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Parikh vectors
2CXC_1 9BEE_1 7ESV_1 Letter Amino acid
14 7 15 R Arginine
15 6 22 G Glycine
14 6 10 I Isoleucine
1 0 11 M Methionine
9 8 18 S Serine
18 10 31 V Valine
7 1 42 A Alanine
1 0 2 C Cysteine
1 2 5 Q Glutamine
13 6 25 L Leucine
3 5 18 T Threonine
4 1 6 Y Tyrosine
14 8 19 E Glutamic acid
1 5 6 H Histidine
12 6 13 K Lycine
4 3 6 F Phenylalanine
1 6 16 P Proline
7 2 11 N Asparagine
5 6 12 D Aspartic acid
0 0 3 W Tryptophan 

2CXC_1|Chain A|NusA|Aeropyrum pernix (272557)
>9BEE_1|Chain A[auth B]|Alpha-crystallin B chain|Homo sapiens (9606)
>7ESV_1|Chains A, B, C|Packaging NTPase|Pseudomonas phage phiYY (1852644)
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 \(w=f(c)\)
2CXC , Knot 68 144 0.78 38 99 136 
MSGDYRITLEELRYISVFHSITGVTAYRCIVDEENNRLIFLVSEGEAGRAIGRGGRLIKLLREALGKNIEVVEYSSDLERIVKNLFPGVKIESINVRERNGVKQVVIKVSEDDKGAAIGKGGKNVKRARLVLSKLFGVEKVVIR
9BEE , Knot 46 88 0.78 34 69 86 
SVNLDVKHFSPEELKVKVLGDVIEVHGKHEERQDEHGFISREFHRKYRIPADVDPLTITSSLSSDGVLTVNGPRKQVSGPERTIPITR
7ESV , Knot 129 291 0.83 40 176 275 
MTDNKRPQLKDKEPAAKPARTRKPKNETVALVVEATTEAEAKKSLREGGLVPAAHEIMIPVGNMILAVDTQVLDKCALALAASDDPGRWFAENESLIHSTVFAPVAKGLHRVYPLLSVRPEVPAGYEASWPTQDHMPGLHLVVGGTGAGKSSYLASQDLTLVIRWGEPAERFDVEGATHAVSDLNEALAVAFVMARAGYRPAIDSFRNLVFGIESAAGEGGISTALYSAMTAINNVCSRLGIVVMVVVNPMATEAKAELVYNNMAASVAGMTVLMDGAVSKQTVRTLSGRT

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 \(c\).

\(|P_{f(2CXC_1)}(2) \setminus P_{f(9BEE_1)}(2)|=68\), \(|P_{f(9BEE_1)}(2) \setminus P_{f(2CXC_1)}(2)|=38\). 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 1:101000101001001011001011010001100000011111001011011101101101100111001011000001001100111110100101000011001110100000111110110010010111001111001110
Pair \(Z_2\) Length of longest common subsequence
2CXC_1,9BEE_1 106 3
2CXC_1,7ESV_1 147 3
9BEE_1,7ESV_1 163 2

Newick tree

 
[
	7ESV_1:84.22,
	[
		2CXC_1:53,9BEE_1:53
	]:31.22
]

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 \((0.85)(0.8)(\frac{232 }{\log_{20} 232}-\frac{88}{\log_{20}88})=46.7\)
Status Protein1 Protein2 d d1/2
Query variables 2CXC_1 9BEE_1 57 45.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]