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A friend of mine is subediting a book. The author of the book died in 1999. The author, at some point, uses the word "trillion" which is, unfortunately, an ambiguous word in the UK: when I was at school it used to mean $10^{18}$ but nowadays it means $10^{12}$. My friend is faced with the following paragraph:

'According to Asimov, the amino acids of the proteins behave in a much freer way than our words do: they can be rearranged in any manner and always retain some meaning. A simple protein is made up of eight amino acids, which can be classified putting the numbers one to eight in a series, changing the order of sequence by one digit each time. Out of the same number of "words" we can construct a little over 40,000 organised "biological phrases" from the same genetic code, each one with its own meaning, which is the mission of every protein. But if the chains become longer, as in the case of more complicated molecules such as insulin, which consists of 30 amino acids, the tally rises to a staggering eight million trillion possibilities.'

The editor doesn't like "eight million trillion" and wants to replace it with "800000..000". The question, of course, is "how many zeros"? More precisely, the question is: does someone with a better understanding of chemistry than me able to reconstruct Asimov's calculation and see whether the answer is approximately $8\times 10^{24}$ or $8\times 10^{18}$?

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Note that there exist 20 different types of amino acids. (Wikipedia's list of 21 includes selenocysteine.) – Larry Wang Aug 13 '10 at 22:48
I thought trillion was $10^{12}$. – anon Aug 13 '10 at 23:18
@muad: the convention differs from country to country. In the UK, billion means (or used to mean?) $10^{12}$ and trillion means (or used to mean?) $10^{18}$. – Qiaochu Yuan Aug 13 '10 at 23:21
I use billion to mean $10^9$. – anon Aug 13 '10 at 23:57
In French, German, Spanish and so on, $10^6$ is million, $10^9$ is Milliard and $10^{12}$ is billion. – user1119 Aug 14 '10 at 0:00
up vote 4 down vote accepted

What I think the author intends in the first part is that some piece of genetic code gives you $8$ distinct amino acids and you want to count the number of ways to rearrange them to get a protein, which is where the $8! \approx 40000$ number comes from. Since there are $20$ amino acids, the best you can do for $30$ amino acids is to use each of them once and ten of them twice, giving an answer of $\frac{30!}{2^{10}} \approx 2.59 \times 10^{29}$.

I am unable to find the actual list of amino acids in insulin (which is ridiculous to me; why isn't this information on a wiki somewhere?), but the above calculation suggests that the $10^{24}$ number is closer. If anyone wants to help me out, the actual polypeptide chain in question is the B chain.

Edit: According to this citation, Asimov's actual estimate is $8 \times 10^{27}$. So it looks like the author misquoted him.

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The original article by Sanger and Tuppy is on pubmed. Or see Michael Lugo's answer if you just want the sequence. – Larry Wang Aug 13 '10 at 23:40
The information is indirectly available through Wikipedia's insulin page -- the "Orthologs" list in the sidebar has links to various protein databases. A slightly random walk via Uniprot gives the human B-chain sequence FVNQHLCGSHLVEALYLVCGERGFFYTPKT -- I haven't checked against other versions, but the sequence is supposed to be quite conserved so there shouldn't be huge differences. – walkytalky Aug 14 '10 at 0:04
There is "the source" for such things: the *.pdb files in the Protein Data Bank ( ) will carry the sequence of amino acids in "one-letter" format which chemical drawing software can subsequently use for drawing. And correct, the insulin in different mammals has "conserved sequences". – J. M. Aug 14 '10 at 0:11
Both valuable references. However, I would assume Asimov went by Sanger's paper. – Larry Wang Aug 14 '10 at 0:28

The second sentence of that paragraph is meaningless, I think. The number of eight-amino-acid polypeptides (those are chains of amino acids) is 20^8, which is much larger than 40,000. And insulin isn't really big enough to be a good example of a typical protein; typical proteins have hundreds of amino acids.

That being said, Qiaochu's interpretation seems correct. I was able to track down the structure of human insulin -- note that the image is for bovine insulin, and human insulin differs by substituting Thr for Ala in the final position.

So the B-chain of human insulin has the sequence

Phe Val Asn Gln His Leu Cys Gly Ser His Leu Val Glu Ala Leu Tyr Leu Val Cys Gly Glu Arg Gly Phe Phe Tyr Thr Pro Lys Ala

which contains

4 Leu

3 each Phe, Val, Gly,

2 each His, Cys, Glu, Ala, Tyr,

1 each Asn, Gln, Ser, Arg, Thr, Pro , Lys

and the number of ways that all of these can be rearranged is

$$ \frac{30!}{4! 3!^3 2!^5} \approx 1.6 \times 10^{27} $$

which is not all that close to either of the proposed replacements. Then again, it's possible that whoever did the calculation was working from the sequence of some insulin other than human, or just made a mistake.

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see the edit to my answer. This is fairly close to the number Asimov seems to have intended. – Qiaochu Yuan Aug 13 '10 at 23:55
+1 in general, but especially for the first paragraph. Asimov seems to have used a pretty daft metric -- proteins do not generally vary by permuting a fixed set of amino acid residues. – walkytalky Aug 14 '10 at 0:09
Indeed the world will be a much messier place if permutation of protein sequences were in fact the case! :) – J. M. Aug 14 '10 at 0:32
Asimov was trained as a chemist. I think he would have known better and someone else made a mistake. – Michael Lugo Aug 14 '10 at 2:07

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