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What is flagellar swimming? Cilia and flagella are whip-like cell appendages that can exhibit regular bending waves. This active process emerges from the non-equilibrium dynamics of molecular motors distributed along the length of cilia and flagella. Eukaryotic cells can possess many cilia and flagella that beat in a coordinated fashion, thus transporting fluids, as in mammalian airways or the ventricular system inside the brain. Many unicellular organisms posses just one or two flagella, rendering them microswimmers that are propelled through fluids by the flagellar beat including sperm cells and the biflagellate green alga Chlamydomonas. Objectives. In this thesis in theoretical biological physics, we seek to understand the nonlinear dynamics of flagellar swimming and synchronization. We investigate the flow fields induced by beating flagella and how in turn external hydrodynamic flows change speed and shape of the flagellar beat. This flagellar load-response is a prerequisite for flagellar synchronization. We want to find the physical principals underlying stable synchronization of the two flagella of Chlamydomonas cells. Results. First, we employed realistic hydrodynamic simulations of flagellar swimming based on experimentally measured beat patterns. For this, we developed analysis tools to extract flagellar shapes from high-speed videoscopy data. Flow-signatures of flagellated swimmers are analysed and their effect on a neighboring swimmer is compared to the effect of active noise of the flagellar beat. We were able to estimate a chemomechanical energy efficiency of the flagellar beat and determine its waveform compliance by comparing findings from experiments, in which a clamped Chlamydomonas is exposed to external flow, to predictions from an effective theory that we designed. These mechanical properties have interesting consequences for the synchronization dynamics of Chlamydomonas, which are revealed by computer simulations. We propose that direct elastic coupling between the two flagella of Chlamydomonas, as suggested by recent experiments, in combination with waveform compliance is crucial to facilitate in-phase synchronization of the two flagella of Chlamydomonas.:1 Introduction 1.1 Physics of cell motility: flagellated swimmers as model system 2 1.1.1 Tissue cells and unicellular eukaryotic organisms have cilia and flagella 2 1.1.2 The conserved architecture of flagella 3 1.1.3 Synchronization in collections of flagella 5 1.2 Hydrodynamics at the microscale 9 1.2.1 Navier-Stokes equation 10 1.2.2 The limit of low Reynolds number 10 1.2.3 Multipole expansion of flow fields 11 1.3 Self-propulsion by viscous forces 13 1.3.1 Self propulsion requires broken symmetries 13 1.3.2 Signatures of flowfields: pusher & puller 15 1.4 Overview of the thesis 16 2 Flow signatures of flagellar swimming 2.1 Self-propulsion of flagellated swimmers 20 2.1.1 Representation of flagellar shapes 20 2.1.2 Computation of hydrodynamic friction forces 21 2.1.3 Material frame and rigid-body transformations 22 2.1.4 The grand friction matrix 23 2.1.5 Dynamics of swimming 23 2.2 The hydrodynamic far field: pusher and puller 26 2.2.1 The flow generated by a swimmer 26 2.2.2 Force dipole characterization 27 2.2.3 Flagellated swimmers alternate between pusher and puller 29 2.2.4 Implications for two interacting Chlamydomonas cells 31 2.3 Inertial screening of oscillatory flows 32 2.3.1 Convection and oscillatory acceleration 33 2.3.2 The oscilet: fundamental solution of unsteady flow 35 2.3.3 Screening length of oscillatory flows 35 2.4 Energetics of flagellar self-propulsion 36 2.4.1 Impact of inertial screening on hydrodynamic dissipation 37 2.4.2 Case study: the green alga Chlamydomonas 38 2.4.3 Discussion: evolutionary optimization and the number of molecular motors 38 2.5 Summary 39 3 The load-response of the flagellar beat 3.1 Experimental collaboration: flagellated swimmers exposed to flows 41 3.1.1 Description of the experimental setup 42 3.1.2 Computed flow profile in the micro-fluidic device 43 3.1.3 Image processing and flagellar tracking 43 3.1.4 Mode decomposition and limit-cycle reconstruction 47 3.1.5 Changes of limit-cycle dynamics: deformation, translation, acceleration 49 3.2 An effective theory of flagellar oscillations 50 3.2.1 A balance of generalized forces 50 3.2.2 Hydrodynamic friction in generalized coordinates 51 3.2.3 Intra-flagellar friction 52 3.2.4 Calibration of active flagellar driving forces 52 3.2.5 Stability of the limit cycle of the flagellar beat 53 3.2.6 Equations of motion 55 3.3 Comparison of theory and experiment 56 3.3.1 Flagellar mean curvature 57 3.3.2 Susceptibilities of phase speed and amplitude 57 3.3.3 Higher modes and stalling of the flagellar beat at high external load 59 3.3.4 Non-isochrony of flagellar oscillations 63 3.4 Summary 63 4 Flagellar load-response facilitates synchronization 4.1 Synchronization to external driving 65 4.2 Inter-flagellar synchronization in the green alga Chlamydomonas 67 4.2.1 Equations of motion for inter-flagellar synchronization 68 4.2.2 Synchronization strength for free-swimming and clamped cells 70 4.2.3 The synchronization strength depends on energy efficiency and waveform compliance 73 4.2.4 The case of an elastically clamped cell 74 4.2.5 Basal body coupling facilitates in-phase synchronization 75 4.2.6 Predictions for experiments 78 4.3 Summary 80 5 Active flagellar fluctuations 5.1 Effective description of flagellar oscillations 84 5.2 Measuring flagellar noise 84 5.2.1 Active phase fluctuations are much larger than thermal noise 84 5.2.2 Amplitude fluctuations are correlated 85 5.3 Active flagellar fluctuations result in noisy swimming paths 86 5.3.1 Effective diffusion of swimming circles of sperm cell 86 5.3.2 Comparison of the effect of noise and hydrodynamic interactions 87 5.4 Summary 88 6 Summary and outlook 6.1 Summary of our results 89 6.2 Outlook on future work 90 A Solving the Stokes equation A.1 Multipole expansion 95 A.2 Resistive-force theory 96 A.3 Fast multipole boundary element method 97 B Linearized Navier-Stokes equation B.1 Linearized Navier-Stokes equation 101 B.2 The case of an oscillating sphere 102 B.3 The small radius limit 103 B.4 Greens function 104 C Hydrodynamic friction C.1 A passive particle 107 C.2 Multiple Particles 107 C.3 Generalized coordinates 108 D Data analysis methods D.1 Nematic filter 111 D.1.1 Nemat 111 D.1.2 Nematic correlation 111 D.2 Principal-component analysis 112 D.3 The quality of the limit-cycle projections of experimental data 113 E Adler equation F Sensitivity analysis for computational results F.1 The distance function of basal coupling 117 F.2 Computed synchronization strength for alternative waveform 118 F.3 Insensitivity of computed load-response to amplitude correlation time 118 List of Symbols List of Figures Bibliography

Subject of study: Cilia and flagella are slender appendages of eukaryotic cells. They are actively bending structures and display regular bending waves. These active flagellar bending waves drive fluid flows on cell surfaces like in the case of the ciliated trachea or propels single cell micro-swimmers like sperm or alga. Objective: The axoneme is the evolutionarily conserved mechanical apparatus within cilia and flagella. It is comprised of a cylindrical arrangement of microtubule doublets, which are the elastic elements and dyneins, which are the force generating elements in the axonemal structure. Dyneins collectively bend the axoneme and must be specifically regulated to generate symmetric and highly asymmetric waveforms. In this thesis, I address the question of the molecular origin of the asymmetric waveform and test different theoretical descriptions for motor regulation. Approach: I introduce the isolated and reactivated Chlamydomonas axoneme as an experimental model for the symmetric and asymmetric flagellar beat. This system allows to study the beat in a controlled and cell free environment. I use high-speed microscopy to record shapes with high spatial and temporal resolution. Through image analysis and shape parameterization I extract a minimal set of parameters that describe the flagellar waveform. Using Chlamydomonas, I make use of different structural and motor mutants to study their effect on the shape in different reactivation conditions. Although the isolated axoneme system has many advantages compared to the cell-bound flagellum, to my knowledge, it has not been characterized yet. Results: I present a shape parameterization of the asymmetric beat using Fourier decomposition methods and find, that the asymmetric waveform can be understood as a sinusoidal beat around a circular arc. This reveals the similarities of the two different beat types: the symmetric and the asymmetric beat. I investigate the origin of the beat-asymmetry and find evidence for a specific dynein motor to be responsible for the asymmetry. I furthermore find experimental evidence for a strong sliding restriction at the basal end of the axoneme, which is important to establish a static bend. In collaboration with P. Sartori and F. Jülicher, I compare theoretical descriptions of different motor control mechanisms and find that a curvature controlled motor-regulation mechanism describes the experimental data best. We furthermore find, that in the dynamic case an additional sliding restriction at the base is unnecessary. By comparing the waveforms of intact cells and isolated reactivated axonemes, I reveal the effect of hydrodynamic loading, and the influence of boundary conditions on the shape of the beating flagella.:Contents 1 Introduction. . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.1 Biology of Cilia and Flagella . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.1.1 The dimensions of flagellated micro-swimmers . . . . . . . . . . . . . . . . . 4 1.1.2 The symmetric and the asymmetric beat . . . . . . . . .. . . . . . . . . . . . 5 1.1.3 Chlamydomonas reinhardtii as a flagella model . . . . . . . . . . 5 1.2 The axoneme is the internal structure in eukaryotic cilia and flagella . . 6 1.3 Structure and function of microtubules and dyneins . . . . . . . . . . . 9 1.3.1 Microtubules: The structural elements in the axoneme . . . . . . 9 1.3.2 Dyneins: The force generators that drive the axonemal beat . . . 10 1.3.3 The asymmetries in the axoneme and consequences for the beat 17 1.4 Axonemal waveform models and mechanisms: from sliding to bending to beating . . . . . . . . . . . . . . 20 1.5 Geometrical representation and parameterization of the axonemal beat . . . . . . . . . . . . . . . 23 2 Questions addressed in this thesis . . . . . . . . . . . . . . 27 3 Material and Methods . . . . . . . . . . . . . . 29 3.1 Chlamydomonas cells: Axoneme preparation and motility assays . . . . 29 3.1.1 Culturing of Chlamydomonas reinhardtii cells . . . . . . . . . . . 29 3.1.2 Isolation, demembranation and storage of axonemes . . . . . . . 33 3.1.3 Reactivation of axonemes in controlled conditions . . . . . . . . . 35 3.1.4 Axoneme gliding assay using kinesin 1 . . . . . . . . . . . . . . . 36 3.2 Imaging and image processing . . . . . . . . . . . . . . . . . . . . . . . . 38 3.2.1 High-speed imaging of the flagella and axonemes . . . . . . . . . 38 3.2.2 Precise tracking of isolated axonemes and the flagella of cells . . 42 3.2.3 High throughput frequency evaluation of isolated axonemes . . . 47 3.2.4 Beat frequency characterization of the reactivated WT axoneme . . . . . . . . . . . . . . 49 4 Results and Discussion . . . . . . . . . . . . . . 53 4.1 The beat of the axoneme propagates from base to tip . . . . . . . . . . . 53 4.1.1 TEM study reveals no sliding at the base of a bend axoneme . . 53 4.1.2 The direction of wave propagation is directly determined from the reactivation of polarity marked axonemes . . . . . . . . . . 57 4.1.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 4.2 The asymmetric beat is the superposition of a static semi-circular arc and a sinusoidal beat . . . . . . .. . . . . . . . . . . . . . . . . 61 4.2.1 The waveform is parameterized by Fourier decomposition in time . . . . . . . . . . . . . . 61 4.2.2 The 0th and 1st Fourier modes describe the axonemal waveform . . . . . . . . . . . . . . 65 4.2.3 General dependence of shape parameters on axoneme length and beat frequency. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 4.2.4 The isolated axoneme is a good model for the intact flagellum . .. . . . . . . . . . . . . . 71 4.2.5 Summary: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 4.3 The circular motion is a consequence of the axonemal waveform . . . . . . . . . . . . . . . . . . . 79 4.3.1 Hydrodynamic relations for small amplitude waves explain the relation between swimming and shape of axonemes . . . . 79 4.3.2 The swimming path can be reconstructed using shape information and a hydrodynamic model . . . . . . . . . . . . . . . . 83 4.3.3 Motor mutations alter the direction of rotation of reactivated axonemes. . . . . . . . . . . . . . . . . . . . . . . . 84 4.3.4 Summary: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 4.4 The molecular origin of the circular mean shape. . . . . . . . . . . . . . 89 4.4.1 Motor Mutations do not abolish the mean shape, a structural mutation does . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 4.4.2 The axoneme is straight in absence of ATP but bend at low ATP concentrations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 4.4.3 Viscous load decreases the mean curvature . . . . . . . . . . . . 99 4.4.4 Summary: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 4.5 Curvature-dependent dynein activation accounts for the shape of the beat of the isolated axoneme . . . . . . . . . . . . . . . . 103 5 Conclusions and Outlook . . . . . . . . . . . . . . . . 109 5.1 Summary of the results . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 5.2 Future directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 Abbreviations . . . . . . . . . . . . . . . . 111 List of figures . . . . . . . . . . . . . . . . 116 List of tables . . . . . . . . . . . . . . . . 118 Bibliography