Uncorrelated changes of subunit stability can generate length-dependent disassembly of treadmilling filaments

The effect of motor-induced shaft dynamics on microtubule stability and length

Control of microtubule abundance, stability, and length is crucial to regulate intracellular transport as well as cell polarity and division. How microtubule stability depends on tubulin addition or removal at the dynamic ends is well studied. However, microtubule rescue, the event when a microtubule switches from shrinking to growing, occurs at tubulin exchange sites along the shaft. Molecular motors have recently been shown to promote such exchanges. Using a stochastic theoretical description, we study how microtubule stability and length depend on motor-induced tubulin exchange and thus rescue. Our theoretical description matches our in vitro experiments on microtubule dynamics in the presence of kinesin-1 molecular motors. Although the overall dynamics of a population of microtubules can be captured by an effective rescue rate, by assigning rescue to exchange sites, we reveal that the dynamics of individual microtubules within the population differ dramatically. Furthermore, we study in detail a transition from bounded to unbounded microtubule growth. Our results provide novel insights into how molecular motors imprint information of microtubule stability on the microtubule network.

Size regulation of multiple organelles competing for a limiting subunit pool

How cells regulate the size of intracellular structures and organelles is a longstanding question. Recent experiments suggest that size control of intracellular structures is achieved through the depletion of a limiting subunit pool in the cytoplasm. While the limiting pool model ensures organelle-to-cell size scaling, it does not provide a mechanism for robust size control of multiple co-existing structures. Here we develop a generalized theory for size-dependent growth of intracellular structures to demonstrate that robust size control of multiple intracellular structures, competing for a limiting subunit pool, is achieved via a negative feedback between the growth rate and the size of the individual structure. This design principle captures size maintenance of a wide variety of subcellular structures, from cytoskeletal filaments to three-dimensional organelles. We identify the feedback motifs for structure size regulation based on known molecular processes, and compare our theory to existing models of size regulation in biological assemblies. Furthermore, we show that positive feedback between structure size and growth rate can lead to bistable size distribution and spontaneous size selection.

Organelle size control is essential for the proper physiological functioning of eukaryotic cells, but the underlying mechanisms of size regulation remain poorly understood. By developing a general theory for organelle size control, we show that robust size control of intracellular structures and organelles is achieved via a negative feedback between individual organelle size and their net growth rates. This design principle not only describes size maintenance of single organelles, but also ensures size stability of multiple co-existing organelles that are built from a limiting pool of subunits. Our results delineate the role of limiting pool as a size scaling mechanism rather than a size control mechanism, supporting the idea that negative feedback control of organelle size via depletion of a limiting subunit pool is not sufficient to maintain the size of multiple competing organelles. In the case of positive feedback between organelle size and growth rate, our model reproduces phenomena such as bistability in organelle size distribution and spontaneous emergence of cell polarity.

Emergence and maintenance of variable-length actin filaments in a limiting pool of building blocks

Actin is one of the key structural components of the eukaryotic cytoskeleton that regulates cellular architecture and mechanical properties. Dynamic regulation of actin filament length and organization is essential for the control of many physiological processes including cell adhesion, motility and division. While previous studies have mostly focused on the mechanisms controlling the mean length of individual actin filaments, it remains poorly understood how distinct actin filament populations in cells maintain different lengths using the same set of molecular building blocks. Here we develop a theoretical model for the length regulation of multiple actin filaments by nucleation and growth rate modulation by actin binding proteins in a limiting pool of monomers. We first show that spontaneous nucleation of actin filaments naturally leads to heterogeneities in filament length distribution. We then investigate the effects of filament growth inhibition by capping proteins and growth promotion by formin proteins on filament length distribution. We find that filament length heterogeneity can be increased by growth inhibition, whereas growth promoters do not significantly affect length heterogeneity. Interestingly, a competition between filament growth inhibitors and growth promoters can give rise to bimodal filament length distribution as well as a highly heterogeneous length distribution with large statistical dispersion. We quantitatively predict how heterogeneity in actin filament length can be modulated by tuning F-actin nucleation and growth rates in order to create distinct filament subpopulations with different lengths. SIGNIFICANCE: Actin filaments organize into different functional network architectures within eukaryotic cells. To maintain distinct actin network architectures, it is essential to regulate the lengths of actin filaments. While the mechanisms controlling the lengths of individual actin filaments have been extensively studied, the regulation of length heterogeneity in actin filament populations is not well understood. Here we show that the modulation of actin filament growth and nucleation rates by actin binding proteins can regulate actin length distribution and create distinct sub-populations with different lengths. In particular, by tuning concentrations of formin, profilin and capping proteins, various aspects of actin filament length distribution can be controlled. Insights gained from our results may have significant implications for the regulation of actin filament length heterogeneity and architecture within a cell.

Cooperative motor action to regulate microtubule length dynamics

Motivated by the recent experimental observations on motor induced cooperative mechanism controlling the length dynamics of microtubules (MTs), we examine how plus-end-targeted proteins of the kinesin family regulate MT polymerization and depolymerization routines. Here, we study a stochastic mathematical model capturing the unusual form of collective motor interaction on MT dynamics originating due to the molecular traffic near the MT tip. We provide an extensive analysis of the joint effect of motor impelled MT polymerization and complete depolymerization. The effect of the cooperative action is included by modifying the intrinsic depolymerization rate. We analyze the model within the framework of continuum mean-field theory and the resultant steady-state analytic solution is expressed in terms of Lambert W functions. Four distinct steady-state phases including a shock phase have been reported. The significant features of the shock including its position and height have been analyzed. Theoretical outcomes are supported by extensive Monte Carlo simulations. To explore the system alterations between the regime of growth and shrinkage phase, we consider kymographs of the microtubule along with the length distributions. Finally, we investigated the dependence of MT length kinetics both on modifying factor of depolymerization rate and motor concentration. The overall extensive study reveals that the flux of molecular traffic at the microtubule plus end initiates a cooperative mechanism, resulting in significant change in MT growth and shrinkage regime as also observed experimentally.

FtsZ filaments have the opposite kinetic polarity of microtubules

FtsZ is the ancestral homolog of tubulin and assembles into the Z ring that organizes the division machinery to drive cell division in most bacteria. In contrast to tubulin that assembles into 13 stranded microtubules that undergo dynamic instability, FtsZ assembles into single-stranded filaments that treadmill to distribute the peptidoglycan synthetic machinery at the septum. Here, using longitudinal interface mutants of FtsZ, we demonstrate that the kinetic polarity of FtsZ filaments is opposite to that of microtubules. A conformational switch accompanying the assembly of FtsZ generates the kinetic polarity of FtsZ filaments, which explains the toxicity of interface mutants that function as a capper and reveals the mechanism of cooperative assembly. This approach can also be employed to determine the kinetic polarity of other filament-forming proteins.

Design Principles of Length Control of Cytoskeletal Structures

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.

Antenna Mechanism of Length Control of Actin Cables

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.

Based on published cell experiments, we propose a novel mechanism of length control of actin cables in budding yeast cells. The key feature of this “antenna mechanism” is negative feedback of the cable length on the activity of formins, which are proteins that attach to the growing ends of actin filaments and catalyse their polymerization. We recently showed that the protein Smy1 is critical for maintaining proper cable length in yeast cells. Smy1 proteins are delivered to the formins by directed motion of myosin motors toward the growing end, and they transiently inhibit actin cable polymerization when bound to the formins. This provides negative feedback resulting in an average rate of cable assembly that diminishes with cable length. Here we incorporate this antenna mechanism into a physical model of cable polymerization and provide experimentally testable predictions for the dependence of the length distribution of cables on the concentration of Smy1, and on mutations that affect its affinity to formins.

Treadmilling and length distributions of active polar filaments

The cytoskeleton is a network of filamentous proteins, notably, actin filaments and microtubules. These filaments are active as their assembly is driven by the hydrolysis of nucleotides bound to the constituting protomers. In addition, the assembly kinetics differs at the two respective ends, making them active polar filaments. Experimental evidence suggests, that, in vivo, actin filaments and microtubules can grow at one and shrink at the other end at the same rate, a state that is known as treadmilling. In this work, we use a generic discrete two-state model for active polar filaments to analyze the conditions leading to treadmilling. We find that a single filament can self-organize into the treadmilling state for a broad range of monomer concentrations. In this regime the corresponding length distribution has a pronounced maximum at a finite value. We then extend our description to consider specifically the dynamics of actin filaments. We show that actin treadmilling should be observable in vitro in the presence of appropriate depolymerization promoting factors.

Continuum descriptions of cytoskeletal dynamics

This tutorial presents an introduction into continuum descriptions of cytoskeletal dynamics. In contrast to discrete models in which each molecule keeps its identity, such descriptions are given in terms of averaged quantities per unit volume like the number density of a certain molecule. Starting with a discrete description for the assembly dynamics of cytoskeletal filaments, we derive the continuity equation, which serves as the basis of many continuum theories. We illustrate the use of this approach with an investigation of spontaneous cytoskeletal polymerization waves. Such waves have by now been observed in various cell types and might help to orchestrate cytoskeletal dynamics during cell spreading and locomotion. Our analysis shows how processes at the scale of single molecules, namely, the nucleation of new filaments and filament treadmilling, can lead to the spontaneous appearance of coherent traveling waves on scales spanning many filament lengths. For readers less familiar with calculus, we include an informal introduction to the Taylor expansion.

Theoretical analysis of microtubules dynamics using a physical-chemical description of hydrolysis

Microtubules are cytoskeleton multifilament proteins that support many fundamental biological processes such as cell division, cellular transport, and motility. They can be viewed as dynamic polymers that function in nonequilibrium conditions stimulated by hydrolysis of GTP (guanosine triphosphate) molecules bound to their monomers. We present a theoretical description of microtubule dynamics based on discrete-state stochastic models that explicitly takes into account all relevant biochemical transitions. In contrast to previous theoretical analysis, a more realistic physical-chemical description of GTP hydrolysis is presented, in which the hydrolysis rate at a given monomer depends on the chemical composition of the neighboring monomers. This dependence naturally leads to a cooperativity in the hydrolysis. It is found that this cooperativity significantly influences all dynamic properties of microtubules. It is suggested that the dynamic instability in cytoskeleton proteins might be observed only for weak cooperativity, while the strong cooperativity in hydrolysis suppresses the dynamic instability. The presented microscopic analysis is compared with existing phenomenological descriptions of hydrolysis processes. Our analytical calculations, supported by computer Monte Carlo simulations, are also compared with available experimental observations.

Impact of motor molecules on the dynamics of treadmilling filaments

We study a driven lattice gas model for the length dynamics of treadmilling filaments in the presence of molecular motors. A treadmilling filament grows by subunit addition at one end and shrinks by subunit removal at the other. Molecular motors can attach to the filament, move towards the shrinking end, and detach from the filament. We consider motors that are also capable of inducing subunit removal at the shrinking filament end. Stochastic simulations reveal a phase of unimodal length distribution and a phase of unbounded growth. Exploiting a condition on the motor flux, we explore the system's phase diagram. In certain limits we can define random walks that allow us to estimate the full length distribution. The width of steady state distributions decreases with increasing motor activity. Our analysis indicates possible ways that cells can use to regulate the size of cytoskeletal structures such as mitotic spindles by controlling various motor properties.

Microtubule length regulation by molecular motors

Length regulation of microtubules (MTs) is essential for many cellular processes. Molecular motors like kinesin-8, which move along MTs and also act as depolymerases, are known as key players in MT dynamics. However, the regulatory mechanisms of length control remain elusive. Here, we investigate a stochastic model accounting for the interplay between polymerization kinetics and motor-induced depolymerization. We determine the dependence of MT length and variance on rate constants and motor concentration. Moreover, our analyses reveal how collective phenomena lead to a well-defined MT length.

Length regulation of active biopolymers by molecular motors

For biopolymers like cytoskeletal actin filaments and microtubules, assembly and disassembly are inherently dissipative processes. Molecular motors can affect the rates of subunit removal at filament ends. We introduce a driven lattice-gas model to study the effects of motor-induced depolymerization on the length of active biopolymers and find that increasing motor activity sharpens unimodal steady-state length distributions. Furthermore, for sufficiently fast moving motors, the relative width of the length distribution is determined only by the attachment rate of motors. Our results show how established molecular processes can be used to robustly regulate the size of cytoskeletal structures like mitotic spindles.

Theoretical modeling of aging effects in microtubule dynamics

The microtubule (MT) network, an important part of the cytoskeleton, is constantly remodeled by alternating phases of growth and shrinkage of individual filaments. Plus-end tracking proteins (+TIPs) interact with the MT and in many cases alter its dynamics. Although it is established that some +TIPs modify MT dynamics by increasing rescues, the plus-end tracking mechanism is still under debate. We present a model for MT dynamics in which a rescue factor is dynamically added to the filament during growth. As a consequence, the filament shows aging behavior that should be experimentally accessible and thus allow one to exclude some hypothesized models regarding the inclusion of rescue factors at the MT plus end. This result is not limited to +TIPs and can be extended to any kind of mechanism shifting the parameters of dynamic instability. Additionally, we show that the cell geometry has a strong influence on the quantitative results.