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Abstract
Earlier Monte Carlo studies on a single-helix model of the GTP cap at the end of a microtubule are extended here to a more realistic five-start helix model of the microtubule end. As in the earlier work, phase changes occur at the microtubule end: the end is either capped with GTP and growing slowly or uncapped and shortening rapidly, and these two regimes alternate (at a given tubulin concentration) at steady state. Macroscopic rate constants for the two-phase model are deduced from the Monte Carlo results. The macroscopic rate constants lead to properties that are in semiquantitative agreement with related experiments of Mitchison and Kirschner.
... 4 A recent model for MT dynamic instability by Hinow et al. [17] describes MT growth and shortening using 5 a similar advection process as Dogterom and Leibler [11]. However, unlike the approach used in [11], Hinow 6 et al. suppose that the growth rate is dependent on the free tubulin concentration, a phenomenon that has been 7 observed in experiments [24]. The latter experiments have shown that when the free tubulin concentration is 8 low, MTs do not grow; at a lower threshold tubulin concentration, MTs grow at a rate that depends linearly 9 on the free tubulin concentration; finally, past an upper threshold for tubulin concentration, the MT growth 10 rate is constant [24]. ...
... These parameters, as well as their range of values and meaning are listed in Table 1. 6 From biological observations of MTs dynamics, it is possible to define ranges for some of the model parameters 7 described in Table 1. In particular, values for the parameters p c , γ depol , a s , and a c can be found in the literature. ...
... However, if 6 we change this parameter simultaneously with γ young hydro or a s , more significant changes are noted. In particular, 7 if we increase the parameter γ old hydro while increasing γ young hydro or decreasing a s , we see that almost all simulated 8 observables change to a large extent. Also, in Table 5, we see significant changes in γ pol (p ∞ ) and γ av hydro , but 9 the value of the relative difference between γ pol and γ av hydro remains at 13% (similar to the control test). ...
Microtubules (MTs) are protein polymers that exhibit a unique type of behavior referred to as dynamic instability. That is, they undergo periods of growth (through the addition of GTP-tubulin) and shortening (through the subtraction of GDP-tubulin). Shortening events are very fast, where this transition is referred to as a catastrophe. There are many processes that regulate MT dynamic instability, however, recent experiments show that MT dynamics may be highly regulated by a MTs age, where young MTs are less likely to undergo shortening events than older ones. In this paper, we develop a novel modeling approach to describe how the age of a MT affects its dynamic properties. In particular, we extend on a previously developed model that describes MT dynamics, by proposing a new concept for GTP-tubulin hydrolysis (the process by which newly incorporated GTP-tubulin is hydrolyzed to lower energy GDP-tubulin). In particular, we assume that hydrolysis is mainly vectorial, age-dependent and delayed according to the GTP-tubulin incorporation into the MT. Through numerical simulation, we are able to show how MT age affects certain properties that define MT dynamics. For example, simulations illustrate how the aging process leads to an increase in the rate of GTP-tubulin hydrolysis for older MTs, as well as increases in catastrophe frequency. Also, since it has been found that MT dynamic instability is affected by chemotherapy microtubule-targeting agents (MTAs), we highlight the fact that our model can be used to investigate the action of MTAs on MT dynamics by varying certain model parameters.
... Over the past few decades, since the discovery of MT dynamic instability ( Mitchison and Kirschner, 1984 ), many mathematical and computational studies have been developed to better understand this dynamical process ( Chen and Hill, 1985;Dogterom and Leibler, 1993;Flyvbjerg et al., 1994Flyvbjerg et al., , 1996Hinow et al., 2009;Martin et al., 1993;Mishra et al., 2005 ). Most computational models are designed to study MTs at the microscopic level, taking into consideration the addition and subtraction of individual tubulin dimers ( Chen and Hill, 1985;Flyvbjerg et al., 1994Flyvbjerg et al., , 1996Martin et al., 1993 ). ...
... Over the past few decades, since the discovery of MT dynamic instability ( Mitchison and Kirschner, 1984 ), many mathematical and computational studies have been developed to better understand this dynamical process ( Chen and Hill, 1985;Dogterom and Leibler, 1993;Flyvbjerg et al., 1994Flyvbjerg et al., , 1996Hinow et al., 2009;Martin et al., 1993;Mishra et al., 2005 ). Most computational models are designed to study MTs at the microscopic level, taking into consideration the addition and subtraction of individual tubulin dimers ( Chen and Hill, 1985;Flyvbjerg et al., 1994Flyvbjerg et al., , 1996Martin et al., 1993 ). Deterministic models have been developed to understand this process at a macroscopic level ( Dogterom and Leibler, 1993;Hinow et al., 2009;Mishra et al., 2005 ). ...
Microtubules (MTs) play a key role in normal cell development and are a primary target for many cancer chemotherapy MT targeting agents (MTAs). As such, understanding MT dynamics in the presence of such agents, as well as other proteins that alter MT dynamics, is extremely important.
... Since the discovery of MT dynamic instability, theoretical modeling of tubulin polymerization played a critical role in interpreting experimental observations and in defining the properties of the GTP-cap (6,11). The proposed models differ significantly in their frameworks, postulates, and mathematical approaches (11)(12)(13)(14)(15)(16)(17)(18)(19)(20). Earlier models represented the MT with single or multiple protofilaments, describing their dynamics with kinetic constants that were different for GTP versus GDP-tubulin dimers. ...
... The two-state model by Hill (21), which additionally assumed random hydrolysis of the tubulin-bound GTP, simulated well the rate of tubulin de/ polymerization as a function of soluble tubulin concentration. However, this and other kinetic models with random GTP hydrolysis (12,22) predicted a large GTP-tubulin cap and suggested that the cap-promoted stabilization should depend strongly on tubulin concentration (23). As a result, it has been difficult with this modeling framework to recapitulate two key experimental dependencies: a moderate suppression of MT catastrophe by increasing tubulin concentration (24,25); and the delay times in dilution experiments, in which MTs were first polymerized using different tubulin concentrations, then the free tubulin was removed rapidly (26,27). ...
Tubulin polymers, microtubules, can switch abruptly from the assembly to shortening. These infrequent transitions, termed "catastrophes", affect numerous cellular processes but the underlying mechanisms are elusive. We approached this complex stochastic system using advanced coarse-grained molecular dynamics modeling of tubulin-tubulin interactions. Unlike in previous simplified models of dynamic microtubules, the catastrophes in this model arise owing to fluctuations in the composition and conformation of a growing microtubule tip, most notably in the number of protofilament curls. In our model, dynamic evolution of the stochastic microtubule tip configurations over a long timescale, known as the system's "aging", gives rise to the nonexponential distribution of microtubule lifetimes, consistent with experiment. We show that aging takes place in the absence of visible changes in the microtubule wall or tip, as this complex molecular-mechanical system evolves slowly and asymptotically toward the steady-state level of the catastrophe-promoting configurations. This new, to our knowledge, theoretical basis will assist detailed mechanistic investigations of the mechanisms of action of different microtubule-binding proteins and drugs, thereby enabling accurate control over the microtubule dynamics to treat various pathologies.
... The first multiple-protofilament model [38] appeared a year after Mitchison and Kirschner's seminal paper [1] and is a computational model. Three years later, Walker et al. [2] wrote ''the model of Chen and Hill [38] is perhaps the most complete formal description of cap dynamics . . . ...
... The first multiple-protofilament model [38] appeared a year after Mitchison and Kirschner's seminal paper [1] and is a computational model. Three years later, Walker et al. [2] wrote ''the model of Chen and Hill [38] is perhaps the most complete formal description of cap dynamics . . . This model can easily fit our catastrophe data for a single end, using only two variations in hydrolysis rate . . .'' ...
A key question in understanding microtubule dynamics is how GTP hydrolysis leads to catastrophe, the switch from slow growth to rapid shrinkage. We first provide a review of the experimental and modeling literature, and then present a new model of microtubule dynamics. We demonstrate that vectorial, random, and coupled hydrolysis mechanisms are not consistent with the dependence of catastrophe on tubulin concentration and show that, although single-protofilament models can explain many features of dynamics, they do not describe catastrophe as a multistep process. Finally, we present a new combined (coupled plus random hydrolysis) multiple-protofilament model that is a simple, analytically solvable generalization of a single-protofilament model. This model accounts for the observed lifetimes of growing microtubules, the delay to catastrophe following dilution and describes catastrophe as a multistep process.
... The relative timing of subunit binding and GTPhydrolysis creates a theorized 'cap' of GTP-bound subunits at the microtubule plus end trailed by GDP-bound subunits in the remainder of the microtubule (Desai and Mitchison, 1997). Since subunit binding and GTP hydrolysis are both rates that are subject to random fluctuations, it is postulated that the GTP-tubulin cap will on occasion be lost due to a lull in polymerization and/or an advance in GTP-hydrolysis (Carlier et al., 1984;Chen and Hill, 1985b;Desai and Mitchison, 1997;Dogterom and Leibler, 1993;Gliksman et al., 1993;Hill, 1984;Hill and Carlier, 1983;Mandelkow et al., 1995;Mitchison and Kirschner, 1987;Walker et al., 1988). This event is theorized to correspond with the observed switch from growth to shortening (termed catastrophe) owing to the lower free subunit affinity of the GDP-bound subunits at the exposed end (Brun et al., 2009;Desai and Mitchison, 1997;. ...
... For this work, we develop a system of linear models relating dynamic instability and nucleation behaviors to tubulin concentration. We then use Monte Carlo methods to simulate microtubule arrays through time within a specified cellular context Hill, 1983, 1985a;Chen and Hill, 1985b;Gliksman et al., 1993;Goodson and Gregoretti, 2010;Gregoretti et al., 2005;Martin et al., 1993;. The results from these simulations provide an important foundation for predicting how specific microtubule dynamics or nucleation parameters should generally affect the microtubule array under a variety of assumptions about the cellular context. ...
Microtubule polymers typically function through their collective organization into a patterned array. The formation of the pattern, whether it is a relatively simple astral array or a highly complex mitotic spindle, relies on controlled microtubule nucleation and the basal dynamics parameters governing polymer growth and shortening. We have investigated the interaction between the microtubule nucleation and dynamics parameters, using macroscopic Monte Carlo simulations, to determine how these parameters contribute to the underlying microtubule array morphology (i.e. polymer density and length distribution). In addition to the well-characterized steady state achieved between free tubulin subunits and microtubule polymer, we propose that microtubule nucleation and extinction constitute a second, interdependent steady state process. Our simulation studies show that the magnitude of both nucleation and extinction additively impacts the final steady state free subunit concentration. We systematically varied individual microtubule dynamics parameters to survey the effects on array morphology and find specific sensitivity to perturbations of catastrophe frequency. Altering the cellular context for the microtubule array, we find that nucleation template number plays a defining role in shaping the microtubule length distribution and polymer density.
... Nucleotide exchange at the plus-end can alleviate protofilament 'poisoning' by GDP-tubulin Recent work (Luo et al., 2023;Piedra et al., 2016) has reinforced early results (Chen and Hill, 1983;Chen and Hill, 1985;Mitchison, 1993) that pointed to the potential role of nucleotide exchange in microtubule dynamics at the plus-end. We implemented a finite rate of nucleotide exchange in the model (see Methods) to determine whether exchange might allow the simulations to better recapitulate the magnitude by which GDP-tubulin super-stoichiometrically decreased plus-end growth ( Figure 5A). ...
GTP-tubulin is preferentially incorporated at growing microtubule ends, but the biochemical mechanism by which the bound nucleotide regulates the strength of tubulin:tubulin interactions is debated. The ‘self-acting’ (cis) model posits that the nucleotide (GTP or GDP) bound to a particular tubulin dictates how strongly that tubulin interacts, whereas the ‘interface-acting’ (trans) model posits that the nucleotide at the interface of two tubulin dimers is the determinant. We identified a testable difference between these mechanisms using mixed nucleotide simulations of microtubule elongation: with a self-acting nucleotide, plus- and minus-end growth rates decreased in the same proportion to the amount of GDP-tubulin, whereas with interface-acting nucleotide, plus-end growth rates decreased disproportionately. We then experimentally measured plus- and minus-end elongation rates in mixed nucleotides and observed a disproportionate effect of GDP-tubulin on plus-end growth rates. Simulations of microtubule growth were consistent with GDP-tubulin binding at and ‘poisoning’ plus-ends but not at minus-ends. Quantitative agreement between simulations and experiments required nucleotide exchange at terminal plus-end subunits to mitigate the poisoning effect of GDP-tubulin there. Our results indicate that the interfacial nucleotide determines tubulin:tubulin interaction strength, thereby settling a longstanding debate over the effect of nucleotide state on microtubule dynamics.
... Recent work (Luo et al., 2023 ;Piedra et al., 2016 ) has reinforced early results (Chen & Hill, 1983 ;Chen & Hill, 1985 ;Mitchison, 1993 ) that pointed to the potential role of nucleotide exchange in microtubule dynamics at the plus-end. We implemented a finite rate of nucleotide exchange in the model (see Methods) to determine whether exchange might allow the simulations to better recapitulate the magnitude by which GDP-tubulin super-stoichiometrically decreased plus-end growth ( Figure 5A ). ...
GTP-tubulin is preferentially incorporated at growing microtubule ends, but the biochemical mechanism by which the bound nucleotide regulates the strength of tubulin:tubulin interactions is debated. The ‘self-acting’ (cis) model posits that the nucleotide (GTP or GDP) bound to a particular tubulin dictates how strongly that tubulin interacts, whereas the ‘interface-acting’ (trans) model posits that the nucleotide at the interface of two tubulin dimers is the determinant. We identified a testable difference between these mechanisms using mixed nucleotide simulations of microtubule elongation: with self-acting nucleotide, plus- and minus-end growth rates decreased in the same proportion to the amount of GDP-tubulin, whereas with interface-acting nucleotide, plus-end growth rates decreased disproportionately. We then experimentally measured plus- and minus-end elongation rates in mixed nucleotides and observed a disproportionate effect of GDP-tubulin on plus-end growth rates. Simulations of microtubule growth were consistent with GDP-tubulin binding at and ‘poisoning’ plus-ends but not at minus-ends. Quantitative agreement between simulations and experiments required nucleotide exchange at terminal plus-end subunits to mitigate the poisoning effect of GDP-tubulin there. Our results indicate that the interfacial nucleotide determines tubulin:tubulin interaction strength, thereby settling a longstanding debate over the effect of nucleotide state on microtubule dynamics.
... Recent work (Luo et al., 2023 ;Piedra et al., 2016 ) has reinforced early results (Chen & Hill, 1983 ;Chen & Hill, 1985 ;Mitchison, 1993 ) that pointed to the potential role of nucleotide exchange in microtubule dynamics at the plus-end. We implemented a finite rate of nucleotide exchange in the model (see Methods) to determine whether exchange might allow the simulations to better recapitulate the magnitude by which GDP-tubulin super-stoichiometrically decreased plus-end growth ( Figure 5A ). ...
GTP-tubulin is preferentially incorporated at growing microtubule ends, but the biochemical mechanism by which the bound nucleotide regulates the strength of tubulin:tubulin interactions is debated. The ‘self-acting’ (cis) model posits that the nucleotide (GTP or GDP) bound to a particular tubulin dictates how strongly that tubulin interacts, whereas the ‘interface-acting’ (trans) model posits that the nucleotide at the interface of two tubulin dimers is the determinant. We identified a testable difference between these mechanisms using mixed nucleotide simulations of microtubule elongation: with self-acting nucleotide plus- and minus-end growth rates decreased in the same proportion to the amount of GDP-tubulin, whereas with interface-acting nucleotide plus-end growth rates decreased disproportionately. We then experimentally measured plus- and minus-end elongation rates in mixed nucleotides and observed a disproportionate effect of GDP-tubulin on plus-end growth rates. Simulations of microtubule growth were consistent with GDP-tubulin binding at and ‘poisoning’ plus-ends but not at minus-ends. Quantitative agreement between simulations and experiments required nucleotide exchange at terminal plus-end subunits to mitigate the poisoning effect of GDP-tubulin there. Our results indicate that the interfacial nucleotide determines tubulin:tubulin interaction strength, thereby settling a longstanding debate over the effect of nucleotide state on microtubule dynamics.
... Nucleotide exchange at the plus-end can alleviate protofilament 'poisoning' by GDP-tubulin Recent work (Luo et al., 2023;Piedra et al., 2016) has reinforced early results (Chen and Hill, 1983;Chen and Hill, 1985;Mitchison, 1993) that pointed to the potential role of nucleotide exchange in microtubule dynamics at the plus-end. We implemented a finite rate of nucleotide exchange in the model (see Methods) to determine whether exchange might allow the simulations to better recapitulate the magnitude by which GDP-tubulin super-stoichiometrically decreased plus-end growth ( Figure 5A). ...
GTP-tubulin is preferentially incorporated at growing microtubule ends, but the biochemical mechanism by which the bound nucleotide regulates the strength of tubulin:tubulin interactions is debated. The ‘self-acting’ (cis) model posits that the nucleotide (GTP or GDP) bound to a particular tubulin dictates how strongly that tubulin interacts, whereas the ‘interface-acting’ (trans) model posits that the nucleotide at the interface of two tubulin dimers is the determinant. We identified a testable difference between these mechanisms using mixed nucleotide simulations of microtubule elongation: with self-acting nucleotide plus- and minus-end growth rates decreased in the same proportion to the amount of GDP-tubulin, whereas with interface-acting nucleotide plus-end growth rates decreased disproportionately. We then experimentally measured plus- and minus-end elongation rates in mixed nucleotides and observed a disproportionate effect of GDP-tubulin on plus-end growth rates. Simulations of microtubule growth were consistent with GDP-tubulin binding at and ‘poisoning’ plus-ends but not at minus-ends. Quantitative agreement between simulations and experiments required nucleotide exchange at terminal plus-end subunits to mitigate the poisoning effect of GDP-tubulin there. Our results indicate that the interfacial nucleotide determines tubulin:tubulin interaction strength, thereby settling a longstanding debate over the effect of nucleotide state on microtubule dynamics.
... Another possibility is to take into account the two-dimensional tubular lattice in which the subunits of the microtubule are actually arranged. This has been done in [4,5,6,9]. The two-dimensional lattice models have been studied by Monte-Carlo simulation. ...
... The model of MT dynamics has been studied using analytical approaches or lattice-based computational approaches. One of the first lattice-based models was proposed by Chen and Hill [38], in which they considered the 13 protofilaments of MT to propose the GTP-cap hypothesis. Later, Bayley et al proposed the Lateral Cap model, where the addition and loss of tubulin dimers depend on the neighboring tubulin dimers [39]. ...
Microtubule severing enzymes Katanin and Spastin cut the microtubule (MT) into smaller fragments and are being studied extensively using in-vitro experiments due to their crucial role in different cancers and
neurodevelopmental disorders. It has been reported that the severing enzymes are either involved in increasing or decreasing the tubulin mass. Currently, there are a few analytical and computational models for MT amplification and severing. However, these models do not capture the action of microtubule severing explicitly, as these are based on partial differential equations in one dimension. On the other hand, a few discrete lattice-based models were used earlier to understand the activity of severing enzymes only on stabilized MTs. Hence, in this study, discrete lattice-based Monte Carlo models that included microtubule dynamics and severing enzyme activity have been developed to understand the effect of severing enzymes on tubulin mass, MT number, and MT length. It was found that the action of severing enzyme reduces average MT length while increasing their number; however, the total tubulin mass can decrease or increase depending on the concentration of GMPCPP (Guanylyl-(α, β)-methylene-diphosphonate) - which is a slowly hydrolyzable analogue of GTP (Guanosine triphosphate). Further, relative tubulin mass also depends on the detachment ratio of GTP/GMPCPP and GDP (Guanosine diphosphate) tubulin dimers and the binding energies of
tubulin dimers covered by the severing enzyme.
... Due to the complexity of MTs, their large system size ($1,000 tubulin monomers for the purpose of modeling) and the long timescales involved (from seconds to minutes), in silico modeling of MT dynamics is challenging. Different kinetic and mechanical models of MT assembly and disassembly have been reported in the literature [25,28,[36][37][38][39]. These models vary in the types of representation of molecular processes, and the extent to which they agree with experiments [14]. ...
Microtubules (MTs), a cellular structure element, exhibit dynamic instability and can switch stochastically from growth to shortening; but the factors that trigger these processes at the molecular level are not understood. We developed a 3D Microtubule Assembly and Disassembly DYnamics (MADDY) model, based upon a bead-per-monomer representation of the αβ-tubulin dimers forming an MT lattice, stabilized by the lateral and longitudinal interactions between tubulin subunits. The model was parameterized against the experimental rates of MT growth and shortening, and pushing forces on the Dam1 protein complex due to protofilaments splaying out. Using the MADDY model, we carried out GPU-accelerated Langevin simulations to access dynamic instability behavior. By applying Machine Learning techniques, we identified the MT characteristics that distinguish simultaneously all four kinetic states: growth, catastrophe, shortening, and rescue. At the cellular 25 μM tubulin concentration, the most important quantities are the MT length L, average longitudinal curvature κlong, MT tip width w, total energy of longitudinal interactions in MT lattice Ulong, and the energies of longitudinal and lateral interactions required to complete MT to full cylinder Ulongadd and Ulatadd. At high 250 μM tubulin concentration, the most important characteristics are L, κlong, number of hydrolyzed αβ-tubulin dimers nhyd and number of lateral interactions per helical pitch nlat in MT lattice, energy of lateral interactions in MT lattice Ulat, and energy of longitudinal interactions in MT tip ulong. These results allow greater insights into what brings about kinetic state stability and the transitions between states involved in MT dynamic instability behavior.
... Here we report our examination of a Brownian dynamics model for MT assembly and disassembly that accommodates the curved shape of GTP-tubulins 21-24 and our finding of curved PFs at the tips of growing MTs 25 . Most previous models have assumed straight PFs at growing MT ends, so tubulin dimer attachment or detachment involved simultaneous formation or breakage of lateral and longitudinal bonds 34,[56][57][58][59][60] . An exception is the model of Margolin et al. 61 , which considered lateral cracks between PFs without breakage of longitudinal bonds. ...
Microtubules are dynamic tubulin polymers responsible for many cellular processes, including the capture and segregation of chromosomes during mitosis. In contrast to textbook models of tubulin self-assembly, we have recently demonstrated that microtubules elongate by addition of bent guanosine triphosphate tubulin to the tips of curving protofilaments. Here we explore this mechanism of microtubule growth using Brownian dynamics modeling and electron cryotomography. The previously described flaring shapes of growing microtubule tips are remarkably consistent under various assembly conditions, including different tubulin concentrations, the presence or absence of a polymerization catalyst or tubulin-binding drugs. Simulations indicate that development of substantial forces during microtubule growth and shortening requires a high activation energy barrier in lateral tubulin-tubulin interactions. Modeling offers a mechanism to explain kinetochore coupling to growing microtubule tips under assisting force, and it predicts a load-dependent acceleration of microtubule assembly, providing a role for the flared morphology of growing microtubule ends. Microtubules are dynamic tubulin polymers which elongate by addition of bent guanosine triphosphate tubulin to the tips of curving protofilaments. Here authors use Brownian dynamics modeling and electron cryotomography to show that the lateral activation energy barrier in tubulin-tubulin interactions is a key parameter for this process, controlling the development of high pulling forces.
... Microtubules are polymers of ab-tubulin, and dynamic instability is the nonequilibrium behavior in which the polymers stochastically switch between periods of growth and shrinkage. This complex, nonequilibrium phenomenon was first simulated numerically in the 1980s (21,22) and has remained a subject of considerable interest for computational biologists, who have developed increasingly sophisticated models (3,4,23,24). The long-term goal of these collective efforts is to develop a powerfully predictive yet minimal model that can be used to explain microtubule physiology. ...
Physical models of biological systems can become difficult to interpret when they have a large number of parameters. But the models themselves actually depend on (i.e., are sensitive to) only a subset of those parameters. This phenomenon is due to parameter space compression (PSC), in which a subset of parameters emerges as "stiff" as a function of time or space. PSC has only been used to explain analytically solvable physics models. We have generalized this result by developing a numerical approach to PSC that can be applied to any computational model. We validated our method against analytically solvable models of a random walk with drift and protein production and degradation. We then applied our method to a simple computational model of microtubule dynamic instability. We propose that numerical PSC has the potential to identify the low-dimensional structure of many computational models in biophysics. The low-dimensional structure of a model is easier to interpret and identifies the mechanisms and experiments that best characterize the system.
... Microtubules are polymers of αβ-tubulin and dynamic instability is the non-equilibrium behavior in which the polymers stochastically switch between periods of growth and shrinkage. This complex, nonequilibrium phenomenon was first simulated numerically in the 1980s [23,24] and has remained a subject of considerable interest for computational biologists, who have developed increasingly sophisticated models [3,4,9,25]. The long-term goal of these collective efforts is to develop a powerfully-predictive yet minimal model that can be used to explain microtubule physiology. ...
Physical models of biological systems can become difficult to interpret when they have a large number of parameters. But the models themselves actually depend on (i.e. are sensitive to) only a subset of those parameters. Rigorously identifying this subset of "stiff" parameters has been made possible by the development of parameter space compression (PSC). However, PSC has only been applied to analytically-solvable physical models. We have generalized this powerful method by developing a numerical approach to PSC that can be applied to any computational model. We validated our method against analytically-solvable models of random walk with drift and protein production and degradation. We then applied our method to an active area of biophysics research, namely to a simple computational model of microtubule dynamic instability. Such models have become increasingly complex, perhaps unnecessarily. By adding two new parameters that account for prominent structural features of microtubules, we identify one that can be "compressed away" (the "seam" in the microtubule) and another that is essential to model performance (the "tapering" of microtubule ends). Furthermore, we show that the microtubule model has an underlying, low-dimensional structure that explains the vast majority of our experimental data. We argue that numerical PSC can identify the low-dimensional structure of any computational model in biophysics. The low-dimensional structure of a model is easier to interpret and identifies the mechanisms and experiments that best characterize the system.
... Lattice geometry may therefore have caused an underestimation of the importance of considering lateral interactions. Chen and Hill [1985] performed Monte Carlo simulations of 5-start 13-protofilament microtubules in which GTP hydrolysis was incorporated but in which only helical growth was allowed (no changes in surface roughness). These studies form the ba- For each variable-length time step of the simulation, the rate k E for each potential association or dissociation event E was computed, and the actual time t E for the potential event to occur was determined from Table 4.3, in which the cases A, B, and C correspond to those of this figure. ...
During cell division, chromosomes must faithfully segregate to maintain genome integrity,
and this dynamic mechanical process is driven by the macromolecular machinery
of the mitotic spindle. However, little is known about spindle mechanics.
For example, spindle microtubules are organized by numerous cross-linking proteins
yet the mechanical properties of those cross-links remain unexplored. To examine
the mechanical properties of microtubule cross-links we applied optical trapping to
mitotic asters that form in mammalian mitotic extracts. These asters are foci of
microtubules, motors, and microtubule-associated proteins that reflect many of the
functional properties of spindle poles and represent centrosome-independent spindlepole
analogs. We observed bidirectional motor-driven microtubule movements, showing
that microtubule linkages within asters are remarkably compliant (mean stiffness
0.025 pN/nm) and mediated by only a handful of cross-links. Depleting the motor
Eg5 reduced this stiffness, indicating that Eg5 contributes to the mechanical properties
of microtubule asters in a manner consistent with its localization to spindle poles
in cells. We propose that compliant linkages among microtubules provide a mechanical
architecture capable of accommodating microtubule movements and distributing
force among microtubules without loss of pole integrity???a mechanical paradigm that
may be important throughout the spindle.
Microtubule assembly and disassembly are vital for many fundamental cellular
processes. Our current understanding of microtubule assembly kinetics is based on
a one-dimensional assembly model, which assumes identical energetics for subunits
exchanging at the tip. In this model, the subunit disassociation rate from a microtubule
tip is independent of free subunit concentration. Using total-internal-reflection
fluorescence (TIRF) microscopy and an optical tweezers assay to measure in vitro microtubule
assembly with nanometer resolution, we find that the subunit dissociation
rate from a microtubule tip increases at higher free subunit concentrations. This
is because, as predicted by Hill, there is a shift in microtubule tip structure from
relatively blunt at low free subunit concentrations to relatively tapered at high concentrations,
which we confirmed experimentally by TIRF microscopy. Because both
the association and the dissociation rates increase with free subunit concentrations,
we find that the kinetics of microtubule assembly are an order of magnitude faster
than currently estimated in the literature.
... In addition to these analytic models, there are computational models in which growth speed and catastrophe frequency are deduced from simulations [30,[70][71][72][73][74][75]. These models often take into account the complex 3-D arrangement of subunits in the microtubule lattice, with the dissociation and hydrolysis rates depending on the whether the neighboring subunits (of which there are up to six) have GTP or GDP bound to them. ...
Recent studies have found that microtubule-associated proteins (MAPs) can regulate the dynamical properties of microtubules in unexpected ways. For most MAPs, there is an inverse relationship between their effects on the speed of growth and the frequency of catastrophe, the conversion of a growing microtubule to a shrinking one. Such a negative correlation is predicted by the standard GTP-cap model, which posits that catastrophe is due to loss of a stabilizing cap of GTP-tubulin at the end of a growing microtubule. However, many other MAPs, notably Kinesin-4 and combinations of EB1 with XMAP215, contradict this general rule. In this review, we show that a more nuanced, but still simple, GTP-cap model, can account for the diverse regulatory activities of MAPs.
... Existing cap models can be divided into two classes : (a) uncoupled destabilization models, which specify no direct link between tubulin subunit addition and destabilization ; and (b) coupled destabilization models, which link destabilization to subunit addition. For uncoupled models, destabilization has been postulated to be stochastic, i .e., any GTP-tubulin may be destabilized, regardless of its location in the polymer lattice (9,26) ; vectorial, i .e., destabilization is promoted at a GDP-tubulin :GTP-tubulin interface within the microtubule lattice (5, 10) or a combination of stochastic and vectorial mechanisms (15). In contrast, coupled destabilization models propose that destabilization occurs whenthe subunit becomes buried in the microtubule lattice through addition of another subunit (1,27,49) . ...
Although the mechanism of microtubule dynamic instability is thought to involve the hydrolysis of tubulin-bound GTP, the mechanism of GTP hydrolysis and the basis of microtubule stability are controversial. Video microscopy of individual microtubules and dilution protocols were used to examine the size and lifetime of the stabilizing cap. Purified porcine brain tubulin (7-23 microM) was assembled at 37 degrees C onto both ends of isolated sea urchin axoneme fragments in a miniature flow cell to give a 10-fold variation in elongation rate. The tubulin concentration in the region of microtubule growth could be diluted rapidly (by 84% within 3 s of the onset of dilution). Upon perfusion with buffer containing no tubulin, microtubules experienced a catastrophe (conversion from elongation to rapid shortening) within 4-6 s on average after dilution to 16% of the initial concentration, independent of the predilution rate of elongation and length. Based on extrapolation of catastrophe frequency to zero tubulin concentration, the estimated lifetime of the stable cap after infinite dilution was less than 3-4 s for plus and minus ends, much shorter than the approximately 200 s observed at steady state (Walker, R. A., E. T. O'Brien, N. K. Pryer, M. Soboeiro, W. A. Voter, H. P. Erickson, and E. D. Salmon. 1988. J. Cell Biol. 107:1437-1448.). We conclude that during elongation, both plus and minus ends are stabilized by a short region (approximately 200 dimers or less) and that the size of the stable cap is independent of 10-fold variation in elongation rate. These results eliminate models of dynamic instability which predict extensive "build-up" stabilizing caps and support models which constrain the cap to the elongating tip. We propose that the cell may take advantage of such an assembly mechanism by using "catastrophe factors" that can promote frequent catastrophe even at high elongation rates by transiently binding to microtubule ends and briefly inhibiting GTP-tubulin association.
... Various experimental and theoretical studies were aimed at extracting characteristic features of life histories of MTs in order to build mathematical models (Mitchison and Kirschner 1984a;Chen and T.L. 1985;Horio and Hotani 1986;Walker et al. 1988;Cassimeris et al. 1988;Bayley et al. 1991;Martin et al. 1993;Vandecandelaere et al. 1995). The most common statistical tools included the construction of histograms and correlation functions (Walker et al. 1988;Gliksman et al. 1993). ...
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.
GTP-tubulin is preferentially incorporated at growing microtubule ends, but the biochemical mechanism by which the bound nucleotide regulates the strength of tubulin:tubulin interactions is debated. The "self-acting" (cis) model posits that the nucleotide (GTP or GDP) bound to a particular tubulin dictates how strongly that tubulin interacts, whereas the "interface-acting" (trans) model posits that the nucleotide at the interface of two tubulin dimers is the determinant. We identified a testable difference between these mechanisms using mixed nucleotide simulations of microtubule elongation: with self-acting nucleotide plus- and minus-end growth rates decreased in the same proportion to the amount of GDP-tubulin, whereas with interface-acting nucleotide plus-end growth rates decreased disproportionately. We then experimentally measured plus- and minus-end elongation rates in mixed nucleotides and observed a disproportionate effect of GDP-tubulin on plus-end growth rates. Simulations of microtubule growth were consistent with GDP-tubulin binding at and "poisoning" plus-ends but not at minus-ends. Quantitative agreement between simulations and experiments required nucleotide exchange at terminal plus-end subunits to mitigate the poisoning effect of GDP-tubulin there. Our results indicate that the interfacial nucleotide determines tubulin:tubulin interaction strength, thereby settling a longstanding debate over the effect of nucleotide state on microtubule dynamics.
Microtubules are linear intracellular polymers that self-assemble from subunits of αβ-tubulin heterodimers, and serve as essential mediators of cellular processes including cell division, migration, as well as intracellular transport. Each of these functions is dependent on the microtubule structure and self-assembly dynamics. Here we discuss the dynamics of microtubule structure and subunit addition and loss in the process of microtubule self-assembly, highlighting how these nanometer/molecular-scale dynamics manifest themselves at the micrometer/cellular-scale.
Microtubules (MTs) are protein polymers found in all eukaryotic cells. They are crucial for normal cell development, providing structural support for the cell and aiding in the transportation of proteins and organelles. In order to perform these functions, MTs go through periods of relatively slow polymerization (growth) and very fast depolymerization (shortening), where the switch from growth to shortening is called a catastrophe and the switch from shortening to growth is called a rescue. Although MT dynamic instability has traditionally been described solely in terms of growth and shortening, MTs have been shown to pause for extended periods of time, however the reason for pausing is not well understood. Here, we present a new mathematical model to describe MT dynamics in terms of growth, shortening, and pausing. Typically, MT dynamics are defined by four key parameters which include the MT growth rate, shortening rate, frequency of catastrophe, and the frequency of rescue. We derive a mathematical expression for the catastrophe frequency in the presence of pausing, as well as expressions to describe the total time that MTs spend in a state of growth and pause. In addition to exploring MT dynamics in a control-like setting, we explore the implicit effect of stabilizing MT associated proteins (MAPs) and stabilizing and destabilizing chemotherapeutic drugs that target MTs on MT dynamics through variations in model parameters.
The kinetics of microtubule reassembly was studied in vitro by quasi-elastic light scattering (QELS). When microtubules assembled in the absence of microtubule-associated proteins (MAPs) were sheared, they rapidly depolymerized, recovered, and reassembled. The mean length of the recovered microtubules was the same as that observed just before shearing, implying that on average one fragment per original microtubule survived the fragmentation and recovery. When microtubules that contained 25 percent brain MAP were sheared, the fragments did not depolymerize extensively and the average length of the fragments decreased by a factor of 3 relative to the unsheared sample. The results support the dynamic instability model, which predicts that cellular microtubules are latently unstable structures protected on their ends by stabilizing caps.
Microtubules are essential parts of the cytoskeleton that are built by polymerization of tubulin heterodimers into a hollow tube. Regardless that their structures and functions have been comprehensively investigated in a modern soft matter, it is unclear how properties of tubulin heterodimer influence and promote the self-assembly. A detailed knowledge of such structural mechanisms would be helpful in drug design against neurodegenerative diseases, cancer, diabetes etc. In this work atomistic molecular dynamics simulations were used to investigate the fundamental dynamics of tubulin heterodimers in a sheet and a short microtubule utilizing well-equilibrated structures. The breathing motions of the tubulin heterodimers during assembly show that the movement at the lateral interface between heterodimers (wobbling) dominates in the lattice. The simulations of the protofilament curvature agrees well with recently published experimental data, showing curved protofilaments at polymerization of the microtubule plus end. The tubulin heterodimers exposed at the microtubule minus end were less curved and displayed altered interactions at the site of sheet closure around the outmost heterodimers, which may slow heterodimer binding and polymerization, providing a potential explanation for the limited dynamics observed at the minus end.
Microtubules are dynamic cytoskeletal filaments composed of αβ-tubulin heterodimers. Historically, the dynamics of single tubulin interactions at the growing microtubule tip have been inferred from steady-state growth kinetics. However, recent advances in the production of recombinant tubulin and in high-resolution optical and cryo-electron microscopies have opened new windows into understanding the impacts of specific intermolecular interactions during growth. The microtubule lattice is held together by lateral and longitudinal tubulin–tubulin interactions, and these interactions are in turn regulated by the GTP hydrolysis state of the tubulin heterodimer. Furthermore, tubulin can exist in either an extended or a compacted state in the lattice. Growing evidence has led to the suggestion that binding of microtubule-associated proteins (MAPs) or motors can induce changes in tubulin conformation and that this information can be communicated through the microtubule lattice. Progress in understanding how dynamic tubulin–tubulin interactions control dynamic instability has benefitted from visualizing structures of growing microtubule plus ends and through stochastic biochemical models constrained by experimental data. Here, we review recent insights into the molecular basis of microtubule growth and discuss how MAPs and regulatory proteins alter tubulin–tubulin interactions to exert their effects on microtubule growth and stability.
The concept of critical concentration (CC) is central to understanding behaviors of microtubules and other cytoskeletal polymers. Traditionally, these polymers are understood to have one CC, measured multiple ways and assumed to be the subunit concentration necessary for polymer assembly. However, this framework does not incorporate dynamic instability (DI), and there is work indicating that microtubules have two CCs. We use our previously established simulations to confirm that microtubules have (at least) two experimentally relevant CCs and to clarify the behaviors of individuals and populations relative to the CCs. At free subunit concentrations above the lower CC (CC Elongation ), growth phases of individual filaments can occur transiently; above the higher CC (CC NetAssembly ), the population's polymer mass will increase persistently. Our results demonstrate that most experimental CC measurements correspond to CC NetAssembly , meaning “typical” DI occurs below the concentration traditionally considered necessary for polymer assembly. We report that [free tubulin] at steady state does not equal CC NetAssembly , but instead approaches CC NetAssembly asymptotically as [total tubulin] increases and depends on the number of stable microtubule seeds. We show that the degree of separation between CC Elongation and CC NetAssembly depends on the rate of nucleotide hydrolysis. This clarified framework helps explain and unify many experimental observations.
Microtubules are cylindrical polymers of αβ-tubulin that play critical roles in fundamental processes like chromosome segregation and vesicular transport. Microtubules display dynamic instability, switching stochastically between growing and rapid shrinking as a consequence of GTPase activity in the lattice. The molecular mechanisms behind microtubule catastrophe, the switch from growing to rapid shrinking, remain poorly defined. Indeed, two-state stochastic models that seek to describe microtubule dynamics purely in terms of the biochemical properties of GTP- and GDP-bound αβ-tubulin incorrectly predict the concentration-dependence of microtubule catastrophe. Recent studies provided evidence for three distinct conformations of αβ-tubulin in the lattice that likely correspond to GTP, GDP.Pi, and GDP. The incommensurate lattices observed for these different conformations raises the possibility that in a mixed nucleotide state lattice, neighboring tubulin dimers might modulate each other's conformations and hence their biochemistry. We explored whether incorporating a GDP.Pi state or the likely effects of conformational accommodation can improve predictions of catastrophe. Adding a GDP.Pi intermediate did not improve the model. In contrast, adding neighbor-dependent modulation of tubulin biochemistry improved predictions of catastrophe. Because this conformational accommodation should propagate beyond nearest-neighbor contacts, our modeling suggests that long-range, through-lattice effects are important determinants of microtubule catastrophe.
Significance
Microtubule polymerization dynamics are fundamental to cell migration and cell division, where they are targets for chemotherapy drugs. Despite significant progress, the precise structural and biochemical events occurring at growing microtubule tips are not well defined, and better understanding is necessary for discriminating mechanisms of microtubule dynamics regulation in cells. Here, we visualize individual tubulin subunits reversibly and irreversibly interacting with dynamic microtubule tips, and thereby directly measure tubulin plus-tip kinetics. By analyzing plus-tip residence times of wild-type and mutant tubulin, we characterize the relative contributions of longitudinal (along protofilaments) and lateral (between protofilaments) bond energies to microtubule growth. This work provides insight into microtubule tip structure and potential modes of microtubule dynamics regulation.
In this work, a comprehensive review has been done on previous research on the mechanical properties of microtubules. For this aim, the review was conducted in three general categories, experimental studies, theoretical studies, and dynamic instability studies of microtubules. Also, we suggested a nanobiomechanical model for microtubules that can be used for all cell members in future works.
At the molecular level, the dynamic instability (random growth and shrinkage) of the microtubule (MT) is driven by the nucleotide state (GTP vs. GDP) in the β subunit of the tubulin dimers at the MT cap. Here, we use large-scale molecular dynamics (MD) simulations and normal mode analysis (NMA) to characterize the effect of a single GTP cap layer on tubulin octamers composed by two neighboring protofilaments (PFs). We utilize recently reported high-resolution structures of dynamic MTs to simulate a GDP octamer both with and without a single GTP cap layer. We perform multiple replicas of long-time atomistic MD simulations (3 replicas, ~0.33μs for each replica, ~0.9 μs for each octamer system, and ~1.8 μs total) of both octamers. We observe that a single GTP cap layer induces structural differences in neighboring PFs, finding that one PF possesses a gradual curvature, compared to the second PF which possesses a kinked conformation. This results in either curling or splaying between these PFs. We suggest that this is due to asymmetric strengths of longitudinal contacts between the two PFs. Furthermore, using NMA, we calculate mechanical properties of these octamer systems and find that octamer system with a single GTP cap layer possesses a lower flexural rigidity.
This chapter1 and the next deal explicitly with simple multi-stranded polymers. Hitherto some multi-stranded polymers have been included only as “effectively” single stranded. We use simple, illustrative models rather than realistic ones (e.g., a microtubule); even some of these simple models require Monte Carlo calculations.
In the remainder of the book, except for Chapters 5 and 6 and Section 24, we shall treat linear polymers formally as if they consist of a single strand only, as in Fig. 2-1(a). To be more precise: we shall assume that there is only a single subunit attachment or departure site at a polymer end or if there are several such sites, that they are all equivalent. In effect, then, there is a single overall on rate constant for a polymer end and a single off rate constant, and these rate constants are constant (see below). This model would be exact for the structure in Fig. 2-1(a) and it would also be exact in Figs. 2-l(b) and 2-1(c) if the intersubunit interactions in the polymer are so strong that there is always only one significant addition site (see the arrows in the figure) and only one significant departure site despite the fact that there is more than one strand. Figure 2-1(b) illustrates a 1-start, 2-strand helical structure (as in actin) and Fig. 2-1 (c) shows a 1-start, 3-strand helical structure (flattened). In the limiting case just mentioned, both structures would behave kinetically like a single helix (i.e., in effect, a single strand). This would be true of any 1-start tubular helical polymer in the strong-interaction limit.
The principal topic in Chapter 7 was a treatment of aggregation of NTP subunits accompanied by fast NTPase activity at the polymer tips. Actin and microtubules were believed, at one time, to behave in this relatively simple way. As of this writing, it seems clear that, in both cases, NTP subunits actually penetrate (or survive) into the polymer ends by virtue of addition of subsequent subunits: conversion of an added NTP subunit into an NDP subunit is in fact not fast compared to on-off transitions.1 Thus there may be a collection or “cap” of surviving NTP subunits at each polymer end. Though this much seems clear, further details are in the process of being worked out and are by no means generally agreed upon. For this reason, in this chapter we bypass biochemical details (except for illustrative examples in Section 24) and devote the bulk of the chapter to a consideration of two-phase behavior at the polymer ends. The two phases referred to are a polymer end either with or without an NTP cap. This subject can be dealt with without a commitment to a particular detailed biochemical model; we merely assume that some unspecified biochemical mechanism exists that generates two-phase activity (see Section 24 for examples). Actually, so far there is evidence for two-phase activity in microtubules2–4 but not in actin.
Microtubules are dynamic polymers of αβ-tubulin that have essential roles in chromosome segregation and organizing the cytoplasm. Catastrophe - the switch from growing to shrinking - occurs when a microtubule loses its stabilizing GTP cap. Recent evidence indicates that the nucleotide on the microtubule end controls how tightly an incoming subunit will be bound (trans-acting GTP), but most current models do not incorporate this information. We implemented trans-acting GTP into a computational model for microtubule dynamics. In simulations, growing microtubules often exposed terminal GDP-bound subunits without undergoing catastrophe. Transient GDP exposure on the growing plus end slowed elongation by reducing the number of favorable binding sites on the microtubule end. Slower elongation led to erosion of the GTP cap and an increase in the frequency of catastrophe. Allowing GDP to GTP exchange on terminal subunits in simulations mitigated these effects. Using mutant αβ-tubulin or modified GTP, we showed experimentally that a more readily exchangeable nucleotide led to less frequent catastrophe. Current models for microtubule dynamics do not account for GDP to GTP exchange on the growing microtubule end, so our findings provide a new way of thinking about the molecular events that initiate catastrophe.
The tubulin monomers of brain microtubules reassembled in vitro are arranged on a 3-start helix, irrespective of whether the number of protofilaments is 13 or 14. The dimer packing is that of the B-lattice described for flagellar microtubules. This implies that the tubulin core of microtubules contains at least one helical discontinuity. Neither 5-start nor 8-start helices have a physical significance and thus cannot be implicated in models of microtubule elongation, but the structure is compatible with elongation of protofilaments by dimers or protofilamentous oligomers. The inner and outer surfaces of the microtubule wall can be visualized by propane jet freezing, freeze fracturing, and metal replication, at a resolution of at least 4 nm. The 3-start helix is left-handed, in contrast to a previous study based on negative staining and shadowing. The reasons for this discrepancy are discussed.
The structure and free energy of multistranded linear polymer ends evolves as individual subunits are added and lost. Thus, the energetic state of the polymer end is not constant, as assembly theory has assumed. Here we utilize a Brownian dynamics approach to simulate the addition and loss of individual subunits at the polymer tip. Using the microtubule as a primary example, we examined how the structure of the polymer tip dictates the rate at which units are added to and lost from individual protofilaments. We find that freely diffusing subunits arrive less frequently to lagging protofilaments but bind more efficiently, such that there is no kinetic difference between leading and lagging protofilaments within a tapered tip. However, local structure at the nanoscale has up to an order-of-magnitude effect on the rate of addition. Thus, the kinetic on-rate constant, integrated across the microtubule tip (kon,MT), is an ensemble average of the varying individual protofilament on-rate constants (kon,PF). Our findings have implications for both catastrophe and rescue of the dynamic microtubule end, and provide a subnanoscale framework for understanding the mechanism of action of microtubule-associated proteins and microtubule-directed drugs. Although we utilize the specific example of the microtubule here, the findings are applicable to multistranded polymers generally.
The new field of protein-based nano-technology takes advantage of the complex interactions between proteins to form unique structures with properties that cannot be achieved with traditional components. Microtubules (MTs), self assembled proteinaceous hollow filaments, offer promise in the development of MT-based nano-systems. The compelling need for the controlled assembly of 3D MT arrays is the fundamental motivation for the first part of this research. We report on the morphology of MTs grown in a crowded environment in the form of high viscosity fluids containing agarose and a novel process that enables the assembly of MTs supported by gel-based 3D scaffolds. Our research on MTs and their interaction with other molecules lead us to discover extraordinary spherulitic structures that changed the course of the project. The novel subject situate us into a complicated dilemma that question the nature of MT asters reported in experiments carried out in cells. The second part of this research is focused in the crystallization of Taxol, a MT stabilizing molecule used as anti-cancer drug. It was confirmed via fluorescent and differential interference contrast microscopy that Taxol crystals can be decorated with fluorescent proteins and fluorochromes without perturbing their morphology. We used theoretical calculations to further investigate Taxol-fluorescent agent interactions. Furthermore, the crystallization of Taxol was studied in pure water, aqueous solutions containing tubulin proteins and tubulin-containing agarose gels. We demonstrated that tubulin is able to heterogeneously nucleate Taxol spherulites. To explain the formation of tubulin-Taxol nuclei a new, secondary Taxol-binding site within the tubulin heterodimer is suggested. Results presented in this work are important for in vivo and in vitro microtubule studies due to the possibility of mistaking these Taxol spherulites for microtubule asters. Thus, we are confirming the need for careful interpretation of fluorescence microscopy observations of MT structures when large concentrations of Taxol are used as stabilizing agent in cells.
A cellular automaton-based model is presented to investigate the influence of an electromagnetic field on microtubule formation dynamics.The cellular space is assumed to represent a portion of the cytoplasm through which the electromagnetic field propagates. Tubulin dimers are seen as dipoles having two states of polarization, and the field is seen as a stream of photons. Both kinds of particles move all over the cellular space. Their interactions are represented as local elementary processes of photon absorption and emission. The effects of the ponderomotive force exerted by the field on the dimers and diffusion are taken into account.Simulation experiments show that a monochromatic electromagnetic beam with a bell-shaped transverse profile can give rise to filamentary regions where the dimer concentration is increased, thus creating a precondition for polymerization.
Althoughthemechanism ofmicrotubuledy- namic instability isthoughttoinvolvethehydrolysis oftubulin-bound GTP, themechanismofGTP hy- drolysis and thebasisofmicrotubulestability are controversial .Videomicroscopyofindividual microtu- bulesand dilution protocolswereusedtoexaminethe sizeand lifetime ofthestabilizing cap.Purifiedpor- cinebraintubulin(7-23p,M)was assembledat37"C ontobothendsofisolated seaurchinaxoneme frag- mentsina miniatureflowcelltogivea 10-foldvaria- tioninelongation rate.The tubulinconcentration in theregionofmicrotubulegrowthcouldbedilutedrap- idly(by84% within3 softheonsetofdilution) . Upon perfusionwithbuffercontaining no tubulin, microtubules experienceda catastrophe (conversion fromelongation torapidshortening) within4-6 s on averageafterdilution to16% oftheinitial concentra- tion,independentofthepredilution rateofelongation and length.Basedon extrapolation ofcatastrophe fre- quencytozerotubulinconcentration, theestimated
A microtubule of a given length undergoes all possible scenarios of transitions between growing and shrinking phases, so-called microtubule dynamic instability. In this paper we utilize a minimal two-state model proposed by Hill [Proc. Natl. Acad. Sci. USA 81, 6728 (1984)] that is equivalent to a two-state random walk. Using a technique for classifying discrete random walk configurations by introducing a counting variable in evolution equations, we have derived expressions for probability densities (which contain information about all transition histories) of phase transitions before the complete disappearance of a microtubule. As a result, the mean lifetime of a microtubule turns out to be equal to the total lifetime of growing and shrinking phases times the average number of transitions. An attractive feature of this simple model is that elementary formulas relating statistical averages to rate parameters are obtained.
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.
Microtubules are linear polymers of the cytoskeleton that serve to organize the cytoplasm of eukaryotic cells. Understanding how microtubule polymers self-assemble is important in biotechnology, including the development of novel cancer therapies and proper guidance of regenerating neurons. The assembly of microtubules occurs by a unique process whereby an individual microtubule undergoes abrupt and apparently stochastic switching between alternating steady states of growth and shrinkage, a phenomenon known as microtubule dynamic instability. To characterize these oscillations spectral (frequency-domain) analysis, commonly used in engineering for system identification, was applied. Power spectra of the individual microtubule-length life histories revealed oscillations within growth phases, directly reflecting acceleration and deceleration in the growth process. These fluctuations were not accounted for by the standard two-state model commonly used in the analysis of microtubule assembly, despite the inclusion of simulated measurement error in the model. Thus, the spectral analysis of microtubule assembly permitted characterization of assembly process dynamics independent of particular assembly models, and as such represents a powerful analytical framework within which to study microtubule dynamic instability and assess its function in vivo.
Morphology of microtubules (MT) grown in crowded environment in the form of high viscosity fluids containing agarose were presented. Agarose influences the diffusion coefficient of molecules in high viscosity solutions and gels. The morphology of MTs were evolved from numerous short, straight MTs to longer MTs to fewer long bundles of MTs that were curved or contorted. Field emission scanning electron microscope (FESEM) showed that the porosity of gel was characterized by large pores separated by a denser network of agarose filaments. The large pores exhibited diameters in the order of hundreds of nanometers, whereas the denser network had small pores with diameters that were not greater than a few tens of nanometers. Experimental investigation demonstrated that the morphology of MTs grown in gels was directly dependent on the tubulin diffusion coefficient.
The recent dramatic progress in the application of diffraction and spectroscopic techniques of structural biology has provided
high resolution structural information on many individual proteins and their complexes. In cases where such proteins form
biological polymers, this information provides a firm basis for consideration of structural factors affecting supramolecular
assembly. However information from additional sources and techniques is frequently necessary in defining the mechanisms of
the assembly process. In this article, structural and kinetic evidence on the assembly of microtubules and the formation of
amyloid fibrils is presented, and compared as examples of two biophysical processes with contrasting biological roles with
respect to cellular health and survival.
Keywords: amyloids, dynamic instability, oligomers, prion protein, protofilaments, tubulin
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.
Understanding the complex self-assembly of biomacromolecules is a major
outstanding question. Microtubules are one example of a biopolymer that
possesses characteristics quite distinct from standard synthetic polymers that
are derived from its hierarchical structure. In order to understand how to
design and build artificial polymers that possess features similar to those of
microtubules, we have initially studied the self-assembly of model monomers
into a tubule geometry. Our model monomer has a wedge shape with lateral and
vertical binding sites that are designed to form tubules. We used molecular
dynamics simulations to study the assembly process for a range of binding site
interaction strengths. In addition to determining the optimal regime for
obtaining tubules, we have calculated a diagram of the structures that form
over a wide range of interaction strengths. Unexpectedly, we find that the
helical tubules form, even though the monomer geometry is designed for
nonhelical tubules. We present the detailed dynamics of the tubule
self-assembly process and show that the interaction strengths must be in a
limited range to allow rearrangement within clusters. We extended previous
theoretical methods to treat our system and to calculate the boundaries between
different structures in the diagram.
Microtubule (MT) dynamic instability is fundamental to many cell functions, but its mechanism remains poorly understood, in part because it is difficult to gain information about the dimer-scale events at the MT tip. To address this issue, we used a dimer-scale computational model of MT assembly that is consistent with tubulin structure and biochemistry, displays dynamic instability, and covers experimentally relevant spans of time. It allows us to correlate macroscopic behaviors (dynamic instability parameters) with microscopic structures (tip conformations) and examine protofilament structure as the tip spontaneously progresses through both catastrophe and rescue. The model's behavior suggests that several commonly held assumptions about MT dynamics should be reconsidered. Moreover, it predicts that short, interprotofilament "cracks" (laterally unbonded regions between protofilaments) exist even at the tips of growing MTs and that rapid fluctuations in the depths of these cracks influence both catastrophe and rescue. We conclude that experimentally observed microtubule behavior can best be explained by a "stochastic cap" model in which tubulin subunits hydrolyze GTP according to a first-order reaction after they are incorporated into the lattice; catastrophe and rescue result from stochastic fluctuations in the size, shape, and extent of lateral bonding of the cap.
A link between dimer-scale processes and microtubule (MT) dynamics at macroscale is studied by comparing simulations obtained using computational dimer-scale model with its mean-field approximation. The novelty of the mean-field model (MFM) is in its explicit representation of inter-protofilament cracks, as well as in the direct incorporation of the dimer-level kinetics. Due to inclusion of both longitudinal and lateral dimer interactions, the MFM is two dimensional, in contrast to previous theoretical models of MTs. It is the first analytical model that predicts and quantifies crucial features of MT dynamics such as (i) existence of a minimal soluble tubulin concentration needed for the polymerization (with concentration represented as a function of model parameters), (ii) existence of steady-state growth and shortening phases (given with their respective velocities), and (iii) existence of an unstable pause state near zero velocity. In addition, the size of the GTP cap of a growing MT is estimated. Theoretical predictions are shown to be in good agreement with the numerical simulations.
This chapter discusses the bioenergetics and kinetics of microtubule and actin filament assembly–disassembly. The chapter focuses on an important property of actin filaments and microtubules: their utilization of the free energy of adenosine triphosphate (ATP) or guanosine triphosphate (GTP) hydrolysis. To interpret the function of nucleoside triphosphate hydrolysis, a number of other properties are considered that include the structural and kinetic polarity of the filaments; their interaction at the ends with specific components; and the action of forces on the filaments. The important boundary conditions, such as forces acting in various ways on the filaments and the effect of materials that interact at the ends of filaments, are discussed in the chapter. The chapter also focuses on the biological implications of these properties in terms of capacity to do work, regulate length, and regulate spatial distribution.
We report here that microtubules in vitro coexist in growing and shrinking populations which interconvert rather infrequently. This dynamic instability is a general property of microtubules and may be fundamental in explaining cellular microtubule organization.
Microtubules are involved in the morphogenesis of most cells and are the structural basis of the mitotic spindle. We report here that purified centrosomes nucleate the assembly of microtubules with unusual dynamic properties. This may have important implications for the mechanism by which microtubule arrays are organized and stabilized in cells.
A model is presented for the interference of GTP hydrolysis in the mechanism of microtubule assembly. This model is suggested by previous results showing that both GTP and GDP are present at microtubule ends because of GTP hydrolysis and that tubulin does not bind to a GDP-bound end. The analytical theory developed here is aimed at calculation of the steady-state subunit flux at one end of the polymer. The GTP/GDP features just mentioned result in a nonlinear plot of the flux versus tubulin concentration. Microtubules are predicted to exhibit a different kinetic behavior below and above the critical concentration, which can be considered as a transition between two regimes.
This paper reports an experimental study of the interference of GTP hydrolysis in the mechanism of microtubule assembly, following the model and theory previously published [Hill, T. L. & Carlier, M.-F. (1983) Proc. Natl. Acad. Sci. USA 80, 7234-7238]. Results from dilution experiments show that microtubules depolymerize faster below the critical concentration than expected with a reversible polymerization model. The experimental plot of flux versus tubulin concentration exhibits a slope discontinuity at the critical concentration, in agreement with the theory. Theoretical points calculated by the Monte Carlo method can be fitted qualitatively to the data. A consequence of this peculiar dynamic behavior of microtubules is that the ratio of tubulin dissociation and association rate constants measured, respectively, below and above the critical concentration does not yield the true value of the critical concentration. It is emphasized that the presence of GTP at microtubule ends is necessary to maintain the stability of the polymer.
GTP-tubulin forms a cap on microtubule ends during aggregation. The bulk of the microtubule is GDP-tubulin. This complicates the usual simple kinetic theory of subunit exchange at microtubule ends to such an extent that Monte Carlo calculations are needed to handle the complications, except in special cases. The Monte Carlo method is introduced here, for this problem, and illustrated with steady-state and transient examples. Monte Carlo transients are needed to simulate dilution experiments. Preliminary results (with M. F. Carlier) have been obtained applying these theoretical procedures to experimental data.
Examination of Monte Carlo kinetic simulations, based on a realistic set of microscopic rate constants that apply to the end of a microtubule with a GTP cap, suggests that the end of a microtubule alternates between two quasimacroscopic phases. In one phase, the microtubule end has a GTP cap that fluctuates in size; in the other phase, the GTP cap has been lost. These repeated phase changes take place at any given tubulin concentration in a wide range of concentrations. While in the first phase, the microtubule grows slowly; while in the second phase, it shortens rapidly and may disappear completely. These results are closely related to the recent experimental work of Mitchison and Kirschner [Mitchison, T. & Kirschner, M.W. (1984) Nature (London), in press].
An introductory analysis is provided for the two-phase macroscopic kinetic model of the end of a microtubule. Some general relations are derived for one end of a very long microtubule in solution but the main results refer to the steady-state properties of microtubules grown on nucleated sites, as in the experiments of Mitchison and Kirschner [Mitchison, T. & Kirschner, M. W. (1984) Nature (London), in press]. The two-phase model makes it possible to understand qualitatively how long microtubules can grow well below the critical concentration and also how grown microtubules can rapidly disappear from a nucleated site by shortening following a phase change.
A quantitative analysis of the interplay between guanosine 5'-triphosphate (GTP) and guanosine 5'-diphosphate (GDP) in microtubule assembly and accompanying GTP hydrolysis has been performed when tubulin was polymerized in the presence of microtubule-associated proteins (MAPs) which display an interfering GTPase activity. The use of adenylyl beta-imidodiphosphate, which specifically inhibits the MAPs GTPase activity, and of vinblastine (or podophyllotoxin), which specifically inhibits GTP hydrolysis due to tubulin, made possible a study of the extensive GTP hydrolysis associated to microtubule assembly. The results indicate that GDP binds to microtubule ends with an affinity comparable to GTP, thus strongly inhibiting both the elongation process and the steady-state GTP hydrolysis at microtubule ends. GDP shifts the equilibrium between tubulin and microtubules toward disassembly. The MAPs which are released from the microtubules during the GDP-driven depolymerization cluster on the remaining microtubules. The resulting increased stability of microtubules is quantitatively consistent with the decrease in the critical concentration of the polymerizing species GTP-tubulin.
- T Mitchison
- M W Kirschner
Mitchison, T. & Kirschner, M. W. (1984) Nature (London)
312, 232-237.
- M F Carlier
- D Pantaloni
Carlier, M. F. & Pantaloni, D. (1982) Biochemistry 21, 1215-
1224.