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Parikh vectors
3ZNU_1 1QAN_1 4FVL_1 Letter Amino acid
5 9 17 T Threonine
4 9 18 Y Tyrosine
3 6 19 A Alanine
3 8 6 Q Glutamine
4 6 27 G Glycine
4 30 25 K Lycine
3 19 27 F Phenylalanine
9 9 13 R Arginine
4 26 15 N Asparagine
8 12 17 E Glutamic acid
12 21 29 L Leucine
6 17 23 S Serine
6 11 15 V Valine
4 9 15 H Histidine
3 26 23 I Isoleucine
2 5 8 M Methionine
6 5 27 P Proline
3 1 9 W Tryptophan
5 13 33 D Aspartic acid
0 2 2 C Cysteine 

3ZNU_1|Chains A, B, C, D, E, F, G, H, I, J|5-CHLOROMUCONOLACTONE DEHALOGENASE|RHODOCOCCUS OPACUS (37919)
>1QAN_1|Chain A|ERMC' METHYLTRANSFERASE|Bacillus subtilis (1423)
>4FVL_1|Chains A, B|Collagenase 3|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 \(w=f(c)\)
3ZNU , Knot 48 94 0.77 38 78 92 
MLYLVRMTVNLPRNLDPREEERLKASEKARSRTLQEQGQWRYLWRTTGKYGNISVFDVNSHDELHEILWSLPFFPYLTIDVEPLSHHPARVGKD
1QAN , Knot 110 244 0.82 40 159 234 
MNEKNIKHSQNFITSKHNIDKIMTNIRLNEHDNIFEIGSGKGHFTLELVQRCNFVTAIEIDHKLCKTTENKLVDHDNFQVLNKDILQFKFPKNQSYKIFGNIPYNISTDIIRKIVFDSIADEIYLIVEYGFAKRLLNTKRSLALFLMAEVDISILSMVPREYFHPKPKVNSSLIRLNRKKSRISHKDKQKYNYFVMKWVNKEYKKIFTKNQFNNSLKHAGIDDLNNISFEQFLSLFNSYKLFNK
4FVL , Knot 157 368 0.84 40 210 350 
YNVFPRTLKWSKMNLTYRIVNYTPDMTHSEVEKAFKKAFKVWSDVTPLNFTRLHDGIADIMISFGIKEHGDFYPFDGPSGLLAHAFPPGPNYGGDAHFDDDETWTSSSKGYNLFLVAAHAFGHSLGLDHSKDPGALMFPIYTYTGKSHFMLPDDDVQGIQSLYGPGDEDPNPKHPKTPDKCDPSLSLDAITSLRGETMIFKDRFFWRLHPQQVDAELFLTKSFWPELPNRIDAAYEHPSHDLIFIFRGRKFWALNGYDILEGYPKKISELGLPKEVKKISAAVHFEDTGKTLLFSGNQVWRYDDTNHIMDKDYPRLIEEDFPGIGDKVDAVYEKNGYIYFFNGPIQFEYSIWSNRIVRVMPANSILWC

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 \(c\).

\(|P_{f(3ZNU_1)}(2) \setminus P_{f(1QAN_1)}(2)|=40\), \(|P_{f(1QAN_1)}(2) \setminus P_{f(3ZNU_1)}(2)|=121\). 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 1:1101101010110010100000101000100001000101001100010010101101000001001110111110101010110001101100
Pair \(Z_2\) Length of longest common subsequence
3ZNU_1,1QAN_1 161 3
3ZNU_1,4FVL_1 184 3
1QAN_1,4FVL_1 163 3

Newick tree

 
[
	4FVL_1:88.94,
	[
		3ZNU_1:80.5,1QAN_1:80.5
	]:8.44
]

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 \((0.85)(0.8)(\frac{338 }{\log_{20} 338}-\frac{94}{\log_{20}94})=76.0\)
Status Protein1 Protein2 d d1/2
Query variables 3ZNU_1 1QAN_1 97 66.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]

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