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Parikh vectors
2RIQ_1 1YME_1 2KDW_1 Letter Amino acid
11 16 8 D Aspartic acid
6 8 1 H Histidine
5 16 7 F Phenylalanine
8 26 5 T Threonine
3 7 8 W Tryptophan
5 2 2 C Cysteine
6 12 12 Q Glutamine
6 21 7 I Isoleucine
14 24 10 L Leucine
10 21 7 A Alanine
4 11 16 R Arginine
5 23 11 G Glycine
25 15 9 K Lycine
3 20 6 Y Tyrosine
12 16 16 V Valine
5 13 6 N Asparagine
13 13 8 E Glutamic acid
3 3 6 M Methionine
6 10 8 P Proline
10 32 11 S Serine 

2RIQ_1|Chain A|Poly [ADP-ribose] polymerase 1|Homo sapiens (9606)
>1YME_1|Chain A|CARBOXYPEPTIDASE A ALPHA|Bos taurus (9913)
>2KDW_1|Chain A|RNA pyrophosphohydrolase|Escherichia coli (83333)
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 \(w=f(c)\)
2RIQ , Knot 73 160 0.77 40 112 153 
MVDEVAKKKSKKEKDKDSKLEKALKAQNDLIWNIKDELKKVCSTNDLKELLIFNKQQVPSGESAILDRVADGMVFGALLPCEECSGQLVFKSDAYYCTGDVTAWTKCMVKTQTPNRKEWVTPKEFREISYLKKLKVKKQDRIFPPETSASVALEHHHHHH
1YME , Knot 137 309 0.84 40 194 301 
ARSTNTFNYATYHTLDEIYDFMDLLVAEHPQLVSKLQIGRSYEGRPIYVLKFSTGGSNRPAIWIDLGIHSREWITQATGVWFAKKFTEDYGQDPSFTAILDSMDIFLEIVTNPDGFAFTHSQNRLWRKTRSVTSSSLCVGVDANRNWDAGFGKAGASSSPCSETYHGKYANSEVEVKSIVDFVKDHGNFKAFLSIHSYSQLLLYPYGYTTQSIPDKTELNQVAKSAVAALKSLYGTSYKYGSIITTIYQASGGSIDWSYNQGIKYSFTFELRDTGRYGFLLPASQIIPTAQETWLGVLTIMEHTVNNLY
2KDW , Knot 82 164 0.85 40 130 162 
MIDDDGYRPNVGIVICNRQGQVMWARRFGQHSWQFPQGGINPGESAEQAMYRALFEEVGLSRKDVRILASTRNWLRYKLPKRLVRWDTKPVCIGQKQKWFLLQLVSGDAEINMQTSSTPEFDGWRWVSYWYPVRQVVSFKRDVYRRVMKEFASVVMSLQENTPK

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 \(c\).

\(|P_{f(2RIQ_1)}(2) \setminus P_{f(1YME_1)}(2)|=49\), \(|P_{f(1YME_1)}(2) \setminus P_{f(2RIQ_1)}(2)|=131\). 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 1:1100110000000000000100110100011101000100100000100111100001101001110011011111111100000101110001000010101100011000010000110100100100100101000001111000101110000000
Pair \(Z_2\) Length of longest common subsequence
2RIQ_1,1YME_1 180 4
2RIQ_1,2KDW_1 154 4
1YME_1,2KDW_1 192 4

Newick tree

 
[
	1YME_1:97.81,
	[
		2RIQ_1:77,2KDW_1:77
	]:20.81
]

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 \((0.85)(0.8)(\frac{469 }{\log_{20} 469}-\frac{160}{\log_{20}160})=91.1\)
Status Protein1 Protein2 d d1/2
Query variables 2RIQ_1 1YME_1 117 87
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]