The 1D Model Predicts Fewer Large Shortening Events at Higher Tubulin Concentrations

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 (left), this model can be characterized by the number of tubulin subunit arrivals to the microtubule tip (λ

_{+}) in a time interval Δt, which is determined by the on rate constant (

*k*_{on,MT}) and the free tubulin concentration ([Tub]). In this model, the number of departure events from the tip (λ

_{−}) in a time interval Δt depends only on a fixed off rate constant (

*k*_{off,MT}), and thus remains constant regardless of the free tubulin concentration.

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 (, 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 (, 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 (, 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.

Microtubule Shortening Events Increase at Higher Tubulin Concentrations in TIRF Experiments

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 (). 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}, ) (

Demchouk et al., 2011).

As expected, the rate of net assembly increased as the GMPCPP-tubulin concentration increased (). We found that microtubule growth trajectories in dilution experiments and at 0.8 μM GMPCPP-tubulin (, green and red, respectively) showed steady growth or shortening with small length fluctuations, while microtubule growth trajectories at 1.5 μM GMPCPP-tubulin (, 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 ().

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, ). 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.

Microtubule Shortening Events Increase at Higher Tubulin Concentrations in Laser Tweezers Experiments

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 (). We observed variability in microtubule growth at three different GMPCPP-tubulin concentrations, with variability increasing substantially at higher tubulin concentrations (). 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 (). 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.

Lateral and Longitudinal Bonds Influence the Concentration-Dependent Subunit Dissociation Rates

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

_{off,PF} will also change. Since the number of lateral bonds between subunits at the tip depends on the tip structure, k

_{off,PF} averaged over time and all protofilaments yields k

_{off,MT}. This value for k

_{off,MT} could increase at higher tubulin concentrations, consistent with our experimental observations ( and ). Therefore, we asked whether the 2D model predicts an increase in k

_{off,MT} at higher tubulin concentrations.

The 2D simulation accounts for (1) subunit arrivals at the tip of the microtubule, and (2) subunit departures from the tip (). For arrivals, the subunit arrival rate at the tip of a protofilament (

*k*_{on,PF}^{*}) is given by:

Where *k*_{on,PF} is the on rate constant per individual protofilament (μM^{−1}s^{−1}, such that *k*_{on,MT}=13**k*_{on,PF}), *k*_{on,PF}^{*} is the arrival rate per individual protofilament (s^{−1}, similarly *k*_{on,MT}^{*}=13**k*_{on,PF}^{*}), and [Tub] is the free tubulin subunit concentration. This expression for tubulin subunit arrival rate to each protofilament is identical to the 1D model. However, to account for energetic differences between tubulin subunits at the microtubule tip, in the 2D model the subunit departure rate from a given protofilament, *k*_{off,PF}, depends on the equilibrium constant, where

Therefore, by rearrangement:

where

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

^{−1}),

*k*_{B} is Boltzmann’s constant,

*T* is absolute temperature, and

*ΔG*^{0}_{tota}_{l} is the total free energy of the stabilizing bonds on a specific subunit, given by

Tip subunits will (by definition) have one longitudinal bond (), and therefore the total value for *ΔG*^{0*}_{Longitudinal} will be identical for all subunits at the tip. However, the *lateral* bond energy (*ΔG*^{0}_{Lateral}) will vary, such that subunits with two lateral neighbors will have a two-fold more negative value for *ΔG*^{0}_{Lateral} than subunits with only one lateral neighbor.

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 (, blue) results from the relatively small difference between a large subunit arrival rate that increases with free tubulin concentration (, green), and a large subunit departure rate that

*also* increases with increasing free tubulin concentration (, 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 (), which must on average be shifted toward less stable configurations as the tubulin concentration increases. For example, tubulin subunits with two lateral neighbors (, 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 (, 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

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)},

*k*_{off,PF}^{(1)}, and

*k*_{off,PF}^{(0)} are the individual protofilament dissociation rate constants for a tubulin subunit with two, one, and zero lateral neighbors, respectively (

Stukalin and Kolomeisky, 2004).

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 (, 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 (, right, blue), and a corresponding decrease in the probability of relatively stable tubulin subunits having 2 lateral neighbors (, 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.

The 1D and 2D Models Predict Different Kinetic Rate Constants for Microtubule Assembly

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

^{−1} s

^{−1} (range 1–9 μM

^{−1} s

^{−1}) and average dissociation rates of

*k*_{off,MT} ≈ 15 s

^{−1} (range 0.1–44 s

^{−1}).

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 (). From the slope and intercept, the 1D model microtubule association rate constant for GMPCPP-Tubulin was

*k*_{on,MT} = 5.1 μM

^{−1}s

^{−1}, and the dissociation rate constant was

*k*_{off,MT} = 3.9 s

^{−1}. Thus, our observed growth velocity dependence on tubulin concentration was within the range previously observed and yields similar 1D model rate constants to those previously reported (

Table S1).

The 2D model also predicts a linear relationship between microtubule growth velocity and tubulin concentration (). However, the 2D model predicts that the average dissociation rate increases approximately linearly with free tubulin concentration due to evolving tip structures (). 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 ( and ). 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

^{−1}s

^{−1} with a 2D model, see

Supplemental Fig. S1 and

Table S2) (). This 2D model association rate constant, when considered on a per protofilament basis (

*k*_{on,PF}=

*k*_{on,MT}/13 = 4 μM

^{−1} s

^{−1}), is consistent with theoretical predictions for rotational-translational diffusion-limited protein-protein association reactions (

Northrup and Erickson, 1992) and is similar to that estimated for F-actin (2 protofilament) self-assembly on a per protofilament basis,

*k*_{on,filament}= 11.6 μM

^{−1}s

^{−1} or

*k*_{on,PF}= 5.8 μM

^{−1}s

^{−1} (

Pollard, 1986).

The Rapid On-Off Kinetics Required in the 2D Model accurately predict the Variability in Growth Rate

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 (). 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:

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

*D*_{p} is the effective diffusion coefficient for the microtubule polymerization, which provides a quantitative measure of the variability in microtubule growth increments. Finally, σ

^{2} is the experimental measurement noise.

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

*D*_{p} (constrained by mean growth rate data to

*k*_{on,MT} =5.1 μM

^{−1} s

^{−1} and

*k*_{off,MT}=3.9 s

^{−1}, as in ) is an order of magnitude lower than the experimental diffusion coefficient (). In contrast, the 2D model, constrained by the mean growth rate experiments to a 10-fold higher association rate constant than in the 1D model (

*k*_{on,MT}=52 μM

^{−1} s

^{−1}), predicts a diffusion coefficient

*D*_{p}, that is similar to the experimental data (, model parameters as in

Supplementary Table S2). Thus, microtubule assembly is too variable to be consistent with the 1D model, but is consistent with the 2D model.

Experiments Confirm Rapid Tubulin On-Off Kinetics as Predicted by the 2D Model

The tubulin subunit on rate (*k*_{on,MT}^{*}) and the off rate (*k*_{off,MT}) from the microtubule tip can be directly estimated from the mean-squared displacement versus time data (). The diffusion coefficient, D_{p}, relates the microtubule length mean squared displacement due to diffusion alone, <Δl_{D}^{2}>, to the time interval, Δt, via 2

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,

In addition, by definition,

Thus, through these two

equations (9) and

(10), the two unknowns,

*k*_{on,MT} and

*k*_{off,MT}, can be directly calculated at a given tubulin concentration using the experimentally estimated values of

*D*_{p} and

*v*_{g}.

To calculate the values for *k*_{on,MT} and *k*_{off,MT} from the experimental data, and to confirm concentration-dependent tubulin subunit dissociation rates, we completed the mean-squared

displacement analysis as described above for a range of free tubulin concentrations in the TIRF analysis (). Detailed results are summarized in

Table S3 and shown in . 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 (). 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

^{−1} (green), and a large off rate,

*k*_{off,MT} ≈ 75 s

^{−1} (red). In addition, the dilution experimental results demonstrate that the off rate collapses to very near the 1D model estimate when microtubule tips are blunt (1D model

*k*_{off,MT} ≈4 s

^{−1}, estimated experimental

*k*_{off,MT} in dilution experiment, ≈8 s

^{−1}).

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 , the 1D model, constrained by mean growth rate data to

*k*_{on,MT} =5.1 μM

^{−1} s

^{−1} and

*k*_{on,MT} =3.9 s

^{−1} (), predicts a microtubule growth rate variance in the laser tweezer assay of σ

^{2}_{Assembly} = 1.0 nm

^{2} at 0.5 μM tubulin (

*k*_{on,MT} ^{*}=2.5 s

^{−1}) and σ

^{2}_{Assembly} = 1.7 nm

^{2} at 1.5 μM tubulin (

*k*_{on,MT} ^{*}=7.6 s

^{−1}). Our laser tweezers assay has a measurement variance due to Brownian motion of the microtubule-bead complex of σ

^{2}_{Thermal} =(3.5 nm)

^{2} = 12 nm

^{2} (at 10 Hz) (

Schek et al., 2007). Thus, the 1D model predicts that the experimentally observed variance, σ

^{2}_{Observed} = σ

^{2}_{Assembly} + σ

^{2}_{Thermal} = 1 nm

^{2} + 12 nm

^{2} = 13 nm

^{2}, will be dominated by the thermal measurement noise and so will not depend on free subunit concentration (, orange line). Contrary to this expectation, the microtubule growth increment distribution from the laser-tweezer assay increases with the tubulin concentration, and is inconsistent with the 1D model at 1.0 μM (p<10

^{−8}, Two-tailed F-test) and 1.5 μM (p<10

^{−8}, Two-tailed F-test).

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 (, 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.

Experiments confirm that Microtubule Tips Are More Tapered as Tubulin Concentrations Increase

In shifting from low tubulin concentration (i.e., favoring 2 lateral neighbors, ) to high concentration (i.e., favoring 1 lateral neighbor, ), 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 (). 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 (, 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 (, 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}) () and the standard deviation of protofilament lengths (σ

_{tip}) () (

Demchouk et al., 2011). Specifically, we expect that tips with a smaller σ

_{tip} have relatively “blunt” tips (, left), while tips with a larger σ

_{tip} have more tapered tips (, 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 (). 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) (). 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.

Rapid GTP-tubulin on-off Kinetics in vitro at physiological concentrations

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

*k*_{off,MT} will increase at higher physiologic GTP-tubulin concentrations (). According to the 2D model, the increment variance will then increase proportionate to the tubulin concentration. Therefore, in our previous study with ~5 μM GTP-tubulin in an

*in vitro* laser tweezers microtubule assembly assay (

Schek et al., 2007) we should have observed a variance of ~3-fold larger than in our present 1.5 μM GMPCPP-tubulin experiments (5 μM/1.5 μM ~3). As shown in , the variance is ~3-fold higher at ~3-fold higher tubulin concentration (5 μM GTP-tubulin vs. 1.5 μM GMPCPP tubulin both sampled at 0.5 s intervals).

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 (), 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 (). Using this method, the tubulin subunit on rate (

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

*k*_{off,MT}) from the microtubule tip can be directly estimated from the data (,

Table S3). Consistent with the GMPCPP tubulin experiments, the experimentally estimated tubulin dissociation rate increases substantially as the GTP-tubulin concentration is increased, and the net experimental microtubule assembly rate (blue) represents the small difference between a large experimental on rate (green) and a large off rate (red), similar to the 2D model prediction (Compare and ).

In Vivo Microtubule Growth Variability occurs with Rapid Tubulin Subunit On-Off Kinetics

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) () (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 , 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 (). 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.