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

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.

Modelling microtubule dynamic instability: Microtubule growth, shortening and pause

Microtubules (MTs) are protein polymers found in all eukaryotic cells. They are crucial for normal cell development, providing structural support for the cell and aiding in the transportation of proteins and organelles. In order to perform these functions, MTs go through periods of relatively slow polymerization (growth) and very fast depolymerization (shortening), where the switch from growth to shortening is called a catastrophe and the switch from shortening to growth is called a rescue. Although MT dynamic instability has traditionally been described solely in terms of growth and shortening, MTs have been shown to pause for extended periods of time, however the reason for pausing is not well understood. Here, we present a new mathematical model to describe MT dynamics in terms of growth, shortening, and pausing. Typically, MT dynamics are defined by four key parameters which include the MT growth rate, shortening rate, frequency of catastrophe, and the frequency of rescue. We derive a mathematical expression for the catastrophe frequency in the presence of pausing, as well as expressions to describe the total time that MTs spend in a state of growth and pause. In addition to exploring MT dynamics in a control-like setting, we explore the implicit effect of stabilizing MT associated proteins (MAPs) and stabilizing and destabilizing chemotherapeutic drugs that target MTs on MT dynamics through variations in model parameters.

A Growth-Fragmentation Approach for Modeling Microtubule Dynamic Instability

Microtubules (MTs) are protein filaments found in all eukaryotic cells which are crucial for many cellular processes including cell movement, cell differentiation, and cell division. Due to their role in cell division, they are often used as targets for chemotherapy drugs used in cancer treatment. Experimental studies of MT dynamics have played an important role in the development and administration of many novel cancer drugs; however, a complete description of MT dynamics is lacking. Here, we propose a new mathematical model for MT dynamics, that can be used to study the effects of chemotherapy drugs on MT dynamics. Our model consists of a growth-fragmentation equation describing the dynamics of a length distribution of MTs, coupled with two ODEs that describe the dynamics of free GTP- and GDP-tubulin concentrations (the individual dimers that comprise of MTs). Here, we prove the well-posedness of our system and perform a numerical exploration of the influence of certain model parameters on the systems dynamics. In particular, we focus on a qualitative description for how a certain class of destabilizing drugs, the vinca alkaloids, alter MT dynamics. Through variation of certain model parameters which we know are altered by these drugs, we make comparisons between simulation results and what is observed in in vitro studies.

Microtubule dynamic instability: a new model with coupled GTP hydrolysis and multistep catastrophe

A key question in understanding microtubule dynamics is how GTP hydrolysis leads to catastrophe, the switch from slow growth to rapid shrinkage. We first provide a review of the experimental and modeling literature, and then present a new model of microtubule dynamics. We demonstrate that vectorial, random, and coupled hydrolysis mechanisms are not consistent with the dependence of catastrophe on tubulin concentration and show that, although single-protofilament models can explain many features of dynamics, they do not describe catastrophe as a multistep process. Finally, we present a new combined (coupled plus random hydrolysis) multiple-protofilament model that is a simple, analytically solvable generalization of a single-protofilament model. This model accounts for the observed lifetimes of growing microtubules, the delay to catastrophe following dilution and describes catastrophe as a multistep process.

Toward the virtual cell: automated approaches to building models of subcellular organization "learned" from microscopy images

We review state-of-the-art computational methods for constructing, from image data, generative statistical models of cellular and nuclear shapes and the arrangement of subcellular structures and proteins within them. These automated approaches allow consistent analysis of images of cells for the purposes of learning the range of possible phenotypes, discriminating between them, and informing further investigation. Such models can also provide realistic geometry and initial protein locations to simulations in order to better understand cellular and subcellular processes. To determine the structures of cellular components and how proteins and other molecules are distributed among them, the generative modeling approach described here can be coupled with high throughput imaging technology to infer and represent subcellular organization from data with few a priori assumptions. We also discuss potential improvements to these methods and future directions for research.

Boundary-induced orientation of dynamic filament networks and vesicle agglomerations

We find a statistical mechanism that can adjust orientations of intracellular filaments to cell geometry in the absence of organizing centers. The effect is based on random and isotropic filament (de-)polymerization dynamics and is independent of filament interactions and explicit regulation. It can be understood by an analogy to electrostatics and appears to be induced by the confining boundaries; for periodic boundary conditions, no orientational bias emerges. Including active transport of particles, the model reproduces experimental observations of vesicle accumulations in transected axons.

Active biopolymers confer fast reorganization kinetics

Many cytoskeletal biopolymers are "active," consuming energy in large quantities. In this Letter, we identify a fundamental difference between active polymers and passive, equilibrium polymers: for equal mean lengths, active polymers can reorganize faster than equilibrium polymers. We show that equilibrium polymers are intrinsically limited to linear scaling between mean lifetime (or mean first-passage time, or MFPT) and mean length, MFPT∼<L>, by analogy to 1D Potts models. By contrast, we present a simple active-polymer model that improves upon this scaling, such that MFPT∼<L>(1/2). Since, to be biologically useful, structural biopolymers must typically be many monomers long yet respond dynamically to the needs of the cell, the difference in reorganization kinetics may help to justify the active polymers' greater energy cost.

Rapid assembly and collective behavior of microtubule bundles in the presence of polyamines

Microtubules (MTs) are cylindrical cytoskeleton polymers composed of α-β tubulin heterodimers whose dynamic properties are essential to fulfill their numerous cellular functions. In response to spatial confinement, dynamic MTs, even in the absence of protein partners, were shown to self-organize into higher order structures (spindle or striped structures) which lead to interesting dynamical properties (MT oscillations). In this study, we considered the assembly and sensitivity of dynamic MTs when in bundles. To perform this study, spermine, a natural tetravalent polyamine present at high concentrations in all eukaryote cells, was used to trigger MT bundling while preserving MT dynamics. Interestingly, we first show that, near physiological ionic strengths, spermine promotes the bundling of MTs whereas it does not lead to aggregation of free tubulin, which would have been detrimental to MT polymerization. Experimental and theoretical results also indicate that, to obtain a high rate of bundle assembly, bundling should take place at the beginning of assembly when rapid rotational movements of short and newly nucleated MTs are still possible. On the other hand, the bundling process is significantly slowed down for long MTs. Finally, we found that short MT bundles exhibit a higher sensitivity to cold exposure than do isolated MTs. To account for this phenomenon, we suggest that a collective behavior takes place within MT bundles because an MT entering into a phase of shortening could increase the probability of the other MTs in the same bundle to enter into shortening phase due to their close proximity. We then elaborate on some putative applications of our findings to in vivo conditions including neurons.

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.

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.

Mathematical modeling of microtubule dynamics: insights into physiology and disease

Computer models of microtubule dynamics have provided the basis for many of the theories on the cellular mechanics of the microtubules, their polymerization kinetics, and the diffusion of tubulin and tau. In the three-dimensional model presented here, we include the effects of tau concentration and the hydrolysis of GTP-tubulin to GDP-tubulin and observe the emergence of microtubule dynamic instability. This integrated approach simulates the essential physics of microtubule dynamics in a cellular environment. The model captures the structure of the microtubules as they undergo steady state dynamic instabilities in this simplified geometry, and also yields the average number, length, and cap size of the microtubules. The model achieves realistic geometries and simulates cellular structures found in degenerating neurons in disease states such as Alzheimer disease. Further, this model can be used to simulate microtubule changes following the addition of antimitotic drugs which have recently attracted attention as chemotherapeutic agents.