Homework 5.2: Microtubule catastrophe (75 pts)
In this problem, we use data from Gardner, Zanic, et al., Depolymerizing kinesins Kip3 and MCAK shape cellular microtubule architecture by differential control of catastrophe, Cell, 147, 1092-1103, 2011. The authors investigated the dynamics of microtubule catastrophe, the switching of a microtubule from a growing to a shrinking state. In particular, they were interested in the time between the start of growth of a microtubule and the catastrophe event. They monitored microtubules in a single-molecule TIRF assay by using tubulin (the monomer that comprises a microtubule) that was labeled with a fluorescent marker. As a control to make sure that fluorescent labels and exposure to laser light did not affect the microtubule dynamics, they performed a similar experiment using differential interference contrast (DIC) microscopy. They measured the time until catastrophe with labeled and unlabeled tubulin. We will carefully analyze the data and make some conclusions about the processes underlying microtubule catastrophe.
In the file gardner_mt_catastrophe_only_tubulin.csv (which you can download here; the file is already present in the data folder if you are using AWS), we have observed catastrophe times of microtubules with different concentrations of tubulin. We will consider the experiment run with a tubulin concentration of 12 µM. So, our data set consists of a set of measurements of the amount of time to
catastrophe. We will consider four models for microtubule catastrophe.
Model 1: The time to catastrophe is Exponentially distributed.
Model 2: The time to catastrophe is Gamma distributed.
Model 3: The time to catastrophe is Weibull distributed.
Model 4: Catastrophe happens upon the arrival of the second of two Poisson processes, each of which arrive at different rates.
Note that these descriptions are for the likelihood; we have not specified priors.
a) Describe the first three models as stories. Give physical descriptions of the meanings of their parameters. Describe how these models are related to each other. Recall that we worked out the likelihood for model 4 last term. How is it related to the other three models?
b) Build full Bayesian generative models for each of the models. Be sure to perform prior predictive checks. After building the model, perform parameter estimates, and then do posterior predictive checks.
c) Based on the graphical analysis, do you prefer one model versus another? We will be discussing quantitative model comparison in coming weeks to enable to more detailed analysis.