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Parikh vectors
7NBY_1 4QFT_1 2VDA_1 Letter Amino acid
17 7 48 D Aspartic acid
16 12 41 S Serine
27 16 57 V Valine
14 6 33 Q Glutamine
26 14 51 G Glycine
29 16 74 L Leucine
11 9 45 K Lycine
13 9 27 P Proline
24 13 40 T Threonine
3 2 7 W Tryptophan
11 5 24 Y Tyrosine
17 9 72 A Alanine
11 6 60 R Arginine
7 7 20 H Histidine
10 6 30 M Methionine
17 7 25 F Phenylalanine
21 5 33 N Asparagine
12 2 1 C Cysteine
9 19 87 E Glutamic acid
11 16 53 I Isoleucine 

7NBY_1|Chains A, B|Main Protease|Severe acute respiratory syndrome coronavirus 2 (2697049)
>4QFT_1|Chain A|COP9 signalosome complex subunit 6|Homo sapiens (9606)
>2VDA_1|Chain A|TRANSLOCASE SUBUNIT SECA|ESCHERICHIA COLI (562)
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 \(w=f(c)\)
7NBY , Knot 134 306 0.83 40 196 290 
SGFRKMAFPSGKVEGCMVQVTCGTTTLNGLWLDDVVYCPRHVICTSEDMLNPNYEDLLIRKSNHNFLVQAGNVQLRVIGHSMQNCVLKLKVDTANPKTPKYKFVRIQPGQTFSVLACYNGSPSGVYQCAMRPNFTIKGSFLNGSCGSVGFNIDYDCVSFCYMHHMELPTGVHAGTDLEGNFYGPFVDRQTAQAAGTDTTITVNVLAWLYAAVINGDRWFLNRFTTTLNDFNLVAMKYNYEPLTQDHVDILGPLSAQTGIAVLDMCASLKELLQNGMNGRTILGSALLEDEFTPFDVVRQCSGVTFQ
4QFT , Knot 85 186 0.79 40 124 179 
GPLGSMACGVTGSVSVALHPLVILNISDHWIRMRSQEGRPVQVIGALIGKQEGRNIEVMNSFELLSHTVEEKIIIDKEYYYTKEEQFKQVFKELEFLGWYTTGGPPDPSDIHVHKQVCEIIESPLFLKLNPMTKHTDLPVSVFESVIDIINGEATMLFAELTYTLATEEAERIGVDHVARMTATGS
2VDA , Knot 314 828 0.85 40 291 756 
VFGSRNDRTLRRMRKVVNIINAMEPEMEKLSDEELKGKTAEFRARLEKGEVLENLIPEAFAVVREASKRVFGMRHFDVQLLGGMVLNERCIAEMRTGEGKTLTATLPAYLNALTGKGVHVVTVNDYLAQRDAENNRPLFEFLGLTVGINLPGMPAPAKREAYAADITYGTNNEYGFDYLRDNMAFSPEERVQRKLHYALVDEVDSILIDEARTPLIISGPAEDSSEMYKRVNKIIPHLIRQEKEDSETFQGEGHFSVDEKSRQVNLTERGLVLIEELLVKEGIMDEGESLYSPANIMLMHHVTAALRAHALFTRDVDYIVKDGEVIIVDEHTGRTMQGRRWSDGLHQAVEAKEGVQIQNENQTLASITFQNYFRLYEKLAGMTGTADTEAFEFSSIYKLDTVVVPTNRPMIRKDLPDLVYMTEAEKIQAIIEDIKERTAKGQPVLVGTISIEKSELVSNELTKAGIKHNVLNAKFHANEAAIVAQAGYPAAVTIATNMAGRGTDIVLGGSWQAEVAALENPTAEQIEKIKADWQVRHDAVLEAGGLHIIGTERHESRRIDNQLRGRSGRQGDAGSSRFYLSMEDALMRIFASDRVSGMMRKLGMKPGEAIEHPWVTKAIANAQRKVESRNFDIRKQLLEYDDVANDQRRAIYSQRNELLDVSDVSETINSIREDVFKATIDAYIPPQSLEEMWDIPGLQERLKNDFDLDLPIAEWLDKEPELHEETLRERILAQSIEVYQRKEEVVGAEMMRHFEKGVMLQTLDSLWKEHLAAMDYLRQGIHLRGYAQKDPKQEYKRESFSMFAAMLESLKYEVISTLSKVQVRMPEE

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 \(c\).

\(|P_{f(7NBY_1)}(2) \setminus P_{f(4QFT_1)}(2)|=132\), \(|P_{f(4QFT_1)}(2) \setminus P_{f(7NBY_1)}(2)|=60\). 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 1:011001111010101011010010001011110011001001100000110100001110000001110110101011100100011010100101001000110101100101110001010110001101010101011010010111010000101001001011011011001010101111000010111000010101111101111010011100100010010111100000110000101111101001111101010100110011010011101110001011011000011010
Pair \(Z_2\) Length of longest common subsequence
7NBY_1,4QFT_1 192 3
7NBY_1,2VDA_1 181 3
4QFT_1,2VDA_1 205 4

Newick tree

 
[
	4QFT_1:10.06,
	[
		7NBY_1:90.5,2VDA_1:90.5
	]:11.56
]

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 \((0.85)(0.8)(\frac{492 }{\log_{20} 492}-\frac{186}{\log_{20}186})=89.1\)
Status Protein1 Protein2 d d1/2
Query variables 7NBY_1 4QFT_1 115 90.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]