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Parikh vectors
3AUG_1 7PLS_1 8PIH_1 Letter Amino acid
4 23 5 F Phenylalanine
1 33 10 S Serine
2 25 5 V Valine
5 16 6 R Arginine
3 19 5 N Asparagine
1 14 1 Q Glutamine
2 17 3 I Isoleucine
2 37 9 L Leucine
1 25 11 D Aspartic acid
0 27 2 W Tryptophan
19 20 4 A Alanine
2 5 10 C Cysteine
0 25 11 T Threonine
4 19 8 Y Tyrosine
9 32 5 P Proline
1 24 5 E Glutamic acid
7 32 11 G Glycine
0 13 7 H Histidine
2 25 12 K Lycine
0 5 3 M Methionine 

3AUG_1|Chains A, B[auth C]|Bovine pancreatic trypsin inhibitor|Bos taurus (9913)
>7PLS_1|Chains A, B, C, D|Tissue alpha-L-fucosidase|Homo sapiens (9606)
>8PIH_1|Chain A|Phospholipase A2|Apis mellifera (7460)
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 \(w=f(c)\)
3AUG , Knot 34 65 0.72 32 49 59 
RPAFCLEPPYAGPGKARIIRYFYNAAAGAAQAFVYGGVRAKRNNFASAADALAACAAAGGPPPPP
7PLS , Knot 186 436 0.86 40 246 419 
GQPPRRYTPDWPSLDSRPLPAWFDEAKFGVFIHWGVFSVPAWGSEWFWWHWQGEGRPQYQRFMRDNYPPGFSYADFGPQFTARFFHPEEWADLFQAAGAKYVVLTTKHHEGFTNWPSPVSWNWNSKDVGPHRDLVGELGTALRKRNIRYGLYHSLLEWFHPLYLLDKKNGFKTQHFVSAKTMPELYDLVNSYKPDLIWSDGEWECPDTYWNSTNFLSWLYNDSPVKDEVVVNDRWGQNCSCHHGGYYNCEDKFKPQSLPDHKWEMCTSIDKFSWGYRRDMALSDVTEESEIISELVQTVSLGGNYLLNIGPTKDGLIVPIFQERLLAVGKWLSINGEAIYASKPWRVQWEKNTTSVWYTSKGSAVYAIFLHWPENGVLNLESPITTSTTKITMLGIQGDLKWSTDPDKGLFISLPQLPPSAVPAEFAWTIKLTGVK
8PIH , Knot 70 133 0.85 40 115 131 
IYPGTLWCGHGNKSSGPNELGRFKHTDACCRTHDMCPDVMSAGESKHGLTNTASHTRLSCDCDDKFYDCLKNSADTISSYFVGKMYFNLIDTKCYKLEHPVTGCGERTEGRCLHYTVDKSKPKVYQWFDLRKY

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 \(c\).

\(|P_{f(3AUG_1)}(2) \setminus P_{f(7PLS_1)}(2)|=17\), \(|P_{f(7PLS_1)}(2) \setminus P_{f(3AUG_1)}(2)|=214\). 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 1:01110101101111010110010011111101110111010000110110111101111111111
Pair \(Z_2\) Length of longest common subsequence
3AUG_1,7PLS_1 231 4
3AUG_1,8PIH_1 134 2
7PLS_1,8PIH_1 213 3

Newick tree

 
[
	7PLS_1:12.30,
	[
		3AUG_1:67,8PIH_1:67
	]:55.30
]

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 \((0.85)(0.8)(\frac{501 }{\log_{20} 501}-\frac{65}{\log_{20}65})=132.\)
Status Protein1 Protein2 d d1/2
Query variables 3AUG_1 7PLS_1 173 97.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]

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