Homework 10.2: Caulobacter growth (55 pts)
In this problem, we will study the growth and division of Caulobacter crescentus over time. The lab of Norbert Scherer at the University of Chicago acquired these data and published the work in PNAS, which you can download here.
The clever experimental set-up allows imaging of single dividing cells in conditions that are identical through time. This is accomplished by taking advantage of a unique morphological feature of Caulobacter. The mother cell is adherent to the a surface through its stalk. Upon division, one of the daughter cells does not have a stalk and is mobile. The system is part of a microfluidic device that gives a constant flow. So, every time a mother cell divides, the un-stalked daughter cell gets washed away. In such a way, the dividing cells are never in a crowded environment and the buffer is always fresh. This also allows for easier segmentation.
We define a growth event as a time period during which a bacterium grows before dividing. After a division, a new growth event begins. Based on image data kindly provided by Charlie Wright and Sri Iyer-Biswas, I processed the images to get a set of growth events. The data are available here: https://s3.amazonaws.com/bebi103.caltech.edu/data/caulobacter_growth_events.csv.
It is well known that in ideal conditions, bacteria grow exponentially. One bacterium divides to form two, those two divide to form four, those four divide to form eight, and so on. But what about each individual growth event? How does a single cell grow?
a) One theoretical model we will consider is that each the growth of an individual bacterium 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?