Sequence Dependent Elasticity of DNA Nils Becker Dissertation zur Erlangung des akademischen Grades Doctor rerum naturalium (Dr.rer.nat.) vorgelegt der Fakultät Mathematik und Naturwissenschaften Technische Universität Dresden Abstract The DNA contained in every living cell not only stores the genetic information; it functions in a complex molecular network that can condense, transcribe, replicate and repair genes. The essential role played by the sequence dependent structure and deformability of DNA in these basic processes of life, has received increasing attention over the past years. The present work aims at better understanding sequence dependent elasticity of double stranded DNA elasticity, across biologically relevant length scales. A theoretical description is developed that makes is possible to relate structural, biochemical and biophysical experiments and simulation. It is based on the rigid base–pair chain (rbc) model which captures all basic deformation modes on the scale of individual base–pair (bp) steps. Existing microscopic parametrizations of the rbc model rely on indirect meth- ods. A way to relate them to biochemical experiments is provided by the indirect readoutmechanism,whereDNAelasticitydeterminesprotein–DNAcomplexation affinities. Bycorrelatingtheoreticalaffinitypredictionswithinvitromeasurements in a well–studied test case, different parameter sets were evaluated. As a result a new, hybrid parameter set is proposed which greatly reduces prediction errors. Indirect readout occurs mostly at particular binding subsites in a complex. A sta- tistical marker is developed which localizes indirect readout subsites, bydetecting elastically optimized sub-sequences. By a systematic coarse–graining of the rbc to the well–characterized worm–like chain (wlc) model, a quantitative connection between microscopic and kbp scale elasticityisestablished. Thegeneralhelicalrbcgeometryismappedtoaneffective, linear‘on-axis’version,yieldingthefullsetofwlcelasticparametersforanygiven sequence repeat. In the random sequence case, structural variability adds confor- mational fluctuations which are correlated by sequence continuity. The sequence disorder correction to entropic elasticity in the rbc model is shown to coincide with the conformational correction. The results show remarkable overall agree- mentofthecoarse–grainedwiththemesoscalewlcparameters,lendingsupportto the model and to the microscopic parameter sets. A continuum version of the rbc is formulated as Brownian motion on the rigid motiongroup. Analyticexpressionsforangularcorrelationfunctionsandmoments of the end–to–end distance distribution are given. In an equivalent Lagrangian approach, conserved quantities along, and the linear response around, a general equilibrium shape are explored. iii Zusammenfassung Die in jeder lebenden Zelle enthaltene DNS speichert nicht nur die genetische Information; Sie funktioniert innerhalb eines komplexen molekularen Netzwerks, das in der Lage ist, Gene zu kondensieren, transkribieren, replizieren und reparie- ren. Die zentrale Rolle, welche der sequenzabhängigen Struktur und Deformier- barkeit von DNS in diesen grundlegenden Lebensprozessen zukommt, erregte in den letzten Jahren zunehmendes Interesse. DievorliegendeArbeithatein besseresVerständnisder sequenzabhängigen ela- stischen Eigenschaften von DNS auf biologisch relevanten Längenskalen zum Ziel. Es wird eine theoretische Beschreibung entwickelt, die es ermöglicht, strukturbio- logische, biochemische und biophysikalische Experimente und Simulationen in Beziehung zu setzen. Diese baut auf dem Modell einer Kette aus starren Basenpaa- ren (rbc) auf, das alle wichtigen Deformationsmoden von DNS auf der Ebene von einzelnen Basenpaar (bp)–Schritten abbildet. Bestehende Parametersätze des rbc-Modells beruhen auf indirekten Methoden. Eine direkte Beziehung zu biochemischen Experimenten kann mithilfe des in- direkten Auslese-Mechanismus hergestellt werden. Hierbei bestimmt die DNS– Elastizität Komplexierungsaffinitäten von Protein–DNS–Komplexen. Durch eine Korrelation von theoretischen Vorhersagen mit in vitro Messungen in einem gut untersuchten Beispielfall werden verschiedene Parametersätze bewertet. Als Resul- tat wird ein neuer Hybrid–Parametersatz vorgeschlagen, der die Vorhersagefehler stark reduziert. Indirektes Auslesen tritt meistens an speziellen Teilbindungsstellen innerhalbeinesKomplexesauf.EswirdeinestatistischeKenngrößeentwickelt,die indirektes Auslesen durch Detektion elastisch optimierter Subsequenzen erkennt. Durch ein systematisches Coarse–Graining des rbc-Modells auf das gut charak- terisierte Modell der wurmartigen Kette (wlc) wird eine quantitative Beziehung zwischen der mikroskopischen und der Elastizität auf einer kbp-Skala hergestellt. Die allgemeine helikale Geometrie wird auf eine effektive, lineare Version der Kette ‘auf der Achse’ abgebildet. Dies führt zur Berechnung des vollen Satzes von wlc-elastischen Parameters für eine beliebig vorgegebene periodische Sequenz. Im Fall zufälliger Sequenz führt die Strukturvariabilität zu zusätzlichen Konformati- onsfluktuationen,diedurchdieKontinuitätderSequenzkurzreichweitigkorreliert sind. Es wird gezeigt, daß die Sequenzunordnungs-Korrektur zur entropischen Elastizität im rbc-Modell identisch ist zur Korrektur der Konformationsstatistik. Die Ergebnisse zeigen eine bemerkenswerte Übereinstimmung der hochskalierten mikroskopischen mit den mesoskopischen wlc-Parameter und bestätigen so die Wahl des Modells und seiner mikroskopischen Parametrisierung. Eine Kontinuumsversion des rbc-Modells wird formuliert als Brownsche Be- wegung auf der Gruppe der Starrkörpertransformationen. Analytische Ausdrücke für Winkelkorrelationsfunktionen und Momente der Verteilung des End-zu-End– Vektors werden angegeben. In einem äquivalenten Lagrange-Formalismus werden ErhaltungsgrößenentlangvonGleichgewichtskonformationenunddielineareAnt- wort in ihrer Umgebung untersucht. iv Acknowledgments First of all, I would like Ralf Everaers for his supervision and his encouragement during this work. His unerring physical intuition was invaluable and he never failed to remind me that theoretical physics is about: the real world, and real data. Frank Jülicher kindly admitted me into his group. I am very grateful for his teaching and for the extremely pleasant and stimulating scientific crowd he has gathered at the Max-Planck-Institute for Complex Systems. Myfellowstudentshavehelpedmeinmanydifferentways. Iwouldliketothank the ‘first generation’, Andreas Hilfinger, Gernot Klein, Peter Borowski, Frank Pollmann1 and Tobias Bollenbach for the team spirit and also Christian Simm, Thomas Bittig, Eva-Maria Schötz, Elisabeth Fischer, Benjamin Friedrich, Lars Wolff and Kai Dierkes for their interest, their support and their encouragement. IthasbeenagreatlearningexperienceandalotoffuntointeractwithBenLind- ner, Karsten Kruse, Simon Tolic-Norrelykke, Eric Galburt and John Maddocks. Like everyone in the biological physics department, I am very much indebted to Nadine Baldes who has‘run the place’ and still had the timefor allmy administra- tive problems. My deepest gratitude belongs to my parents for their love and constant support in these last three years, again. 1honorarygroupmember v Contents 1 DNA at the base pair level 7 1.1 Sequence dependent DNA elasticity . . . . . . . . . . . . . . . . . . 7 1.2 Rigid base–pair elasticity . . . . . . . . . . . . . . . . . . . . . . . . 10 1.3 Fluctuations of rigid base–pair steps. . . . . . . . . . . . . . . . . . 13 1.4 Fluctuations of rigid base–pair chains . . . . . . . . . . . . . . . . . 16 1.5 Linear elasticity of rigid base–pair steps . . . . . . . . . . . . . . . . 19 1.6 Microscopic parametrization of rbp potentials . . . . . . . . . . . . 21 2 Indirect Readout in Protein-DNA complexes 25 2.1 DNA-protein recognition . . . . . . . . . . . . . . . . . . . . . . . . 25 2.2 Indirect readout in 434 repressor . . . . . . . . . . . . . . . . . . . 29 3 Local elastic optimization 36 3.1 Local elasticity in 434 repressor . . . . . . . . . . . . . . . . . . . . 36 3.2 Elastic optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 3.3 Origins of specificity . . . . . . . . . . . . . . . . . . . . . . . . . . 45 3.4 Elastic consensus sequences . . . . . . . . . . . . . . . . . . . . . . 46 3.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 4 Rigid base–pair chains 53 4.1 Linear elastic response of a rigid base–pair chain . . . . . . . . . . . 53 4.2 Basic properties of the rigid motion group . . . . . . . . . . . . . . 55 4.3 Rigid base–pair elasticity revisited . . . . . . . . . . . . . . . . . . . 66 5 Coarse graining of helical DNA 72 5.1 DNA elasticity is scale dependent . . . . . . . . . . . . . . . . . . . 72 5.2 Thermal fluctuations in a rigid base–pair chain . . . . . . . . . . . . 73 5.3 Effective semiflexible polymer for homogeneous chains . . . . . . . 74 5.4 Coarse–graining relations . . . . . . . . . . . . . . . . . . . . . . . 79 vi Contents 5.5 Anisotropic bending . . . . . . . . . . . . . . . . . . . . . . . . . . 82 5.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 6 Coarse graining of random DNA 85 6.1 Mapping a random sequence rbc to a homogeneous wlc . . . . . . 85 6.2 Random sequence chain conformations and numerical test . . . . 93 6.3 Response to external forces . . . . . . . . . . . . . . . . . . . . . . 94 6.4 Effective worm–like chain parameters . . . . . . . . . . . . . . . . 97 6.5 Limits of applicability of the wlc model . . . . . . . . . . . . . . . . 100 6.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 7 Random walks on the rigid motion group 104 7.1 Continuous models for DNA . . . . . . . . . . . . . . . . . . . . . . 104 7.2 The worm–like chain limit . . . . . . . . . . . . . . . . . . . . . . . 104 7.3 Continuum limit of the rigid base–pair chain . . . . . . . . . . . . . 108 7.4 Moment odes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 8 Lagrangian mechanics on the rigid motion group 126 8.1 Lagrangian approach to random paths . . . . . . . . . . . . . . . . 126 8.2 Euler–Lagrange equations . . . . . . . . . . . . . . . . . . . . . . . 128 8.3 Conservation laws . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 8.4 Linear response of the crbc. . . . . . . . . . . . . . . . . . . . . . . 132 8.5 Fluctuations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 9 Outlook 141 9.1 Superhelical looping . . . . . . . . . . . . . . . . . . . . . . . . . . 141 9.2 More on indirect readout . . . . . . . . . . . . . . . . . . . . . . . 143 9.3 Forces and torques in crystal structures . . . . . . . . . . . . . . . . 149 A Appendix 155 A.1 Robustness to parametrization errors . . . . . . . . . . . . . . . . . 155 A.2 The kernel of the adjoint map . . . . . . . . . . . . . . . . . . . . . 155 A.3 Finite matrix power series . . . . . . . . . . . . . . . . . . . . . . . 156 A.4 The differential of the exponential map . . . . . . . . . . . . . . . . 157 A.5 Lie algebra automorphisms of se . . . . . . . . . . . . . . . . . . . 159 A.6 Partial diagonal forms of the se stiffness matrix. . . . . . . . . . . . 160 A.7 Volume element . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 vii Contents A.8 Conversion from 3DNA coordinates . . . . . . . . . . . . . . . . . . 162 A.9 Dimensional structure of the rigid base–pair chain . . . . . . . . . . 162 A.10Explicit expression for the generator . . . . . . . . . . . . . . . . . . 164 Bibliography 165 Glossary 179 viii Introduction The implementation of the genome Whenaskedtonamethemostimportantbiomolecule,onewouldprobablysayit’s DNA, deoxyribonucleic acid. DNA is present in every living cell, with a chemical structurethathasbeenconservedoverbillionsofyears. Itfunctionsasthephysical implementation of the genome, preserving the genetic information of any living organism with unmatched storage density and reliability. Our DNA base sequence defines if not who we are, so at least what we are, by encoding for the protein components all cells are made of. After completion of the Human Genome Project [Int03b, Int03a], the genetic information of man is readily available, and more and more species are being sequenced. Given the rapid progress in efficiency, it will soon be possible to sequence entire genomes of individuals for an affordable price. So in a way, one could think that all secrets that have surrounded DNA are finally resolved, and one should move on to study something else. However,neitherthecompletegenomesequencenortheatomicstructureofthe double helix discovered 50 years earlier [Wat53b] can explain how the molecule really works. How exactly is DNA able to perform the enormous tasks of stor- ing gigabytes of genetic information in an error–tolerant way, repairing inevitable damage? How can the appropriate bits of that information be read out with ap- propriate frequency? How does the machinery work that allows DNA to replicate itself faithfully, then to condense and separate before cell division and finally to de-condense in the nucleus afterward? Like any component of a complex system, DNA does not function on its own. Understanding DNA means understanding its interactions with a multitude of co- evolved proteins, whoseintricate biochemical network performsessential molecu- lar processes of life collectively. 1 Contents DNA as a physical object In all of these interactions, the physical properties of the DNA molecule as a complex polymer are essential. Here, thinking in terms of physics can give insight oftheconstraintsunderwhichthebiologicalsystemworks. Someexamplesfollow. In a stereotyped eucaryotic cell 10µm in length, between divisions, DNA is concentrated in the nucleus of 1µm radius. The total contour length of DNA is of the order of 1cm, less than could be fit into the nucleus by tight packing. So is DNA really compressed at all? From polymer physics one knows that the bending persistence length of 50nm sets the scale for the extension of a coil of DNA free in solution. The result is at least 50µm radius for 1cm of DNA, indicating that confinement into the nucleus does require work. Separating such a highly condensed coil of threadlike polymer for cell division is a nontrivial task, since the inevitable entanglement of strands poses topological constraints[Sch04]. Cellsdealwiththemononehandbyasetofenzymesthatcan activelychangethelinkingofDNAcoils,andontheotherhandbyawholehierar- chy of organized packing structures which compact DNA and limit entanglement at the same time (see e.g. [Sin94, Alb02]). Ofthispackinghierarchy,thelowestlevelisbestunderstood. Thebasicpacking motif is called the nucleosome core particle. It consists of about 50nm of DNA wrappedin1.7turnsaroundacylindricalspoolwithabout10nmdiameter[Ric03]. ThehistonesthatformthespoolandotherDNA–associatedproteinsactuallymake up more than half of the material in the cell nucleus. The tight bending of DNA ontothe5nmouterradiusofthehistonespoolcostsenergy,andthereexistsafree energy balance between chemical bonds of DNA with the histone surface, and its wrapping. In effect, histones are bound strongly enough to occupy DNA almost densely but still not too strongly to block transcription [Sch03]. Protein levels in the cell are regulated in response to cell fate and to environ- mental conditions. One of the involved feedback mechanisms works at the level of transcription of DNA to RNA (ribonucleic acid). Here, depending on protein product concentration, a regulatory protein binds DNA at a specific sequence of several base–pairs, close to the transcription initiation site, thereby modifying the rate of transcription. In crystal structures of such complexes, DNA is often deformed from its equilibrium shape. As a result, the base–sequence dependent deformability of DNA affects the binding strength of the complex and thus also the resulting protein levels [Kou06, Kou87, Heg02]. 2