Cells assemble microns-long filamentous structures from protein monomers that are nanometers in size. These structures are often highly dynamic, yet in order for them to function properly, cells maintain them at a precise length. Here we investigate length-dependent depolymerization as a mechanism of length control. This mechanism has been recently proposed for flagellar length control in the single cell organisms Chlamydomonas and Giardia. Length dependent depolymerization can arise from a concentration gradient of a depolymerizing protein, such as kinesin-13 in Giardia, along the length of the flagellum. Two possible scenarios are considered: a linear and an exponential gradient of depolymerizing proteins. We compute analytically the probability distributions of filament lengths for both scenarios and show how these distributions are controlled by key biochemical parameters through a dimensionless number that we identify. In Chlamydomonas cells, the assembly dynamics of its two flagella are coupled via a shared pool of molecular components that are in limited supply, and so we investigate the effect of a limiting monomer pool on the length distributions. Finally, we compare our calculations to experiments. While the computed mean lengths are consistent with observations, the noise is two orders of magnitude smaller than the observed length fluctuations.

Intracellular protein gradients serve a variety of functions, such as the establishment of cell polarity or to provide positional information for gene expression in developing embryos. Given that cell size in a population can vary considerably, for the protein gradients to work properly they often have to be scaled to the size of the cell. Here, we examine a model of protein gradient formation within a cell that relies on cytoplasmic diffusion and cortical transport of proteins toward a cell pole. We show that the shape of the protein gradient is determined solely by the cell geometry. Furthermore, we show that the length scale over which the protein concentration in the gradient varies is determined by the linear dimensions of the cell, independent of the diffusion constant or the transport speed. This gradient provides scale-invariant positional information within a cell, which can be used for assembly of intracellular structures whose size is scaled to the linear dimensions of the cell, such as the cytokinetic ring and actin cables in budding yeast cells.

5Cells assemble microns-long filamentous structures from protein monomers that are nanometers 6 in size. These structures are often highly dynamic, yet in order for them to function properly, 7 cells maintain precise filament lengths. In this paper, we investigate length dependent 8 depolymerization as a mechanism of length control. This mechanism has been recently proposed 9 for flagellar length control in the single cell organisms Chlamydomonas and Giardia. Length 10 dependent depolymerization can arise from a concentration gradient of a depolymerizing protein, 11 such as kinesin-13 in Giardia, along the length of the flagellum. Two possible scenarios are 12 considered, that of a linear and an exponential gradient of depolymerizing proteins. We compute 13 analytically the probability distributions of filament lengths for both scenarios and show how 14 these distributions are controlled by key biochemical parameters through a dimensionless 15 number that we identify. In Chlamydomonas cells it has been well documented that the assembly 16 dynamics of its two flagella are coupled via a shared pool of molecular components, and so we 17 investigate the effect of a limiting monomer pool on the length distributions. Finally, we 18 compare our calculations to experiments. While the computed mean lengths are consistent with 19 observations, the noise is two orders of magnitude smaller than the observed length fluctuations. 20 21 I. Introduction 22All eukaryotic cells contain nanometer-sized proteins that polymerize to form filaments that are 23 hundreds of nanometers to tens of microns in length. These filaments often associate with other 24 proteins to form larger structures that perform critical roles in cell division, cell motility, and 25 intracellular transport. They are often highly dynamic, experiencing a high turnover rate of their 26 constitutive protein subunits [1,2]. Despite this, in order to function properly some of these 27 structures, like the mitotic spindle, or actin cables in budding yeast[2,3] must maintain specific 28 and well-defined sizes. 29When considering dynamics of filament length, the two basic processes are the addition of 30 monomer units, characterized by the rate of assembly, and the removal of these units, which is 31 described by the rate of disassembly. For the dynamics to reach a steady state with a well-32 defined length, one or both rates must be length dependent. Therefore, the search for the 33 molecular mechanism of length regulation often boils down to finding out whether the rate of 34 2 assembly or disassembly (or both) is length dependent, and what molecular-scale interactions 35 produce this length dependence [4,5]. 36 Many mechanisms for length regulation have been proposed based on careful experiments in 37 cells and on reconstituted filamentous structures in vitro[3,6-9]. One well studied mechanism 38 [10] of length control is to limit the number of subunits available for assembly. For example, in 39 vitro experiments have demonstrated how the size of the mitoti...

Intracellular protein gradients serve a variety of functions, such as the establishment of cell polarity and to provide positional information for gene expression in developing embryos. Given that cell size in a population can vary considerably, for the protein gradients to work properly they often have to be scaled to the size of the cell. Here we examine a model of protein gradient formation within a cell that relies on cytoplasmic diffusion and cortical transport of proteins toward a cell pole. We show that the shape of the protein gradient is determined solely by the cell geometry. Furthermore, we show that the length scale over which the protein concentration in the gradient varies is determined by the linear dimensions of the cell, independent of the diffusion constant or the transport speed. This gradient provides scale-invariant positional information within a cell, which can be used for assembly of intracellular structures whose size is scaled to the linear dimensions of the cell, as was recently reported for the cytokinetic ring and actin cables in budding yeast cells.

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