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Parikh vectors
4HLG_1 8VNX_1 9ESM_1 Letter Amino acid
12 42 12 L Leucine
7 19 16 T Threonine
9 36 19 V Valine
5 11 0 C Cysteine
7 21 9 Q Glutamine
20 21 13 G Glycine
11 15 4 F Phenylalanine
1 4 2 W Tryptophan
8 22 7 Y Tyrosine
11 18 4 R Arginine
16 12 0 H Histidine
10 34 7 K Lycine
16 37 10 S Serine
7 29 8 D Aspartic acid
11 26 14 I Isoleucine
6 17 3 M Methionine
3 26 2 P Proline
7 27 17 A Alanine
11 20 9 N Asparagine
13 27 8 E Glutamic acid 

4HLG_1|Chains A, C[auth B]|Tankyrase-2|Homo sapiens (9606)
>8VNX_1|Chain A|Mitogen-activated protein kinase 10|Homo sapiens (9606)
>9ESM_1|Chains A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T, U, V, W, X, Y|Archaellin|Halobacterium salinarum (2242)
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 \(w=f(c)\)
4HLG , Knot 91 191 0.83 40 146 184 
MHHHHHHSSGVDLGTENLYFQSMLNTSGSGTILIDLSPDDKEFQSVEEEMQSTVREHRDGGHAGGIFNRYNILKIQKVCNKKLWERYTHRRKEVSEENHNHANERMLFHGSPFVNAIIHKGFDERHAYIGGMFGAGIYFAENSSKSNQYVYGIGGGTGCPVHKDRSCYICHRQLLFCRVTLGKSFLQFSAM
8VNX , Knot 196 464 0.86 40 261 448 
MSLHFLYYCSEPTLDVKIAFCQGFDKQVDVSYIAKHYNMSKSKVDNQFYSVEVGDSTFTVLKRYQNLKPIGSGAQGIVCAAYDAVLDRNVAIKKLSRPFQNQTHAKRAYRELVLMKCVNHKNIISLLNVFTPQKTLEEFQDVYLVMELMDANLCQVIQMELDHERMSYLLYQMLCGIKHLHSAGIIHRDLKPSNIVVKSDCTLKILDFGLARTAGTSFMMTPYVVTRYYRAPEVILGMGYKENVDIWSVGCIMGEMVRHKILFPGRDYIDQWNKVIEQLGTPCPEFMKKLQPTVRNYVENRPKYAGLTFPKLFPDSLFPADSEHNKLKASQARDLLSKMLVIDPAKRISVDDALQHPYINVWYDPAEVEAPPPQIYDKQLDEREHTIEEWKELIYKEVMNSEEKTKNGVVKGQPSPSGAAVNSSESLPPSSSVNDISSMSTDQTLASDTDSSLEASAGPLGCCR
9ESM , Knot 85 193 0.77 38 127 173 
MFEFITDEDERGQVGIGTLIVFIAMVLVAAIAAGVLINTAGYLQSKGSATGEEASAQVSNRINIVSAYGNVXXXXVDYVNLTVRQAAGADNINLXKSTIQWIGPDXATTLTYXXXXXXXXXENFTTXSIKGXXXXVLVDQSDRIKVIMYAXXVXXXLXXGXEVQLTVTTQYGSKTTYWAQVPESLKDKNAVXL

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 \(c\).

\(|P_{f(4HLG_1)}(2) \setminus P_{f(8VNX_1)}(2)|=41\), \(|P_{f(8VNX_1)}(2) \setminus P_{f(4HLG_1)}(2)|=156\). 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 1:10000000011011000101001100010101110101000010010001000100000110111110000110100100001100000000010000000100011101011101110011000010111111111011000000000101111101011000000010000111001011001101011
Pair \(Z_2\) Length of longest common subsequence
4HLG_1,8VNX_1 197 3
4HLG_1,9ESM_1 169 3
8VNX_1,9ESM_1 220 4

Newick tree

 
[
	8VNX_1:11.24,
	[
		4HLG_1:84.5,9ESM_1:84.5
	]:25.74
]

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 \((0.85)(0.8)(\frac{655 }{\log_{20} 655}-\frac{191}{\log_{20}191})=131.\)
Status Protein1 Protein2 d d1/2
Query variables 4HLG_1 8VNX_1 169 118
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]