How can a protein folding transition state have zero lifetime?

Question

I'm doing a module on my Biochemistry course looking at protein folding, and in a discussion of [folding] transition states I was a little confused at the thought of a zero-lifetime transition state - especially as for folding "the TS is probably an ensemble of states (a population) with a lifetime of 0".

In searching for anything with mention of protein folding TS with lifetime of 0 I found this old discussion ported to Google Groups, in which this is described as an 'inherent contradiction' to TS theory.

..."capital K, double dagger" is defined as a *thermodynamic equilibrium constant*. So, for example, a Transition State Theory applied to a bimolecular reaction involves a term corresponding to the *thermodynamic equilibrium constant* for the transition state [AB]**. At the same time, the transition state itself is presumed to have *zero lifetime*.

In other words, the Transition State Theory is wonderfully self-contradictory in its very essence (it contains an *equilibrium* constant for something that has zero lifetime). (via)

I'm just wondering if anyone can enlighten me / agree / disagree with this, and maybe suggest some of the original literature I might go to for an understanding of the meaning to this.

One of my chemistry textbooks has a small section on Hammond's postulate but without explanation to why transient is defined as zero lifetime:

Further, it's possible I've brought myself to the wrong equations but if I understand correctly the mass balance equation (described from the source above as the reason TS is only a 'quasi-steady state') says here the dm/dt does not tend to zero, i.e. zero lifetime is overextrapolation of the theory :

one can assume steady state for a species as long as its production rate and its lifetime t have both remained approximately constant for a time period much longer than t. When the production rate and t both vary but on time scales longer than t, the steady-state assumption is still applicable even though the concentration of the species keeps changing; such a situation is called quasi steady state or dynamic equilibrium. The way to understand steady state in this situation is to appreciate that the loss rate of the species is limited by its production rate, so that production and loss rates remain roughly equal at all times. Even though dm/dt never tends to zero, it is always small relative to the production and loss rates. (via)

Please let me know in comments if I can make this question more clear, thanks.

Update: This is a question about protein folding transition states - please can you not give me answers or comments directing me to resources covering the basics of transition state theory.

Answer 1

I think that most treatments of TS miss the point. I'll try to organize my answer stressing some points. At first, proteins have nothing special in this context.

On the definition of the transition state.

1º - What kind of state are we talking about? A quantum mechanical state? A classical one? As we need to fall in Born-Oppenheimer approximation, we are dealing (according to Copenhagen interpretation) with a kind of mix of them. We treat particles (atoms) following classical mechanics in some PES given for 'quantum' mechanics. It is important because in classical mechanics we define a state with the positions and momentum of particles (as will do in Transition State Theory).

2º - In this context, we have the same situation for the definition of an equilibrium state and for the definition of a TS, just because we need to define it in terms of positions and momentum.

3º - For a given time $t$, is almost sure that there is not even a single molecule in the exact position corresponding to the minimum of the potential energy surface, and the same is true for the TS. This is simply because the atomic positions $\mathbf{x}=\{x_1,x_2,...,x_n\}$ and $t$ correspond to a continuous variables (that have infinity possible values), that is, these two state are two points of a infinite probable points. I mean, it is like to say, for example, what is the probability of picking the number (say...) 0.45 when (truly) randomly someone choose a number between 0 and 1 with an uniform distribution.

4º - So to speak about equilibrium states or TS, we need to refer them at least as "a" state that is truly many different states, all those with $\mathbf{x}$ values in the range $\{x_1 \pm \delta x_1,x_2\pm \delta x_2,...,x_n\pm \delta x_n\}$

On the usefulness of the definition.

In some sense it is all about probability. It essence lies in statistical physics. If we want to know how fast a reaction is, and we assume that it is carried out thought the TS, we need to know how probable (relative to something) is the occurrence of a TS.

So the things go like that: we define a TS as those states in some volume in phase space. Recall that knowing energy distributions and degeneracies you can get the relative amount of states. So we use the equilibrium states (defined for volumes in phase space) because we can know much of them,like the concentration.

Next steps are irrelevant here, but we should recognize the necessity of defining these states by volumes and not just points in the phase space. By doing that (and recalling point 3º) we are establishing that it has no sense (physical or practical) to define TS giving them zero life time quality. (notice that momentum are finite)

On the equilibrium constant

To my understanding, there is nothing wrong with call it an well defined equilibrium constant. We are always said that it validity requires equilibrium (like everything equilibrium thermodynamics), but it has few sense if we recognize that we can derive it from statistical physics. An equilibrium state just needs to satisfy a constant density in phase, so if it is recovered fast enough we are safe working with thermodynamic equations. States that doesn't change in the time .. .. is just a way of be (almost) completely safe.

Bibliography

I would not recommend the original bibliography (Eyring papers). I think a better idea would be for example:

Variational Transition State Theory Annual Review of Physical Chemistry Vol. 35: 159-189 (Volume publication date October 1984) DOI: 10.1146/annurev.pc.35.100184.001111

Other good source for some points can be: http://www.amazon.com/Chemical-Dynamics-Papers-Advances-Physics/dp/0471400661 but only if you really have time to read a lot of articles...

"The principles of statistical mechanics" by Tolman is (imho) a book that can make you feel it intuitively (although it is rigorous enough).

Finally, for a "short" description (based in statistical physics) of the TST, I would recommend "Statistical mechanics, the theory of the properties of matter in equilibrium", by Fowler.

Final comment:

I'm not completely sure if I address your dude. Honestly it seems to me that this kinds of confusions can be avoided thinking in terms of statistical physics, from where the answers to questions like the life time of a TS arises naturally.

Please, let me know if I didnt answer your question or if something needs more details. Few years ago this theory filled me of doubts and I know that it is not easy find answers about it.

Answer 2

I don’t know where the person from the Google Groups you quote got the information that a transition state is "presumed to have zero lifetime", but a far as I can tell it is wrong. On the contrary, the existence of transition states is postulated in transition state theory (TST), which evidently implies that within the TST framework transition states have non-zero lifetime. Existence clearly contradicts zero lifetime: if some object has zero lifetime, what do you mean then by its existence?

Next, while the lifetime of a transition state in TST is postulated to be not zero, it is also proposed to be quite small, on the order of the time of a single molecular vibration ($10^{-13}$ seconds). Why do we have to make such a proposition? Well, transition state can be defined as the state corresponding to the highest potential energy between the reactants and the products (i.e. along a reaction coordinate) such that the probabilities of forming the reactants or products are equal. On potential energy surface (PES) then transition states correspond to the so-called saddle points (first-order saddle points, to be more precise), which, loosely speaking, can be thought of as a maximum in one direction (along the reaction coordinate) and minimum in all other directions. And structures that correspond to maxima (even in one direction) on PES) are unstable: once a system is at a maximum on PES it will almost instantly fall from it to one of the two neighbouring minima (either to reactants or to the products side).

Due to small assumed lifetime of a transition state, it is often said that "no physical or spectroscopic method is available to directly observe the structure of the transition state" (quote from Wikipedia article on TST). Although, with femtosecond IR spectroscopy one can actually observe molecular structures extremely close to the transition state structures. The practical problem is that quite often along the reaction coordinate in addition to a maximum corresponding to transition state between the reactants and the products there exist a minimum (or few) corresponding to the so-called reactive intermediates, and it is quite difficult to distinguish between transition states and reactive intermediates.

Note here that reactive intermediates correspond to minimum in all directions on PES, and are thus stable, or, to be more precise, metastable, since they are high-energy molecules which are quickly converted into more stable molecules (either another reactive intermediate or eventually the products). Reactive intermediates are also short-lived but with lifetimes appreciably longer than a molecular vibration, so they cannot usually be isolated, but they can be observed with the already mentioned femtosecond IR spectroscopy.

To summarise:

1. One should not in the first place think about transition states being real (and consequently detectable). A transition state is a part of a model description of a chemical reaction introduced by TST, where we just postulate their existence, and say no word on are they really exist or not. Irregardles of their real existence, transition states exist in our model description, and their existence implies non-zero lifetime.

2. Besides, with femtosecond IR spectroscopy it is possible to directly observe a transition states for some chemical reactions and it was actually done.

Answer 3

This didn't get any answers specific to proteins as I was looking for - I don't think this site has a big biochem. community - though I've found what may be the answer. For any other bioscientists reading this, from what I've read it seems to not be the case, but rather a product of current experimental resolution in measuring protein folding transition states.

A lecturer told me that if there was non-zero lifetime the species at the top of the free energy barrier would make it a non-two state system, and with the proliferation of potential protein conformations (into an 'ensemble' to be sampled as per Wolynes's energy landscape theory) this would increase still further, depleting concentrations on either side of the barrier, paradoxically slowing the reaction... $\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ ^{(or\ words\ to\ that\ effect)}$

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In a recent issue of Protein Science, Weikl and Paul describe "Decoupling and temporal ordering of conformational changes and binding events for small binding transition times"

In general, both conformational changes and binding/unbinding events are thermally activated processes that require the crossing of free-energy barriers. A characteristic feature of thermally activated processes is that the actual time for crossing the free-energy barrier, the "transition time," is significantly smaller than the dwell times in the states befre and affter the barrier crossing. In single-molecule experiments that probe conformational changes of proteins, the transition times are typically beyond experimental resolution, and the conformational changes appear as "sudden jumps" between two or more conformational states (Kim 2013; Schuler 2008; Chung 2012; Zoldak 2013; Woodside 2014).

To me this indicates the 'jump' is not actually instantaneous, but an artefact of experimental limitations. Zero lifetime is however permitted as a simplication in such a setup. The suggestion is though that it would be in reality finite, and open to determination with better methods.

Chung (2012) go into such a [statistical, Gopich-Szabo maximum likelihood] method in an entire paper:

Theory predicts that folding mechanisms are heterogeneous, so that an individual unfolded molecule can self-assemble to form its iologically active, folded structure by means of many different sequences of conformational changes. The distribution of these folding pathways can now be calculated from atomistic molecular dynamics simulations. Information on pathway distributions from experiments must come from measurements on single molecules, because only average propertiew are obtained in experiments on the large ensemble of molecules in bulk experiments. A single-molecule, equilibrium protein folding-unfolding trajectory is illustrated in Fig. 1, as monitored by FRET spectroscopy, and its relation the free-energy barrier as it crosses between the folded and unfolded states is shown.

The most interesting part of the trajectory is contained in what appears to be an instantaneous jump between the two states, called the transition path, which contains all of the information on the mechanism of folding and unfolding. The first step toward observing transition paths in protein folding, which we report here, is the determination of its average duration (transition-path time) for a fast-folding, all-β protein shown to be two-state in ensemble studies, as well as a markedly reduced upper bound compared with our previous study for α/β protein GB1 (the B1 Ig-binding domain of protein G from Streptococcus. In contrast to a rate coefficient, which measures the frequency of a transition, the transition-path time is the duration of a successful barrier-crossing event.

The paper defines "instantaneous" according to the limit of a two state mathematical model, ${\lim\limits_{\tau_s \rightarrow 0}L(\tau_s)}$, "i.e. it occurs faster than the shortest photon interval", and goes on to compare the likelihood (L) plots for cases of instantaneous vs. finite transition path time, to try and assess what might be going on beneath this limit of experimental resolution.

They found a significant peak in the likelihood function in the case of finite transition-path time — "Fig. 4B at 16 (±3) μs. (The error is the standard deviation obtained from the curvature of the peak.)"

They also found that a fast- and a slow-folding protein take almost the same time to fold when folding actually happens.

Our determination of an average transition-path time is a first step toward the goal of obtaining information on the distribution of folding pathways from measurements of interdye distance versus time trajectories during transition paths. However, the result of this first step by itself has turned out to be extremely interesting. Folding involves a complex and intricate rearrangement of a polypeptide chain to form a unique structure, yet the time for this nontrivial self-assembly process is almost the same for two proteins with different topologies and vastly different folding rates.

Of biochemical interest, "the diffusion coefficient of the model depends on the roughness of the underlying energy landscape and could therefore differ substantially among proteins."

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Googling the Gopich/Szabo maximum likelihood method I found a workshop presentation given by Chung at ETH Zurich, which explicitly states that "transition-paths for barrier crossings have never been observed experimentally for any molecular system in solution. Because it is a single molecule property, even determining the average transition-path time is challenging".

I'm still not 100% sure I've landed at the right conclusion from all this — there's no explicit statements in these papers that the species in the "barrier top population" (a phrase to recognise that the transition state may actually be an ensemble) have a finite lifetime, rather that the "transition path" traversing their formation has a finite lifetime.

Personally, I think figures 2B and 2C are trying to infer an answer to this question: whether the TS species lifetime is finite or zero/'instantaneous', and as such, I think the fact that it's the former may have disproven the zero-lifetime TS species model (though given how recently it became possible to do so, 0-lifetime TS species may continue to be described).

Edit: reading a 2003 paper from prominent protein scientist Martin Karplus, Catalysis and Specificity in Enzymes: A Study of Triosephosphate Isomerase and Comparison with Methyl Glyoxal Synthase within Adv Prot Chem, I'm not so sure if I am right to equate non-zero transition trajectory with non-instantaneous TS species: Karplus describes the former in a consistent manner with the "instantaneous barrier located between the transition state and the products state and vice versa". In the excerpt below he describes crossing the barrier as having a finite time scale:

In that case, still would appreciate an answer from a protein chemist!

Answer 4

Think of the molecule negotiating a saddle point in potential energy as an athlete doing a high jump.

There is no stopping and lingering at the highest point. If you call the transition state a given conformation (not a range of conformations along the path), and consider some transition speed (on the order of a bond vibration), the lifetime will be approaching zero (in the sense of a limit). This might be surprising because the reaction takes finite time. You resolve this by realizing that if you define states as a given conformation, the path contains an infinite number of states is has to go through.

So this is a question of understanding the math, not the chemistry.

If you want to consider an even weirder situation, there are some mechanism that involve hydrogen tunneling. In those cases, there is an interupted path from reactant to product, and the transition state is not populated.

Whatever experimental method you pursue to get a signal from the transition state will also observe signals from the very similar states neighboring the transition state. Thus, you might observe the transition state, but you will never observe it exclusively. The signal the method gives you, rather, will be a combination of signals from many transition-like states.

If you switch from a single-molecule to a bulk methods, things will get more fuzzy because you don't expect complicated systems like proteins to have identical potential energy landscapes for each individual protein (because their conformations are subtly distinct). Instead, there will be an ensemble of transition states, plus near-transition states, that all will contribute to the "transition-state signal" you supposedly are observing.

None of this gives you insights about what is happening with the protein. It is just a property of how the transition state is defined.

[OP] Update: This is a question about protein folding transition states - please can you not give me answers or comments directing me to resources covering the basics of transition state theory.

I am not aware that to apply transition-state theory to protein folding, you have any new emerging behavior. It is just a chemical reaction with a couple more degrees of freedom.