Models of assembly and disassembly of individual microtubules: stochastic and averaged equations

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.

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.

The size of the EB cap determines instantaneous microtubule stability

The function of microtubules relies on their ability to switch between phases of growth and shrinkage. A nucleotide-dependent stabilising cap at microtubule ends is thought to be lost before this switch can occur; however, the nature and size of this protective cap are unknown. Using a microfluidics-assisted multi-colour TIRF microscopy assay with close-to-nm and sub-second precision, we measured the sizes of the stabilizing cap of individual microtubules. We find that the protective caps are formed by the extended binding regions of EB proteins. Cap lengths vary considerably and longer caps are more stable. Nevertheless, the trigger of instability lies in a short region at the end of the cap, as a quantitative model of cap stability demonstrates. Our study establishes the spatial and kinetic characteristics of the protective cap and provides an insight into the molecular mechanism by which its loss leads to the switch from microtubule growth to shrinkage.

DOI:http://dx.doi.org/10.7554/eLife.13470.001

Much like the skeleton supports the human body, a structure called the cytoskeleton provides support and structure to cells. Part of this cytoskeleton is made up of small tubes called microtubules that – unlike bones – can shrink and grow very quickly. This allows the cell to change shape, move and split into two new cells.

Exactly how the microtubules switch between growing and shrinking was not clear. One suggestion is that a protective cap at the end of microtubule allows it to keep growing and prevents it from shrinking. However, the nature and size of this cap have been debated.

Now, Duellberg et al. have measured the caps of microtubules with high precision by combining the techniques of microfluidics, TIRF microscopy and recently developed image analysis tools. This revealed that the cap sizes change, with longer caps being more stable. In addition, proteins called end-binding proteins can destabilize the cap by binding to it. This allows microtubules to switch from a growing to a shrinking state more often.

Future work could now investigate how changes in cap length cause the microtubules to switch from growing to shrinking. It also remains to be seen whether other proteins also influence the cap to control this switching behaviour.

DOI:http://dx.doi.org/10.7554/eLife.13470.002

A deterministic oscillatory model of microtubule growth and shrinkage for differential actions of short chain fatty acids

Short-chain fatty acids have distinct effects on cytoskeletal proteins at the level of expression and organisation. We report a new oscillatory, deterministic model which accounts for different actions and predicts response according to fatty acid chain length.

An integrated multidisciplinary model describing initiation of cancer and the Warburg hypothesis

Background

In this paper we propose a chemical physics mechanism for the initiation of the glycolytic switch commonly known as the Warburg hypothesis, whereby glycolytic activity terminating in lactate continues even in well-oxygenated cells. We show that this may result in cancer via mitotic failure, recasting the current conception of the Warburg effect as a metabolic dysregulation consequent to cancer, to a biophysical defect that may contribute to cancer initiation.

Model

Our model is based on analogs of thermodynamic concepts that tie non-equilibrium fluid dynamics ultimately to metabolic imbalance, disrupted microtubule dynamics, and finally, genomic instability, from which cancers can arise. Specifically, we discuss how an analog of non-equilibrium Rayleigh-Benard convection can result in glycolytic oscillations and cause a cell to become locked into a higher-entropy state characteristic of cancer.

Conclusions

A quantitative model is presented that attributes the well-known Warburg effect to a biophysical mechanism driven by a convective disturbance in the cell. Contrary to current understanding, this effect may precipitate cancer development, rather than follow from it, providing new insights into carcinogenesis, cancer treatment, and prevention.

Simulation of the effects of microtubules in the cortical rotation of amphibian embryos in normal and zero gravity

This paper reports the results of computer modeling of microtubules that end up in the cortical region of a one-cell amphibian embryo, prior to the first cell division. Microtubules are modeled as initially randomly oriented semi-flexible rods, represented by several lines of point-masses interacting with one another like masses on springs with longitudinal and transverse stiffness. They are also considered to be space-filling rods floating in a viscous fluid (cytoplasm) experiencing drag forces and buoyancy from the fluid under a variable gravity field to test gravitational effects. Their randomly distributed interactions with the surrounding spherical container (the cell membrane) have a statistical nonzero average that creates a torque causing a rotational displacement between the cytoplasm and the rigid cortex. The simulation has been done for zero and normal gravity and it validates the observation that cortical rotation occurs in microgravity as well as on Earth. The speed of rotation depends on gravity, but is still substantial in microgravity.

Modeling the effects of drug binding on the dynamic instability of microtubules

We propose a stochastic model that accounts for the growth, catastrophe and rescue processes of steady-state microtubules assembled from MAP-free tubulin in the possible presence of a microtubule-associated drug. As an example of the latter, we both experimentally and theoretically study the perturbation of microtubule dynamic instability by S-methyl-D-DM1, a synthetic derivative of the microtubule-targeted agent maytansine and a potential anticancer agent. Our model predicts that among the drugs that act locally at the microtubule tip, primary inhibition of the loss of GDP tubulin results in stronger damping of microtubule dynamics than inhibition of GTP tubulin addition. On the other hand, drugs whose action occurs in the interior of the microtubule need to be present in much higher concentrations to have visible effects.

Continuous model for microtubule dynamics with catastrophe, rescue, and nucleation processes

Microtubules are a major component of the cytoskeleton distinguished by highly dynamic behavior both in vitro and in vivo referred to as dynamic instability. We propose a general mathematical model that accounts for the growth, catastrophe, rescue, and nucleation processes in the polymerization of microtubules from tubulin dimers. Our model is an extension of various mathematical models developed earlier formulated in order to capture and unify the various aspects of tubulin polymerization. While attempting to use a minimal number of adjustable parameters, the proposed model covers a broad range of behaviors and has predictive features discussed in the paper. We have analyzed the range of resultant dynamical behavior of the microtubules by changing each of the parameter values at a time and observing the emergence of various dynamical regimes that agree well with the previously reported experimental data and behavior.

Microtubule assembly of isotypically purified tubulin and its mixtures

Numerous isotypes of the structural protein tubulin have now been characterized in various organisms and their expression offers a plausible explanation for observed differences affecting microtubule function in vivo. While this is an attractive hypothesis, there are only a handful of studies demonstrating a direct influence of tubulin isotype composition on the dynamic properties of microtubules. Here, we present the results of experimental assays on the assembly of microtubules from bovine brain tubulin using purified isotypes at various controlled relative concentrations. A novel data analysis is developed using recursive maps which are shown to be related to the master equation formalism. We have found striking similarities between the three isotypes of bovine tubulin studied in regard to their dynamic instability properties, except for subtle differences in their catastrophe frequencies. When mixtures of tubulin isotypes are analyzed, their nonlinear concentration dependence is modeled and interpreted in terms of lower affinities of tubulin dimers belonging to the same isotype than those that represent different isotypes indicating hitherto unsuspected influences of tubulin dimers on each other within a microtubule. Finally, we investigate the fluctuations in microtubule assembly and disassembly rates and conclude that the inherent rate variability may signify differences in the guanosine-5'-triphosphate composition of the growing and shortening microtubule tips. It is the main objective of this article to develop a quantitative model of tubulin polymerization for individual isotypes and their mixtures. The possible biological significance of the observed differences is addressed.

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.

Compartment volume influences microtubule dynamic instability: a model study

Microtubules (MTs) are cytoskeletal polymers that exhibit dynamic instability, the random alternation between growth and shrinkage. MT dynamic instability plays an essential role in cell development, division, and motility. To investigate dynamic instability, simulation models have been widely used. However, conditions under which the concentration of free tubulin fluctuates as a result of growing or shrinking MTs have not been studied before. Such conditions can arise, for example, in small compartments, such as neuronal growth cones. Here we investigate by means of computational modeling how concentration fluctuations caused by growing and shrinking MTs affect dynamic instability. We show that these fluctuations shorten MT growth and shrinkage times and change their distributions from exponential to non-exponential, gamma-like. Gamma-like distributions of MT growth and shrinkage times, which allow optimal stochastic searching by MTs, have been observed in various cell types and are believed to require structural changes in the MT during growth or shrinkage. Our results, however, show that these distributions can already arise as a result of fluctuations in the concentration of free tubulin due to growing and shrinking MTs. Such fluctuations are possible not only in small compartments but also when tubulin diffusion is slow or when many MTs (de)polymerize synchronously. Volume and all other factors that influence these fluctuations can affect MT dynamic instability and, consequently, the processes that depend on it, such as neuronal growth cone behavior and cell motility in general.

Polymerization dynamics of double-stranded biopolymers: chemical kinetic approach

The polymerization dynamics of double-stranded polymers, such as actin filaments, is investigated theoretically using simple chemical kinetic models that explicitly take into account some microscopic details of the polymer structure and the lateral interactions between the protofilaments. By considering all possible molecular configurations, the exact analytical expressions for the growth velocity and dispersion for two-stranded polymers are obtained in the case of the growing at only one end, and for the growth from both polymer ends. Exact theoretical calculations are compared with the predictions of approximate multilayer models that consider only a finite number of the most relevant polymer configurations. Our theoretical approach is applied to analyze the experimental data on the growth and fluctuations dynamics of individual single actin filaments.

Monte Carlo simulations of rigid biopolymer growth processes

Rigid biopolymers, such as actin filaments, microtubules, and intermediate filaments, are vital components of the cytoskeleton and the cellular environment. Understanding biopolymer growth dynamics is essential for the description of the mechanisms and principles of cellular functions. These biopolymers are composed of N parallel protofilaments which are aligned with arbitrary but fixed relative displacements, thus giving rise to complex end structures. We have investigated rigid biopolymer growth processes by Monte Carlo simulations by taking into account the effects of such "end" properties and lateral interactions. Our simulations reproduce analytical results for the case of N = 2, which is biologically relevant for actin filaments. For the case of N = 13, which applies to microtubules, the simulations produced results qualitatively similar to the N = 2 case. The simulation results indicate that polymerization events are evenly distributed among the N protofilaments, which imply that both end-structure effects and lateral interactions are significant. The effect of different splittings in activation energy has been investigated for the case of N = 2. The effects of activation energy coefficients on the specific polymerization and depolymerization processes were found to be unsubstantial. By expanding the model, we have also obtained a force-velocity relationship of microtubules as observed in experiments. In addition, a range of lateral free-energy parameters was found that yields growth velocities in accordance with experimental observations and previous simulation estimates for the case of N = 13.

Microtubule Dynamics may Embody a Stationary Bipolarity Forming Mechanism Related to the Prokaryotic Division Site Mechanism (Pole-to-Pole Oscillations)

Cell division mechanisms in eukaryotes and prokaryotes have until recently been seen as being widely different. However, pole-to-pole oscillations of proteins like MinE in prokaryotes are now known to determine the division plane. These protein waves arise through spontaneous pattern forming reaction-diffusion mechanisms, based on cooperative binding of the proteins to a quasistationary matrix (like the cell membrane or DNA). Rather than waves, stationary bipolar pattern formation may arise as well. Some of the involved proteins have eukaryotic homologs (e.g. FtsZ and tubulin), pointing to a possible ancient shared mechanism. Tubulin polymerizes to microtubules in the spindle. Mitotic microtubules are in a highly dynamical state, frequently undergoing rapid shortening (catastrophe), and fragments formed from the microtubule ends are inferred to enhance the destabilization. Here, we show that cooperative binding of such fragments to microtubules may set up a similar pattern forming mechanism as seen in prokaryotes. The result is a spontaneously formed, well controllable, bipolar state of microtubule dynamics in the cell, which may contribute to defining the bipolar spindle.