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Parikh vectors
7JHB_1 2DVY_1 8RQO_1 Letter Amino acid
0 9 12 T Threonine
0 11 17 D Aspartic acid
0 20 15 I Isoleucine
0 19 16 K Lycine
0 9 16 P Proline
0 7 13 F Phenylalanine
0 17 22 S Serine
0 1 3 W Tryptophan
0 14 3 Y Tyrosine
5 10 17 A Alanine
0 9 5 N Asparagine
3 1 6 C Cysteine
0 3 10 Q Glutamine
0 18 32 L Leucine
0 4 12 M Methionine
0 22 17 V Valine
0 15 14 R Arginine
0 20 24 E Glutamic acid
4 12 14 G Glycine
0 5 16 H Histidine 

7JHB_1|Chain A|DNA (5'-D(*GP*AP*GP*CP*AP*GP*AP*CP*CP*AP*GP*A)-3')|synthetic construct (32630)
>2DVY_1|Chains A, B, C, D, E, F|Restriction endonuclease PabI|Pyrococcus abyssi (29292)
>8RQO_1|Chain A|Isoform Alpha-1 of Thyroid hormone receptor alpha|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 \(w=f(c)\)
7JHB , Knot 6 12 0.41 6 6 8 
GAGCAGACCAGA
2DVY , Knot 103 226 0.82 40 149 216 
MIHLTSVEASVSFENGKIVVRLPITRPTSKIRVKKIENGVGIPVSTRKKSFPSDENLRDYYIEWQISYARDGKYDYELSRMVRLAHEHGILTYNDIYELLKFADDVKSYLEDKGIRRESTNEELYGFNIYEDVYPVAKKELPSGEFIGIVLKHKQRAVGYQSMVYVCIPLTNVEPSLAGRVARRNEVVKYEVPVDLMKELLKAFIIASETHKNDIVKFLRSIIGTS
8RQO , Knot 130 284 0.86 40 193 273 
MGSSHHHHHHSSGLVPRGSHMEEMIRSLQQRPEPTPEEWDLIHIATEAHRSTNAQGSHWKQRRKFLPDDIGQSPIVSMPDGDKVDLEAFSEFTKIITPAITRVVDFAKKLPMFSELPCEDQIILLKGCCMEIMSLRAAVRYDPESDTLTLSGEMAVKREQLKNGGLGVVSDAIFELGKSLSAFNLDDTEVALLQAVLLMSTDRSGLLCVDKIEKSQEAYLLAFEHYVNHRKHNIPHFWPKLLMKVTDLRMIGACHASRFLHMKVECPTELFPPLFLEVFEDQEV

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 \(c\).

\(|P_{f(7JHB_1)}(2) \setminus P_{f(2DVY_1)}(2)|=5\), \(|P_{f(2DVY_1)}(2) \setminus P_{f(7JHB_1)}(2)|=148\). 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 1:111011100111
Pair \(Z_2\) Length of longest common subsequence
7JHB_1,2DVY_1 153 2
7JHB_1,8RQO_1 191 3
2DVY_1,8RQO_1 152 3

Newick tree

 
[
	7JHB_1:89.75,
	[
		2DVY_1:76,8RQO_1:76
	]:13.75
]

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 \((0.85)(0.8)(\frac{238 }{\log_{20} 238}-\frac{12}{\log_{20}12})=78.7\)
Status Protein1 Protein2 d d1/2
Query variables 7JHB_1 2DVY_1 101 52.5
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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