# Introduction to protein folding for mathematicians

My background is mostly in (applied) math with healthy doses of physics and computer science. Are there any good introductions to protein folding and its challenges for someone with that kind of quantitative background but very little knowledge of biology, chemistry, etcetera?

• Welcome to our site! (I liked "...healthy doses of physics...", is that possible?)
– Camps
Dec 4 '20 at 17:36

Great question! Protein folding has been in open question for decades. Just recently, there's been a lot of discussion regarding DeepMind's AlphaFold project, which was discussed at length on our very own site here.

My answer will be complementary to the one above, but the references I will provide will be closer to the physics side of the problem.

First thing's first, what you should be studying is Statistical Mechanics both in and out of equilibrium. How deep you want to dive into Statistical Mechanics will depend on your research interests or requirements.

As an introduction, a text that would be suitable for you should also be suitable for a biochemist wanting to come to the quantitative side. A "meet halfway" book.

This is it: "Molecular Driving Forces: Statistical Thermodynamics in Biology, Chemistry, Physics, and Nanoscience" by Ken A. Dill and Sarina Bromberg

If you've had healthy doses of physics, this book is a great introduction to basic statistical physics, thermodynamics and biophysics. I definitely recommend it. The problems are not difficult and facilitate the development of the needed physical intuition.

The author, Ken Dill is well-known for his contributions in this field. One of his most cited papers is: "A lattice statistical mechanics model of the conformational and sequence spaces of proteins"

An alternative text on this subject is Lectures on "Statistical Physics and Protein Folding", by Kerson Huang, late professor emeritus at MIT.

Depending on how much Statistical Mechanics you want to learn, I would also recommend David Chandler's "Introduction to Modern Statistical Mechanics" as a supporting text. Other classics are great too.

A great amount of literature is around, some focusing even on topolgy in protein folding, others more focused on the computational-bio aspect, but I think these are the basics from the physics side.

Hopefully a good start!

• Is statistical mechanics a great model for such events? Is it accurate, or is it like taking shots in the dark and getting most of them right? Dec 6 '20 at 0:09
• I definitely would not describe using Statistical Mechanics as "taking shots in the dark"! Dec 6 '20 at 21:43
• I mean, it's probability based, right? The model may be good, but is it perfect? Dec 7 '20 at 0:15
• There is no such thing as a "perfect" model. All models will have some error to account for that will be related to the amount of information we have about a system or measurement... statistical mechanics deals with that too, actually! Basically, what I'm saying is that statistical mechanics is robust enough to get very, very accurate answers. The second law of thermodynamics in its very nature is probabilistic... and has never been violated. So I'd say it's as perfect as one can hope for! Dec 10 '20 at 1:08
• We have a pretty much perfect model for Newtonian physics. Probability based models seem like a good starting point, but could there really not be a perfect solution for such problems? Like say we know the probability distribution of some events; could there not be an equation that correctly models the event, where in that equation the events happen with the same probability as predicted by the probability distribution? Is that not one of the goals of chaos theory? Dec 10 '20 at 22:46

Someone more familiar with the problem might have a better suggestion, but I recently came across Daniel B. Dix' notes on Mathematical Models of Protein Folding. This is not my field, so I won't guarantee correctness. However, to a layman at least, these notes seem well suited for someone with your background. The abstract reads

We present an elementary introduction to the protein folding problem directed toward applied mathematicians. We do not assume any prior knowledge of statistical mechanics or of protein chemistry. Our goal is to show that as well as being an extremely important physical, chemical, and biological problem, protein folding can also be precisely formulated as a set of mathematics problems. We state several problems related to protein folding and give precise definitions of the quantities involved. We also attempt to give physical derivations justifying our model.

• Thanks! I want to give others some more time to answer but this looks really promising. Dec 4 '20 at 18:29

### Preamble

Since I don't know your specific background, this is a generic answer for any applied mathematician wishing to enter the field of protein folding. Not everything will apply specifically to you, and please don't feel offended if there's something I assume you don't already know or do!

First of all, as a fellow mathematician (I was trained in university as a mathematician and made the switch over to computational chemistry/biochemistry later in life), congratulations on making the first step towards working on (more) real-world problems. One of the first things you might notice will be that work in this field can be much "higher impact", so your papers may get cited and mentioned in the media much more often than you're accustomed, but also the field can be far more competitive and developing familiarity with how things work here can take some time. While papers in chemistry and biology get more citations (on average) than in applied math, you might still find it equally difficult to find funding because there's more people working in the field. Contrary to many math fields, conference papers are worth almost nothing compared to journal articles, and even arXiv papers can be totally ignored by many departments (when there's so many papers in the field, some people just don't want to spend the time reading things that are not published in refereed journals unless they know the author(s) or the paper was recommended to someone who they trust). While talks in math are often done "at the chalkboard", talks in computational chemistry are almost always done with a slide-show presentation. Finally, get used to having to read a larger volume of papers and learning the skills required to be able to capture the important details efficiently, as opposed to in applied math where you might be encouraged to read a smaller number of papers but much more slowly.

Even more importantly, stop thinking about "existence and uniqueness" for solutions to the relevant differential equations in computational biology, and suppress all temptations to be "overly" rigorous about everything. Do not tell the reader that your conclusion is only valid on a "simply connected bounded open subset whose boundary is a simple piece-wise $$C_2$$ closed curve" and avoid using too much jargon or mathematical symbols (just write "$$a$$ is between 0 and 1" rather than $$a\in[0,1]$$, the readers are smart enough to know from the context whether or not "between" should be extrapolated to imply "between an including", and do not say "$$\forall x \in \mathbb{R}$$" because any smart reader will assume that $$x$$ is a real number (or a complex number or Hermitian operator for much of quantum-mechanical modeling). Stop worrying about which Banach space you're using.

I already said "Finally..." and "Even more importantly..." so I'm running out of ways to start sentences now, but there's still more advice to give: Be patient and respectful to others working in the field (remember the opening to my answer: I'm not making any assumptions about you, just giving advice for applied mathematicians in general). There's just so much literature out there (compared to applied math at least), so not everyone will know everything, and everyone will certainly have a different academic history. Many people will not know as much math as you, but many will in fact surprisingly know even more than you.

This is what I'd recommend for an applied mathematician that has taken a healthy dose of physics and computer science, but hasn't yet had their booster shots biology and chemistry to review and renew their knowledge in these fields:

• Actually spend some time to learn the very basics of biochemistry, rather than trying to use mathematics to model something for which high school (and some primary school students) these days know more than you. For protein folding, this simply means to learn the answer to:
• Which elements are found in proteins (most of biology has H,C,O,N, but proteins differ from nucleic acids in that proteins have S and nucleic acids have P)?
• What is a "functional group", in particular: amino group and carboxyl group?
• What is an amino acid, peptide bond, dipeptide, polypepdide, polypeptide chain, and finally: protein?
• What is electronegativity and electropositivity? What is a hydrogen bond? How does a hydrogen bond work. What are Van der Waals forces? Why do hydrophobic and hydrophilic mean? What are steric interactions?
• What is primary/secondary/tertiary/quaternary structure in the context of proteins? What is a structural motif? What is an alpha helix, beta sheet, and disulfide bridge?
• What is a Ramachandran plot?

The above could be covered in a 1-week crash course. Email me at nike@hpqc.org if you think there would be enough interest for me to organize one.

Now for some of the techniques and theory:

• Molecular dynamics is just Newton's 2nd law applied to a large number of particles, and is implemented in a lot of open source software.
• Statistical mechanics is what Etienne's answer discusses, but you don't have to learn everything or read that whole textbook. Just review Gibbs free energy, Boltzman distributions, and things such as $$e^{-\beta H}$$.
• Contrary to what some people might tell you, quantum mechanics can almost entirely be avoided for protein folding simulations, but do review the concept of a classical potential energy surface, and a classical Hamiltonian. Many people forcefully interject quantum mechanics into computational chemistry that can avoid quantum mechanics, largely because their area of expertise is in the quantum realm and they want to try to raise the impact of their work. It's true that quantum mechanics is absolutely essential for understanding most of molecular structure and how the potential energy surfaces used are actually calculated, using what is colloquially known as "quantum chemistry" or "electronic structure theory", however a beginner applied mathematician trying to take their first steps in protein folding, might wish to just use potential energy surfaces that comes from X-ray experiments or from other people's ab initio calculations.

The above could also be covered in a 1-week crash course. Email me at nike@hpqc.org if you think there would be enough interest for me to organize one.

Now you might want to read some seminal papers on the topic of protein folding, to get an idea of what has already been done. There's too many papers, but some papers by certain authors have such titles and citation counts that you might not want to miss them, for example (others can feel free to add more lines if I have missed anyone):

• As I'm sure you can see, I have already selected "the" answer but I wanted to comment that I really appreciated your thoroughness. I'm not in school/academia right now ... it was really starting to bother me how non-real world math can be, so I left my math PhD program with an MS. I'm actually looking for a way back into research that's a little more impactful in a practical sense and I am exploring options now. Point being, I appreciate the guidance. Dec 7 '20 at 2:24
• Send me an email at nike@hpqc.org for research opportunities. Dec 7 '20 at 2:25