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Parikh vectors
7HSK_1 2HLJ_1 3ZUZ_1 Letter Amino acid
19 10 24 D Aspartic acid
13 9 6 H Histidine
15 6 28 I Isoleucine
9 5 11 P Proline
8 7 19 Y Tyrosine
11 14 11 A Alanine
23 20 35 L Leucine
14 3 25 K Lycine
21 7 24 S Serine
19 9 14 R Arginine
15 10 21 Q Glutamine
21 8 39 E Glutamic acid
7 4 10 M Methionine
6 6 14 F Phenylalanine
22 10 18 V Valine
4 1 2 C Cysteine
21 12 16 G Glycine
28 10 20 T Threonine
7 2 6 W Tryptophan
12 4 22 N Asparagine 

7HSK_1|Chains A, B|Oleoyl-acyl carrier protein thioesterase 1, chloroplastic|Arabidopsis thaliana (3702)
>2HLJ_1|Chain A|Hypothetical protein|Pseudomonas putida (160488)
>3ZUZ_1|Chain A|PROTEIN SHQ1|SACCHAROMYCES CEREVISIAE (559292)
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 \(w=f(c)\)
7HSK , Knot 131 295 0.84 40 189 276 
MGSLTEDGLSYKEKFVVRSYEVGSNKTATVETIANLLQEVGCNHAQSVGFSTDGFATTTTMRKLHLIWVTARMHIEIYKYPAWGDVVEIETWCQSEGRIGTRRDWILKDSVTGEVTGRATSKWVMMNQDTRRLQKVSDDVRDEYLVFCPQEPRLAFPEENNRSLKKIPKLEDPAQYSMIGLKPRRADLDMNQHVNNVTYIGWVLESIPQEIVDTHELQVITLDYRRECQQDDVVDSLTTTTSEIGGTNGSATSGTQGHNDSQFLHLLRLSGDGQEINRGTTLWRKKPSSHHHHHH
2HLJ , Knot 78 157 0.83 40 121 152 
GMPALITYRTTVQEDWVDYNGHLRDAFYLLIFSYATDALMDRIGLDADSRGQSGNSLFTLEAHINYLHEVKLGTEVWVQTQILGFDRKRLHVYHSLHRAGFDEVLAASEQMLLHVDLAGPQSAPFGHTTVCRLNHLVEQQEGAQAPQYMGRTIKLPA
3ZUZ , Knot 152 365 0.82 40 208 343 
GQMGEGFNWEIEQKMDSSTNNGILKTKYGFDNLYDTVISVSTSNGNDINELDDPEHTDANDRVIERLRKENLKFDPEYYVSEYMTHKYGNEEDLEINGIKELLKFTPSIVKQYLQWYKDSTNPNLVMPIEFTDEEQKQMQDNLPKKSYLVEDIKPLYVTILSVLFSYVFEQIENEGTHTTESAWTMGKLCPQISFLDQQLKQVNELQDGMKEISKVNKDSSLIKIAIITGIRRALSYPLHRNYDLAMKAWTFVYYILRGGKRLVIRALLDIHETFRFHDVYYVYDKVLLDDLTAWFISQGSENVIRSLALEMRKEQESLSKQDIEFECIASFNEQTGEPEWETLNIREMEILAESEYREQQQNPQ

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 \(c\).

\(|P_{f(7HSK_1)}(2) \setminus P_{f(2HLJ_1)}(2)|=119\), \(|P_{f(2HLJ_1)}(2) \setminus P_{f(7HSK_1)}(2)|=51\). 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 1:1101000110000011100001100001010011011001100010011100011100001001011110101010100011110110100100001011000011100010101010100011110000001001000100001110100101111000000100110100110001111010010101000100100111110011001100001011010000000000110010000001110010100100100000110110101010010010011000100000000
Pair \(Z_2\) Length of longest common subsequence
7HSK_1,2HLJ_1 170 3
7HSK_1,3ZUZ_1 153 3
2HLJ_1,3ZUZ_1 187 3

Newick tree

 
[
	2HLJ_1:93.24,
	[
		7HSK_1:76.5,3ZUZ_1:76.5
	]:16.74
]

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 \((0.85)(0.8)(\frac{452 }{\log_{20} 452}-\frac{157}{\log_{20}157})=87.3\)
Status Protein1 Protein2 d d1/2
Query variables 7HSK_1 2HLJ_1 112 85
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]