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Theoretical Aspects of Motor Protein Induced Filament Depolymerisation

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Theoretical Aspects of Motor Protein Induced Filament Depolymerisation Institut f¨ur Theoretische Physik Fakult¨at Mathematik und Naturwissenschaft Technische Universit¨at Dresden Theoretical aspects of motor protein induced filament depolymerisation Dissertation Zur Erlangung des akademischen Grades Doktor der Naturwissenschaften (Dr. rer. nat.) angefertigt am Max Planck Institut f¨ur Physik komplexer Systeme Dresden vorgelegt von Gernot A. Klein Dresden, 2005 Contents 1 Introduction 1 2 The cytoskeleton and motor proteins 5 2.1 Cytoskeletalfilaments . ... 5 2.1.1 Structure and organisation of actin and intermediate filaments . 6 2.1.2 Assembly and structure of microtubules . ..... 8 2.2 Cytoskeletal motor proteins . ..... 10 2.2.1 Myosinanddynein............................. 11 2.2.2 Kinesin ................................... 12 2.3 Microtubuledynamicsandstructures. ....... 14 2.3.1 Force generation by polymerising filaments . ...... 14 2.3.2 Dynamic instability of growing microtubules . ....... 16 2.3.3 Reorganisation of microtubules during mitosis . ......... 17 2.4 Microtubule-depolymerising motor proteins . .......... 18 2.4.1 Properties of kinesin-13 motors . 19 2.4.2 Mechanism of microtubule depolymerisation . ....... 20 2.4.3 Cellularfunctions. .. .. .. .. .. .. .. .. 22 2.5 Theoretical description of motor proteins . ......... 23 2.5.1 Phenomenological approach . 23 2.5.2 Isothermalratchetmodels . 25 2.5.3 Stochastic models on discrete lattices . ....... 26 3 General description of motor induced filament depolymerisation 29 3.1 Filament depolymerisation . ..... 29 i ii Contents 3.1.1 Dynamics of motor densities . 30 3.1.2 Motor induced filament depolymerisation . ..... 31 3.1.3 Processivesubunitremoval . 32 3.2 Steady-statesolutions . 32 3.2.1 Density profiles of motor proteins . 33 3.2.2 Stabilityofsolutions . 34 3.3 Dynamic accumulation of motor proteins . ....... 36 3.3.1 Mechanisms of motor accumulation at filament ends . ...... 36 3.3.2 Influence of collective effects . 37 3.3.3 Roleofprocessivity ............................ 41 3.4 Instabilities due to collective effects . .......... 42 3.5 Summary ...................................... 45 4 Microscopic mechanisms and fluctuations 47 4.1 Lattice description of motor dynamics . ....... 47 4.1.1 Microscopic interactions . 48 4.1.2 Influenceofcrowding. 49 4.2 Masterequationapproach . 51 4.3 Mean-fieldapproximation . 52 4.3.1 Continuum limit and mean-field approximation . ...... 52 4.3.2 Connection to generic description . ..... 54 4.4 Depolymerisation mechanisms influencing accumulation ............ 56 4.4.1 Crowding hindering depolymerisation . ..... 57 4.4.2 Crowdingreducingprocessivity . 58 4.5 Effectsoffluctuations ............................. 60 4.6 Summary ...................................... 60 5 Effects of multiple protofilaments 63 5.1 Coupling of sublattices by motor dynamics . ....... 63 5.2 Effectsonaccumulation . .. .. .. .. .. .. .. .. 66 5.2.1 Effects of inter-sublattice processivity . ........ 67 5.2.2 Regimes of accumulation . 68 5.3 Edge structure of multiple sublattices . ........ 69 5.3.1 Uncoupledsublattices . 70 Contents iii 5.3.2 Coupledsublattices . .. .. .. .. .. .. .. 71 5.4 Ringcurrentsofmotors ............................ 75 5.5 Mean-field description with effective parameters . ........... 76 5.6 Summary ...................................... 80 6 Motor induced filament depolymerisation under applied forces 81 6.1 Genericdescription.............................. 81 6.1.1 Collective effects of bound motors . 82 6.1.2 Dynamicsofunboundmotors . 84 6.1.3 Forcebalance................................ 84 6.2 Force-velocity relationships . ....... 85 6.2.1 Depolymerisation velocity . 85 6.2.2 Critical detachment force . 88 6.3 Roleoffluctuations............................... 88 6.3.1 Evolutionequations . .. .. .. .. .. .. .. 88 6.3.2 Comparison with generic theory . 91 6.4 Summary ...................................... 94 7 Summary and Perspectives 97 A Parameter determination in the generic description 101 B Master and evolution equations for the one-dimensional discrete model 107 C Subunit labelling and rate determination on an array of sublattices 109 D Master equations for the two-dimensional discrete model 113 E Random deposition model 117 Bibliography 119 1 Introduction Many active biological processes, for example cell locomotion, the transport of organelles inside the cell, and cell division, are driven by highly specialised motor proteins which interact with the cytoskeleton. The cytoskeleton is a network of filamentous structures which are linear aggregates of proteins, for example actin and tubulin. In a living cell, actin filaments and microtubules are dynamic structures and can rapidly change their lengths by addition and removal of subunits at the ends, allowing their use for a wide range of different tasks. Because they have two structurally distinguishable ends, these filaments exhibit an asymmetry along their lattice, determining the direction of motion of, and force generation by bound motor proteins. These motor proteins are able to transduce the chemical energy of a fuel called ATP to do mechanical work while interacting with a filament. In addition to generating forces along filaments, it is now a well-established fact that cer- tain motor proteins can also interact with filament ends, where they influence polymerisation and depolymerisation. Examples of depolymerising proteins are certain members of the ki- nesin family, with members of the KIN-13 subfamily being the best-studied so far, which are capable of depolymerising microtubules and regulating microtubule lengths in a cell. One role for KIN-13 family members is thought to be in the mitotic spindle, a microtubule structure that is formed in the process of cell division and is responsible for separation and distribution of the duplicated genetic material to the forming daughter cells. The accurate distribution of the genetic material to daughter cells is essential for the survival of the cells and the whole organism. During the process of division, shortening microtubules generate forces which pull the chromosomes to which they are attached towards the opposing poles of the cell. Recent experimental studies have started to illuminate the essential contributions of mem- bers of the KIN-13 subfamily to the assembly of the mitotic spindle and subsequent chromo- some segregation, for example correcting the erroneous attachment ofmicrotubules to chro- mosomes and producing chromosome movement towards the poles of the cell. In addition, in vitro assays have elucidated kinetic properties of these proteins and have produced aston- ishing results, notably protein accumulation at depolymerised filament ends, the possibility of processive subunit removal and surprisingly fast filament end finding by these proteins. These and further aspects of the biological background to this work is given in chapter 2. 1 2 Introduction Chapter 1 In contrast to the ongoing experimental engagement, no theoretical attempts to find un- derlying mechanisms and principles, which could help explain these phenomena, has been made so far. The aim of this work therefore is to develop a theoretical framework capable of describing experimentally observed behaviour, shed light on underlying principles of motor induced subunit removal and, going a step further, make predictions about how these pro- teins may behave in situations not yet experimentally investigated. As these motor proteins consume energy in order to induce subunit removal, they represent a non-equilibrium system and the observed phenomena cannot be described with the standard concepts of equilibrium statistical mechanics. In the following work we will therefore use two different theoretical approaches to describe motor dynamics, introduced in chapter 3 and chapter 4. In chapter 3 we will describe motor dynamics by phenomenological continuum equations. Here, instead of describing individual motor proteins, motor densities are considered and fluctuations due to the discrete number of proteins are neglected. In general, this kind of description is applicable for spatially extended systems and for describing dynamic behaviour on length scales which are large compared to the underlying molecular length scales. One of the main advantages of this approach is the minor amount of microscopic information needed, in order to formulate equations which correctly describe the system. Therefore, the continuum equations themselves are to a large extent independent of the underlying molecular details of the system, whereas the specific values of the macroscopic parameters entering the equations are determined by these details. In chapter 4 on the other hand, we take the discrete nature of the motor proteins explicitly into account, leading to a stochastic description of motors on a filament. Here, motor proteins are represented by identical particles moving stochastically on a discrete lattice. This kind of description is part of a class of driven lattice gas models which are used to study a broad range of transport phenomena, including the movement of motor proteins on filaments, the diffusion of ions in channels and the traffic flow of cars on highways. This kind of description enables us to discuss the effects of different microscopic mechanisms of subunit removal and to investigate the role of fluctuations on the dynamic behaviour of motor proteins. These are expected to become important if motor dynamics are observed on length scales comparable to the underlying molecular length scales, or if the number of observed proteins becomes small. A discrete stochastic
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