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. Although 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.

Actin cables are linear cytoskeletal structures that serve as tracks for myosin-based intracellular transport of vesicles and organelles in both yeast and mammalian cells. In a yeast cell undergoing budding, cables are in constant dynamic turnover yet some cables grow from the bud neck toward the back of the mother cell until their length roughly equals the diameter of the mother cell. This raises the question: how is the length of these cables controlled? Here we describe a novel molecular mechanism for cable length control inspired by recent experimental observations in cells. This “antenna mechanism” involves three key proteins: formins, which polymerize actin, Smy1 proteins, which bind formins and inhibit actin polymerization, and myosin motors, which deliver Smy1 to formins, leading to a length-dependent actin polymerization rate. We compute the probability distribution of cable lengths as a function of several experimentally tuneable parameters such as the formin-binding affinity of Smy1 and the concentration of myosin motors delivering Smy1. These results provide testable predictions of the antenna mechanism of actin-cable length control.

The single-celled green algae Chlamydomonas reinhardtii with its two flagella—microtubule-based structures of equal and constant lengths—is the canonical model organism for studying size control of organelles. Experiments have identified motor-driven transport of tubulin to the flagella tips as a key component of their length control. Here we consider a class of models whose key assumption is that proteins responsible for the intraflagellar transport (IFT) of tubulin are present in limiting amounts. We show that the limiting-pool assumption is insufficient to describe the results of severing experiments, in which a flagellum is regenerated after it has been severed. Next, we consider an extension of the limiting-pool model that incorporates proteins that depolymerize microtubules. We show that this ‘active disassembly’ model of flagellar length control explains in quantitative detail the results of severing experiments and use it to make predictions that can be tested in experiments.

Summary How the size of micron-scale cellular structures like the mitotic spindle, cytoskeletal filaments, the nucleus, the nucleolus and other non-membrane bound organelles is controlled despite a constant turnover of their constituent parts is a central problem in biology. Experiments have implicated the limiting-pool mechanism: structures grow by stochastic addition of molecular subunits from a finite pool until the rates of subunit addition and removal are balanced, producing a structure of well-defined size. Here, we consider these dynamics when multiple filamentous structures are assembled stochastically from a shared pool of subunits. Using analytical calculations and computer simulations, we show that robust size control can be achieved when only one filament is assembled at a time. When multiple filaments compete for monomers, filament lengths exhibit large fluctuations. These results extend to three-dimensional structures and reveal the physical limitations of the limiting pool mechanism of size-control when multiple organelles are assembled from a shared pool of subunits.

Control of organelle size is a problem that has intrigued cell biologists for at least a 9 century. The single-celled green algae Chlamydomonas reinhardtii with its two flagella has proved to 10 be a very useful model organism for studies of size control. Numerous experiments have identified 11 motor-driven transport of tubulin to the growing ends of microtubules at the tip of the flagella as 12 the key component of the machinery responsible for controlling their length. Here we consider a 13 model of flagellar length control whose key assumption is that proteins responsible for the 14 intraflagellar transport (IFT) of tubulin are present in limiting amounts. We show that this 15 limiting-pool assumption and simple reasoning based on the law of mass action leads to an inverse 16 relationship between the rate at which a flagellum grows and its length, which has been observed 17 experimentally, and has been shown theoretically to provide a mechanism for length control. 18 Experiments in which one of the two flagella are severed have revealed the coupled nature of the 19 growth dynamics of the two flagella, and we extend our length-control model to two flagella by 20 considering different mechanisms of their coupling. We describe which coupling mechanisms are 21 capable of reproducing the observed dynamics in severing experiments, and why some that have 22 been proposed previously are not. Within our theoretical framework we conclude that if tubulin 23 and IFT proteins are freely exchanged between flagella simultaneous length control is not possible 24 if the disassembly rate is constant. However, if disassembly depends on the concentration of IFT 25 proteins at the tip of the flagellum, simultaneous length control can be achieved. Finally, we make 26 quantitative predictions for experiments that could test this model. 27 28 55 Buisson et al. (2013). IFT particles containing tubulin are transported along the flagellum by two 56 different motor proteins: kinesin-2 transports IFT particles from the flagellar base to tip (the 57 anterograde direction) whereas dynein carries IFT particles from the tip to the base (the retrograde 58 direction). 59 These observations motivated the development of mathematical models of flagellar length 60 dynamics such as the balance point model Marshall and Rosenbaum (2001); Marshall et al. (2005). 61 In the balance point model, the steady-state flagellar length is achieved when the assembly and 62 disassembly processes, which are in continual competition, come into a balance set by the finite pool 63 of tubulin shared between the flagellum and basal body. The balance point model yields a length-64 dependent assembly rate by assuming a constant number of transport complexes moving along the 65 flagellum. This is consistent with experimental evidence that the flagellar assembly rate decreases 66 with length Rosenbaum and Child (1967); Marshall and Rosenbaum (2001). Notably, however, 67 the balance-point does not explain how the cell could maintain a constant number of transport 68 458 A...