Design Principles of Length Control of Cytoskeletal Structures

Design Principles of Length Control of Cytoskeletal Structures Lishibanya Mohapatra1, Bruce L. Goode2, Predrag Jelenkovic3, Rob Phillips4, Jane Kondev5,* 1Department of Physics, Brandeis University, Waltham, MA, USA lishi87@brandeis.edu 2Department of Biology and Rosenstiel Basic Medical Sciences Research Center, Brandeis University, Waltham, MA, USA goode@brandeis.edu 3Department of Electrical Engineering, Columbia University, New York, NY predrag@ee.columbia.edu 4Department of Applied Physics and Division of Biology and Biological Engineering, California Institute of Technology, Pasadena, CA, USA phillips@pboc.caltech.edu 5Department of Physics, Brandeis University, Waltham, MA, USA kondev@brandeis.edu

• Corresponding author; Department of Physics, MS 057, Brandeis University, Waltham, MA 02454. Phone (781) 736-2812. Fax (781) 736-2915. E-mail kondev@brandeis.edu Shortened running title Length Control of Cytoskeletal Structures

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Table of contents

1. Introduction 1.1 Case studies of cytoskeletal structures 1.2 Length dependent assembly and disassembly rates 1.3 Master equation for filament length 1.4 Solution of the master equation
2. Mechanisms of length control 2.1 Unregulated filament 2.2 Length control by assembly
   2.2.1 Finite subunit pool mechanism
   2.2.2 Elongators and dampers 2.2.3 Active transport of monomers
   2.3 Length control by disassembly 2.3.1 Depolymerizers 2.3.2 Severing
3. Discussion 3.1 Experimental signatures of different length control mechanisms 3.2 The problem of multiple cytoskeleton structures 3.3 Size control of non-cytoskeletal structures in the cell

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Keywords: Organelle size, Actin filaments, Microtubules, Living Polymers, Master equation Abstract: Cells contain elaborate and interconnected networks of protein polymers which make up the cytoskeleton. The cytoskeleton governs the internal positioning and movement of vesicles and organelles, and controls dynamic changes in cell polarity, shape and movement. Many of these processes require tight control of the size and shape of cytoskeletal structures, which is achieved despite rapid turnover of their molecular components. Here we review mechanisms by which cells control the size of filamentous cytoskeletal structures from the point of view of simple quantitative models that take into account stochastic dynamics of their assembly and disassembly. Significantly, these models make experimentally testable predictions that distinguish different mechanisms of length-control. While the primary focus of this review is on cytoskeletal structures, we believe that the broader principles and mechanisms discussed herein will apply to a range of other subcellular structures whose sizes are tightly controlled and are linked to their functions.

1. Introduction
   A remarkable feature of all living cells is that they have a variety of distinguishable subcellular parts (organelles) with characteristic sizes and shapes. These structures have been observed since the dawn of microscopy and yet it is only recently that we have developed experimental tools to address key questions, such as: How do organelles obtain their specific shapes, and how do cells control their number and size? For example, how does a cell ‘decide’ how many mitochondria or centrioles should it have? Or, how does a cell construct structures with precisely arranged parts, 4

such as sarcomeres in muscle with its regimented arrays of actin filaments interdigitated with myosin fibers? The cytoskeleton provides a particularly fruitful arena to develop quantitative models that address these questions of morphology, in light of the wealth of quantitative information about its structure and dynamics at the molecular level. In this review, we use theory as a guide and a common language for describing the various size control mechanisms that have been proposed recently for diverse cytoskeleton structures. By reviewing the field from the point of view of simple models we hope to identify fruitful directions for new experiments.

The cytoskeleton consists of a number of organelles and substructures that seem to be designed with a precise size and geometry, suggesting that these physical properties are intimately tied to their biological functions. Examples include cytoskeletal structures such as the mitotic spindle, actin cables, and cilia. The majority of cytoskeleton structures are comprised of protein polymers such as microtubules and actin filaments, which are themselves made up of simple building block proteins such as tubulin dimers and actin monomers, respectively. How these structures are able to maintain a remarkably constant size despite undergoing highly dynamic turnover of their components is still not well understood.

1.1 Case studies of cytoskeletal structures In cells, we find numerous examples of cytoskeletal structures with sizes that are dictated by the particular cellul

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