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# An Analogy For Protein Unfolding

Lets play a game. Imagine you have some dice. The rules of the game are as follows:

1) Roll all your dice at once

2) Remove any dice that land on a 6

3) Pick up your remaining dice and repeat until no dice remain

As an illustration, say you start with four dice. After the first roll, one six appears. You have three dice remaining. The second roll yields no sixes. Same for the third. The next roll, luckily, you obtain two sixes. After a series of discouraging attempts, your last die turns-up on the seventh roll. As a convenient representation, we can sum up this game in a time-series plot.

This graph provides us with a 'history' of our little game.

Now, say you wanted a complete description of this game. You want to be able to predict, to the best of your ability, how many dice will remain after some number of trials. So you repeat this process. You repeat it a lot of times. After many iterations, you average your results and obtain a smooth-looking curve. I illustrate this repetitive process below.

The blue line shows the running average of our game, while the faded red lines are individual histories of each iteration. If we normalize this curve, make its maximum value equal to 1, we can express things in terms of *probabilities*. Additionally, I'm going to show you the results of this process after it has been repeated 1000 times.

This probability tells us that if we repeat our process for 10 trials, we have around a 60% chance of ridding all our dice. Now lets take this one step further. If you repeat this process an infinite number of times, and then fit an exponential to this average curve you obtain, you can extract from that the probability of a single dice turning up a 6. This convoluted methods allows you extract a fundamental property of the system, in this case the probability of a single dice rolling a 6, without *directly* measuring it.

This is all fine and dandy, but you're probably asking yourself by now: "Kirill, how does this relate to proteins?" So here's the kicker. When we roll these dice, we are "sampling the system." We're preforming some action that forces these dice to take a stand, to make a decision. We can, effectively, do the same to proteins.

By applying mechanical force to proteins we can similarly "sample the system." We may construct a protein to consist of many tangled up subunits (pictured below). When exposed to force, each subunit is continuously "sampled." That is, at every instance each subunit is forced to take a stand: should I unravel myself, or not? If it unravels itself, the protein becomes longer.

We can probe the process of protein unfolding by applying force to a protein and measuring its length extension.

With dice, we "sampled the system" only when we roll the dice. Nature, however, does not play such games. Rather, at every infinitesimal instant, nature "samples the system" and decides if the protein should unfold. Pulling on proteins for an extended time produces traces similar to those shown above. Then, by aggregating these traces like we did with the dice, we may extract the "rate" at which these proteins unfold.

In reality, this phenomena of protein unfolding is not as instantaneous as these unfolding traces suggest. A more technical description of this phenomena would involve modeling the cascade of interactions between the individual molecules upon the application of force. From here, we could extrapolate the actual unfolding behavior, on the order of picoseconds. However, for the purpose of extracting protein unfolding rates, this description suffices.

Hopefully, this analogy I've drawn between out little game and protein unfolding has given you some intuition on the process.