I've heard that there's a relatively new field of science called Mathematical Biology. It will certainly apply well known and less known mathematical techniques to the understanding of some biological phenomena such as strings of DNA knotting together, proteins bending, cell membranes, population dynamics,...

1) Which interesting advances

from the point of view of a mathematicianare there that have been inspired by Biology and related areas?

For example, I heard of "Membrane computation", but I don't know if it's genuinely inspired by biology in some nontrivial way or if it just bears that name by a loose analogy...

2) Which fields of Mathematics are currently used in the discipline called Mathematical Biology?

If I were asked about Economy and Mathematical Finance I would loosely quote probability theory, stochastic differential equations, Ito integrals, ... So, what about Biology?