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Parikh vectors
6CAI_1 6TDH_1 5VMZ_1 Letter Amino acid
21 13 8 I Isoleucine
5 8 4 P Proline
24 15 4 G Glycine
24 17 9 L Leucine
17 10 11 K Lycine
6 11 12 Y Tyrosine
16 12 6 D Aspartic acid
4 3 3 M Methionine
11 4 4 F Phenylalanine
14 10 13 S Serine
23 16 7 V Valine
24 13 5 E Glutamic acid
15 11 13 R Arginine
6 5 3 N Asparagine
0 5 8 C Cysteine
4 4 6 Q Glutamine
10 3 10 H Histidine
15 13 3 T Threonine
2 2 1 W Tryptophan
23 15 4 A Alanine 

6CAI_1|Chains A, B|Kemp Eliminase KE07|synthetic construct (32630)
>6TDH_1|Chain A|Glucosamine 6-phosphate N-acetyltransferase|Aspergillus fumigatus Af293 (330879)
>5VMZ_1|Chain A|Transcriptional regulator Kaiso|Homo sapiens (9606)
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 \(w=f(c)\)
6CAI , Knot 115 264 0.81 38 158 244 
MALAKRIDAALILKDGRVVKGSNFENLRDSGDPVELGKFYSEIGIDELSFWDITASVEKRKTMLELVEKVAEQIDIPITVGGGIYDFETASELILRGADKVEINTAAVENPSLITQIAQTFGSQAVVVYIAAKRVDGEFMVFTYSGTKNTGILLRDWVVEVEKRGAGEIVLGSIDRLGTKSGYDTEMIRFVRPLTTLPIIAHRGAGKTEHFLEAFLAGADAAKADSVFHSREIDVRELKEYLKKHGVNVRLEGLGSLEHHHHHH
6TDH , Knot 87 190 0.80 40 142 184 
MTNATIAPTTTAAPVTKSVDAPTADENTPLFSPSLISPDVLAVLPADYTIRPLCRSDYKRGYLDVLRVLTTVGDINEEQWNSRYEWIRARSDEYYLLVVCDGEGRIVGTGSLVVERKFIHSLGMVGHIEDIAVEKGQQGKKLGLRIIQALDYVAEKVGCYKTILDCSEANEGFYIKCGFKRAGLEMAHYY
5VMZ , Knot 70 134 0.85 40 112 131 
MANKRMKVKHDDHYELIVDGRVYYICIVCKRSYVCLTSLRRHFNIHSWEKKYPCRYCEKVFPLAQYRTKHEIHHTGERRYQCLACGKSFINYQFMSSHIKSVHSQDPSGDSKLYRLHPCRSLQIRQYAYLSDRS

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 \(c\).

\(|P_{f(6CAI_1)}(2) \setminus P_{f(6TDH_1)}(2)|=78\), \(|P_{f(6TDH_1)}(2) \setminus P_{f(6CAI_1)}(2)|=62\). 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 1:111100101111100101101001001000101101101000111001011010101000001101100110010111011111001001001110110010100111001011001100110011110111001010111100010000111100111010001110111101001100010000110110110011111001110000110111111011010011000010100100010001101010111010000000
Pair \(Z_2\) Length of longest common subsequence
6CAI_1,6TDH_1 140 4
6CAI_1,5VMZ_1 174 4
6TDH_1,5VMZ_1 162 3

Newick tree

 
[
	5VMZ_1:88.24,
	[
		6CAI_1:70,6TDH_1:70
	]:18.24
]

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 \((0.85)(0.8)(\frac{454 }{\log_{20} 454}-\frac{190}{\log_{20}190})=77.3\)
Status Protein1 Protein2 d d1/2
Query variables 6CAI_1 6TDH_1 96 82
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]