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Parikh vectors
5RUP_1 4ONN_1 2FFL_1 Letter Amino acid
9 5 34 D Aspartic acid
19 12 99 L Leucine
11 11 57 S Serine
16 11 12 N Asparagine
3 5 24 Q Glutamine
5 13 48 P Proline
6 8 50 T Threonine
7 5 24 Y Tyrosine
15 4 70 A Alanine
6 13 40 E Glutamic acid
6 7 38 I Isoleucine
13 9 22 K Lycine
2 8 14 M Methionine
0 3 8 W Tryptophan
2 9 39 R Arginine
3 1 17 C Cysteine
13 14 54 G Glycine
6 0 23 H Histidine
5 3 32 F Phenylalanine
22 12 51 V Valine 

5RUP_1|Chains A, B|Non-structural protein 3|Severe acute respiratory syndrome coronavirus 2 (2697049)
>4ONN_1|Chain A|Ubiquitin-conjugating enzyme E2 variant 2|Homo sapiens (9606)
>2FFL_1|Chains A, B, C, D|Dicer|Giardia intestinalis (5741)
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 \(w=f(c)\)
5RUP , Knot 80 169 0.81 38 119 162 
SMVNSFSGYLKLTDNVYIKNADIVEEAKKVKPTVVVNAANVYLKHGGGVAGALNKATNNAMQVESDDYIATNGPLKVGGSCVLSGHNLAKHCLHVVGPNVNKGEDIQLLKSAYENFNQHEVLLAPLLSAGIFGADPIHSLRVCVDTVRTNVYLAVFDKNLYDKLVSSFL
4ONN , Knot 78 153 0.85 38 123 149 
GPLGSPEFMAVSTGVKVPRNFRLLEELEEGQKGVGDGTVSWGLEDDEDMTLTRWTGMIIGPPRTNYENRIYSLKVECGPKYPEAPPSVRFVTKINMNGINNSSGMVDARSIPVLAKWQNSYSIKVVLQELRRLMMSKENMKLPQPPEGQTYNN
2FFL , Knot 284 756 0.83 40 289 685 
GAMHALGHCCTVVTTRGPSHWLLLLDTHLGTLPGFKVSAGRGLPAAEVYFEAGPRVSLSRTDATIVAVYQSILFQLLGPTFPASWTEIGATMPHNEYTFPRFISNPPQFATLAFLPLLSPTSPLDLRALMVTAQLMCDAKRLSDEYTDYSTLSASLHGRMVATPEISWSLYVVLGIDSTQTSLSYFTRANESITYMRYYATAHNIHLRAADLPLVAAVRLDDLKDHQIPAPGSWDDLAPKLRFLPPELCLLLPDEFDLIRVQALQFLPEIAKHICDIQNTICALDKSFPDCGRIGGERYFAITAGLRLDQGRGRGLAGWRTPFGPFGVSHTDVFQRLELLGDAVLGFIVTARLLCLFPDASVGTLVELKMELVRNEALNYLVQTLGLPQLAEFSNNLVAKSKTWADMYEEIVGSIFTGPNGIYGCEEFLAKTLMSPEHSKTVGSACPDAVTKASKRVCMGEAGAHEFRSLVDYACEQGISVFCSSRVSTMFLERLRDIPAEDMLDWYRLGIQFSHRSGLSGPGGVVSVIDIMTHLARGLWLGSPGFYVEQQTDKNESACPPTIPVLYIYHRSVQCPVLYGSLTETPTGPVASKVLALYEKILAYESSGGSKHIAAQTVSRSLAVPIPSGTIPFLIRLLQIALTPHVYQKLELLGDAFLKCSLALHLHALHPTLTEGALTRMRQSAETNSVLGRLTKRFPSVVSEVIIESHPKIQPDSKVYGDTFEAILAAILLACGEEAAGAFVREHVLPQVVADA

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 \(c\).

\(|P_{f(5RUP_1)}(2) \setminus P_{f(4ONN_1)}(2)|=77\), \(|P_{f(4ONN_1)}(2) \setminus P_{f(5RUP_1)}(2)|=81\). 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 1:0110010101010001010010110010010101110110101001111111100100011010000011001110111001101001100010111101001001011001000100001111111011111101100101010010001011110001000110011
Pair \(Z_2\) Length of longest common subsequence
5RUP_1,4ONN_1 158 3
5RUP_1,2FFL_1 214 4
4ONN_1,2FFL_1 222 4

Newick tree

 
[
	2FFL_1:11.33,
	[
		5RUP_1:79,4ONN_1:79
	]:38.33
]

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 \((0.85)(0.8)(\frac{322 }{\log_{20} 322}-\frac{153}{\log_{20}153})=51.6\)
Status Protein1 Protein2 d d1/2
Query variables 5RUP_1 4ONN_1 66 64.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]