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Parikh vectors
1BQL_1 6RUL_1 8ENX_1 Letter Amino acid
2 5 8 H Histidine
13 9 18 K Lycine
11 11 11 Y Tyrosine
11 21 13 A Alanine
5 4 4 C Cysteine
10 18 16 E Glutamic acid
5 8 4 M Methionine
5 8 10 W Tryptophan
11 23 26 V Valine
10 11 16 N Asparagine
8 11 6 Q Glutamine
9 13 19 I Isoleucine
9 17 19 L Leucine
6 9 15 F Phenylalanine
34 22 24 S Serine
10 18 18 D Aspartic acid
14 28 19 G Glycine
22 11 28 T Threonine
7 18 13 R Arginine
10 14 15 P Proline 

1BQL_1|Chain A[auth L]|HYHEL-5 FAB (LIGHT CHAIN)|Mus musculus (10090)
>6RUL_1|Chain A|GFP-LAMA-F98 a GFP enhancer nanobody with cpDHFR insertion,Dihydrofolate reductase,GFP-LAMA-F98 a GFP enhancer nanobody with cpDHFR insertion,Dihydrofolate reductase,Dihydrofolate reductase,GFP-LAMA-F98 a GFP enhancer nanobody with cpDHFR insertion,Dihydrofolate reductase,GFP-LAMA-F98 a GFP enhancer nanobody with cpDHFR insertion|Lama glama (9844)
>8ENX_1|Chains A, B|Coatomer subunit beta'|Saccharomyces cerevisiae (4932)
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 \(w=f(c)\)
1BQL , Knot 100 212 0.84 40 146 202 
DIVLTQSPAIMSASPGEKVTMTCSASSSVNYMYWYQQKSGTSPKRWIYDTSKLASGVPVRFSGSGSGTSYSLTISSMETEDAATYYCQQWGRNPTFGGGTKLEIKRADAAPTVSIFPPSSEQLTSGGASVVCFLNNFYPKDINVKWKIDGSERQNGVLNSWTDQDSKDSTYSMSSTLTLTKDEYERHNSYTCEATHKTSTSPIVKSFNRNEC
6RUL , Knot 122 279 0.82 40 185 267 
GQVQLVESGGALVQPGGSLRLSCAASGFPVNRYSMRWYRQAPGKEREWVAGMSSAGDRSSYEDSVKGRFTISRDDARNTVYLQMNSLKPEDTAVYYCNVNVGFLPADLAWFKRNTLNKPVIMGRHTWESIGRPLPGRKNIILSSQPGTDDRVTWVKSVDEAIAACGDVPEIMVIGGGRVYEQFLPKAQKLYLTHIDAEVEGDTHFPDYEPDDWESVFSEFHDADAQNSHSYCFEILERRGGGGGMISLIAALAVDRVIGMENAMPWNEYWGQGTQVTVS
8ENX , Knot 128 302 0.80 40 182 277 
SMKLDIKKTFSNRSDRVKGIDFHPTEPWVLTTLASGRVEIWNYETQVEVRSIQVTETPVRAGKFIARKNWIIVGSDDFRIRVFNYNTGEKVVDFEAHPDYIRSIAVHPTKPYVLSGSDDLTVKLWNWENNWALEQTFEGHEHFVMCVAFNPKDPSTFASGCLDRTVKVWSLGQSTPNFTLTTGQERGVNYVDYYPLPDKPYMITASDDLTIKIWDYQTKSCVATLEGHMSNVSFAVFHPTLPIIISGSEDGTLKIWNSSTYKVEKTLNVGLERSWCIATHPTGRKNYIASGFDNGFTVLSLG

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 \(c\).

\(|P_{f(1BQL_1)}(2) \setminus P_{f(6RUL_1)}(2)|=67\), \(|P_{f(6RUL_1)}(2) \setminus P_{f(1BQL_1)}(2)|=106\). 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 1:01110001111010110010100010001001010000010010011000001101111010101010000101001000011000000110010111100101001011101011110000100111011011001010010101010100000111001000000000001000101000000000000001000000011100100000
Pair \(Z_2\) Length of longest common subsequence
1BQL_1,6RUL_1 173 4
1BQL_1,8ENX_1 170 4
6RUL_1,8ENX_1 191 4

Newick tree

 
[
	6RUL_1:93.05,
	[
		1BQL_1:85,8ENX_1:85
	]:8.05
]

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 \((0.85)(0.8)(\frac{491 }{\log_{20} 491}-\frac{212}{\log_{20}212})=80.7\)
Status Protein1 Protein2 d d1/2
Query variables 1BQL_1 6RUL_1 104 90
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]