Mathematical models of running cockroaches and scale-invariance in cells

I often think of myself as an applied mathematician — I even spent a year of grad school in a math department (although it was “Combinatorics and Optimization” not “Applied Math”) — but when the giant systems of ODEs or PDEs come a-knocking, I run and hide. I confine myself to abstract or heuristic models, and for the questions I tend to ask these are the models people often find interesting. These models are built to be as simple as possible, and often are used to prove a general statement (if it is an abstraction) that will hold for any more detailed model, or to serve as an intuition pump (if it is a heuristic). If there are more than a handful of coupled equations or if a simple symmetry (or Mathematica) doesn’t solve them, then I call it quits or simplify.

However, there is a third type of model — an insilication. These mathematical or computational models are so realistic that their parameters can be set directly by experimental observations (not merely optimized based on model output) and the outputs they generate can be directly tested against experiment or used to generate quantitative predictions. These are the domain of mathematical engineers and applied mathematicians, and some — usually experimentalists, but sometimes even computer scientists — consider these to be the only real scientific models. As a prototypical example of an insilication, think of the folks at NASA numerically solving the gravitational model of our solar system to figure out how to aim the next mission to Mars. These models often have dozens or hundreds (or sometimes more!) coupled equations, where every part is known to perform to an extreme level of accuracy.