A Three-Dimensional Computer Simulation Model Reveals the Mechanisms for Self-Organization of Plant Cortical Microtubules into Oblique Arrays

The noncentrosomal cortical microtubules (CMTs) of plant cells self-organize into a parallel three-dimensional (3D) array that is oriented transverse to the cell elongation axis in wild-type plants and is oblique in some of the mutants that show twisted growth. To study the mechanisms of CMT array organization, we developed a 3D computer simulation model based on experimentally observed properties of CMTs. Our computer model accurately mimics transverse array organization and other fundamental properties of CMTs observed in rapidly elongating wild-type cells as well as the defective CMT phenotypes observed in the Arabidopsis mor1-1 and fra2 mutants. We found that CMT interactions, boundary conditions, and the bundling cutoff angle impact the rate and extent of CMT organization, whereas branch-form CMT nucleation did not significantly impact the rate of CMT organization but was necessary to generate polarity during CMT organization. We also found that the dynamic instability parameters from twisted growth mutants were not sufficient to generate oblique CMT arrays. Instead, we found that parameters regulating branch-form CMT nucleation and boundary conditions at the end walls are important for forming oblique CMT arrays. Together, our computer model provides new mechanistic insights into how plant CMTs self-organize into specific 3D arrangements.     

A mechanics model of microtubule buckling in living cells

As the most rigid cytoskeletal filaments, microtubules bear compressive forces in living cells, balancing the tensile forces within the cytoskeleton to maintain the cell shape. It is often observed that, in living cells, microtubules under compression severely buckle into short wavelengths. By contrast, when compressed, isolated microtubules in vitro buckle into single long-wavelength arcs. The critical buckling force of the microtubules in vitro is two orders of magnitude lower than that of the microtubules in living cells. To explain this discrepancy, we describe a mechanics model of microtubule buckling in living cells. The model investigates the effect of the surrounding filament network and the cytosol on the microtubule buckling. The results show that, while the buckling wavelength is set by the interplay between the microtubules and the elastic surrounding filament network, the buckling growth rate is set by the viscous cytosol. By considering the nonlinear deformation of the buckled microtubule, the buckling amplitude can be determined at the kinetically constrained equilibrium. The model quantitatively correlates the microtubule bending rigidity, the surrounding filament network elasticity, and the cytosol viscosity with the buckling wavelength, the buckling growth rate, and the buckling amplitude of the microtubules. Such results shed light on designing a unified experimental protocol to measure various critical mechanical properties of subcellular structures in living cells.     

A quantitative study of growth variability of tumour cell clones in vitro

Abstract.   Objectives : In this study, we quantify growth variability of tumour cell clones from a human leukaemia cell line. Materials and methods : We have used microplate spectrophotometry to measure growth kinetics of hundreds of individual cell clones from the Molt3 cell line. Growth rate of each clonal population has been estimated by fitting experimental data with the logistic equation. Results : Growth rates were observed to vary between different clones. Up to six clones with growth rates above or below mean growth rate of the parent population were further cloned and growth rates of their offspring were measured. Distribution of growth rates of the subclones did not significantly differ from that of the parent population, thus suggesting that growth variability has an epigenetic origin. To explain observed distributions of clonal growth rates, we have developed a probabilistic model, assuming that fluctuation in the number of mitochondria through successive cell cycles is the leading cause of growth variability. For fitting purposes, we have estimated experimentally by flow cytometry the average maximum number of mitochondria in Molt3 cells. The model fits nicely observed distributions in growth rates; however, cells in which mitochondria were rendered non-functional (ρ⁰ cells) showed only 30% reduction in clonal growth variability with respect to normal cells. Conclusions : A tumour cell population is a dynamic ensemble of clones with highly variable growth rates. At least part of this variability is due to fluctuations in the initial number of mitochondria in daughter cells.     

Experimentally Verified Parameter Sets for Modelling Heterogeneous Neocortical Pyramidal-Cell Populations

Models of neocortical networks are increasingly including the diversity of excitatory and inhibitory neuronal classes. Significant variability in cellular properties are also seen within a nominal neuronal class and this heterogeneity can be expected to influence the population response and information processing in networks. Recent studies have examined the population and network effects of variability in a particular neuronal parameter with some plausibly chosen distribution. However, the empirical variability and covariance seen across multiple parameters are rarely included, partly due to the lack of data on parameter correlations in forms convenient for model construction. To addess this we quantify the heterogeneity within and between the neocortical pyramidal-cell classes in layers 2/3, 4, and the slender-tufted and thick-tufted pyramidal cells of layer 5 using a combination of intracellular recordings, single-neuron modelling and statistical analyses. From the response to both square-pulse and naturalistic fluctuating stimuli, we examined the class-dependent variance and covariance of electrophysiological parameters and identify the role of the h current in generating parameter correlations. A byproduct of the dynamic I-V method we employed is the straightforward extraction of reduced neuron models from experiment. Empirically these models took the refractory exponential integrate-and-fire form and provide an accurate fit to the perisomatic voltage responses of the diverse pyramidal-cell populations when the class-dependent statistics of the model parameters were respected. By quantifying the parameter statistics we obtained an algorithm which generates populations of model neurons, for each of the four pyramidal-cell classes, that adhere to experimentally observed marginal distributions and parameter correlations. As well as providing this tool, which we hope will be of use for exploring the effects of heterogeneity in neocortical networks, we also provide the code for the dynamic I-V method and make the full electrophysiological data set available.     

Alteration of microtubule dynamic instability during preprophase band formation revealed by yellow fluorescent protein-CLIP170 microtubule plus-end labeling

At the onset of mitosis, plant cells form a microtubular preprophase band that defines the plane of cell division, but the mechanism of its formation remains a mystery. Here, we describe the use of mammalian yellow fluorescent protein-tagged CLIP170 to visualize the dynamic plus ends of plant microtubules in transfected cowpea protoplasts and in stably transformed and dividing tobacco Bright Yellow 2 cells. Using plus-end labeling, we observed dynamic instability in different microtubular conformations in live plant cells. The interphase plant microtubules grow at 5 micro m/min, shrink at 20 micro m/min, and display catastrophe and rescue frequencies of 0.02 and 0.08 events/s, respectively, exhibiting faster turnover than their mammalian counterparts. Strikingly, during preprophase band formation, the growth rate and catastrophe frequency of plant microtubules double, whereas the shrinkage rate and rescue frequency remain unchanged, making microtubules shorter and more dynamic. Using these novel insights and four-dimensional time-lapse imaging data, we propose a model that can explain the mechanism by which changes in microtubule dynamic instability drive the dramatic rearrangements of microtubules during preprophase band and spindle formation in plant cells.     

Microtubule assembly of isotypically purified tubulin and its mixtures

Numerous isotypes of the structural protein tubulin have now been characterized in various organisms and their expression offers a plausible explanation for observed differences affecting microtubule function in vivo. While this is an attractive hypothesis, there are only a handful of studies demonstrating a direct influence of tubulin isotype composition on the dynamic properties of microtubules. Here, we present the results of experimental assays on the assembly of microtubules from bovine brain tubulin using purified isotypes at various controlled relative concentrations. A novel data analysis is developed using recursive maps which are shown to be related to the master equation formalism. We have found striking similarities between the three isotypes of bovine tubulin studied in regard to their dynamic instability properties, except for subtle differences in their catastrophe frequencies. When mixtures of tubulin isotypes are analyzed, their nonlinear concentration dependence is modeled and interpreted in terms of lower affinities of tubulin dimers belonging to the same isotype than those that represent different isotypes indicating hitherto unsuspected influences of tubulin dimers on each other within a microtubule. Finally, we investigate the fluctuations in microtubule assembly and disassembly rates and conclude that the inherent rate variability may signify differences in the guanosine-5′-triphosphate composition of the growing and shortening microtubule tips. It is the main objective of this article to develop a quantitative model of tubulin polymerization for individual isotypes and their mixtures. The possible biological significance of the observed differences is addressed.     

Top-predator abundance and chaos in tritrophic food chains

Nonlinear models and laboratory experiments suggest that populations can be chaotic, whereas field data show that a fair proportion of observed populations are not too far from being chaotic. Thus, a natural question arises: do ecosystems enjoy special properties at the edge of chaos? By limiting the analysis to three classes of tritrophic food chains and to the role played by the nutrient available to the bottom of the ecosystem, we show that top-predator mean abundance is maximum at the edge of chaos. The geographical variability of species abundance along nutrient gradients and the dynamic complexity of observed natural populations are consistent with our findings. Effective guidelines for the sustainability of exploited ecosystems are also derived.     

A Stochastic Multiscale Model That Explains the Segregation of Axonal Microtubules and Neurofilaments in Neurological Diseases

The organization of the axonal cytoskeleton is a key determinant of the normal function of an axon, which is a long thin projection of a neuron. Under normal conditions two axonal cytoskeletal polymers, microtubules and neurofilaments, align longitudinally in axons and are interspersed in axonal cross-sections. However, in many neurotoxic and neurodegenerative disorders, microtubules and neurofilaments segregate apart from each other, with microtubules and membranous organelles clustered centrally and neurofilaments displaced to the periphery. This striking segregation precedes the abnormal and excessive neurofilament accumulation in these diseases, which in turn leads to focal axonal swellings. While neurofilament accumulation suggests an impairment of neurofilament transport along axons, the underlying mechanism of their segregation from microtubules remains poorly understood for over 30 years. To address this question, we developed a stochastic multiscale model for the cross-sectional distribution of microtubules and neurofilaments in axons. The model describes microtubules, neurofilaments and organelles as interacting particles in a 2D cross-section, and is built upon molecular processes that occur on a time scale of seconds or shorter. It incorporates the longitudinal transport of neurofilaments and organelles through this domain by allowing stochastic arrival and departure of these cargoes, and integrates the dynamic interactions of these cargoes with microtubules mediated by molecular motors. Simulations of the model demonstrate that organelles can pull nearby microtubules together, and in the absence of neurofilament transport, this mechanism gradually segregates microtubules from neurofilaments on a time scale of hours, similar to that observed in toxic neuropathies. This suggests that the microtubule-neurofilament segregation can be a consequence of the selective impairment of neurofilament transport. The model generates the experimentally testable prediction that the rate and extent of segregation will be dependent on the sizes of the moving organelles as well as the density of their traffic.     

Yeast kinesin-8 depolymerizes microtubules in a length-dependent manner

The microtubule cytoskeleton and the mitotic spindle are highly dynamic structures, yet their sizes are remarkably constant, thus indicating that the growth and shrinkage of their constituent microtubules are finely balanced. This balance is achieved, in part, through kinesin-8 proteins (such as Kip3p in budding yeast and KLP67A in Drosophila) that destabilize microtubules. Here, we directly demonstrate that Kip3p destabilizes microtubules by depolymerizing them--accounting for the effects of kinesin-8 perturbations on microtubule and spindle length observed in fungi and metazoan cells. Furthermore, using single-molecule microscopy assays, we show that Kip3p has several properties that distinguish it from other depolymerizing kinesins, such as the kinesin-13 MCAK. First, Kip3p disassembles microtubules exclusively at the plus end and second, remarkably, Kip3p depolymerizes longer microtubules faster than shorter ones. These properties are consequences of Kip3p being a highly processive, plus-end-directed motor, both in vitro and in vivo. Length-dependent depolymerization provides a new mechanism for controlling the lengths of subcellular structures.     

A neurally plausible model of the dynamics of motion integration in smooth eye pursuit based on recursive Bayesian estimation

In this study, we describe a model of motion integration in smooth eye pursuit based on a recursive Bayesian estimation process, which displays a dynamic behaviour qualitatively similar to the dynamics of the motion integration process observed experimentally, both psychophysically in humans and monkeys, and physiologically in monkeys. By formulating the model as an approximate version of a Kalman filter algorithm, we have been able to show that it can be put into a neurally plausible, distributed recurrent form which coarsely corresponds to the recurrent circuitry of visual cortical areas V1 and MT. The model thus provides further support for the notion that the motion integration process is based on a form of Bayesian estimation, as has been suggested by many psychophysical studies, and moreover suggests that the observed dynamic properties of this process are the result of the recursive nature of the motion estimation.     

In vivo selective cytoskeleton dynamics quantification in interphase cells induced by pulsed ultraviolet laser nanosurgery

We report on the manipulation of intracellular filaments using a nanosurgery system based on a subnanosecond pulsed UV laser optimized for the localized severing of biological polymers. By inducing artificial catastrophe of selected microtubules (MTs), we perform shrinkage-rate measurements in interphase Ptk-2 cells throughout the entire cell. We quantify the impact of two labeling methods and three fluorescent markers, showing a 25% faster depolymerization with Alexa-488 tubulin compared with Rhodamine and yellow fluorescent protein (YFP) tubulins and a 20% higher variability induced by microinjection compared with stable transfection. Using EB3-GFP as a tip marker, we establish a new protocol to measure shrinkage rate, growth rate and rescue frequency simultaneously with high temporal and spatial specificity in live cells. As our analysis shows, laser-induced MT dynamics are physiologically relevant. The high statistical efficiency that the method offers in terms of numbers of measured events and therefore reduced standard deviations represents an important quantitative improvement in the measurement of dynamic instability parameters in vivo. We extend the application of the method by demonstrating induced dynamic behavior of actin-stress fibers after severing. This new method enables the quantitative investigation of cytoskeleton dynamics in a local confinement.     

Kinetics of microtubule catastrophe assessed by probabilistic analysis.

Microtubules are cytoskeletal filaments whose self-assembly occurs by abrupt switching between states of roughly constant growth and shrinkage, a process known as dynamic instability. Understanding the mechanism of dynamic instability offers potential for controlling microtubule-dependent cellular processes such as nerve growth and mitosis. The growth to shrinkage transitions (catastrophes) and the reverse transitions (rescues) that characterize microtubule dynamic instability have been assumed to be random events with first-order kinetics. By direct observation of individual microtubules in vitro and probabilistic analysis of their distribution of growth times, we found that while the slower growing and biologically inactive (minus) ends obeyed first-order catastrophe kinetics, the faster growing and biologically active (plus) ends did not. The non-first-order kinetics at plus ends imply that growing microtubule plus ends have an effective frequency of catastrophe that depends on how long the microtubules have been growing. This frequency is low initially but then rises asymptotically to a limiting value. Our results also suggest that an additional parameter, beyond the four parameters typically used to describe dynamic instability, is needed to account for the observed behavior and that changing this parameter can significantly affect the distribution of microtubule lengths at steady state.     

Polarity of microtubules nucleated by centrosomes and chromosomes of Chinese hamster ovary cells in vitro

The structural and growth polarities of centrosomal and chromosomal microtubules were studied by analyzing the kinetics of growth of these microtubules and those initiated by flagellar seeds. By comparing rates of elongation of centrosomal and flagellar-seeded microtubules, we determined whether the centrosomal microtubules were free to grow at their plus ends only, minus ends ony, or at both ends. Our results show that centrosomal microtubules elongate at a rate corresponding to the addition of subunits at the plus end only. The depolymerization rate was also equivalent to that for the plus end only. Chromosomal microtubule elongation was similar to the centrosome-initiated growth. Since the data do not support the hypothesis that both ends of these spindle microtubules are able to interact with monomer in solution, then growth must occur only distal or only proximal to the organizing centers, implying tha the opposite ends in unavailable for exchange of subunits. Experiments with flagellar-seeded microtubules serving as internal controls indicated that the inactivity of the minus end could not be accounted for by a diffusible inhibitor, suggesting a structural explanation. Since there is no apparent way in which the distal ends may be capped, whereas the proximal ends are embedded in the pericentriolar cloud, we conclude that centrosomal microtubules are oriented with their plus ends distal to the site of nucleation. A similar analysis for chromosomal microtubules suggests that they too must be oriented with their plus ends distal to the site of initiation.     

Effects of brain microtubule-associated proteins on microtubule dynamics and the nucleating activity of centrosomes

In this paper, we report on the effect of brain microtubule-associated proteins (MAPs) on the dynamic instability of microtubules as well as on the nucleation activity of purified centrosomes. Under our experimental conditions, tau and MAP2 have similar effects on microtubule nucleation and dynamic instability. Tau increases the apparent elongation rate of microtubules in proportion to its molar ratio to tubulin, and we present evidence indicating that this is due to a reduction of microtubule instability rather than to an increase of the on rate of tubulin subunits at the end of growing microtubules. Increasing the molar ratio of tau over tubulin leads also to an increase in the average number of microtubules nucleated per centrosome. This number remains constant with time. This suggests that the number of centrosome-nucleated microtubules at steady state can be determined by factors that are not necessarily irreversibly bound to centrosomes but, rather, affect the dynamic properties of microtubules.     

Nanomechanical model of microtubule translocation in the presence of electric fields

Research efforts in recent years have been directed toward actively controlling the direction of translocation of microtubules on a kinesin-coated glass surface with E-fields (electric fields), opening up the possibility of engineering controllable nanodevices that integrate microtubules and motor proteins into their function. Here, we present a detailed, biophysical model that quantitatively describes our observations on the steering of microtubules by electric fields. A sudden application of an electric field parallel to the surface and normal to the translocation direction of a microtubule bends the leading end toward the anode, because Coulombic (electrophoretic) forces are dominant on negatively charged microtubules. Modeling this bending as a cantilever deflection with uniform loading requires accurate mechanical and electrical properties of microtubules, including their charge density, viscous drag, and flexural rigidity. We determined the charge density of microtubules from measurements of the electrophoretic mobility in a “zero flow” capillary electrophoresis column and estimate it to be 256 e⁻ per micron of length. Viscous drag forces on deflecting microtubules in electroosmotic flows were studied theoretically and experimentally by directly characterizing flows using a caged dye imaging method. The flexural rigidity of microtubules was measured by applying E-fields to microtubules with biotinylated segments that were bound to streptavidin-coated surfaces. From the calculated loading, and the Bernoulli-Euler curvature and moment equation, we find that the flexural rigidity of microtubules depends on their length, suggesting microtubules are anisotropic. Finally, our model accurately predicts the biophysical properties and behavior of microtubules directed by E-fields, which opens new avenues for the design of biomolecular nanotransport systems.     

Disentangling genetic and epigenetic determinants of ultrafast adaptation

A major rationale for the advocacy of epigenetically mediated adaptive responses is that they facilitate faster adaptation to environmental challenges. This motivated us to develop a theoretical–experimental framework for disclosing the presence of such adaptation‐speeding mechanisms in an experimental evolution setting circumventing the need for pursuing costly mutation–accumulation experiments. To this end, we exposed clonal populations of budding yeast to a whole range of stressors. By growth phenotyping, we found that almost complete adaptation to arsenic emerged after a few mitotic cell divisions without involving any phenotypic plasticity. Causative mutations were identified by deep sequencing of the arsenic‐adapted populations and reconstructed for validation. Mutation effects on growth phenotypes, and the associated mutational target sizes were quantified and embedded in data‐driven individual‐based evolutionary population models. We found that the experimentally observed homogeneity of adaptation speed and heterogeneity of molecular solutions could only be accounted for if the mutation rate had been near estimates of the basal mutation rate. The ultrafast adaptation could be fully explained by extensive positive pleiotropy such that all beneficial mutations dramatically enhanced multiple fitness components in concert. As our approach can be exploited across a range of model organisms exposed to a variety of environmental challenges, it may be used for determining the importance of epigenetic adaptation‐speeding mechanisms in general.     

Structure of growing microtubule ends: two-dimensional sheets close into tubes at variable rates

Observation of microtubule growth at different rates by cryo-electron microscopy reveals that the ends range from blunt to long, gently curved sheets. The mean sheet length increases with the growth rate while the width of the distributions increases with the extent of assembly. The combination of a concentration dependent growth rate of the tubulin sheet with a variable closure rate of the microtubule cylinder, results in a model in which stochastic fluctuations in sheet length and tubulin conformation confine GTP-tubulins to microtubule ends. We propose that the variability of microtubule growth rate observed by video microscopy (Gildersleeve, R. F., A. R. Cross, K. E. Cullen, A. P. Fagen, and R. C. Williams. 1992. J. Biol. Chem. 267: 7995- 8006, and this study) is due to the variation in the rate of cylinder closure. The curvature of the sheets at the end of growing microtubules and the small oligomeric structures observed at the end of disassembling microtubules, indicate that tubulin molecules undergo conformational changes both during assembly and disassembly.     

Effects of dynein on microtubule mechanics and centrosome positioning

To determine forces on intracellular microtubules, we measured shape changes of individual microtubules following laser severing in bovine capillary endothelial cells. Surprisingly, regions near newly created minus ends increased in curvature following severing, whereas regions near new microtubule plus ends depolymerized without any observable change in shape. With dynein inhibited, regions near severed minus ends straightened rapidly following severing. These observations suggest that dynein exerts a pulling force on the microtubule that buckles the newly created minus end. Moreover, the lack of any observable straightening suggests that dynein prevents lateral motion of microtubules. To explain these results, we developed a model for intracellular microtubule mechanics that predicts the enhanced buckling at the minus end of a severed microtubule. Our results show that microtubule shapes reflect a dynamic force balance in which dynein motor and friction forces dominate elastic forces arising from bending moments. A centrosomal array of microtubules subjected to dynein pulling forces and resisted by dynein friction is predicted to center on the experimentally observed time scale, with or without the pushing forces derived from microtubule buckling at the cell periphery.