# 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.

Such models can easily take 15 years to build, at least that is how long Phil Holmes — the second speaker on the second day of the 2nd workshop on Natural Algorithms and the Sciences — and his collaborators have spent building a model of the neuromechanics of running cockroaches. Their model is represented by hundreds of differential equations,in 1998 it started with just the joints of the cockroaches legs, and now include variables at the level of Hodgkin–Huxley model for neuron firing rates. Everything is coupled to help model the dynamics of a running cockroach, or a cockroach that’s been tripped, or had a mini-cannon fired from its back to destabilize it. Although they reached a peak in model complexity around the mid 2000s, and have been able to simplify some of the 100s of equations while still retaining exact numerical predictions, Holmes stresses that:

[We] still don’t understand non-linear dynamics beyond linearizations or general symmetries … [and] it is not clear which details matters a priori.

But even these symmetries, which an abstract or heuristic modeler would just assume, need to be studied in a close tandem of math and experiment if we want to be relevant to the real world. Eduardo Sontag — the second to last speaker of the workshop — looked at general properties of non-steady state response to inputs in cells. He was inspired by the classic Weber-Fechner law of psychophysics that just-noticile difference between two stimuli is proportional to the magnitude of the signal. For example, if I make you hold up a 2 kg dumbbell and start adding weight to it gradually, asking you to tell me when you notice a difference, you might not feel a difference until I add 30 grams. However, if instead you were holding a 20 kg dumbbell then you wouldn’t notice the difference until I added 300 grams. This is a sort of scale-invariance in human perception, do cells have something similar?

In cell response curves, we are interested in two features: the onset-delay and magnitude of response. Thus, there are three types of responses possible to two scaled signals, as show in the above figure. Suppose that our cell is observing a steady signal (a) of magnitude 10 that suddenly jumps to 30, or a steady signal (b) of magnitude 5 that suddenly jumps to 15 (note the same fold increase). If our cell is merely adaptive then its’ response to (a) will quickly spike to some high level and the slowly decay to base levels, and to (b) it will respond slightly slower at to a lower spike before decaying to base levels. A system that follows the cell variant of the Weber-Fechner law will respond to the same magnitude in both (a) and (b) but the on-set of response will be slightly slower for the smaller input (b). A perfectly scale-invariant system will respond to both (a) and (b) identically.

Sontag combined abstraction and insilication in his modeling of cell responses. With Shoval et al. (2010), he proved a general theorem on how to detect fold-response/scale-invariance in your giant ODE model of a cell by checking if an alternative (but smaller) system of equations has a solution. The authors then applied their theorem to specific cells to predict scale-invariance, and were able to do so with such precision that their predictions were confirmed by experiment a few weeks later. For Sontag, however, the biggest excitement is not the numerous times that his models were confirmed by experiment, but the few times they were falsified. It is incredibly rewarding when mathematicians can build models that are realistic enough that they can hope to be falsified. I definitely don’t expect to ever build experimentally falsifiable models, maybe this is why I like to think of science as a narrative and exploration of understanding and not the generation of prediction that engineers prefer.

Note: this is the fourth blogpost of a series (1, 2, and 3) on the recent workshop on Natural Algorithms and the Sciences at Princeton’s Center for Computational Intractability.
Shoval O, Goentoro L, Hart Y, Mayo A, Sontag E, & Alon U (2010). Fold-change detection and scalar symmetry of sensory input fields. Proceedings of the National Academy of Sciences of the United States of America, 107 (36), 15995-6000 PMID: 20729472