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Parikh vectors
9BXM_1 7PLN_1 9MKO_1 Letter Amino acid
12 4 10 R Arginine
11 15 7 N Asparagine
6 12 5 C Cysteine
16 14 23 A Alanine
11 14 2 H Histidine
19 16 7 K Lycine
6 12 6 P Proline
3 2 1 W Tryptophan
9 9 4 Q Glutamine
25 11 10 E Glutamic acid
4 5 3 M Methionine
18 27 14 S Serine
15 14 8 Y Tyrosine
14 21 11 V Valine
14 10 10 D Aspartic acid
17 11 9 I Isoleucine
20 17 12 L Leucine
16 11 5 F Phenylalanine
17 26 5 T Threonine
20 20 12 G Glycine 

9BXM_1|Chains A, B|5-thiohistidine N-methyltransferase OvoM|Sulfuricurvum sp. (2025608)
>7PLN_1|Chains A, B|Sporozoite micronemal protein essential for cell traversal, putative|Plasmodium vivax (5855)
>9MKO_1|Chains A, C, E[auth F], G[auth I], I[auth L], K[auth O]|R-phycoerythrin class I alpha subunit|Ceramium secundatum (193556)
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 \(w=f(c)\)
9BXM , Knot 119 273 0.81 40 171 260 
MGSSHHHHHHSSGLVPRGSHMIFKEKNIYETDESVSQYCDFQYGEDCFGVLNFALACATKAIGYTKETPKNRALDLGCATGRASFELARSYKHVDGVDYSQAFVDAATELQKNGCIGYSQNGEGELKNYKVIDREGYAFRDSFTKVEFFQGDACNLTPQFKEYDLIMATNLIDRLYEPRLFLENIHKRINEKGYLILTSPYTWREEYTAKKFWIGGYVDENGKEVSTLEGLKEILEIHFELVATEDIPFVIRETSRKFQHTISQMSVWKVIST
7PLN , Knot 122 271 0.84 40 180 260 
ETGKDIVKILTASTTVTKTGPPPISAECPHNMVVLFGFVVKQNFWDHTNKLQSYEMEICESGASSCTSKQGNTNKYDVSYTYIECGPQALPFTEQVVSVSGTTYNSVKCPNDYSVLFGFGMATSSGRHQSALYSYFTPCRPGLKSCSLNMNEHDDKSYIYLVCVDATIWTGLNALSMIAKDDLHSAVNRYQQFNDGELVVTCPSEGTILTGFYGETHTSSPYVTVPFGKCAKSLKACSVHGSGQAIGIHNYRTLFTVALCKNNKTHHHHHH
9MKO , Knot 78 164 0.80 40 123 160 
MKSVITTTISAADAAGRFPSSSDLESVQGNIQRAASRLEAAEKLAGNHEAVVKEAGDACFAKYPYLKNPGEAGDSQEKINKCYRDIDHYMRLINYSLVVGGTGPLDEWCIAGAREVYRALNLPGSSYIAAFVFTRDRLCVPRDMSAQAAVEFSGALDYVINSLC

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 \(c\).

\(|P_{f(9BXM_1)}(2) \setminus P_{f(7PLN_1)}(2)|=88\), \(|P_{f(7PLN_1)}(2) \setminus P_{f(9BXM_1)}(2)|=97\). 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 1:110000000000111101001110000100000010000010010001111011110100111000001000110110101010101100000101100001110110010001011000010101000011000101100010010110101001010100001111001100100101110010001000101110010010000010011111010001001001011001101010111000111110000001000100101101100
Pair \(Z_2\) Length of longest common subsequence
9BXM_1,7PLN_1 185 6
9BXM_1,9MKO_1 162 4
7PLN_1,9MKO_1 183 3

Newick tree

 
[
	7PLN_1:95.38,
	[
		9BXM_1:81,9MKO_1:81
	]:14.38
]

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 \((0.85)(0.8)(\frac{544 }{\log_{20} 544}-\frac{271}{\log_{20}271})=77.3\)
Status Protein1 Protein2 d d1/2
Query variables 9BXM_1 7PLN_1 99 99
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]