Synchronous oscillations in microtubule polymerization

Length Regulation Drives Self-Organization in Filament-Motor Mixtures

Cytoskeletal networks form complex intracellular structures. Here we investigate a minimal model for filament-motor mixtures in which motors act as depolymerases and thereby regulate filament length. Combining agent-based simulations and hydrodynamic equations, we show that resource-limited length regulation drives the formation of filament clusters despite the absence of mechanical interactions between filaments. Even though the orientation of individual remains fixed, collective filament orientation emerges in the clusters, aligned orthogonal to their interfaces.

Proteomic analysis and microtubule dynamicity of human sperm in electromagnetic cryopreservation

The proteomic changes, microtubule dynamicity, and quality parameters of human sperm were investigated during cryopreservation in an extremely low electromagnetic field (ELEF) condition. Semen samples were obtained from 210 healthy individuals with normospermia and then were divided into three experimental groups: fresh control, frozen control, and frozen ELEF group. Shotgun proteomics was performed to assess the identification of microtubule proteins of the sperm in experimental groups. Microtubule dynamicity, secondary, and tertiary structure modifications of tubulins, characteristics of transmission electron microscopy of sperm as well as sperm quality parameters were evaluated. The expression ratios of α- and β-tubulins were significantly increased after cryopreservation compared with fresh control while this ratio was not significantly different in frozen ELEF group. The expression ratio of tubulin polymerization-promoting protein was significantly decreased after cryopreservation compared with fresh control. The length, width, and the activity of microtubule, secondary, and tertiary structures of tubulins, motility, and the viability of the sperm were decreased in frozen control as compared with fresh control. The microtubule activity, secondary, and tertiary structures of sperm tubulin in frozen ELEF group were higher than frozen control. Transmission electron microscopy of microtubules showed that the size of the width and length of the microtubules in frozen ELEF group were greater than frozen control. Motility, viability, and reactive oxygen species levels were improved in frozen ELEF group when compared with frozen control. While the microtubule dynamicity of the sperm was affected by the cryopreservation, this trait was improved during the electromagnetic cryopreservation resulted in better motility and viability.

Non-equilibrium assembly of microtubules: from molecules to autonomous chemical robots

Biological systems have evolved to harness non-equilibrium processes from the molecular to the macro scale. It is currently a grand challenge of chemistry, materials science, and engineering to understand and mimic biological systems that have the ability to autonomously sense stimuli, process these inputs, and respond by performing mechanical work. New chemical systems are responding to the challenge and form the basis for future responsive, adaptive, and active materials. In this article, we describe a particular biochemical-biomechanical network based on the microtubule cytoskeletal filament - itself a non-equilibrium chemical system. We trace the non-equilibrium aspects of the system from molecules to networks and describe how the cell uses this system to perform active work in essential processes. Finally, we discuss how microtubule-based engineered systems can serve as testbeds for autonomous chemical robots composed of biological and synthetic components.

Effects of aging in catastrophe on the steady state and dynamics of a microtubule population

Several independent observations have suggested that the catastrophe transition in microtubules is not a first-order process, as is usually assumed. Recent in vitro observations by Gardner et al. [M. K. Gardner et al., Cell 147, 1092 (2011)] showed that microtubule catastrophe takes place via multiple steps and the frequency increases with the age of the filament. Here we investigate, via numerical simulations and mathematical calculations, some of the consequences of the age dependence of catastrophe on the dynamics of microtubules as a function of the aging rate, for two different models of aging: exponential growth, but saturating asymptotically, and purely linear growth. The boundary demarcating the steady-state and non-steady-state regimes in the dynamics is derived analytically in both cases. Numerical simulations, supported by analytical calculations in the linear model, show that aging leads to nonexponential length distributions in steady state. More importantly, oscillations ensue in microtubule length and velocity. The regularity of oscillations, as characterized by the negative dip in the autocorrelation function, is reduced by increasing the frequency of rescue events. Our study shows that the age dependence of catastrophe could function as an intrinsic mechanism to generate oscillatory dynamics in a microtubule population, distinct from hitherto identified ones.

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.

Atomic water channel controlling remarkable properties of a single brain microtubule: correlating single protein to its supramolecular assembly

Microtubule nanotubes are found in every living eukaryotic cells; these are formed by reversible polymerization of the tubulin protein, and their hollow fibers are filled with uniquely arranged water molecules. Here we measure single tubulin molecule and single brain-neuron extracted microtubule nanowire with and without water channel inside to unravel their unique electronic and optical properties for the first time. We demonstrate that the energy levels of a single tubulin protein and single microtubule made of 40,000 tubulin dimers are identical unlike conventional materials. Moreover, the transmitted ac power and the transient fluorescence decay (single photon count) are independent of the microtubule length. Even more remarkable is the fact that the microtubule nanowire is more conducting than a single protein molecule that constitutes the nanowire. Microtubule's vibrational peaks condense to a single mode that controls the emergence of size independent electronic/optical properties, and automated noise alleviation, which disappear when the atomic water core is released from the inner cylinder. We have carried out several tricky state-of-the-art experiments and identified the electromagnetic resonance peaks of single microtubule reliably. The resonant vibrations established that the condensation of energy levels and periodic oscillation of unique energy fringes on the microtubule surface, emerge as the atomic water core resonantly integrates all proteins around it such that the nanotube irrespective of its size functions like a single protein molecule. Thus, a monomolecular water channel residing inside the protein-cylinder displays an unprecedented control in governing the tantalizing electronic and optical properties of microtubule.

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.

Cooperative dynamics of microtubule ensembles: polymerization forces and rescue-induced oscillations

We investigate the cooperative dynamics of an ensemble of N microtubules growing against an elastic barrier. Microtubules undergo so-called catastrophes, which are abrupt stochastic transitions from a growing to a shrinking state, and rescues, which are transitions back to the growing state. Microtubules can exert pushing or polymerization forces on an obstacle, such as an elastic barrier, if the growing end is in contact with the obstacle. We use dynamical mean-field theory and stochastic simulations to analyze a model where each microtubule undergoes catastrophes and rescues and where microtubules interact by force sharing. For zero rescue rate, cooperative growth terminates in a collective catastrophe. The maximal polymerization force before catastrophes grows linearly with N for small N or a stiff elastic barrier, in agreement with available experimental results, whereas it crosses over to a logarithmic dependence for larger N or a soft elastic barrier. For a nonzero rescue rate and a soft elastic barrier, the dynamics becomes oscillatory with both collective catastrophe and rescue events, which are part of a robust limit cycle. Both the average and maximal polymerization forces then grow linearly with N, and we investigate their dependence on tubulin on-rates and rescue rates, which can be involved in cellular regulation mechanisms. We further investigate the robustness of the collective catastrophe and rescue oscillations with respect to different catastrophe models.

On the nature and shape of tubulin trails: implications on microtubule self-organization

Microtubules, major elements of the cell skeleton are, most of the time, well organized in vivo, but they can also show self-organizing behaviors in time and/or space in purified solutions in vitro. Theoretical studies and models based on the concepts of collective dynamics in complex systems, reaction-diffusion processes and emergent phenomena were proposed to explain some of these behaviors. In the particular case of microtubule spatial self-organization, it has been advanced that microtubules could behave like ants, self-organizing by 'talking to each other' by way of hypothetic (because never observed) concentrated chemical trails of tubulin that are expected to be released by their disassembling ends. Deterministic models based on this idea yielded indeed like-looking spatio-temporal self-organizing behaviors. Nevertheless the question remains of whether microscopic tubulin trails produced by individual or bundles of several microtubules are intense enough to allow microtubule self-organization at a macroscopic level. In the present work, by simulating the diffusion of tubulin in microtubule solutions at the microscopic scale, we measure the shape and intensity of tubulin trails and discuss about the assumption of microtubule self-organization due to the production of chemical trails by disassembling microtubules. We show that the tubulin trails produced by individual microtubules or small microtubule arrays are very weak and not elongated even at very high reactive rates. Although the variations of concentration due to such trails are not significant compared to natural fluctuations of the concentration of tubuline in the chemical environment, the study shows that heterogeneities of biochemical composition can form due to microtubule disassembly. They could become significant when produced by numerous microtubule ends located in the same place. Their possible formation could play a role in certain conditions of reaction. In particular, it gives a mesoscopic basis to explain the collective dynamics observed in excitable microtubule solutions showing the propagation of concentration waves of microtubules at the millimeter scale, although we doubt that individual microtubules or bundles can behave like molecular ants.

The tempered polymerization of human neuroserpin

Neuroserpin, a member of the serpin protein superfamily, is an inhibitor of proteolytic activity that is involved in pathologies such as ischemia, Alzheimer's disease, and Familial Encephalopathy with Neuroserpin Inclusion Bodies (FENIB). The latter belongs to a class of conformational diseases, known as serpinopathies, which are related to the aberrant polymerization of serpin mutants. Neuroserpin is known to polymerize, even in its wild type form, under thermal stress. Here, we study the mechanism of neuroserpin polymerization over a wide range of temperatures by different techniques. Our experiments show how the onset of polymerization is dependent on the formation of an intermediate monomeric conformer, which then associates with a native monomer to yield a dimeric species. After the formation of small polymers, the aggregation proceeds via monomer addition as well as polymer-polymer association. No further secondary mechanism takes place up to very high temperatures, thus resulting in the formation of neuroserpin linear polymeric chains. Most interesting, the overall aggregation is tuned by the co-occurrence of monomer inactivation (i.e. the formation of latent neuroserpin) and by a mechanism of fragmentation. The polymerization kinetics exhibit a unique modulation of the average mass and size of polymers, which might suggest synchronization among the different processes involved. Thus, fragmentation would control and temper the aggregation process, instead of enhancing it, as typically observed (e.g.) for amyloid fibrillation.

Yeast prions assembly and propagation: contributions of the prion and non-prion moieties and the nature of assemblies

Yeast prions are self-perpetuating protein aggregates that are at the origin of heritable and transmissible non-Mendelian phenotypic traits. Among these, [PSI+], [URE3] and [PIN+] are the most well documented prions and arise from the assembly of Sup35p, Ure2p and Rnq1p, respectively, into insoluble fibrillar assemblies. Fibril assembly depends on the presence of N- or C-terminal prion domains (PrDs) which are not homologous in sequence but share unusual amino-acid compositions, such as enrichment in polar residues (glutamines and asparagines) or the presence of oligopeptide repeats. Purified PrDs form amyloid fibrils that can convert prion-free cells to the prion state upon transformation. Nonetheless, isolated PrDs and full-length prion proteins have different aggregation, structural and infectious properties. In addition, mutations in the "non-prion" domains (non-PrDs) of Sup35p, Ure2p and Rnq1p were shown to affect their prion properties in vitro and in vivo. Despite these evidences, the implication of the functional non-PrDs in fibril assembly and prion propagation has been mostly overlooked. In this review, we discuss the contribution of non-PrDs to prion assemblies, and the structure-function relationship in prion infectivity in the light of recent findings on Sup35p and Ure2p assembly into infectious fibrils from our laboratory and others.

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

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

Yeast prions assembly and propagation: contributions of the prion and non-prion moieties and the nature of assemblies

Yeast prions are self-perpetuating protein aggregates that are at the origin of heritable and transmissible non-Mendelian phenotypic traits. Among these, [PSI+], [URE3] and [PIN+] are the most well documented prions and arise from the assembly of Sup35p, Ure2p and Rnq1p, respectively, into insoluble fibrillar assemblies. Fibril assembly depends on the presence of N- or C-terminal prion domains (PrDs) which are not homologous in sequence but share unusual amino-acid compositions, such as enrichment in polar residues (glutamines and asparagines) or the presence of oligopeptide repeats. Purified PrDs form amyloid fibrils that can convert prion-free cells to the prion state upon transformation. Nonetheless, isolated PrDs and full-length prion proteins have different aggregation, structural and infectious properties. In addition, mutations in the "non-prion" domains (non-PrDs) of Sup35p, Ure2p and Rnq1p were shown to affect their prion properties in vitro and in vivo. Despite these evidences, the implication of the functional non-PrDs in fibril assembly and prion propagation has been mostly overlooked. In this review, we discuss the contribution of non-PrDs to prion assemblies, and the structure-function relationship in prion infectivity in the light of recent findings on Sup35p and Ure2p assembly into infectious fibrils from our laboratory and others.

Synchrony in reaction-diffusion models of morphogenesis: applications to curvature-dependent proliferation and zero-diffusion front waves

The paper presents the classical age-dependent approach of the morphogenesis in the framework of the von Foerster equation, in which we introduce a new constraint and study a new feature: (i) the new constraint concerns cell proliferation along the contour lines of the cell density, depending on the local curvature such as it favours the amplification of the concavities (like in the gastrulation process) and (ii) the new feature consists of considering, on the cell density surface, a remarkable line (the null mean Gaussian curvature line), on which the normal diffusion vanishes, favouring local coexistence of diffusing morphogens, metabolites or cells, and hence the auto-assemblages of these entities. Two applications to biological multi-agents systems are studied, gastrulation and feather morphogenesis.

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.

Structural insight into the inhibition of tubulin by vinca domain peptide ligands

The tubulin vinca domain is the target of widely different microtubule inhibitors that interfere with the binding of vinblastine. Although all these ligands inhibit the hydrolysis of GTP, they affect nucleotide exchange to variable extents. The structures of two vinca domain antimitotic peptides--phomopsin A and soblidotin (a dolastatin 10 analogue)--bound to tubulin in a complex with a stathmin-like domain show that their sites partly overlap with that of vinblastine and extend the definition of the vinca domain. The structural data, together with the biochemical results from the ligands we studied, highlight two main contributors in nucleotide exchange: the flexibility of the tubulin subunits' arrangement at their interfaces and the residues in the carboxy-terminal part of the beta-tubulin H6-H7 loop. The structures also highlight common features of the mechanisms by which vinca domain ligands favour curved tubulin assemblies and destabilize microtubules.

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.

Liénard systems and potential-Hamiltonian decomposition: applications in biology

In separated notes, we described the mathematical aspects of the potential-Hamiltonian (PH) decomposition, in particular, for n-switches and Liénard systems [J. Demongeot, N. Glade, L. Forest, Liénard systems and potential-Hamiltonian decomposition - I. Methodology, II. Algorithm and III. Applications, C. R. Acad. Sci., Paris, Ser. I, in press]. In the present note, we give some examples of biological regulatory systems susceptible to be decomposed. We show that they can be modelled in terms of 2D ordinary differential equations belonging to n-switches and Liénard system families [O. Cinquin, J. Demongeot, High-dimensional switches and the modeling of cellular differentiation, J. Theor. Biol. 233 (2005) 391-411]. Although simplified, these models can be decomposed into a set of equations combining a potential and a Hamiltonian part. We discuss about the advantage of such a PH-decomposition for understanding the mechanisms involved in their regulatory abilities. We suggest a generalized algorithm to deal with differential systems having a second part of rational-fraction type (frequently used in metabolic systems). Finally, we comment what can be interpreted as a precise signification in biological systems from the dynamical behaviours of both the potential and Hamiltonian parts.

Analysis of a mesoscopic stochastic model of microtubule dynamic instability

A theoretical model of dynamic instability of a system of linear one-dimensional microtubules (MTs) in a bounded domain is introduced for studying the role of a cell edge in vivo and analyzing the effect of competition for a limited amount of tubulin. The model differs from earlier models in that the evolution of MTs is based on the rates of single-mesoscopic-unit (e.g., a heterodimer per protofilament) transformations, in contrast to postulating effective rates and frequencies of larger-scale macroscopic changes, extracted, e.g., from the length history plots of MTs. Spontaneous GTP hydrolysis with finite rate after polymerization is assumed, and theoretical estimates of an effective catastrophe frequency as well as other parameters characterizing MT length distributions and cap size are derived. We implement a simple cap model which does not include vectorial hydrolysis. We demonstrate that our theoretical predictions, such as steady-state concentration of free tubulin and parameters of MT length distributions, are in agreement with the numerical simulations. The present model establishes a quantitative link between mesoscopic parameters governing the dynamics of MTs and macroscopic characteristics of MTs in a closed system. Last, we provide an explanation for nonexponential MT length distributions observed in experiments. In particular, we show that the appearance of such nonexponential distributions in the experiments can occur because a true steady state has not been reached and/or due to the presence of a cell edge.

Compartment volume influences microtubule dynamic instability: a model study

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

Effect of tubulin diffusion on polymerization of microtubules

The dynamics of microtubules (MT's) growing from a nucleation center is simulated with a kinetic Monte Carlo model that includes tubulin diffusion. In the limit of fast diffusion (homogeneous tubulin concentration), MT growth is synchronous and bounded. The microtubules form an aster with a monotonously decreasing long-time distribution of lengths. Slow tubulin diffusion leads to rapid dephasing in the growth dynamics, unbounded growth of some MT's, spatial inhomogeneities, and morphological change toward a morphology with bounded short MT's located in the nucleation center and unbounded long MT's with narrowly distributed lengths. The transition from unbounded to bounded growth is driven by the competition between the reaction rate of the tubulin assembly and the tubulin's diffusion rate. While the present study reports the effect of the tubulin diffusion coefficient on the transition, the results of the simulations are qualitatively comparable to the morphological and dynamical changes of centrosome-nucleated MT's from interphase to mitosis in cellular systems where the transition is regulated by the reaction rates.

Dynamic instability in a DNA-segregating prokaryotic actin homolog

Dynamic instability-the switching of a two-state polymer between phases of steady elongation and rapid shortening-is essential to the cellular function of eukaryotic microtubules, especially during chromosome segregation. Since the discovery of dynamic instability 20 years ago, no other biological polymer has been found to exhibit this behavior. Using total internal reflection fluorescence microscopy and fluorescence resonance energy transfer, we observe that the prokaryotic actin homolog ParM, whose assembly is required for the segregation of large, low-copy number plasmids, displays both dynamic instability and symmetrical, bidirectional polymerization. The dynamic instability of ParM is regulated by adenosine triphosphate (ATP) hydrolysis, and filaments are stabilized by a cap of ATP-bound monomers. ParM is not related to tubulin, so its dynamic instability must have arisen by convergent evolution driven by a set of common constraints on polymer-based segregation of DNA.

Pattern formation driven by nematic ordering of assembling biopolymers

The biopolymers actin and microtubules are often in an ongoing assembling-disassembling state far from thermal equilibrium. Above a critical density this leads to spatially periodic patterns, as shown by a scaling argument and in terms of a phenomenological continuum model, which meets also Onsager's statistical theory of the nematic-to-isotropic transition in the absence of reaction kinetics. This pattern forming process depends much on nonlinear effects and a common linear stability analysis of the isotropic distribution of the filaments is often misleading. The wave number of the pattern decreases with the assembling-disassembling rate and there is an uncommon discontinuous transition between the nematic and periodic states.

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.

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.

Coherent and robust modulation of a metabolic network by cytoskeletal organization and dynamics

In order to investigate the influence of cytoskeletal organization and dynamics on cellular biochemistry, a mathematical model was formulated based on our own experimental evidence. The model couples microtubular protein (MTP) dynamics to the glycolytic pathway and its branches: the Krebs cycle, ethanolic fermentation, and the pentose phosphate (PP) pathway. Results show that the flux through glycolysis coherently and coordinately increases or decreases with increased or decreased levels of polymerized MTP, respectively. The rates of individual enzymatic steps and metabolite concentrations change with the polymeric status of MTP throughout the metabolic network. Negative control is exerted by the PP pathway on the glycolytic flux, and the extent of inhibition depends inversely on the polymerization state of MTP, i.e. a high degree of polymerization relieves the negative control. The stability of the model's steady state dynamics for a wide range of variation of metabolic parameters increased with the degree of polymerized MTP. The findings indicate that the organization of the cytoskeleton bestows coherence and robustness to the coordination of cellular metabolism.

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.

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 metastable intermediate state of microtubule dynamic instability that differs significantly between plus and minus ends

The current two-state GTP cap model of microtubule dynamic instability proposes that a terminal crown of GTP-tubulin stabilizes the microtubule lattice and promotes elongation while loss of this GTP-tubulin cap converts the microtubule end to shortening. However, when this model was directly tested by using a UV microbeam to sever axoneme-nucleated microtubules and thereby remove the microtubule's GTP cap, severed plus ends rapidly shortened, but severed minus ends immediately resumed elongation (Walker, R.A., S. Inoué, and E.D. Salmon. 1989. J. Cell Biol. 108: 931-937). To determine if these previous results were dependent on the use of axonemes as seeds or were due to UV damage, or if they instead indicate an intermediate state in cap dynamics, we performed UV cutting of self-assembled microtubules and mechanical cutting of axoneme-nucleated microtubules. These independent methods yielded results consistent with the original work: a significant percentage of severed minus ends are stable after cutting. In additional experiments, we found that the stability of both severed plus and minus ends could be increased by increasing the free tubulin concentration, the solution GTP concentration, or by assembling microtubules with guanylyl-(alpha,beta)-methylene-diphosphonate (GMPCPP). Our results show that stability of severed ends, particularly minus ends, is not an artifact, but instead reveals the existence of a metastable kinetic intermediate state between the elongation and shortening states of dynamic instability. The kinetic properties of this intermediate state differ between plus and minus ends. We propose a three-state conformational cap model of dynamic instability, which has three structural states and four transition rate constants, and which uses the asymmetry of the tubulin heterodimer to explain many of the differences in dynamic instability at plus and minus ends.

Heterogeneous distribution and organization of cytoskeletal proteins drive differential modulation of metabolic fluxes

On the basis of experimental data obtained in vitro, we propose that differential segregation of actin and tubulin in the cytoplasm may be a regulatory mechanism of metabolic fluxes. The results presented point out that the same enzymes may be differentially modulated at different locations in the cytoplasm, depending on the cytoskeletal protein present at that location, its concentration, polymeric status, or geometric arrangement. Essentially, actin or microtubular protein would exert their effect on enzymatic catalysis through displacement of the equilibrium of enzyme oligomers either to active or less active species. The latter was corroborated by mathematical modeling of the dynamic coupling between microtubular protein assembly-disassembly and pyruvate kinase activity. From these results, a precise biochemical meaning can be given to the putative linkage existing between the mechanisms by which cells rearrange their cytoplasmic architecture and the dynamics of biochemical reactions taking place concomitantly.

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.

Solitary wave dynamics as a mechanism for explaining the internal motion during microtubule growth

Microtubules, which play many diverse and important roles in biological systems, are usually made up of 13 nearly axial protofilaments formed from individual tubulin molecules. In this paper, a nonlinear dynamic model has been developed to elucidate the mechanism of the internal motion occurring during the assembly of microtubules. The results derived from the model indicate that such internal motion is associated with a solitary wave, or kink, excited by the energy released from the hydrolysis of GTP-->GDP in microtubular solutions. As the kink moves forward, the individual tubulin molecules involved in the kink undergo motions that can be likened to the dislocation of atoms within the crystal lattice. Thus, the dynamic instability of microtubules may be characterized by a series of dislocation motions of the tubulin molecules. An energy estimate shows that a kink in the system possesses about 0.36-0.44 eV, which is quite close to but smaller than the 0.49 eV of energy released from the hydrolysis of GTP. Therefore, the relevant energy derived from our model is fully consistent with experimental observations; this finding also suggests that the hydrolysis energy may be responsible for exciting the solitary wave, or kink, leading to tubulin dislocation in microtubules. Our model, and its intrinsic properties, i.e., dynamic nonlinearity, thermodynamic irreversibility, as well as an energy input from a sustained source, implies that the growth of microtubules is a typical dissipative process and that their structure in vivo is typical of dissipative structures.

Thermodynamics of tubulin polymerization into zinc sheets: assembly is not regulated by GTP hydrolysis

The thermodynamics of tubulin assembly into Zn sheets have been studied, with special emphasis on the role of bound nucleotide and of GTP hydrolysis in polymerization. In contrast to microtubules, Zn sheets could be assembled from GDP-tubulin as well as from GTP-tubulin. Accordingly, no appreciable destabilization of the Zn sheets was observed following GTP hydrolysis and P(i) release, indicating that the binding of Zn2+ to tubulin has abolished the regulatory switch role played by GTP hydrolysis in tubulin assembly. As a consequence, the critical concentration for assembly of Zn sheets did not increase with tubulin concentration, a feature characteristic of microtubule assembly. Zn sheets do not bind P(i) analogs, indicating that the gamma-phosphate binding locus of the E-site of tubulin is occluded following GTP hydrolysis in these GDP-tubulin polymers. Nonlinear van't Hoff plots were obtained for assembly of Zn sheets in the presence of either GTP or GDP, consistent with a change in heat capacity. Enthalpy, entropy, and heat capacity changes had values similar to those reported for assembly of microtubules or polymerization of tubulin-colchicine, indicating that hydrophobic tubulin-tubulin interactions are of comparable size in these different polymers.

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.

Nucleotide hydrolysis regulates the dynamics of actin filaments and microtubules

Actin filaments and microtubules are major dynamic components of the cytoskeleton of eukaryotic cells. Assembly of these polymers from monomeric actin or tubulin occurs with expenditure of energy, because ATP (or GTP) tightly bound to actin (or tubulin) is irreversibly hydrolysed during polymerization. Therefore, actin filaments an microtubules are dissipative structures. Our purpose has been to understand how the dissipation of chemical energy perturbs the laws of reversible helical polymerization defined by Oosawa, and affects the dynamics of these polymers. A kinetic study has shown that nucleotide is hydrolysed on the polymer within at least two steps consecutive to the incorporation of the monomer: cleavage of the gamma-phosphoester bond followed by the slower release of Pi; only the second reaction appears reversible. Pi release, and not cleavage of the gamma-phosphate, is linked to the destabilization of protein-protein interactions in the polymer, and therefore plays the role of a conformational switch. The dynamic properties of the polymer in the NTP- and NDP-Pi intermediate states of the assembly process have been investigated using non-hydrolysable analogues of nucleotides and structural analogues of Pi, AlF4- and (BeF3-, H2O). Because nucleotide hydrolysis is uncoupled from polymerization, actin filaments and microtubules grow with a 'cap' of terminal NTP- and NDP-Pi-subunits that interact strongly, and prevent the rapid depolymerization of the unstable core of the polymer formed of NDP-subunits. The fact that the dynamic properties of the polymer are affected by bound nucleotide results in a nonlinear dependence of the rate of elongation on monomer concentration.(ABSTRACT TRUNCATED AT 250 WORDS)

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.

Membrane fluctuations in erythrocytes are linked to MgATP-dependent dynamic assembly of the membrane skeleton

The observation of low-frequency fluctuations of the cell membrane in erythrocytes and in several nucleated cells suggests that this phenomenon may be a general property of the living cell. A study of these fluctuations in human erythrocytes and its ghosts has now been carried out using a novel optical method based on point dark field microscopy. We have demonstrated that the reestablishment of membrane fluctuations in erythrocyte ghosts is dependent on MgATP but does not necessarily require the restoration of the biconcave shape. The results imply that the dominant component of membrane fluctuations are metabolically dependent and suggest the existence of a dynamic mechano-chemical coupling within the membrane skeleton network induced by MgATP.

Kinetics of GTP hydrolysis during the assembly of chick brain MAP2-tubulin microtubule protein

The kinetics of GTP hydrolysis during microtubule assembly have been examined using chick brain MAP2-tubulin microtubule protein in a NaCl-supplemented buffer. The elongating microtubules terminate in a 'GTP cap', since the kinetics of GTP hydrolysis are slower than those of subunit addition. GTP hydrolysis is (a) stoichiometric, (b) occurs as a vectorial wave as the initial rate of hydrolysis is proportional to the molar concentration of microtubule ends and not to the initial rate of subunit addition, and (c) either does not occur, or occurs only at a much lower rate, in the terminal subunits.