From the article
Mathematics Is Biology's Next Microscope, Only Better
by Joel E. Cohen PLOS Biology, 2(12) Dec 2004
Box 1. Challenges

Here are five biological challenges that could stimulate, and benefit
from, major innovations in mathematics.

1.  Understand cells, their diversity within and between organisms, and
their interactions with the biotic and abiotic environments. The complex
networks of gene interactions, proteins, and signaling between the cell
and other cells and the abiotic environment is probably incomprehensible
without some mathematical structure perhaps yet to be invented.

2.  Understand the brain, behavior, and emotion. This, too, is a system
problem. A practical test of the depth of our understanding is this simple
question: Can we understand why people choose to have children or choose
not to have children (assuming they are physiologically able to do so)?

3.  Replace the tree of life with a network or tapestry to represent
lateral transfers of heritable features such as genes, genomes,
and prions (Delwiche and Palmer 1996; Delwiche 1999, 2000a, 2000b;
Li and Lindquist 2000; Margulis and Sagan 2002; Liu et al. 2002;

4.  Couple atmospheric, terrestrial, and aquatic biospheres with global
physicochemical processes.

5.  Monitor living systems to detect large deviations such as natural
or induced epidemics or physiological or ecological pathologies.

Here are five mathematical challenges that would contribute to the
progress of biology.

1.  Understand computation. Find more effective ways to gain insight and
prove theorems from numerical or symbolic computations and agent-based
models. We recall Hamming: The purpose of computing is insight,
not numbers (Hamming 1971, p. 31).

2.  Find better ways to model multi-level systems, for example, cells
within organs within people in human communities in physical, chemical,
and biotic ecologies.

3.  Understand probability, risk, and uncertainty. Despite three
centuries of great progress, we are still at the very beginning of a
true understanding. Can we understand uncertainty and risk better by
integrating frequentist, Bayesian, subjective, fuzzy, and other theories
of probability, or is an entirely new approach required?

4.  Understand data mining, simultaneous inference, and statistical
de-identification (Miller 1981). Are practical users of simultaneous
statistical inference doomed to numerical simulations in each case,
or can general theory be improved? What are the complementary limits of
data mining and statistical de-identification in large linked databases
with personal information?

5.  Set standards for clarity, performance, publication and permanence
of software and computational results.