Dynamics of Microtubule Instabilities

On the critical concentration for net assembly of dynamically unstable polymers

Cytoskeletal and cytomotive filaments are protein polymers that move molecular cargo and organize cellular contents in all domains of life. A key parameter describing the self-assembly of many of these polymers -including actin filaments and microtubules- is the minimum concentration required for polymer formation. This 'critical concentration for net assembly' (cc N ) is easy to calculate for eukaryotic actins but more difficult for dynamically unstable filaments such as microtubules and some bacterial polymers. To better understand how cells (especially bacteria) regulate assembly of dynamically unstable polymers I investigate the microscopic parameters that influence their critical concentrations. Assuming simple models for spontaneous nucleation and catastrophe I derive expressions for the monomer-polymer balance. In the absence of concentration-dependent rescue, fixed catastrophe rates do not produce clear critical concentrations. In contrast, simple ATP-/GTP-cap models with concentration-dependent catastrophe rates, generate phenomenological critical concentrations that increase linearly with the rate of nucleotide hydrolysis and decrease logarithmically with the rate of spontaneous nucleation.

Universal catastrophe time distributions of dynamically unstable polymers

Dynamic instability-the growth, catastrophe, and shrinkage of quasi-one-dimensional filaments-has been observed in multiple biopolymers. Scientists have long understood the catastrophic cessation of growth and subsequent depolymerization as arising from the interplay of hydrolysis and polymerization at the tip of the polymer. Here we show that for a broad class of catastrophe models, the expected catastrophe time distribution is exponential. We show that the distribution shape is insensitive to noise, but that depletion of monomers from a finite pool can dramatically change the distribution shape by reducing the polymerization rate. We derive a form for this finite-pool catastrophe time distribution and show that finite-pool effects can be important even when the depletion of monomers does not greatly alter the polymerization rate.

Microtubule catastrophe under force: mathematical and computational results from a Brownian ratchet model

In the intracellular environment, the intrinsic dynamics of microtubule filaments is often hindered by the presence of barriers of various kind, such as kinetochore complexes and cell cortex, which impact their polymerisation force and dynamical properties such as catastrophe frequency. We present a theoretical study of the effect of a forced barrier, also subjected to thermal noise, on the statistics of catastrophe events in a single microtubule as well as a 'bundle' of two parallel microtubules. For microtubule dynamics, which includes growth, detachment, hydrolysis and the consequent dynamic instability, we employ a one-dimensional discrete stochastic model. The dynamics of the barrier is captured by over-damped Langevin equation, while its interaction with a growing filament is assumed to be hard-core repulsion. A unified treatment of the continuum dynamics of the barrier and the discrete dynamics of the filament is realized using a hybrid Fokker-Planck equation. An explicit mathematical formula for the force-dependent catastrophe frequency of a single microtubule is obtained by solving the above equation, under some assumptions. The prediction agrees well with results of numerical simulations in the appropriate parameter regime. More general situations are studied via numerical simulations. To investigate the extent of 'load-sharing' in a microtubule bundle, and its impact on the frequency of catastrophes, the dynamics of a two-filament bundle is also studied. Here, two parallel, non-interacting microtubules interact with a common, forced barrier. The equations for the two-filament model, when solved using a mean-field assumption, predicts equal sharing of load between the filaments. However, numerical results indicate the existence of a wide spectrum of load-sharing behaviour, which is characterized using a dimensionless parameter.

Structural effects of cap, crack, and intrinsic curvature on the microtubule catastrophe kinetics

Microtubules (MTs) experience an effect called "catastrophe," which is the transition from the MT growth to a sudden dramatic shrinkage in length. The straight guanosine triphosphate (GTP)-tubulin cap at the filament tip and the intrinsic curvature of guanosine diphosphate (GDP)-tubulins are known to be the key thermodynamic factors that determine MT catastrophe, while the hydrolysis of this GTP-cap acts as the kinetic control of the process. Although several theoretical models have been developed, assuming the catastrophe occurs when the GTP-cap shrinks to a minimal stabilizing size, the structural effect of the GTP-cap and GDP-curvature is not explicitly included; thus, their influence on catastrophe kinetics remains less understood. To investigate this structural effect, we apply a single-protofilament model with one GTP-cap while assuming a random hydrolysis mechanism and take the occurrence of a crack in the lateral bonds between neighboring protofilaments as the onset of the catastrophe. Therein, we find the effective potential of the tip along the peel-off direction and formulate the catastrophe kinetics as a mean first-passage time problem, subject to thermal fluctuations. We consider cases with and without a compressive force on the MT tip, both of which give a quadratic effective potential, making MT catastrophe an Ornstein-Uhlenbeck process in our formalism. In the free-standing case, the mean catastrophe time has a sensitive tubulin-concentration dependence, similar to a double-exponential function, and agrees well with the experiment. For a compressed MT, we find a modified exponential function of force that shortens the catastrophe time.

Length-dependent dynamics of microtubules

Certain regulatory proteins influence the polymerization dynamics of microtubules by inducing catastrophe with a rate that depends on the microtubule length. Using a discrete formulation, here we show that, for a catastrophe rate proportional to the microtubule length, the steady-state probability distributions of length decay much faster with length than an exponential decay as seen in the absence of these proteins.

Prolonging assembly through dissociation: a self-assembly paradigm in microtubules

We study a one-dimensional model of microtubule assembly and disassembly in which GTP bound to tubulins within the microtubule undergoes stochastic hydrolysis. In contrast to models that consider only a cap of GTP-bound tubulin, stochastic hydrolysis allows GTP-bound tubulin remnants to exist within the microtubule. We find that these buried GTP remnants enable an alternative mechanism of recovery from shrinkage and enhances fluctuations of filament lengths. Under conditions for which this alternative mechanism dominates, an increasing depolymerization rate leads to a decrease in dissociation rate and thus a net increase in assembly.

Mean-field study of the role of lateral cracks in microtubule dynamics

A link between dimer-scale processes and microtubule (MT) dynamics at macroscale is studied by comparing simulations obtained using computational dimer-scale model with its mean-field approximation. The novelty of the mean-field model (MFM) is in its explicit representation of inter-protofilament cracks, as well as in the direct incorporation of the dimer-level kinetics. Due to inclusion of both longitudinal and lateral dimer interactions, the MFM is two dimensional, in contrast to previous theoretical models of MTs. It is the first analytical model that predicts and quantifies crucial features of MT dynamics such as (i) existence of a minimal soluble tubulin concentration needed for the polymerization (with concentration represented as a function of model parameters), (ii) existence of steady-state growth and shortening phases (given with their respective velocities), and (iii) existence of an unstable pause state near zero velocity. In addition, the size of the GTP cap of a growing MT is estimated. Theoretical predictions are shown to be in good agreement with the numerical simulations.

Role of ATP-hydrolysis in the dynamics of a single actin filament

We study the stochastic dynamics of growth and shrinkage of single actin filaments taking into account insertion, removal, and ATP hydrolysis of subunits either according to the vectorial mechanism or to the random mechanism. In a previous work, we developed a model for a single actin or microtubule filament where hydrolysis occurred according to the vectorial mechanism: the filament could grow only from one end, and was in contact with a reservoir of monomers. Here we extend this approach in two ways--by including the dynamics of both ends and by comparing two possible mechanisms of ATP hydrolysis. Our emphasis is mainly on two possible limiting models for the mechanism of hydrolysis within a single filament, namely the vectorial or the random model. We propose a set of experiments to test the nature of the precise mechanism of hydrolysis within actin filaments.

Dynamical phase transition of a one-dimensional transport process including death

Motivated by biological aspects related to fungus growth, we consider the competition of growth and corrosion. We study a modification of the totally asymmetric exclusion process, including the probabilities of injection alpha and death of the last particle delta . The system presents a phase transition at deltac(alpha), where the average position of the last particle L grows as sqrt[t]. For delta>deltac, a nonequilibrium stationary state exists while for delta<deltac the asymptotic state presents a low density and max current phases. We discuss the scaling of the density and current profiles for parallel and sequential updates.

A theory of microtubule catastrophes and their regulation

Dynamic instability, in which abrupt transitions occur between growing and shrinking states, is an intrinsic property of microtubules that is regulated by both mechanics and specialized proteins. We discuss a model of dynamic instability based on the popular idea that growth is maintained by a cap at the tip of the fiber. The loss of this cap is thought to trigger the transition from growth to shrinkage, called a catastrophe. The model includes longitudinal interactions between the terminal tubulins of each protofilament and "gating rescues" between neighboring protofilaments. These interactions allow individual protofilaments to transiently shorten during a phase of overall microtubule growth. The model reproduces the reported dependency of the catastrophe rate on tubulin concentration, the time between tubulin dilution and catastrophe, and the induction of microtubule catastrophes by walking depolymerases. The model also reproduces the comet tail distribution that is characteristic of proteins that bind to the tips of growing microtubules.

Nonequilibrium self-assembly of a filament coupled to ATP/GTP hydrolysis

We study the stochastic dynamics of growth and shrinkage of single actin filaments or microtubules taking into account insertion, removal, and ATP/GTP hydrolysis of subunits. The resulting phase diagram contains three different phases: two phases of unbounded growth: a rapidly growing phase and an intermediate phase, and one bounded growth phase. We analyze all these phases, with an emphasis on the bounded growth phase. We also discuss how hydrolysis affects force-velocity curves. The bounded growth phase shows features of dynamic instability, which we characterize in terms of the time needed for the ATP/GTP cap to disappear as well as the time needed for the filament to reach a length of zero (i.e., to collapse) for the first time. We obtain exact expressions for all these quantities, which we test using Monte Carlo simulations.

Dynamics of an idealized model of microtubule growth and catastrophe

We investigate a simple dynamical model of a microtubule that evolves by attachment of guanosine triphosphate (GTP) tubulin to its end, irreversible conversion of GTP to guanosine diphosphate (GDP) tubulin by hydrolysis, and detachment of GDP at the end of a microtubule. As a function of rates of these processes, the microtubule can grow steadily or its length can fluctuate wildly. In the regime where detachment can be neglected, we find exact expressions for the tubule and GTP cap length distributions, as well as power-law length distributions of GTP and GDP islands. In the opposite limit of instantaneous detachment, we find the time between catastrophes, where the microtubule shrinks to zero length, and determine the size distribution of avalanches (sequence of consecutive GDP detachment events). We obtain the phase diagram for general rates and verify our predictions by numerical simulations.