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Parikh vectors
3OYP_1 3LQZ_1 7VEJ_1 Letter Amino acid
12 4 24 R Arginine
5 7 35 K Lycine
4 2 12 M Methionine
17 6 28 S Serine
9 11 19 D Aspartic acid
20 8 32 G Glycine
4 16 26 F Phenylalanine
14 11 22 P Proline
7 8 17 Q Glutamine
14 17 48 L Leucine
18 13 19 T Threonine
3 9 21 H Histidine
7 6 32 I Isoleucine
2 4 11 W Tryptophan
5 5 11 Y Tyrosine
13 11 39 A Alanine
4 10 24 N Asparagine
7 3 7 C Cysteine
3 17 49 E Glutamic acid
19 13 29 V Valine 

3OYP_1|Chains A, B|Serine protease NS3|Hepatitis C virus (isolate Japanese) (11116)
>3LQZ_1|Chain A|HLA class II histocompatibility antigen, DP alpha 1 chain|Homo sapiens (9606)
>7VEJ_1|Chains A, B|Glycosyltransferase|Phytolacca americana (3527)
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 \(w=f(c)\)
3OYP , Knot 91 187 0.84 40 139 181 
APITAYSQQTRGLLGCIITSLTGRDKNQVDGEVQVLSTATQSFLATCVNGVCWTVYHGAGSKTLAGPKGPITQMYTNVDQDLVGWPAPPGARSMTPCTCGSSDLYLVTRHADVIPVRRRGDSRGSLLSPRPVSYLKGSSGGPLLCPSGHVVGIFRAAVCTRGVAKAVDFIPVESMETTMRSPVFTDN
3LQZ , Knot 91 181 0.87 40 142 176 
IKADHVSTYAAFVQTHRPTGEFMFEFDEDEMFYVDLDKKETVWHLEEFGQAFSFEAQGGLANIAILNNNLNTLIQRSNHTQATNDPPEVTVFPKEPVELGQPNTLICHIDKFFPPVLNVTWLCNGELVTEGVAESLFLPRTDYSFHKFHYLTFVPSAEDFYDCRVEHWGLDQPLLKHWEAQ
7VEJ , Knot 202 505 0.83 40 260 469 
MNHKVHHHHHHLQENLYFQGMGAEPQQLHVVFFPIMAHGHMIPTLDIARLFAARNVRATIITTPLNAHTFTKAIEMGKKNGSPTIHLELFKFPAQDVGLPEGCENLEQALGSSLIEKFFKGVGLLREQLEAYLEKTRPNCLVADMFFPWATDSAAKFNIPRLVFHGTSFFSLCALEVVRLYEPHKNVSSDEELFSLPLFPHDIKMMRLQLPEDVWKHEKAEGKTRLKLIKESELKSYGVIVNSFYELEPNYAEFFRKELGRRAWNIGPVSLCNRSTEDKAQRGKQTSIDEHECLKWLNSKKKNSVIYICFGSTAHQIAPQLYEIAMALEASGQEFIWVVRNNNNNDDDDDDSWLPRGFEQRVEGKGLIIRGWAPQVLILEHEAIGAFVTHCGWNSTLEGITAGVPMVTWPIFAEQFYNEKLVNQILKIGVPVGANKWSRETSIEDVIKKDAIEKALREIMVGDEAEERRSRAKKLKEMAWKAVEEGGSSYSDLSALIEELRGYHA

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 \(c\).

\(|P_{f(3OYP_1)}(2) \setminus P_{f(3LQZ_1)}(2)|=85\), \(|P_{f(3LQZ_1)}(2) \setminus P_{f(3OYP_1)}(2)|=88\). 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 1:1110100000011110110010100000101010110010001110010110101001110001111011100100010001111111111001010001000101100010111100010001011010110010100111110101011111011100011101101111001000100111000
Pair \(Z_2\) Length of longest common subsequence
3OYP_1,3LQZ_1 173 3
3OYP_1,7VEJ_1 199 3
3LQZ_1,7VEJ_1 194 3

Newick tree

 
[
	7VEJ_1:10.87,
	[
		3OYP_1:86.5,3LQZ_1:86.5
	]:15.37
]

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 \((0.85)(0.8)(\frac{368 }{\log_{20} 368}-\frac{181}{\log_{20}181})=55.9\)
Status Protein1 Protein2 d d1/2
Query variables 3OYP_1 3LQZ_1 74 72
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]