Homework 9.2: Caulobacter growth part 3 (55 pts)¶
Be sure to read problem 9.3 before beginning this problem.
In problem 5.2, you analyzed images of single Caulobacter cells growing and dividing over time, and in problem 6.1, you processed the time series data you acquired to separate the measurements into growth events. Reread that problem statement to refresh yourself about the experiment. You will use this data set in this problem. Please be sure it is committed in your repository as a CSV file so the TAs have access to it while grading.
If you were not able to complete problem 6.1, you may download a CSV file here. If you got any sort of reasonable segmentation and growth event determination results, you should use your own results and not those I have provided.
a) One theoretical model we will consider is that each the growth of individual bacteria is linear. That is,
\begin{align} a(t) = a_0(1 + k t), \end{align}
where \(a\) denotes the area observed in the microscope images. An alternative model is that each individual bacterium grows exponentially, such that
\begin{align} a(t) = a_0\mathrm{e}^{kt}. \end{align}
Considering each growth event to be independent of all others, develop a generative model for each of the two theoretical models. Comment on any considerations you made and concerns you may have with your modeling procedures.
b) Using this model, perform parameter estimates for \(a_0\) and \(k\) for each growth event separately. Think about how to display your results graphically and make informative graphics.
c) Compare the two models. Do you think growth is exponential or linear?