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Parikh vectors
8SMY_1 3IHO_1 7OOX_1 Letter Amino acid
1 6 8 N Asparagine
12 16 38 L Leucine
13 14 11 K Lycine
10 7 8 T Threonine
2 1 6 C Cysteine
3 1 4 M Methionine
4 6 14 F Phenylalanine
3 7 7 Y Tyrosine
5 6 24 S Serine
4 8 20 D Aspartic acid
8 3 11 Q Glutamine
7 9 24 E Glutamic acid
3 6 13 H Histidine
7 7 19 I Isoleucine
7 9 19 P Proline
18 6 14 A Alanine
18 6 21 R Arginine
9 5 23 G Glycine
0 8 7 W Tryptophan
6 7 22 V Valine 

8SMY_1|Chains A, E|Histone H3.1|Homo sapiens (9606)
>3IHO_1|Chain A|Methyl-CpG-binding domain protein 4|Homo sapiens (9606)
>7OOX_1|Chain A|Serine/threonine-protein kinase pim-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 \(w=f(c)\)
8SMY , Knot 68 140 0.80 38 107 132 
GPGHMARTKQTARKSTGGKAPRKQLATKAARKSAPATGGVKKPHRYRPGTVALREIRRYQKSTELLIRKLPFQRLVREIAQDFKTDLRFQSSAVMALQEACEAYLVGLFEDTNLCAIHAKRVTIMPKDIQLARRIRGERA
3IHO , Knot 70 138 0.83 40 116 135 
KWTPPRSPFNLVQETLFHDPWKLLIATIFLNRTSGKMAIPVLWKFLEKYPSAEVARTADWRDVSELLKPLGLYDLRAKTIVKFSDEYLTKQWKYPIELHGIGKYGNDSYRIFCVNEWKQVHPEDHKLNKYHDWLWENH
7OOX , Knot 135 313 0.82 40 196 296 
SMLLSKINSLAHLRAAPCNDLHATKLAPGKEKEPLESQYQVGPLLGSGGFGSVYSGIRVSDNLPVAIKHVEKDRISDWGELPNGTRVPMEVVLLKKVSSGFSGVIRLLDWFERPDSFVLILERPEPVQDLFDFITERGALQEELARSFFWQVLEAVRHCHNCGVLHRDIKDENILIDLNRGELKLIDFGSGALLKDTVYTDFDGTRVYSPPEWIRYHRYHGRSAAVWSLGILLYDMVCGDIPFEHDEEIIGGQVFFRQRVSSECQHLIRWCLALRPSDRPTFEEIQNHPWMQDVLLPQETAEIHLHSLSPGPS

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 \(c\).

\(|P_{f(8SMY_1)}(2) \setminus P_{f(3IHO_1)}(2)|=72\), \(|P_{f(3IHO_1)}(2) \setminus P_{f(8SMY_1)}(2)|=81\). 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 1:11101100000100001101100011001100011101110010000110111001000000001110011100110011001000101000111110010010111110000101101001011100101100101001
Pair \(Z_2\) Length of longest common subsequence
8SMY_1,3IHO_1 153 3
8SMY_1,7OOX_1 181 4
3IHO_1,7OOX_1 184 3

Newick tree

 
[
	7OOX_1:95.66,
	[
		8SMY_1:76.5,3IHO_1:76.5
	]:19.16
]

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 \((0.85)(0.8)(\frac{278 }{\log_{20} 278}-\frac{138}{\log_{20}138})=43.5\)
Status Protein1 Protein2 d d1/2
Query variables 8SMY_1 3IHO_1 55 54
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]