Tubulin oligomers and microtubule oscillations. Antagonistic role of microtubule stabilizers and destabilizers

Evaluation of fendiline treatment in VP40 system with nucleation-elongation process: a computational model of Ebola virus matrix protein assembly

Ebola virus (EBOV) infection is threatening human health, especially in Central and West Africa. Limited clinical trials and the requirement of biosafety level-4 laboratories hinder experimental work to advance our understanding of EBOV and the evaluation of treatment. In this work, we use a computational model to study the assembly and budding process of EBOV and evaluate the effect of fendiline on these processes in the context of fluctuating host membrane lipid levels. Our results demonstrate for the first time that the assembly of VP40 filaments may follow the nucleation-elongation theory, as this mechanism is critical to maintaining a pool of VP40 dimers for the maturation and production of virus-like particles (VLPs). We further find that this nucleation-elongation process is likely influenced by fluctuating phosphatidylserine (PS), which can complicate the efficacy of lipid-targeted therapies like fendiline, a drug that lowers cellular PS levels. Our results indicate that fendiline-induced PS reduction may actually increase VLP production at earlier time points (24 h) and under low fendiline concentrations (≤2 µM). However, this effect is transient and does not change the conclusion that fendiline generally decreases VLP production. In the context of fluctuating PS levels, we also conclude that fendiline can be more efficient at the late stage of VLP budding relative to earlier phases. Combination therapy with a VLP budding step-targeted drug may therefore further increase the treatment efficiency of fendiline. Finally, we also show that fendiline-induced PS reduction more effectively lowers VLP production when VP40 expression is high. Taken together, our results provide critical quantitative information on how fluctuating lipid levels (PS) affect EBOV assembly and egress and how this mechanism can be disrupted by lipid-targeting molecules like fendiline.

IMPORTANCE

Ebola virus (EBOV) infection can cause deadly hemorrhagic fever, which has a mortality rate of ~50%-90% without treatment. The recent outbreaks in Uganda and the Democratic Republic of the Congo illustrate its threat to human health. Though two antibody-based treatments were approved, mortality rates in the last outbreak were still higher than 30%. This can partly be due to the requirement of advanced medical facilities for current treatments. As a result, it is very important to develop and evaluate new therapies for EBOV infection, especially those that can be easily applied in the developing world. The significance of our research is that we evaluate the potential of lipid-targeted treatments in reducing EBOV assembly and egress. We achieved this goal using the VP40 system combined with a computational approach, which both saves time and lowers cost compared to traditional experimental studies and provides innovative new tools to study viral protein dynamics.

Evaluation of Fendiline Treatment in VP40 System with Nucleation-Elongation Process: A Computational Model of Ebola Virus Matrix Protein Assembly

Ebola virus (EBOV) infection is threatening human health, especially in Central and West Africa. Limited clinical trials and the requirement of biosafety level-4 (BSL-4) laboratories hinders experimental work to advance our understanding of EBOV and evaluation of treatment. In this work, we use a computational model to study the assembly and budding process of EBOV and evaluate the effect of fendiline on these processes. Our results indicate that the assembly of VP40 filaments may follow the nucleation-elongation theory, as it is critical to maintain a pool of VP40 dimer for the maturation and production of virus-like particles (VLPs). We further find that the nucleation-elongation process can also be influenced by phosphatidylserine (PS), which can complicate the efficacy of fendiline, a drug that lowers cellular PS levels. We observe that fendiline may increase VLP production at earlier time points (24 h) and under low concentrations (≤ 2 μM). But this effect is transient and does not change the conclusion that fendiline generally decreases VLP production. We also conclude that fendiline can be more efficient at the stage of VLP budding relative to earlier phases. Combination therapy with a VLP budding step-targeted drug may further increase the treatment efficiency of fendiline. Finally, we also show that fendiline has higher efficacy when VP40 expression is high. While these are single-cell level results based on the VP40 system, it points out a potential way of fendiline application affecting EBOV assembly, which can be further tested in experimental studies with multiple EBOV proteins or live virus.

Insights into Chemically Fueled Supramolecular Polymers

Supramolecular polymerization can be controlled in space and time by chemical fuels. A nonassembled monomer is activated by the fuel and subsequently self-assembles into a polymer. Deactivation of the molecule either in solution or inside the polymer leads to disassembly. Whereas biology has already mastered this approach, fully artificial examples have only appeared in the past decade. Here, we map the available literature examples into four distinct regimes depending on their activation/deactivation rates and the equivalents of deactivating fuel. We present increasingly complex mathematical models, first considering only the chemical activation/deactivation rates (i.e., transient activation) and later including the full details of the isodesmic or cooperative supramolecular processes (i.e., transient self-assembly). We finish by showing that sustained oscillations are possible in chemically fueled cooperative supramolecular polymerization and provide mechanistic insights. We hope our models encourage the quantification of activation, deactivation, assembly, and disassembly kinetics in future studies.

Mechanisms of phosphatidylserine influence on viral production: A computational model of Ebola virus matrix protein assembly

Ebola virus (EBOV) infections continue to pose a global public health threat, with high mortality rates and sporadic outbreaks in Central and Western Africa. A quantitative understanding of the key processes driving EBOV assembly and budding could provide valuable insights to inform drug development. Here, we use a computational model to evaluate EBOV matrix assembly. Our model focuses on the assembly kinetics of VP40, the matrix protein in EBOV, and its interaction with phosphatidylserine (PS) in the host cell membrane. It has been shown that mammalian cells transfected with VP40-expressing plasmids are capable of producing virus-like particles (VLPs) that closely resemble EBOV virions. Previous studies have also shown that PS levels in the host cell membrane affects VP40 association with the plasma membrane inner leaflet and that lower membrane PS levels result in lower VLP production. Our computational findings indicate that PS may also have a direct influence on VP40 VLP assembly and budding, where a higher PS level will result in a higher VLP budding rate and filament dissociation rate. Our results further suggest that the assembly of VP40 filaments follow the nucleation-elongation theory, where initialization and oligomerization of VP40 are two distinct steps in the assembly process. Our findings advance the current understanding of VP40 VLP formation by identifying new possible mechanisms of PS influence on VP40 assembly. We propose that these mechanisms could inform treatment strategies targeting PS alone or in combination with other VP40 assembly steps.

Self-Organization and Information Processing: From Basic Enzymatic Activities to Complex Adaptive Cellular Behavior

One of the main aims of current biology is to understand the origin of the molecular organization that underlies the complex dynamic architecture of cellular life. Here, we present an overview of the main sources of biomolecular order and complexity spanning from the most elementary levels of molecular activity to the emergence of cellular systemic behaviors. First, we have addressed the dissipative self-organization, the principal source of molecular order in the cell. Intensive studies over the last four decades have demonstrated that self-organization is central to understand enzyme activity under cellular conditions, functional coordination between enzymatic reactions, the emergence of dissipative metabolic networks (DMN), and molecular rhythms. The second fundamental source of order is molecular information processing. Studies on effective connectivity based on transfer entropy (TE) have made possible the quantification in bits of biomolecular information flows in DMN. This information processing enables efficient self-regulatory control of metabolism. As a consequence of both main sources of order, systemic functional structures emerge in the cell; in fact, quantitative analyses with DMN have revealed that the basic units of life display a global enzymatic structure that seems to be an essential characteristic of the systemic functional metabolism. This global metabolic structure has been verified experimentally in both prokaryotic and eukaryotic cells. Here, we also discuss how the study of systemic DMN, using Artificial Intelligence and advanced tools of Statistic Mechanics, has shown the emergence of Hopfield-like dynamics characterized by exhibiting associative memory. We have recently confirmed this thesis by testing associative conditioning behavior in individual amoeba cells. In these Pavlovian-like experiments, several hundreds of cells could learn new systemic migratory behaviors and remember them over long periods relative to their cell cycle, forgetting them later. Such associative process seems to correspond to an epigenetic memory. The cellular capacity of learning new adaptive systemic behaviors represents a fundamental evolutionary mechanism for cell adaptation.

Microtubule dynamics at low temperature: evidence that tubulin recycling limits assembly

How temperature specifically affects microtubule dynamics and how these lead to changes in microtubule networks in cells have not been established. We investigated these questions in budding yeast, an organism found in diverse environments and therefore predicted to exhibit dynamic microtubules across a broad temperature range. We measured the dynamics of GFP-labeled microtubules in living cells and found that lowering temperature from 37°C to 10°C decreased the rates of both polymerization and depolymerization, decreased the amount of polymer assembled before catastrophes, and decreased the frequency of microtubule emergence from nucleation sites. Lowering to 4°C caused rapid loss of almost all microtubule polymer. We provide evidence that these effects on microtubule dynamics may be explained in part by changes in the cofactor-dependent conformational dynamics of tubulin proteins. Ablation of tubulin-binding cofactors (TBCs) further sensitizes cells and their microtubules to low temperatures, and we highlight a specific role for TBCB/Alf1 in microtubule maintenance at low temperatures. Finally, we show that inhibiting the maturation cycle of tubulin by using a point mutant in β-tubulin confers hyperstable microtubules at low temperatures and rescues the requirement for TBCB/Alf1 in maintaining microtubule polymer at low temperatures. Together, these results reveal an unappreciated step in the tubulin cycle.

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.

Elements of the cellular metabolic structure

A large number of studies have demonstrated the existence of metabolic covalent modifications in different molecular structures, which are able to store biochemical information that is not encoded by DNA. Some of these covalent mark patterns can be transmitted across generations (epigenetic changes). Recently, the emergence of Hopfield-like attractor dynamics has been observed in self-organized enzymatic networks, which have the capacity to store functional catalytic patterns that can be correctly recovered by specific input stimuli. Hopfield-like metabolic dynamics are stable and can be maintained as a long-term biochemical memory. In addition, specific molecular information can be transferred from the functional dynamics of the metabolic networks to the enzymatic activity involved in covalent post-translational modulation, so that determined functional memory can be embedded in multiple stable molecular marks. The metabolic dynamics governed by Hopfield-type attractors (functional processes), as well as the enzymatic covalent modifications of specific molecules (structural dynamic processes) seem to represent the two stages of the dynamical memory of cellular metabolism (metabolic memory). Epigenetic processes appear to be the structural manifestation of this cellular metabolic memory. Here, a new framework for molecular information storage in the cell is presented, which is characterized by two functionally and molecularly interrelated systems: a dynamic, flexible and adaptive system (metabolic memory) and an essentially conservative system (genetic memory). The molecular information of both systems seems to coordinate the physiological development of the whole cell.

On the dynamics of the adenylate energy system: homeorhesis vs homeostasis

Biochemical energy is the fundamental element that maintains both the adequate turnover of the biomolecular structures and the functional metabolic viability of unicellular organisms. The levels of ATP, ADP and AMP reflect roughly the energetic status of the cell, and a precise ratio relating them was proposed by Atkinson as the adenylate energy charge (AEC). Under growth-phase conditions, cells maintain the AEC within narrow physiological values, despite extremely large fluctuations in the adenine nucleotides concentration. Intensive experimental studies have shown that these AEC values are preserved in a wide variety of organisms, both eukaryotes and prokaryotes. Here, to understand some of the functional elements involved in the cellular energy status, we present a computational model conformed by some key essential parts of the adenylate energy system. Specifically, we have considered (I) the main synthesis process of ATP from ADP, (II) the main catalyzed phosphotransfer reaction for interconversion of ATP, ADP and AMP, (III) the enzymatic hydrolysis of ATP yielding ADP, and (IV) the enzymatic hydrolysis of ATP providing AMP. This leads to a dynamic metabolic model (with the form of a delayed differential system) in which the enzymatic rate equations and all the physiological kinetic parameters have been explicitly considered and experimentally tested in vitro. Our central hypothesis is that cells are characterized by changing energy dynamics (homeorhesis). The results show that the AEC presents stable transitions between steady states and periodic oscillations and, in agreement with experimental data these oscillations range within the narrow AEC window. Furthermore, the model shows sustained oscillations in the Gibbs free energy and in the total nucleotide pool. The present study provides a step forward towards the understanding of the fundamental principles and quantitative laws governing the adenylate energy system, which is a fundamental element for unveiling the dynamics of cellular life.

The bacterial cell division proteins FtsA and FtsZ self-organize into dynamic cytoskeletal patterns

Bacterial cytokinesis is commonly initiated by the Z-ring, a cytoskeletal structure that assembles at the site of division. Its primary component is FtsZ, a tubulin superfamily GTPase, which is recruited to the membrane by the actin-related protein FtsA. Both proteins are required for the formation of the Z-ring, but if and how they influence each other's assembly dynamics is not known. Here, we reconstituted FtsA-dependent recruitment of FtsZ polymers to supported membranes, where both proteins self-organize into complex patterns, such as fast-moving filament bundles and chirally rotating rings. Using fluorescence microscopy and biochemical perturbations, we found that these large-scale rearrangements of FtsZ emerge from its polymerization dynamics and a dual, antagonistic role of FtsA: recruitment of FtsZ filaments to the membrane and negative regulation of FtsZ organization. Our findings provide a model for the initial steps of bacterial cell division and illustrate how dynamic polymers can self-organize into large-scale structures.

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 metabolic core and catalytic switches are fundamental elements in the self-regulation of the systemic metabolic structure of cells

BACKGROUND

Experimental observations and numerical studies with dissipative metabolic networks have shown that cellular enzymatic activity self-organizes spontaneously leading to the emergence of a metabolic core formed by a set of enzymatic reactions which are always active under all environmental conditions, while the rest of catalytic processes are only intermittently active. The reactions of the metabolic core are essential for biomass formation and to assure optimal metabolic performance. The on-off catalytic reactions and the metabolic core are essential elements of a Systemic Metabolic Structure which seems to be a key feature common to all cellular organisms.

METHODOLOGY/PRINCIPAL FINDINGS

In order to investigate the functional importance of the metabolic core we have studied different catalytic patterns of a dissipative metabolic network under different external conditions. The emerging biochemical data have been analysed using information-based dynamic tools, such as Pearson's correlation and Transfer Entropy (which measures effective functionality). Our results show that a functional structure of effective connectivity emerges which is dynamical and characterized by significant variations of bio-molecular information flows.

CONCLUSIONS/SIGNIFICANCE

We have quantified essential aspects of the metabolic core functionality. The always active enzymatic reactions form a hub--with a high degree of effective connectivity--exhibiting a wide range of functional information values being able to act either as a source or as a sink of bio-molecular causal interactions. Likewise, we have found that the metabolic core is an essential part of an emergent functional structure characterized by catalytic modules and metabolic switches which allow critical transitions in enzymatic activity. Both, the metabolic core and the catalytic switches in which also intermittently-active enzymes are involved seem to be fundamental elements in the self-regulation of the Systemic Metabolic Structure.

Quantitative analysis of cellular metabolic dissipative, self-organized structures

One of the most important goals of the postgenomic era is understanding the metabolic dynamic processes and the functional structures generated by them. Extensive studies during the last three decades have shown that the dissipative self-organization of the functional enzymatic associations, the catalytic reactions produced during the metabolite channeling, the microcompartmentalization of these metabolic processes and the emergence of dissipative networks are the fundamental elements of the dynamical organization of cell metabolism. Here we present an overview of how mathematical models can be used to address the properties of dissipative metabolic structures at different organizational levels, both for individual enzymatic associations and for enzymatic networks. Recent analyses performed with dissipative metabolic networks have shown that unicellular organisms display a singular global enzymatic structure common to all living cellular organisms, which seems to be an intrinsic property of the functional metabolism as a whole. Mathematical models firmly based on experiments and their corresponding computational approaches are needed to fully grasp the molecular mechanisms of metabolic dynamical processes. They are necessary to enable the quantitative and qualitative analysis of the cellular catalytic reactions and also to help comprehend the conditions under which the structural dynamical phenomena and biological rhythms arise. Understanding the molecular mechanisms responsible for the metabolic dissipative structures is crucial for unraveling the dynamics of cellular life.

Global self-regulation of the cellular metabolic structure

BACKGROUND

Different studies have shown that cellular enzymatic activities are able to self-organize spontaneously, forming a metabolic core of reactive processes that remain active under different growth conditions while the rest of the molecular catalytic reactions exhibit structural plasticity. This global cellular metabolic structure appears to be an intrinsic characteristic common to all cellular organisms. Recent work performed with dissipative metabolic networks has shown that the fundamental element for the spontaneous emergence of this global self-organized enzymatic structure could be the number of catalytic elements in the metabolic networks.

METHODOLOGY/PRINCIPAL FINDINGS

In order to investigate the factors that may affect the catalytic dynamics under a global metabolic structure characterized by the presence of metabolic cores we have studied different transitions in catalytic patterns belonging to a dissipative metabolic network. The data were analyzed using non-linear dynamics tools: power spectra, reconstructed attractors, long-term correlations, maximum Lyapunov exponent and Approximate Entropy; and we have found the emergence of self-regulation phenomena during the transitions in the metabolic activities.

CONCLUSIONS/SIGNIFICANCE

The analysis has also shown that the chaotic numerical series analyzed correspond to the fractional Brownian motion and they exhibit long-term correlations and low Approximate Entropy indicating a high level of predictability and information during the self-regulation of the metabolic transitions. The results illustrate some aspects of the mechanisms behind the emergence of the metabolic self-regulation processes, which may constitute an important property of the global structure of the cellular metabolism.

The number of catalytic elements is crucial for the emergence of metabolic cores

Background

Different studies show evidence that several unicellular organisms display a cellular metabolic structure characterized by a set of enzymes which are always in an active state (metabolic core), while the rest of the molecular catalytic reactions exhibit on-off changing states. This self-organized enzymatic configuration seems to be an intrinsic characteristic of metabolism, common to all living cellular organisms. In a recent analysis performed with dissipative metabolic networks (DMNs) we have shown that this global functional structure emerges in metabolic networks with a relatively high number of catalytic elements, under particular conditions of enzymatic covalent regulatory activity.

Methodology/Principal Findings

Here, to investigate the mechanism behind the emergence of this supramolecular organization of enzymes, we have performed extensive DMNs simulations (around 15,210,000 networks) taking into account the proportion of the allosterically regulated enzymes and covalent enzymes present in the networks, the variation in the number of substrate fluxes and regulatory signals per catalytic element, as well as the random selection of the catalytic elements that receive substrate fluxes from the exterior. The numerical approximations obtained show that the percentages of DMNs with metabolic cores grow with the number of catalytic elements, converging to 100% for all cases.

Conclusions/Significance

The results show evidence that the fundamental factor for the spontaneous emergence of this global self-organized enzymatic structure is the number of catalytic elements in the metabolic networks. Our analysis corroborates and expands on our previous studies illustrating a crucial property of the global structure of the cellular metabolism. These results also offer important insights into the mechanisms which ensure the robustness and stability of living cells.

Microtubule assembly dynamics at the nanoscale

BACKGROUND

The labile nature of microtubules is critical for establishing cellular morphology and motility, yet the molecular basis of assembly remains unclear. Here we use optical tweezers to track microtubule polymerization against microfabricated barriers, permitting unprecedented spatial resolution.

RESULTS

We find that microtubules exhibit extensive nanometer-scale variability in growth rate and often undergo shortening excursions, in some cases exceeding five tubulin layers, during periods of overall net growth. This result indicates that the guanosine triphosphate (GTP) cap does not exist as a single layer as previously proposed. We also find that length increments (over 100 ms time intervals, n = 16,762) are small, 0.81 +/- 6.60 nm (mean +/- standard deviation), and very rarely exceed 16 nm (about two dimer lengths), indicating that assembly occurs almost exclusively via single-subunit addition rather than via oligomers as was recently suggested. Finally, the assembly rate depends only weakly on load, with the average growth rate decreasing only 2-fold as the force increases 7-fold from 0.4 pN to 2.8 pN.

CONCLUSIONS

The data are consistent with a mechanochemical model in which a spatially extended GTP cap allows substantial shortening on the nanoscale, while still preventing complete catastrophe in most cases.

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.

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.

A reassessment of the factors affecting microtubule assembly and disassembly in vitro

Current models of microtubule assembly from pure tubulin involve a nucleation phase followed by microtubule elongation at a constant polymer number. Both the rate of microtubule nucleation and elongation are thought to be tightly influenced by the free GTP-tubulin concentration, in a law of mass action-dependent manner. However, these basic hypotheses have remained largely untested due to a lack of data reporting actual measurements of the microtubule length and number concentration during microtubule assembly.Here, we performed simultaneous measurements of the polymeric tubulin concentration, of the free GTP-tubulin concentration, and of the microtubule length and number concentration in both polymerizing and depolymerizing conditions. In agreement with previous work we find that the microtubule nucleation rate is strongly dependent on the initial GTP-tubulin concentration. But we find that microtubule nucleation persists during microtubule elongation. At any given initial tubulin-GTP concentration, the microtubule nucleation rate remains constant during polymer assembly, despite the wide variation in free GTP-tubulin concentration. We also find a remarkable constancy of the rate of microtubule elongation during assembly. Apparently, the rate of microtubule elongation is intrinsic to the polymers, insensitive to large variations of the free GTP-tubulin concentration. Finally we observe that when, following assembly, microtubules depolymerize below the free GTP-tubulin critical concentration, the rate-limiting factor for disassembly is the frequency of microtubule catastrophe. At all time-points during disassembly, the microtubule catastrophe frequency is independent of the free GTP-tubulin concentration but, as the microtubule nucleation rate, is strongly dependent on the initial free GTP-tubulin concentration. We conclude that the dynamics of both microtubule assembly and disassembly depend largely on factors other than the free GTP-tubulin concentration. We propose that intrinsic structural factors and endogenous regulators, whose concentration varies with the initial conditions, are also major determinants of these dynamics.

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.

Microtubules: strange polymers inside the cell

This paper provides a consistent approach (within a one-dimensional approximation) to the description of the evolution of the microtubule length at both low- and high-density concentrations. We derive general master-type equations which are based on the key chemical reactions involved in the assembly and disassembly of microtubules. The processes included are: polymerization and depolymerization of a single protein dimer, catastrophic disassembly affecting an a piori arbitrary number of dimers, and a rescue event. Solutions of the derived equations are compared with the existing experimental data. Important conclusions linking the emergence of bell-shaped histograms with the nature of catastrophe and rescue phenomena are drawn. Finally, we briefly discuss the emergence of coherent phenomena in microtubule polymerization, i.e., a transition to collective oscillations in the assembly and disassembly effects.

A model of microtubule oscillations

Simulations of microtubule oscillations have been obtained by a kinetic model including nucleation of microtubules, elongation by addition of GTP-loaded tubulin dimers, disassembly into oligomers, and dissolution of oligomers followed by nucleotide exchange at the free dimers. Dynamic instability is described by the on and off rates for dimer association in the growth phase, the rate of rapid shortening, and the transition rates for catastrophe and rescue. The latter are assumed to be completely determined by the current state of the system ("short cap hypothesis"). Microtubule oscillations and normal polymerizations measured by time-resolved X-ray scattering were used to test the model. The model is able to produce oscillations without further assumptions. However, in order to obtain good fits to the experimental data one requires an additional mechanism which prevents rapid desynchronization of the microtubules. One of several possible mechanisms that will be discussed is the destabilization of microtubules by the products of disassembly.

Dynamics of microtubules from erythrocyte marginal bands

Microtubules can adjust their length by the mechanism of dynamic instability, that is by switching between phases of growth and shrinkage. Thus far this phenomenon has been studied with microtubules that contain several components, that is, a mixture of tubulin isoforms, with or without a mixture of microtubule-associated proteins (MAPs), which can act as regulators of dynamic instability. Here we concentrate on the influence of the tubulin component. We have studied MAP-free microtubules from the marginal band of avian erythrocytes and compared them with mammalian brain microtubules. The erythrocyte system was selected because it represents a naturally stable aggregate of microtubules; second, the tubulin is largely homogeneous, in contrast to brain tubulin. Qualitatively, erythrocyte microtubules show similar features as brain microtubules, but they were found to be much less dynamic. The critical concentration of elongation, and the rates of association and dissociation of tubulin are all lower than with brain microtubules. Catastrophes are rare, rescues frequent, and shrinkage slow. This means that dynamic instability can be controlled by the tubulin isotype, independently of MAPs. Moreover, the extent of dynamic behavior is highly dependent on buffer conditions. In particular, dynamic instability is strongly enhanced in phosphate buffer, both for erythrocyte marginal band and brain microtubules. The lower stability in phosphate buffer argues against the hypothesis that a cap of tubulin.GDP.Pi subunits stabilizes microtubules. The difference in dynamics between tubulin isotypes and between the two ends of microtubules is preserved in the different buffer systems.

Cytoskeletal logic: a model for molecular computation via Boolean operations in microtubules and microtubule-associated proteins

Adaptive behaviors and dynamic activities within living cells are organized by the cytoskeleton: intracellular networks of interconnected protein polymers which include microtubules (MTs), actin, intermediate filaments, microtubule associated proteins (MAPs) and other protein structures. Cooperative interactions among cytoskeletal protein subunit conformational states have been used to model signal transmission and information processing. In the present work we present a theoretical model for molecular computing in which Boolean logic is implemented in parallel networks of individual MTs interconnected by MAPs. Conformational signals propagate on MTs as in data buses and in the model MAPs are considered as Boolean operators, either as bit-lines (like MTs) where a signal can be transported unchanged between MTs ('BUS-MAP'), or as bit-lines where a Boolean operation is performed in one of the two MAP-MT attachments ('LOGIC-MAP'). Three logic MAPs have been defined ('NOT-MAP, 'AND-MAP', 'XOR-MAP') and used to demonstrate addition, subtraction and other arithmetic operations. Although our choice of Boolean logic is arbitrary, the simulations demonstrate symbolic manipulation in a connectionist system and suggest that MT-MAP networks can perform computation in living cells and are candidates for future molecular computing devices.

Microtubule dynamics and microtubule caps: a time-resolved cryo-electron microscopy study

Microtubules display the unique property of dynamic instability characterized by phase changes between growth and shrinkage, even in constant environmental conditions. The phases can be synchronized, leading to bulk oscillations of microtubules. To study the structural basis of dynamic instability we have examined growing, shrinking, and oscillating microtubules by time-resolved cryo-EM. In particular we have addressed three questions which are currently a matter of debate: (a) What is the relationship between microtubules, tubulin subunits, and tubulin oligomers in microtubule dynamics?; (b) How do microtubules shrink? By release of subunits or via oligomers?; and (c) Is there a conformational change at microtubule ends during the transitions from growth to shrinkage and vice versa? The results show that (a) oscillating microtubules coexist with a substantial fraction of oligomers, even at a maximum of microtubule assembly; (b) microtubules disassemble primarily into oligomers; and (c) the ends of growing microtubules have straight protofilaments, shrinking microtubules have protofilaments coiled inside out. This is interpreted as a transition from a tense to a relaxed conformation which could be used to perform work, as suggested by some models of poleward chromosome movement during anaphase.

Opposite-end behaviour of dynamic microtubules

Microtubules are dynamic polar structures with different kinetic properties at the two ends. The inherent asymmetry of the microtubule lattice determines that the relationship between the addition reaction of tubulin-GTP and the associated hydrolysis of a tubulin-GTP on the polymer is different at the two ends of the microtubule. We present a unified treatment for both ends of the microtubule, using the principles of the Lateral Cap formulation for microtubule dynamic instability. This shows that the two ends can exhibit significantly different dynamic properties in terms of amplitudes and lifetimes of growth and shrinking, depending on the relative importance of longitudinal and lateral contacts in the coupling of tubulin-GTP hydrolysis. These predictions are readily amenable to experimental verification. This modelling suggests that fine details of the subunit-subunit interactions at the microtubule end can determine the characteristic differences in kinetic behaviour of the opposite ends of dynamic microtubules. Variation of these interactions would provide a potentially sensitive general mechanism for the control of such dynamics, both in vitro and in vivo.

Spatial patterns from oscillating microtubules

Microtubules are fibers of the cytoskeleton involved in the generation of cell shape and motility. They can be highly dynamic and are capable of temporal oscillations in their state of assembly. Solutions of tubulin (the subunit protein of microtubules) and guanosine triphosphate (GTP, the cofactor required for microtubule assembly and oscillations) can generate various dissipative structures. They include traveling waves of microtubule assembly and disassembly as well as polygonal networks. The results imply that cytoskeletal proteins can form dynamic spatial structures by themselves, even in the absence of cellular organizing centers. Thus the microtubule system could serve as a simple model for studying pattern formation by biomolecules in vitro.