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Cell. Author manuscript; available in PMC 2012 August 19.

Published in final edited form as:

PMCID: PMC3171214

NIHMSID: NIHMS319247

Melissa K. Gardner,^{1,}^{2,}^{3} Blake D. Charlebois,^{4} Imre M. Jánosi,^{5} Jonathon Howard,^{2} Alan J. Hunt,^{4,}^{‡} and David J. Odde^{3,}^{‡}

Microtubule assembly is vital for many fundamental cellular processes. Current models for microtubule assembly kinetics assume that the subunit disassociation rate from a microtubule tip is independent of free subunit concentration. Using Total-Internal-Reflection-Fluorescence (TIRF) microscopy and a laser tweezers assay to measure in vitro microtubule assembly with nanometer resolution, we find that the subunit dissociation rate from a microtubule tip increases as the free subunit concentration increases. These data are consistent with a two-dimensional model for microtubule assembly, and are explained by a shift in microtubule tip structure from a relatively blunt shape at low free concentrations to relatively tapered at high free concentrations. Because both the association and the dissociation rates increase at higher free subunit concentrations, we find that the kinetics of microtubule assembly are an order-of-magnitude higher than currently estimated in the literature.

Microtubules are rapidly adaptable multi-stranded linear polymers that self-assemble and function in many crucial cellular processes such as cell division, intracellular transport, cell polarization, and nerve regeneration. To produce the length changes required during these cellular processes, microtubule ends stochastically switch between growing and shortening phases (Mitchison and Kirschner, 1984). Microtubules are most frequently observed as a column of 13 protofilaments, each consisting of αβ-tubulin subunits stacked end-to-end. Thus, microtubules grow and shorten at their ends via the addition and loss of αβ-tubulin subunits from the individual protofilaments, and are stabilized by the combination of lateral and longitudinal bonds between subunits.

Microtubule assembly kinetics are important during cellular morphogenesis and are a target of widely used medical treatments. As a result, they have been the subject of intensive analysis for over 3 decades (Desai and Mitchison, 1997). Microtubule assembly is altered by drugs used to treat diseases such as cancer, gout, and artery restenosis. In addition, in vivo microtubule assembly dynamics are regulated by microtubule-associated proteins (MAPs), which act to alter tubulin subunit addition and/or loss rates (Howard and Hyman, 2007).

The tubulin subunit association and dissociation rate constants, *k*_{on,MT} (μM^{−1} s^{−1}) and *k*_{off,MT} (s^{−1}), respectively, have been previously estimated from in vitro measurements of the net microtubule assembly rate, v_{g} (subunits/s; with ~1625 subunits/μm of MT length), as a function of the free tubulin concentration [Tub] (μM) (Supplemental Table S1). These rate constants have been estimated by fitting the microtubule assembly data to a linear one-dimensional (1D) model, initially applied to biological polymers by Oosawa (1970). A key, but untested, assumption in the 1D model is that the tubulin subunit dissociation rate is constant regardless of free tubulin concentration. This assumption cannot be strictly correct since it requires that the microtubule tip structure is on average constant over varying tubulin concentrations, which is not thermodynamically tenable (Hill, 1986). In addition, microtubule tips shift from relatively blunt tip structures at low tubulin concentration to relatively tapered tip structures at high concentrations (Chretien et al., 1995). However, the 1D model might still serve as a reasonable approximation if the microtubule tip structure is only weakly dependent on free tubulin subunit concentration. Thus, the canonical view of microtubule polymerization implicitly rests on the assumption that all tubulin subunits at the tip of a growing microtubule are on average energetically similar to one another, so that *k*_{off,MT} is independent of free subunit concentration.

To explicitly test whether thermodynamic differences between tubulin subunits at the microtubule tip are important for interpreting growth data, we used total internal reflection fluorescence (TIRF) microscopy, and our high temporal and spatial resolution nanoscale laser tweezer assay (Schek et al., 2007) to study microtubule assembly at near-molecular resolution. We show (1) that tubulin subunit dissociation rates from the tip of a growing microtubule depend on the free tubulin concentration, and (2) that dissociation rates increase substantially at higher free tubulin concentrations, which requires that the tubulin subunit on-off kinetics during microtubule growth occur 10X more rapidly than previously appreciated. Together, these data lead to a new perspective on the regulation of microtubule assembly via MAPs and drugs, where weak and infrequent interactions with tubulin subunits at the growing microtubule tip can strongly affect net assembly.

In a linear (1D) model, a key assumption is that the departure rate of tubulin subunits from the microtubule tip (*k _{off,MT}*) is constant regardless of the free tubulin concentration (Oosawa, 1970). As shown in Fig. 1A (left), this model can be characterized by the number of tubulin subunit arrivals to the microtubule tip (λ

Given the number of arrivals, λ_{+}, and the number of departures, λ_{−}, the resulting difference, (λ_{+}-λ_{−}), gives the distribution of incremental microtubule length changes (Fig. 1A, right). An untested prediction of the 1D model is that the likelihood of large microtubule shortening events will *decrease* as the subunit arrival rate (λ_{+}) becomes larger (Fig. 1A, right). This is because microtubule growth is favored when the rate of subunit departure remains constant and the rate of subunit arrival increases in the presence of increasing [Tub]. Therefore, the probability of a large shortening event (see dashed vertical line located at |λ_{+}-λ_{−}|=|−4|) decreases with increasing free tubulin concentration (Fig 1A, right). To directly test this prediction of the 1D model, we quantified the distribution of microtubule length increments as a function of free tubulin concentration, asking if the frequency of large shortening events does in fact *decrease* at higher tubulin concentrations.

To test the 1D model predictions, we performed in vitro GMPCPP-tubulin experiments to measure microtubule assembly dynamics at nanometer scale resolution. Using GMPCPP allowed us to focus on the intrinsic variability in the microtubule growth rate without the potentially confounding effects of nucleotide hydrolysis. Assembly of GMPCPP-tubulin labeled with Alexa-488 (55% labeled, Invitrogen Corp.) onto Rhodamine-labeled GMPCPP-tubulin seeds was imaged by TIRF microscopy (Fig 1B). The positions of microtubule ends were estimated by fitting the error function to the fluorescence intensity at microtubule end, where the microtubule end was defined as the position at which the error function fell to 50% of the signal (μ_{tip}, Fig. 1B) (Demchouk et al., 2011).

As expected, the rate of net assembly increased as the GMPCPP-tubulin concentration increased (Fig. 1C). We found that microtubule growth trajectories in dilution experiments and at 0.8 μM GMPCPP-tubulin (Fig. 1C, green and red, respectively) showed steady growth or shortening with small length fluctuations, while microtubule growth trajectories at 1.5 μM GMPCPP-tubulin (Fig. 1C, blue) appeared to grow “noisily” with larger length fluctuations. These fluctuations were quantified by calculating the net length change increments for each 1.4 sec interval during the first 30 seconds of growth for each microtubule (Fig. 1D).

Using the TIRF system, we found that the frequency of large shortening events (e.g. <−32 nm, equivalent to 4 dimer lengths) *increases* at higher tubulin concentrations (from 1.8 +/− 0.4 % in dilution experiments to 9.1 +/− 0.8 % at 1.5 μM tubulin, Fig. 1D). We note that for fixed (non-dynamic) microtubules the frequency of “shortening” events arising from measurement noise alone is approximately independent of tubulin concentration, ranging from 1.4 +/− 0.4 % at low concentration to 1.8 +/− 0.4 % at high concentration. Thus, the increase in the frequency and magnitude of shortening events at higher tubulin concentrations strongly suggests that the tubulin subunit dissociation rate from the microtubule tip increases at higher free tubulin concentrations, which invalidates a basic assumption of the 1D model.

To examine microtubule growth rates with higher resolution, we measured the variability in growth rates using a nanoscale laser tweezers assay (Charlebois et al., 2010; Schek et al., 2007), which achieves a spatial resolution of <3.5 nm at 10 Hz temporal resolution by measuring the position of microtubule-attached beads as microtubules grow into a microfabricated barrier (Fig. 2A). We observed variability in microtubule growth at three different GMPCPP-tubulin concentrations, with variability increasing substantially at higher tubulin concentrations (Fig. 2B). By quantitatively comparing the growth increment sizes at 0.1 sec time intervals, we found that the frequency of large shortening events (e.g. < −8 nm) increased from 1.4±0.3% of all events at 0.5 μM tubulin, to 6.3±0.5% of all events at 1.5 μM tubulin (Fig. 2C). Because of the higher spatial-temporal resolution in the tweezers experiment, we tested for the addition of oligomers (Kerssemakers et al., 2006; Schek et al., 2007). We never observed increments equal to or larger than 16 nm (i.e. two subunits) at 0.5 μM and 1.0 μM (N=4998 increments), and only rarely at 1.5 μM (0.93%, N=2567 increments). These rare instances of additions larger than 16 nm could be explained by rapid successive addition events of single subunits (Schek et al., 2007). Thus, similar to the TIRF data, the optical tweezers data indicates that the tubulin subunit dissociation rate from the microtubule tip increases at higher tubulin concentrations, which is inconsistent with the 1D model.

Because the tubulin dissociation rate from the microtubule tip increases at higher tubulin concentrations, we asked if the state of the microtubule tip is on average shifted to a less stable configuration at higher tubulin concentrations. To ask this question, we performed computer simulations using our previously published “2D” model for microtubule assembly (VanBuren et al., 2002). In contrast to the 1D model, where protofilaments are regarded as independent with a single *k _{off,PF}*, the 2D model explicitly accounts for the bonds in both the lateral and longitudinal directions. Together, these bonds stabilize a subunit. As the number of these bonds varies, k

The 2D simulation accounts for (1) subunit arrivals at the tip of the microtubule, and (2) subunit departures from the tip (Fig. 3A). For arrivals, the subunit arrival rate at the tip of a protofilament (*k _{on,PF}^{*}*) is given by:

$${{k}_{on,PF}}^{}$$

(1)

Where *k _{on,PF}* is the on rate constant per individual protofilament (μM

$${K}_{eq}\frac{{k}_{on,PF}}{{k}_{\mathit{off},PF}}={e}^{-\mathrm{\Delta}{{G}^{0}}_{\mathit{total}}/{k}_{B}T}$$

(2a)

Therefore, by rearrangement:

$${k}_{\mathit{off},PF}\frac{{k}_{on,PF}}{{e}^{-\mathrm{\Delta}{{G}^{0}}_{\mathit{total}}/{k}_{B}T}}$$

(2b)

where *k _{off,PF}* is the off rate per individual protofilament (s

$$\mathrm{\Delta}{{G}^{0}}_{\mathit{total}}=\sum \mathrm{\Delta}{{G}^{0}\mathit{Longitudial}}_{}+\sum \mathrm{\Delta}{{G}^{0}}_{\mathit{Lateral}}$$

(3)

Tip subunits will (by definition) have one longitudinal bond (Fig. 3A), and therefore the total value for *ΔG ^{0*}_{Longitudinal}* will be identical for all subunits at the tip. However, the

As the stochastic 2D model computer simulation proceeds (see animated supplemental movies S1 and S2), the tip structure evolves as subunits arrive and depart from the tip of each protofilament, and the average behavior can be calculated for a given tubulin concentration (simulation parameters in Supplemental Table S2 and Fig. S1). The steady-state subunit off rate from the microtubule tip (averaged over time and summed over all protofilaments) can then be calculated as a function of the free subunit concentration. The 2D model predicts that net microtubule assembly (Fig. 3B, blue) results from the relatively small difference between a large subunit arrival rate that increases with free tubulin concentration (Fig. 3B, green), and a large subunit departure rate that *also* increases with increasing free tubulin concentration (Fig. 3B, red). Thus, the 2D model assumes that subunits at the tip of a growing microtubule are not energetically identical, and therefore predicts that the average *k _{off,MT}* increases at higher tubulin concentrations. This result provides an explanation for the discrepancy between our experimental observations and the 1D model.

Why do the subunit dissociation rates increase at higher free tubulin concentrations? The protofilament dissociation rate constant is dependent on the number of neighboring protofilaments available for forming stabilizing lateral bonds (Fig. 3C), which must on average be shifted toward less stable configurations as the tubulin concentration increases. For example, tubulin subunits with two lateral neighbors (Fig. 3C, left) have a low dissociation rate constant, while subunits with one lateral neighbor have a moderate dissociation rate constant, and tubulin subunits with no lateral neighbors have a high dissociation rate constant due to the lack of any stabilizing lateral bonds (Fig. 3C, left). Thus, if the probability of the configurations was known, the dissociation rate constant averaged over the entire microtubule tip (*k _{off,MT}*) would be given by

$${k}_{\mathit{off},MT}\approx {f}_{2}{k}_{\mathit{off},PF}^{(2)}+{f}_{1}{k}_{\mathit{off},PF}^{(1)}+{f}_{0}{k}_{\mathit{off},PF}^{(0)}$$

(4)

Where *f*_{2}, *f*_{1}, and *f*_{0} are the mean probabilities of tubulin subunits at the tip with two, one, and zero lateral neighbors, respectively, and *k _{off,PF}^{(2)}*,

We examined how the relative fractions of tubulin subunits with 0, 1, or 2 lateral neighbors change with increasing free subunit concentrations in the 2D model and found that the fraction of tubulin subunits at microtubule tips with 0 lateral neighbors remains relatively constant at all concentrations (Fig. 3C, right, yellow). However, there is an increase in the probability of relatively unstable 1 lateral neighbor subunits at the microtubule tip with increasing tubulin concentration (Fig. 3C, right, blue), and a corresponding decrease in the probability of relatively stable tubulin subunits having 2 lateral neighbors (Fig. 3C, right, magenta). Thus, the off rate is indirectly dependent upon the on rate. This is because the tip becomes more extended (i.e. protofilament lengths are more variable with increasing tendency toward only one lateral neighbor) as the on rate increases, which in turn increases the off rate. Thus, the off and on rates rise together.

Numerous *in vitro* studies have confirmed the predicted linear relationship between growth velocity and free concentration for growing GTP-tubulin microtubules and for GMPCPP-tubulin microtubules. These studies, summarized in Table S1, have consistently yielded average association rate constants *k _{on,MT}* of ≈ 5 μM

Similar to these studies, we experimentally measured the microtubule growth velocity as a function of GMPCPP-tubulin concentration using both the *in vitro* TIRF and the nanoscale laser tweezers assays. We then calculated the 1D model association and dissociation rate constants using the combined results from both experiments (Fig. 4A). From the slope and intercept, the 1D model microtubule association rate constant for GMPCPP-Tubulin was *k _{on,MT}* = 5.1 μM

The 2D model also predicts a linear relationship between microtubule growth velocity and tubulin concentration (Fig. 4B). However, the 2D model predicts that the average dissociation rate increases approximately linearly with free tubulin concentration due to evolving tip structures (Fig. 3B). This in turn requires that the association rate constant is substantially larger than in a 1D model in order to produce the observed net microtubule growth rate (Figs. 3B and and4B).4B). As a result, the 2D model association rate constant must be approximately an *order of magnitude* higher than the association rate constant estimated for the 1D model (*k _{on,MT}* ≈ 52 μM

For a stochastic process the variance in the number of events per time interval is equal to the mean number of events. Therefore the variability in microtubule growth rate should reflect the underlying rates of subunit addition and loss. Since the 2D model predicts rate constants that are an order of magnitude higher than the 1D model, it also predicts a much larger variance in the microtubule assembly rate. To quantitatively compare the predictions of the 1D and 2D models to our experimental results, we calculated the mean squared displacement of microtubule length increments for increasing time steps, and plotted the results for each model with the 1.5 μM TIRF experimental data (Fig. 4C). By plotting the results in this manner, the length fluctuations of filament growth can be described by a diffusion with drift equation, as given by:

$$<\mathrm{\Delta}{L}^{2}>={v}_{g}^{2}\mathrm{\Delta}{t}^{2}+2{D}_{p}\mathrm{\Delta}t+{\sigma}^{2}$$

(6)

where *ΔL* is the microtubule length change increment (in nm) over a given time step *Δt* (in sec), *v _{g}* is the net growth rate (nm/s), and

By fitting the experimental data and both models’ predicted data to a quadratic equation, the diffusion coefficient (*D _{p}*) can be estimated, which provides a quantitative measure of the microtubule growth variability. The 1D model diffusion coefficient

The tubulin subunit on rate (*k _{on,MT}^{*}*) and the off rate (

$$<\mathrm{\Delta}{{l}_{D}}^{2}>=2{D}_{p}\mathrm{\Delta}t$$

(7)

and the growth rate variance due to diffusion is described by a Skellam distribution (Oosawa, 1970; Skellam, 1946), so that

$$<\mathrm{\Delta}{{l}_{D}}^{2}>={a}^{2}({k}_{on,MT}[\mathit{Tub}]+{k}_{\mathit{off},MT})\mathrm{\Delta}t$$

(8)

where α is the change in microtubule length contributed by a single dimer, which is on average 0.615 nm. Combining Eqs. 7 and 8 and solving for *D _{p}* we obtain,

$${D}_{p}=\frac{{a}^{2}}{2}({k}_{on,MT}[\mathit{Tub}]+{k}_{\mathit{off},MT})$$

(9)

In addition, by definition,

$${v}_{g}=a({k}_{on,MT}[\mathit{Tub}]-{k}_{\mathit{off},MT})$$

(10)

Thus, through these two equations (9) and (10), the two unknowns, *k _{on,MT}* and

To calculate the values for *k _{on,MT}* and

displacement analysis as described above for a range of free tubulin concentrations in the TIRF analysis (Fig. 4D). Detailed results are summarized in Table S3 and shown in Fig. 4E. The experimentally estimated tubulin dissociation rate increases substantially as the tubulin concentration is increased, and the relatively slow net experimental microtubule assembly rate (blue) represents the difference between a large experimental on rate (green) and a large off rate (red), similar to the 2D model prediction (Fig. 3B). For example, at 1.5 μM, the net microtubule assembly rate (blue) represents the difference between a large on rate, *k _{on,MT}*[Tub] ≈ 77 s

We then asked whether, consistent with the TIRF assays, the variance in the growth rate measured using the optical tweezers assay is quantitatively larger than expected for the 1D model. As shown in Fig. 4F, the 1D model, constrained by mean growth rate data to *k _{on,MT}* =5.1 μM

The 2D model, constrained by the mean growth rate experiments to a 10-fold higher association rate constant than the 1D model, predicts a substantial increase in growth rate variance with increasing tubulin concentration. For the 2D model at 0.1 sec time intervals, the growth rate variance is predicted to increase from 15 nm^{2} at 0.5 μM tubulin to 25 nm^{2} at 1.5 μM tubulin (Fig. 4F, blue, 2D model parameters as in Supplementary Table S2) which is consistent with experimental results (1.0 μM, p=0.47; 1.5 μM, p=0.31; Two-tailed F-Test). Thus, the experimentally measured microtubule growth variance at higher GMPCPP-Tubulin concentrations is inconsistent with the relatively slow kinetics predicted by the 1D model, but is well described using a 10-fold higher tubulin association rate constant.

In shifting from low tubulin concentration (i.e., favoring 2 lateral neighbors, Fig. 3C) to high concentration (i.e., favoring 1 lateral neighbor, Fig. 3C), the microtubule tip structure predicted from the 2D model shifts from being relatively blunt, where protofilaments are typically the same length, to a more tapered tip in which protofilament lengths are more variable (Fig. 5A). This is because at high tubulin concentration (i.e. when subunits arrive rapidly to the tip), protofilaments will tend to grow independently of their neighbors, which results in tip subunits that are more likely to have only 1 lateral neighbor. Specifically, the tip structures at higher free tubulin concentrations will exhibit greater disparity in protofilament length (i.e., higher protofilament length standard deviation), with more single-neighbor protofilament extensions (Fig. 5B, blue). In contrast, at low tubulin concentrations, the relatively higher off rate will favor blunt ends (i.e., lower protofilament length standard deviation), where two lateral neighbors predominate (Fig. 5B, red). These predictions are consistent with cryo-electron microscopy images of microtubules formed from GTP-tubulin, where the mean microtubule tip taper lengths increase with increasing GTP-tubulin concentration (Chretien et al., 1995).

To test this prediction for GMPCPP microtubules, we estimated tip structures using TIRF microscopy by fitting the error function to the green Alexa-488 fluorescence intensity at microtubule ends, which yields both the mean protofilament length (μ_{tip}) (Fig. 1B) and the standard deviation of protofilament lengths (σ_{tip}) (Fig. 5C) (Demchouk et al., 2011). Specifically, we expect that tips with a smaller σ_{tip} have relatively “blunt” tips (Fig. 5C, left), while tips with a larger σ_{tip} have more tapered tips (Fig. 5C, right). Using this approach, we quantified the distribution of tip standard deviations (σ_{tip}) for similar lengths of Alexa-488 microtubules grown at 0.8 μM and 1.5 μM GMPCPP-Tubulin (Fig. 5D). We found that the mean tip standard deviation, <σ_{tip}>, is 99±22 nm (mean±sd; n=700) at a concentration of 0.8 μM GMPCPP-Tubulin, while <σ_{tip}> at 1.5 μM GMPCPP-Tubulin is 140±55 nm (mean±sd; n=1120) (Fig. 5D). This difference in <σ_{tip}> measurements (p<10^{−15}) suggests that microtubule tips are relatively blunt (with low protofilament length variation) in 0.8 μM GMPCPP-Tubulin, while at 1.5 μM they are more extended, with a larger variation in protofilament lengths. These results are consistent with the prediction that a decrease in the mean number of subunit lateral neighbors at the microtubule tip causes the mean subunit dissociation rate to accelerate with increasing free tubulin concentration.

We then tested whether the conclusions from our GMPCPP tubulin experiments are consistent with GTP-tubulin experiments at physiological concentrations (Supplemental Fig. S2).

As is the case with GMPCPP-tubulin, the 2D model predicts that both *k _{on,MT}^{*}* and

We performed TIRF experiments with GTP-tubulin at 7, 9, and 12 μM, and we observed an increase in microtubule growth variability and in negative growth excursions in going from 7 μM tubulin to 12 μM tubulin (Fig. 6C), similar to the GMPCPP tubulin data. As described above, we then fit the mean-squared-displacement data to the diffusion-drift model for each concentration (Fig. 6D). Using this method, the tubulin subunit on rate (*k _{on,MT}^{*}*) and the off rate (

To assess how our *in vitro* data compare with microtubule growth *in vivo* we measured microtubule growth at 0.5 sec intervals near to the periphery of live LLC-PK1α cells, as previously described (Demchouk et al., 2011) (Fig. 6F) (tubulin concentration estimate 5–15 μM, see Fig. S2). To assess the measurement error, we also measured the length of microtubules in fixed LLC-PK1α cells at 0.5 sec intervals. As shown in Fig. 6G, there is high variability in live-cell microtubule growth rates, consistent with previous reports (Shelden and Wadsworth, 1993). With our high-resolution tracking accuracy (~36 nm) (Demchouk et al., 2011), we can now readily detect nanoscale changes in microtubule length, including periods where the growth rate abruptly shifts (Fig. 6G). That the net rate is commonly ~600 s^{−1}, at least transiently, strongly suggests that the *in vivo* on-off dynamics are on the scale of 1 kHz. The *in vivo* variance in growth rate at 0.5 sec intervals is ~1600 nm^{2} (after subtracting the fixed microtubule variance), which is nearly 5-fold larger than the *in vitro* growth rate variance for GTP-tubulin microtubules at ~5 μM tubulin (downsampled to 0.5 s intervals). Thus, the kinetic rate constants *in vivo* are at least as large as we estimate for *in vitro* assembly.

The finding that tubulin association and dissociation rates are an order of magnitude higher than has been previously recognized, and nearly equal to each other at all concentrations, significantly changes the conceptualization of microtubule growth (Fig. 7). Consequently, our thinking about regulation of assembly by MAPs and therapeutic drugs (such as taxol), also needs reassessment.

The near equality of on and off rates implies that infrequent and/or weak suppression of the off-rate should lead to dramatic shifts in net assembly for single microtubules *in vivo* (Fig 6G), and also implies that a slight tipping of the balance by a single MAP at the microtubule tip might significantly affect catastrophe and rescue *in vivo*. For example, weak suppression of the off rate by a stabilizing MAP would lead to a rapid and significant increase in the microtubule GTP-cap size, and a corresponding substantial decrease in catastrophe frequency. Furthermore, a much larger off rate allows a substantial range of growth regulation to be exerted purely through suppression of tubulin loss from the microtubule tip. This has important implications for explaining how a MAP such as XMAP215/chTOG, which is over-expressed in colon and hepatic tumors, promotes assembly. Growth rates *in vivo,* and with this MAP *in vitro* are commonly 2–10 fold higher than with pure tubulin at ~10 μM GTP-tubulin (Cassimeris et al., 1988; Charrasse et al., 1998; Pryer et al., 1992; Rusan et al., 2001; Shelden and Wadsworth, 1993; Vasquez et al., 1994). In the context of the 1D model, a 10-fold increase in growth rate can only be achieved by increasing the on rate because, for example, suppressing tubulin dissociation entirely (i.e. decreasing the off rate from 10 s^{−1} to 0 s^{−1}) would only achieve a 25% increase in net assembly rate. However, in the 2D model, a ten-fold increase in net growth rate by a MAP can be achieved simply by suppressing the off rate by a factor of two (see Supplemental Figure S4). Such a mechanism of growth enhancement is similar to that proposed by Brouhard et al. (2008) and VanBuren et al. (2002) to account for the 5-fold acceleration of growth rate by XMAP215.

Conversely, even a weakly destabilizing MAP would be expected to tip the balance strongly toward catastrophe. The best understood catastrophe-promoting MAP, MCAK, seems to at least partly fulfill this description. The catalytic effect of MCAK at saturation is to increase the tubulin off-rate by 100-fold (Hunter et al., 2003), which in the 2D model corresponds to a modest free energy increase of ΔG°=+ln(100)=+4.6 k_{B}T. Interestingly, this energy turns out to be nearly equal in magnitude to our estimate of the lateral bond energy in the 2D model (−4.5–5.0 k_{B}T; Supplementary Table S2). In addition, MCAK’s catalytic effect on the off-rate is only weakly cooperative, so that even a single molecule is able to affect the off-rate (Hunter et al., 2003). The *in vivo* expression level of MCAK is crucial to maintenance of proper ploidy, which is important to avoid progression into a cancerous phenotype (Bakhoum et al., 2009a; Bakhoum et al., 2009b).

An in vivo example of a weakly destabilizing MAP that tips the balance strongly toward catastrophe is the yeast kinesin-5 motor Cin8 (and to a lesser extent, Kip1; (Gardner et al., 2008)). During yeast mitosis, Cin8 promotes microtubule catastrophe in a length-dependent manner to mediate chromosome congression. Interestingly, the motor:microtubule ratio in vivo is very low, ~50 motors for ~40 microtubules in the yeast spindle, or nearly 1:1 overall ((Gardner et al., 2008; Winey et al., 1995)). Such a low ratio implies that the presence of even a single Cin8 motor at the plus end has a significant impact on the microtubule assembly state. From modeling studies, it was estimated that the presence of a single motor at the plus end was sufficient to promote catastrophe ~10-fold (Gardner et al., 2008). Given that metaphase yeast kinetochore microtubules contain on average about 600 tubulin dimers, the effect of the motor can be regarded as even more potent, with about one motor per ~600 tubulin dimers. Based on our present results, we suggest that such potency at low stoichiometry could be achieved with modest enhancement of the off rate, and/or suppression of the on rate, by one or a few motors.

Hypersensitivity of assembly can also potentially facilitate rapid cell-wide microtubule array reorganization. A prime example of a dramatic microtubule reorganization process is Drosophila embryo mitosis, where a mitotic spindle forms, segregates a genome, and subsequently disassembles in a matter of minutes (Stiffler et al., 1999). Such dramatic transitions *in vivo* occur under conditions of fixed total tubulin concentration, which serves to buffer perturbations to net assembly (Mitchison and Kirschner, 1987). For example, if an assembly-promoting MAP is upregulated, then the initial response will be potent. However, as net assembly is promoted, the free tubulin concentration will drop, which will mitigate the assembly-promoting activity. Thus, constant total tubulin concentration will act to buffer and mitigate perturbations to microtubule dynamics, and confer a degree of robustness and homeostasis on the microtubule cytoskeleton. To overcome the built-in cell-level buffering effect and perform highly dynamic operations such as mitosis, the microtubule array needs to be hypersensitive.

To summarize, it will be important to revisit current models for microtubule regulation *in vivo* via MAPs and therapeutic drugs in light of our findings. In particular, because both on and off rates are rapid and nearly equal, MAPs that promote net microtubule assembly or disassembly could work through a modest changes in tubulin subunit kinetics (Brouhard et al., 2008; Murphy et al., 1977; Pryer et al., 1992; VanBuren et al., 2005; VanBuren et al., 2002).

More generally, our results identify a fundamental theoretical limitation to the 1D model of Oosawa, which has been used for nearly 40 years to interpret kinetic self-assembly data of microtubules, F-actin, viruses, and amyloid aggregates (Cannon et al., 2004; Collins et al., 2004). It will be important to revisit earlier 1D model-based studies to confirm that the assumption of constant subunit dissociation rates holds for self-assembled polymers containing both longitudinal and lateral interactions, since presently established kinetic estimates for these biological assemblies may need to be revised as we have found with microtubules.

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The authors thank Dominique Seetapun and Dr. Chris Gell for technical assistance. We thank Aleksey Demchouk for providing the microtubule tip tracking code. The LLC-PK1α strain was kindly provided by Dr. Pat Wadsworth. This work was supported by NIH GM076177 to A.J.H. and D.J.O, NIH GM071522 to D.J.O and NSF 615568 to D.J.O. M.K.G. was supported in part by a Whitaker International Scholar Fellowship.

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