Compartment volume influences microtubule dynamic instability: a model study

A model for generating differences in microtubules between axonal branches depending on the distance from terminals

In the remodeling of axonal arbor, the growth and retraction of branches are differentially regulated within a single axon. Although cell-autonomously generated differences in microtubule (MT) turnover are thought to be involved in selective branch regulation, the cellular system whereby neurons generate differences of MTs between axonal branches has not been clarified. Because MT turnover tends to be slower in longer branches compared with neighboring shorter branches, feedback regulation depending on branch length is thought to be involved. In the present study, we generated a model of MT lifetime in axonal terminal branches by adapting a length-dependent model in which parameters for MT dynamics were constant in the arbor. The model predicted that differences in MT lifetime between neighboring branches could be generated depending on the distance from terminals. In addition, the following points were predicted. Firstly, destabilization of MTs throughout the arbor decreased the differences in MT lifetime between branches. Secondly, differences of MT lifetime existed even before MTs entered the branch point. In axonal MTs in primary neurons, treatment with a low concentration of nocodazole significantly decreased the differences of detyrosination (deTyr) and tyrosination (Tyr) of tubulins, indicators of MT turnover. Expansion microscopy of the axonal shaft before the branch point revealed differences in deTyr/Tyr modification on MTs. Our model recapitulates the differences in MT turnover between branches and provides a feedback mechanism for MT regulation that depends on the axonal arbor geometry.

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.

Mathematical models of neuronal growth

The establishment of a functioning neuronal network is a crucial step in neural development. During this process, neurons extend neurites—axons and dendrites—to meet other neurons and interconnect. Therefore, these neurites need to migrate, grow, branch and find the correct path to their target by processing sensory cues from their environment. These processes rely on many coupled biophysical effects including elasticity, viscosity, growth, active forces, chemical signaling, adhesion and cellular transport. Mathematical models offer a direct way to test hypotheses and understand the underlying mechanisms responsible for neuron development. Here, we critically review the main models of neurite growth and morphogenesis from a mathematical viewpoint. We present different models for growth, guidance and morphogenesis, with a particular emphasis on mechanics and mechanisms, and on simple mathematical models that can be partially treated analytically.

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.

Theoretical Models of Neural Development

Constructing a functioning nervous system requires the precise orchestration of a vast array of mechanical, molecular, and neural-activity-dependent cues. Theoretical models can play a vital role in helping to frame quantitative issues, reveal mathematical commonalities between apparently diverse systems, identify what is and what is not possible in principle, and test the abilities of specific mechanisms to explain the data. This review focuses on the progress that has been made over the last decade in our theoretical understanding of neural development.

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.

Identification of microtubule growth deceleration and its regulation by conserved and novel proteins

Live imaging of microtubule dynamics in Caenorhabditis elegans muscle cells reveals a novel microtubule behavior characterized by an abrupt change in growth rate, named “microtubule growth deceleration.” The conserved protein ZYG-9^TOGp and two novel ORFs, cylc-1 and cylc-2, are involved in the regulation of this novel microtubule behavior.

Microtubules (MTs) are cytoskeletal polymers that participate in diverse cellular functions, including cell division, intracellular trafficking, and templating of cilia and flagella. MTs undergo dynamic instability, alternating between growth and shortening via catastrophe and rescue events. The rates and frequencies of MT dynamic parameters appear to be characteristic for a given cell type. We recently reported that all MT dynamic parameters vary throughout differentiation of a smooth muscle cell type in intact Caenorhabditis elegans. Here we describe local differences in MT dynamics and a novel MT behavior: an abrupt change in growth rate (deceleration) of single MTs occurring in the cell periphery of these cells. MT deceleration occurs where there is a decrease in local soluble tubulin concentration at the cell periphery. This local regulation of tubulin concentration and MT deceleration are dependent on two novel homologues of human cylicin. These novel ORFs, which we name cylc-1 and -2, share sequence homology with stathmins and encode small, very basic proteins containing several KKD/E repeats. The TOG domain–containing protein ZYG-9^TOGp is responsible for the faster polymerization rate within the cell body. Thus we have defined two contributors to the molecular regulation for this novel MT behavior.

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.

How Cells Measure Length on Subcellular Scales

Cells are not just amorphous bags of enzymes, but precise and complex machines. With any machine, it is important that the parts be of the right size, yet our understanding of the mechanisms that control size of cellular structures remains at a rudimentary level in most cases. One problem with studying size control is that many cellular organelles have complex 3D structures that make their size hard to measure. Here we focus on linear structures within cells, for which the problem of size control reduces to the problem of length control. We compare and contrast potential mechanisms for length control to understand how cells solve simple geometry problems.

The dynamics of growth cone morphology

BACKGROUND

Normal brain function depends on the development of appropriate patterns of neural connections. A critical role in guiding axons to their targets during neural development is played by neuronal growth cones. These have a complex and rapidly changing morphology; however, a quantitative understanding of this morphology, its dynamics and how these are related to growth cone movement, is lacking.

RESULTS

Here we use eigenshape analysis (principal components analysis in shape space) to uncover the set of five to six basic shape modes that capture the most variance in growth cone form. By analysing how the projections of growth cones onto these principal modes evolve in time, we found that growth cone shape oscillates with a mean period of 30 min. The variability of oscillation periods and strengths between different growth cones was correlated with their forward movement, such that growth cones with strong, fast shape oscillations tended to extend faster. A simple computational model of growth cone shape dynamics based on dynamic microtubule instability was able to reproduce quantitatively both the mean and variance of oscillation periods seen experimentally, suggesting that the principal driver of growth cone shape oscillations may be intrinsic periodicity in cytoskeletal rearrangements.

CONCLUSIONS

Intrinsically driven shape oscillations are an important component of growth cone shape dynamics. More generally, eigenshape analysis has the potential to provide new quantitative information about differences in growth cone behaviour in different conditions.

Live visualizations of single isolated tubulin protein self-assembly via tunneling current: effect of electromagnetic pumping during spontaneous growth of microtubule

As we bring tubulin protein molecules one by one into the vicinity, they self-assemble and entire event we capture live via quantum tunneling. We observe how these molecules form a linear chain and then chains self-assemble into 2D sheet, an essential for microtubule, —fundamental nano-tube in a cellular life form. Even without using GTP, or any chemical reaction, but applying particular ac signal using specially designed antenna around atomic sharp tip we could carry out the self-assembly, however, if there is no electromagnetic pumping, no self-assembly is observed. In order to verify this atomic scale observation, we have built an artificial cell-like environment with nano-scale engineering and repeated spontaneous growth of tubulin protein to its complex with and without electromagnetic signal. We used 64 combinations of plant, animal and fungi tubulins and several doping molecules used as drug, and repeatedly observed that the long reported common frequency region where protein folds mechanically and its structures vibrate electromagnetically. Under pumping, the growth process exhibits a unique organized behavior unprecedented otherwise. Thus, “common frequency point” is proposed as a tool to regulate protein complex related diseases in the future.

A one-dimensional moving-boundary model for tubulin-driven axonal growth

A one-dimensional continuum-mechanical model of axonal elongation due to assembly of tubulin dimers in the growth cone is presented. The conservation of mass leads to a coupled system of three differential equations. A partial differential equation models the dynamic and the spatial behaviour of the concentration of tubulin that is transported along the axon from the soma to the growth cone. Two ordinary differential equations describe the time-variation of the concentration of free tubulin in the growth cone and the speed of elongation. All steady-state solutions of the model are categorized. Given a set of the biological parameter values, it is shown how one easily can infer whether there exist zero, one or two steady-state solutions and directly determine the possible steady-state lengths of the axon. Explicit expressions are given for each stationary concentration distribution. It is thereby easy to examine the influence of each biological parameter on a steady state. Numerical simulations indicate that when there exist two steady states, the one with shorter axon length is unstable and the longer is stable. Another result is that, for nominal parameter values extracted from the literature, in a large portion of a fully grown axon the concentration of free tubulin is lower than both concentrations in the soma and in the growth cone.

Competitive dynamics during resource-driven neurite outgrowth

Neurons form networks by growing out neurites that synaptically connect to other neurons. During this process, neurites develop complex branched trees. Interestingly, the outgrowth of neurite branches is often accompanied by the simultaneous withdrawal of other branches belonging to the same tree. This apparent competitive outgrowth between branches of the same neuron is relevant for the formation of synaptic connectivity, but the underlying mechanisms are unknown. An essential component of neurites is the cytoskeleton of microtubules, long polymers of tubulin dimers running throughout the entire neurite. To investigate whether competition between neurites can emerge from the dynamics of a resource such as tubulin, we developed a multi-compartmental model of neurite growth. In the model, tubulin is produced in the soma and transported by diffusion and active transport to the growth cones at the tip of the neurites, where it is assembled into microtubules to elongate the neurite. Just as in experimental studies, we find that the outgrowth of a neurite branch can lead to the simultaneous retraction of its neighboring branches. We show that these competitive interactions occur in simple neurite morphologies as well as in complex neurite arborizations and that in developing neurons competition for a growth resource such as tubulin can account for the differential outgrowth of neurite branches. The model predicts that competition between neurite branches decreases with path distance between growth cones, increases with path distance from growth cone to soma, and decreases with a higher rate of active transport. Together, our results suggest that competition between outgrowing neurites can already emerge from relatively simple and basic dynamics of a growth resource. Our findings point to the need to test the model predictions and to determine, by monitoring tubulin concentrations in outgrowing neurons, whether tubulin is the resource for which neurites compete.

Cell Cycle Regulation of the Centrosome and Cilium

Centrosomes and cilia are conserved microtubule-based organelles whose structure and function depend on cell cycle stages. In dividing cells, centrosomes organize mitotic spindle poles, while in differentiating cells, centrosomes template ciliogenesis. Classically, this functional dichotomy has been attributed to regulation by cell cycle-dependent post-translational modifications, and recently PLK1, Nek2, Aurora A, and tubulin deacetylase were implicated in regulating the transition from cilia to centrosome. However, other recent studies suggest that tubulin dimers, the core structural components of centrosomes and cilia, also have a regulatory role. These regulatory mechanisms can be a target for chemotherapeutic intervention.

Modeling transport of a pulse of radiolabeled organelles in a Drosophila unipolar motor neuron

Based on published experimental evidence, this paper develops a model for the transport of a pulse of radiolabeled organelles in a unipolar Drosophila motor neuron. In particular, since published data indicate that no microtubules (MTs) travel from the primary neurite into the dendrite, it is investigated how organelles are transported into the dendrite. Analytical solutions describing concentrations of kinesin- and dynein-driven organelles in the primary neurite, axon, and dendrite are obtained. The effects of increasing the width of the pulse and increasing the rate of organelle transition rate from the kinesin-driven to the dynein-driven state are investigated.

Intrinsic microtubule GTP-cap dynamics in semi-confined systems: kinetochore-microtubule interface

In order to quantify the intrinsic dynamics associated with the tip of a GTP-cap under semi-confined conditions, such as those within a neuronal cone and at a kinetochore-microtubule interface, we propose a novel quantitative concept of critical nano local GTP-tubulin concentration (CNLC). A simulation of a rate constant of GTP-tubulin hydrolysis, under varying conditions based on this concept, generates results in the range of 0-420 s(-1). These results are in agreement with published experimental data, validating our model. The major outcome of this model is the prediction of 11 random and distinct outbursts of GTP hydrolysis per single layer of a GTP-cap. GTP hydrolysis is accompanied by an energy release and the formation of discrete expanding zones, built by less-stable, skewed GDP-tubulin subunits. We suggest that the front of these expanding zones within the walls of the microtubule represent soliton-like movements of local deformation triggered by energy released from an outburst of hydrolysis. We propose that these solitons might be helpful in addressing a long-standing question relating to the mechanism underlying how GTP-tubulin hydrolysis controls dynamic instability. This result strongly supports the prediction that large conformational movements in tubulin subunits, termed dynamic transitions, occur as a result of the conversion of chemical energy that is triggered by GTP hydrolysis (Satarić et al., Electromagn Biol Med 24:255-264, 2005). Although simple, the concept of CNLC enables the formulation of a rationale to explain the intrinsic nature of the "push-and-pull" mechanism associated with a kinetochore-microtubule complex. In addition, the capacity of the microtubule wall to produce and mediate localized spatio-temporal excitations, i.e., soliton-like bursts of energy coupled with an abundance of microtubules in dendritic spines supports the hypothesis that microtubule dynamics may underlie neural information processing including neurocomputation (Hameroff, J Biol Phys 36:71-93, 2010; Hameroff, Cognit Sci 31:1035-1045, 2007; Hameroff and Watt, J Theor Biol 98:549-561, 1982).

Tubulin nucleotide status controls Sas-4-dependent pericentriolar material recruitment

Regulated centrosome biogenesis is required for accurate cell division and for maintaining genome integrity¹. Centrosomes consist of a centriole pair surrounded by a protein network known as pericentriolar material (PCM)¹. PCM assembly is a tightly regulated, critical step that determines a centrosome’s size and capability^2–4. Here, we report a role for tubulin in regulating PCM recruitment via the conserved centrosomal protein Sas-4. Tubulin directly binds to Sas-4; together they are components of cytoplasmic complexes of centrosomal proteins^5,6. A Sas-4 mutant, which cannot bind tubulin, enhances centrosomal protein complex formation and has abnormally large centrosomes with excessive activity. These suggest that tubulin negatively regulates PCM recruitment. Whereas tubulin-GTP prevents Sas-4 from forming protein complexes, tubulin-GDP promotes it. Thus, tubulin’s regulation of PCM recruitment depends on its GTP/GDP-bound state. These results identify a role for tubulin in regulating PCM recruitment independent of its well-known role as a building block of microtubules⁷. Based on its guanine bound state, tubulin can act as a molecular switch in PCM recruitment.

The mechanisms of microtubule catastrophe and rescue: implications from analysis of a dimer-scale computational model

ETOC: The behavior of a dimer-scale computational model predicts that short interprotofilament “cracks” (laterally unbonded regions between protofilaments) exist even at the tips of growing MTs and that rapid fluctuations in the depths of these cracks govern both catastrophe and rescue.

Microtubule (MT) dynamic instability is fundamental to many cell functions, but its mechanism remains poorly understood, in part because it is difficult to gain information about the dimer-scale events at the MT tip. To address this issue, we used a dimer-scale computational model of MT assembly that is consistent with tubulin structure and biochemistry, displays dynamic instability, and covers experimentally relevant spans of time. It allows us to correlate macroscopic behaviors (dynamic instability parameters) with microscopic structures (tip conformations) and examine protofilament structure as the tip spontaneously progresses through both catastrophe and rescue. The model's behavior suggests that several commonly held assumptions about MT dynamics should be reconsidered. Moreover, it predicts that short, interprotofilament “cracks” (laterally unbonded regions between protofilaments) exist even at the tips of growing MTs and that rapid fluctuations in the depths of these cracks influence both catastrophe and rescue. We conclude that experimentally observed microtubule behavior can best be explained by a “stochastic cap” model in which tubulin subunits hydrolyze GTP according to a first-order reaction after they are incorporated into the lattice; catastrophe and rescue result from stochastic fluctuations in the size, shape, and extent of lateral bonding of the cap.

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 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.

The stochastic search dynamics of interneuron migration

Migration is a dynamic process in which a cell searches the environment and translates acquired information into somal advancement. In particular, interneuron migration during development is accomplished by two distinct processes: the extension of neurites tipped with growth cones; and nucleus translocation, termed nucleokinesis. The primary purpose of our study is to investigate neurite branching and nucleokinesis using high-resolution time-lapse confocal microscopy and computational modeling. We demonstrate that nucleokinesis is accurately modeled by a spring-dashpot system and that neurite branching is independent of the nucleokinesis event, and displays the dynamics of a stochastic birth-death process. This is in contrast to traditional biological descriptions, which suggest a closer relationship between the two migratory mechanisms. Our models are validated on independent data sets acquired using two different imaging protocols, and are shown to be robust to alterations in guidance cues and cellular migratory mechanisms, through treatment with brain-derived neurotrophic factor, neurotrophin-4, and blebbistatin. We postulate that the stochastic branch dynamics exhibited by interneurons undergoing guidance-directed migration permit efficient exploration of the environment.

Quantifying neurite growth mediated by interactions among secretory vesicles, microtubules, and actin networks

Neurite growth is a fundamental process of neuronal development, which requires both membrane expansions by exocytosis and cytoskeletal dynamics. However, the specific contribution of these processes has not been yet assessed quantitatively. To study and quantify the growth process, we construct a biophysical model in which we relate the overall neurite outgrowth rate to the vesicle dynamics. By considering the complex motion of vesicles in the cell soma, we demonstrate from biophysical consideration that the main step of finding the neurite initiation site relies mainly on a two-dimensional diffusion/sequestration/fusion at the cell surface and we obtain a novel formula for the flux of vesicles at the neurite base. In the absence of microtubules, we show that a nascent neurite initiated by vesicular delivery can only reach a small length. By adding the microtubule dynamics to the secretory pathway and using stochastic analysis and simulations, we study the complex dynamics of neurite growth. Within this model, depending on the coupling parameter between the microtubules and the neurite, we find different regimes of growth, which describe dendritic and axonal growth. To validate one aspect of our model, we demonstrate that the experimental flux of TI-VAMP but not Synaptobrevin 2 vesicles contributes to the neurite growth. We conclude that although vesicles can be generated randomly in the cell body, the search for the neurite position using the microtubule network and diffusion is quite fast. Furthermore, when the TI-VAMP vesicular flow is large enough, the interactions between the microtubule bundle and the neurite control the growth process. In addition, all of these processes intimately cooperate to mediate the various modes of neurite growth: the model predicts three different growing modes including, in addition to the stable axonal growth and the stochastic dendritic growth, a fast oscillatory regime. Finally our study demonstrates that cytoskeletal dynamics is necessary to generate long protrusion, while vesicular delivery alone can only generate small neurite.

Methods for simulating the dynamics of complex biological processes

In this chapter, we provide the basic information required to understand the central concepts in the modeling and simulation of complex biochemical processes. We underline the fact that most biochemical processes involve sequences of interactions between distinct entities (molecules, molecular assemblies), and also stress that models must adhere to the laws of thermodynamics. Therefore, we discuss the principles of mass-action reaction kinetics, the dynamics of equilibrium and steady state, and enzyme kinetics, and explain how to assess transition probabilities and reactant lifetime distributions for first-order reactions. Stochastic simulation of reaction systems in well-stirred containers is introduced using a relatively simple, phenomenological model of microtubule dynamic instability in vitro. We demonstrate that deterministic simulation [by numerical integration of coupled ordinary differential equations (ODE)] produces trajectories that would be observed if the results of many rounds of stochastic simulation of the same system were averaged. In Section V, we highlight several practical issues with regard to the assessment of parameter values. We draw some attention to the development of a standard format for model storage and exchange, and provide a list of selected software tools that may facilitate the model building process, and can be used to simulate the modeled systems.

Three-dimensional preparation and imaging reveal intrinsic microtubule properties

We present an experimental investigation of microtubule dynamic instability in three dimensions, based on laser light-sheet fluorescence microscopy. We introduce three-dimensional (3D) preparation of Xenopus laevis egg extracts in Teflon-based cylinders and provide algorithms for 3D image processing. Our approach gives experimental access to the intrinsic dynamic properties of microtubules and to microtubule population statistics in single asters. We obtain evidence for a stochastic nature of microtubule pausing.

Model based dynamics analysis in live cell microtubule images

Background

The dynamic growing and shortening behaviors of microtubules are central to the fundamental roles played by microtubules in essentially all eukaryotic cells. Traditionally, microtubule behavior is quantified by manually tracking individual microtubules in time-lapse images under various experimental conditions. Manual analysis is laborious, approximate, and often offers limited analytical capability in extracting potentially valuable information from the data.

Results

In this work, we present computer vision and machine-learning based methods for extracting novel dynamics information from time-lapse images. Using actual microtubule data, we estimate statistical models of microtubule behavior that are highly effective in identifying common and distinct characteristics of microtubule dynamic behavior.

Conclusion

Computational methods provide powerful analytical capabilities in addition to traditional analysis methods for studying microtubule dynamic behavior. Novel capabilities, such as building and querying microtubule image databases, are introduced to quantify and analyze microtubule dynamic behavior.

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.

Modelling microtubule patterns

The cellular cytoskeleton is well studied in terms of its biological and physical properties, making it an attractive subject for systems approaches. Here, we describe the experimental and theoretical strategies used to study the collective behaviour of microtubules and motors. We illustrate how this led to the beginning of an understanding of dynamic cellular patterns that have precise functions.

Mathematical modelling and numerical simulation of the morphological development of neurons

Background

The morphological development of neurons is a very complex process involving both genetic and environmental components. Mathematical modelling and numerical simulation are valuable tools in helping us unravel particular aspects of how individual neurons grow their characteristic morphologies and eventually form appropriate networks with each other.

Methods

A variety of mathematical models that consider (1) neurite initiation (2) neurite elongation (3) axon pathfinding, and (4) neurite branching and dendritic shape formation are reviewed. The different mathematical techniques employed are also described.

Results

Some comparison of modelling results with experimental data is made. A critique of different modelling techniques is given, leading to a proposal for a unified modelling environment for models of neuronal development.

Conclusion

A unified mathematical and numerical simulation framework should lead to an expansion of work on models of neuronal development, as has occurred with compartmental models of neuronal electrical activity.

Microtubule stability studied by three-dimensional molecular theory of solvation

We study microtubular supramolecular architectures of tubulin dimers self-assembling into linear protofilaments, in turn forming a closed tube, which is an important component of the cytoskeleton. We identify the protofilament arrangements with the lowest free energy using molecular dynamics to optimize tubulin conformations. We then use the three-dimensional molecular theory of solvation to obtain the hydration structure of protofilaments built of optimized tubulins and the solvent-mediated effective potential between them. The latter theoretical method, based on first principles of statistical mechanics, is capable of predicting the structure and thermodynamics of solvation of supramolecular architectures. We obtained a set of profiles of the potential of mean force between protofilaments in a periodic two-dimensional sheet in aqueous solution. The profiles were calculated for a number of amino acid sequences, tubulin conformations, and spatial arrangements of protofilaments. The results indicate that the effective interaction between protofilaments in aqueous solution depends little on the isotypes studied; however, it strongly depends on the M loop conformation of beta-tubulin. Based on the analysis of the potential of mean force between adjacent protofilaments, we found the optimal arrangement of protofilaments, which is in good agreement with other studies. We also decomposed the potential of mean force into its energetic and entropic components, and found that both are considerable in the free-energy balance for the stabilized protofilament arrangements.

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.