Microtubule dynamic instability plays a fundamental role in cell biology, enabling microtubules to find and interact with randomly distributed cargo and spatially localized signals. In vitro, microtubules transition between growth and shrinkage symmetrically, consistent with the theoretical understanding of the mechanism of dynamic instability. In vivo, however, microtubules commonly exhibit asymmetric dynamic instability, growing persistently in the cell interior and experiencing catastrophe near the cell edge. What is the origin of this behavior difference? One answer is that the cell edge causes the asymmetry by inducing catastrophe in persistently growing microtubules. However, the origin of the persistent growth itself is unclear. Using a simplified coarse-grained stochastic simulation of a system of dynamic microtubules, we provide evidence that persistent growth is a predictable property of a system of nucleated, dynamic, microtubules containing sufficient tubulin in a confined space – MAP activity is not required. Persistent growth occurs because cell-edge-induced catastrophe increases the concentration of free tubulin at steady-state. Our simulations indicate that other aspects of MT dynamics thought to require temporal or spatial changes in MAP activity are also predictable, perhaps unavoidable, outcomes of the `systems nature' of the cellular microtubule cytoskeleton. These include the mitotic increase in microtubule dynamics and the observation that defects in nucleation cause changes in the behavior of microtubule plus ends. These predictions are directly relevant to understanding of the microtubule cytoskeleton, but they are also attractive from an evolutionary standpoint because they provide evidence that apparently complex cellular behaviors can originate from simple interactions without a requirement for intricate regulatory machinery.
Microtubules (MTs) are components of the cytoskeleton, the network of proteinaceous fibers that endows the cell with structural integrity, motile properties, and internal organization. MTs play a particularly important role in cell organization: they pull the chromosomes apart at mitosis, act as a `railroad system' for intracellular transport, and define the localization and structure of internal membrane systems (Kline-Smith and Walczak, 2004; Musch, 2004; Rogers and Gelfand, 2000). Two characteristics of MTs are particularly significant for these functions. First, MT nucleation is regulated, and the limitation of nucleation to an organelle near the nucleus (the centrosome) endows most cell types with a radial organization. Second, and perhaps more importantly, the MT cytoskeleton is dynamic: individual MTs in the same cell (or same test tube) constantly change in length, either growing or shrinking with random transitions between these phases (Desai and Mitchison, 1997).
This counterintuitive behavior is termed dynamic instability and is fundamental to MT function. First, dynamic instability is a mechanism for exploring cellular space, bringing MT railroads into contact with poorly diffusible cargo, such as chromosomes for subsequent transport (Hill, 1985; Holy and Leibler, 1994; Mitchison and Kirschner, 1984; Wollman et al., 2005). Second, this turnover ensures rapid response of the cytoskeleton to internal and external signals. Selective stabilization of dynamic MTs probably plays a key role in morphogenesis and appears to play a central role in the self-organizing properties of the mitotic spindle (Heald et al., 1996; Kirschner and Mitchison, 1986; Wollman et al., 2005).
Understanding the role of MTs in cell organization requires a detailed understanding of dynamic instability and its regulation. MTs are noncovalent polymers of the protein tubulin. Dynamic instability originates in conformational changes that occur in tubulin subunits after polymerization. Briefly, tubulin subunits (which are obligate dimers of the polypeptides α- and β-tubulin) bind GTP. Upon polymerization, this GTP is hydrolyzed to GDP, but only after a short delay. This delay is thought to result in a `GTP cap', which predisposes the MT to continued growth. The idea is that if this cap is lost (via hydrolysis or other mechanism), the exposed GDP-tubulin subunits rapidly depolymerize in an event termed catastrophe (reviewed in Desai and Mitchison, 1997). This elegant explanation has recently been modified to include the possibility that the conformational state of the tubulin subunits in the cap may be more important than the state of the bound nucleotide (Arnal et al., 2000; Chretien et al., 1995; Wang and Nogales, 2005). Dynamic instability occurs in solutions of pure tubulin and GTP, but it can be modulated by a variety of MT-binding proteins (also called microtubule-associated proteins or MAPs), including MAP4, stathmin, and CLIP-170 (Andersen, 2000; Arnal et al., 2004; Desai and Mitchison, 1997).
Although this molecular-level explanation has been extremely useful, many aspects of MT behavior remain poorly understood. In particular, what determines dynamic-instability parameters? One expects them to be a function of fundamental chemical parameters (association constants, dissociation constants and hydrolysis rates) and environmental parameters (the concentration of tubulin, the number of nucleation sites and the presence of spatial constraints), but no complete mathematical description of the relationship between these quantities has been derived. A major reason for this incomplete understanding is that dynamic instability is an emergent phenomenon – a behavior that arises from the independent interaction of many individual components, resulting in system-level properties that are not obviously predictable from the characteristics of the components.
One approach that has proven useful for studying such complex systems is computational modeling, in which the behavior of a system is simulated by allowing components of the system to interact according to defined rules. Insight into the system is obtained by seeing how the system changes when the rules are altered, and comparing these observations to expectation or experiment. Significant efforts have been made to predict and explain MT behavior using both deterministic models (systems of interacting equations) and stochastic simulations (Markov chain/Monte Carlo approaches) (e.g. Bayley et al., 1989; Bolterauer et al., 1999; Dogterom and Leibler, 1993; Flyvbjerg et al., 1996,Flyvbjerg et al., 1996; Freed, 2002; Gliksman et al., 1993; Govindan and Spillman, Jr, 2004; Hill and Chen, 1984; VanBuren et al., 2002).
These studies have provided insight into numerous aspects of MT behavior, including the origin of dynamic instability, the nature of the stabilizing cap, and the effect of physical boundaries on MT length distributions. However, many of these efforts examine the behavior of single MTs. Those with multiple MTs occur in semi-infinite space and/or have variables, such as tubulin concentration or transition frequencies, defined to be constants. None of these studies examines how the behavior of MTs is influenced by the constraints of a cell-like environment, in which multiple MTs (but not an infinite number) compete for a limited pool of tubulin subunits and microtubule growth is spatially confined.
To begin to address these issues, we have performed a series of Monte-Carlo simulations of a system of dynamic MT in such a cell-like environment. In these simulations, MTs (nucleated by a defined number of seeds) compete with each other for free GTP tubulin subunits in a `cell' of defined size and shape. MTs can be followed visually or statistically at the level of an individual or the population. Dynamic instability parameters (transition frequencies, growth rates, and shrinkage rates) are not set by the user but instead evolve from the interactions of the different parts of the system as the simulation proceeds. A recent article by Janulevicius et al. (Janulevicius et al., 2006) describes a model that has similarities to the one we used here, but which was used to address different questions.
Using our model, we find that several cellular phenomena that were thought to require complex regulatory machinery are instead predictable outcomes of interactions between a system of dynamic MTs and its physical environment. More specifically, we find that the surprisingly persistent growth of MTs observed in vivo is a predictable property of a nucleated system of dynamic MTs containing sufficient tubulin and polymerizing in a constrained space. This perturbation of MT dynamics is an outcome of the increase in the steady-state concentration of free tubulin, resulting from interactions of MTs with the cell boundary. Similarly, changes in nucleation activity are expected to have major effects on MT length and transition frequencies, dictating changes such as those seen at the interphase-mitosis transition (Piehl et al., 2004; Rusan et al., 2001). These observations do not exclude MAP involvement (obviously, MAPs play central roles in these processes), but they imply that MAPs modulate these behaviors instead of creating them. These studies indicate that the classic concept of critical concentration requires revision when applied to cellular systems, and they provide a foundation for quantitative understanding of MT dynamics in vitro and in vivo.
Results and Discussion
Recapitulation of dynamic instability by our Monte Carlo model
Using the rules outlined in the Materials and Methods, we have built a simplified coarse-grained stochastic simulation of MT dynamics that incorporates the major elements of the MT polymerization process in vivo, including spatially constrained nucleation, competition between MTs for tubulin subunits, and the imposition of physical limitations to MT polymerization by the cell edge. Adjustable parameters are the total tubulin concentration, cell size, rate of GTP hydrolysis, and association and dissociation rate constants for GTP and GDP tubulin. MT dynamic instability parameters (rate of growth, rate of depolymerization, catastrophe frequency, and rescue frequency) are not set by the modeler but instead change with time and conditions, emerging from the dynamic interactions of the system. This model recapitulates the obvious qualitative features of dynamic instability (Fig. 1, Movie 1 in supplementary material). When appropriate adjustable parameters are chosen (see Materials and Methods), it can approximate the quantitative features, including growth rate and transition frequencies (Table 1). Unless otherwise indicated, all data presented in this manuscript are obtained from simulations run under a set of reference parameters chosen to approximate the behavior of tubulin in vivo in interphase (Rusan et al., 2001), and are from steady-state (see Materials and Methods for details).
This model is similar to the stochastic simulations used by Hill and Chen (Hill and Chen, 1984), but there are two key differences. First, in our model, a system of dynamic MTs is simulated instead of a single MT, and the concentration of soluble tubulin ([Tu]soluble) in this system changes as MTs grow and compete for subunits (in most previous simulations, including those of Hill, [Tu]soluble is a set parameter). Second, we have imposed MT-length limits (analogous to the limitations imposed by the cell boundary). The sum of these characteristics suggests that this model is useful for investigating the constraints on MT dynamics imposed by cell-like systems.
Persistent growth of MTs in vivo and in silico
Perturbation of the simulation by changing the concentration of total tubulin ([Tu]total), leaving all other parameters unaltered, reveals an important relationship between tubulin concentration and MT behavior. At low [Tu]total, MTs in the simulation mimic MTs observed in vitro at steady-state: catastrophe is frequent, rescue is relatively rare and MT lengths decay exponentially (short MTs significantly outnumber long MTs) (Fig. 2A,E, Table 1, Movie 2 in supplementary material). However, at relatively high [Tu]total, dramatically different behavior is observed: MTs begin to grow persistently, meaning that many MTs reach the cell edge without ever undergoing catastrophe (Fig. 2B and Table 1, Movie 3 in supplementary material). Most catastrophes occur near the cell edge and rescue becomes more frequent, resulting in an accumulation of MT ends near the cell edge (Fig. 2F). This behavior mimics that observed in vivo (Komarova et al., 2002b).
What is the origin of this change from an in-vitro-like behavior to an in-vivo-like behavior? The observation that catastrophe frequency displays position dependence in vivo has been addressed previously: Komarova et al. and Maly et al. have suggested that the cell edge induces catastrophe in persistently growing MTs, causing the catastrophe asymmetry and resulting in the MT length distribution observed in vivo (Komarova et al., 2002b; Maly, 2002). Induction of catastrophe by physical boundaries has been observed experimentally (Janson et al., 2003). We have incorporated these ideas into our model by stipulating that the cell edge prevents the addition of new tubulin subunits, resulting in loss of the GTP cap, which then leads to catastrophe.
However, although the `edge-induced depolymerization' explanation is attractive, it is not complete. It explains the catastrophe asymmetry, but fails to address the key issue of the cause of the persistent growth. Komarova and colleagues proposed that MAP activities are involved, and provided evidence that the MT plus end tracking protein CLIP-170 plays a role in persistent growth (Komarova et al., 2002a). However, the only difference between the simulations in Fig. 2A,B (one exhibiting persistent growth and one not) is the amount of total tubulin in the system.
Consideration of the behavior of our computational model leads to an alternative hypothesis for the origin of the persistent growth: (1) the cell boundary induces catastrophe prematurely (Komarova et al., 2002b; Maly, 2002); (2) this catastrophe induction causes [Tu]soluble at steady-state to be higher than it would be in the absence of the barrier; (3) this higher availability of [Tu]soluble is what allows the MTs to grow persistently, experiencing few catastrophes and undergoing more frequent rescues. This hypothesis predicts that a physical boundary can cause the [Tu]soluble steady-state to rise above its natural steady-state level, and this increase in [Tu]soluble can be sufficient to cause persistent MT growth.
Relationship between total tubulin and soluble tubulin: expected behavior according to the classic model
As a first test of this hypothesis, we investigated the relationship between [Tu]total and [Tu]soluble at steady-state. Before examining our system, we wanted to first understand what was expected from the existing literature. The classic understanding of monomer/polymer partitioning as defined by Oosawa and colleagues, and refined by Johnson and Borisy is shown in Fig. 3A (Johnson and Borisy, 1977; Oosawa and Asakura, 1975; Oosawa and Kasai, 1962), see also Howard for a more recent discussion (Howard, 2001). Examination of this figure shows that tubulin put into a system is expected to remain unpolymerized until the concentration of unpolymerized tubulin reaches a critical concentration, after which point all additional tubulin is incorporated into polymer. Under this model, [Tu]soluble remains at the critical concentration no matter how much additional tubulin is added (assuming that all of it is active).
Deviation from this expected behavior
This Oosawa model has been the dominant framework for understanding bulk MT polymerization. Our model reproduces this expected behavior when MT length is unconstrained (compare red solid curves in Fig. 3A,B)1. However, when MT length is physically constrained – as it would be in a cell – the concentration of soluble tubulin deviates significantly from the behavior according to the classic model (Fig. 3B, dashed curve): there is no clearly observed critical concentration and, instead of plateauing after the initial rise, [Tu]soluble continues to rise with [Tu]total, at first slowly and then more quickly. Because the two curves of Fig. 3B are produced under conditions that are identical except for the presence or absence of catastrophe induction by the cell edge, these observations suggest that the `edge effect' predicted above can cause [Tu]soluble to rise above that expected from standard considerations of critical concentration2. Moreover, the effect increases as more tubulin is added to the system.
Effect of small changes in the [Tu]soluble on MT behavior
We next examined the effect of such changes in [Tu]soluble on MT behavior. The differences seen between the solid and dashed lines of the curves in Fig. 3B are relatively small: when [Tu]total rises from 5 μM to 14 μM, [Tu]soluble rises only from ∼4 to ∼6 μM, respectively. Although this increase may seem insignificant, Fig. 4 shows that it is predicted to have a major effect on the behavior of the simulated MTs: Changing [Tu]soluble from 4.1 μM to 6.4 μM shifts the behavior of the simulated MTs from classic dynamic instability to, apparently, persistent growth (Fig. 4A). Note that, to test the dependence of MT behavior on [Tu]soluble as in Fig. 4A, it is necessary to hold [Tu]soluble constant (i.e. we altered the simulation so that in these trials the MTs do not compete for tubulin. In addition, there is no spatial constraint).
To better understand this transition to persistent growth, we examined the relationship between [Tu]soluble and average MT length in more detail. When [Tu]soluble is held constant at low to moderate levels, the length of MTs at steady-state increases as soluble tubulin increases, but it is finite across this concentration range (Fig. 4B). However, as [Tu]soluble increases more, there is a narrow range in which the average steady-state length rises steeply, appearing to approach infinity at some threshold (between 5 and 6 μM soluble tubulin under these conditions) (Fig. 4B). This threshold is the transition to persistent growth – as can be seen in a plot of net MT growth rate as a function of [Tu]soluble (Fig. 4C). When this threshold is passed, MTs still experience catastrophes but the balance between catastrophe and rescue is such that, the amount of polymer in such a system increases constantly with time (Fig. 4A,C). On the basis of this analysis, small increases in the steady-state concentration of soluble tubulin (like those caused by interaction of MTs with the cell edge) should induce persistent growth if the initial [Tu]soluble is close enough to the persistent growth threshold. Note that the transition to persistent growth occurs at a concentration empirically similar to the critical concentration observed in the absence of spatial constraint (compare the dotted line in Fig. 3B with the x-axis intercept of the dashed line in Fig. 4C), but we have not yet developed a formal demonstration of this relationship.
These observations are consistent with previous analyses indicating that average MT length should increase with [Tu]soluble, and that MTs should show a transition from `bounded' to `unbounded' growth when [Tu]soluble passes a threshold (Dogterom and Leibler, 1993; Verde et al., 1992). In a physiological system, the total mass of tubulin is constant, and competition between MTs for this tubulin would be expected to keep [Tu]soluble below the transition to persistent growth. However, if a physical boundary limits polymerization and the edge effect (Fig. 3B) causes [Tu]soluble to rise above the threshold, persistent growth should occur.
One conclusion of this work is that the classic concept of critical concentration must be used carefully in cellular systems – [Tu]soluble at steady-state is not simply equal to the critical concentration. Instead, the steady-state [Tu]soluble is a dynamic parameter that depends on a host of variables including the number of nucleation sites, the total amount of tubulin and the cell size. Transition frequencies and MT length distribution are coupled to these same variables through [Tu]soluble. As originally suggested by Mitchison and Kirschner (Mitchison and Kirschner, 1987), the increase of [Tu]soluble with [Tu]total is probably a major part of the mechanism, allowing total tubulin levels to be tightly regulated in vivo.
On the basis of these simulations and the reasoning above, we propose that the persistent growth of MTs observed in vivo is an expected outcome of placing a sufficient amount of tubulin in a spatially confined system under conditions where nucleation is limited. This reasoning does not exclude the involvement of MT-binding proteins in persistent growth in vivo, but argues that they tune the behavior instead of generating it. Although physiological systems differ from our simulations (and from each other) in their quantitative details, this analysis suggests that persistent growth in the presence of an appropriate spatial constraint is an intrinsic property of any system of MTs or, indeed, of any nucleated two-state polymer system.
Effect of MT nucleation on MT dynamics
A key aspect of MT growth, both in vivo and in our model, is that it normally occurs from stable nuclei or `seeds'. We were interested to see whether our simulations could provide insight into two experimental observations related to nucleation: (1) the changes in MT length and dynamics seen together with increases in nucleation during mitosis; (2) the ability of mutations in S. pombe nucleation factors to alter MT length and dynamics. In both cases, the effects on MT length and dynamics have been proposed to result from the direct action of MAPs.
Implications for the transition to mitosis
When cells transition into mitosis, the MT cytoskeleton changes from a relatively stable interphase array that has long, persistently growing MTs into an early mitotic array that has shorter, more numerous MTs that are also more dynamic (Piehl et al., 2004; Rusan et al., 2001). Using our computational model, we investigated the relationship between these characteristics. We found that, simply increasing the number of MT seeds in an interphase-like simulation causes the system to adopt mitosis-like dynamic instability characteristics: MTs become significantly shorter and more dynamic, undergoing more spontaneous (not-edge-induced) catastrophes and fewer rescues (Fig. 2C, compare with 2B, Table 1).
These changes in length and dynamics result from the reduction in [Tu]soluble that occurs as the number of nucleation sites is increased (supplementary material Fig. S2). This reduction in [Tu]soluble results from the decreased likelihood of edge-induced catastrophe: when a given mass of tubulin is split between a greater number of microtubules, the MTs get shorter, eventually reaching a natural length that is too short to interact significantly with the cell boundary (compare Fig. 2B with 2C). These considerations suggest that, the increase in MT dynamics observed in vivo during mitosis is an obligatory consequence of the observed increase in MT nucleation. Note that, this assumes that nucleation is controlled directly. An alternative hypothesis is that MAPs could increase the frequency of catastrophes, causing [Tu]soluble to rise. Given that MT growth from centrosomes is reported to increase with [Tu]soluble, an increase in [Tu]soluble could cause the observed increase in the number of MTs. The main point of this analysis is to stress the obligatory connection between [Tu]total, [Tu]soluble, microtubule number and transition frequencies.
Implications for the interpretation of nucleation mutants
Similarly, reducing the number of seeds can be expected to have profound effects on MT length and dynamics (Table 1, Fig. 2, supplementary material Fig. S2). Mutations in γ-tubulin-complex proteins, which cause defects in MT nucleation, also cause abnormalities in MT dynamics, including inappropriately continuous growth, abnormally long MTs, increased growth rate, and catastrophe defects (Zimmerman and Chang, 2005). A number of explanations have been proposed for these effects, including inappropriate regulation of MT-binding proteins, such as Tip1p. Our analysis suggests that, these effects are expected outcomes of the increase in the steady-state concentration of soluble tubulin that is predicted to result from reduction in the number of MTs (Fig. 2D, compare with 2B; Table 1; supplementary material Fig. S2).
Implications for the study of MAPs
Although we did not directly study MAPs in these simulations, our analysis has implications for the understanding of MAP function. Experimental characterization of MAPs has often produced conflicting results (e.g. Cassimeris, 2002; McNally, 2003). We propose that some of the discrepancies stem from failure to take into account the fact that a solution of dynamic MTs in vitro is an evolving system. For example, a protein that suppresses catastrophe without altering other parameters under pre-steady-state conditions (when [Tu]soluble is near the initial concentration) could have have little or no effect on catastrophe at steady-state because the protein would reduce [Tu]soluble at steady-state, and catastrophe frequency increases as [Tu]soluble decreases. Indeed, such a protein could have the paradoxical effect of slowing MT growth under steady-state conditions. These issues highlight the systems nature of the MT cytoskeleton and suggest that comprehensive understanding of MAP function is likely to require integration between experimental and computational approaches.
We have provided evidence that several aspects of MT dynamics that are often thought to be imposed by MAP activity (or changes in MAP activity) are instead simple, perhaps unavoidable, outcomes of the systems nature of the cellular MT cytoskeleton. More specifically, we find that the surprising persistent growth of MTs observed in vivo is a predictable property of a nucleated system of dynamic MTs polymerizing in a constrained space. Moreover, our simulations indicate that the increase in MT dynamics seen in mitosis is an expected outcome of the mitotic increase in nucleation activity. Similarly, our work provides an explanation for the observation that defects in nucleation cause abnormally stable MT plus ends, without invoking action of nucleation factors at MT plus ends. Our simulations indicate that these effects are mediated by changes in the concentration of free tubulin in the system at steady-state.
We do not mean to imply that regulatory proteins are not involved in behaviors such as persistent growth and the transition to mitosis – it is clear that MAPs play central roles in these processes. However, we suggest that, instead of causing these behaviors, this regulatory machinery tunes them, for example by shifting the amount of free tubulin available or the concentration of free tubulin at which persistent growth occurs.
A pressing question in the history of eukaryotic life is how complex cellular processes arise. Our work indicates that certain behaviors are intrinsic to a system of dynamic microtubules in a confined space. It seems plausible that cells with a rudimentary microtubule array could then, over time, develop machinery to modulate these intrinsic behaviors. This hypothesis is attractive from an evolutionary perspective because it provides evidence that complex cellular behaviors can have simple origins.
Materials and Methods
Essential features of the simulation
MTs are modeled as simple linear polymers of tubulin subunits, with addition and loss of subunits from the tip occurring according to probabilities defined by the following rules: (1) Tubulin subunits have two states that are assumed to be GTP- and GDP-bound, but these states could represent other conformations. (2) Subunits polymerize (add to the end of a MT or seed) in the GTP-bound state, then undergo GTP hydrolysis after polymerization. (3) Hydrolysis of a given subunit occurs according to a set probability that is independent of the nucleotide state of the surrounding subunits, i.e. hydrolysis is a stochastic first order process, producing a shifting `cap' of GTP subunits of variable length. (4) The probability of subunit-addition or -loss on a MT depends on the nucleotide state of the tubulin subunit exposed at the tip. (5) Subunit addition at the tip depends on the concentration of [Tu]soluble, whereas subunit loss is independent of [Tu]soluble. (6) The total number of tubulin subunits in the system is fixed (unless otherwise indicated), resulting in competition between microtubules for tubulin subunits and evolution of dynamic-instability parameters as the simulation proceeds. (7) The maximum length of microtubules is limited by a boundary (analogous to the cell edge). This is summarized in the following mathematical terms: (1) and
Note that, because the model is stochastic, probability transition constants (K) are used in place of kinetic rate constants (k).
The number of MT-nucleation sites, the size of the pool of tubulin subunits and the size of the cell are set by the user. Unless otherwise indicated, the simulations were run at [Tu]total=14 μM, 128 stable seeds (nucleation sites) and cell radius =10 μm. These values were arbitrarily chosen within the range that is biologically plausible (while mammalian cells have many MTs, yeast cells have few) and computationally rapid. Probability transition constants can be varied, but for the purpose of these simulations they were set to constant values chosen to qualitatively approximate the behavior observed in mammalian cells during interphase (Rusan et al., 2001). Specific constants used were μM–1 seconds–1, μM–1 second–1, seconds–1, seconds–1, seconds–1. Note that, because the microtubules are modeled as simple linear polymers with subunits that are larger than tubulin (20 nm vs 8 nm), these rate constants do not correspond to experimentally determined values. We emphasize that, the purpose of these simulations is to explore the relationships common to microtubule systems in general, not to quantitatively replicate specific experimental observations. All other parameters of the system (growth rates, depolymerization rates, transition frequencies, concentrations of polymeric and soluble tubulin) are emergent properties. Varying the user-defined constants over a large range changed the specific values of the emergent parameters but did not alter the basic relationships discussed in the text (data not shown).
Other aspects of the model
Induction of catastrophe by the simulation boundary (cell edge) is an indirect result of inhibition of new subunit addition at the boundary, which leads to loss of the GTP cap and eventual depolymerization. In some cases (indicated in the text), simulations were run without spatial constraint, allowing unlimited MT length. Some simulations (as indicated) were also run without constraints on the mass of total tubulin (i.e. constant soluble tubulin). Both tubulin diffusion and regeneration of GTP-tubulin from detached GDP-tubulin were assumed to be fast and, therefore, modeled as instantaneous (Brylawski and Caplow, 1983; Odde, 1997). Therefore, although we present the shape of the cell as rectangular in Fig. 1 and as radial in the movies (see supplementary material, Movies 1-3), the actual cell geometry is undefined. Investigations into the effects of finite diffusion rates or alternative geometries are important avenues for future study. We refer to recent work by Janulevicius et al. for analysis of the effects of limited compartment volume, such as those that might be found in cell processes, on microtubule dynamics (Janulevicius et al., 2006).
Computation and analysis of the simulations
The algorithm was coded in C language and run on PCs using Linux operating systems. The output was analyzed and visualized with Matlab 7.0 SP2 (MathWorks, Natick, MA); movies were made from Matlab output with QuickTime Pro (Apple Computer, Inc., Cupertino, CA). Calculation of transition-frequencies and -rates was performed with a Matlab script that filtered out the noise, and detected persistent growth and shortening phases. Phase transitions occurring within 1% of cell radius from MT seeds or cell edge were excluded from the calculations. Time conversion is: first simulation step equals 0.02 seconds. Source code for all programs is available upon request. Error bars depicted in figures represent the ± standard deviation of 50 repetitions of each simulation.
Classical relationship between [Tu]total and [Tu]soluble
The prediction depicted in Fig. 3A was plotted as described by Johnson and Borisy (Johnson and Borisy, 1975). Briefly, the relationship between Co ([Tu]total) and C1 ([Tu]soluble) is expected to satisfy (2)
For Fig. 3A, we used a MT nucleus size (s) of 5, in agreement with previous analysis (Johnson and Borisy, 1975) and chose K=2.86×10–1 and K′=1.43 ×10–2 to produce an appropriate shape and plateau position. K and K′ have units of inverse concentration.
Limitations of the analysis
This model is an approximation chosen to be general (i.e. not dependent on unknown details of the polymerization process) – consistent with MT behavior observed microscopically – and computationally fast, while still incorporating key aspects of tubulin biochemistry. Necessarily, the model has simplifications, the most obvious of which is that MTs were modeled as simple linear polymers instead of tubes composed of protofilaments. Incorporation of structural detail is an important goal for future work, but we do not expect this simplification to alter our basic conclusions because the rules governing the simulation are largely independent of this structural detail (i.e. the rate of growth is expected to be a linear function of subunit concentration (Flyvbjerg et al., 1995), regardless of the size of the subunits or the number of protofilaments). A second simplification is that, our simulation assumes that tubulin has two states, with a stochastic transformation between them. It is important to notice that the identity of the two states (GTP and GDP, GDP-Pi and GDP, or open sheet and closed tube) is irrelevant – what is important is that there are two states with different characteristics. There may in fact be several conformational states at the end of the MTs but we assume that one of the transitions is rate-limiting, and it is this rate-limiting transition that we are simulating. We chose stochastic transitions instead of the commonly assumed vectorial hydrolysis because previous modeling efforts have indicated that vectorially catalyzed transitions do not fit the data well (Flyvbjerg et al., 1996). Given the similarity of the simulation to the behavior of MT systems observed experimentally (Table 1, Figs 1, 2), we believe that this model is an informative approximation.
We thank David Odde, Sid Shaw, and members of the Chicago Cytoskeleton group for insightful discussions, and the Goodson and Alber groups for both helpful discussions and comments on the manuscript. Adam Willis assisted in early stages of this work. This work was supported by NIH grant to HVG (GM065420), and by a NIH Supplement for the Study of Complex Biological Systems to MA and HVG. Simulations were performed on computers of the NSF-supported BEOWULF Biocomplexity cluster (NSF-DBI-0420980).
↵1 The sharper transition to the plateau of the curve in panel A (compared with the curve in panel B) results from the fact that the model used to yield curve A incorporates a very unfavorable nucleation step, which normally exists when tubulin polymerizes in vitro. Our model, which is meant to simulate microtubule polymerization from stable nucleation sites in vivo, produces a similarly sharp transition if the difficulty in initial growth from the stable seeds is increased (supplementary material Fig. S1).
↵2 In 1987, Mitchison and Kirschner performed a theoretical analysis suggesting that [Tu]soluble increases with [Tu]total in mass- and number-limited systems of dynamic microtubules, such as those in a cell, even in the absence of spatial constraint (Mitchison and Kirschner, 1987). Some of their conclusions are similar to those discussed here. However, their conclusions depend on the assumption that rescue does not occur. Because rescue is common in cellular systems, and because our simulations behave according to their predictions only when both rescue and spatial constraint are `turned off' (supplementary material Fig. S1), we believe that the conclusions of their analysis were prescient but the equations leading to these conclusions are of limited use.
Supplementary material available online at http://jcs.biologists.org/cgi/content/full/119/22/4781/DC1
- Accepted September 4, 2006.
- © The Company of Biologists Limited 2006