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Parikh vectors
2JDG_1 7VLX_1 4AQS_1 Letter Amino acid
4 9 22 T Threonine
4 4 5 W Tryptophan
14 32 43 G Glycine
18 5 25 R Arginine
10 7 15 Q Glutamine
7 11 11 M Methionine
4 41 25 A Alanine
10 3 19 H Histidine
2 3 22 K Lycine
6 9 21 F Phenylalanine
9 10 30 P Proline
14 3 21 Y Tyrosine
6 23 24 V Valine
7 3 46 C Cysteine
11 8 37 D Aspartic acid
10 6 37 E Glutamic acid
10 35 23 I Isoleucine
18 34 37 L Leucine
13 13 34 S Serine
6 9 28 N Asparagine 

2JDG_1|Chain A|GAMMA CRYSTALLIN B|HOMO SAPIENS (9606)
>7VLX_1|Chains A, C, E[auth Y]|Mannose/fructose/sorbose family PTS transporter subunit IIC|Listeria monocytogenes (1639)
>4AQS_1|Chain A|LAMININ SUBUNIT BETA-1|MUS MUSCULUS (10090)
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 \(w=f(c)\)
2JDG , Knot 84 183 0.79 40 127 175 
MGSIIFLEDRAFQGRIYGCTTDCPNLQPYFSRCNSIVVQSGCWMIYERPNYQGHQYFLRRGEYPDYQQWMGLSDSIRSCCLIPPHSGAYRMKIYDRDELRGQMSELTDDCLSVQDRFHLTEIHSLNVLEGSWILYEMPNYRGRQYLLRPGEYRRFLDWGAPNAKVGSLRRVMDLYLEHHHHHH
7VLX , Knot 109 268 0.75 40 146 234 
MSVISIILVVLIAFLAGIEGILDEFQFHQPLIACTLIGLVTGNLTACIILGGTLQMIALGWANIGAAVAPDAALASVASAIILVLGGQGVAGIPSAIAIAIPLAVAGLFLTMIVRTLAVPIVHLMDRAAEKGNIRSVEWLHISAICMQGIRIAIPAAALLFIPADSVQSFLEAMPAWLTDGMAIGGGMVVAVGYALVINMMATKEVWPFFVIGFVVAAISQLTLIAIGALGVALALIYLNLSKMGGGNSNGGGGGNSRDPLGDILNDY
4AQS , Knot 215 525 0.85 40 269 487 
APLAQEPEFSYGCAEGSCYPATGDLLIGRAQKLSVTSTCGLHKPEPYCIVSHLQEDKKCFICDSRDPYHETLNPDSHLIENVVTTFAPNRLKIWWQSENGVENVTIQLDLEAEFHFTHLIMTFKTFRPAAMLIERSSDFGKAWGVYRYFAYDCESSFPGISTGPMKKVDDIICDSRYSDIEPSTEGEVIFRALDPAFKIEDPYSPRIQNLLKITNLRIKFVKLHTLGDNLLDSRMEIREKYYYAVYDMVVRGNCFCYGHASECAPVDGVNEEVEGMVHGHCMCRHNTKGLNCELCMDFYHDLPWRPAEGRNSNACKKCNCNEHSSSCHFDMAVFLATGNVSGGVCDNCQHNTMGRNCEQCKPFYFQHPERDIRDPNLCEPCTCDPAGSENGGICDGYTDFSVGLIAGQCRCKLHVEGERCDVCKEGFYDLSAEDPYGCKSCACNPLGTIPGGNPCDSETGYCYCKRLVTGQRCDQCLPQHWGLSNDLDGCRPCDCDLGGALNNSCSEDSGQCSCLPHMIGRQCNE

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 \(c\).

\(|P_{f(2JDG_1)}(2) \setminus P_{f(7VLX_1)}(2)|=72\), \(|P_{f(7VLX_1)}(2) \setminus P_{f(2JDG_1)}(2)|=91\). 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 1:110111100011010101000001010101000001110010111000100010001100100100001111000100001111001100101000001010100100001010001010010010110101110011000100011011000011011110101101001101010000000
Pair \(Z_2\) Length of longest common subsequence
2JDG_1,7VLX_1 163 3
2JDG_1,4AQS_1 226 3
7VLX_1,4AQS_1 211 4

Newick tree

 
[
	4AQS_1:11.12,
	[
		2JDG_1:81.5,7VLX_1:81.5
	]:35.62
]

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 \((0.85)(0.8)(\frac{451 }{\log_{20} 451}-\frac{183}{\log_{20}183})=78.7\)
Status Protein1 Protein2 d d1/2
Query variables 2JDG_1 7VLX_1 94 81.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]