How does mathematics fit into modern biology research? Long gone are the frog dissections and the classification of birds. The new biology establishment, with an army of white-lab-coated experts and billions of dollars in funding from enormous funding agencies such as the NIH and the NSF, has found formidable opponents in the form of microscopic bacteria, transparent worms and fruit flies. The challenge: to understand, at the molecular level, the very mechanisms that allow such organisms to function. New technologies such as mRNA microarrays, as well as the increasing sophistication of older ones like X-ray crystallography, have made increasingly feasible what seemed far-fetched just a few years ago. Recent efforts are made towards an integrative approach, modeling quantitatively the entire molecular process once the key players have been identified and once the basic interactions have been found. Classic examples of such molecular processes are the digestion of lactose in \emph{E. Coli} bacteria, and the way that a protein binding to the membrane of skin cells triggers cell division. Some of the effects that are observed lie outside of the expertise of the biologist, and fall within that of the control theorist, physicist or engineer. An abstraction of some of these models and problems can also become the field of study for a mathematician.
A parallel problem of a more philosophical nature is the following. There are obvious structural similarities among different gene networks (as these `molecular processes' are also called), for instance the facts that they are usually based on the expression of proteins and use mRNA molecules as intermediaries (being often regulated through so-called transcription factors, etc). But the dynamics produced by these systems seems to be quite varied depending on the task at hand, and the way that Nature has chosen to solve a given problem sometimes appears to be quite arbitrary. The problem is whether gene networks are biased towards a certain underlying structure, beyond that provided by their physical implementation. This is very related to a more practical question, namely what mathematical tools can be best used for a formal analysis of gene network models; this is also why a mathematician may have a useful point of view to address that question.
There is a common metaphor comparing modern biology with astronomy. During the late 16th century, the Danish astronomer Tycho Brahe spent much of his life making the most accurate astronomical measurements of his time. After his death, Johannes Kepler spent many years studying these measurements, and he finally proposed three simple laws according to which the motion of the planets could be predicted. Later on, Newton would be able to provide general physical laws that imply Kepler's as a particular case. In the same way in biology, a challenge for a mathematician can be to participate in the search for such an `underlying order', at the same time as obtaining inspiration from biology to create new mathematics with an interest of their own.
It is important to realize that in spite of the new tools and the available information, quantitative modeling continues to be a controversial undertaking in mainstream biology. The reason: only the dynamics of the simplest and very best understood processes can be reliably predicted quantitatively at this point. The mainstream biologist still concentrates on the foundations: what genes and proteins participate in what processes, what is the overall effect of the over- or underexpression of a protein, and so on. The same sobering comment can be said about the search for some underlying order in the dynamics of molecular systems: we may well still be in Tycho Brahe's time rather than in Kepler's (let alone Newton's) and many physicists and biologists are skeptic about whether such general principles exist at all.
Finally, one can argue that the reason why quantitative modeling doesn't replicate the behavior of most systems is not that it is fundamentally the wrong tool, but rather that we don't have the necessary information to incorporate into the models. Thus as a theoretician one can nevertheless prepare the ground by finding general tools and algorithms, to be applied when the time is ripe in a few decades. Such a strategy doesn't sound nearly as misguided if one considers that the average drug takes about 15 years to develop from the lab into the market.
(From my dissertation preface)
Back to