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Parikh vectors
7ZOY_1 3QMG_1 5VGS_1 Letter Amino acid
8 11 2 R Arginine
11 2 5 N Asparagine
21 2 10 I Isoleucine
6 3 5 M Methionine
15 4 7 F Phenylalanine
18 5 15 G Glycine
4 7 5 H Histidine
8 0 4 W Tryptophan
19 3 9 A Alanine
3 3 4 D Aspartic acid
6 5 18 E Glutamic acid
17 6 10 S Serine
10 3 6 Y Tyrosine
16 0 11 V Valine
2 8 5 C Cysteine
6 5 5 Q Glutamine
29 3 10 L Leucine
9 6 11 K Lycine
5 2 6 P Proline
21 1 9 T Threonine 

7ZOY_1|Chain A|Synechocystis halorhodopsin|Synechocystis sp. PCC 7509 (927677)
>3QMG_1|Chain A|CpG-binding protein|Homo sapiens (9606)
>5VGS_1|Chain A|Lachrymatory-factor synthase|Allium cepa (4679)
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 \(w=f(c)\)
7ZOY , Knot 107 234 0.83 40 155 223 
MAQIWLWIGVIGMALGSIFFGIGAHNAKNERWKILFTINFFICAIATGLYLSMALGQGRSVIAGRPTVWVRYITWFLSTPLLILDLTFLGKTSLPITASLLGANAYMIATGFVATISADRTIGHIWYVVSCFAFLATVYLLVNQYRKQAERNYPQAKKVFRKLLSVHLVLWTLYPIVWLLGNTGFNAVNQGTETMFYTILDITSKVGFGFLSLNSMHTLEKNTESVSSYESSTI
3QMG , Knot 42 79 0.77 36 65 74 
MHHHHHHSSRENLYFQGQIKRSARMCGECEACRRTEDCGHCDFCRDMKKFGGPNKIRQKCRLRQCQLRARESYKYFPSS
5VGS , Knot 78 157 0.83 40 125 149 
GSPGISGGGGGKVHALLPNTKPEQAWTLLKDFINLHKVMPSLSVCELVEGEANVVGCVRYVKGIMHPIEEEFWAKEKLVALDNKNMSYSYIFTECFTGYEDYTATMQIVEGPEHKGSRFDWSFQCKYIEGMTESAFTEILQHWATEIGQKIEEVCSA

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 \(c\).

\(|P_{f(7ZOY_1)}(2) \setminus P_{f(3QMG_1)}(2)|=139\), \(|P_{f(3QMG_1)}(2) \setminus P_{f(7ZOY_1)}(2)|=49\). 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 1:110111111111111101111111001000010111010111011101101011110100111101011100101110011111010111000111010111101011101111010100011011011001111101011100000010000101001100110101111010111111100110110010001100110100011111101001001000000100000001
Pair \(Z_2\) Length of longest common subsequence
7ZOY_1,3QMG_1 188 2
7ZOY_1,5VGS_1 170 3
3QMG_1,5VGS_1 140 2

Newick tree

 
[
	7ZOY_1:95.25,
	[
		5VGS_1:70,3QMG_1:70
	]:25.25
]

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 \((0.85)(0.8)(\frac{313 }{\log_{20} 313}-\frac{79}{\log_{20}79})=74.1\)
Status Protein1 Protein2 d d1/2
Query variables 7ZOY_1 3QMG_1 97 64.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]