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Parikh vectors
5XSU_1 2HBR_1 5MYO_1 Letter Amino acid
20 11 7 A Alanine
11 7 8 R Arginine
16 9 9 N Asparagine
18 12 9 E Glutamic acid
39 14 18 L Leucine
10 6 9 F Phenylalanine
14 11 22 T Threonine
25 10 13 D Aspartic acid
14 6 9 Q Glutamine
25 10 15 G Glycine
9 4 3 H Histidine
9 7 3 M Methionine
17 9 11 P Proline
30 16 30 S Serine
1 2 3 W Tryptophan
2 5 5 C Cysteine
33 14 13 K Lycine
25 14 9 I Isoleucine
16 2 9 Y Tyrosine
21 9 14 V Valine 

5XSU_1|Chains A, B, C, D|Poly [ADP-ribose] polymerase 1|Homo sapiens (9606)
>2HBR_1|Chain A|Caspase-1|Homo sapiens (9606)
>5MYO_1|Chains A, C|Fab c#6 light chain|Mus musculus (10090)
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 \(w=f(c)\)
5XSU , Knot 153 355 0.84 40 206 336 
GPLGSKSKLPKPVQDLIKMIFDVESMKKAMVEYEIDLQKMPLGKLSKRQIQAAYSILSEVQQAVSQGSSDSQILDLSNRFYTLIPHDFGMKKPPLLNNADSVQAKAEMLDNLLDIEVAYSLLRGGSDDSSKDPIDVNYEKLKTDIKVVDRDSEEAEIIRKYVKNTHATTHNAYDLEVIDIFKIEREGECQRYKPFKQLHNRRLLWHGSRTTNFAGILSQGLRIAPPEAPVTGYMFGKGIYFADMVSKSANYCHTSQGDPIGLILLGEVALGNMYELKHASHISKLPKGKHSVKGLGKTTPDPSANISLDGVDVPLGTGISSGVNDTSLLYNEYIVYDIAQVNLKYLLKLKFNFKT
2HBR , Knot 88 178 0.85 40 146 176 
NPAMPTSSGSEGNVKLCSLEEAQRIWKQKSAEIYPIMDKSSRTRLALIICNEEFDSIPRRTGAEVDITGMTMLLQNLGYSVDVKKNLTASDMTTELEAFAHRPEHKTSDSTFLVFMSHGIREGICGKKHSEQVPDILQLNAIFNMLNTKNCPSLKDKPKVIIIQACAGDSPGVVWFKD
5MYO , Knot 102 219 0.83 40 159 212 
DVVMTQTPLTLSVTIGQPASISCKSSQSLLYSDGKTYLNWLLQRPGQSPMRLIYLVSKLDSGVPDRFTGSGSGTDFTLKISRVEAEDLGVYYCVQGTHFPFTFGSGTKLEIKRADAAPTVSIFPPSSEQLTSGGASVVCFLNNFYPKDINVKWKIDGSERQNGVLNSWTDQDSKDSTYSMSSTLTLTKDEYERHNSYTCEATHKTSTSPIVKSFNRNEC

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 \(c\).

\(|P_{f(5XSU_1)}(2) \setminus P_{f(2HBR_1)}(2)|=111\), \(|P_{f(2HBR_1)}(2) \setminus P_{f(5XSU_1)}(2)|=51\). 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 1:1111000011011001101110100100111000101001111010000101100110010011001000001101000100111001110011110010010101011001101011001101100000001101000010001011000000101100010000100001001011011010001000000110010000111010000011111001101111011101011101101101100010000000101111111101111010010010010011010001011100010101010101101111011001100001100001100110101001101010100
Pair \(Z_2\) Length of longest common subsequence
5XSU_1,2HBR_1 162 3
5XSU_1,5MYO_1 169 4
2HBR_1,5MYO_1 173 4

Newick tree

 
[
	5MYO_1:86.95,
	[
		5XSU_1:81,2HBR_1:81
	]:5.95
]

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 \((0.85)(0.8)(\frac{533 }{\log_{20} 533}-\frac{178}{\log_{20}178})=102.\)
Status Protein1 Protein2 d d1/2
Query variables 5XSU_1 2HBR_1 131 98
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]