This semester I’ve helped teach a biology-meets-math sort of a course. There’s been some debate among my labmates about what such a course ought to include and about the merits of math in biology more generally. It’s true, I’ve never explicitly used control theory to calculate how much liquid I should pipette into my experiment. And I very seldom think about the principle of detailed balance when I peer into a microscope. So why would we take a room full of experimentalists and teach them about dynamical systems on abstract graphs—how could this possible improve their biology?
My students were wondering the same thing when they showed up to the first section this semester. That week we were discussing a paper by van Oudenaarden on yeast’s ability to regulate and maintain its internal osmotic pressure despite living in a varying external environment. The technique they used was indirect but powerful. Assault the cell with pulses of carefully measured salt concentrations and then watch as a judiciously chosen protein read-out accumulate in the nucleus in response. A mess of genes and proteins and other factors have been associated with osmoregulation and the trick was to pick out which of these players are most important and when.
By matching their salt pulse inputs with fluorescently labeled protein outputs, the researchers were able to come up with a simplified model of the cell’s vastly complicated internal logic. And while their model is a cartoon of reality, it was extremely good at predicting previously unmeasured behavior when different cells and different cell types were subjected to new concentrations of salt. Combing the literature helped the researchers identify the parts that popped up in their model. They pared down a hair ball of chemical interactions and were left with a relatively simple mechanism.
This experiment is wholly unlike the classic experiments I usually think to do as a biologist. For example, how would a biologist determine whether the moon affects the tides? Now, hold on. I know that the moon and tides don’t normally reside within the realm of biological experimentation. This shouldn’t worry you. Biology, just like any other discipline, has its own methodologies. And these methodologies make some knowledge easy to dig up and verify and others hard. So, what does biology have to say about the tides?
First off, you’d do a knock-out study. Blow up the moon, but keep everything else the same, and see how the tides changed—if at all. Perfect. They stopped. What next? Well, there’s an obvious follow-up study: over-express the moon. Put two of them up there; maybe three, just in case. Now that the tides are back, we have some strong evidence that the moon in some way seems to influence the tides.
Part of the beauty of the van Oudenaarden input-output approach is that it didn’t require us to muck about with the genetics of the yeast. No knock-outs, no knock-ins. They kept the cells normal and genetically intact. Instead, they did what my friends in the cognitive neuroscience labs do with children subjects. They asked the cells a question: “What will you do with this salt concentration?” And then they listened for the response. From the answers the cells gave them, the researchers were able to infer something about the decision-making process. In this case, they drew it up as second-order linear, time invariant system. Here is an example where math allowed a biologist to do something very surprising. Math was used to talk to cells, ask them questions in their own language (so to speak), and learn something about them from their answers. Doesn’t that sound nice?
The moon never went through its full rotation–
How would a mathematician/child psychologist ask the moon what its doing to the tide?