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Parikh vectors
3AMB_1 7QES_1 6OXW_1 Letter Amino acid
16 6 4 N Asparagine
18 7 4 D Aspartic acid
2 9 2 C Cysteine
9 5 1 H Histidine
25 16 2 F Phenylalanine
6 7 2 W Tryptophan
25 11 3 A Alanine
16 11 4 R Arginine
14 6 5 Q Glutamine
21 20 13 I Isoleucine
17 13 0 S Serine
13 14 8 T Threonine
27 11 4 E Glutamic acid
32 19 12 L Leucine
35 17 7 K Lycine
6 7 2 M Methionine
14 8 6 P Proline
19 23 6 V Valine
22 11 13 G Glycine
14 9 1 Y Tyrosine 

3AMB_1|Chain A|cAMP-dependent protein kinase catalytic subunit alpha|Homo sapiens (9606)
>7QES_1|Chains A, B|Gap junction beta-2 protein|Homo sapiens (9606)
>6OXW_1|Chains A, B|Protease NL4-3|Human immunodeficiency virus 1 (11676)
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 \(w=f(c)\)
3AMB , Knot 150 351 0.83 40 205 336 
MGNAAAAKKGSEQESVKEFLAKAKEDFLKKWESPAQNTAHLDQFERIRTLGTGSFGRVMLVKHKETGNHYAMKILDKQKVVKLKQIEHTLNEKRIQQAVNFPFLVKLEFSFKDNSNLYMVLEYAPGGEMFSHLRRIGRFSEPHARFYAAQIVLTFEYLHSLDLIYRDLKPENLLIDQQGYIKVADFGFAKRVKGRTWTLCGTPEYLAPEIILSKGYNKAVDWWALGVLIYEMAAGYPPFFADQPIQIYEKIVSGKVRFPSHFSSDLKDLLRNLLQVDLTKRFGNLKNGVNDIKNHKWFATTDWIAIYQRKVEAPFIPKFKGPGDTSNFDDYEEEEIRVSINEKCGKEFSEF
7QES , Knot 106 230 0.83 40 170 224 
MDWGTLQTILGGVNKHSTSIGKIWLTVLFIFRIMILVVAAKEVWGDEQADFVCNTLQPGCKNVCYDHYFPISHIRLWALQLIFVSTPALLVAMHVAYRRHEKKRKFIKGEIKSEFKDIEEIKTQKVRIEGSLWWTYTSSIFFRVIFEAAFMYVFYVMYDGFSMQRLVKCNAWPCPNTVDCFVSRPTEKTVFTVFMIAVSGICILLNVTELCYLLIRYCSGKSKKPVLVPR
6OXW , Knot 52 99 0.80 38 79 94 
PQITLWKRPLVTIKIGGQLKEALLDTGADDTVLEEMNLPGRWKPKMIGGIGGFIKVRQYDQILIEICGHKAIGTVLVGPTPVNIIGRNLLTQIGCTLNF

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 \(c\).

\(|P_{f(3AMB_1)}(2) \setminus P_{f(7QES_1)}(2)|=109\), \(|P_{f(7QES_1)}(2) \setminus P_{f(3AMB_1)}(2)|=74\). 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 1:110111100100000100111010001100100110001010010010011010110111100000100011011000011010010001000010011011111010101000001011100111101100100110100101010110111010010010110001010011100010101101111001010010101010011101110010001101111111100111101111100110100011010101100100010011001101010001101001100100001110001111000010111110101110000100000001010100001001001
Pair \(Z_2\) Length of longest common subsequence
3AMB_1,7QES_1 183 3
3AMB_1,6OXW_1 186 4
7QES_1,6OXW_1 163 3

Newick tree

 
[
	3AMB_1:95.56,
	[
		7QES_1:81.5,6OXW_1:81.5
	]:14.06
]

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 \((0.85)(0.8)(\frac{581 }{\log_{20} 581}-\frac{230}{\log_{20}230})=99.7\)
Status Protein1 Protein2 d d1/2
Query variables 3AMB_1 7QES_1 128 105.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]