Movements of Molecular Motors: Diffusion and Directed Walks
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Movements of Molecular Motors: Diffusion and Directed Walks Dissertation zur Erlangung des akademischen Grades Doktor der Naturwissenschaften (Dr. rer. nat.) in der Wissenschaftsdisziplin Theoretische Physik eingereicht an der Mathematisch{Naturwissenschaftlichen Fakult¨at der Universit¨at Potsdam angefertigt am Max{Planck{Institut fur¨ Kolloid- und Grenzfl¨achenforschung in Golm von Stefan Klumpp geboren am 29. September 1973 in Waiblingen Potsdam, im April 2003 Zusammenfassung Bewegungen von prozessiven molekularen Motoren des Zytoskeletts sind durch ein Wechsel- spiel von gerichteter Bewegung entlang von Filamenten und Diffusion in der umgebenden L¨osung gekennzeichnet. Diese eigentumlichen¨ Bewegungen werden in der vorliegenden Arbeit untersucht, indem sie als Random Walks auf einem Gitter modelliert werden. Ein weiterer Gegenstand der Untersuchung sind Effekte von Wechselwirkungen zwischen den Motoren auf diese Bewegungen. Im einzelnen werden vier Transportph¨anomene untersucht: (i) Random Walks von einzel- nen Motoren in Kompartimenten verschiedener Geometrien, (ii) station¨are Konzentrations- profile, die sich in geschlossenen Kompartimenten infolge dieser Bewegungen einstellen, (iii) randinduzierte Phasenuberg¨ ¨ange in offenen r¨ohrenartigen Kompartimenten, die an Motoren- reservoirs gekoppelt sind, und (iv) der Einfluß von kooperativen Effekten bei der Motor{ Filament-Bindung auf die Bewegung. Alle diese Ph¨anomene sind experimentell zug¨anglich, und m¨ogliche experimentelle Realisierungen werden diskutiert. Abstract Movements of processive cytoskeletal motors are characterized by an interplay between directed motion along filament and diffusion in the surrounding solution. In the present work, these peculiar movements are studied by modeling them as random walks on a lattice. An additional subject of our studies is the effect of motor{motor interactions on these movements. In detail, four transport phenomena are studied: (i) Random walks of single motors in compartments of various geometries, (ii) stationary concentration profiles which build up as a result of these movements in closed compartments, (iii) boundary-induced phase transitions in open tube-like compartments coupled to reservoirs of motors, and (iv) the influence of cooperative effects in motor–filament binding on the movements. All these phenomena are experimentally accessible and possible experimental realizations are discussed. Contents 1 Introduction 1 1.1 Brownian Motion . 1 1.2 Molecular motors . 3 1.2.1 Recent experimental developments . 5 1.2.2 Modeling | separation of scales . 5 1.2.3 Bound and unbound movements . 7 1.3 Overview . 8 2 Lattice models for the walks of molecular motors 9 2.1 Lattice models . 9 2.1.1 Basic lattice model . 10 2.1.2 Master equations . 11 2.1.3 Mapping parameters of motion to the hopping rates . 12 2.1.4 Modified lattice model . 12 2.2 General features of the models . 14 2.3 Interacting motors . 15 2.4 Continuum equations . 16 2.5 Two-state approximation . 17 2.6 Summary . 18 3 Random walks of single motors in open compartments 19 3.1 Anomalous drift: Scaling arguments and simulations . 19 3.2 Analytical solution in two and three dimensions . 22 3.2.1 The two-dimensional case . 23 3.2.2 The three-dimensional case . 27 3.2.3 Sticking probability . 28 3.3 Return to the filament and asymptotic displacement . 30 3.4 Continuum model and open compartments . 31 3.4.1 The two-dimensional case . 31 3.4.2 The three-dimensional case . 32 3.4.3 Open compartments . 33 3.5 Effective diffusion coefficient . 37 3.6 Arrays of parallel filaments . 37 3.7 Summary . 38 4 Concentration profiles in closed compartments 39 4.1 Density profile in a closed tube . 39 4.2 Two-state approximation . 43 4.2.1 Continuous two-state equations . 43 4.2.2 Discrete model . 47 4.3 Depletion layer . 49 4.4 Some modifications of the model . 51 4.5 Density profiles in asters . 53 4.5.1 Two-state equations . 53 4.5.2 Numerical results . 55 4.6 Summary . 56 5 Motor traffic through open tube-like compartments 59 5.1 Open tube systems . 59 5.2 Periodic boundary conditions . 61 5.3 Open boundaries with radial equilibrium . 63 5.3.1 Phase diagram . 63 5.3.2 Density profiles . 66 5.4 Diffusive injection and extraction of motors . 72 5.4.1 Diffusive bottlenecks . 72 5.4.2 Phase diagrams with and without maximal current phase . 73 5.4.3 Active versus thermal unbinding from the 'last' filament site . 75 5.5 Summary . 76 6 Motors with attractive interactions 77 6.1 Interactions of motors . 77 6.2 Attractive interactions . 78 6.3 Two species of motors . 80 6.3.1 Two kinds of phase transitions . 81 6.3.2 Mean field theory . 85 6.3.3 General remarks and extensions . 88 6.4 Summary . 89 7 Conclusions and outlook 91 Appendix 95 A Details of the computer simulations 97 B Theory of random walks 99 B.1 Random walks on homogeneous cubic lattices . 99 B.2 Return to the origin . 100 B.3 Tauberian theorems . 101 B.4 Integrals for the random walks of molecular motors . 102 C Several motors transporting one cargo 103 D More about motor traffic in tube-like compartments 105 D.1 Radial equilibrium for periodic boundary conditions . 105 D.2 Discrete mean field theory for open boundaries with radial equilibrium . 106 D.3 Diffusive injection and extraction of motor particles: The one-dimensional case 110 List of symbols 111 Bibliography 115 Acknowledgments 123 Chapter 1 Introduction Movement is one of the signatures of life and can be found on various length scales in the living world, from motion of whole organisms to movements of single biomolecules. In the following we will study movements of biomolecules within the cell or in vitro. There are two fundamentally distinct kinds of movements of biomolecules, organelles and cells: passive diffusion or Brownian motion and active motion powered by chemical free energy. Brownian motion affects every sufficiently small particle and is ubiquitous in biological systems [1]. In contrast, active motion depends on specific molecules, called molecular motors, which are able to convert chemical free energy into movement or mechanical work [2,3]. In the following chapters, we will focus on processive cytoskeletal motors. These motors move unidirectionally along filaments of the cytoskeleton, hydrolyzing the fuel molecule adeno- sine triphosphate (ATP) and converting the released free energy into directed motion. The best studied examples are kinesins, which walk along microtubules, and certain types of myosins, which move along actin filaments. In the cell, these motors are involved in transport of vesicles and segregation of chromosomes. In addition to the distinction between active and passive movements, these movements can further be classified according to their dimensionality, i.e. whether they occur in three- dimensional space, on two-dimensional surfaces, or along one-dimensional lines. The latter cases are relevant for particles that can bind to a surface or line, in which case their move- ments are restricted to these low-dimensional manifolds. In principle, both active and passive movements are possible in all three cases. Cytoskeletal motors thus perform active movement along one-dimensional filaments. However, if such a motor is observed over a sufficiently long time, it will detach from its filamentous track, since its binding energy can be overcome by thermal fluctuations. The unbound motor then performs passive diffusion in the surrounding fluid, until it eventually reattaches to the same or another filament. This interplay between directed active movement along filaments and free diffusion will be the central subject of the following chapters, where we will discuss several transport phenomena related to this combined type of motion. In the remaining sections of this introduction, we will introduce the molecular motors and their passive and active movements in more detail and give an overview of the following chapters. 1.1 Brownian Motion In 1828 Robert Brown observed that small particles extracted from pollen of various plants performed erratic movements in water. He first thought that he had observed self-animated 2 Introduction particles or "active molecules" [4], which at that time were believed by some biologists to be the constituents of organic bodies. Motion of these self-animated active particles was believed to be driven by a kind of internal living force specific for the organic world. During his further studies, however, Brown realized that particles which moved, when they were suspended in water, could be obtained from almost every substance, from dried plants and vegetable products to various kinds of minerals. Hence, contrary to his first conjecture, his observations showed that Brownian motion is a generic physical phenomenon rather than a biological one [5]. It was soon proposed that Brownian motion is related to heat, since it is affected by temperature changes [5], but quantitative explanations of the phenomenon failed until the works of Einstein [6] and Smoluchowsky [7] at the beginning of the 20th century. The crucial point of the explanation is the identification of these movements as fluctuation phenomena: while the random collisions with the thermally moving particles of the surrounding fluid do not cause motion on average, the fluctuations due to these collisions are visible. The variance of the position of the Brownian particle rather than its average position increases linearly with time. This linear behaviour is governed by the diffusion coefficient D, which is given by the famous Stokes{Einstein relation [1], k T D = B : (1.1) 6πηRhyd It depends on the thermal energy kBT , the viscosity η of the fluid and the effective hydrody- namic radius Rhyd of the Brownian particle. Since a Brownian particle explores the whole accessible space, reduction of the dimen- sionality can enhance diffusion-driven processes such as collisions of Brownian particles or diffusion-controlled chemical reactions [8]. In biological systems, reduction of the dimension- ality occurs if the Brownian particle binds to a surface such as the membrane of a cell or to one-dimensional objects such as cytoskeletal filaments or DNA, so that its motion is restricted to directions along these manifolds.