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Parikh vectors
3BXC_1 1DMQ_1 1WHZ_1 Letter Amino acid
18 9 7 E Glutamic acid
2 2 1 W Tryptophan
10 4 1 N Asparagine
7 10 0 Q Glutamine
25 13 7 G Glycine
19 7 7 L Leucine
13 3 1 S Serine
11 3 1 Y Tyrosine
4 3 0 C Cysteine
10 9 9 R Arginine
10 8 2 D Aspartic acid
13 3 4 H Histidine
11 7 3 M Methionine
10 5 4 F Phenylalanine
11 8 6 P Proline
10 12 3 A Alanine
19 1 4 K Lycine
19 4 3 T Threonine
14 13 5 V Valine
9 7 2 I Isoleucine 

3BXC_1|Chains A, B, C, D, E, F, G, H|Far-red fluorescent protein mKate|Entacmaea quadricolor (6118)
>1DMQ_1|Chain A|STEROID DELTA-ISOMERASE|Pseudomonas putida (303)
>1WHZ_1|Chain A|Hypothetical protein|Thermus thermophilus (274)
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 \(w=f(c)\)
3BXC , Knot 113 245 0.84 40 170 235 
MRGSHHHHHHGSMSELITENMHMKLYMEGTVNNHHFKCTSEGEGKPYEGTQTMRIKVVEGGPLPFAFDILATSFMYGSKTFINHTQGIPDFFKQSFPEGFTWERVTTYEDGGVLTATQDTSLQDGCLIYNVKIRGVNFPSNGPVMQKKTLGWEASTEMLYPADGGLEGRSDMALKLVGGGHLICNLKTTYRSKKPAKNLKMPGVYYVDRRLERIKEADKETYVEQHEVAVARYCDLPSKLGHKLN
1DMQ , Knot 65 131 0.80 40 110 125 
MNLPTAQEVQGLMARYIELVDVGDIEAIVQMFADDATVEDPFGQPPIHGREQIAAFYRQGLGGGKVRACLTGPVRASHNGCGAMPFRVEMVWNGQPCALDVIDVMRFDEHGRIQTMQAYWSEVNLSVREPQ
1WHZ , Knot 38 70 0.76 36 61 68 
MWMPPRPEEVARKLRRLGFVERMAKGGHRLYTHPDGRIVVVPFHSGELPKGTFKRILRDAGLTEEEFHNL

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 \(c\).

\(|P_{f(3BXC_1)}(2) \setminus P_{f(1DMQ_1)}(2)|=118\), \(|P_{f(1DMQ_1)}(2) \setminus P_{f(3BXC_1)}(2)|=58\). 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 1:10100000001010011000101010101010000100000101010010001010110111111110111001101000110000111011000110110100100000111101000001001011001010110110011110000111010001101101110100011101111101100100000000110010111100100010010010000010000111100001100110010
Pair \(Z_2\) Length of longest common subsequence
3BXC_1,1DMQ_1 176 3
3BXC_1,1WHZ_1 163 3
1DMQ_1,1WHZ_1 133 3

Newick tree

 
[
	3BXC_1:90.09,
	[
		1WHZ_1:66.5,1DMQ_1:66.5
	]:23.59
]

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 \((0.85)(0.8)(\frac{376 }{\log_{20} 376}-\frac{131}{\log_{20}131})=74.4\)
Status Protein1 Protein2 d d1/2
Query variables 3BXC_1 1DMQ_1 95 73
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]