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Microtubule Catastrophe

Amanda Li, Maggie Sui, Cindy Cao

Abstract

Microtubule catastrophe is an important cellular event that allows rapid reorganization of cellular structure. Tubulin monomers that make up microtubules dissociate rapidly in sudden events termed “catastrophe”. These events are crucial for microtubule function. Despite the importance of microtubule catastrophe, the exact mechanism by which catastrophe occurs is not well understood. To gain an understanding of the mechanism, we used data collected from Garner et. al to model microtubule catastrophe in two distinct models. The first model is a gamma distribution model that assumes that microtubule catastrophe can be modeled as N poisson processes with the same rate of arrival; whereas our second model assumes microtubule catastrophe can be modeled as 2 successive poisson processes with different rates of arrival. We found that microtubule catastrophe is best modeled as a gamma distribution with rate parameter \(\beta \approx\) steps / second and increasing step parameter \(\alpha\) for increasing tubulin concentration. This suggests that at higher tubulin concentrations, more steps of the same rate are required for microtubule catastrophe to occur; that is, microtubule catastrophe takes longer at higher tubulin concentrations.

Microtubule Catastrophe in Labeled and Unlabeled Tubulin

Garner et. al used fluorescent markers to monitor microtubules in order to measure time to catastrophe. As a control to ensure that fluorescent labeling did not affect catastrophe times, they also performed a similar experiment using differential interference contrast (DIC) microscopy. The first dataset we worked with includes measured times to microtubule catastrophe for both labeled and unlabeled tubulin. We first visualized the ECDFs of the data:

We observe that there is no obvious difference between the ECDFs for labeled and unlabeled tubulin and confirm that the 95% confidence intervals of the times to catastrophe for labeled and unlabeled tubulin overlap. To supplement our graphical analysis, we computed bootstrap confidence intervals of the plug-in estimate for mean time to catastrophe (i.e. the sample mean) for labeled and unlabeled tubulin and found heavy overlap. Finally, we performed a permutation hypothesis test with the difference of means as our test statistic to test whether the distribution of time to catastrophe is the same for labeled and unlabeled tubulin. If so, then the means of the distributions should be the same. Since the permutation test concatenates and shuffles the data as if the labels are irrelevant, it is a natural choice for testing whether two distributions are the same. We report a p-value of approximately $0.77$; that is, $77\%$ of the permuted samples had at least the difference between means present in the original data set. Combined, our results suggest that labeling the tubulin does not significantly affect the microtubule dynamics. Thus, in the remaining analysis, we only work with labeled tubulin.

Simulating Distributions

In their paper, Gardner et al. paper assumed that the microtubule catatrophe times are Gamma distributed. That is, we have a multi-step process with the same rate per step. As an alternative model, we assert that two biochemical processes with different rates must occur in succession to trigger catastrophe. We model the two processes as independent Poisson processes with arrivals rates of $\beta_1$ and $\beta_2$, respectively, where $\beta_1 \neq \beta_2$. Then, we simulated the experiment according to this model.

From the resulting plot, we observe that increasing $\beta_2$ decreases time to catastrophe if we hold $\beta_1$ (and consequently the time until the first biochemical process) constant. This makes sense because increasing $\beta_2$ decreases $1/\beta_2$, which is the mean time between the first and second biochemical processes. The PDF for this model, which we will refer to hereafter as the “custom” model, is: \begin{align} f(t;\beta_1, \beta_2) = \frac{\beta_1 \beta_2}{\beta_2 - \beta_1}\left(\mathrm{e}^{-\beta_1 t} - \mathrm{e}^{-\beta_2 t}\right) \end{align} The analytical CDF of the time to catastrophe (t) is given as: \begin{align} F(t; \beta_1, \beta_2) = \frac{\beta_1 \beta_2}{\beta_2-\beta_1}\left[ \frac{1}{\beta_1}\left(1-\mathrm{e}^{- \beta_1 t}\right)- \frac{1}{\beta_2}\left(1-\mathrm{e}^{-\beta_2 t}\right) \right]. \end{align} This was overlayed with our simulated ECDF with $\beta_1 = 1, \beta_2 = 2$.

It looks like our analytical CDF and simulated ECDF match fairly well. This suggests that the custom model could potentially fit the data well.

MLEs of Microtubule Catastrophe Model

We now have two potential models for microtubule catastrophe. To further analyze them both, we obtained maximum likelihood estimates as well as confidence intervals for $\beta_1$ and $\beta_2$ in the two successive Poisson process (custom) model as well as $\alpha$ and $\beta$ for the Gamma model. We observe that modeling the times to catastrophe as Gamma-distributed means that we find the MLES of the number of steps until catastrophe to be about $\alpha = $2−3 steps 95% of the time, and MLE of the rate to be about $\beta = $0.0045−0.0068 steps per second 95% of the time. This means time to catastrophe would be about 300−660 seconds, which matches our data fairly well. Similarly, we obtained confidence intervals of the MLEs for $\beta_1$ and $\beta_2$ for the custom model and have plotted these values below to compare rates between the two models.

Compared to the Gamma model, which models catastrophe as a $2-3$ step process with the same rate, our custom model models catastrophe as a $2$-step process where each step has a different rate. One of our parameters ($\beta_1$) is smaller than the $\beta$ from the Gamma model, while the other rate is about the same. This is expected because our custom model has fewer steps, so on average each step should take more time and the rate should be slower.

Model Comparison

So that still leaves the question: which is the better model? For a graphical model assessment, we drew samples from the generative distributions parametrized by the MLEs and created Q-Q plots for both models using the samples.

It appears that for the Gamma model, the generative quantiles match the observed quantiles well for low times to catastrophe but deviate from the observed quantiles more for higher times to catastrophe. With our custom model, we see deviations between the observed and generative quantiles at both low and medium values of time to catastrophe. We will also produce predictive ECDFs for both models, which will allow us to see how data generated from the distributions parametrized by our MLEs would look.

We see that for the Gamma model, the ECDF of the data matches the 95% confidence interval of the predictive ECDFs well for lower times to catastrophe. Again, there is some deviation for higher times, but the deviation does not appear severe. We observe that the deviation of the measured vs predictive ECDFs is much more severe for the custom model than the Gamma model. We can confirm these observations by examining the differences between the predictive and measured ECDFs:

From our graphical model assessment, it seems clear that the Gamma model is preferred. We also examined the Akaike information criterion, or AIC, for model selection. Given two models, the one with the lesser AIC is likely closer to the true generative distribution. The AIC for a model is given by \begin{align} AIC = −2l(\theta^{MLE};data)+2p \end{align} where $\theta^{MLE}$ represents the MLEs of the parameters $\theta$ of the model, $l(\theta;data)$ is the log-likelihood of the model, and $p$ is the number of free parameters in the model. The Akaike weight of model $i$ among many models is \begin{align} w_i = \frac{e^{−(AIC_i−AIC_{max})/2}}{\sum_j e^{−(AIC_j−AIC_{max})/2}} \end{align} Computing the Akaike weights of the two models, we find that $w_{gamma} \approx 1$ and $w_{custom} \approx 0$. Thus, it is clear from both the graphical model assessment and the AIC that the Gamma model is preferred. We use the Gamma model in the remaining analysis.

Effects of Tubulin Concentration

We then investigate the effect of tubulin concentrations on microtubule catastrophe. Using the Gamma model, we obtained parameter estimates for various tubulin concentrations ranging from $7$ to $14 \mu M $and plotted the MLEs and confidence intervals for α and $\beta$ for each concentration:

We observe that in general, higher tubulin concentration results in a larger $\alpha$ value (more steps in the multi-step process). In particular, there is almost no overlap between the confidence intervals for $7 \mu M$ and $14 \mu M$. This makes sense given that microtubules polymerize faster at higher concentrations. We hypothesize that the larger number of steps for higher tubulin concentration occurs because more tubulin have to come off during catastrophe when there is higher concentration. There does not appear to be a significant difference between the rates $\beta$ as tubulin concentration increases; the confidence intervals for all concentrations overlap a lot. So for higher concentrations, the ratio $\alpha / \beta$ is generally larger, meaning time to catastrophe takes longer.

Summary

We confirmed that GFP fluorescent tagged microtubule does not significantly impact the rate to catastrophe and thus we can measure microtubule catastrophe with GFP labeling. Then, we compared two models for microtubule distribution and found that time between microtubule catastrophe events can be best modeled as a gamma distribution. In addition, as the concentration of tubulin increases, the number of events between microtubule catastrophes increase as well, though all events still occur at the approximately the same rate. The Github repo for this site can be found here.

Acknowledgments

This work was done for BE/Bi103a 2020 at Caltech. We thank Garner et. al for their data, Griffin Chure for this website template, and Justin Bois and the BE/Bi103a staff for making this course possible.