Dr. David Morton's lab at OHSU was involved in understanding how hypoxic conditions are detected using fruitfly grubs. The grubs were placed into a clear box with a controlled concentration of oxygen and were monitored until they stopped feeding and started their escape behavior. The time at which each grub began its escape behavior was recorded. The experiment ended either when all grubs were escaping, or after 15 minutes, whichever came first.
The lab has expertise in in the genetics and biochemistry of this process, but not in probabilistic modeling or dynamics. Their initial data analysis used differences in mean escape time to determine whether certain mutants or treatments caused changes in escape behavior.
The time to escape histograms had a characteristic shape, with a long tail to the right. In certain mutants many maggots simply never escaped in the 15 minutes alloted, leading to censoring of the results. In talking to the lab members, it became clear that a causal mechanism to explain the histogram shape would be useful.
To respond to this need, we developed a coupled set of differential equations, one of which described the build-up of a hypothesized active signalling molecule, and the other of which described the rate of change in the total probability of escape as a function of the current level of the signal molecule. Solving these equations led to an analytic expression for the CDF and PDF of the time-to-escape data as a function of various rate constants related to the genetic potency of different genotypes. Now the entire dataset could be used to fit the latent potency variables and determine how each mutant or treatment affected the build-up of the internal signal. Also, after fitting to determine the coefficients, predictions could be made for dynamically changing oxygen levels, these predictions, if observed in real dynamic experiments would provide a sensitive test for the accuracy of the internal mechanistic model.
By incorporating knowledge about chemical and physical processes together with mathematical tools from differential equations and probability theory, the Biologists were able to use this "mathematical microscope" to infer how certain biochemical processes differed between different conditions. Instead of using simple differences in means the entire shape of the histogram became a sensitive instrument for more accurately determining the unknown values associated with a clear mechanistic causal mechanism.