What math can teach us about drug discovery and biology (and all of science, really)

Columbia University mathematical physicist (and slayer of string theory myths) Peter Woit recently gave an interesting talk at Rutgers in which he made an appeal for theoretical physicists working on string theory to 'reform' by learning from mathematicians. Woit thinks the physicists have lost their way in their drive toward empirically unsupportable and mathematically dubious claims. He says that unlike physics, mathematics is alive and robust, and unlike the physicists' barren exploits in the realm of fundamental physics, mathematicians have been proving many important fundamental theorems in the last few years (Fermat's last theorem, the Poincare conjecture, the twin prime conjecture etc.)

Woit's main point seems to be that mathematics has very clear protocols and boundaries for achieving its goals, and these kinds of guidelines would serve as useful pointers to physicists who seem to have abandoned rigor and logic in certain highly speculative areas. On one of his slides Woit had a listing of the steps mathematicians adopt as reality checks, and it struck me that one could apply the same lessons to a lot of the modeling - molecular as well as statistical - that goes on in drug discovery. In fact one could apply these lessons to much of empirical science as a whole. Here's the slide:

The first point about being very clear about your assumptions is spot on. My graduate school advisor Jim Snyder always used to emphasize this point; often when I breathlessly extolled the virtues of a fine paper with impressive graphics and methodology published in Science or Nature, he would calmly quip, "Looks interesting...but there's an assumption here...". The point was that no matter how elegant a study might be and no matter how impressive the technical manipulations in it might seem, if it's built on a foundation of flawed assumptions the whole structure can collapse.

As we have often seen in molecular modeling for instance, it's not hard at all to present fancy-looking computer models based on dubious assumptions. We don't have to look too far to realize this; a lot of docking and molecular dynamics is predicated on the assumption of an accurate force field. The even more basic assumption there is that we can accurately model the complicated intermolecular forces causing a small molecule to bind to a protein of interest. As we have found out over the years, both those assumptions can be suspect and they can turn the whole process into a crapshoot. No amount of computing power, beautiful graphics or elegant software can turn conjecture into reality if the basic assumptions underlying these things are questionable or inaccurate.

Woit's second point is also very valid. The chain of logic extending from one argument to another can often break down in complex, multilevel fields like chemistry and biology. We see this in drug discovery all the time. For instance, much of pre-clinical drug discovery depends on the accuracy of assays that are supposed to mimic the conditions inside a diseased human being. In reality this is almost never true; assays reveal themselves under conditions of high dilution in simple test tubes, diseases under chaotic regimes of intense molecular crowding inside complicated bodies. Extrapolating from one assay to another is a constant and daily headache for drug hunters, and extrapolating from assays to mice or human models is usually a great step into the unknown. Thus Woit's caveat to constant double-check the logical progression from one step to another - from one assay to another to mice to monkeys to humans - is absolutely critical before you declare that a dug molecule is actually working.

Lastly, one can never underestimate the staggering relevance of the third admonition - to separate what's known from what's murky, ill-defined and simply dark. Casually striding over the boundaries of these two domains has gotten scientists into a lot of trouble. Especially in a highly complicated endeavor like biology where emergent chaos play havoc on reductionist elan, mistaking what's unknown for what's known will ensure that you're in for some very rude shocks, and will likely consign your project to a swift end. At every step in a project involving complex biological or physical systems, it is imperative to make sure you clearly understand the known knowns, known unknowns and the unknown unknowns.

In 1931 the 25 year old Kurt Gödel taught us that even in that most exalted realm of mathematics which we are talking about, what we can know is fundamentally limited. It's a lesson which we should take very seriously to heart in the far more uncertain world of empirical science in general and biology in particular, otherwise we may very well end up taking it to our grave.