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Parikh vectors
2AIQ_1 4RBL_1 2RPB_1 Letter Amino acid
23 22 4 G Glycine
4 4 2 M Methionine
15 18 5 P Proline
11 21 3 S Serine
13 19 13 D Aspartic acid
5 24 8 E Glutamic acid
5 12 2 H Histidine
10 10 6 K Lycine
12 5 1 C Cysteine
19 18 12 I Isoleucine
14 7 5 T Threonine
7 7 3 Y Tyrosine
11 22 17 V Valine
16 9 6 A Alanine
18 6 4 N Asparagine
6 11 4 Q Glutamine
21 32 8 L Leucine
5 14 1 F Phenylalanine
4 7 1 W Tryptophan
12 21 8 R Arginine 

2AIQ_1|Chain A|Protein C activator|Agkistrodon contortrix contortrix (8713)
>4RBL_1|Chain A|Serine/threonine-protein kinase pim-1|Homo sapiens (9606)
>2RPB_1|Chain A|hypothetical membrane protein|Pyrococcus horikoshii (53953)
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 \(w=f(c)\)
2AIQ , Knot 106 231 0.83 40 164 226 
VIGGDECNINEHRFLALVYANGSLCGGTLINQEWVLTARHCDRGNMRIYLGMHNLKVLNKDALRRFPKEKYFCLNTRNDTIWDKDIMLIRLNRPVRNSAHIAPLSLPSNPPSVGSVCRIMGWGTITSPNATLPDVPHCANINILDYAVCQAAYKGLAATTLCAGILEGGKDTCKGDSGGPLICNGQFQGILSVGGNPCAQPRKPGIYTKVFDYTDWIQSIISGNTDATCPP
4RBL , Knot 126 289 0.82 40 183 272 
GPHMKEKEPLESQYQVGPLLGSGGFGSVYSGIRVSDNLPVAIKHVEKDRISDWGELPNGTRVPMEVVLLKKVSSGFSGVIRLLDWFERPDSFVLILERPEPVQDLFDFITERGALQEELARSFFWQVLEAVRHCHNCGVLHRDIKDENILIDLNRGELKLIDFGSGALLKDTVYTDFDGTRVYSPPEWIRYHRYHGRSAAVWSLGILLYDMVCGDIPFEHDEEIIRGQVFFRQRVSSECQHLIRWCLALRPSDRPTFEEIQNHPWMQDVLLPQETAEIHLHSLSPGPSK
2RPB , Knot 54 113 0.75 40 86 106 
GSDHVDLREHVIDVPPQEVICKDNVVVTVDAVVYYQVIDPVKAVYNVSDFLMAIVKLAQTNLRAIIGEMELDETLSGRDIINARLREELDKITDRWGVKITRVEIQRIDPPKD

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 \(c\).

\(|P_{f(2AIQ_1)}(2) \setminus P_{f(4RBL_1)}(2)|=79\), \(|P_{f(4RBL_1)}(2) \setminus P_{f(2AIQ_1)}(2)|=98\). 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 1:111100001000011111010101011011000111010000010101011100101100011001100001010000001100011110100110001011110110011011010011111010010101101100101011001100110011110010111101100000100111110010101110111010101001110001100001100110100010011
Pair \(Z_2\) Length of longest common subsequence
2AIQ_1,4RBL_1 177 3
2AIQ_1,2RPB_1 170 3
4RBL_1,2RPB_1 155 3

Newick tree

 
[
	2AIQ_1:89.64,
	[
		2RPB_1:77.5,4RBL_1:77.5
	]:12.14
]

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 \((0.85)(0.8)(\frac{520 }{\log_{20} 520}-\frac{231}{\log_{20}231})=82.9\)
Status Protein1 Protein2 d d1/2
Query variables 2AIQ_1 4RBL_1 105 95
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]