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Parikh vectors
7JRX_1 2UWD_1 1OEV_1 Letter Amino acid
0 7 9 Y Tyrosine
0 8 19 R Arginine
0 27 30 E Glutamic acid
0 17 23 K Lycine
1 16 24 S Serine
2 15 14 G Glycine
0 7 11 M Methionine
2 6 22 P Proline
2 11 16 V Valine
0 16 18 D Aspartic acid
0 10 14 F Phenylalanine
0 1 6 W Tryptophan
1 11 14 Q Glutamine
0 4 11 H Histidine
1 20 18 I Isoleucine
2 18 29 L Leucine
0 19 13 T Threonine
1 15 13 A Alanine
0 8 11 N Asparagine
1 0 6 C Cysteine 

7JRX_1|Chains A, E[auth a]|Chymotrypsin A chain A|Bos taurus (9913)
>2UWD_1|Chain A|HEAT SHOCK PROTEIN HSP 90-ALPHA|HOMO SAPIENS (9606)
>1OEV_1|Chain A|PROTEIN-TYROSINE PHOSPHATASE, NON-RECEPTOR TYPE 1|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 \(w=f(c)\)
7JRX , Knot 11 13 0.72 18 12 11 
CGVPAIQPVLSGL
2UWD , Knot 104 236 0.80 38 157 227 
MPEETQTQDQPMEEEEVETFAFQAEIAQLMSLIINTFYSNKEIFLRELISNSSDALDKIRYESLTDPSKLDSGKELHINLIPNKQDRTLTIVDTGIGMTKADLINNLGTIAKSGTKAFMEALQAGADISMIGQFGVGFYSAYLVAEKVTVITKHNDDEQYAWESSAGGSFTVRTDTGEPMGRGTKVILHLKEDQTEYLEERRIKEIVKKHSQFIGYPITLFVEKERDKEVSDDEAE
1OEV , Knot 141 321 0.84 40 207 309 
MEMEKEFEQIDKSGSWAAIYQDIRHEASDFPCRVAKLPKNKNRNRYRDVSPFDHSRIKLHQEDNDYINASLIKMEEAQRSYILTQGPLPNTCGHFWEMVWEQKSRGVVMLNRVMEKGSLKCAQYWPQKEEKEMIFEDTNLKLTLISEDIKSYYTVRQLELENLTTQETREILHFHYTTWPDFGVPESPASFLNFLFKVRESGSLSPEHGPVVVHCSAGIGRSGTFCLADTCLLLMDKRKDPSSVDIKKVLLEMRKFRMGLIQTADQLRFSYLAVIEGAKFIMGDSSVQDQWKELSHEDLEPPPEHIPPPPRPPKRILEPHN

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 \(c\).

\(|P_{f(7JRX_1)}(2) \setminus P_{f(2UWD_1)}(2)|=9\), \(|P_{f(2UWD_1)}(2) \setminus P_{f(7JRX_1)}(2)|=154\). 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 1:0111110111011
Pair \(Z_2\) Length of longest common subsequence
7JRX_1,2UWD_1 163 2
7JRX_1,1OEV_1 197 3
2UWD_1,1OEV_1 184 3

Newick tree

 
[
	1OEV_1:99.48,
	[
		7JRX_1:81.5,2UWD_1:81.5
	]:17.98
]

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 \((0.85)(0.8)(\frac{249 }{\log_{20} 249}-\frac{13}{\log_{20}13})=81.6\)
Status Protein1 Protein2 d d1/2
Query variables 7JRX_1 2UWD_1 100 53
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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