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Parikh vectors
1YTP_1 6EPT_1 8JNS_1 Letter Amino acid
2 5 13 Y Tyrosine
6 6 30 A Alanine
3 11 37 I Isoleucine
5 6 24 P Proline
0 0 3 W Tryptophan
2 12 44 E Glutamic acid
5 1 18 G Glycine
0 2 18 M Methionine
1 8 36 V Valine
2 6 41 S Serine
4 13 32 R Arginine
0 4 18 Q Glutamine
3 7 30 K Lycine
2 5 30 F Phenylalanine
1 13 62 L Leucine
3 6 25 T Threonine
3 3 22 N Asparagine
5 16 38 D Aspartic acid
6 3 15 C Cysteine
0 3 13 H Histidine 

1YTP_1|Chain A|proteinase inhibitor|Nicotiana alata (4087)
>6EPT_1|Chain A|ATPase family AAA domain-containing protein 2|Homo sapiens (9606)
>8JNS_1|Chains A, B, C, D, E, F, G, H, I|Cell death protein 4|Caenorhabditis elegans (6239)
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 \(w=f(c)\)
1YTP , Knot 29 53 0.72 32 43 48 
DRIATNCCAGTKGCKYFSDDGTFVCEGESDPRNPKACPRNADPRIAYGICPLA
6EPT , Knot 63 130 0.78 38 97 126 
SMQEEDTFRELRIFLRNVTHRLAIDKRFRVFTKPVDPDEVPDYVTVIKQPMDLSSVISKIDLHKYLTVKDYLRDIDLICSNALEYNPDRDPGDRLIRHRACALRDTAYAIIKEELDEDFEQLCEEIQESR
8JNS , Knot 224 549 0.85 40 262 522 
MLCEIECRALSTAHTRLIHDFEPRDALTYLEGKNIFTEDHSELISKMSTRLERIANFLRIYRRQASELGPLIDFFNYNNQSHLADFLEDYIDFAINEPDLLRPVVIAPQFSRQMLDRKLLLGNVPKQMTCYIREYHVDRVIKKLDEMCDLDSFFLFLHGRAGSGKSVIASQALSKSDQLIGINYDSIVWLKDSGTAPKSTFDLFTDILLMLKSEDDLLNFPSVEHVTSVVLKRMICNALIDRPNTLFVFDDVVQEETIRWAQELRLRCLVTTRDVEISNAASQTCEFIEVTSLEIDECYDFLEAYGMPMPVGEKEEDVLNKTIELSSGNPATLMMFFKSCEPKTFEKMAQLNNKLESRGLVGVECITPYSYKSLAMALQRCVEVLSDEDRSALAFAVVMPPGVDIPVKLWSCVIPVDICSNEEEQLDDEVADRLKRLSKRGALLSGKRMPVLTFKIDHIIHMFLKHVVDAQTIANGISILEQRLLEIGNNNVSVPERHIPSHFQKFRRSSASEMYPKTTEETVIRPEDFPKFMQLHQKFYDSLKNFACC

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 \(c\).

\(|P_{f(1YTP_1)}(2) \setminus P_{f(6EPT_1)}(2)|=27\), \(|P_{f(6EPT_1)}(2) \setminus P_{f(1YTP_1)}(2)|=81\). 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 1:00110000110010001000101100100010010101001010110110111
Pair \(Z_2\) Length of longest common subsequence
1YTP_1,6EPT_1 108 2
1YTP_1,8JNS_1 249 3
6EPT_1,8JNS_1 203 4

Newick tree

 
[
	8JNS_1:12.39,
	[
		1YTP_1:54,6EPT_1:54
	]:73.39
]

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 \((0.85)(0.8)(\frac{183 }{\log_{20} 183}-\frac{53}{\log_{20}53})=44.3\)
Status Protein1 Protein2 d d1/2
Query variables 1YTP_1 6EPT_1 56 38
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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