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Parikh vectors
6YST_1 3WUA_1 5AIL_1 Letter Amino acid
7 11 2 R Arginine
2 3 11 Q Glutamine
4 12 6 G Glycine
1 2 6 P Proline
5 10 12 S Serine
5 7 11 T Threonine
5 12 13 A Alanine
1 14 9 N Asparagine
6 6 18 K Lycine
2 3 3 Y Tyrosine
3 7 5 D Aspartic acid
4 1 7 H Histidine
0 3 11 F Phenylalanine
4 6 16 V Valine
0 8 3 C Cysteine
1 2 13 E Glutamic acid
2 6 14 I Isoleucine
3 8 17 L Leucine
2 2 7 M Methionine
0 6 2 W Tryptophan 

6YST_1|Chain A[auth 0]|50S ribosomal protein L32|Escherichia coli (562)
>3WUA_1|Chain A|Lysozyme C|Gallus gallus (9031)
>5AIL_1|Chains A, B|POLY [ADP-RIBOSE] POLYMERASE 9|HOMO SAPIENS (9606)
Protein code \(c\) LZ-complexity \(\mathrm{LZ}(w)\) Length \(n=|w|\) \(\frac{\mathrm{LZ}(w)}{n /\log_{20} n}\) \(p_w(1)\) \(p_w(2)\) \(p_w(3)\) Sequence \(w=f(c)\)
6YST , Knot 34 57 0.80 34 51 55 
MAVQQNKPTRSKRGMRRSHDALTAVTSLSVDKTSGEKHLRHHITADGYYRGRKVIAK
3WUA , Knot 66 129 0.82 40 104 127 
KVFGRCELAAAMKRHGLDNYRGYSLGNWVCAAKFESNFNTQATNRNTDGSTDYGILQINSRWWCNDGRTPGSRNLCNIPCSALLSSDITASVNCAKKIVSDGNGMNAWVAWRNRCKGTDVQAWIRGCRL
5AIL , Knot 90 186 0.84 40 141 183 
SMNAMVVNNLTLQIVQGHIEWQTADVIVNSVNPHDITVGPVAKSILQQAGVEMKSEFLATKAKQFQRSQLVLVTKGFNLFCKYIYHVLWHSEFPKPQILKHAMKECLEKCIEQNITSISFPALGTGNMEIKKETAAEILFDEVLTFAKDHVKHQLTVKFVIFPTDLEIYKAFSSEMAKRSKMLSLN

Let \(P_w(n)\) be the set of distinct subwords (intervals) in a word \(w\). Let \(p_w(n)\) be the cardinality of \(P_w(n)\). Let \(f(c)\) be the sequence in FASTA with 4-symbol Protein Data Bank code \(c\).

\(|P_{f(6YST_1)}(2) \setminus P_{f(3WUA_1)}(2)|=34\), \(|P_{f(3WUA_1)}(2) \setminus P_{f(6YST_1)}(2)|=87\). Let \( Z_k(x,y)=|P_x(k)\setminus P_y(k)|+|P_y(k)\setminus P_x(k)| \) be a LZ76 style (set of subwords) Jaccard distance numerator for \(P(k)\).Hydrophobic-polar version of Sequence 1:111000010000011000001101100101000010001000101010001001110
Pair \(Z_2\) Length of longest common subsequence
6YST_1,3WUA_1 121 3
6YST_1,5AIL_1 142 3
3WUA_1,5AIL_1 169 3

Newick tree

 
[
	5AIL_1:83.07,
	[
		6YST_1:60.5,3WUA_1:60.5
	]:22.57
]

Let d be the Otu--Sayood distance d.
Let d1 be the Otu--Sayood distance d1. (This makes the 4TYN sequence AAAAAA a close match...)
A roughly speaking expected distance is \((0.85)(0.8)(\frac{186 }{\log_{20} 186}-\frac{57}{\log_{20}57})=43.7\)
Status Protein1 Protein2 d d1/2
Query variables 6YST_1 3WUA_1 55 39
Was not able to put for d
Was not able to put for d1

In notation analogous to [Theorem 16, Kjos-Hanssen, Niraula and Yoon (2022)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]

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