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Parikh vectors
5DAU_1 6RHS_1 9DMJ_1 Letter Amino acid
4 31 42 I Isoleucine
7 10 15 Y Tyrosine
20 22 26 K Lycine
4 9 15 M Methionine
4 15 28 P Proline
1 2 10 W Tryptophan
6 16 14 Q Glutamine
14 27 23 E Glutamic acid
2 6 13 H Histidine
12 29 44 L Leucine
15 45 16 A Alanine
5 12 12 R Arginine
10 36 22 G Glycine
4 21 34 S Serine
9 15 28 T Threonine
6 21 45 V Valine
8 20 17 N Asparagine
8 20 23 D Aspartic acid
0 0 7 C Cysteine
4 13 23 F Phenylalanine 

5DAU_1|Chain A|Thermonuclease|Staphylococcus aureus (1280)
>6RHS_1|Chain A|Putative L-lactate oxidase|Pediococcus acidilactici DSM 20284 (862514)
>9DMJ_1|Chains A, B[auth C]|Acetylcholine receptor subunit alpha|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 \(w=f(c)\)
5DAU , Knot 67 143 0.77 38 102 134 
ATSTKKLHKEPATLIKAIDGDTVKLMYKGQPMTFRLLLEDTPEFNEKYGPEASAFTKKMVENAKKIEVEFDKGQRTDKYGRGLAYIYADGKMVNEALERQGLAKVAYVYKGNNTHEQLLRKAEAQAKKEKLNNWSEDNADSGQ
6RHS , Knot 156 370 0.83 38 203 348 
SMTMINGYEQSDREEKIDILNLESLEKQAEEIIPAGGFGYIAGGSEDEWTLKQNRMAFHHRQIAPKALSGIEKPELNTEIFGIPLNTPVMMAPAAAQGLAHSQGEKDTARGLAAVGGLMAQSTYSSVSIAETAAAGGDAPQFFQLYMSKDWNFNESLLDEAKKANVKAIILTVDATVDGYREADIKNKFTFPLPMANLIKFSEGNGQGKGIEEIYASAAQNIRPEDVKRIADYTNLPVIVKGIQTPEDAIRAIDAGAAGIYVSNHGGRQLNGGPASFDVLEDIATAVNKQVPIIFDSGVRRGSDVFKALASGADLVALGRPVIYGLALGGAKGVQSVFEHLNHELEIVMQLAGTKTIEDVKNNSLLNIKY
9DMJ , Knot 185 457 0.82 40 241 434 
MEPWPLLLLFSLCSAGLVLGSEHETRLVAKLFKDYSSVVRPVEDHRQVVEVTVGLQLIQLINVDEVNQIVTTNVRLKQQWVDYNLKWNPDDYGGVKKIHIPSEKIWRPDLVLYNNADGDFAIVKFTKVLLQYTGHITWTPPAIFKSYCEIIVTHFPFDEQNCSMKLGTWTYDGSVVAINPESDQPDLSNFMESGEWVIKESRGWKHSVTYSCCPDTPYLDITYHFVMQRLPLYFIVNVIIPCLLFSFLTGLVFYLPTDSGEKMTLSISVLLSLTVFLLVIVELIPSTSSAVPLIGKYMLFTMVFVIASIIITVIVINTHHRSPSTHVMPNWVRKVFIDTIPNIMFFSTMKRPSREKQDKKIFTEDIDISDISGKPGPPPMGFHSPLIKHPEVKSAIEGIKYIAETMKSDQESNNAAAEWKYVAMVMDHILLGVFMLVCIIGTLAVFAGRLIELNQQG

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 \(c\).

\(|P_{f(5DAU_1)}(2) \setminus P_{f(6RHS_1)}(2)|=32\), \(|P_{f(6RHS_1)}(2) \setminus P_{f(5DAU_1)}(2)|=133\). 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 1:10000010001101101101001011001011010111000101000011010110001100100101010010000001011101010101100110001110110100100000011001010100001001000010010
Pair \(Z_2\) Length of longest common subsequence
5DAU_1,6RHS_1 165 4
5DAU_1,9DMJ_1 201 3
6RHS_1,9DMJ_1 162 4

Newick tree

 
[
	5DAU_1:95.31,
	[
		6RHS_1:81,9DMJ_1:81
	]:14.31
]

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 \((0.85)(0.8)(\frac{513 }{\log_{20} 513}-\frac{143}{\log_{20}143})=108.\)
Status Protein1 Protein2 d d1/2
Query variables 5DAU_1 6RHS_1 135 92.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]