While mathematics has long been at the heart of the physical sciences, "computational biology" is just now coming into its own as a branch of science. Biologists are realizing the utility of mathematics to their research, and mathematicians are finding applications for their work in biology. The possibilities are limitless.
One of the earliest applications of mathematics to biology is the use of differential equations to model population dynamics. The rate of change of a population is expressed as the difference between the birthrate and the death rate, resulting in a differential equation that can be solved- exactly or approximately, as the case may be- to express the population as a function of time. Because birth and death rates are often complex functions themselves, modeling populations in an accurate, meaningful way is not simple and is an active area of mathematical research.
Differential equations are also used to solve fluid flow problems involving blood vessels and to model the movement of an electrical impulse through a nerve fiber. Most of these equations cannot be solved analytically, which means that understanding and applying these models becomes an exercise in numerical analysis. To truly model the body mathematically, it is necessary, but not necessarily sufficient, to have both a thorough command of the underlying mathematics and some strong computer skills.
Statistics have always been important to biological research. In order to test a new drug, for example, a pharmaceutical company hires statisticians to analyze the data and determine whether any improvements or side effects are "statistically significant". Drawing a conclusion about the safety and effectiveness of a new drug is not trivial since there will always be some patients who get better, some who get worse, and a few who experience unfortunate health problems during the study. In late-stage research, the test groups are often small, which means that statisticians must extract as much meaning as possible from a very small amount of data. Concepts such as hypothesis testing and correlation are key to drawing correct conclusions.
The growing research area of bioinformatics deals with statistically analyzing large volumes of data like those generated by sequencing of DNA and RNA. Comparing DNA and RNA samples has applications in forensics as well as medical testing and basic research.
As Albert Einstein said, "The universe is written in the language of mathematics." This is as true for living systems as it is for the stars in the sky or the molecules in a test tube. As computational biology opens more and more doors, it will continue to be a growing field with amazing potential.