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Parikh vectors
8ESM_1 1JGN_1 3TOF_1 Letter Amino acid
14 5 2 M Methionine
33 8 1 S Serine
8 0 0 C Cysteine
30 6 13 G Glycine
12 3 1 H Histidine
18 2 4 D Aspartic acid
14 7 5 Q Glutamine
17 8 4 E Glutamic acid
15 5 7 K Lycine
20 0 1 Y Tyrosine
54 13 5 A Alanine
34 2 4 R Arginine
19 2 3 N Asparagine
26 8 6 P Proline
13 5 8 T Threonine
16 0 2 W Tryptophan
34 6 6 V Valine
23 3 15 I Isoleucine
57 14 10 L Leucine
31 1 2 F Phenylalanine 

8ESM_1|Chains A, B|Diacylglycerol O-acyltransferase 1|Homo sapiens (9606)
>1JGN_1|Chain A|polyadenylate-binding protein 1|Homo sapiens (9606)
>3TOF_1|Chains A, B|Gag-Pol polyprotein|Human immunodeficiency virus type 1 (BRU ISOLATE) (11686)
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 \(w=f(c)\)
8ESM , Knot 194 488 0.82 40 244 452 
MGDRGSSRRRRTGSRPSSHGGGGPAAAEEEVRDAAAGPDVGAAGDAPAPAPNKDGDAGVGSGHWELRCHRLQDSLFSSDSGFSNYRGILNWCVVMLILSNARLFLENLIKYGILVDPIQVVSLFLKDPYSWPAPCLVIAANVFAVAAFQVEKRLAVGALTEQAGLLLHVANLATILCFPAAVVLLVESITPVGSLLALMAHTILFLKLFSYRDVNSWCRRARAKAASAGKKASSAAAPHTVSYPDNLTYRDLYYFLFAPTLCYELNFPRSPRIRKRFLLRRILEMLFFTQLQVGLIQQWMVPTIQNSMKPFKDMDYSRIIERLLKLAVPNHLIWLIFFYWLFHSCLNAVAELMQFGDREFYRDWWNSESVTYFWQNWNIPVHKWCIRHFYKPMLRRGSSKWMARTGVFLASAFFHEYLVSVPLRMFRLWAFTGMMAQIPLAWFVGRFFQGNYGNAAVWLSLIIGQPIAVLMYVHDYYVLNYEAPAAEA
1JGN , Knot 46 98 0.71 34 75 96 
GPLGSPLTASMLASAPPQEQKQMLGERLFPLIQAMHPTLAGKITGMLLEIDNSELLHMLESPESLRSKVDEAVAVLQAHQAKEAAQKAVNSATGVPTV
3TOF , Knot 53 99 0.82 38 77 95 
PQITLWKRPLVTIKIGGQLKEALLDTGADDTVIEEMSLPGRWKPKMIGGIGGFIKVRQYDQIIIEIAGHKAIGTVLVGPTPVNIIGRNLLTQIGATLNF

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 \(c\).

\(|P_{f(8ESM_1)}(2) \setminus P_{f(1JGN_1)}(2)|=190\), \(|P_{f(1JGN_1)}(2) \setminus P_{f(8ESM_1)}(2)|=21\). 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 1:11001000000010010001111111100010011111011111011111100010111101010100001000110000110000111010111111001011100110011110110110111001001111011111011111110100011111100011111011011011011111111100101110111111001111011000010010001010110110010011110010010010000100111110100010110010100011100110111100101111001111010001011001000011001101111001111111011100010111011011000100011000010011001011100101001001110010001110011111011100011011101101111011110111111110110100101111101111011111101000011000111101
Pair \(Z_2\) Length of longest common subsequence
8ESM_1,1JGN_1 211 4
8ESM_1,3TOF_1 217 3
1JGN_1,3TOF_1 110 3

Newick tree

 
[
	8ESM_1:11.41,
	[
		1JGN_1:55,3TOF_1:55
	]:64.41
]

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 \((0.85)(0.8)(\frac{586 }{\log_{20} 586}-\frac{98}{\log_{20}98})=143.\)
Status Protein1 Protein2 d d1/2
Query variables 8ESM_1 1JGN_1 181 106.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]