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Parikh vectors
6NNJ_1 2YJM_1 8VCM_1 Letter Amino acid
3 1 0 C Cysteine
2 10 9 F Phenylalanine
11 11 7 S Serine
7 0 1 W Tryptophan
3 9 4 Y Tyrosine
6 11 5 R Arginine
5 9 12 D Aspartic acid
8 18 5 E Glutamic acid
9 11 8 I Isoleucine
2 5 9 P Proline
12 16 9 A Alanine
10 6 8 N Asparagine
16 6 2 Q Glutamine
5 13 16 K Lycine
7 2 7 T Threonine
8 12 10 V Valine
12 7 12 G Glycine
1 5 13 H Histidine
23 19 10 L Leucine
3 5 5 M Methionine 

6NNJ_1|Chain A[auth B]|Envelope glycoprotein gp41|Human immunodeficiency virus 1 (11676)
>2YJM_1|Chain A|TTRD|ARCHAEOGLOBUS FULGIDUS (2234)
>8VCM_1|Chains A, B|Galactose-binding lectin|Mytilus californianus (6549)
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 \(w=f(c)\)
6NNJ , Knot 71 153 0.77 40 110 145 
AVGIGAVFLGFLGAAGSTMGAASMTLTVQARNLLSGIVQQQSNLLRAPEAQQHLLKLTVWGIKQLQARVLAVERYLRDQQLLGIWGCSGKLICCTNVPWNSSWSNRNLSEIWDNMTWLQWDKEISNYTQIIYGLLEESQNQQEKNEQDLLALD
2YJM , Knot 81 176 0.79 38 126 169 
GAMTIGRAKVYATLSKIFYHLFYDEAIPKDCREIIEKFGEIDFNLRSVLVRELRGSVLIKDMPQSLAEVYESVMKDFYERYGFQASELHADHIAVELAFMSKLVEREISLAQQMKEEELYKIRAAQHRFIKAHLQPLVKNLPSAPLLNFVRDFVREDAKYLYSSLVGEKNEGADNN
8VCM , Knot 68 152 0.75 38 107 134 
GHMTTFLIKHKASGKFLHPKGGSSNPANDTNLVLHSDIHERMYFQFDVVDERWGYIKHAASGKIVHPLGGKADPPNETKLVLHQDRHDRALFAMDFFNDNIIHKAGKYIHPKGGSTNPPNETLTVMHGDKHKAMEFIFVSPKDKDKRVLVYV

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 \(c\).

\(|P_{f(6NNJ_1)}(2) \setminus P_{f(2YJM_1)}(2)|=63\), \(|P_{f(2YJM_1)}(2) \setminus P_{f(6NNJ_1)}(2)|=79\). 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 1:111111111111111100111101010101001101110000011011010001101011110010101111000100001111110010110000111000100001001100101101000100000110111000000000000011110
Pair \(Z_2\) Length of longest common subsequence
6NNJ_1,2YJM_1 142 4
6NNJ_1,8VCM_1 153 4
2YJM_1,8VCM_1 157 3

Newick tree

 
[
	8VCM_1:79.55,
	[
		6NNJ_1:71,2YJM_1:71
	]:8.55
]

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 \((0.85)(0.8)(\frac{329 }{\log_{20} 329}-\frac{153}{\log_{20}153})=53.6\)
Status Protein1 Protein2 d d1/2
Query variables 6NNJ_1 2YJM_1 69 63.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]