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Parikh vectors
8GAS_1 2BFA_1 3QKK_1 Letter Amino acid
11 21 19 T Threonine
2 7 5 C Cysteine
1 8 10 H Histidine
6 16 14 P Proline
1 6 13 M Methionine
13 20 16 S Serine
4 11 15 Y Tyrosine
2 43 20 A Alanine
5 16 19 R Arginine
13 26 33 L Leucine
6 8 26 K Lycine
4 6 21 F Phenylalanine
2 2 3 W Tryptophan
6 25 20 V Valine
7 4 10 Q Glutamine
3 12 31 E Glutamic acid
6 10 13 I Isoleucine
1 12 8 N Asparagine
7 13 22 D Aspartic acid
12 22 23 G Glycine 

8GAS_1|Chains A, E[auth F], I[auth L]|vFP48.02 light chain|Mus musculus (10090)
>2BFA_1|Chains A, B, C, D|PTERIDINE REDUCTASE 1|LEISHMANIA MAJOR (5664)
>3QKK_1|Chain A|RAC-alpha serine/threonine-protein kinase|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 \(w=f(c)\)
8GAS , Knot 58 112 0.81 40 93 108 
DVVMTQTPLTLSVTIGQPASISCKSSQSLLDSDGKTYLNWLLQRPGQSPKRLIYLVSKLDSGVPDRFTGSGSGTDFTLKISRVEAEDLGIYYCWQGTHFPQTFGGGTRLEIK
2BFA , Knot 123 288 0.80 40 172 271 
MTAPTVPVALVTGAAKRLGRSIAEGLHAEGYAVCLHYHRSAAEANALSATLNARRPNSAITVQADLSNVATAPVSGADGSAPVTLFTRCAELVAACYTHWGRCDVLVNNASSFYPTPLLRNDEDGHEPCVGDREAMETATADLFGSNAIAPYFLIKAFAHRVAGTPAKHRGTNYSIINMVDAMTNQPLLGYTIYTMAKGALEGLTRSAALELAPLQIRVNGVGPGLSVLVDDMPPAVWEGHRSKVPLYQRDSSAAEVSDVVIFLCSSKAKYITGTCVKVDGGYSLTRA
3QKK , Knot 153 341 0.87 40 212 334 
GAMARVTMNEFEYLKLLGKGTFGKVILVKEKATGRYYAMKILKKEVIVAKDEVAHTLTENRVLQNSRHPFLTALKYSFQTHDRLCFVMEYANGGELFFHLSRERVFSEDRARFYGAEIVSALDYLHSEKNVVYRDLKLENLMLDKDGHIKITDFGLCKEGIKDGATMKTFCGTPEYLAPEVLEDNDYGRAVDWWGLGVVMYEMMCGRLPFYNQDHEKLFELILMEEIRFPRTLGPEAKSLLSGLLKKDPKQRLGGGSEDAKEIMQHRFFAGIVWQHVYEKKLSPPFKPQVTSETDTRYFDEEFTAQMITITPPDQDDSMECVDSERRPHFPQFDYSASSTA

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 \(c\).

\(|P_{f(8GAS_1)}(2) \setminus P_{f(2BFA_1)}(2)|=40\), \(|P_{f(2BFA_1)}(2) \setminus P_{f(8GAS_1)}(2)|=119\). 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 1:0111000110101011011010000000110001000101110011001001101100100111001010101001010100101001110001010011001111001010
Pair \(Z_2\) Length of longest common subsequence
8GAS_1,2BFA_1 159 3
8GAS_1,3QKK_1 175 4
2BFA_1,3QKK_1 178 3

Newick tree

 
[
	3QKK_1:90.98,
	[
		8GAS_1:79.5,2BFA_1:79.5
	]:11.48
]

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 \((0.85)(0.8)(\frac{400 }{\log_{20} 400}-\frac{112}{\log_{20}112})=87.6\)
Status Protein1 Protein2 d d1/2
Query variables 8GAS_1 2BFA_1 108 74.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]