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Parikh vectors
4KQY_1 4OTX_1 8XWN_1 Letter Amino acid
0 6 19 R Arginine
0 11 19 D Aspartic acid
0 16 70 L Leucine
0 4 6 W Tryptophan
33 13 22 G Glycine
0 23 21 T Threonine
0 14 15 K Lycine
0 3 9 M Methionine
0 11 16 Y Tyrosine
40 10 26 A Alanine
0 10 13 E Glutamic acid
0 2 8 H Histidine
0 7 28 I Isoleucine
0 11 15 P Proline
0 31 38 S Serine
0 12 30 V Valine
0 12 17 N Asparagine
27 4 10 C Cysteine
0 12 8 Q Glutamine
0 7 26 F Phenylalanine 

4KQY_1|Chain A|YitJ S box/SAM-I riboswitch|Bacillus subtilis (1192196)
>4OTX_1|Chains A[auth L], C[auth M]|N203 light chain|Mus musculus (10090)
>8XWN_1|Chain A[auth R]|C-X-C chemokine receptor type 2|Homo sapiens (9606)
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 \(w=f(c)\)
4KQY , Knot 35 119 0.46 8 16 54 
GUUCUUAUCAAGAGAAGCAGAGGGACUGGCCCGACGAAGCUUCAGCAACCGGUGUAAUGGCGAAAGCCAUGACCAAGGUGCUAAAUCCAGCAAGCUCGAACAGCUUGGAAGAUAAGAAC
4OTX , Knot 105 219 0.86 40 158 212 
QIVMTQSPSSLTVTAGEKVTMSCKSSQSLLNSGNQKNYLTWYQQKPGQPPKLLIYWASTRESGVPDRFTGSGSGTDFTLTISSVQAEDLAVYYCQNDYSYPLTFGAGTKLELKRADAAPTVSIFPPSSEQLTSGGASVVCFLNNFYPKDINVKWKIDGSERQNGVLNSWTDQDSKDSTYSMSSTLTLTKDEYERHNSYTCEATHKTSTSPIVKSFNRNE
8XWN , Knot 170 416 0.82 40 220 381 
MGKTIIALSYIFCLVFADYKDDDDAANFTPVNGSSGNQSVRLVTSSSLEVLFQGPGSEDFNMESDSFEDFWKGEDLSNYSYSSTLPPFLLDAAPCEPESLEINKYFVVIIYALVFLLSLLGNSLVMLVILYSRVGRSVTDVYLLNLALADLLFALTLPIWAASKVNGWIFGTFLCKVVSLLKEVNFYSGILLLACISVDRYLAIVHATRTLTQKRYLVKFICLSIWGLSLLLALPVLLFRRTVYSSNVSPACYEDMGNNTANWRMLLRILPQSFGFIVPLLIMLFCYGFTLRTLFKAHMGQKHRAMRVIFAVVLIFLLCWLPYNLVLLADTLMRTQVIQETCERRNHIDRALDATEILGILHSCLNPLIYAFIGQKFRHGLLKILAIHGLISKDSLPKDSRPSFVGSSSGHTSTTL

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 \(c\).

\(|P_{f(4KQY_1)}(2) \setminus P_{f(4OTX_1)}(2)|=12\), \(|P_{f(4OTX_1)}(2) \setminus P_{f(4KQY_1)}(2)|=154\). 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 1:10000010011111111011111110011000110111100001101100110101101101111100101100111101001110001101110001110110001111110111110
Pair \(Z_2\) Length of longest common subsequence
4KQY_1,4OTX_1 166 3
4KQY_1,8XWN_1 232 2
4OTX_1,8XWN_1 182 4

Newick tree

 
[
	8XWN_1:11.43,
	[
		4KQY_1:83,4OTX_1:83
	]:27.43
]

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 \((0.85)(0.8)(\frac{338 }{\log_{20} 338}-\frac{119}{\log_{20}119})=67.5\)
Status Protein1 Protein2 d d1/2
Query variables 4KQY_1 4OTX_1 101 67.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]

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