monte carlo methods for simulation of protein folding and...

MonteCarloMethodsfor Simulationof ProteinFolding andTitration

Inauguraldissertationzur ErlangungdesGrades

einesDoktorsderNaturwissenschaften

beidemFachbereichsBiologie– Chemie– Pharmazie

derFreienUniversitatBerlin

vorgelegt von

Bjorn RabensteinDiplom-BiochemikerausBerlin

Berlin, August31,2000

1. Gutachter:Prof.Dr. E.W.Knapp,FreieUniversitat Berlin

2. Gutachter:Dr. H. Sklenar, Max-Delbruck-Centrum,Berlin

TagdermundlichenPrufung: 25. Oktober2000

Preface

Foldingandtitration of proteinsarethetwo majortopicsof this work. Seeminglyunconnected,theyawait their unificationat thevery endof the lastchapter. Anotherhiddenconnectionis theconceptof continuumelectrostatics,which is usedfor both. Therefore,I summarizesomeaspectsaboutthemethodsof continuumelectrostaticsin the first chapterincluding a brief derivation of themostimportantequations,hopingthat this will give thereadera first aid for understandingtheunderlyingprinciples.Thefirst chapteris merelyatoolboxusedin laterchapters,wheremy actualresearchworkis presented.

Thesecondchapterdealswith thelong-termdynamicsof proteins,whichmeansmainly to tacklethe folding problem. This is one of the two greatchallengesof doing computersimulationsofbiomolecules. (The other is to understandthe function of enzymesin detail by combiningquan-tum mechanicsandmolecularmechanics.)A Monte Carlo methodfor the long-termdynamicsofproteinshasbeendevelopedby theKnappgroupfor many years.My own contribution reportedinchapter2 is thefurther improvementof thefolding of peptidesin vacuoandfinally theadditionof asolventmodel.

Thethird chaptercomprisesthemajorpartof my work. Titratingproteinsin thecomputermeansto calculatetheprotonationstateof all titratablesiteswithin aproteinfor differentpH values.Redoxactive sitescan be treatedby very similar methodsandare thereforeincludedin the calculations.I appliedthe electrostaticmethodsfor the calculationof titration behavior successfullyto bacterialphotosyntheticreactioncentersandto myoglobin.Theresultsdid notonly show thereliability of themethodsby reproducingexperimentalmeasurements,but providedalsonew insightsinto thestudiedprocessesand enablednew interpretationsof experimentalfindings that were previously not wellunderstood.Theseresultsareanexamplethatnowadaystheoryin biosciencecandomuchmorethanmerelyreproducingresultseverybodyhasknown before.

Sciencewithout collaborationis impossible. And so I am especiallyhappy to presentthe fol-lowing long list of acknowledgmentsfor all thosehelpful peopleandinstitutionswho supportedmeduringmy work. At first, thereis Ernst-WalterKnapp,whowasanexcellentDoktorvaterfor me,andhisgroup,whereI foundperfectworkingconditionsandplentyof inspiringdiscussionthroughoutallfieldsof science,from the foggy swampsof biology to thewindy crestsof mathematics.Specially,I am grateful to G. MatthiasUllmann (now working at the ScrippsInstitute) for his collaborationin thecalculationof electron-transferandprotonationreactionsof thebacterialphotosyntheticreac-tion center(seechapter3) and to Bjorn Kleier for the collaborationin preparingenergy functionsfor CAMLAB++ (seechapter2). The KARLSBERG program(seeappendixE) would not exist, ifBenediktDietrich hadnot introducedme to C++. It wasa pleasureto take part in otherprojectsofthe group,especiallyPeterVagedes’work on the deacylation stepof acetylcholinesterase, DraganPopovic’s work on thecomputermodelingof artificial cytochromeb, andTimm Essigke’s studyonusingCAMLAB++ for solvingNMR structures.To take carefor thetechnicalbasisof our scientificwork, a runningnetwork of computers,wasa rewarding, informative, andmostly enjoyable taskIwasallowedto do togetherwith BerndMelchers,Timm Essigke andDanielWinkelmann.I hadthehonorto contributesometheoreticalpartsto theexperimentalwork of Jorg WischhusenandIoakim

I

II

Spyridopoulosat theuniversityof Tubingen.I got financialsupportfrom theDeutscheForschungs-gemeinschaft,Sfb 312 and Sfb 498. I was allowed to usenumerouscomputerprogramsfree ofcharge: Specially, I thankDonaldBashfordfor providing MEAD andDanielHoffmannfor providingCAMLAB. The typesettingof this thesiswasdoneby LATEX. Moleculargraphicsaredrawn withMOLSCRIPT. PlotsweredoneusingXMGR. Mostof thefigureswerepreparedor postprocessedwithXFIG.

At last, let me appreciatea changeof paradigmthat has taken placeduring my work in theKnappgroupandis sometimescalledthe“opensourcerevolution”. Thanksto theGNU projectandexpeciallytheUnix-likeoperatingsystemLinux, wearenow ableto turncheapoff-the-shelfpersonalcomputersfreeof charge into all-roundprofessionalworkstations,well equippedandstableenoughto run our programsfor weeksandmonths.We arenow goingto assemblelots of themto a massiveparallel supercomputer, not more expensive than one or two of the workstationswe usedwhen IjoinedtheKnappgroupseveralyearsago.Theopensourceparadigmis deeplyscientificin its desireto shareinformation,ideasandimprovements.Thesuccessof opensourcesoftwaremadetheworlda little bit betterby makingit a little bit morescientific.Sciencebenefitsfrom opensourcesoftware,but by doingsoit shouldalsofeelexhortednot to forgetthevirtue of its own openprinciples.

BJORN RABENSTEIN

FreieUniversitat BerlinJuly 2000

Contents

1 An intr oduction to continuum electrostatics 11.1 ThePoisson-Boltzmannequation. . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.1.1 Derivation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.1.2 Numericalsolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.1.2.1 Finite differencemethod. . . . . . . . . . . . . . . . . . . . . . . 31.1.2.2 Focusing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.1.2.3 Grid artifact . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.2 TheAnalytical ContinuumSolvent(ACS) . . . . . . . . . . . . . . . . . . . . . . . 51.2.1 Integral formulationof theelectrostaticenergy . . . . . . . . . . . . . . . . 51.2.2 Solvationenergy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.2.3 GeneralizedBornapproximation. . . . . . . . . . . . . . . . . . . . . . . . 81.2.4 Pairwiseself energy potential . . . . . . . . . . . . . . . . . . . . . . . . . 91.2.5 Atomic chargeandvolumerepresentation. . . . . . . . . . . . . . . . . . . 101.2.6 Evaluationof theself energy . . . . . . . . . . . . . . . . . . . . . . . . . . 101.2.7 Non-polarsolvationterm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2 Folding 132.1 An off-lattice MonteCarlomethodfor proteinfolding . . . . . . . . . . . . . . . . . 13

2.1.1 Theproteinmodel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.1.2 Tuningtheenergy function . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.1.3 Thewindow move . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.1.4 TheCAMLAB program . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.2 Folding in vacuo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.2.1 To beinvestigated:theAGA andVGV modelpeptides . . . . . . . . . . . . 172.2.2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.2.2.1 Determinationof secondarystructure. . . . . . . . . . . . . . . . 172.2.2.2 Treatmentof sidechains. . . . . . . . . . . . . . . . . . . . . . . 192.2.2.3 Mixing window movesandsimplemoves . . . . . . . . . . . . . 202.2.2.4 Parametersof theMC simulation . . . . . . . . . . . . . . . . . . 20

2.2.3 ResultsandDiscussion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.2.3.1 Investigatingthetrajectories. . . . . . . . . . . . . . . . . . . . . 212.2.3.2 Loosingof timescales. . . . . . . . . . . . . . . . . . . . . . . . 242.2.3.3 Possiblesolventandvacuumeffects . . . . . . . . . . . . . . . . 24

2.3 Folding in solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.3.1 To beinvestigated:theC-terminalhelix of RNaseA andthesyntheticpeptide

BH8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.3.2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.3.3 ResultsandDiscussion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.3.3.1 18-alanine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

III

IV CONTENTS

2.3.3.2 PeptideRN24 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272.3.3.3 PeptideBH8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3 Titration 313.1 Titrationof asingle,rigid proteinstructure. . . . . . . . . . . . . . . . . . . . . . . 31

3.1.1 To beinvestigated:bacterialphotosyntheticreactioncenters . . . . . . . . . 313.1.2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.1.2.1 Preparationof structures. . . . . . . . . . . . . . . . . . . . . . . 333.1.2.2 Atomic partialcharges. . . . . . . . . . . . . . . . . . . . . . . . 353.1.2.3 Theenergy of aprotonationstate . . . . . . . . . . . . . . . . . . 353.1.2.4 Calculationof protonationprobabilitiesby MC titration . . . . . . 413.1.2.5 Energeticsof electrontransferandprotonationof thequinones . . 43

3.1.3 ResultsandDiscussion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443.1.3.1 Totalprotonationandprotonationpatterns . . . . . . . . . . . . . 443.1.3.2 FirstElectronTransferfrom QA to QB. . . . . . . . . . . . . . . . 483.1.3.3 SecondElectronTransferfrom QA to QB andfirst protonationof QB. 513.1.3.4 SecondProtonationof QB. . . . . . . . . . . . . . . . . . . . . . 523.1.3.5 Sequenceof electron-transferandprotonationreactions . . . . . . 533.1.3.6 Conformationalgating. . . . . . . . . . . . . . . . . . . . . . . . 53

3.2 Conformationalrelaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 583.2.1 To beinvestigated:againthebRCfrom Rps.viridis . . . . . . . . . . . . . . 583.2.2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

3.2.2.1 Structuralrelaxation . . . . . . . . . . . . . . . . . . . . . . . . . 583.2.2.2 Calculatingtheenergy of electrontransferfor the relaxed confor-

mations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 593.2.3 ResultsandDiscussion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

3.2.3.1 ImprovedMC samplingwith triple moves . . . . . . . . . . . . . 623.2.3.2 Relaxedselfconsistentstructures . . . . . . . . . . . . . . . . . . 623.2.3.3 Totalprotonationandindividual siteprotonation . . . . . . . . . . 633.2.3.4 Energeticsof theelectrontransfers . . . . . . . . . . . . . . . . . 643.2.3.5 Thedielectricconstantwithin theprotein . . . . . . . . . . . . . . 65

3.2.4 Concludingremarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 663.3 Titrationof anensembleof proteinconformations. . . . . . . . . . . . . . . . . . . 66

3.3.1 To beinvestigated:carbonmonoxymyoglobin . . . . . . . . . . . . . . . . . 663.3.2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

3.3.2.1 Coordinatesandparameters. . . . . . . . . . . . . . . . . . . . . 683.3.2.2 Titrationof singleconformers. . . . . . . . . . . . . . . . . . . . 693.3.2.3 Titrationof aconformationalensemble. . . . . . . . . . . . . . . 693.3.2.4 Paralleltempering . . . . . . . . . . . . . . . . . . . . . . . . . . 703.3.2.5 Conformationalreferenceenergy . . . . . . . . . . . . . . . . . . 71

3.3.3 ResultsandDiscussion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 723.3.3.1 Populationof theconformers . . . . . . . . . . . . . . . . . . . . 723.3.3.2 Titrationbehavior of individual sites . . . . . . . . . . . . . . . . 743.3.3.3 ThestronglycoupledpairHis-24/His-119 . . . . . . . . . . . . . 753.3.3.4 Comparisonof calculatedpKa valueswith experimentalvaluesand

othertheoreticalstudies . . . . . . . . . . . . . . . . . . . . . . . 753.3.3.5 Implicationsfor theassignmentof taxonomicsubstates . . . . . . 78

CONTENTS V

4 Outlook 794.1 Improving theMonteCarlofolding . . . . . . . . . . . . . . . . . . . . . . . . . . . 794.2 Improving theMonteCarlotitration . . . . . . . . . . . . . . . . . . . . . . . . . . 80

4.2.1 ThegeneralizedMC methodof PaulBeroza. . . . . . . . . . . . . . . . . . 804.2.2 CombiningMonteCarlodynamicsandMonteCarlotitration . . . . . . . . . 81

A Abstract 83

B Zusammenfassung(German abstract) 85

C Glossary 87

D CAM L AB++ Manual 89D.1 Commands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

D.1.1 Subcommandsof read energy terms . . . . . . . . . . . . . . . . . . . 89D.1.1.1 lj pair . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89D.1.1.2 noe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89D.1.1.3 quadratic repulsion . . . . . . . . . . . . . . . . . . . . . 90D.1.1.4 sartori . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

D.1.2 Subcommandsof read move . . . . . . . . . . . . . . . . . . . . . . . . . 92D.1.2.1 dihedral window . . . . . . . . . . . . . . . . . . . . . . . . 92

D.2 Utilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94D.2.1 Generationof Input Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

D.2.1.1 MakeSartoriPot entia ls . . . . . . . . . . . . . . . . . . . 94D.2.1.2 MakeSartoriEte rms . . . . . . . . . . . . . . . . . . . . . . 94D.2.1.3 MakePivot . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94D.2.1.4 MakeDihedrals . . . . . . . . . . . . . . . . . . . . . . . . . 95

D.2.2 Processingof OutputData . . . . . . . . . . . . . . . . . . . . . . . . . . . 95D.2.2.1 clt2dcd . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

E K ARL SBERG Manual 97E.1 Whatis KARLSBERG? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97E.2 Installation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97E.3 RunningKARLSBERG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

E.3.1 Samplefiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98E.3.2 Syntaxof input to stdin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98E.3.3 Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

E.4 Postprocessingof output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101E.4.1 make curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101E.4.2 make protonation table . . . . . . . . . . . . . . . . . . . . . . . . 101

F Curriculum vitae of the author 103

Bibliography 109

VI CONTENTS

List of Figures

1.1 A moleculein aheterogenousdielectricmedium. . . . . . . . . . . . . . . . . . . . 11.2 Partof thegrid usedto solve thePBE. . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.1 A singleamideplaneof theproteinmodel.. . . . . . . . . . . . . . . . . . . . . . . 142.2

�φ � ψ � torsionpotentialsfor alanineandglycine. . . . . . . . . . . . . . . . . . . . . 15

2.3 Sketchof awindow of threeamideplanes.. . . . . . . . . . . . . . . . . . . . . . . 162.4 Areasof secondarystructuremotifs in theRamachandranplot. . . . . . . . . . . . . 192.5 Snapshotsfrom theMC dynamicsof theAGA modelpeptide. . . . . . . . . . . . . 222.6 Snapshotsfrom theMC dynamicsof theVGV modelpeptide. . . . . . . . . . . . . 232.7 Helix formationandmeltingof 18-alanine. . . . . . . . . . . . . . . . . . . . . . . 262.8 Lowestenergy structureof thepeptideRN24. . . . . . . . . . . . . . . . . . . . . . 272.9 Lowestenergy structureof thepeptideBH8. . . . . . . . . . . . . . . . . . . . . . . 29

3.1 Cartoonof thebRC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323.2 QB bindingsiteof differentbRCstructures. . . . . . . . . . . . . . . . . . . . . . . 343.3 Atom namesusedfor MQ andUQ in Tables3.1and3.2. . . . . . . . . . . . . . . . 353.4 Atom namesusedfor bacteriochlorophyll-b andbacteriopheophytin-b in Table3.4. . 383.5 Treatmentof stronglycoupledsitesin MonteCarlotitration. . . . . . . . . . . . . . 423.6 Schemeof thepossibleelectron-transferandprotonationreactions . . . . . . . . . . 493.7 Possibleelectrontransferreactionsandconformationalgatingin thebRC. . . . . . . 543.8 Overview of thecompleterelaxationprocedure.. . . . . . . . . . . . . . . . . . . . 603.9 Thermodynamiccycle to calculatetheconformationalenergy. . . . . . . . . . . . . 613.10 Heme,CO ligand,andsidechainsof His-64andHis-93from the threeMbCO x-ray

structuresat pH 4, 5, and6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 673.11 Schemeof anMC simulationwith paralleltempering.. . . . . . . . . . . . . . . . . 713.12 Calculatedpopulationprobabilitiesof thedifferentconformersin thepH rangefrom

3 to 7. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 733.13 Populationprobabilityof His-64 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 743.14 Protonationprobabilityof His-81andHis-82. . . . . . . . . . . . . . . . . . . . . . 753.15 Structuralrepresentationof theof thestronglycoupledpairHis-24/His-119.. . . . . 763.16 Populationprobabilitiesof thethreepossiblestatesof thestronglycoupledpair His-

24/His-119. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

VII

VIII LIST OF FIGURES

List of Tables

2.1 Amino acidpropensitiesfor helix, strandandturn. . . . . . . . . . . . . . . . . . . . 18

3.1 Atomic partialchargesfor menaquinone.. . . . . . . . . . . . . . . . . . . . . . . . 363.2 Atomic partialchargesfor ubiquinone.. . . . . . . . . . . . . . . . . . . . . . . . . 373.3 Partial chargesof theiron center. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383.4 Atomic partialchargesfor bacteriochlorophyll-b andbacteriopheophytin-b. . . . . . 393.5 Groupsconsideredastitratablewith thepKa valuesof their modelcompounds. . . . 403.6 Totalprotonationandsomesingle-siteprotonationsatpH 7.5. . . . . . . . . . . . . 453.7 Summaryof the computedresultsat pH7.0 for the dark-adaptedandlight-exposed

x-ray structures.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473.8 Examplesof residueswith MC samplingproblems. . . . . . . . . . . . . . . . . . . 623.9 Rootmeansquaredeviationsof therelaxedstructures.. . . . . . . . . . . . . . . . . 623.10 Totalprotonationof thebRCandindividual siteprotonations.. . . . . . . . . . . . . 633.11 Atomic partialchargesof titratablegroups. . . . . . . . . . . . . . . . . . . . . . . 693.12 Rootmeansquaredeviationsin A for thefivex-ray structuresof MbCO. . . . . . . . 733.13 ExperimentalandcalculatedpKa valuesof histidinesin MbCO. . . . . . . . . . . . 77

4.1 Comparisonof differentmethodsfor titrationwith conformationalflexibility. . . . . 81

IX

X LIST OF TABLES

Appendix A

Abstract

The presentwork is divided into two parts: The first part presentsthe folding of peptidesusingaMonte Carlo methodfor the long-termdynamicsof proteins. The secondpart containsstudiesofthetitration andredoxbehavior of thebacterialphotosyntheticreactioncenterandof myoglobin.Bycombiningbothparts,I suggesta new methodto calculatetitration behavior takinginto accountfullconformationalflexibility.

The Knappgrouphasbeenworking on a Monte Carlo methodfor the long-termdynamicsofproteinsfor a long time. I improved this methodfor the first part of this work. After that, it waspossibleto fold not only a modelpeptideto a helix–turn–helixmotif, asalreadydonebefore,butalsoto simulatethe folding of a β-hairpin. However, so far thesecalculationsall lacked a solventmodel.Therefore,theAnalytical ContinuumSolventmodelof MichaelSchaeferwasintegratedintoourmethod.Usingthismodel,theformationandmeltingof apolyalaninehelix couldbesimulatedinagreementwith resultsfrom a moleculardynamicssimulationwith anexplicit solventmodel. Also,thefolding of a fragmentof ribonucleaseA wassuccessful,but thefolding of theβ-hairpinformingpeptideBH8 failed. This unveiledsofar unknown problemsof our proteinmodelwith rigid peptideplanesandspecialeffective torsionpotentialsfor thebackbonetorsionangles.The folding of BH8wassuccessfulusinga fully flexible proteinmodel.However, usingtheflexible modelthefolding ismuchlessefficient asexpectedfor therigid model.

Thecalculationof protonationandredoxstatesof titratableandredox-active groupsof proteinsis doneon the basisof continuumelectrostatics,similar to the solvent model usedin the foldingsimulationdescribedabove. In continuumelectrostatics,the solvent is representedby a mediumof a high dielectricconstant.By solvingthePoisson-Boltzmannequationnumericallyon a grid, theresultingelectrostaticpotentials(andthusalsotheinteractionenergies)canbecalculated.In thisway,the total electrostaticenergy of a specificprotonationandredoxstateof a molecularsystemcanbedetermined.A proteincontainsusuallyalot of titratableor redox-activegroups.For n of suchgroups,thetotal numberof possiblestatesis extremelylarge(2n) sothatthecalculationof thermodynamicalaverages,whicharenecessaryto determinetitrationcurvesandredoxpotentials,by exactsummationis computationallyinfeasible.Therefore,I applyin thepresentwork aMonteCarlomethodwhereanimportancesamplingof thehugenumberof possiblestatesis performedaccordingto theMetropoliscriterion. Themethodyieldsgoodresultsafter relatively shortsamplingtimes. Thestatisticalerrorof theseresultscanbeprobedby evaluatingacorrelationfunction.

By applyingthesemethods,I couldgaina lot of valuableinsights.For thefirst time,theenergy oftheelectrontransferfrom Q� �A to QB in thebacterialreactioncenterwascalculatedcorrectly. Thecal-culationsalsoyield importanthintsfor thesofarunsettledsequenceof thefollowing electrontransferandprotonationeventsinvolving thequinonesof thereactioncenter. In addition,theconformationalgatinghypothesisof the electrontransferfrom Q� �A to QB could be supportedfrom the viewpointof theory, and the insight into the relatedprocesseswasdeepened.I presentin this work several

83

84 APPENDIXA. ABSTRACT

approachesto includeconformationalensemblesandconformationalflexibility in thecalculationoftitration behavior. By explicit considerationof conformationalrelaxation,thedielectricconstantoftheproteininteriorcouldbedecreasedremarkablyin accordancewith fundamentalprinciples.By in-cludinganensembleof x-raystructures,thepH inducedconformationalchangesof myoglobincouldbe reproduced.The pKa valuescalculatedin this studyare in betteragreementwith experimentalvaluesthanany otherresultobtainedby non-trivial methodsbefore.

In the concludingoutlook, I discussthe future improvementsof the Monte Carlo methodforthe long-termdynamicsof proteins,andthe assetsanddrawbacksof existing methodsto calculatetitration behavior consideringconformationalflexibility. I show how to avoid mostof thedrawbacksby a new methodon the basisof a combinationof the Monte Carlo dynamicswith the titrationcalculation.Thismethodrealizestheunrestrictedandunbiasedsamplingof theconformationalspaceandthespaceof titration statesat thesametime.

Appendix B

Zusammenfassung(German abstract)

Die vorliegendeArbeit gliedertsich in zwei Teile. Der ersteTeil behandeltdie Faltungvon Pepti-denmit einerMonte-Carlo-Methodezur Langzeitdynamikvon Proteinen. Der zweiteTeil enthaltStudienzumTitrations-undRedoxverhaltendesbakteriellenphotosynthetischenReaktionszentrumsund von Myoglobin. Durch die KombinationbeiderTeile skizziereich am Schlußder Arbeit eineneueMethodezur Titrationsberechnungvon Proteinenmit vollstandigerKonformationsflexibilit at.

Die ArbeitsgruppeKnapparbeitetbereitsseit langerZeit anderEntwicklungeinerMonte-Carlo-Methodezur Langzeitdynamikvon Proteinen. DieseMethodewurde von mir fur den erstenTeilmeinerArbeit weiterentwickelt. Es gelangdaraufhinnicht nur, wie bereitszuvor, ein ModellpeptidzueinemHelix–Turn–Helix-Motif zufalten,sondernauchdieFaltungeinesβ-Hairpinszusimulieren.DieseBerechnungenbeinhaltetenjedochnochkein Modell fur dasLosungsmittel.In einemweite-ren Schritt wurdedaherdasAnalytical ContinuumSolvent-Modell von Michael Schaeferin unsereMethodeintegriert. Mit diesemModell konntederAufbauunddasSchmelzeneinerα-Helix ausPo-lyalanin in Ubereinstimmungmit ErgebnisseneinerMolekulardynamikmit explizitem Losungsmit-telmodellsimuliertwerden.Auch die FaltungeinesFragmentesderRibonucleaseA war erfolgreich,wohingegendie Faltungdesβ-Hairpin-bildendenPeptidsBH8 mißlang. Hier zeigtensich bislangnicht erkannteSchwachenunseresProteinmodellsmit starrerGeometrieder Peptidebeneund spe-ziellen effektiven Potentialenfur die Torsionswinkel. Die Faltungvon BH8 konnteerfolgreichmiteinemvoll flexiblen Proteinmodellsimuliertwerden,allerdingsweitauswenigereffizient alsmanesbeiVerwendungdesstarrenProteinmodellserwartenwurde.

Die BerechnungderProtonierungs-undRedoxzustandevontitrierbarenbzw. redoxaktivenGrup-penin Proteinenerfolgte,wie auchschonbeim fur die Simulationder ProteinfaltungverwendetenLosungsmittelmodell,auf BasisderKontinuumselektrostatik, bei derdasLosungsmitteldurcheinenBereicherhohter Dielektrizitatskonstante simuliert wird. Durch numerischesLosender Poisson-Boltzmann-GleichungaufeinemGitter lassensichdieresultierendenelektrostatischenPotentialeunddamit auchdie elektrostatischenInteraktionsenergien berechnen.Damit kannmandie elektrostati-scheEnergie einesbestimmtenProtonierungs-undRedoxzustandesdesGesamtproteinsbestimmen.Ein Proteinverfugt in der Regel ubereinegroßeZahl n von titrierbarenbzw. redoxaktiven Grup-pen. Die Gesamtzahlan moglichenZustanden(2n) ist extrem groß,so daßsich thermodynamischeMittelwerte,die zur Bestimmungvon Titrationskurven und Redoxpotentialennotig sind, praktischnicht exakt berechnenlassen. Daherverwendeich in der vorliegendenArbeit eine Monte-Carlo-Methode,mit dereineTeilmengederVielzahlmoglicherZustandegemaßdesMetropoliskriteriumsdurchmustertwird. DurchdieseMethodeerlangtmannachkurzerZeit ErgebnissehoherstatistischerGenauigkeit, wasdurchAuswerteneinerKorrelationsfunktiongezeigtwerdenkann.

Mit dergenantenVorgehensweisewar esmir moglich, eineVielzahlwertvoller Erkenntnissezusammeln.So konnteerstmaligdie Energie desElektronentransfersvon Q� �A zu QB im bakteriellenReaktionszentrumkorrekt berechnetwerden. Die Berechnungenergabenauchwichtige Hinweise

85

86 APPENDIXB. ZUSAMMENFASSUNG(GERMAN ABSTRACT)

fur die zuvor ungeklarteweitereReihenfolgeder Elektronentransfer- und ProtonierungsschrittederChinoneim Reaktionszentrum.Desweiterenkonnteauchvon theoretischerSeitedie conformationalgating-HypothesebetreffenddenElektronentransfersvonQ� �A zuQB untermauertunddasVerstandnisderzugehorigenVorgangeim Reaktionszentrumvertieft werden.Ich entwickele in dervorliegendenArbeit mehrereAnsatzezurEinfuhrungmultiplerKonformationenundvonKonformationsflexibilit atin die Titrationsberechnungen. In einemFall konntedurchdie explizit berucksichtigtestrukturelleRelaxiationdie Dielektrizitatskonstante fur dasInneredesProteinsentsprechendallgemeinerPrinzi-pienstarkverringertwerden.In einemanderenFall wurdedie Energetik pH-induzierterstrukturellerAnderungenvon Myoglobin durchEinbezieheneinesEnsemblesvon Kristallstrukturenverstanden.Die dabeiberechnetenpKa-Werte zeigeneine Ubereinstimmungmit dem Experiment,die in die-serGenauigkeit niemalszuvor mit nicht-trivialenMethodenzur pKa-Wertberechnungerzieltwordensind.

In einemAusblick erortereich die in Zukunft moglichenVerbesserungenderMonte-Carlo-Me-thodezur Langzeitdynamikvon Proteinenund die Vor- und Nachteileder existierendenMethodenzurTitrationsberechnungmit Konformationsflexibilit at. Ich zeigeauf,wie sichdurcheinezukunftigeKombinationderMonte-Carlo-Dynamikmit derTitrationsberechnungdiemeistenNachteilederexi-stierendenMethodenumgehenlassen.DieseMethodeverwirklicht dasuneingeschrankteundverzer-rungsfreiegleichzeitigeDurchmusterndesKonformationsraumesunddesRaumesderTitrationszu-stande.

Appendix C

Glossary

Thefollowing abbreviationsandsymbolsareusedin thiswork:

∇ Nablaoperator, gradient�A surfaceintegral�V volumeintegral

α occupationprobabilityβ

�kBT � � 1

∆ Laplaceoperatorε dielectricconstantεp dielectricconstantof theproteininteriorεs dielectricconstantof thesolventρ density(usuallychargedensity)φ electrostaticpotentialA area,surfaceACS analyticalcontinuumsolventbRC bacterial(photosynthetic)reactioncenterc concentrationCM CartesianmoveCPU centralprocessingunit�D electrostaticdisplacementDSSP dictionaryof secondarystructuresof proteins

(methodfor secondarystructureassignment)�E electrostaticfieldFTIR Fouriertransforminfrared(spectroscopy)G freeenergy, Gibb’s energyI ionic strengthkB BoltzmannconstantMbCO carbonmonoxymyoglobinMD moleculardynamicsMQ menaquinoneP specialpair (of thebRC)PBE Poisson-BoltzmannequationPDB BrookhavenProteinDataBank

87

88 APPENDIXC. GLOSSARY

q chargeQB primaryquinoneacceptorin thebRCQB secondaryquinoneacceptorin thebRCR molargasconstant�r coordinatesof apoint in three-dimensionalspaceRb. sphaeroides RhodobactersphaeroidesRP Ramachandranplot basedmethodof secondarystructureassignmentRps.viridis RhodopseudomonasviridisSM simplemoveT absolutetemperatureu (electrostatic)energy densityUQ ubiquinoneV volumeWM window move

Appendix D

CAM L AB++ Manual

This is a manualof theadditionalcommandsandutilities of CAMLAB++ comparedto thenormalCAMLAB program.To useCAMLAB++ , youwill alsoneedthestandardCAMLAB manual,whichcanbefoundin theWWW:http://cartan. gmd. de/ co mpch em/ho ff mann/C AMLab/manual/m anual. ht mlThe documentationof the ACS partsof CAMLAB++, which weremainly implementedby BjornKleier, arestill in apreliminarystateandnot includedhere.

D.1 Commands

D.1.1 Subcommandsof read energy terms

Thesecommandscanbeusedastype-specifierin theread energy terms -command.

D.1.1.1 lj pair

Format of additional data:n pairs numberOfLJPairsmolname1 atomnum1 molname2 atomnum2 LJeLJr. . .

Description:lj pair is usedto introducespecialLennard-Jonesinteractionsbetweentwo individual atoms.Ineachline two atomsarespecifiedby moleculename(molname) andatomnumber(atomnum). TheLennard-Jonesinteractionis givenby thetwousualLennard-Jonesparametersenergy (LJe) andradius(LJr).

D.1.1.2 noe

Format of additional data:temperature tempParamscaling scaleFactorn distances numberOfNOEConstraintsa numberOfAtomsInGroupAOfConstraint1m1aa1am2aa2a. . .b numberOfAtomsInGroupBOfConstraint1

89

90 APPENDIXD. CAMLAB++ MANUAL

m1ba1bm2ba2b. . .para kminrmin kmaxrmaxfmaxa numberOfAtomsInGroupAOfConstraint2. . .. . .

Description:noe is an energy function to useNOE constraintswithin CAMLab. It is very similar to dis-tance 1, but all energy termsaremultiplied by tempParam andscaleFactor. If you usethesameparameterfor tempParamandfor thetemperatureparameterin themetropoliscriterion,temperaturewill have no effecton theNOE energy termsin MC sampling.In addition to distance 1, noe allows to give more than one atom on both side of the NOEconstraint.All possibledistancesarethenaveragedby thefollowing equation:

r � 1

6

�∑i j 1

r6i j

D.1.1.3 quadratic repulsion

Format of additional data:scaling scaleFactorn atoms numberOfAtomsmoleculeNamenumberOfAtom1radiusOfAtom1forceConstant1moleculeNamenumberOfAtom2radiusOfAtom2forceConstant2. . .

quadratic repulsion is anenergy functionwhichcalculatesthequadraticrepulsionof atom iwith eachotheratom j basedon its radii r i andr j andtheir distancer i j . Theatomicforceconstantski andk j arescaledby thescalingfactors.

E � s∑i j � kik j

�r i j � 1

2

�r i r j �� 2

D.1.1.4 sartori

Format of additional data:epsilonfactor numberepsilonfunctio n funtionStringepsilon14facto r numbernumber of torsion potentials numberOfTorsionPotentialstorsion potential nameOfPotential 1dimension n ! numberof valuesin onedimensionenergyValue phi 1 psi 1energyValue phi 1 psi 2. . .energyValue phi 1 psi nenergyValue phi 2 psi 1

D.1. COMMANDS 91

energyValue phi 2 psi 2. . .energyValue phi n psi ntorsion potential nameOfPotantial 2dimension numberOfValuesInOneDimension. . .number of molecules numbermolecule nameOfMolecule1bonds numberOfBondTermsInMolec1atomnumber1 atomnumber2 forceConstantequilibriumDistance. . .bondangles numberOfBondangleTermsInMolec1atomnum1 atomnum2 atomnum3 forceConstantequilAngle. . .torsions numberOfTorsionTermsInMolec1atomnum1 atomnum2 atomnum3 atomnum4 forceConstmultiplicity equilTorsion. . .impropers numberOfImproperTermsInMolec1atomnum1 atomnum2 atomnum3 atomnum4 forceConstequilImproper. . .nonbondeds numberOfNonbondedsInMolec 1atomNummasscharge LJeLJr LJ14eLJ14rdeltaR. . .excluded nonbonded pairs numberOfExcludedNBPairsatomnum1 atomnum2. . .torsions with potential numberOfTorsionsWithPotentialInMolec 1nameOfPotentialphi 1 phi 2 phi 3 phi 4 psi 1 psi 2 psi 3 psi 4. . .fixed bonds numberOfFixedBondsInMolec1atomnum1 atomnum2 bondLength. . .fixed angles numberOfFixedAnglesInMolec1atomnum1 atomnum2 atomnum3 angle. . .fixed torsions numberOfFixedTorsionsInMolec1atomnum1 atomnum2 atomnum3 atomnum4 torsionAngle. . .fixed impropers numberOfFixedImpropersInMolec 1atomnum1 atomnum2 atomnum3 atomnum4 improperTorsionAngle. . .molecule nameOfMolecule1bonds numberOfBondTermsInMolec2. . .. . .

Description:

sartori is very similar to md force field . It doeseverythingwhatmd force field does.In addition, it is possibleto readin twodimensionaltorsionpotentialsfor pairsof dihedralangles.


Also parametersfor fixedbondlengthsandbondanglescanbegiven.Hereonly thedifferencesto themd force file subcommandaredescribed.To switchoff thecalculationof Coulombinteraction,setepsilonfactor to zero.(Thatis neces-saryif youwantto substitutetheCoulombpartby anotherforcefield,e.g.acs .)The epsilon14factor is applied to Coulomb interactionof 1-4-linked atoms. It is 0.5 forthe Charmmparameter-set of MSI, 0.4 for the CharmmPARAM19 set, and 1.0 for the CharmmPARAM22 set.Torsionpotentialsaregiven asenergy valueson a twodimensionalgrid with n grid points in eachdimension.The grid is torus-like. The first grid point in a grid row is identicalto the last. Hence,with a 10� grid n is 37. energyValue phi 1 psi 1 is theenergy valueat φ � � 180� andψ � � 180� .energyValue phi n psi n is theenergy valueatφ � 180� andψ � 180� . Bothgridpointsandvaluesareidentical. CAMLAB++ usesthe torus-like form of thepotentialto checkfor consistency of thegivendata.Theinput datafor thetorsionpotentialscanbegeneratedusingthestand-aloneprogram� MakeSartoriPot enti al s .In theenergy functionof FredoSartori,thereis acorrectionparameterfor theCoulombenergy:

ECoulomb � 14πεε0

q1q2

r ∆R

Theparameter∆R (in A) canbegivenfor eachindividualatomin thenonbondeds section(deltaR).Usually, it is only appliedto H atoms.Thetwodimensionaltorsionpotentialsusuallyincludesomenon-bondedtermsimplicitly. Thesenon-bondedinteractionsmustbe switchedoff individually usingthe excluded nonbonded pairssubcommand.Theorderof thepairsis critical. Thefirst atomnumbermustalwaysbegreaterthanthesecond.Thefirst atomnumberin row n 1 mustbe lessor equalthanthefirst atomnumberinrow n. If thefirst atomnumbersin row n 1 andn areequal,thesecondatomnumberin row n 1mustbelessthanthesecondatomnumberin row n.With the torsions with potential subcommandpairsof dihedralanglescanbeassignedtothetwodimensionaltorsionpotentialsdefinedby thetorsion potential subcommand.The remainingsubcommandsareusedto definevaluesfor fixed degreesof freedom. Thesevaluesareonly usedfor thingslike chainrebuilding. They don’t make surethat thesedegreesof freedomare really fixed. If you apply for examplea cartesianmove, the “fix ed” degreesof freedomwillprobablychange.Thus,you have to take careto keepthe“fix ed” degreesof freedomreally fixedbyanappropriateselectionof moves.Sincethechainrebuilding procedureis not yet implementedin CAMLAB++, thesecommandsareactuallyuselesswith theimportantexceptionof fixed bonds . Theparametersof thissubcommandin connectionwith thoseof thebonds subcommandareusedto createthebondedlist. Eachbondinthemoleculemustberepresentatedeitherin thebonds sectionor in the fixed bonds section.The parameters for a molecule can be generated using the stand-alone program� MakeSartoriEte rms .

D.1.2 Subcommandsof read move

D.1.2.1 dihedral window

Format of additional data:old energy EoldParaNamenew energy EnewParaNameselection energy Eselectionnumber of tries NumberOfTries ! (optional)conformational change ConfParaName! (optional)

D.1. COMMANDS 93

acceptance AcceptParaNamebackbone dihedral scaling BBDihedralScaleFactsidechain dihedral scaling SCDihedralScaleFactprerotation dihedral scaling PRDihedralScaleFact ! (optional)cartesian displacement Displacement! (optional)int list WListenergy subset index EnergySubsetParaNameatom subset index Asubsetcriterion CritType. . . (criteriondata)end criterionnumber of molecules n molecsmolecule molecNamenumber of chains ncsstart chain chaintypeaStart! (chaintypeis fixed or free or cyclic )sc s1s2s3s4DeltaPhiSl nlsa first 1 a last 1. . .a first nl a last nlr nrs. . .a first nr a last nr. . .cartesian nCart ! (optional)a first 1 a last 1. . .a first nCart a last nCartbb pivot apivotbb b1 b2 b3b4 DeltaPhiBl nlb. . .r nrb. . .sc s1s2s3s4DeltaPhiS. . .cartesian nCart ! (optional). . .bb pivot apivot. . .end chain chaintypeaEndstart chain chaintypeaStart. . .

Description:

Thedihedral window commandof standardCAMLab hasbeenreplacedin CAMLAB++. Thenew dihedral window commandhassomeadditional featuresbut is neverthelesscompletelycompatibleto the old version. All new keywordsareoptionalandcanbe omittedin the input file.However, if you want to useoneof thenew features,you have to insertthenew keyword exactly at


theplacegivenin thesyntaxdescriptionabove.All keywordsnotmentionedbelow havethesamemeaningasin theold versionof dihedral window .number of tries : If afterapplicationof theprerotationsno geometricalsolutionis possible,theprogramwill try it againwith new prerotations,until the numberof tries reachesthe valueof thisparameter.conformational change : If anew conformation(i. e. differentfrom theoriginalconformation)is selected,thevalueof thisparameteris setto 1 (if not, it is setto 0). Thevalueof thisparameterwillbe1 even if thenew conformationis rejectedby the respective criterion . With this parameter,you candistinguishbetween“acceptancedueto unchangedconformation”(i. e. unchangedenergy)and“real acceptance”(of a new conformation).prerotation dihedral scaling : This parameterscalesonly theprerotations.If it is given,dihedral scaling scalesonly unconstrainedmovesata freeendor startof achainanddihedralchangesthatarenotprerotations.If yousetprerotation dihedral scaling to asmallvalueanddihedral scaling to alargevalue,youwill geteasilyalot of geometricalpossiblesolutionsof thewindow move. It is, however, questionablewetherdetailedbalanceis maintainedor not.cartesian displacement : This parametergives the maximal possibledisplacementfor thecartesianmove.cartesian : In this section,atomsfor thecartesiandisplacement,which is appliedif a new con-formationhasbeenselescted,areselected.To selectatomsin thesidechain,this block mustappearbetweenthetwo dihedralanglesat thepivot atom.

D.2 Utilities

Theitemsdescribedherearenot CAMLAB++ commandsbut stand-aloneprograms.

D.2.1 Generationof Input Data

D.2.1.1 MakeSartoriPotentials

Syntax:

MakeSartoriPot enti al s inputfilepotential-name[ factor ]

This tool convertsa two dimensionalpotentialfrom a raw-ASCII format to the CAMLAB++ inputformat.All valuesaremultiplied by factor.

D.2.1.2 MakeSartoriEterms

Syntax:

MakeSartoriEte rms masses.rtfparm.prmpsf-filedelta-Rmolecule-nameoutput-file

A CHARMM psf -file is convertedto thegeneralizedinput formatof CAMLAB++. For thisprocess,alsothe CHARMM parameterfile andrtf -file is needed.delta-Ris thehydrogen-bondcorrection-value.

D.2.1.3 MakePivot

Syntax:

MakePivot dihedral-intervall offsetaa1[ aa2. . . ]

D.2. UTILITIES 95

This is a perl script that writes a CAMLab pivot file to stdout . The script is very specializedandwill work only underspecialconditions.Thepeptidefor which thepivot file is createdmustbebuilt with CHARMM19usingtheNACTandCMAMpatch.Somecustomizationcanbedoneby settinga suitableoffset for the atomnumbers.However, you will have to edit the first residue. dihedral-intervall is themaximalchangeof adihedralangleduringawindow move. aan denotestherespectiveresidue.CurrentlysupportedareALA, ARG, ASN, GLU, GLY, HSC(protonatedHis), ILE , LEU, LYS,PHE, THR, TYR, VAL. Timm Essigke haswritten thecorrespondingprogramfor CHARMM22.

D.2.1.4 MakeDihedrals

Syntax:

MakeDihedrals dihedral-intervall offsetaa1[ aa2. . . ]

ThisisanalogoustoMakePivot , but doesnotcreateapivot file, but adihedralfile for theapplicationof simpledihedralmoves.

D.2.2 Processingof Output Data

D.2.2.1 clt2dcd

Syntax:

clt2dcd [-h ] [-f steps-per-record] [-n stepoffset] [-o dcd-out-file] [-s frame-skip] [-swap ]clt-in-file(s). . .

This programconvertsCAMLab trajectoryfiles into CHARMM dcd trajectoryfiles. All input filesaremergedto oneoutputfile in theordergivenat thecommandline. They mustbeall for thesamemolecule. The stepoffset is only appliedonceat the beginning, but steps-per-record mustbe validfor all input files. If no namefor the dcd-out-fileis given, a nameis generatedfrom the first inputfile name.With the-swap optiongiven,thebyteorderis swappedsothat thedcd file is usableonmachineswith differentendian(only).

Appendix E

K ARL SBERG Manual

This is theorginal versionof themanualfor KARLSBERG 0.4. I did notapplyany changesto preventconfusion.Soyou will find a kind of self-referenceto my PhDthesiswithin themanual.I hopethiswill notbethesourceof anotherkind of confusion.I assureyou thatthereferredPhDthesisis reallyjust thebookyou arereadingnow. Themanual(perhapsevena newer version)canalsobefoundattheKARLSBERG homepage:http://lie.che mi e. fu- berl in .d e/k ar ls berg /

E.1 What is K ARL SBERG?

KARLSBERG is aprogramfor Monte-Carlo(MC) calculationof titrationpatternsof proteins.It needsintrinsicpKa valuesof thetitratableresiduesandaninteractionmatrixbetweenthemasinput. It willcalculatetheprotonationprobabilitiesfor thesetitratablesitesat givenpH values.This input usuallycomesfrom electrostaticprograms,e. g. MEAD by DonaldBashford(Bashford& Karplus,1990).So far, KARLSBERG is very similar to the MCTI programof Paul Beroza(Berozaet al., 1991).All featuresof MCTI arealsopresentin KARLSBERG: doublemoves, reducedsite titration, errorestimationby correlationfunctions. However, therearesomeadditions:KARLSBERG is ableto doparallel tempering(Hansmann,1997), triple moves(Rabensteinet al., 1998a),energy-biasedMC(Berozaet al., 1995)and,mostimportant,samplingof multiple conformationsusinganMC versionof themethoddescribedby Rabensteinet al. (1998a).PaperswhereKARLSBERG wasapplied,areRabensteinet al. (2000),RabensteinandKnapp(2000a),Vagedeset al. (2000).Especiallyin RabensteinandKnapp(2000a)youwill findmostof thescientificbackgrounddescribedat oneplace.An evenmorecompletesourceis my PhDthesis,which will beavailableassoonasit is publishedvia my WWW publicationlist:http://lie.che mi e. fu- berl in .d e/˜ ra be/s ci enc e/ publ ic ati ons. ht ml

KARLSBERG itself is alsoavailablein theWWW:http://lie.che mi e. fu- berl in .d e/k ar ls berg /

E.2 Installation

Untarthetarball.Binaries for Linux and IRIX are part of the distribution (karlsberg.Linux andkarlsberg.IRIX ). Move the correctbinary to your favored binary directory and renameit tokarlsberg . Theutilities (make curves andmake protonation table ) arePerlprograms,soyouneedPerlto run them.

97

98 APPENDIXE. KARLSBERG MANUAL

KARLSBERG is written in C++, so you needan appropriatecompiler, if you want to compile ityourself.Go to the src subdirectoryandtype make (you mustuseGNU-make!). This will work on LinuxandIRIX. Theresultingbinaryis calledkarlsberg.Linu x or karlsberg.IRIX , respectively.(They arecreatedin the samedirectoryasthe sourcefiles.)For otherplatforms,you mustunfortu-natelyhelpyourselfandmodify theMakefile.

E.3 Running K ARL SBERG

Therearenocommandline parameters.All programparametersandspecialcommandsarereadfromstdinline by line. Usually, youwill pipeacontrolfile with all theparametersandcommandsinto theprogram:

karlsberg < input-file > log-file

KARLSBERG tells you a lot aboutwhat it is doing to stdout. This outputshouldbeself-explaining.I/O from andto files is controlledby thecommandsgivento stdin. As input you needat leasta filewith intrinsic pKa valuesanda file with the interactionmatrix. The real resultsarewritten to files,not to stdout.

E.3.1 Samplefiles

In the directorysample of the distribution, you find a setof samplefiles. Run KARLSBERG bytypingkarlsberg < myoglobin.in > myoglobin.outTo understandthecommandsin myoglobin.in , readthefollowing section.

E.3.2 Syntaxof input to stdin

Eachline startseitherwith a keyword or with the commentsign “#”. Lines startingwith “#” areignored.You canusethecommentsignalsowithin a line. Thepartof the line after “#” is ignored.After a keyword,parametersmayfollow. If you give too many parameters,KARLSBERG will ignorethe surplus. If you give too few, KARLSBERG will assigndefault valuesif thereareany. Input isfinishedby a line startingwith thekeywordend . Everythingafterthis line is ignored.If nosuchlineis present,inputwill stopwheneof is detected.Orderof linesmaybearbitrarywith two exceptions:The oneis the end line (of course),and the otherarerepeatedkeywords. Most keywordsshouldonly appearoncein theinput. If sucha keyword is usedmorethanonce,only thefirst appearanceisrecognized,all othersareignored.In thefollowing, all keywordswith their respective parametersarelisted(in alphabeticalorder)andexplained.In thesyntaxdescription,squarebrackets( [. . . ]) markoptionalparameters.Default valuesfor theseparametersaregivenin thedescription.

bias site nameenergy [unit]This keyword canbeusedmorethanonce. It definesa biasenergy that is addedto theproto-nationenergy of thespecifiedsite. (For detailsseeBerozaet al. (1995).) Thedefault unit ise2/A. Otherunitsmaybespecifiedasunit: kJ/mol , kcal/mol , meV, eV.

conformation pkint file interaction matrix file [referenceenergy [unit]]This keyword canbeusedmorethanonce.Eachoccurrencegeneratesa new conformer. Datafor intrinsic pKa valuesandthe interactionmatrix arereadfrom the given files. The defaultvaluefor thereferenceenergy is 0 (zero),thedefault unit is e2/A. Otherunitsmaybespecified

E.3. RUNNING KARLSBERG 99

asunit: kJ/mol , kcal/mol , meV, eV.Formatfor pkint file: Onerow pertitratablesite.Eachrow hasthreeitemsseparatedby spaces.First item: intrinsic pKa value. Seconditem: oneletter, A (anionic)or C (cationic),anionicmeansuncharged referencestateis protonated,cationic meansuncharged referencestateisunprotonated.The restof the line is third item: nameof site. It is recommendedto usesitenameswithout spaces,sinceyou cannot give sitenameswith spacesin thecommandsbiasandpH independent .Formatof interaction matrix file: Onerow per interaction(total numberof rows is numberoftitratablesitessquared).Eachrow hasthreeitemsseparatedby spaces(itemsafterthethird areignored).First item: numberof first site(sitesarenumberedfrom 1 to n in theorderthey occurin pkint file). Seconditem: numberof secondsite. Third item: interactionenergy in unitsofe2/A.

conformation moves per scan [real number]Numberof conformation-changing movesperMC scan.Thesemovesarerandomlymixedwiththeothermoves. This parameteris a statisticalexpectationvalue. It maybesmallerthanone.Default valueis 1.

conformation reference state site namestateSometimestheconformationalreferenceenergy cannot bedeterminedwith all titratablesitesin their uncharged referencestate. The titration stateof individual titratablesitesduring thecalculationof the conformationalreferenceenergy canbe given using this keyword, if it isdifferentfrom theunchargedstateusedasreferencestateduringthecalculationof theintrinsicpKa values.stateis eitherP (protonated)or U (unprotonated).Thiskeyword canbeusedmorethanonce.

correlation limit [factor]Thisis thefractionof thevariancethathastobereachedby thecorrelationfunctionto determinecorrelationtime. Must be greaterthanzeroandsmallerthanone. Default valueis 0.1. (FordetailsseeBerozaet al. (1991).)

end (noparameters)Thiskeyword tells KARLSBERG to stopreadingfrom stdinandto startthecalculation.

full scans [numberof scans]Numberof MC scanswith all titratablesites.Default valueis 10,000.Onescancomprisesasmany randomattemptsto changethetitrationstatesasdifferenttitrationmovesareeligible(sin-gle,double,andtriple movestogether)plusthegivennumberof conformationalandtemperingmoves(seeconformation moves per scan andtempering moves per scan ). Inadditionto thenumberof full scansgivenhere,KARLSBERG will apply10 equilibrationscansbeforetheproductionrun is started.

min int pairs [interaction-energy]Minimal interactionenergy in pKa unitsto considera pair of titratablegroupsasstronglycou-pled. For suchpairs,doublemovesareappliedin additionto simplemoves. Default valueis2.5pKa units. If 0 (zero)is given asparameter, no doublemovesaredoneat all. If you areusingmultiple conformations,for eachconformationtheinteractionenergiesmaybedifferent.KARLSBERG will alwayslook for the strongestavailableinteractionenergy of a specificsitepair.

min int triples [interaction-energy]Minimal interactionenergy in pKa units to considera triple of titratablegroupsas strongly


coupled.For suchpairs,triple movesareappliedin additionto simplemoves. Default valueis 5pKa units. If 0 (zero)is given asparameter, no triple movesaredoneat all. Pleasenote,thatfor threegroupsA, B, andC thetriple criterionis fulfilled if A andB arecoupledstronglyenoughandB andC, too. A andC donotneedto bestronglycoupled.If youareusingmultipleconformations,for eachconformationtheinteractionenergiesmaybedifferent. KARLSBERG

will alwayslook for thestrongestavailableinteractionenergy of aspecificsitepair.

output [output file prefix]All outputfilesaregeneratedusingthisprefix. Default valueis karlsberg-outpu t .

pH end [pH value]At thispH,titrationstops.MustbegreaterthanorequalpH end . DefaultisequaltopH start .

pH incr [incrementvalue]Incrementfor pH value.Must bepositive. Default is 0.1.

pH independent site nameNot only titratablegroups,but alsoredoxgroupscanbetreatedby KARLSBERG. Theredoxpo-tentialof thesegroupsis independentonpH.Thiskeyword,whichcanbeusedmorethanonce,marksa siteasa pH-independentredoxgroup. Which redoxstateis thepseudo-“protonated”andpseudo-“unprotonated” andwhichof themis thereferencestate,dependson your input.

pH start [pH value]At this pH, titration begins.Mustbesmallerthanor equalpH end . Default is 7.0

reduced scans [numberof scans]Number of MC scans with the reduced set of titratable sites (see alsoreduced set tolerance ). Defaultvalueis 100,000.Onescancomprisesasmany randomattemptsto changethe titration statesasdifferenttitration movesareeligible (single,double,andtriple movestogether).

reduced set tolerance [tolerancevalue]After thefull MC calculationhasfinished,all titratablesiteswhoseprotonationprobabilityhasa differenceto zeroor unity smallerthan this value arefixed in their respective state(fullyprotonatedor unprotonated).Thedifferencemustbesmallenoughin all conformations(thatweresampledat leastonce)andatall temperatures(of paralleltempering).After that,reducedMC calculationis done,wherethe fixed titratablesitesarenot involved any longer. Defaultvalueis 0.000001.

seed [positiveinteger]Seednumberfor the randomgenerator. Must be a positive integer. If 0 (zero)or nothingisgiven,theseedis takenfrom thepresentsystemtime.

temperature [temperature]Temperaturefor MC simulationin K. Default valueis 300K. This keyword canbegivenmorethanonce.If this is thecase,paralleltemperingis donewith onecopy pergiventemperature.The lowest temperaturegiven in this way is assumedto be the temperature,whereyou havecalculatedtheintrinsicpKa values(KARLSBERG usesthistemperaturefor convertingpKa unitsto kJ/mol).

tempering moves per scan [real number]Numberof temperingmovesperMC scan.Thetemperingmovesarerandomlymixedwith theothermoves.This parameteris a statisticalexpectationvalue.It maybesmallerthanone.Thedefault valueis 1.

E.4. POSTPROCESSINGOF OUTPUT 101

E.3.3 Output

KARLSBERG generatesoneoutputfile for eachcombinationof pH, temperature,conformationandfull / reducedsampling.File nameslook like

prefix [full � red ] pHpH tempK conf conf-nr

Conformationnumber0 (zero)meansthat all conformationsareconsideredtogether. If only oneconformationis considered,the last part of the filenameis omitted. In eachfile, thereis a singlenumberin thefirst line. This numberis therelative occupancy of therespective conformationduringtheMC simulation.After that,a tablewith fivecolumnsfollows. Thefirst columnis thesitenumber,thesecondthesitename,the third thebiasedprotonationprobability, the fourth thestatisticalerror(standarddeviation),andthefifth therebiasedprotonationprobability. If youhavenotappliedbiasedMC, thethird andfifth columnareequal.Statisticalerror for therebiasedprotonationprobability isnot calculated.In thefirst row of thetable(sitenumber0 (zero)),thesumof all sitesis given.

E.4 Postprocessingof output

E.4.1 make curves

Syntax:

make curves prefix� prefix� must includethe full or red part of the filenames.The programwill try to find allusableoutputfilesof KARLSBERG by itself.This postprocessingprogramwrites onefile for eachsite, onefor the sumof all sites,andonefortheoccupancy of thedifferentconformations.It takesdatafrom thelastcolumnof theKARLSBERG

outputfiles (rebiasedprotonation)andputsthemfor all pH valuestogether. It will try to calculatethepKa valueandtheslopeof theHill plot. The resultis written to thecommentsectionof theoutputfiles. After thiscommentsectiona tablefollows. In thefirst columnof this tableyouwill find thepHvalueandin thefollowing columnstheprotonationprobability for thedifferentconformations.Thefile with theoccupanciesof thedifferentconformationshasaformatsimilar to thatof theprotonationprobabilityfiles. In thefirst column,you will find againthepH. In the following columnsyou willfind theoccupanciesof theconformations.Theoutputfiles canbe visualizedusinga programlikexmgr or gnuplot . Thetreatmentof histidinesis special(Bashfordet al., 1993).

E.4.2 make protonation table

Syntax:

make protonation table file1 [file2 [. . . ] ] [ start-confend-conf]

This postprocessingprogramwritesa LATEX file of a protonationtablefor thegivenfiles. It is a Perlscript. Seethe script file itself for details. The treatmentof histidinesis special(Bashfordet al.,1993).

Appendix F

Curriculum vitae of the author

Bjorn Rabenstein

1971/10/20bornin Berlin, Germany

1978–1984primaryschool:Annedore-Leber-Grundschule, Berlin-Tempelhof

1984–1991high school:Georg-Buchner-Oberschule (Gymnasium), Berlin-Tempelhof

1991 Abitur ( � highschooldiploma,grade1.0,sehrgut)

1991–1992personalstudiumgenerale at theFreieUniversitat Berlin in musicologyandtheology

1992–1997studyof biochemistryat the Freie Universitat Berlin , including externalresearchpar-ticipationsat the Max-Volmer-Institut of theTechnische Universitat Berlin andtheWeizmannInstituteof Science, Rehovot, Israel

1992–1997scholarof the Germannational scholarship foundation(StudienstiftungdesdeutschenVolkes)

1994/07/04Vordiplom in biochemistry( � bachelor, grade:sehrgut)

1997/01/14Diplom in biochemistry( � masterwith thesis: Protonierungs-und RedoxzustandederChinoneim bakteriellenphotosynthetischen Reaktionszentrumvon Rps. viridis, grade: sehrgut)

1997–2000PhDstudentin themacromolecularmodellinggroupof Prof. E. W. Knapp

July 2000 PhDthesis:Monte-CarloMethodsfor Simulationof ProteinFoldingandTitration

Parents:

Ir eneRabenstein (administrationsecretary, programmer, catechist)

Ulf Rabenstein (constructionengineer)

Teaching:

1995–1999supervisingseveralcoursesin basiclipid biochemistryandcomputationalbiochemistry

103

104 APPENDIXF. CURRICULUM VITAE OF THE AUTHOR

Presentaddress:

office:Institut fur Chemie(Kristallographie)Takustraße6D-14195BerlinGermany

e-mail: [email protected]:+49-30-838-53484fax: +49-30-838-53464

pri vate:Mehringdamm80D-10965BerlinGermany

phone:+49-30-78954895

105

Publications

Journal articles:

� B. Rabenstein,G. M. Ullmann, E. W. Knapp (1998). Energeticsof ElectronTransferandProtonationReactionsin the PhotosyntheticReactionCenterof Rhodopseudomonasviridis.Biochemistry37, 2488-2495

� B. Rabenstein,G. M. Ullmann, E. W. Knapp (1998).Calculationof ProtonationPatternsinProteinswith StructuralRelaxationandMolecularEnsembles— Applicationto thePhotosyn-theticReactionCenter. Eur. Biophys.J. 27, 626-637

� B. Rabenstein,G. M. Ullmann, E. W. Knapp (2000).ElectronTransferbetweentheQuinonesin thePhotosyntheticReactionCenterandits Couplingto ConformationalChanges.Biochem-istry published on web

� P. Vagedes,B. Rabenstein,J. Aqvist, J. Marelius, E. W. Knapp (2000). The DeacylationStepof Acetylcholinesterase:ComputerSimulationStudies.J. Am.Chem.Soc.in press

� B. Rabenstein,E. W. Knapp (2000). CalculatedpH-DependentPopulationandProtonationof CO-MyoglobinConformers.Biophys.J. submitted

� I. Spyridopoulos, J. Wischhusen,B. Rabenstein,D. Axel, K.-U. Fr ohlich, K. R. Karsch(2000). Alcohol EnhancesOxysterol-MediatedApoptosisby a Calcium-DependentMecha-nism.Arterioscl.Throm.Vas. in press

� D. Popovic, S.Zari c, B. Rabenstein,E. W. Knapp (2000).ComputerModelingof ArtificialCytochromeb: Evaluationof RedoxPotentialsin preparation

Theses:

� B. Rabenstein(1997).Protonierungs-undRedoxzustandederChinoneim bakteriellenphoto-synthetischenReaktionszentrumvon Rhodopseudomonasviridis. Masterthesis,FreieUniver-sitat Berlin.

� B. Rabenstein(2000).MonteCarloMethodsfor Simulationof ProteinFolding andTitration.PhDthesis,FachbereichBiologie/Chemie/Pharmazie— Institut fur Chemie,FreieUniversitatBerlin.

Talks at international conferences:

� B. Rabenstein,G. M. Ullmann, E. W. Knapp. ProtonationandRedoxStatesof theQuinonesin theReactionCenterof Rps.viridis. Talkatthe2ndEuropeanBiophysicsCongress,EBSA97,OrleansJuly 13-171997.Abstractpublishedin Eur. Biophys.J. 26, 122.

� B. Rabenstein,B. Kleier, D. Hoffmann, M. Schafer, E. W. Knapp. Foldingof α-Helicesandβ-HairpinsUsinga Monte-CarloMethodandan Implicit Solvation Model. Talk (andposter)at theInternationalScientificCongressof theBelgianBiophysicalSociety“Foldingof SolubleandMembraneProteins”,GemblouxJune14-171999.


Book articles, conferenceproceedingsetc.� B. Rabenstein,D. Hoffmann, E. W. Knapp (1999). Simulationof oligopeptidefolding orhow do residuestalk. In Biological Physics- Third InternationalSymposium, AIP ConferenceProceedings487,AmericanInstituteof Physics,Melville, New York, pp. 54-68.

� B. Rabenstein,E. W. Knapp (2000). ProblemsEvaluatingEnergeticsof ElectronTransferfrom Q�A to QB: theLight-ExposedandDark-AdaptedBacterialReactionCenter. submitted

Posterpresentations,talks of co-workers etc.� B. Rabenstein,G. M. Ullmann, E. W. Knapp. Energeticsof ElectronandProtonTransferReactionsof theQuinonesin thePhotosyntheticReactionCenterof Rps.viridis. Posterat theInternationalMeetingof theDeutscheBunsen-Gesellschaft“HydrogenTransfer:ExperimentandTheory”,Berlin September10-131997.

� B. Rabenstein,G. M. Ullmann, E. W. Knapp. Triple Moves in Monte Carlo Titration ofSystemswith StronglyCoupledTitratableGroups— Applicationto theReactionCenterof Rps.viridis with aLow DielectricConstant.Posterat theHLRZ-Workshop“Monte CarloApproachto BiopolymersandProteinFolding”, ForschungszentrumJulich December3-51997.

� B. Rabenstein,D. Hoffmann, F. Sartori, E. W. Knapp. Folding of β-structureelementswith a Monte-Carlomethod.Posterat theEuroconference“Dynamicsof Complex MolecularLiquids - ComputerSimulationsandExperiments”,Vaalsbroek(NL) May 24-281998.

� B. Rabenstein,G. M. Ullmann, E. W. Knapp. ProtonationPatternsof the PhotosyntheticReactionCenterandEnergeticsof theDifferentChargeandProtonationStatesof theQuinones.Posterat theXIth InternationalCongressonPhotosynthesis,BudapestAugust17-221998.

� B. Rabenstein,D. Hoffmann, E. W. Knapp. Simulationof oligopeptidefolding or how doresiduestalk. Talk (speaker: E. W. Knapp)at theThird InternationalSymposiumonBiologicalPhysics,SantaFeSeptember20-241998.

� B. Rabenstein,D. Hoffmann, F. Sartori, E. W. Knapp. Folding of helix-turn-helixmotifsand β-hairpinswith a Monte-Carlomethod. Posterat the annualmeetingof the DeutscheGesellschaftfur Biophysik,Frankfurt/MainSeptember21-231998.

� B. Rabenstein,B. Kleier, D. Hoffmann, M. Schafer, E. W. Knapp. Foldingof α-Helicesandβ-HairpinsUsing a Monte-CarloMethodandan Implicit Solvation Model. Poster(andtalk)at theInternationalScientificCongressof theBelgianBiophysicalSociety“Foldingof SolubleandMembraneProteins”,GemblouxJune14-171999.

� E. W. Knapp, B. Rabenstein,G. M. Ullmann. Energeticsof theElectronTransferfrom Q�Ato QB for the Light andDark AdaptedReactionCenters.Talk (speaker E. W. Knapp)at the218th AmericanChemicalSocietyNationalMeeting,New OrleansAugust22-261999.

� E. W. Knapp, B. Rabenstein,G. M. Ullmann. ElectrostaticInteractionsin Proteins. Talk(speaker E. W. Knapp)at the XIII InternationalBiophysicsCongress,New Delhi September19-241999.

� P. Vagedes,D. Popovic, B. Rabenstein,E. W. Knapp. ElectrostaticInteractionsin Proteins:RedoxReactionsandEnzymeCatalysis.Talk (speaker E. W. Knapp)at the Third EuropeanConferenceonComputationalChemistry, BudapestSeptember4-82000.

107

� B. Rabenstein,E. W. Knapp. CalculatedpH-DependentPopulationandProtonationof CO-MyoglobinConformers.Posterat the3rdEuropeanBiophysicsCongress,MunchenSeptember9-132000..

� P. Vagedes,D. Popovic, B. Rabenstein,E. W. Knapp. ElectrostaticInteractionsin Proteins:RedoxReactionsandEnzymeCatalysis.Talk (speaker E. W. Knapp)at theInternationalMeet-ing “UnderstandingProteinElectrostatics”,Huddinge(Sweden)September15-182000.

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