Interesting mathematical topics arising from biology 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,...


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*Which interesting advances from the point of view of a mathematician are 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...



*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?
 A: If you haven't seen it yet, this article by M. Gromov has, at the very least, a lot of references.
A: A very thorough introduction to some now classical topics can be found in James D. Murray's
now two-volume book published by Springer. Expect lots of ODE's and PDE's in that one.
As far as more exotic math is concerned, a complete overview would be difficult: it seems people throw everything they have and see what works. I've seen some interesting talks involving combinatorics, others involving algebraic geometry.
A: You might also look at this article by Avner Friedman from the Notices of the AMS which gives a survey of the field.
A: For a broad overview, an entry point seems to be: this Wikipedia page
To partially address your second question, here is a very short list:

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*Stochastic process; stochastic differential equations etc.

*Graph theory and combinatorics

*PDEs

A: In the 1930's Etherington developed several nonassociative algebras that express much of (what was known at the time about) genetics and inheritance.  Here's a survey from the early 1990's by Mary Lynn Reed.
A: Knot theory has been used to understand the mechanism by which enzymes like gyrase> relieve tension on DNA molecules.  See here, jstor.
A: Also try to partially answer your second question here, I would say:
1: Differential Equations including Ordinary Differential Equations, Difference Equations, Delay Differential Equations, Integral-Differential or Integral-Difference Equations, 
2: Dynamical Systems;
3: Game Theory;
4: Numerical Analysis;
5: Programming techniques: MatLab, Maple, Python, etc. 
In the workshop I recently attended, I heard talks about using Algebraic Geometry and Lie Theory in this field as well. 
A: I wound up with two copies of a book, Math & Bio 2010 edited by Lynn Arthur Steen. It was mainly aimed at improving undergraduate interdisciplinary education. But it also has a huge bibliography on research topics, several pages of researchers with websites. It says ISBN 0-88385-818-5. Alright, table of contents HERE (Wayback Machine)
A: Bernd Sturmfels wrote a book called Algebraic Statistics for Computational Biology.
A variety (no pun intended) of mathematical topics occur.  Phylogenetic trees are studied by means of "tropical" algebraic geometry.  It's been long enough since I've looked at it that I'm not going to trust myself to give more examples yet.  Here's the amazon.com page: https://www.amazon.com/Algebraic-Statistics-Computational-Biology-Pachter/dp/0521857007
A: Differential geometry. For example, see this answer. This also relates to your first question- study of writhe was heavily influenced by mathematical biology, with many central papers having been written by biologists.
A: Warren Ewens wrote this book: Mathematical Population Genetics 1: Theoretical Introduction
He is the eponym of the Ewens sampling formula: https://en.wikipedia.org/wiki/Ewens%27s_sampling_formula
A: There has been an interplay among ideas in physics, math, and biology in the exploration via nonlinear differential equations of synchronicity, order/disorder, chaos, and similar phenomena apparent over a wide span of systems from purely mechanical such as coupled pendulum clocks to individual organisms and ecosystems. See, e.g., Sync: How Order Emerges from Chaos in the Universe, Nature and Daily Life by Strogatz. This has a long history going at least as far back as Huygens (cf. "Huygens' clocks revisited" by Willms, et al.)
Some more interplay, receiving less attention in the popular science literature, has been among combinatorics, topology, and the study of the configurations and dynamics of RNA and DNA.

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*"On the distribution of cycles and paths in multichromosomal breakpoint graphs and the expected value of rearrangement distance]" by Feijao, Martinez, and Thevenin (related to OEIS A039683).


*"Enumeration of chord diagrams on many intervals and their non-orientable analogs" by Alexeev, Andersen, Penner, and Zograf with applications to models of interactions among several RNA molecules.


*"Link Nomenclature, Random Grid Diagrams, and Markov Chain Methods in Knot Theory" by Witte. Excerpt: "Of primary importance in knot theory is determining the equivalence or inequivalence of knots and links. This is usually done through the use of topological invariants (Section 2.3.6). Many of
these invariants do not distinguish knots or links that differ only by a mirroring, orientation change,
or relabeling of components, i.e. different isotopy classes of a link. This is okay for a broad study
of knots and links, but these can be crucial considerations for particular applications." One application is to the study of double stranded, circular bacterial DNA.
A: I think this is quite relevant,
https://liorpachter.wordpress.com/2014/12/30/the-two-cultures-of-mathematics-and-biology/
especially the references to Haussler's work,
Mapping to a Reference Genome Structure and
Efficient haplotype matching and storage using the positional Burrows–Wheeler transform (PBWT).
(Incidentally, I'd say "David Haussler ... studied mathematics" is quite an understatement, as he's made seminal contributions to computational geometry and combinatorics.)
A: I guess you might find this article of Bernd Sturmfels interesting: CAN BIOLOGY LEAD TO NEW THEOREMS?
A: As regards interesting mathematics arising in biology: 


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*A mathematically fascinating class of integro-PDEs arise in the study of age-structured population models. The independent variables are age $a$ and time $t$ ; the systems are first-order PDE; and the boundary conditions on the curve age=0 are given in terms of integrals of the dependent variables, $u$. That is, $u(0,t)= \int_{a=0}^T \phi(u(s,t)) ds$ where $\phi$ may be a nonlinear function. Such models arise frequently in physiology. It's my impression that this is a field with many interesting open mathematical questions to be asked.

*PDEs arising in pattern-forming systems in biology exhibit interesting mathematical behavior; questions about long-time regularity of such PDE are mathematically interesting. One may wish, for example, to characterize finite-time blow-up, or development of geometric singularities on interfaces.

*Dynamical systems with delays (functional differential equations) for the form $\frac{dy}{dt} = A(y(t-\tau),t)$ arise naturally in biology. This is a field which is not as mathematically developed as the theory of ODE.
A: Mathematical biology is a huge area which is not so young.
Statistics is a major research tool in biology (as in most other areas of natural and social science) so biology questions rely and have led to substantial progress in mathematical statistics and related probability theory.
Like in most natural sciences differential equations of various types (and some unique types of euations) arose in biology. It makes sense to mention in particular the pioneering work of Alan Turing in his paper entitled  The Chemical Basis of Morphogenesis. Nilima Nigam's answer mentioned several related connections and this is just the tip of the iceberg. 
Biology have led to vast questions regarding algorithms, computation, and optimization. The fields of computational biology and bioinformatics are closely related.
Mathematical biology have led to Evolutionary game theory which had strong impact on mathematical game theory.
Sometimes a single question in biology is related to a large number of mathematical disciplines. For example, Amit Singer and Yoel Shkolnisky are involved in a long term project aiming to determine the 3-dimensional shape of a certain molecule based on noisy 2-dimensional pictures from an electronic microscope. Their research is related to fascinating questions in harmonic analysis (and wavelets), graph theory, representation theory, semidefinite programmings, probability theory, and even the notion of unique games fron theoretical computer science enter.
A last example: John Bush and David Hu had remarlable mathematical models explaining how insects walk on waters. Bush's homepage is a good source for various issues in mathematical biology. 
A: I don't know if my answer fits with your question, please let me to know it that I can to delete my post if it isn't interesting in this thread.
First, a book that seems very inspiring and nice is [1]. The website Princeton University Press includes information about the book, and you can to read about the book from the label Praise. I've read the book, and it's incredible not only for the technical details it gives, but also for the stories of the scientists it tells (it is pure civilization and heroism; the efforts of a philanthropist, the person who the author refers, are also heroic, and those efforts of volunteers that he refers in his book).
On the other hand I think that maybe other interesting/inspiring mathematical topic could to arise from the paper [2].
Finally other inspiring book that I've read is (the Spanish edition of) [3], Wikipedia has the article Quantum biology, the book is very interesting but I don't know to tell you about what problems in quantum biology could be approachable/interesting by mathematicians. On YouTube is edited from the official channel Oxford Philosophy of Physics, seven months ago, the talk with slides with title Jim Al-Khalili: Life on the Edge - the dawn of quantum biology by professor Jim Al-Khalili.
References:
[1] Sean B. Carroll, The Serengeti Rules: The Quest to Discover How Life Works and Why It Matters, Princeton University Press (2016).
[2] Pasquale Cirillo and Nassim Nicholas Taleb, Tail risk of contagious diseases, Nature Physics, Volume 16, pages 606–613 (2020).
[3] Jim Al-Khalili and Johnjoe McFadden , Life on the Edge: The Coming of Age of Quantum Biology, Bantam Press (2014).
