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Parikh vectors
3BKZ_1 5AXK_1 9CBI_1 Letter Amino acid
17 11 37 P Proline
9 16 51 S Serine
3 4 10 W Tryptophan
13 21 34 R Arginine
10 8 28 D Aspartic acid
9 17 24 I Isoleucine
21 24 76 L Leucine
4 5 18 M Methionine
10 17 20 F Phenylalanine
12 5 28 Q Glutamine
4 26 51 E Glutamic acid
5 1 18 C Cysteine
17 14 44 G Glycine
7 8 20 H Histidine
7 21 30 K Lycine
9 9 31 T Threonine
7 7 19 Y Tyrosine
20 18 52 A Alanine
8 10 28 N Asparagine
9 9 57 V Valine 

3BKZ_1|Chain A|Alpha-ketoglutarate-dependent dioxygenase alkB|Escherichia coli K12 (83333)
>5AXK_1|Chains A, B|tRNA(His)-5'-guanylyltransferase (Thg1) like protein|Methanosarcina acetivorans (2214)
>9CBI_1|Chain A|Polyamine deacetylase HDAC10|Danio rerio (7955)
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 \(w=f(c)\)
3BKZ , Knot 92 201 0.81 40 145 195 
PLAAGAVILRRFAFNAAEQLIRDINDVASQSPFRQMVTPGGYTMSVAMTNCGHLGWTTHRQGYLYSPIDPQTNKPWPAMPQSFHNLCQRAATAAGYPDFQPDACLINRYAPGAKLSLHQDKCEPDLRAPIVSVSLGLPAIFQFGGLKRNDPLKRLLLEHGDVVVWGGESRLFYHGIQPLKAGFHPLTIDCRYNLTFRQAGK
5AXK , Knot 110 251 0.80 40 159 236 
MKTREIYAEMRCIPPVVLRADGRNFKNTLSGLGFEKPYDKTFARAMADTAELFIKKSGLSPLFAYTFSDEISFLFTDLPFDGRVEKIDSVVASFLGSALTIKLRLEEPIAFDSRLVALQKEEIPEYFHRRQLEAWRNFVASWGYYALRNEGMGRNEAAKYLKRKKESEIHEMLFERGINLATLPSWQRRGVIISKEAREIQGFNPVSGKEEKSLRRKITQNWEIPKFKSEKGIPFLEKLINRNLEHHHHHH
9CBI , Knot 266 676 0.85 40 280 626 
AASGSALIFDEEMSRYKLLWTDPECEIEVPERLTVSYEALRTHGLAQRCKAVPVRQATEQEILLAHSEEYLEAVKQTPGMNVEELMAFSKKYNAVYFHQNIYHCAKLAAGATLQLVDSVMKREVRNGMALVRPPGHHSQRSAANGFCVFNNVAFAALYAKKNYNLNRILIVDWDVHHGQGIQYCFEEDPSVLYFSWHRYEHQSFWPNLPESDYSSVGKGKGSGFNINLPWNKVGMTNSDYLAAFFHVLLPVAYEFDPELVIVSAGFDSAIGDPEGEMCALPEIFAHLTHLLMPLAAGKMCVVLEGGYNLTSLGQSVCQTVHSLLGDPTPRISGLGTACDSALESIQNVRNVQSSYWSSFKHLAQSETNPKRPRLDATNGGPKESSEPASESNPKKTAQDIVWPEPLKRMPASVRTVVVPPPGVELTLPKNCQHSGDISESTAKEVQRIRDKHFHDLTDQNILRSLGNIISVLDRMMRSDEVCNGCVVVSDLSVSVQCALQHALTEPAERVLVVYVGDGELPVKTNDGKVFLVQICTKETEDKCVNRLTLCLREGESLTAGFMQALLGLILPVAYEFNPALVLGIVEETAAKTRLMRVWGHMTCLIQGLARGRMLTLLQGYDKDLLELTVSALSGASISPLGPLRAPKPEDVEMMEKQRQRLQERWGLLRCTVSESW

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 \(c\).

\(|P_{f(3BKZ_1)}(2) \setminus P_{f(5AXK_1)}(2)|=79\), \(|P_{f(5AXK_1)}(2) \setminus P_{f(3BKZ_1)}(2)|=93\). 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 1:111111111001110110011001001100011001101110010111000101110000010100110100001111110010010001101110101010101100011110101000000101011110101111111011110000110011100101111110001100110110111011010000010100110
Pair \(Z_2\) Length of longest common subsequence
3BKZ_1,5AXK_1 172 4
3BKZ_1,9CBI_1 199 5
5AXK_1,9CBI_1 197 4

Newick tree

 
[
	9CBI_1:10.97,
	[
		3BKZ_1:86,5AXK_1:86
	]:16.97
]

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 \((0.85)(0.8)(\frac{452 }{\log_{20} 452}-\frac{201}{\log_{20}201})=73.3\)
Status Protein1 Protein2 d d1/2
Query variables 3BKZ_1 5AXK_1 93 83.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]