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Parikh vectors
3BHN_1 6VRF_1 1NMN_1 Letter Amino acid
7 26 10 R Arginine
12 13 6 Q Glutamine
10 8 2 H Histidine
6 11 6 P Proline
6 9 4 N Asparagine
29 29 13 L Leucine
12 19 7 K Lycine
12 11 8 T Threonine
16 19 7 D Aspartic acid
3 5 0 C Cysteine
21 25 15 G Glycine
14 17 10 I Isoleucine
18 21 7 S Serine
6 8 3 Y Tyrosine
17 21 8 V Valine
14 11 12 A Alanine
15 15 10 E Glutamic acid
7 9 2 M Methionine
9 18 6 F Phenylalanine
2 4 2 W Tryptophan 

3BHN_1|Chain A|ThiJ/PfpI domain protein|Shewanella loihica PV-4 (323850)
>6VRF_1|Chains A, B|Tau-tubulin kinase 2|Homo sapiens (9606)
>1NMN_1|Chains A, B|Hypothetical protein yqgF|Escherichia coli (562)
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 \(w=f(c)\)
3BHN , Knot 110 236 0.85 40 165 226 
MGSDKIHHHHHHENLYFQGMYKVGIVLFDDFTDVDFFLMNDLLGRTSDSWTVRILGTKPEHHSQLGMTVKTDGHVSEVKEQDVVLITSGYRGIPAALQDENFMSALKLDPSRQLIGSICAGSFVLHELGLLKGKKLTTNPDAKAVLQGMGGDVQDLPLVIEGNIATAGGCLSLLYLVGWLAERLFDSVKRKQIQNQLIPAGQMEIFETLISETIQSAESAYEYRSACESDAESLVV
6VRF , Knot 133 299 0.84 40 190 283 
MSGGGEQPDILSVGILVKERWKVLRKIGGGGFGEIYDALDMLTRENVALKVESAQQPKQVLKMEVAVLKKLQGKDHVCRFIGCGRNDRFNYVVMQLQGRNLADLRRSQSRGTFTISTTLRLGRQILESIESIHSVGFLHRDIKPSNFAMGRFPSTCRKCYMLDFGLARQFTNSCGDVRPPRAVAGFRGTVRYASINAHRNREMGRHDDLWSLFYMLVEFVVGQLPWRKIKDKEQVGSIKERYDHRLMLKHLPPEFSIFLDHISSLDYFTKPDYQLLTSVFDNSIKTFGVIESDPFDWEK
1NMN , Knot 65 138 0.77 38 101 131 
MSGTLLAFDFGTKSIGVAVGQRITGTARPLPAIKAQDGTPDWNIIERLLKEWQPDEIIVGLPLNMDGTEQPLTARARKFANRIHGRFGVEVKLHDERLSTVEARSGLFEQGGYRALNKGKVDSASAVIILESYFEQGY

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 \(c\).

\(|P_{f(3BHN_1)}(2) \setminus P_{f(6VRF_1)}(2)|=66\), \(|P_{f(6VRF_1)}(2) \setminus P_{f(3BHN_1)}(2)|=91\). 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 1:11000100000000101011001111110010010111100111000001010111001000001110100010100100001111001001111110000110110101000111010110111001111010010001010111011110100111110101101110101101111110011001000010001111101011001100010010010000010000100111
Pair \(Z_2\) Length of longest common subsequence
3BHN_1,6VRF_1 157 3
3BHN_1,1NMN_1 140 3
6VRF_1,1NMN_1 163 4

Newick tree

 
[
	6VRF_1:83.08,
	[
		3BHN_1:70,1NMN_1:70
	]:13.08
]

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 \((0.85)(0.8)(\frac{535 }{\log_{20} 535}-\frac{236}{\log_{20}236})=85.4\)
Status Protein1 Protein2 d d1/2
Query variables 3BHN_1 6VRF_1 107 95.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]