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Parikh vectors
5QKM_1 4PYR_1 4WQO_1 Letter Amino acid
12 12 30 E Glutamic acid
13 26 21 G Glycine
3 1 12 H Histidine
2 2 3 W Tryptophan
5 7 6 I Isoleucine
8 7 6 Y Tyrosine
21 32 18 V Valine
2 0 2 C Cysteine
12 35 21 L Leucine
7 18 20 P Proline
9 8 7 T Threonine
11 5 5 F Phenylalanine
17 55 10 A Alanine
3 25 21 R Arginine
7 7 9 N Asparagine
12 19 11 D Aspartic acid
14 15 8 Q Glutamine
17 7 3 K Lycine
6 7 4 M Methionine
8 17 16 S Serine 

5QKM_1|Chains A, B|Thiol:disulfide interchange protein|Escherichia coli K-12 (83333)
>4PYR_1|Chain A|Putative branched-chain amino acid ABC transporter|Chromobacterium violaceum (243365)
>4WQO_1|Chain A|Von Hippel-Lindau disease tumor suppressor|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 \(w=f(c)\)
5QKM , Knot 86 189 0.79 40 134 185 
AQYEDGKQYTTLEKPVAGAPQVLEFFSFFCPHCYQFEEVLHISDNVKKKLPEGVKMTKYHVNFMGGDLGKDLTQAWAVAMALGVEDKVTVPLFEGVQKTQTIRSASDIRDVFINAGIKGEEYDAAWNSFVVKSLVAQQEKAAADVQLRGVPAMFVNGKYQLNPQGMDTSNMDVFVQQYADTVKYLSEKK
4PYR , Knot 123 305 0.77 38 148 270 
KVRIGVILPAESSALGEAAAVVRSGVEAAAQVDQSAELYSVDATGDNVVERYRAAVADGVNVVIGPLSRDSIVKLAPSVTVPTLALNSVGREAAANPKLYSLSLIVEGEARQLARLMRDDSRAAPLLVVGGDALSQRLGKAFADEWRAAAGKPVRQMAFDANDMAPLLQAAGQADAVALALDVAQAARLKSALTPDVPVYGTSQLNVGGMQPELAGVRFIDMPWFLMPAHPAVQRYPRPAAPLTRQTERLYALGIDAYRLAVQLAGSRSGAAVRLDGVTGDLKLGRDRAFERQLPAGVMGGNALQ
4WQO , Knot 102 233 0.79 40 143 209 
MGSSHHHHHHSSGLVPRGSHMPRRAENWDEAEVGAEEAGVEEYGPEEDGGEESGAEESGPEESGPEELGAEEEMEAGRPRPVLRSVNSREPSQVIFCNRSPRVVLPVWLNFDGEPQPYPTLPPGTGRRIHSYRGHLWLFRDAGTHDGLLVNQTELFVPSLNVDGQPIFANITLPVYTLKERCLQVVRSLVKPENYRRLDIVRSLYEDLEDHPNVQKDLERLTQERIAHQRMGD

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 \(c\).

\(|P_{f(5QKM_1)}(2) \setminus P_{f(4PYR_1)}(2)|=70\), \(|P_{f(4PYR_1)}(2) \setminus P_{f(5QKM_1)}(2)|=84\). 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 1:100001000001001111110110110110100001001101000100011011010000101111011001001111111111000101111011000001001001001110111010000111001110011100001110101011111110100010101100001011100010010010000
Pair \(Z_2\) Length of longest common subsequence
5QKM_1,4PYR_1 154 4
5QKM_1,4WQO_1 173 3
4PYR_1,4WQO_1 131 4

Newick tree

 
[
	5QKM_1:86.66,
	[
		4PYR_1:65.5,4WQO_1:65.5
	]:21.16
]

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 \((0.85)(0.8)(\frac{494 }{\log_{20} 494}-\frac{189}{\log_{20}189})=88.7\)
Status Protein1 Protein2 d d1/2
Query variables 5QKM_1 4PYR_1 105 86
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]