# Does quantum mechanics play a role in protein folding?

Protein folding takes a very long time (relatively speaking) when thinking of quantum mechanical effect. However, for the initial micro-steps of folding, when an atom, or a configuration of atoms, can go (freely speaking) right or left (that is, to other possible configurations), in processes that take less than a picosecond, isn't QM relevant? that is, the atom has a probability to go to either places, in a QM sense, that it goes to both places, and due to interaction with the world, it's wave-function collapses or localizes and that is how the folding progresses?

• – Mithoron Jun 17 '16 at 13:29

There's several aspects to your question.

Does quantum mechanics play a role in protein folding? Yes. The origin of the van der Waals interaction is ultimately a quantum mechanical one. (At least the induced dipole portions: how the electrons move with respect to each other and an external electrical field is driven by quantum mechanics.) Also, while part of hydrogen bonding can be described by classical dipole-dipole interactions, there's a non-negligible portion which is due to non-classical electron interaction. (How much this last bit contributes, if at all, is a bit controversial, though.)

But that's all electrons. The other part of your question is asking implicitly about the quantum behavior of nuclei.

In molecular modeling we often use a viewpoint termed "the Born-Oppenheimer approximation". That is, while we allow the electrons to delocalize and exchange and be their quantum mechanical self, we treat the nuclei pretty much like classical particles. Using this approximation, we can get pretty decent correspondence with experimental results.

So the best we can tell, the quantum nature of atoms as a whole (that is, nuclei as opposed to electrons) isn't needed for things like protein folding. The nuclei are probably not substantially delocalized, they're probably not interfering with one another, and they're probably not quantum tunneling.

Probably. There's some evidence that quantum tunneling of nuclei in the process of enzyme catalyzed reactions. This is mainly hydrogens, though. In most of these experiments the quantum tunneling drops off dramatically when you go to deuterium or tritium nuclei. I'm unaware of any system where carbon/oxygen/nitrogen tunneling is seriously proposed - they're just too heavy.

So there may be some contribution to the quantum nature of nuclei to protein folding, but all indications at this point indicate that it would be minor. The quantum properties of electrons, though, are definitely important.

• Here is a lecture containing examples of reactions where heavier atoms like nitrogen and carbon tunnel and the author stresses on the fact that not only the individual atoms that tunnels but the system containing these atoms is tunneling as well. princeton.edu/chemistry/macmillan/group-meetings/… – M.ghorab Jun 18 '16 at 23:42

Protein folding takes a very long time (relatively speaking) when thinking of quantum mechanical effect.

First note that, in principle, for the time being, quantum mechanics is considered to be universally valid at all size and time scales. So, in general, it does not really matter how big a system is or how long a process takes: every system is a quantum system and every process is a quantum process.

Yes, in some circumstances we use classical theories instead for the description of physical systems and processes. But we do so only for the reasons that in some cases quantum theories provide unnecessarily complicated descriptions of such systems and processes, while simpler classical theories lead to the very same predictions (taking the desired accuracy into account), and thus, can serve as good approximations. For example, as the number of particles in a system grows, the smaller "quantum effects" become and the better classical mechanics describes such a system. In the infinite number of particles limit quantum mechanics simply reduces to classical.

Note, however, that it is quite difficult to say in general when you can safely switch from quantum description to classical one1, so that in practice we often do the switch not when we can, but when we are forced to, i.e. the classical modeling is usually used when the quantum one is simply impossible due to enormous computational cost and not necessarily when the former is certainly a good approximation to the later. This is indeed the case of any simulations at atomic level that are performed classically, including protein folding simulations: currently we have computers at which at best we can do only semi-classical/semi-quantum treatment (AIMD) and of only very small proteins. This is just a technical limitation.

To summarize, there is no distinct boundary between classical and quantum worlds simply because classical world is not real, it is just a mental construction that serves occasionally as a good approximation to quantum reality. The world we live in is a quantum world.

that is, the atom has a probability to go to either places, in a QM sense, that it goes to both places, and due to interaction with the world, it's wave-function collapses or localizes and that is how the folding progresses?

This is an interpretational question that, strictly speaking, lies outside of the domain of science. Protein folding is described by the laws of quantum mechanics, the way you interpret these laws is irrelevant.

1) At best you can have some fuzzy criteria such as, for instance, that quantum theories are unavoidable when action is of the order of the Planck constant.

• This answer, however nice, could use a little of more direct approach ;) – Mithoron Jun 17 '16 at 13:31

I was curious to see what kind of research had been done on this topic, and didn't expect to find much, but I was quite wrong. So, I'll summarize some articles that I found and give the references at the bottom.

Why would we ever consider quantum mechanics in protein folding?

First, an introduction to why someone would even try to come up with a quantum mechanical model for protein folding. Well, protein folding can be thought of as a chemical reaction, and should thus be described by the various laws we have for rates of chemical reactions. Specifically, the Arrhenius rate equation relates temperature to the rate of a reaction. It is often seen written as follows, $$k(T)=Ae^{\frac{-E_a}{RT}}$$ where $k(T)$ is the rate constant, $A$ is the pre-exponential factor (not a very descriptive name), $E_a$ is the activation energy, $R$ is the gas constant and $T$ is the temperature. This equation is where we get the general idea that as temperature gets larger, the rate constant will also get larger.

Another key is that this equation predicts that if we were to plot $ln(k)$ vs. $\frac1T$, we would get a straight line. This is indeed true for most all chemical reactions. These so-called Arrhenius plots have been made for many proteins, and they are very unusual. Here are a whole lot of these plots which I found in an article on non-Arrhenius kinetics of folding and unfolding.

The details of these plots are beyond my understanding of the experiment, but you can read more in my first citation. One detail which should be pointed out is that they are using some type of absorbance parameter to make the experiment possible but this absorbance is directly related to the rate of at that temperature, so this still demonstrates non-Arrhenius kinetics. The point is, something weird is going on here (rate decreasing with temperature non-linearly)! What makes it even weirder is that something else which defies chemical intuition happens: the unfolding process is not simply the reverse of the folding process. In fact, the unfolding process generally produces a fairly linear Arrhenius plot, but the folding process does not.

Finally, the last reason one might expect quantum mechanics to be important in a process like this is the very fact that it ever happens! It's been pointed out (I can't find where I read this, but it was in a journal article) that if a protein found the minimum energy configuration by sampling all possible states, it would never finish folding! Even if every sample was less than a femtosecond, proteins of any reasonable length (say 100 amino acid) could not fold nearly as quickly as they do because the number of possible combinations is astronomical. Thus, this cannot just be random motion like how we normally think of these types of things.

How do we explain this?

Needless to say, lots of people have tried to explain this phenomenon with varying levels of success. Most of the early interpretations ascribed it some kind of solvent effect which has to do with the non-linear change in hydrophobicity (I hate that word) of portions of the protein with temperature. Alas, the physicists caught wind of this, and they were not convinced. Somewhere in China, a physicist said to himself, this sounds like quantum mechanics to me! Why not? Super weird things happening? Probably quantum mechanics.

So, he came up with a quantum mechanical model for protein folding. How is this even possible? Well, I'll explain the various assumptions he made (as far as I understand them), and why this could even work.

First, he makes a model which can only describe two-state proteins. These are proteins which have two minima on the potential energy surface where the protein is almost always found, namely, folded or unfolded (see ref. 2).

Second, he knows from other literature that the most time consuming portion of protein folding is any kind of torsional motion. Thus, this is the portion of the motion which he treats explicitly.

Third, because one must have a potential energy function to describe these torsions, he assumes that they behave like harmonic oscillators.

Finally, because the exact form of the partition function for a harmonic oscillator is known, he is able to put this quantum mechanical model in terms of various thermodynamic quantities such as Gibbs and entropy. This is key because the model must be compared to previous experiments.

Feel free to look at the math in references 3 and 4. It's not impossible to follow, but isn't easy either...

Now, I will show two comparisons of their quantum mechanical theory with experiments performed in the literature (this is from ref. 4). The first figure shows Arrhenius plots for protein folding (notice their unusual shape), while the second shows Arrhenius plots for unfolding (mostly linear). It is very hard for me to understand this difference between folding and unfolding, but their model predicts it, so there is precisely no reason to reject the quantum mechanical aspect of this. Of course, one can never prove this sort of thing (nothing is proven in science), but they've basically shown that any truly predictive theory of protein-folding must account for quantum mechanical effects somehow.

Figure 1 (protein folding):

Figure 2 (protein unfolding):

Conclusions:

Well, despite what we might have expected, when one devises a quite simple model using quantum mechanics (simple compared to what it could've been), one gets very good comparison between theory and experiment. Interpret that however you please, but there's no denying that it's true. Expect for people to make models which are mor robust, or ae capable of describing multi-state proteins. Should be interesting to watch this whole thing... unfold.

References:

1. Matagne, A., Jamin, M., Chung, E. W., Robinson, C. V., Radford, S. E., & Dobson, C. M. (2000). Journal of molecular biology, 297(1), 193-210.
2. Zwanzig, R. (1997). Proceedings of the National Academy of Sciences, 94(1), 148-150.
3. Luo, L. (2014). Quantum theory on protein folding. Science China Physics, Mechanics and Astronomy, 57(3), 458-468.
4. Lv, J., & Luo, L. (2014). Science China Life Sciences, 57(12), 1197-1212.
5. Onuchic, J. N., Luthey-Schulten, Z., & Wolynes, P. G. (1997). Annual review of physical chemistry, 48(1), 545-600.
• I don't see how the papers you reference and then show pictures tell us anything about how quantum mechanics influences protein folding. First folding is not ' a chemical reaction' in the normal sense of the word and there seems no reason why the Arrhenius equation would describe such a complex process. It is observed in numerous simple reactions A+B->C etc. where one barrier is crossed from reactants to products. But proteins are much more complex and many barriers are involved between the many unfolded states and folded protein. Arrhenius does not work, it should not, its not applicable. – porphyrin Jun 19 '16 at 11:36
• The comment you make about proteins searching configuration space and never finding the folded state as this takes too long is called the Leventhal paradox. It means that, since proteins can fold spontaneously after being unfolded, folding must be a co-operative process. – porphyrin Jun 19 '16 at 11:40
• @porphyrin They tell you how quantum mechanics could potentially be involved because someone made a purely quantum mechanical model for protein folding (solid lines) and compared with actual experimental rate data and found quite impressive agreement. And clearly the Arrhenius equation does sometimes apply because many of those unfolding plots are nearly linear. I believe that it works because all you're comparing is the initial and final state, so they're being intentionally naive about what happens in the middle, so one should still expect Arrhenius behavior. – jheindel Jun 19 '16 at 16:56
• And also protein folding as far as considering it a chemical reaction is nothing but very complicated isomerization. And isomerization is definitely a chemical reaction. – jheindel Jun 19 '16 at 16:59
• Hi jheidel, I looked at your reference 1, it describes non-arrhenius behaviour of the Lysozyme. The fits to the data come not from any quantum calculation but initially from normal (phenomenological) chemical kinetics equations (their eqn. 1) which leads to their eqn. 7. They use the Arrhenius/Eyring equation on the way to this result. They fit the data by varying several parameters unit a fit is found, not from any QM calculation. The Arrhenius equation is not based on quantum mechanics per se but is based on the Boltzmann equation and classical thermodynamics. It was developed before QM. – porphyrin Jun 19 '16 at 21:03

Yes it is true that protein folding takes a long time compared to bond vibrations (few femtosecond) or rotations of a amino acid (picoseconds to nanoseconds) and for small proteins can take of the order of a few tens of nanoseconds. Typically proteins with many amino acid residues, say 100, this time can reach to microseconds and seconds and in fact cannot be characterised by a single lifetime but a range of these.

As to quantum mechanics, I would reiterate other answers in that as far as we know quantum mechanics describes the world we observe, but it does not mean that every thing has to be analysed using QM in order to make sense of it. Having said that, in proteins the place where classical behaviour breaks down is always to be found in tunnelling, which the H atom in hydrogen bonds does. So in the sense I think you mean, hydrogen bonding is the place where it is most apparent.

However, experiments suggest that in folding, van-der waals interaction in the normal usage of polarisable electrons, as well as pi-pi interactions play an important part in forming intermediate (temporary, short lived) structures that allow further folding to take place by reducing the configuration space available. They also play a vital part in the final structure.

The pi-pi interaction is quantum in nature and can, for example, be between pi bond electrons in the peptide C=O bond and those in aromatic residues such as tryptophan. We should not also forget that the chemical bond is inherently quantum in nature; bonds would not exist in a classical world.

If you don't restrict yourself to protein folding but to proteins in general then in photosynthesis there is overtly quantum behaviour both in energy transfer and in electron transfer.