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Parikh vectors
3NRM_1 7HFN_1 5RYI_1 Letter Amino acid
9 16 17 N Asparagine
4 3 6 C Cysteine
10 6 17 H Histidine
19 13 32 K Lycine
12 7 16 Y Tyrosine
20 15 17 A Alanine
10 9 21 D Aspartic acid
17 13 29 G Glycine
4 2 7 M Methionine
4 0 8 W Tryptophan
11 3 23 Q Glutamine
21 2 16 R Arginine
24 6 37 E Glutamic acid
13 6 31 I Isoleucine
34 19 43 L Leucine
12 5 24 F Phenylalanine
14 5 20 P Proline
20 11 38 S Serine
11 6 39 T Threonine
14 22 22 V Valine 

3NRM_1|Chain A|Serine/threonine-protein kinase 6|Homo sapiens (9606)
>7HFN_1|Chains A, B|Non-structural protein 3|Severe acute respiratory syndrome coronavirus 2 (2697049)
>5RYI_1|Chain A|Phosphatidylinositol 3,4,5-trisphosphate 5-phosphatase 1|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 \(w=f(c)\)
3NRM , Knot 128 283 0.85 40 179 275 
GAMGSRQWALEDFEIGRPLGKGKFGNVYLAREKQSKFILALKVLFKAQLEKAGVEHQLRREVEIQSHLRHPNILRLYGYFHDATRVYLILEYAPLGTVYRELQKLSKFDEQRTATYITELANALSYCHSKRVIHRDIKPENLLLGSAGELKIADFGWSVHAPSSRRAALCGTLDYLPPEMIEGRMHDEKVDLWSLGVLCYEFLVGKPPFEANTYQETYKRISRVEFTFPDFVTEGARDLISRLLKHNPSQRPMLREVLEHPWITANSSKPSNCQNKESASKQS
7HFN , Knot 80 169 0.81 38 119 162 
SMVNSFSGYLKLTDNVYIKNADIVEEAKKVKPTVVVNAANVYLKHGGGVAGALNKATNNAMQVESDDYIATNGPLKVGGSCVLSGHNLAKHCLHVVGPNVNKGEDIQLLKSAYENFNQHEVLLAPLLSAGIFGADPIHSLRVCVDTVRTNVYLAVFDKNLYDKLVSSFL
5RYI , Knot 195 463 0.86 40 252 447 
SMEQPEPDMITIFIGTWNMGNAPPPKKITSWFLSKGQGKTRDDSADYIPHDIYVIGTQEDPLSEKEWLEILKHSLQEITSVTFKTVAIHTLWNIRIVVLAKPEHENRISHICTDNVKTGIANTLGNKGAVGVSFMFNGTSLGFVNSHLTSGSEKKLRRNQNYMNILRFLALGDKKLSPFNITHRFTHLFWFGDLNYRVDLPTWEAETIIQKIKQQQYADLLSHDQLLTERREQKVFLHFEEEEITFAPTYRFERLTRDKYAYTKQKATGMKYNLPSWCDRVLWKSYPLVHVVCQSYGSTSDIMTSDHSPVFATFEAGVTSQFVSKNGPGTVDSQGQIEFLRCYATLKTKSQTKFYLEFHSSCLESFVKSQEGENEEGSEGELVVKFGETLPKLKPIISDPEYLLDQHILISIKSSDSDESYGEGCIALRLEATETQLPIYTPLTHHGELTGHFQGEIKLQTSQ

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 \(c\).

\(|P_{f(3NRM_1)}(2) \setminus P_{f(7HFN_1)}(2)|=115\), \(|P_{f(7HFN_1)}(2) \setminus P_{f(3NRM_1)}(2)|=55\). 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 1:1111000111001011011101011010110000001111101110101001110001000101000100101101010100100101110011110100010010010000010010011011000000011000101001111011010110111010110000111010100111011010100001011011110001111011101000000000100101011011001100110011000100011100110011101000010000000010000
Pair \(Z_2\) Length of longest common subsequence
3NRM_1,7HFN_1 170 5
3NRM_1,5RYI_1 175 4
7HFN_1,5RYI_1 183 4

Newick tree

 
[
	5RYI_1:90.97,
	[
		3NRM_1:85,7HFN_1:85
	]:5.97
]

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 \((0.85)(0.8)(\frac{452 }{\log_{20} 452}-\frac{169}{\log_{20}169})=83.4\)
Status Protein1 Protein2 d d1/2
Query variables 3NRM_1 7HFN_1 107 84
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]