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Parikh vectors
5SUX_1 3HRW_1 4JEJ_1 Letter Amino acid
4 1 12 Q Glutamine
24 11 17 K Lycine
1 6 12 P Proline
15 10 16 V Valine
30 2 28 I Isoleucine
9 1 3 M Methionine
10 11 17 G Glycine
25 17 31 L Leucine
19 7 7 F Phenylalanine
11 7 11 T Threonine
8 20 20 A Alanine
4 1 0 C Cysteine
15 9 12 D Aspartic acid
16 4 16 E Glutamic acid
1 11 5 H Histidine
35 14 16 S Serine
4 1 1 W Tryptophan
13 3 5 Y Tyrosine
12 3 4 R Arginine
20 2 11 N Asparagine 

5SUX_1|Chain A|AraC family transcriptional regulator|Vibrio cholerae (666)
>3HRW_1|Chains A, C|Hemoglobin subunit alpha|Mus musculus (10090)
>4JEJ_1|Chain A|Geranylgeranylglyceryl phosphate synthase|Flavobacterium johnsoniae (376686)
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 \(w=f(c)\)
5SUX , Knot 122 276 0.82 40 166 262 
MIGKKSFQTNVYRMSKFDTYIFNNLYINDYKMFWIDSGIAKLIDKNCLVSYEINSSSIILLKKNSIQRFSLTSLSDENINVSVITISDSFIRSLKSYILGDLMIRNLYSENKDLLLWNCEHNDIAVLSEVVNGFREINYSDEFLKVFFSGFFSKVEKKYNSIFITDDLDAMEKISCLVKSDITRNWRWADICGELRTNRMILKKELESRGVKFRELINSIRISYSISLMKTGEFKIKQIAYQSGFASVSYFSTVFKSTMNVAPSEYLFMLTGVAEK
3HRW , Knot 67 141 0.78 40 104 134 
VLSGEDKSNIKAAWGKIGGHGAEYGAEALERMFASFPTTKTYFPHFDVSHGSAQVKGHGKKVADALASAAGHLDDLPGALSALSDLHAHKLRVDPVNFKLLSHCLLVTLASHHPADFTPAVHASLDKFLASVSTVLTSKYR
4JEJ , Knot 108 244 0.81 38 153 235 
MEQKILTTIHQQILEAKKNGQKLLAILLDPDKIVWENLDHLLLKINQSPATHIFVGGSIVESTIIEDLIAQLKQKTRLPVVIFPGDPSQISPKADAILFLSLLSGRNPDYLIEYQVQAAPILKKTNLEVISTGYILIESGNETAVARVSKTEPLNRENFDLALATAQAGEMLGSKLIYLEAGSGAKKPVPLEMISVISQNVEIPIIVGGGIVDLHGIKKAYNAGADLVVIGTAFENDSHFFDSL

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 \(c\).

\(|P_{f(5SUX_1)}(2) \setminus P_{f(3HRW_1)}(2)|=113\), \(|P_{f(3HRW_1)}(2) \setminus P_{f(5SUX_1)}(2)|=51\). 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 1:111000100010010010001100101000011110011101100001100010000111100001001010010000101011010001100100011101110010000001111000000111100110110010000011011101110010000001110001011001001100010001011010101000011100010001101001100101000101100101010011000111010010011000101110001111011100
Pair \(Z_2\) Length of longest common subsequence
5SUX_1,3HRW_1 164 4
5SUX_1,4JEJ_1 165 4
3HRW_1,4JEJ_1 147 3

Newick tree

 
[
	5SUX_1:84.96,
	[
		3HRW_1:73.5,4JEJ_1:73.5
	]:11.46
]

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 \((0.85)(0.8)(\frac{417 }{\log_{20} 417}-\frac{141}{\log_{20}141})=82.7\)
Status Protein1 Protein2 d d1/2
Query variables 5SUX_1 3HRW_1 106 78.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]