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Parikh vectors
2IJX_1 2GRR_1 4TWV_1 Letter Amino acid
12 7 2 N Asparagine
2 3 11 H Histidine
23 13 17 L Leucine
13 7 7 F Phenylalanine
5 16 4 P Proline
25 6 7 V Valine
25 11 16 E Glutamic acid
20 14 16 K Lycine
23 9 5 S Serine
0 4 2 W Tryptophan
0 4 0 C Cysteine
19 9 9 I Isoleucine
8 6 7 T Threonine
13 12 15 A Alanine
8 8 2 R Arginine
18 6 8 D Aspartic acid
8 7 6 Q Glutamine
11 10 15 G Glycine
4 4 3 M Methionine
7 5 2 Y Tyrosine 

2IJX_1|Chains A, B, C, D|DNA polymerase sliding clamp A|Sulfolobus solfataricus (273057)
>2GRR_1|Chain A|Ubiquitin-conjugating enzyme E2 I|Homo sapiens (9606)
>4TWV_1|Chain A|Myoglobin|Equus caballus (9796)
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 \(w=f(c)\)
2IJX , Knot 109 244 0.81 36 149 234 
MKVVYDDVRVLKDIIQALARLVDEAVLKFKQDSVELVALDRAHISLISVNLPREMFKEYDVNDEFKFGFNTQYLMKILKVAKRKEAIEIASESPDSVIINIIGSTNREFNVRNLEVSEQEIPEINLQFDISATISSDGFKSAISEVSTVTDNVVVEGHEDRILIKAEGESEVEVEFSKDTGGLQDLEFSKESKNSYSAEYLDDVLSLTKLSDYVKISFGNQKPLQLFFNMEGGGKVTYLLAPKV
2GRR , Knot 78 161 0.82 40 125 156 
GSHMSGIALSRLAQERKAWRKDHPFGFVAVPTKNPDGTMNLMNWECAIPGKKGTPWEGGLFKLRMLFKDDYPSSPPKCKFEPPLFHPNVYPSGTVCLSILEEDKDWRPAITIKQILLGIQELLNEPNIQSPAQAEAYTIYCQNRVEYEKRVRAQAKKFAPS
4TWV , Knot 75 154 0.81 38 117 150 
MGLSDGEWQQVLNVWGKVEADIAGHGQEVLIRLFTGHPETLEKFDEFKHLKTEAEMKASEDLKEHGTVVLTALGGILKKKGHHEAELKPLAQSHATEHKIPIKYLEFISDAIIHVLHSKHPGDFGADAQGAMTKALELFRNDIAAKYKELGFQG

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 \(c\).

\(|P_{f(2IJX_1)}(2) \setminus P_{f(2GRR_1)}(2)|=88\), \(|P_{f(2GRR_1)}(2) \setminus P_{f(2IJX_1)}(2)|=64\). 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 1:1011000101100110111011001110100001011110010101101011001100001000101110000110110110000110110001001110111000001010010100001101010101010100011001100100100011101000011101010001010100001110010100000000010010011010010001010110001101110101110100111101
Pair \(Z_2\) Length of longest common subsequence
2IJX_1,2GRR_1 152 3
2IJX_1,4TWV_1 140 4
2GRR_1,4TWV_1 156 3

Newick tree

 
[
	2GRR_1:79.20,
	[
		2IJX_1:70,4TWV_1:70
	]:9.20
]

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 \((0.85)(0.8)(\frac{405 }{\log_{20} 405}-\frac{161}{\log_{20}161})=72.8\)
Status Protein1 Protein2 d d1/2
Query variables 2IJX_1 2GRR_1 91 75.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]