How cold is really cold enough?

[This is another guest post by Doug Clow — thanks Doug! I asked a question on LongeCity: is the 29 kJ/mol figure for the activation energy of the “catalase reaction” given in How Cold Is Cold Enough correct? Doug was kind enough to give a detailed answer, and permission to edit it a little and reproduce here.]

I did do a chemistry degree, with a lot of biochemistry in it. And even as a postgraduate student attended a research seminar by another postgrad who was investigating catalase analogues, which almost certainly touched directly on the question. But that was long ago and I haven’t done this stuff in anger for decades.

Alas, I don’t have good data books to hand and can’t answer the direct question (“What is the activation energy of the decomposition of hydrogen peroxide when catalysed by catalase”) authoritatively.

Partly, it’s because there isn’t a single answer, and anyone who tells you there is is fibbing. There are shedloads of different catalases (more if you include general peroxidases). They are indeed legendarily fast, and they are more or less ubiquitous in oxygen-metabolising species. The story goes that it’s as perfectly evolved an enzyme as you can hope for. It’s not a bad choice for the worst-case scenario for this context, although I wouldn’t go as far as to say it was the very worst without checking up for other very-fast enzymes in metabolic pathways and signal transduction (e.g. acetylcholinesterase is also legendarily fast). Which would be overkill.

I’d say using any value between 1 kJ/mol and 20 kJ/mol is not unreasonable, and if you pressed me for a value, I’d probably settle on 10 kJ/mol as a round value. (See e.g. http://www.ncbi.nlm.nih.gov/pubmed/8320233 which gave 10 kJ/mol for a catalase from a halophile bacterium — no reason for choice except I alighted on it quickly, or http://www.sciencedirect.com/… which found 11 kJ/mol but looks odd for several reasons.)

At the very top end, a value of 50 kJ/mol for a reaction that happens at a reasonable rate for practical experimental purposes at room temperature is fairly typical. There’s a (sorely abused) rule of thumb that says that reaction rate doubles with an increase of 10 C, which only applies under fairly restrictive conditions, one of which is that the Ea is 50 kJ/mol.

This does, of course, yield materially different results. I tried duplicating that big table in ‘How Cold is Cold Enough’ in a toy spreadsheet, and couldn’t quite reproduce his results, but did get within an order of magnitude which is close enough for these purposes. I played around looking at his ‘Rate relative to liquid N2’ column, for different values for the activation energy.

• 50 kJ/mol -> 2.2 x 10²⁵ times faster at 37C than at LN2
• 20 kJ/mol -> 1.4 x 10¹⁰ times faster
• 10 kJ/mol -> 1.2 x 10⁵ times faster
• 8 kJ/mol -> 1.1 x 10⁴
• 5 kJ/mol -> 340 times faster
• 2 kJ/mol -> 10 times faster
• 1 kJ/mol -> 3.2 times faster

For the question at hand, this makes a huge difference — to this analysis.

This analysis is likely to be wrong, anyway.

A quick look at the Arrhenius equation:

k = A e^-E_a/RT

Let’s take a very, very simplified reaction, where one molecule of reactant hits one catalyst to produce one product. The pre-exponential factor A represents the number of collisions that occur; the bit in the exponent tells you what proportion of those collisions have energy above the activation energy for the reaction.

Now, mathematical instinct might tell you that the bit in the exponent will give you all the action, but that’s not necessarily true. For practical purposes, the rate of catalase in vivo is limited by the rate at which molecules collide, not by the proportion of the molecules colliding which have greater than the activation energy needed for the reaction. Essentially, if a molecule of hydrogen peroxide bumps in to catalase, it’s breaking down. My biochemical intuition is likely to be sorely astray at LN2 temperatures, but I’d guess the same situation applied.

The pre-exponential factor A is probably more key: it’s the rate at which collisions occur between molecules that might react. If you have a perfectly efficient catalyst, this is the main factor affecting the rate of reaction — which makes sense, since they’d reduce the activation energy to a negligible value. Some enzymes — catalase is an excellent example — have been under geological periods of selection pressure in that direction. The Arrhenius equation is a simplification that works (better than it ought to) across a lot of practically-important situations. (One simplification is that the activation energy is not temperature-dependent. It sometimes is.)

If you get a phase change to solid — vitrification at very low temperatures — then you’ll get a staggering-number-of-orders-of-magnitude change in A. Those molecules are going nowhere fast, and so are flat out not going to bump in to each other. Never mind how much energy they’ve got when they do.

So I think that all is not lost for cryonics on this point.