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Parikh vectors
5KCU_1 7ZWZ_1 5JTF_1 Letter Amino acid
14 4 3 K Lycine
15 6 2 M Methionine
9 3 11 P Proline
12 9 16 R Arginine
9 7 3 N Asparagine
9 4 10 Q Glutamine
15 6 16 E Glutamic acid
13 9 13 I Isoleucine
11 7 6 T Threonine
4 3 11 Y Tyrosine
13 10 18 V Valine
13 3 12 H Histidine
7 7 7 F Phenylalanine
3 0 3 W Tryptophan
13 8 10 D Aspartic acid
20 9 11 S Serine
15 8 20 A Alanine
4 5 0 C Cysteine
10 5 17 G Glycine
48 15 18 L Leucine 

5KCU_1|Chains A, B|Estrogen receptor|Homo sapiens (9606)
>7ZWZ_1|Chain A|B-cell lymphoma 6 protein|Homo sapiens (9606)
>5JTF_1|Chains A, B|Putative Phosphinothricin N-acetyltransferase|Pseudomonas putida (160488)
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 \(w=f(c)\)
5KCU , Knot 117 257 0.84 40 165 246 
IKRSKKNSLALSLTADQMVSALLDAEPPILYSEYDPTRPFSEASMMGLLTNLADRELVHMINWAKRVPGFVDLTLHDQVHLLECAWLEILMIGLVWRSMEHPGKLLFAPNLLLDRNQGKCVEGMVEIFDMLLATSSRFRMMNLQGEEFVCLKSIILLNSGVYTFLSSTLKSLEEKDHIHRVLDKITDTLIHLMAKAGLTLQQQHQRLAQLLLILSHIRHMSNKGMEHLYSMKCKNVVPLSDLLLEMLDAHRLHAPTS
7ZWZ , Knot 65 128 0.82 38 98 125 
GPGADSCIQFTRHASDVLLNLNRLRSRDILTDVVIVVSREQFRAHKTVLMACSGLFYSIFTDQLKCNLSVINLDPEINPEGFCILLDFMYTSRLNLREGNIMAVMATAMYLQMEHVVDTCRKFIKASE
5JTF , Knot 92 207 0.79 38 138 193 
MHSGIDIRVARPEDAEEIQIIYAPIVLNTAISFEEAVPSVEQMRERISTTLQTYPYLVAVREGRVVGYAYASQHRARAAYRWAVDVTVYVAEGQRRSGIARQLYDVLLPVLKRLGYRSAYAGIALPNEGSVGLHERLGFQHIGTFPQVGFKLDAWHDVGYWRFDFGDEGLHPEAPLGFLSQIPLGPEQKLISEEDLNSAVDHHHHHH

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 \(c\).

\(|P_{f(5KCU_1)}(2) \setminus P_{f(7ZWZ_1)}(2)|=111\), \(|P_{f(7ZWZ_1)}(2) \setminus P_{f(5KCU_1)}(2)|=44\). 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 1:10000000111010100110111010111100000100110010111110011000110110110011111010100010110011101111111100100110111110111000010010111011011110000101101010011010011110011001100010010000010011001000110111011101000000110111110010010001100100100001111001110110100101100
Pair \(Z_2\) Length of longest common subsequence
5KCU_1,7ZWZ_1 155 3
5KCU_1,5JTF_1 169 3
7ZWZ_1,5JTF_1 166 4

Newick tree

 
[
	5JTF_1:85.73,
	[
		5KCU_1:77.5,7ZWZ_1:77.5
	]:8.23
]

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 \((0.85)(0.8)(\frac{385 }{\log_{20} 385}-\frac{128}{\log_{20}128})=78.0\)
Status Protein1 Protein2 d d1/2
Query variables 5KCU_1 7ZWZ_1 102 75
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]