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Parikh vectors
8ANW_1 4PRX_1 4TNT_1 Letter Amino acid
15 34 11 G Glycine
6 19 8 H Histidine
15 27 6 K Lycine
20 25 9 S Serine
16 12 3 Y Tyrosine
25 35 6 V Valine
16 20 6 R Arginine
2 2 10 C Cysteine
11 11 4 Q Glutamine
14 24 4 I Isoleucine
5 9 2 M Methionine
12 14 3 F Phenylalanine
23 12 2 P Proline
25 31 6 A Alanine
12 31 3 E Glutamic acid
28 22 3 T Threonine
4 2 0 W Tryptophan
14 17 6 N Asparagine
18 25 4 D Aspartic acid
19 26 7 L Leucine 

8ANW_1|Chain A|Capsid protein, VP1|Human poliovirus 3 (12086)
>4PRX_1|Chain A|GyrB|Escherichia coli (562)
>4TNT_1|Chains A, B|Mineralocorticoid receptor|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 \(w=f(c)\)
8ANW , Knot 133 300 0.84 40 195 290 
GIEDLITEVAQGALTLSLPKQQDSLPDTKASGPAHSKEVPALTAVETGATNPLVPSDTVQTRHVIQRRSRSESTIESFFARGACVAIIEVDNEEPTTRAQKLFAMWRITYKDTVQLRRKLEFFTYSRFDMELTFVVTANFTNTNNGHALNQVYQIMYIPPGAPTPKSWDDYTWQTSSNPSIFYTYGAAPARISVPYVGLANAYSHFYDGFAKVPLKTDANDQIGDSLYSAMTVDDFGVLAIRVVNDHNPTKVTSKVRIYMKPKHVRVWCPRPPRAVPYYGPGVDYKDNLNPLSEKGLTTY
4PRX , Knot 166 398 0.83 40 213 369 
MHHHHHHSNSYDSSSIKVLKGLDAVRKRPGMYIGDTDDGTGLHHMVFEVVDNAIDEALAGHCKEIIVTIHADNSVSVQDDGRGIPTGIHPEEGVSAAEVIMTVLHAGGKFDDNSYKVSGGLHGVGVSVVNALSQKLELVIQREGKIHRQIYEHGVPQAPLAVTGETEKTGTMVRFWPSLETFTNVTEFEYEILAKRLRELSFLNSGVSIRLRDKRDGKEDHFHYEGGIKAFVEYLNKNKTPIHPNIFYFSTEKDGIGVEVALQWNDGFQENIYCFTNNIPQRDGGTHLAGFRAAMTRTLNAYMDKEGYSKKAKVSATGDDAREGLIAVVSVKVPDPKFSSQTKDKLVSSEVKSAVEQQMNELLAEYLLENPTDAKIVVGKIIDAARAREAARRAREMT
4TNT , Knot 54 103 0.81 38 86 98 
MHHHHHHSSGVDLGTENLYFQSNASTGSSRPSKICLVCGDEASGCHYGVVTCGSCKVFFKRAVEGQHNYLCAGRNDCIIDKIRRKNCPACRLQKCLQAGMNLG

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 \(c\).

\(|P_{f(8ANW_1)}(2) \setminus P_{f(4PRX_1)}(2)|=70\), \(|P_{f(4PRX_1)}(2) \setminus P_{f(8ANW_1)}(2)|=88\). 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 1:110011001101110101100000110001011100001111011001100111100010000110000000001001110110111101000010001001111101000001010001011000010101011101010000010110010011011111101001000010000010110001111101011011110100010011101110001000110010011010011111101100001001000101010100101101011011100111100000101100011000
Pair \(Z_2\) Length of longest common subsequence
8ANW_1,4PRX_1 158 4
8ANW_1,4TNT_1 205 3
4PRX_1,4TNT_1 195 8

Newick tree

 
[
	4TNT_1:10.11,
	[
		8ANW_1:79,4PRX_1:79
	]:27.11
]

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 \((0.85)(0.8)(\frac{698 }{\log_{20} 698}-\frac{300}{\log_{20}300})=109.\)
Status Protein1 Protein2 d d1/2
Query variables 8ANW_1 4PRX_1 136 120.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]