A chemical kinetics model for microtubule oscillations

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.

Oscillations, travelling fronts and patterns in a supramolecular system

Supramolecular polymers, such as microtubules, operate under non-equilibrium conditions to drive crucial functions in cells, such as motility, division and organelle transport¹. In vivo and in vitro size oscillations of individual microtubules^2,3 (dynamic instabilities) and collective oscillations⁴ have been observed. In addition, dynamic spatial structures, like waves and polygons, can form in non-stirred systems⁵. Here we describe an artificial supramolecular polymer made of a perylene diimide derivative that displays oscillations, travelling fronts and centimetre-scale self-organized patterns when pushed far from equilibrium by chemical fuels. Oscillations arise from a positive feedback due to nucleation-elongation-fragmentation, and a negative feedback due to size-dependent depolymerization. Travelling fronts and patterns form due to self-assembly induced density differences that cause system-wide convection. In our system, the species responsible for the nonlinear dynamics and those that self-assemble are one and the same. In contrast, other reported oscillating assemblies formed by vesicles⁶, micelles⁷ or particles⁸ rely on the combination of a known chemical oscillator and a stimuli-responsive system, either by communication through the solvent (for example, by changing pH^7-9), or by anchoring one of the species covalently (for example, a Belousov-Zhabotinsky catalyst^6,10). The design of self-oscillating supramolecular polymers and large-scale dissipative structures brings us closer to the creation of more life-like materials¹¹ that respond to external stimuli similarly to living cells, or to creating artificial autonomous chemical robots¹².

Spinodal decomposition and the emergence of dissipative transient periodic spatio-temporal patterns in acentrosomal microtubule multitudes of different morphology

We have studied a spontaneous self-organization dynamics in a closed, dissipative (in terms of guansine 5'-triphosphate energy dissipation), reaction-diffusion system of acentrosomal microtubules (those nucleated and organized in the absence of a microtubule-organizing centre) multitude constituted of straight and curved acentrosomal microtubules, in highly crowded conditions, in vitro. Our data give experimental evidence that cross-diffusion in conjunction with excluded volume is the underlying mechanism on basis of which acentrosomal microtubule multitudes of different morphologies (straight and curved) undergo a spatial-temporal demix. Demix is constituted of a bifurcation process, manifested as a slow isothermal spinodal decomposition, and a dissipative process of transient periodic spatio-temporal pattern formation. While spinodal decomposition is an energy independent process, transient periodic spatio-temporal pattern formation is accompanied by energy dissipative process. Accordingly, we have determined that the critical threshold for slow, isothermal spinodal decomposition is 1.0 ± 0.05 mg/ml of microtubule protein concentration. We also found that periodic spacing of transient periodic spatio-temporal patterns was, in the overall, increasing versus time. For illustration, we found that a periodic spacing of the same pattern was 0.375 ± 0.036 mm, at 36 °C, at 155th min, while it was 0.540 ± 0.041 mm at 31 °C, and at 275th min after microtubule assembly started. The lifetime of transient periodic spatio-temporal patterns spans from half an hour to two hours approximately. The emergence of conditions of macroscopic symmetry breaking (that occur due to cross-diffusion in conjunction with excluded volume) may have more general but critical importance in morphological pattern development in complex, dissipative, but open cellular systems.

Models of the collective behavior of proteins in cells: tubulin, actin and motor proteins

One of the most important issues of molecular biophysics is the complex and multifunctional behavior of the cell's cytoskeleton. Interiors of living cells are structurally organized by the cytoskeleton networks of filamentous protein polymers: microtubules, actin and intermediate filaments with motor proteins providing force and directionality needed for transport processes. Microtubules (MT's) take active part in material transport within the cell, constitute the most rigid elements of the cell and hence found many uses in cell motility (e.g. flagella andcilia). At present there is, however, no quantitatively predictable explanation of how these important phenomena are orchestrated at a molecular level. Moreover, microtubules have been demonstrated to self-organize leading to pattern formation. We discuss here several models which attempt to shed light on the assembly of microtubules and their interactions with motor proteins. Subsequently, an overview of actin filaments and their properties isgiven with particular emphasis on actin assembly processes. The lengths of actin filaments have been reported that were formed by spontaneous polymerization of highly purified actin monomers after labeling with rhodamine-phalloidin. The length distributions are exponential with a mean of about 7 μm. This length is independent of the initial concentration of actin monomer, an observation inconsistent with a simple nucleation-elongation mechanism. However, with the addition of physically reasonable rates of filament annealing and fragmenting, a nucleation-elongation mechanism can reproduce the observed average length of filaments in two types of experiments: (1) filaments formed from a wide range of highly purified actin monomer concentrations, and (2) filaments formed from 24 mM actin over a range of CapZ concentrations. In the final part of the paper we briefly review the stochastic models used to describe the motion of motor proteins on protein filaments. The vast majority of these models are based on ratchet potentials with the presence of thermal noise and forcing due to ATP binding and a subsequent hydrolysis. Many outstanding questions remain to be quantitatively addressed on a molecular level in order to explain the structure-to-function relationship for the key elements of the cytoskeleton discussed in this review.

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

In this paper we present solutions of the master equations for the microtubule length and show that the local probability for rescues or catastrophes can lead to bell-shaped length histograms. Conversely, as already known, non-local probabilities for these events result in exponential length histograms. We also derive master equations for a stabilizing cap and obtain a new boundary condition which provides an explanation of the results obtained in dilution and cutting experiments.

The zinc dyshomeostasis hypothesis of Alzheimer's disease

Alzheimer's disease (AD) is the most common form of dementia in the elderly. Hallmark AD neuropathology includes extracellular amyloid plaques composed largely of the amyloid-β protein (Aβ), intracellular neurofibrillary tangles (NFTs) composed of hyper-phosphorylated microtubule-associated protein tau (MAP-tau), and microtubule destabilization. Early-onset autosomal dominant AD genes are associated with excessive Aβ accumulation, however cognitive impairment best correlates with NFTs and disrupted microtubules. The mechanisms linking Aβ and NFT pathologies in AD are unknown. Here, we propose that sequestration of zinc by Aβ-amyloid deposits (Aβ oligomers and plaques) not only drives Aβ aggregation, but also disrupts zinc homeostasis in zinc-enriched brain regions important for memory and vulnerable to AD pathology, resulting in intra-neuronal zinc levels, which are either too low, or excessively high. To evaluate this hypothesis, we 1) used molecular modeling of zinc binding to the microtubule component protein tubulin, identifying specific, high-affinity zinc binding sites that influence side-to-side tubulin interaction, the sensitive link in microtubule polymerization and stability. We also 2) performed kinetic modeling showing zinc distribution in extra-neuronal Aβ deposits can reduce intra-neuronal zinc binding to microtubules, destabilizing microtubules. Finally, we 3) used metallomic imaging mass spectrometry (MIMS) to show anatomically-localized and age-dependent zinc dyshomeostasis in specific brain regions of Tg2576 transgenic, mice, a model for AD. We found excess zinc in brain regions associated with memory processing and NFT pathology. Overall, we present a theoretical framework and support for a new theory of AD linking extra-neuronal Aβ amyloid to intra-neuronal NFTs and cognitive dysfunction. The connection, we propose, is based on β-amyloid-induced alterations in zinc ion concentration inside neurons affecting stability of polymerized microtubules, their binding to MAP-tau, and molecular dynamics involved in cognition. Further, our theory supports novel AD therapeutic strategies targeting intra-neuronal zinc homeostasis and microtubule dynamics to prevent neurodegeneration and cognitive decline.

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.

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.

Penrose-Hameroff orchestrated objective-reduction proposal for human consciousness is not biologically feasible

Penrose and Hameroff have argued that the conventional models of a brain function based on neural networks alone cannot account for human consciousness, claiming that quantum-computation elements are also required. Specifically, in their Orchestrated Objective Reduction (Orch OR) model [R. Penrose and S. R. Hameroff, J. Conscious. Stud. 2, 99 (1995)], it is postulated that microtubules act as quantum processing units, with individual tubulin dimers forming the computational elements. This model requires that the tubulin is able to switch between alternative conformational states in a coherent manner, and that this process be rapid on the physiological time scale. Here, the biological feasibility of the Orch OR proposal is examined in light of recent experimental studies on microtubule assembly and dynamics. It is shown that the tubulins do not possess essential properties required for the Orch OR proposal, as originally proposed, to hold. Further, we consider also recent progress in the understanding of the long-lived coherent motions in biological systems, a feature critical to Orch OR, and show that no reformation of the proposal based on known physical paradigms could lead to quantum computing within microtubules. Hence, the Orch OR model is not a feasible explanation of the origin of consciousness.

Effective potential of longitudinal interactions between microtubule protofilaments

An effective potential for longitudinal interactions between adjacent protofilaments in a microtubule is introduced. Our proposed interaction potential is a periodic and continuous function of the offset between two protofilaments, which also incorporates the bending energy of protofilaments. This potential produces the results of atomistic simulations. Further, using the potential, a Monte Carlo simulation gives results for the skew angles of observed structures that are in good agreement with experiments.

New horizons of adenosinetriphosphate energetics arising from interaction with magnesium cofactor

MD DFT:B3LYP (6-31G** basis set, T = 310 K) method is used to study interactions [singlet (S) and triplet (T) reaction paths] between adenosinetriphosphate, ATP(4-), and [Mg(H(2)O)(6)](2+) in water environment, modeled with 78 water molecules. Computations reveal the appearance of low and high-energy states (stable, quasi-stable, and unstable), assigned to different spin symmetries. At the initial stage of interaction, ATP donates a part of its negative charge to the Mg complex making the Mg slightly charged. As a result, the original octahedral Mg complex loses two (S state) or four (T state) water molecules. Moving along S or T potential energy surfaces (PESs), Mg(H(2)O)(4 )or Mg(H(2)O)(2) display different ways of complexation with ATP. S path favors the formation of a stable chelate with the O1-O2 fragment of ATP triphosphate tail, whereas T path favors producing a single-bonded complex with the O2. The latter, being unstable, undergoes a further conversion into a spin-separated complex, also unstable, and two metastable S complexes, which finally arise in two stable, low-energy and high-energy, chelates. The spin-separated complex experiences rapid decomposition resulting in the production of a highly reactive adenosinemonophosphate ion-radical *AMP, early observed in the CIDNP experiment (Tulub 2006). Biological consequences of the findings are discussed.

Integrated modeling methodology for microtubule dynamics and Taxol kinetics with experimentally identifiable parameters

Microtubule dynamics play a critical role in cell function and stress response, modulating mitosis, morphology, signaling, and transport. Drugs such as paclitaxel (Taxol) can impact tubulin polymerization and affect microtubule dynamics. While theoretical methods have been previously proposed to simulate microtubule dynamics, we develop a methodology here that can be used to compare model predictions with experimental data. Our model is a hybrid of (1) a simple two-state stochastic formulation of tubulin polymerization kinetics and (2) an equilibrium approximation for the chemical kinetics of Taxol drug binding to microtubule ends. Model parameters are biologically realistic, with values taken directly from experimental measurements. Model validation is conducted against published experimental data comparing optical measurements of microtubule dynamics in cultured cells under normal and Taxol-treated conditions. To compare model predictions with experimental data requires applying a "windowing" strategy on the spatiotemporal resolution of the simulation. From a biological perspective, this is consistent with interpreting the microtubule "pause" phenomenon as at least partially an artifact of spatiotemporal resolution limits on experimental measurement.

A multistranded polymer model explains MinDE dynamics in E. coli cell division

In Escherichia coli, the location of the site for cell division is regulated by the action of the Min proteins. These proteins undergo a periodic pole-to-pole oscillation that involves polymerization and ATPase activity of MinD under the controlling influence of MinE. This oscillation suppresses division near the poles while permitting division at midcell. Here, we propose a multistranded polymer model for MinD and MinE dynamics that quantitatively agrees with the experimentally observed dynamics in wild-type cells and in several well-studied mutant phenotypes. The model also provides new explanations for several phenotypes that have never been addressed by previous modeling attempts. In doing so, the model bridges a theoretical gap between protein structure, biochemistry, and mutant phenotypes. Finally, the model emphasizes the importance of nonequilibrium polymer dynamics in cell function by demonstrating how behavior analogous to the dynamic instability of microtubules is used by E. coli to achieve a sufficiently rapid timescale in controlling division site selection.

Activation of tubulin assembly into microtubules upon a series of repeated femtosecond laser impulses

Tubulin, a globular protein, mostly distributed in nature in the dimeric alpha, beta form, can polymerize in vivo and in vitro into microtubules-longitudinal dynamic assemblies, involved in numerous cellular functions, including cell division and signaling. Tubulin polymerization starts upon binding Mg(2+) with the tubulin guanosine triphosphate (GTP) site. In the current study we show that a series of repeated femtosecond laser impulses activate the same site without adding Mg(2+). GTP site activation (without GTP no polymerization occurs) produces hydrated electrons (they are detected by the UV spectra), which are trapped in the shell of biological water, surrounding the tubulin. These electrons generate an additional, nonlinear by nature, polarization effect, responsible for the second harmonic generation at lambda=365 nm (the first harmonic is centered at lambda=730 nm) and manyfold increase in strength of the initial electric field. The results are supported by model calculations, based on the assumption of positive (negative) feedback, appearing on interaction of charge transfer exciton dipoles with the applied electromagnetic field.

Comparative autoregressive moving average analysis of kinetochore microtubule dynamics in yeast

To elucidate the regulation of kinetochore microtubules (kMTs) by kinetochore proteins in Saccharomyces cerevisiae, we need tools to characterize and compare stochastic kMT dynamics. Here we show that autoregressive moving average (ARMA) models, combined with a statistical framework for testing the significance of differences between ARMA model parameters, provide a sensitive method for identifying the subtle changes in kMT dynamics associated with kinetochore protein mutations. Applying ARMA analysis to G1 kMT dynamics, we found that 1), kMT dynamics in the kinetochore protein mutants okp1-5 and kip3Delta are different from those in wild-type, demonstrating the regulation of kMTs by kinetochore proteins; 2), the kinase Ipl1p regulates kMT dynamics also in G1; and 3), the mutant dam1-1 exhibits three different phenotypes, indicating the central role of Dam1p in maintaining the attachment of kMTs and regulating their dynamics. We also confirmed that kMT dynamics vary with temperature, and are most likely differentially regulated at 37 degrees C. Therefore, when elucidating the role of a protein in kMT regulation using a temperature-sensitive mutant, dynamics in the mutant at its nonpermissive temperature must be compared to those in wild-type at the same temperature, not to those in the mutant at its permissive temperature.

Understanding intracellular transport processes pertinent to synthetic gene delivery via stochastic simulations and sensitivity analyses

A major challenge in synthetic gene delivery is to quantitatively predict the optimal design of polymer-based gene carriers (polyplexes). Here, we report a consistent, integrated, and fundamentally grounded computational methodology to address this challenge. This is achieved by accurately representing the spatio-temporal dynamics of intracellular structures and by describing the interactions between gene carriers and cellular components at a discrete, nanoscale level. This enables the applications of systems tools such as optimization and sensitivity analysis to search for the best combination of systems parameters. We validate the approach using DNA delivery by polyethylenimine as an example. We show that the cell topology (e.g., size, circularity, and dimensionality) strongly influences the spatiotemporal distribution of gene carriers, and consequently, their optimal intracellular pathways. The model shows that there exists an upper limit on polyplexes' intracellular delivery efficiency due to their inability to protect DNA until nuclear entry. The model predicts that even for optimally designed polyethylenimine vectors, only approximately 1% of total DNA is delivered to the nucleus. Based on comparison with gene delivery by viruses, the model suggests possible strategies to significantly improve transfection efficiencies of synthetic gene vectors.

Yeast kinetochore microtubule dynamics analyzed by high-resolution three-dimensional microscopy

We have probed single kinetochore microtubule (k-MT) dynamics in budding yeast in the G1 phase of the cell cycle by automated tracking of a green fluorescent protein tag placed proximal to the centromere on chromosome IV and of a green fluorescent protein tag fused to the spindle pole body protein Spc42p. Our method reliably distinguishes between different dynamics in wild-type and mutant strains and under different experimental conditions. Using our methods we established that in budding yeast, unlike in metazoans, chromosomes make dynamic attachments to microtubules in G1. This makes it possible to interpret measurements of centromere tag dynamics as reflecting k-MT dynamics. We have examined the sensitivity of our assay by studying the effect of temperature, exposure to benomyl, and a tubulin mutation on k-MT dynamics. We have found that lowering the temperature and exposing cells to benomyl attenuate k-MT dynamics in a similar manner. We further observe that, in contrast to previous reports, the mutant tub2-150 forms k-MTs that depolymerize faster than wild type. Based on these findings, we propose high-resolution light microscopy of centromere dynamics in G1 yeast cells as a sensitive assay for the regulation of single k-MT dynamics.

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.

Organization of the cytokeratin network in an epithelial cell

The cytoskeleton is a dynamic three-dimensional structure mainly located in the cytoplasm. It is involved in many cell functions such as mechanical signal transduction and maintenance of cell integrity. Among the three cytoskeletal components, intermediate filaments (the cytokeratin in epithelial cells) are the best candidates for this mechanical role. A model of the establishment of the cytokeratin network of an epithelial cell is proposed to study the dependence of its structural organization on extracellular mechanical environment. To implicitly describe the latter and its effects on the intracellular domain, we use mechanically regulated protein synthesis. Our model is a hybrid of a partial differential equation of parabolic type, governing the evolution of the concentration of cytokeratin, and a set of stochastic differential equations describing the dynamics of filaments. Each filament is described by a stochastic differential equation that reflects both the local interactions with the environment and the non-local interactions via the past history of the filament. A three-dimensional simulation model is derived from this mathematical model. This simulation model is then used to obtain examples of cytokeratin network architectures under given mechanical conditions, and to study the influence of several parameters.

Models of spatial and orientational self-organization of microtubules under the influence of gravitational fields

Tabony and co-workers [C. Papaseit, N. Pochon, and J. Tabony, Proc. Natl. Acad. Sci. U.S.A. 97, 8364 (2000)] showed that the self-organization of microtubules from purified tubulin solutions is sensitive to gravitational conditions. In this paper, we propose two models of spatial and orientational self-organization of microtubules in a gravitational field. First, the spatial model is based on the dominant chemical kinetics. The pattern formation of microtubule concentration is obtained (1) in terms of a moving kink in the limit when the disassembly rate is negligible, and (2) for the case of no free tubulin and only assembled microtubules present. Second, the orientational pattern of striped microtubule domains is consistent with predictions from a phenomenological Landau-Ginzburg free energy expansion in terms of an orientational order parameter.

Modeling oscillatory microtubule polymerization

Polymerization of microtubules is ubiquitous in biological cells and under certain conditions it becomes oscillatory in time. Here, simple reaction models are analyzed that capture such oscillations as well as the length distribution of microtubules. We assume reaction conditions that are stationary over many oscillation periods, and it is a Hopf bifurcation that leads to a persistent oscillatory microtubule polymerization in these models. Analytical expressions are derived for the threshold of the bifurcation and the oscillation frequency in terms of reaction rates, and typical trends of their parameter dependence are presented. Both, a catastrophe rate that depends on the density of guanosine triphosphate liganded tubulin dimers and a delay reaction, such as the depolymerization of shrinking microtubules or the decay of oligomers, support oscillations. For a tubulin dimer concentration below the threshold, oscillatory microtubule polymerization occurs transiently on the route to a stationary state, as shown by numerical solutions of the model equations. Close to threshold, a so-called amplitude equation is derived and it is shown that the bifurcation to microtubule oscillations is supercritical.

Microtubule self-organization is gravity-dependent

Although weightlessness is known to affect living cells, the manner by which this occurs is unknown. Some reaction-diffusion processes have been theoretically predicted as being gravity-dependent. Microtubules, a major constituent of the cellular cytoskeleton, self-organize in vitro by way of reaction-diffusion processes. To investigate how self-organization depends on gravity, microtubules were assembled under low gravity conditions produced during space flight. Contrary to the samples formed on an in-flight 1 x g centrifuge, the samples prepared in microgravity showed almost no self-organization and were locally disordered.

A Landau-Ginzburg Model of the Co-existence of Free Tubulin and Assembled Microtubules in Nucleation and Oscillations Phenomena

A link is shown between reaction-diffusion kinetics for microtubuleassembly and time-dependent Landau-Ginzburg phenomenology. In the latter,microtubule assembly is treated as a first-order phase transition using apostulated Landau-Ginzburg free energy expansion. The results establish aconnection between the oscillations observed in experiment and the phasediagram for microtubule assembly. The model also predicts a specific heatbehavior which could be verified experimentally.

Model for spatial microtubule oscillations

Under particular in vitro conditions, oscillating spatial and temporal waves of assembled microtubules can be observed. A reaction-diffusion model is presented to reproduce these results. This model is based on a set of chemical reaction equations and extended to include spatial dependence and diffusion. The basic properties of the model are presented and the results are demonstrated to connect the observable waves with turbidimetric measurements. The results of the model are consistent with experimental findings.