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Parikh vectors
1SJV_1 3NCE_1 4NKR_1 Letter Amino acid
6 12 6 R Arginine
2 4 2 C Cysteine
5 10 12 K Lycine
5 10 3 N Asparagine
1 8 6 H Histidine
11 17 5 S Serine
9 7 12 T Threonine
8 7 4 Y Tyrosine
3 8 9 D Aspartic acid
6 18 15 E Glutamic acid
13 4 14 G Glycine
2 6 1 M Methionine
5 5 7 F Phenylalanine
2 6 6 P Proline
8 10 12 A Alanine
8 8 9 Q Glutamine
3 12 10 I Isoleucine
7 23 20 L Leucine
2 2 1 W Tryptophan
8 9 12 V Valine 

1SJV_1|Chain A|r9|Lama glama (9844)
>3NCE_1|Chain A|Prolactin|Homo sapiens (9606)
>4NKR_1|Chains A, B, C, D, E|Molybdopterin-guanine dinucleotide biosynthesis protein B|Bacillus subtilis subsp. spizizenii (655816)
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 \(w=f(c)\)
1SJV , Knot 58 114 0.80 40 97 111 
QVQLQESGGGLVQAGESLKLSCAASGNTFSGGFMGWYRQAPGKQRELVATINSRGITNYADFVKGRFTISRDNAKKTVYLEMNSLEPEDTAVYYCYTHYFRSYWGQGTQVTVSS
3NCE , Knot 91 186 0.85 40 142 179 
MLRDLFDRAVVLSAYIHNLSSEMFSEFDKRYTHGRGFITKAINSCHTSSLATPEDKEQAQQMNQKDFLSLIVSILRSWNEPLYHLVTEVRGMQEAPEAILSKAVEIEEQTKRLLERMELIVSQVHPETKENEIYPVWSGLPSLQMADEESRLSAYYNLLHCLRRDSHKIDNYLKLLKCRIIHNNNC
4NKR , Knot 82 166 0.84 40 123 162 
FPIVQVVGFQNSGKTTFIERILEKASEQGLNLGCLKHHGHGGEPQTFTEGKDTDRYQAAGADVTAVEGAGVLQLTARRLWDLTRLIELYQFLETDCLLIEGFKKAPYPKVVILSEKEDLEALKTVNTIAIIYRKKEHMTEHQGLPIFHADDPVAVDLVLSQLKGES

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 \(c\).

\(|P_{f(1SJV_1)}(2) \setminus P_{f(3NCE_1)}(2)|=53\), \(|P_{f(3NCE_1)}(2) \setminus P_{f(1SJV_1)}(2)|=98\). 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 1:010100011111011001010011010010111111000111000011101000110001011010101000010001010100101000110000000100011010010100
Pair \(Z_2\) Length of longest common subsequence
1SJV_1,3NCE_1 151 3
1SJV_1,4NKR_1 140 3
3NCE_1,4NKR_1 153 3

Newick tree

 
[
	3NCE_1:77.89,
	[
		1SJV_1:70,4NKR_1:70
	]:7.89
]

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 \((0.85)(0.8)(\frac{300 }{\log_{20} 300}-\frac{114}{\log_{20}114})=58.1\)
Status Protein1 Protein2 d d1/2
Query variables 1SJV_1 3NCE_1 72 59
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]