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Parikh vectors
6PWF_1 5IIM_1 3NGF_1 Letter Amino acid
1 8 13 N Asparagine
7 22 21 E Glutamic acid
7 16 10 I Isoleucine
13 19 6 K Lycine
4 13 15 D Aspartic acid
6 14 14 P Proline
0 4 3 W Tryptophan
6 18 18 V Valine
18 28 28 A Alanine
1 7 3 C Cysteine
3 5 9 M Methionine
10 18 9 T Threonine
3 10 10 Y Tyrosine
6 26 9 S Serine
18 25 18 R Arginine
8 18 8 Q Glutamine
7 26 20 G Glycine
2 12 14 H Histidine
12 34 25 L Leucine
4 11 16 F Phenylalanine 

6PWF_1|Chains A, E|Histone H3|Drosophila melanogaster (7227)
>5IIM_1|Chain A|DNA polymerase lambda|Homo sapiens (9606)
>3NGF_1|Chains A, B|AP endonuclease, family 2|Brucella melitensis biovar Abortus (359391)
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 \(w=f(c)\)
6PWF , Knot 65 136 0.78 38 104 128 
MARTKQTARKSTGGKAPRKQLATKAARKSAPATGGVKKPHRYRPGTVALREIRRYQKSTELLIRKLPFQRLVREIAQDFKTDLRFQSSAVMALQEASEAYLVGLFEDTNLCAIHAKRVTIMPKDIQLARRIRGERA
5IIM , Knot 145 334 0.84 40 211 323 
AQPSSQKATNHNLHITEKLEVLAKAYSVQGDKWRALGYAKAINALKSFHKPVTSYQEACSIPGIGKRMAEKIIEILESGHLRKLDHISESVPVLELFSNIWGAGTKTAQMWYQQGFRSLEDIRSQASLTTQQAIGLKHYSDFLERMPREEATEIEQTVQKAAQAFNSGLLCVACGSYRRGKATCGDVDVLITHPDGRSHRGIFSRLLDSLRQEGFLTDDLVSQEENGQQQKYLGVCRLPGPGRRHRRLDIIVVPYSEFACALLYFTGSAHFNRSMRALAKTKGMSLSEHALSTAVVRNTHGCKVGPGRVLPTPTEKDVFRLLGLPYREPAERDW
3NGF , Knot 123 269 0.85 40 179 262 
MAHHHHHHMPRFAANLSTMFNEVPFLERFRLAAEAGFGGVEFLFPYDFDADVIARELKQHNLTQVLFNMPPGDWAAGERGMAAISGREQEFRDNVDIALHYALALDCRTLHAMSGITEGLDRKACEETFIENFRYAADKLAPHGITVLVEPLNTRNMPGYFIVHQLEAVGLVKRVNRPNVAVQLDLYHAQIMDGDLTRLIEKMNGAFSHVQIASVPDRHEPDEGELNYPYLFSVLESVGYRGWVGCEYNPRGKTESGLAWFAPYRDQSA

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 \(c\).

\(|P_{f(6PWF_1)}(2) \setminus P_{f(5IIM_1)}(2)|=30\), \(|P_{f(5IIM_1)}(2) \setminus P_{f(6PWF_1)}(2)|=137\). 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 1:1100000100001101100011001100011101110010000110111001000000001110011100110011001000101000111110010010111110000101101001011100101100101001
Pair \(Z_2\) Length of longest common subsequence
6PWF_1,5IIM_1 167 3
6PWF_1,3NGF_1 165 3
5IIM_1,3NGF_1 174 4

Newick tree

 
[
	5IIM_1:86.17,
	[
		6PWF_1:82.5,3NGF_1:82.5
	]:3.67
]

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 \((0.85)(0.8)(\frac{470 }{\log_{20} 470}-\frac{136}{\log_{20}136})=99.2\)
Status Protein1 Protein2 d d1/2
Query variables 6PWF_1 5IIM_1 129 88.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]