# Why is statistical mechanics relevant to RNA and protein folding?

This is a very naive question. As far as I understand the folding of a molecule is governed by the electromagnetic forces between its atoms and also between its atoms and the atoms in the surrounding environment (so basically a many body problem). So I don't understand how statistical mechanics, such as Boltzmann's law, come to play.

To my knowledge statistical mechanics and thermodynamics exist when we have an ensemble of particle such as in a liquid or a gas. So what is this ensemble for the RNA or protein folding problem?

• There are too many possiblities. And it still would need supercomputers to calculate how a protein is folded. As well as it is still unkown how the complete mechanisms are working.... therefore statistics are necessary... Commented Apr 21, 2016 at 14:57
• Consider a polypeptide, newly synthesized, prior to folding up into its 3-dimensional shape, or structure. Each substituent amino acid, contributes to the unique sequence of the polymer, and each affects, in some small way, the number of different states that the polymer could be found in, at any given moment. Given the constraints imposed by the length and geometry of the chemical bonds, there are some states the polymer cannot adopt, but there are many, many states that are potentially available, and each has a unique energy. Over time you can plot these energies yielding a distribution. Commented Apr 22, 2016 at 23:14

I’m no physicist, but your statement “To my knowledge statistical mechanics and thermodynamics exist when we have an ensemble of particle such as in a liquid or a gas” is surely incorrect. The problem of protein folding is one of thermodynamics — finding the structure of lowest free energy, and the path by which it is reached from a random coil.

One of the difficulties relating to the path is that of getting ‘stuck’ in a position in the energy landscape from which the protein cannot escape. This is called ‘frustration’ and there is apparently a quantitative treatment of ‘minimal frustration’ using the statistical mechanics of spin gasses. This is mentioned in a review by Onuchic and Wolynes, available on-line, which you will probably be able to understand better than I can.

If you want to relate statistical mechanics to RNA and protein folding, you're on the right track with ensembles. Macromolecules are very dynamic and exist in many physical states. These macrostates are what are described by David on an energy landscape. The macrostates can be further broken down into microstates which are closely related movements of the macrostates. It depends how specific you want to get when describing your system. Statistics come into play when describing the populations of the macromolecule in its different macrostates. Typically, the macrostate with the most favorable energy conformation is the most populated.

You seem to have confused two different ways of discussing a thing with two different things. Electromechanics always applies, whether talking about liquids or gases or what have you. All molecules are subject to electromagnetic forces. And all matter has populations of molecules, which we can talk about with statistical mechanics. In statistical mechanics we abstract away some of the details of electromechanics and treat them statistically, just like how in economics we abstract away some of the details of individual psychology. So both apply, in every case. The problem is that RNA folding is a particularly tricky physical system, so abstracting it away is with statistical mechanics is far harder than it would be for say, a noble gas.

Indeed, statistical mechanics is applicable to the systems with many degrees of freedom (typically of the order of the Avogadro constant, $$N_A\sim 10^{23}$$). Gases, liquids, and solids routinely fill this definition, as they consist of huge numbers of atoms or molecules, each of which has at least three degrees of freedom (for molecules we also need incorporate their rotation and elastic movement).

There are two ways how statistical mechanics can be applied to proteins (see, e.g., the discussion in the book by Doi):

• one may consider a solution containing a macroscopic (i.e. $$\sim N_A$$) number of polymer molecules
• one may consider a single polymer molecule as a statistical ensemble, because a polimer with many links can adopt huge number of configurations and therefore satisfies most of the criteria necessary for applying statistical phsyics. This is specially true for polymers with $$10^5-10^6$$ links, but provides useful guidelines for relatively short protein molecules, with typical lengths of a few hundreds od amino-acids.

As an example we may consider protein folding (see, e.g., this question): a protein would fold spontaneously even in a vacuum, and even if the forces between its parts were non-existent. The reason is entropy: a chain of molecules can adopt billions of configurations in space, of which the stretched configurations would form only a small fraction. Statistical physics tells us that all these configurations are equiprobable, and we therefore are likely (what practically amounts to certainty) to find protein in a folded configuration. (This equiprobable state is referred to as maximal entropy or maximum of an appropriate thermodynamic potential, if we are not dealing with a closed system.)

Adding interactions between protein links and between the protein and the solvent molecules makes the problem even more amenable to statistical physics analysis (although now one may have to deal with folding-unfolding phase transition).

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