Abstract
The genetic code specifies $20$ amino acids with $64$ codons. This paper shows that the surplus is not distributed evenly——the same one-letter error costs $15.75$ times more at one position than at another. No new theorem or law is claimed.
First, over the $61$ sense codons there are $\mathbf{183}$ single-base substitutions at each position (Section 2).
Second, those that leave the amino acid unchanged number $8$ at the first position and $126$ at the third (Section 2).
Third, and this is the core of the paper. At the second position it is exactly $\mathbf{0}$: all $183$ substitutions change the amino acid (Section 2).
Fourth, the ratio is $126/8=\mathbf{15.75}$, a fraction that divides exactly (Section 2).
Fifth, of the $134$ silent errors, $\mathbf{94.03}$ per cent are gathered at the third position (Section 3).
Sixth, the separator is which position was hit, not how many bases changed (Section 5).
Scope (what is not claimed)
No new theorem or law is claimed——the standard codon table, degeneracy, the wobble hypothesis and robustness to error are all known in molecular biology. Origins are not discussed——why the code took this form (frozen accident, coevolution, optimisation) is not the subject. Only the numbers in the present table are counted. No optimality is claimed——it is not asserted that this code is the best. No comparison with random codes is made. Mutation rates are not treated——whether real mutations are equally likely at each position is not the subject. Only counts of possible substitutions. Only the standard code——variant codes such as the mitochondrial one are not the subject. Translation is not treated——tRNA, the ribosome and wobble base pairing are not entered. Relation to earlier papers: Paper 379 showed that “twice as hot” has three answers——here “a one-base error” has three weights. Paper 391 showed that the significant wave height is neither the mean nor the maximum——here a “mutation rate” quoted as a mean hides the positions. Paper 388 showed two different quantities going to one constant——here one operation becomes different quantities by position. Paper 300 showed that same-root and different-root can be decided——the three positions are different-root by that criterion. What is added is an exhaustive count of substitutions by position, showing that the second position is exactly zero, showing that the ratio $126/8=15.75$ divides exactly, obtaining the $94.03$ per cent concentration at the third position, and placing the separator at “position”.
1. Notation and the Question
Setting: take the standard genetic code and the $61$ codons that are not stop codons (UAA, UAG, UGA). At each position of each codon, count substitutions to the other three bases.
$$61\times3=183\quad\text{(substitutions at each position)},\qquad 183\times3=549\quad\text{(in total)}$$Observation 1.1 $64$ codons specify $21$ outcomes ($20$ amino acids plus stop). A surplus of $43$——there is redundancy somewhere.
Observation 1.2 redundancy works as insurance. An error that leaves the amino acid unchanged leaves the protein unchanged——the question is where the insurance was bought.
Observation 1.3 it could have been spread evenly. A third at each position——whether it was is a matter of counting.
Question: how heavy is a one-letter error?
2. At the Second Position It Is Exactly Zero
Computation 2.1: single-base substitutions and synonymous rate by position (counted exhaustively in this paper).
| Position | Substitutions | Synonymous (amino acid unchanged) | Rate |
|---|---|---|---|
| First | $183$ | $\mathbf{8}$ | $0.043716$ |
| Second | $183$ | $\mathbf{0}$ | $\mathbf{0.000000}$ |
| Third | $183$ | $\mathbf{126}$ | $\mathbf{0.688525}$ |
| Total | $\mathbf{549}$ | $\mathbf{134}$ | $0.244080$ |
Observation 2.2 this is the core of the paper. The synonymous count at the second position is exactly zero——all $183$ substitutions change the amino acid; not one policy of insurance was bought here.
Observation 2.3 at the third position $68.85$ per cent change nothing. $126$ of them——more than two thirds pass in silence.
Observation 2.4 the ratio is $126/8=15.75$. It divides——cancel the common denominator $183$ and an exact fraction remains.
Observation 2.5 the $8$ at the first position is not zero but small. $4.37$ per cent——it comes from the six-fold degeneracy of leucine and arginine.
Observation 2.6 overall, $134$ of $549$ are synonymous, $24.41$ per cent. A quarter——and this average describes none of the three positions.
3. Ninety-Four Per Cent of the Silent Errors Are in One Place
Computation 3.1: breakdown of synonymous substitutions by position (computed in this paper).
| Position | Synonymous substitutions | Share of the $134$ |
|---|---|---|
| First | $8$ | $5.97$ % |
| Second | $0$ | $0.00$ % |
| Third | $\mathbf{126}$ | $\mathbf{94.03}$ % |
Observation 3.2 the redundancy is gathered in one place. $94.03$ per cent——an even spread would give $33.3$ per cent each.
Observation 3.3 of the $61$ sense codons, $32$ are four-fold degenerate at the third position. Any of the four bases gives the same amino acid——with $24$ two-fold, $3$ three-fold and $2$ not degenerate.
Observation 3.4 so for more than half the codons the third position carries no information. $32$ out of $61$——one letter in three is frequently free.
Observation 3.5 the shape is that of Paper 391. There the mean of the “wave height” was not representative——here a “mutation rate of $24.41$ per cent” fits none of the positions. An average conceals three.
Observation 3.6 they are different-root by the criterion of Paper 300. The three positions play different roles in tRNA recognition——one phrase, “a single-base substitution”, names three things with different roots.
4. And When It Does Change, the Manner of the Change Also Differs
Computation 4.1: among non-synonymous substitutions, the fraction that preserves the polarity class (non-polar, polar, charged) (computed in this paper).
| Position | Non-synonymous | Fraction preserving the class |
|---|---|---|
| First | $166$ | $0.469880$ |
| Second | $\mathbf{176}$ | $\mathbf{0.375000}$ |
| Third | $50$ | $0.440000$ |
Observation 4.2 the second position not only changes things but changes them most crudely. $0.375000$, the lowest of the three——zero synonymous substitutions, and the lowest preservation of character as well.
Observation 4.3 even so, $37.5$ per cent preserve the class. Not wholly defenceless——with no first line of insurance, a second line remains.
Observation 4.4 the third position has only $50$ non-synonymous substitutions. Out of $183$——it rarely changes anything in the first place.
Observation 4.5 the defence is two-tiered. First “do not change it”, then “change it to something close”——and which tier acts differs by position.
5. Separator: Which Position Was Hit
Observation 5.1 separator——not how many bases changed, but which position changed.
| Position | Synonymous rate | Class preserved | Overall weight |
|---|---|---|---|
| First | $0.0437$ | $0.4699$ | middle |
| Second | $\mathbf{0.0000}$ | $\mathbf{0.3750}$ | heaviest |
| Third | $\mathbf{0.6885}$ | $0.4400$ | lightest |
Observation 5.2 the two middle columns point the same way. The position with the lowest synonymous rate also preserves polarity least——by chance they could have pointed opposite ways.
Observation 5.3 so “one mutation occurred” is not enough information. Without the position the weight is undetermined——a range of $15.75$.
Observation 5.4 this separator can be tested. Whether real sequences accumulate substitutions preferentially at the third position between species——counting synonymous and non-synonymous rates separately settles it.
Observation 5.5 to be honest——both $15.75$ and $94.03$ per cent are counts of possibilities under an assumption of equally likely substitutions. Real mutations are biased by base type——what does not move is that the second position has exactly zero synonymous substitutions, and that $126/8$ divides exactly.
Observation 5.6 one last thing——three letters is one letter more than $20$ outcomes require. $4^{2}=16<20\le64=4^{3}$——the surplus was not discarded; it was placed, all in one position, as margin against error.
6. Roll Call and Closing
- $183$ per position, $549$ in total (Section 1): over the $61$ sense codons.
- $8$ at the first position (Computation 2.1): $4.37$ per cent.
- Exactly $0$ at the second (Computation 2.1): not one policy of insurance.
- $126$ at the third (Computation 2.1): $68.85$ per cent.
- The ratio is $126/8=15.75$ (Observation 2.4): it divides.
- $24.41$ per cent overall (Observation 2.6): describing none of the three.
- $94.03$ per cent of silent errors at the third (Computation 3.1): an even spread gives $33.3$.
- $32$ of $61$ are four-fold degenerate (Observation 3.3): more than half.
- The second preserves polarity least (Observation 4.2): $0.375000$.
- A two-tiered defence (Observation 4.5): do not change; change to something close.
Closing: the genetic code specifies $20$ amino acids with $64$ codons, leaving a surplus of $43$——and that surplus is not distributed evenly. Over the $61$ sense codons there are $183$ single-base substitutions at each position and $549$ in total——of which those leaving the amino acid unchanged number $\mathbf{8}$ at the first position, $\mathbf{0}$ at the second and $\mathbf{126}$ at the third. At the second position it is exactly zero——all $183$ substitutions change the amino acid; not one policy of insurance was bought there. At the third, $68.85$ per cent change nothing and more than two thirds pass in silence. The ratio is $126/8=\mathbf{15.75}$, which divides exactly once the common denominator $183$ is cancelled. The $8$ at the first position is small but not zero ($4.37$ per cent), coming from the six-fold degeneracy of leucine and arginine. The redundancy is gathered in one place——of the $134$ silent errors, $\mathbf{94.03}$ per cent are at the third position, where an even spread would give $33.3$ per cent. Of the $61$ sense codons, $32$ are four-fold degenerate at the third position ($24$ two-fold, $3$ three-fold, $2$ not degenerate), so for more than half the codons the third letter carries no information. As in Paper 391, where the mean of the “wave height” was not representative, a “mutation rate of $24.41$ per cent” fits none of the three positions. And when it does change, the manner of the change also differs——among non-synonymous substitutions the fraction preserving the polarity class is $\mathbf{0.375000}$ at the second position, the lowest of the three. Zero synonymous substitutions, and the lowest preservation of character as well. Even so $37.5$ per cent preserve the class, so with no first line of insurance a second line remains. The defence is two-tiered——first “do not change it”, then “change it to something close”——and which tier acts differs by position. One thing does the separating——not how many bases changed, but which position changed. The position with the lowest synonymous rate also preserves polarity least; by chance the two could have pointed opposite ways. So “one mutation occurred” is not enough information: without the position the weight is undetermined, across a range of $15.75$. By the criterion of Paper 300 the three positions are different-root, playing different roles in tRNA recognition. To be honest——both $15.75$ and $94.03$ per cent are counts of possibilities under an assumption of equally likely substitutions, and real mutations are biased by base type. What does not move is that the second position has exactly zero synonymous substitutions, and that $126/8$ divides exactly. One last thing——three letters is one letter more than $20$ outcomes require ($4^{2}=16<20\le64=4^{3}$). The surplus was not discarded; it was placed, all in one position, as margin against error.
References
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- Nirenberg, M., Leder, P. (1964). RNA Codewords and Protein Synthesis. Science.
- Woese, C. R. (1965). On the evolution of the genetic code. Proceedings of the National Academy of Sciences.
- Freeland, S. J., Hurst, L. D. (1998). The Genetic Code Is One in a Million. Journal of Molecular Evolution.
- Yamagishi, Y. (2026). The Three Hundredth Census: Whether Two Things Share a Root Is Decidable [Paper 300]. Zenodo. 10.5281/zenodo.22171914
- Yamagishi, Y. (2026). “Twice as Hot” Has Three Answers, and Only One Gives a Ratio of $\mathbf{2}$ [Paper 379]. Zenodo. 10.5281/zenodo.22698802
- Yamagishi, Y. (2026). With a Million Candidates the Chance of Picking the Best Stays at $0.3679$ [Paper 388]. Zenodo. 10.5281/zenodo.22699147
- Yamagishi, Y. (2026). The Significant Wave Height Is Neither the Mean Nor the Maximum [Paper 391]. Zenodo. 10.5281/zenodo.22699212
Reference
Yamagishi, Y. (2026). The Same One-Letter Typo Costs $\mathbf{15.75}$ Times More at One Position Than Another, and at the Second It Is Exactly $\mathbf{0}$ [Paper 398]. YOI: https://www.yamagishiyuuki.com/yoi/398
DOI: 10.5281/zenodo.22699326
ORCID: 0009-0006-8587-3450
PDF (English) · PDF (Japanese) · Date proof of the English PDF (OpenTimestamps .ots) · SHA256SUMS
The record page for this paper (in Japanese, with the PDFs in each language and the date proof): YOI 398. This paper is a preprint and has not been peer reviewed.