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Parikh vectors
2YUO_1 6NCN_1 7MNQ_1 Letter Amino acid
0 5 5 M Methionine
4 15 17 A Alanine
7 7 15 D Aspartic acid
7 23 12 E Glutamic acid
6 26 19 L Leucine
10 13 10 S Serine
2 4 3 W Tryptophan
6 19 10 R Arginine
2 8 13 H Histidine
4 1 10 F Phenylalanine
2 4 13 P Proline
1 0 3 C Cysteine
9 10 17 G Glycine
5 8 18 K Lycine
1 4 8 Y Tyrosine
3 12 23 V Valine
2 0 10 N Asparagine
1 18 9 Q Glutamine
5 0 10 I Isoleucine
1 8 11 T Threonine 

2YUO_1|Chain A|RUN and TBC1 domain containing 3|Mus musculus (10090)
>6NCN_1|Chain A|Apolipoprotein E|Homo sapiens (9606)
>7MNQ_1|Chain A|GTP-binding nuclear protein Ran|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 \(w=f(c)\)
2YUO , Knot 41 78 0.76 38 63 73 
GSSGSSGRRAKALLDFERHDDDELGFRKNDIITIISQKDEHCWVGELNGLRGWFPAKFVEVLDERSKEYSIASGPSSG
6NCN , Knot 80 185 0.75 34 108 171 
GSSHHHHHHSSGLVPRGSHMKVEQAVETEPEPELRQQTEWQSGQRWELALGRFWDYLRWVQTLSEQVQEELLSSQVTQELRALMDETMKELKAYKSELEEQLTPVAEETRARLSKELQAAQARLGADMEDVRGRLVQYRGEVQAMLGQSTEELRVRLASHLRKLRKRLLRDADDLQKRLAVYQAG
7MNQ , Knot 109 236 0.84 40 167 226 
MGSSHHHHHHSSGLVPRGSHMAAQGEPQVQFKLVLVGDGGTGKTTFVKRHLTGESEKKYVATLGVEVHPLVFHTNRGPIKFNVWDTAGQEKFGGLRDGYYIQAQCAIIMFDVTSRVTYKNVPNWHRDLVRVCENIPIVLCGNKVDIKDRKVKAKSIVFHRKKNLQYYDISAKSNYNFEKPFLWLARKLIGDPNLEFVAMPALAPPEVVMDPALAAQYEHDLEVAQTTALPDEDDDL

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 \(c\).

\(|P_{f(2YUO_1)}(2) \setminus P_{f(6NCN_1)}(2)|=38\), \(|P_{f(6NCN_1)}(2) \setminus P_{f(2YUO_1)}(2)|=83\). 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 1:100100100101110100000001110000110110000000111010110111110110110000000011011001
Pair \(Z_2\) Length of longest common subsequence
2YUO_1,6NCN_1 121 3
2YUO_1,7MNQ_1 166 3
6NCN_1,7MNQ_1 159 20

Newick tree

 
[
	7MNQ_1:87.09,
	[
		2YUO_1:60.5,6NCN_1:60.5
	]:26.59
]

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 \((0.85)(0.8)(\frac{263 }{\log_{20} 263}-\frac{78}{\log_{20}78})=59.6\)
Status Protein1 Protein2 d d1/2
Query variables 2YUO_1 6NCN_1 71 50.5
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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