphas

generallyhighermap-modelcorrelationbutthedifferencevariessomewhatbylocationalong

thechain.

Effect of applying multiple cycles of density modification

Supplementary Fig. S7 illustrates the effect of multiple cycles of density modification on the average

map-model FSC obtained. Panel A corresponds to Fig. 2B after 5 cycles of density modification.

Panel B compares the average map-model FSC after one cycle with the same metric after 5 cycles.

It can be seen that additional cycles lead to a substantial improvement in a number of cases, while in

many other cases there is little effect after additional cycles, and in 2 cases there is substantial

worsening.

Softwareavailability

AlltheproceduresdescribedinthisworkareavailableusingthePhenixtoolresolve_cryo_emin

versions3689andlaterofthePhenixsoftwaresuite20.

Acknowledgements

ThisworkwassupportedbytheNIH(grantGM063210toPDA,RJRandTTandgrantR01-

GM080139toSJL),theWellcomeTrust(grant20947/Z/17/ZtoRJR),andthePhenixIndustrial

Consortium.ThisworkwassupportedinpartbytheUSDepartmentofEnergyunderContract

No.DE-AC02-05CH11231atLawrenceBerkeleyNationalLaboratory.

Authorcontributions

SLcarriedoutimageprocessingoftestdatasetstoevaluatevaryingreconstructionprocedures,

RJRandTCTcontributedideasontheformoferrorsincryo-EM,PAdevelopedtoolsforthe

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Page 35: Improvement of cryo-EM maps by density modificationIn macromolecular crystallography, the amplitudes of Fourier coefficients are generally measured accurately and the phases are poorly

testinginfrastructure,TCTdevelopedthesoftwareforerroranalysis,andPDAandTCT

supervisedthework.

Authorinformation

Theauthorsdeclarenocompetingfinancialinterests.

Correspondenceandrequestsformaterialsshouldbeaddressedto:

[email protected].

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SupplementaryFigures

SupplementaryFig.S1.Analysisofnon-density-modifiedmapstoexaminewhethersimplemodifications(sharpening,usingdepositedmaps)wouldyieldmapsthatappearsimilartothedensity-modifiedmaps.A.DepositedmapshowingthesameregionasdepictedinFig.2C.Thismapalsoshowspoordensityfortheloop.VaryingthesharpeningofthemapdidnotyieldconnecteddensityasinFig.2D.B.SharpenedversionoftheoriginalmapshowninFig.2E.

SupplementaryFig.S2.Histogrammatchingofemd-20026.A.Averageofhalf-maps,sharpenedtomatchdensity-modifiedmapinC.B,histogram-matchedversionofmapinA,sharpenedasinA.C,Density-modifiedmap.Allcontourssettoencloseequalvolumes.Mapsmaskedaroundatomsinmodel.

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SupplementaryFig.S3.Comparisonofhalf-maps1and2fordensitymodificationshowninFig.S2.Allmapsmaskedandsharpenedtomatchdensity-modifiedmapinFig.S2C.AandC,halfmaps1and2.BandD,density-modifiedhalf-maps1and2.

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SupplementaryFig.S4.EffectofdensitymodificationresolutiononactualandestimatedchangeinresolutionwhereFSCis½.Half-datasetmapsforEMD-20028weredensitymodifiedasinFig.1,butvaryingtheresolutionfordensitymodificationfrom2to2.9Å.Theestimatechangeinresolution(calculatedfromtheerroranalysis)andtheactualchangeinresolution(calculatedfromtheFSCtothehigh-resolutionEMD-20026map)areshown.

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SupplementaryFig.S5.Changeinmodel/mapmetricsafterdensitymodification.Analysesareplottedasafunctionofresolutionforthe104datasetsshowninFig.2.A.Map-modelcorrelation,B.Rotameroutliers,C.ClashScore,D.Ramachandranpercentageinfavoredregion,E,EMRingerscores.

SupplementaryFig.6.Map-modelresiduecorrelationsmoothedin10-residuewindowsforEMD-7544(PDBentry6coy).

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SupplementaryFig.7.Effectofapplying5cyclesofdensitymodificationvs1.A.Averagemap-modelFSCafter5cyclesofdensitymodification(comparewithFig.2B).B.Changeinaveragemap-modelFSCbetweenonecycleofdensitymodificationand5cycles.

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