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Parikh vectors
8POD_1 6HGZ_1 2ABH_1 Letter Amino acid
20 29 34 G Glycine
12 16 15 P Proline
13 13 14 Q Glutamine
21 22 20 D Aspartic acid
15 28 15 E Glutamic acid
34 33 24 L Leucine
10 14 12 F Phenylalanine
27 22 19 S Serine
23 15 21 T Threonine
16 38 35 A Alanine
14 7 17 N Asparagine
18 19 17 I Isoleucine
17 22 31 K Lycine
26 21 22 V Valine
20 15 4 R Arginine
9 5 1 H Histidine
9 10 0 M Methionine
8 1 8 W Tryptophan
10 14 12 Y Tyrosine
8 6 0 C Cysteine 

8POD_1|Chain A|Activin receptor type-1|Homo sapiens (9606)
>6HGZ_1|Chains A, B|ADP-ribosylhydrolase like 2|Latimeria chalumnae (7897)
>2ABH_1|Chain A|PHOSPHATE-BINDING PROTEIN|Escherichia coli (562)
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 \(w=f(c)\)
8POD , Knot 148 330 0.86 40 206 318 
SMTTNVGDSTLADLLDHSCTSGSGSGLPFLVQRTVARQITLLECVGKGRYGEVWRGSWQGENVAVKIFSSRDEKSWFRETELYNTVMLRHENILGFIASDMTSRHSSTQLWLITHYHEMGSLYDYLQLTTLDTVSCLRIVLSIASGLAHLHIEIFGTQGKPAIAHRDLKSKNILVKKNGQCCIADLGLAVMHSQSTNQLDVGNNPRVGTKRYMAPEVLDETIQVDCFDSYKRVDIWAFGLVLWEVARRMVSNGIVEDYKPPFYDVVPNDPSFEDMRKVVCVDQQRPNIPNRWFSDPTLTSLAKLMKECWYQNPSARLTALRIKKTLTKID
6HGZ , Knot 152 350 0.84 40 204 340 
GPMVSLAQVRGALCGALLGDCMGAEFEGSDAVELPDVLEFVRLLEKEKKAGTLFYTDDTAMTRAVIQSLIAKPDFDEVDMAKRFAEEYKKEPTRGYGAGVVQVFKKLLSPKYSDVFQPAREQFDGKGSYGNGGAMRVASIALAYPNIQDVIKFARRSAQLTHASPLGYNGAILQALAVHFALQGELKRDTFLEQLIGEMERIEGGEMSASDAGEHDRPNEVKLPFCSRLKKIKEFLASSNVPKADIVDELGHGIAALESVPTAIYSFLHCMESDPDIPDLYNNLQRTIIYSISLGGDTDTIATMAGAIAGAYYGMDQVTPSWKRSCEAIVETEESAVKLYELYCKQLKTP
2ABH , Knot 138 321 0.82 36 177 306 
EASLTGAGATFPAPVYAKWADTYQKETGNKVNYQGIGSSGGVKQIIANTVDFGASDAPLSDEKLAQEGLFQFPTVIGGVVLAVNIPGLKSGELVLDGKTLGDIYLGKIKKWDDEAIAKLNPGLKLPSQNIAVVRRADGSGTSFVFTSYLAKVNEEWKNNVGTGSTVKWPIGLGGKGNDGIAAFVQRLPGAIGYVEYAYAKQNNLAYTKLISADGKPVSPTEENFANAAKGADWSKTFAQDLTNQKGEDAWPITSTTFILIHKDQKKPEQGTEVLKFFDWAYKTGAKQANDLDYASLPDSVVEQVRAAWKTNIKDSSGKPLY

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 \(c\).

\(|P_{f(8POD_1)}(2) \setminus P_{f(6HGZ_1)}(2)|=91\), \(|P_{f(6HGZ_1)}(2) \setminus P_{f(8POD_1)}(2)|=89\). 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 1:010001100011011000000101011111100011001011001101001011010101001110110000000110000100011100001111110010000000011110000011010001010010010010111011011101010111001011110001000011100010001101111110000000101100101100001110110001010010000010111111111011001100111000011100111001010010011010000101100110010100110110001000101010110100010010
Pair \(Z_2\) Length of longest common subsequence
8POD_1,6HGZ_1 180 3
8POD_1,2ABH_1 177 3
6HGZ_1,2ABH_1 157 4

Newick tree

 
[
	8POD_1:92.56,
	[
		2ABH_1:78.5,6HGZ_1:78.5
	]:14.06
]

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 \((0.85)(0.8)(\frac{680 }{\log_{20} 680}-\frac{330}{\log_{20}330})=96.4\)
Status Protein1 Protein2 d d1/2
Query variables 8POD_1 6HGZ_1 125 120.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]