This is a framing practicum post. We’ll talk about what turnover time is, how to recognize when turnover time is relevant in the wild, and what questions to ask when you find it. Then, we’ll have a challenge to apply the idea.

Today’s challenge: come up with 3 examples of approximately-independent turnover which do not resemble any you’ve seen before, and estimate their turnover times. They don’t need to be good, they don’t need to be useful, they just need to be novel (to you).

Expected time: ~15-30 minutes at most, including the Bonus Exercise.

What’s Turnover Time?

The proteins in biological cells are often damaged by radiation or uncontrolled chemical reactions. To avoid damaged proteins building up, organisms turn over their proteins regularly: they’re constantly broken down and replaced with fresh proteins. As long as the stressors which damage proteins are at a constant level, the proportion of damaged proteins reaches an equilibrium, at which turnover balances damage rate.

The typical time it takes a protein to turn over is its turnover time. (This might be an average time, median time, half-life, etc… it’s a Fermi estimate, so it doesn’t need to be exact.)

Another example, with a more abstract (but more general) notion of “turnover”: air molecules in a box. We can draw an imaginary divider down the middle, and then think about the typical time it takes a molecule on one side to bounce over to the other side. This is a turnover time.

What To Look For

In general, turnover time should come to mind whenever we have many parts which change, are created and destroyed, or which move between states roughly-independently over time.

Useful Questions To Ask

Imagine a biological cell is hit with a pulse of radiation or a splash of harsh chemical, so that a bunch of its proteins are damaged. Assuming the cell is still functional, how long will it take the proportion-of-damaged-proteins to return to equilibrium? The turnover time is a good Fermi estimate: we’re asking how long it will take for the damaged proteins to be replaced by new proteins, and the turnover time is the typical time it takes proteins to be replaced by new proteins.

Note that this is just a rough estimate - the cell might detect the extra damage and upregulate protein turnover, or a lot of the protein-turnover-machinery may itself be damaged; either of these would throw off the estimate, but probably not by many orders of magnitude. If we just want to know whether the process will take milliseconds, minutes or months, then the turnover time estimate will likely be fine.

What about the box-of-air example? If there’s a pulse of air on one side of the box, then the turnover time is potentially a good estimate of the time for pressures to equilibrate. BUT, this example  highlights the key assumption: turnover time is a good estimate of equilibration time mainly when the parts turn over independently. If the air is at low enough pressure that the molecules aren’t colliding too often (i.e. ideal gas approximation holds), then the estimate is probably good. But if the molecules are colliding often, then the equilibrium will mostly come from molecules pushing each other around, rather than moving around independently - and our turnover time estimate potentially breaks down.

When parts change states roughly independently, turnover time tells us roughly the timescale on which the system equilibrates.

The Challenge

Come up with 3 examples of approximately-independent turnover which do not resemble any you’ve seen before, and estimate their turnover times. They don’t need to be good, they don’t need to be useful, they just need to be novel (to you).

Any answer must include at least 3 to count. That’s the challenge. We’re here to challenge ourselves, not just review examples we already know.

However, they don’t have to be very good answers or even correct answers. Posting wrong things on the internet is scary, but a very fast way to learn, and I will enforce a high bar for kindness in response-comments. I will personally default to upvoting every complete answer, even if parts of it are wrong, and I encourage others to do the same.

Post your answers inside of spoiler tags. (How do I do that?)

Celebrate others’ answers. This is really important, especially for tougher questions. Sharing exercises in public is a scary experience. I don’t want people to leave this having back-chained the experience “If I go outside my comfort zone, people will look down on me”. So be generous with those upvotes. I certainly will be.

If you comment on someone else’s answers, focus on making exciting, novel ideas work — instead of tearing apart worse ideas. Yes, And is encouraged.

I will remove comments which I deem insufficiently kind, even if I believe they are valuable comments. I want people to feel encouraged to try and fail here, and that means enforcing nicer norms than usual.

If you get stuck, look for:

• Systems in which the parts change states more-or-less independently over time.
• Systems in which the parts are created and destroyed over time.
• If turnover is not independent, remember that it might be independent conditional on holding some variable fixed (like a feedback signal).

Bonus Exercise: for each of your three examples from the challenge, what kinds of things might interact with the system on timescales much shorter than the turnover time - i.e. on timescales at which the state-distribution is roughly constant? What kinds of things might interact with the system on timescales much longer than the turnover time - i.e. on timescales at which the system is (approximately) in equilibrium?

This bonus exercise is great blog-post fodder!

Motivation

Much of the value I get from math is not from detailed calculations or elaborate models, but rather from framing tools: tools which suggest useful questions to ask, approximations to make, what to pay attention to and what to ignore.

Using a framing tool is sort of like using a trigger-action pattern: the hard part is to notice a pattern, a place where a particular tool can apply (the “trigger”). Once we notice the pattern, it suggests certain questions or approximations (the “action”). This challenge is meant to train the trigger-step: we look for novel examples to ingrain the abstract trigger pattern (separate from examples/contexts we already know).

The Bonus Exercise is meant to train the action-step: apply whatever questions/approximations the frame suggests, in order to build the reflex of applying them once we estimate a turnover time.

Hopefully, this will make it easier to notice when a turnover frame can be applied to a new problem you don’t understand in the wild, and to actually use it.