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This question is to everyone, but especially to whoever is doing a PhD in Mathematical Biology, to who is teaching Mathematical Biology and finally to who works or researches in this area.

What are the most important challenges of Mathematical Biology today and for the future?

By most important I mean those areas where most of the funding is going, areas where research could lead to great discoveries or improvements of say drugs, medicines, therapies, or what have you.

I would like to have a feel of the problems that mathematicians are trying to tackle in Mathematical Biology nowadays, but again concentrating on the most important ones, the ones that could bring a very positive contribution.

If you can answer, can you please list the most important challenges, together with what could be achieved if something was modelled correctly, what impact it could have?

At my university the movement of proteins in cells and the development of cancer are the main areas, however I am sure there are a lot more and diverse.

EDIT: I am not restricting myself to pure mathematics at all! Indeed I am talking about modelling, which can be numerical, probabilistic, computational, etc.

Did
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Euler_Salter
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    Protein folding is one of the more important tasks to accomplish of the science (to me this is more relevant than cold fusion, by example) and it have some bioinformatics involved but I dont know if this problem can be approach from a pure mathematical point of view. Other related things is data analysis of sequence in genetics, analysis of chaotic complex relations between species in ecosystems, a physic model for semipermeable membranes, etc... But, in general, all these tasks are closer to physics or chemistry than pure mathematics. –  Dec 09 '16 at 19:31
  • @Masacroso thank you for your comment! Just one thing: I am not talking of pure mathematics, by mathematics I mean also applied mathematics, so modelling with PDEs and other techniques! I know many people study biology using group theory or related pure maths techniques, but I'm talking in general! Anyway, thank you! Why is protein folding so important? What could be achieve by having a complete understanding of it? – Euler_Salter Dec 09 '16 at 19:36
  • @Masacroso Actually, there is a project joint between CalTech and Aarhus University (QGM) which uses some pretty high-powered machinery from geometry and (quantum)topology to study precisely protein folding. – Tobias Kildetoft Dec 09 '16 at 19:40
  • Ask your prof/advisor who are the leaders in the field of math biology and what conference they attend. pickup a copy of the list of topics on the conference, look at what people are working on. Ask you advisor whether there is anyone in your campus or any of his/her /PhD you can talk to . The opinion of an expert in a specific field is more useful than any info you can gather here. – achille hui Dec 09 '16 at 19:40
  • @Euler_Salter because protein folding can give us a lot of power to manipulate biology in general, and then health. Maybe I exaggerated a bit, cold fusion is very important :p –  Dec 09 '16 at 19:46
  • @achillehui I talked to my lecturer of applied mathematics and to the lecturer studying protein folding and development of cancer. He told me that those two are the most important problems $\textit{for him}$. So I guess that his answer was quite personal - hence I decided to write here. Thank you anyway! However, there aren't many experts, so I would like to have a broader range, as I would like to have a full picture. Mathematical Biology interests me, however I want to know that the results could actually make an impact! – Euler_Salter Dec 09 '16 at 19:47
  • @Masacroso and how likely is it that we will manage to model protein folding? By how likely I mean, compare it to the quantum computer. People are researching it, but they do not know if it is actually possible to build. Is it the same here? People are studying protein folding, but do we know that once we can model it, we can actually use it? – Euler_Salter Dec 09 '16 at 19:49
  • @Masacroso don't worry, it happens when you are excited about a topic! – Euler_Salter Dec 09 '16 at 19:53
  • A good model of protein folding allow one to have more efficient simulation of the folding process and significantly shorten the time to get results. That's means money for big pharmaceutical companies during the drug designing/testing process, that why research in protein folding get a lot of funding (at least ten years ago, no idea now). – achille hui Dec 09 '16 at 19:58
  • @achillehui it makes sense. So it wouldn't have one main effect, but rather it would improve drug designing/testing processes independently of what kind of drugs or treatments we are talking about? – Euler_Salter Dec 09 '16 at 20:32
  • The key of a good model is identify what are the important factors. If some factor is important in one instance, it is likely to be important in similar instances. One result may not improve all kind of drugs of treatments but it do help one make good educated guesses.... – achille hui Dec 09 '16 at 21:03
  • @TobiasKildetoft Is there anything concerning the study of protein folding at the Centre for Quantum Geometry of Moduli Spaces, which would be more substantial than the mention Possible spin-off applications: • The protein folding problem in Biology • Quantum computing in computer science/physics which is found there? This is the only hit that the search "protein folding" produces on the whole QGM website... – Did Dec 12 '16 at 16:57
  • @TobiasKildetoft OK, the query "protein" also returns this, announcing a meeting held at IHÉS in 2013... – Did Dec 12 '16 at 17:01
  • @Did I don't really know much about it. I think there was a publication about it at some point, but I am not sure if it even mentions QGM. There was also some collaboration with a Japanese university, but I forgot which one. If you find a paper in biology with Jørgen Ellegaard Andersen as one of the authors, that would be it (actually, thinking back to a joint workshop with that Japanese university, it might be RNA folding rather than protein). – Tobias Kildetoft Dec 12 '16 at 18:14
  • @TobiasKildetoft Thanks for the info. Found (but not read yet) this: http://www.nature.com/articles/ncomms6803 – Did Dec 12 '16 at 19:58
  • @Did Good that you found it based on that (I would not have known where to look). I found the concept interesting, but as I said, it is not really something I know much about (having been in the "algebraic" part of QGM). – Tobias Kildetoft Dec 13 '16 at 08:08

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This question is a cold case and the OP may have moved on to be a professional in some area - perhaps even in the particular topic of the question posed. All responses were comments and carefully avoided the fact that this question has no answer unless the specific intentions are analyzed.

It appears to me that the OP wishes to find an area that in some which way is "profitable" (see e.g. "... areas where research could lead to great discoveries or improvements of say drugs, medicines, therapies, or what have you" or "... but again concentrating on the most important ones, the ones that could bring a very positive contribution.", or "By most important I mean those areas where most of the funding is going ...").

To me this seem incompatible with the question posed in the highlighted box: "What are the most important challenges of Mathematical Biology today and for the future?"

These are divergent trajectories. The first quotes basically ask "is there some area where I can make a career?", while the latter quote sounds like a grand challenges question; Mathematical Biology could be replaced by Physics, Biology, Mathematics, Chemistry and so forth.

Nobody can really say whether one can make a career out of something. The factor influencing this turn-out are extrinsic to the field. Also, there is strong dependence on personal "features", including drive, excitement and, in particular, guts and tenacity.

If one wishes to do something for the field, in this case Mathematical Biology, then one must be willing to risk something - be it short- to medium-term lack of "popularity". Furthermore, one must be willing to talk to those whose problems one proposes to solve. Biologists are the audience to court.

Therein lies the core to building notoriety, not in what is popular now. Interests can shift very quickly in biology and, as a result, the only thing that the OP should be looking for - and asking help for - are the areas currently at the fringes - both present and past.

Foundations is the most important challenge in mathematical biology, if this area of scientific activity wishes to be called a field and be able distinguish itself from "biomathematics" or "computational biology" or "Systems theory" or any of such terms that flourish like weeds (nothing wrong with weeds).

For a young person to engage in mathematical biology I can only suggest to become clear about the meaning of the term by asking: Am I a mathematician who enjoys thinking about life phenomena or am I a biologist who seriously uses mathematical tools to encrypt and interpret data.

In the first case, very characteristically for a mathematician, there should be the drive to prove. In fact, more than that: Clarity of definition, choice of logic within which to reason, language to formulate propositions and structure to deliver proof. Proficiency in the modern structural theories that combine mathematics and computer science is an absolute must and I would suggest reading just one of the many writings of John Carlos Baez. He is not the only one who should be mentioned, but he writes and speaks very well.

In the second case, meaning where you identify as a biologist who seriously uses mathematical tools to compress experimental data into interpretable forms, then becoming an experimental biologist who shines due to proficiencies in some parts of mathematics and who can program his way into and out of tight corners is the direction to go. Mathematics is a sought-after skill in this case, but is not the profession.

To come to an end, I would like to note that my motivation was less to answer a cold case question that was posed almost seven years ago and remained unanswered, but foremost to point to the fact that this very question has crossed my desk regularly over the decades and I would like to share my answer more broadly.

Hans Sieburg
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