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Parikh vectors
3ZUL_1 7ONM_1 5IMM_1 Letter Amino acid
8 8 8 R Arginine
10 9 4 N Asparagine
14 26 6 L Leucine
10 12 5 F Phenylalanine
12 17 14 S Serine
14 8 9 Y Tyrosine
15 17 9 V Valine
12 14 11 A Alanine
4 2 2 C Cysteine
16 12 5 E Glutamic acid
20 24 5 K Lycine
11 17 2 P Proline
16 19 6 D Aspartic acid
12 8 2 I Isoleucine
6 2 2 M Methionine
5 11 8 Q Glutamine
20 23 16 G Glycine
9 12 6 H Histidine
7 12 7 T Threonine
2 7 4 W Tryptophan 

3ZUL_1|Chains A, B, C, D, E, F|FLUORESCENT PROTEIN DRONPA|ECHINOPHYLLIA SP. SC22 (301887)
>7ONM_1|Chain A|Carbonic anhydrase 2|Homo sapiens (9606)
>5IMM_1|Chain A[auth B]|Nanobody|Camelidae (9835)
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 \(w=f(c)\)
3ZUL , Knot 105 223 0.84 40 163 216 
SVIKPDMKIKLRMEGAVNGHPFAIEGVGLGKPFEGKQSMDLKVKEGGPLPFAYDILTMAFCYGNRVFAKYPENIVDYFKQSFPEGYSWERSMIYEDGGICNATNDITLDGDCYIYEIRFDGVNFPANGPVMQKRTVKWELSTEKLYVRDGVLKSDGNYALSLEGGGHYRCDFKTTYKAKKVVQLPDYHSVDHHIEIKSHDKDYSNVNLHEHAEAHSELPRQAK
7ONM , Knot 112 260 0.79 40 176 249 
MAHHWGYGKHNGPEHWHKDFPIAKGERQSPVDIDTHTAKYDPSLKPLSVSYDQATSLRILNNGHAFGVRFDDSQDKAVLKGGPLDGTYRLCQFHFHWGSLDGQGSEHTVDKKKYAAELHLVHWNTKYGDFGKAVQQPDGLAVLGIFLKVGSAKPGLQKVVDVLDSIKTKGKSADFTNFDPRGLLPESLDYWTYPGSLTTPPLLECVTWIVLKEPISVSSEQVLKFRKLNFNGEGEPEELMVDNWRPAQPLKNRQIKASFK
5IMM , Knot 62 131 0.77 40 102 124 
QVQLVESGGGLVQAGGSLRLSCAASGRTFSSYGMGWFRQAPGKEREFVAAIRWNGGSTYYADSVKGRFTISRDNAKNTVYLQMNSLKPEDTAVYYCAAGRWDKYGSSFQDEYDYWGQGTQVTVSSHHHHHH

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 \(c\).

\(|P_{f(3ZUL_1)}(2) \setminus P_{f(7ONM_1)}(2)|=78\), \(|P_{f(7ONM_1)}(2) \setminus P_{f(3ZUL_1)}(2)|=91\). 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 1:0110101010101011101011110111110110100010101001111111001101110010011100100110010001101001000110001110010001010100010010101101110111100001010100001010011100010011010111000001000001001101100001000101000000000101000101000110010
Pair \(Z_2\) Length of longest common subsequence
3ZUL_1,7ONM_1 169 4
3ZUL_1,5IMM_1 187 3
7ONM_1,5IMM_1 158 4

Newick tree

 
[
	3ZUL_1:92.23,
	[
		7ONM_1:79,5IMM_1:79
	]:13.23
]

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 \((0.85)(0.8)(\frac{483 }{\log_{20} 483}-\frac{223}{\log_{20}223})=75.1\)
Status Protein1 Protein2 d d1/2
Query variables 3ZUL_1 7ONM_1 96 89.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]