Steady states of a microtubule assembly in a confined geometry

Minimal Mechanisms of Microtubule Length Regulation in Living Cells

The microtubule cytoskeleton is responsible for sustained, long-range intracellular transport of mRNAs, proteins, and organelles in neurons. Neuronal microtubules must be stable enough to ensure reliable transport, but they also undergo dynamic instability, as their plus and minus ends continuously switch between growth and shrinking. This process allows for continuous rebuilding of the cytoskeleton and for flexibility in injury settings. Motivated by in vivo experimental data on microtubule behavior in Drosophila neurons, we propose a mathematical model of dendritic microtubule dynamics, with a focus on understanding microtubule length, velocity, and state-duration distributions. We find that limitations on microtubule growth phases are needed for realistic dynamics, but the type of limiting mechanism leads to qualitatively different responses to plausible experimental perturbations. We therefore propose and investigate two minimally-complex length-limiting factors: limitation due to resource (tubulin) constraints and limitation due to catastrophe of large-length microtubules. We combine simulations of a detailed stochastic model with steady-state analysis of a mean-field ordinary differential equations model to map out qualitatively distinct parameter regimes. This provides a basis for predicting changes in microtubule dynamics, tubulin allocation, and the turnover rate of tubulin within microtubules in different experimental environments.

Coupling of mitochondrial population evolution to microtubule dynamics in fission yeast cells: a kinetic Monte Carlo study

Mitochondrial populations in cells are maintained by cycles of fission and fusion events. Perturbation of this balance has been observed in several diseases such as cancer and neurodegeneration. In fission yeast cells, the association of mitochondria with microtubules inhibits mitochondrial fission [Mehta et al., J. Biol. Chem., 2019, 294, 3385], illustrating the intricate coupling between mitochondria and the dynamic population of microtubules within the cell. In order to understand this coupling, we carried out kinetic Monte Carlo (KMC) simulations to predict the evolution of mitochondrial size distributions for different cases; wild-type cells, cells with short and long microtubules, and cells without microtubules. Comparisons are made with mitochondrial distributions reported in experiments with fission yeast cells. Using experimentally determined mitochondrial fission and fusion frequencies, simulations implemented without the coupling of microtubule dynamics predicted an increase in the mean number of mitochondria, equilibrating within 50 s. The mitochondrial length distribution in these models also showed a higher occurrence of shorter mitochondria, implying a greater tendency for fission, similar to the scenario observed in the absence of microtubules and cells with short microtubules. Interestingly, this resulted in overestimating the mean number of mitochondria and underestimating mitochondrial lengths in cells with wild-type and long microtubules. However, coupling mitochondria's fission and fusion events to the microtubule dynamics effectively captured the mitochondrial number and size distributions in wild-type and cells with long microtubules. Thus, the model provides greater physical insight into the temporal evolution of mitochondrial populations in different microtubule environments, allowing one to study both the short-time evolution as observed in the experiments (<5 minutes) as well as their transition towards a steady-state (>15 minutes). Our study illustrates the critical role of microtubules in mitochondrial dynamics and coupling microtubule growth and shrinkage dynamics is critical to predicting the evolution of mitochondrial populations within the cell.

Critical threshold for microtubule amplification through templated severing

The cortical microtubule array of dark-grown hypocotyl cells of plant seedlings undergoes a striking, and developmentally significant, reorientation on exposure to light. This process is driven by the exponential amplification of a population of longitudinal microtubules, created by severing events localized at crossovers with the microtubules of the pre-existing transverse array. We present a dynamic one-dimensional model for microtubule amplification through this type of templated severing. We focus on the role of the probability of immediate stabilization-after-severing of the newly created lagging microtubule, observed to be a characteristic feature of the reorientation process. Employing stochastic simulations, we show that in the dynamic regime of unbounded microtubule growth, a finite value of this probability is not required for amplification to occur but does strongly influence the degree of amplification and hence the speed of the reorientation process. In contrast, in the regime of bounded microtubule growth, we show that amplification only occurs above a critical threshold. We construct an approximate analytical theory, based on a priori limiting the number of crossover events considered, which allows us to predict the observed critical value of the stabilization-after-severing probability with reasonable accuracy.

Microtubule-based actin transport and localization in a spherical cell

The interaction between actin filaments and microtubules is crucial for many eukaryotic cellular processes, such as, among others, cell polarization, cell motility and cellular wound healing. The importance of this interaction has long been recognized, yet very little is understood about both the underlying mechanisms and the consequences for the spatial (re)organization of the cellular cytoskeleton. At the same time, understanding the causes and the consequences of the interaction between different biomolecular components are key questions for in vitro research involving reconstituted biomolecular systems, especially in the light of current interest in creating minimal synthetic cells. In this light, recent in vitro experiments have shown that the actin-microtubule interaction mediated by the cytolinker TipAct, which binds to actin lattice and microtubule tips, causes the directed transport of actin filaments. We develop an analytical theory of dynamically unstable microtubules, nucleated from the centre of a spherical cell, in interaction with actin filaments. We show that, depending on the balance between the diffusion of unbound actin filaments and propensity to bind microtubules, actin is either concentrated in the centre of the cell, where the density of microtubules is highest, or becomes localized to the cell cortex.

Shape and Size Control of Artificial Cells for Bottom-Up Biology

Bottom-up biology is an expanding research field that aims to understand the mechanisms underlying biological processes via in vitro assembly of their essential components in synthetic cells. As encapsulation and controlled manipulation of these elements is a crucial step in the recreation of such cell-like objects, microfluidics is increasingly used for the production of minimal artificial containers such as single-emulsion droplets, double-emulsion droplets, and liposomes. Despite the importance of cell morphology on cellular dynamics, current synthetic-cell studies mainly use spherical containers, and methods to actively shape manipulate these have been lacking. In this paper, we describe a microfluidic platform to deform the shape of artificial cells into a variety of shapes (rods and discs) with adjustable cell-like dimensions below 5 μm, thereby mimicking realistic cell morphologies. To illustrate the potential of our method, we reconstitute three biologically relevant protein systems (FtsZ, microtubules, collagen) inside rod-shaped containers and study the arrangement of the protein networks inside these synthetic containers with physiologically relevant morphologies resembling those found in living cells.

Monte Carlo simulations of microtubule arrays: The critical roles of rescue transitions, the cell boundary, and tubulin concentration in shaping microtubule distributions

Microtubules are dynamic polymers required for a number of processes, including chromosome movement in mitosis. While regulators of microtubule dynamics have been well characterized, we lack a convenient way to predict how the measured dynamic parameters shape the entire microtubule system within a cell, or how the system responds when specific parameters change in response to internal or external signals. Here we describe a Monte Carlo model to simulate an array of dynamic microtubules from parameters including the cell radius, total tubulin concentration, microtubule nucleation rate from the centrosome, and plus end dynamic instability. The algorithm also allows dynamic instability or position of the cell edge to vary during the simulation. Outputs from simulations include free tubulin concentration, average microtubule lengths, length distributions, and individual length changes over time. Using this platform and reported parameters measured in interphase LLCPK1 epithelial cells, we predict that sequestering ~ 15-20% of total tubulin results in fewer microtubules, but promotes dynamic instability of those remaining. Simulations also predict that lowering nucleation rate will increase the stability and average length of the remaining microtubules. Allowing the position of the cell's edge to vary over time changed the average length but not the number of microtubules and generated length distributions consistent with experimental measurements. Simulating the switch from interphase to prophase demonstrated that decreased rescue frequency at prophase is the critical factor needed to rapidly clear the cell of interphase microtubules prior to mitotic spindle assembly. Finally, consistent with several previous simulations, our results demonstrate that microtubule nucleation and dynamic instability in a confined space determines the partitioning of tubulin between monomer and polymer pools. The model and simulations will be useful for predicting changes to the entire microtubule array after modification to one or more parameters, including predicting the effects of tubulin-targeted chemotherapies.

Tracking of plus-ends reveals microtubule functional diversity in different cell types

Many cellular processes are tightly connected to the dynamics of microtubules (MTs). While in neuronal axons MTs mainly regulate intracellular trafficking, they participate in cytoskeleton reorganization in many other eukaryotic cells, enabling the cell to efficiently adapt to changes in the environment. We show that the functional differences of MTs in different cell types and regions is reflected in the dynamic properties of MT tips. Using plus-end tracking proteins EB1 to monitor growing MT plus-ends, we show that MT dynamics and life cycle in axons of human neurons significantly differ from that of fibroblast cells. The density of plus-ends, as well as the rescue and catastrophe frequencies increase while the growth rate decreases toward the fibroblast cell margin. This results in a rather stable filamentous network structure and maintains the connection between nucleus and membrane. In contrast, plus-ends are uniformly distributed along the axons and exhibit diverse polymerization run times and spatially homogeneous rescue and catastrophe frequencies, leading to MT segments of various lengths. The probability distributions of the excursion length of polymerization and the MT length both follow nearly exponential tails, in agreement with the analytical predictions of a two-state model of MT dynamics.

Effects of aging in catastrophe on the steady state and dynamics of a microtubule population

Several independent observations have suggested that the catastrophe transition in microtubules is not a first-order process, as is usually assumed. Recent in vitro observations by Gardner et al. [M. K. Gardner et al., Cell 147, 1092 (2011)] showed that microtubule catastrophe takes place via multiple steps and the frequency increases with the age of the filament. Here we investigate, via numerical simulations and mathematical calculations, some of the consequences of the age dependence of catastrophe on the dynamics of microtubules as a function of the aging rate, for two different models of aging: exponential growth, but saturating asymptotically, and purely linear growth. The boundary demarcating the steady-state and non-steady-state regimes in the dynamics is derived analytically in both cases. Numerical simulations, supported by analytical calculations in the linear model, show that aging leads to nonexponential length distributions in steady state. More importantly, oscillations ensue in microtubule length and velocity. The regularity of oscillations, as characterized by the negative dip in the autocorrelation function, is reduced by increasing the frequency of rescue events. Our study shows that the age dependence of catastrophe could function as an intrinsic mechanism to generate oscillatory dynamics in a microtubule population, distinct from hitherto identified ones.

Dynamics and length distribution of microtubules under force and confinement

We investigate the microtubule polymerization dynamics with catastrophe and rescue events for three different confinement scenarios, which mimic typical cellular environments: (i) The microtubule is confined by rigid and fixed walls, (ii) it grows under constant force, and (iii) it grows against an elastic obstacle with a linearly increasing force. We use realistic catastrophe models and analyze the microtubule dynamics, the resulting microtubule length distributions, and force generation by stochastic and mean field calculations; in addition, we perform stochastic simulations. Freely growing microtubules exhibit a phase of bounded growth with finite microtubule length and a phase of unbounded growth. The main results for the three confinement scenarios are as follows: (i) In confinement by fixed rigid walls, we find exponentially decreasing or increasing stationary microtubule length distributions instead of bounded or unbounded phases, respectively. We introduce a realistic model for wall-induced catastrophes and investigate the behavior of the average length as a function of microtubule growth parameters. (ii) Under a constant force, the boundary between bounded and unbounded growth is shifted to higher tubulin concentrations and rescue rates. The critical force f(c) for the transition from unbounded to bounded growth increases logarithmically with tubulin concentration and the rescue rate, and it is smaller than the stall force. (iii) For microtubule growth against an elastic obstacle, the microtubule length and polymerization force can be regulated by microtubule growth parameters. For zero rescue rate, we find that the average polymerization force depends logarithmically on the tubulin concentration and is always smaller than the stall force in the absence of catastrophes and rescues. For a nonzero rescue rate, we find a sharply peaked steady-state length distribution, which is tightly controlled by microtubule growth parameters. The corresponding average microtubule length self-organizes such that the average polymerization force equals the critical force f(c) for the transition from unbounded to bounded growth. We also investigate the force dynamics if growth parameters are perturbed in dilution experiments. Finally, we show the robustness of our results against changes of catastrophe models and load distribution factors.

Microtubules interacting with a boundary: mean length and mean first-passage times

I formulate a dynamical model for microtubules interacting with a catastrophe-inducing boundary. In this model microtubules are either waiting to be nucleated, actively growing or shrinking, or stalled at the boundary. I first determine the steady-state occupation of these various states and the resultant length distribution. Next, I formulate the problem of the mean first-passage time to reach the boundary in terms of an appropriate set of splitting probabilities and conditional mean first-passage times and solve explicitly for these quantities using a differential equation approach. As an application, I revisit a recently proposed search-and-capture model for the interaction between microtubules and target chromosomes [M. Gopalakrishnan and B. S. Govindan, Bull. Math. Biol. 73, 2483 (2011)]. I show how my approach leads to a direct and compact solution of this problem.

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.

A first-passage-time theory for search and capture of chromosomes by microtubules in mitosis

The mitotic spindle is an important intermediate structure in eukaryotic cell division, in which each of a pair of duplicated chromosomes is attached through microtubules to centrosomal bodies located close to the two poles of the dividing cell. Several mechanisms are at work toward the formation of the spindle, one of which is the 'capture' of chromosome pairs, held together by kinetochores, by randomly searching microtubules. Although the entire cell cycle can be up to 24 hours long, the mitotic phase typically takes only less than an hour. How does the cell keep the duration of mitosis within this limit? Previous theoretical studies have suggested that the chromosome search and capture is optimized by tuning the microtubule dynamic parameters to minimize the search time. In this paper, we examine this conjecture. We compute the mean search time for a single target by microtubules from a single nucleating site, using a systematic and rigorous theoretical approach, for arbitrary kinetic parameters. The result is extended to multiple targets and nucleating sites by physical arguments. Estimates of mitotic time scales are then obtained for different cells using experimental data. In yeast and mammalian cells, the observed changes in microtubule kinetics between interphase and mitosis are beneficial in reducing the search time. In Xenopus extracts, by contrast, the opposite effect is observed, in agreement with the current understanding that large cells use additional mechanisms to regulate the duration of the mitotic phase.

Providing positional information with active transport on dynamic microtubules

Microtubules (MTs) are dynamic protein polymers that change their length by switching between growing and shrinking states in a process termed dynamic instability. It has been suggested that the dynamic properties of MTs are central to the organization of the eukaryotic intracellular space, and that they are involved in the control of cell morphology, but the actual mechanisms are not well understood. Here, we present a theoretical analysis in which we explore the possibility that a system of dynamic MTs and MT end-tracking molecular motors is providing specific positional information inside cells. We compute the MT length distribution for the case of MT-length-dependent switching between growing and shrinking states, and analyze the accumulation of molecular motors at the tips of growing MTs. Using these results, we show that a transport system consisting of dynamic MTs and associated motor proteins can deliver cargo proteins preferentially to specific positions within the cell. Comparing our results with experimental data in the model organism fission yeast, we propose that the suggested mechanisms could play important roles in setting length scales during cellular morphogenesis.

Insights into cytoskeletal behavior from computational modeling of dynamic microtubules in a cell-like environment

Microtubule dynamic instability plays a fundamental role in cell biology, enabling microtubules to find and interact with randomly distributed cargo and spatially localized signals. In vitro, microtubules transition between growth and shrinkage symmetrically, consistent with the theoretical understanding of the mechanism of dynamic instability. In vivo, however, microtubules commonly exhibit asymmetric dynamic instability, growing persistently in the cell interior and experiencing catastrophe near the cell edge. What is the origin of this behavior difference? One answer is that the cell edge causes the asymmetry by inducing catastrophe in persistently growing microtubules. However, the origin of the persistent growth itself is unclear. Using a simplified coarse-grained stochastic simulation of a system of dynamic microtubules, we provide evidence that persistent growth is a predictable property of a system of nucleated, dynamic, microtubules containing sufficient tubulin in a confined space--MAP activity is not required. Persistent growth occurs because cell-edge-induced catastrophe increases the concentration of free tubulin at steady-state. Our simulations indicate that other aspects of MT dynamics thought to require temporal or spatial changes in MAP activity are also predictable, perhaps unavoidable, outcomes of the "systems nature" of the cellular microtubule cytoskeleton. These include the mitotic increase in microtubule dynamics and the observation that defects in nucleation cause changes in the behavior of microtubule plus ends. These predictions are directly relevant to understanding of the microtubule cytoskeleton, but they are also attractive from an evolutionary standpoint because they provide evidence that apparently complex cellular behaviors can originate from simple interactions without a requirement for intricate regulatory machinery.

Analysis of a mesoscopic stochastic model of microtubule dynamic instability

A theoretical model of dynamic instability of a system of linear one-dimensional microtubules (MTs) in a bounded domain is introduced for studying the role of a cell edge in vivo and analyzing the effect of competition for a limited amount of tubulin. The model differs from earlier models in that the evolution of MTs is based on the rates of single-mesoscopic-unit (e.g., a heterodimer per protofilament) transformations, in contrast to postulating effective rates and frequencies of larger-scale macroscopic changes, extracted, e.g., from the length history plots of MTs. Spontaneous GTP hydrolysis with finite rate after polymerization is assumed, and theoretical estimates of an effective catastrophe frequency as well as other parameters characterizing MT length distributions and cap size are derived. We implement a simple cap model which does not include vectorial hydrolysis. We demonstrate that our theoretical predictions, such as steady-state concentration of free tubulin and parameters of MT length distributions, are in agreement with the numerical simulations. The present model establishes a quantitative link between mesoscopic parameters governing the dynamics of MTs and macroscopic characteristics of MTs in a closed system. Last, we provide an explanation for nonexponential MT length distributions observed in experiments. In particular, we show that the appearance of such nonexponential distributions in the experiments can occur because a true steady state has not been reached and/or due to the presence of a cell edge.

Dynamic instability of microtubules: effect of catastrophe-suppressing drugs

Microtubules are stiff filamentary proteins that constitute an important component of the cytoskeleton of cells. These are known to exhibit a dynamic instability. A steadily growing microtubule can suddenly start depolymerizing very rapidly; this phenomenon is known as a "catastrophe." However, often a shrinking microtubule is "rescued" and starts polymerizing again. Here we develop a model for the polymerization-depolymerization dynamics of microtubules in the presence of catastrophe-suppressing drugs. Solving the dynamical equations in the steady state, we derive exact analytical expressions for the length distributions of the microtubules tipped with drug-bound tubulin subunits as well as those of the microtubules, in the growing and shrinking phases, tipped with drug-free pure tubulin subunits. We also examine the stability of the steady-state solutions.