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Parikh vectors
1ODW_1 3BAM_1 4NRM_1 Letter Amino acid
2 0 10 F Phenylalanine
13 2 15 G Glycine
1 0 7 H Histidine
12 0 17 I Isoleucine
6 0 12 K Lycine
2 0 2 M Methionine
1 0 8 Y Tyrosine
7 0 21 V Valine
3 4 12 A Alanine
4 0 12 D Aspartic acid
2 2 5 C Cysteine
6 0 15 P Proline
8 4 5 T Threonine
12 0 15 L Leucine
1 0 14 S Serine
2 0 1 W Tryptophan
4 0 21 R Arginine
3 0 4 N Asparagine
6 0 13 Q Glutamine
4 0 21 E Glutamic acid 

1ODW_1|Chains A, B|HIV-1 PROTEASE|Human immunodeficiency virus 1 (11676)
>3BAM_1|Chain A[auth C]|DNA (5'-D(*TP*AP*TP*GP*GP*AP*TP*CP*CP*AP*TP*A)-3')|
>4NRM_1|Chain A|RNA demethylase ALKBH5|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 \(w=f(c)\)
1ODW , Knot 51 99 0.79 40 81 94 
PQVTLWQRPLVTIKIGGQLKEALLDTGADDTVLEEMSLPGRWKPKMIGGIGGFIKVRQYDQILIEICGHKAIGTVLVGPTPVNIIGRNLLTQIGCTLNF
3BAM , Knot 7 12 0.48 8 8 10 
TATGGATCCATA
4NRM , Knot 105 230 0.82 40 160 220 
GHMPERSDYEEQQLQKEEEARKVKSGIRQMRLFSQDECAKIEARIDEVVSRAEKGLYNEHTVDRAPLRNKYFFGEGYTYGAQLQKRGPGQERLYPPGDVDEIPEWVHQLVIQKLVEHRVIPEGFVNSAVINDYQPGGCIVSHVDPIHIFERPIVSVSFFSDSALCFGCKFQFKPIRVSEPVLSLPVRRGSVTVLSGYAADEITHCIRPQDIKERRAVIILRKTRLDAPRL

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 \(c\).

\(|P_{f(1ODW_1)}(2) \setminus P_{f(3BAM_1)}(2)|=78\), \(|P_{f(3BAM_1)}(2) \setminus P_{f(1ODW_1)}(2)|=5\). 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 1:101011001110101110100111001100011001011101010111111111010000011101010011101111101101110011001100101
Pair \(Z_2\) Length of longest common subsequence
1ODW_1,3BAM_1 83 2
1ODW_1,4NRM_1 165 3
3BAM_1,4NRM_1 162 2

Newick tree

 
[
	4NRM_1:91.30,
	[
		1ODW_1:41.5,3BAM_1:41.5
	]:49.80
]

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 \((0.85)(0.8)(\frac{111 }{\log_{20} 111}-\frac{12}{\log_{20}12})=38.1\)
Status Protein1 Protein2 d d1/2
Query variables 1ODW_1 3BAM_1 48 26.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]