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Parikh vectors
2BLP_1 3EYF_1 5OHX_1 Letter Amino acid
12 15 31 A Alanine
4 10 14 R Arginine
10 11 42 K Lycine
9 14 25 V Valine
5 8 28 D Aspartic acid
8 5 11 C Cysteine
5 12 30 E Glutamic acid
4 3 9 H Histidine
15 27 32 S Serine
10 19 28 T Threonine
0 3 8 W Tryptophan
7 13 18 Q Glutamine
3 14 35 G Glycine
3 7 56 I Isoleucine
2 18 40 L Leucine
10 5 30 N Asparagine
4 2 14 M Methionine
3 9 11 F Phenylalanine
4 13 23 P Proline
6 8 19 Y Tyrosine 

2BLP_1|Chain A|RIBONUCLEASE PANCREATIC PRECURSOR|BOS TAURUS (9913)
>3EYF_1|Chains A, C|8f9 Fab|Homo sapiens (9606)
>5OHX_1|Chains A, B|Cystathionine beta-synthase|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 \(w=f(c)\)
2BLP , Knot 64 124 0.83 38 107 121 
KETAAAKFERQHMDSSTSAASSSNYCNQMMKSRNLTKDRCKPVNTFVHESLADVQAVCSQKNVACKNGQTNCYQSYSTMSITDCRETGSSKYPNCAYKTTQANKHIIVACEGNPYVPVHFDASV
3EYF , Knot 97 216 0.80 40 151 203 
MEIVLTQSPATLSLSPGERATLSCRASQSVGGYLTWYQHKPGQAPRLLIFDASIRATGIPARFSGSGSGTDFTLTITRLEPEDFAVYYCQQRSMWPPVTFGQGTKLEIKRTVAAPSVFIFPPSDEQLKSGTASVVCLLNNFYPREAKVQWKVDNALQSGNSQESVTEQDSKDSTYSLSSTLTLSKADYEKHKVYACEVTHQGLSSPVTKSFNRGEC
5OHX , Knot 208 504 0.85 40 248 472 
MEFKQPNRPSYCTWELNATNSPHTCRTKNGDYTKIMPDILTAIGQTPLIKLNNIPKSYGIKCEIYAKCEFLNPGGSVKDRIAYRMIQDAEDKGLLKPGCTIIEPTSGNTGIGLAMAAAVRGYKCIIVMPEKMSDEKISTLYALGAKIIRTPTEASWHSPEAHISVAQKLQKEIPNSIILDQYTNPGNPLAHYDQTAIEIWKQCEGKIDYLVAGAGTGGTISGIGRKLKELSPNIKIIAVDPKGSILDPSSDSQNEVGFYEVEGIGYDFIPTVLDRNVIDKWIKTEDNESLNAARMLIRQEGLLCGGSSGAALIAALKIAKDIPEEKRMVIILPDGIRNYLTKFVSEYWMETRGFLQPVCQNEMNKWWWNMKISNLSFDKQSLLKENTVTCQEAMHMLKNADSQLLVISDDNIHIKGVISLNKLTSYVISGIVKCTDFVDKAMVKQYVKVKHSATLGYISRVLEKEPYVIILDDEHDDAFIGIVNQFHILQFITKNGTSNNYLIN

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 \(c\).

\(|P_{f(2BLP_1)}(2) \setminus P_{f(3EYF_1)}(2)|=59\), \(|P_{f(3EYF_1)}(2) \setminus P_{f(2BLP_1)}(2)|=103\). 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 1:0001110100001000001100000000110000100000011001100011010110000011000100000000001010000001000010010000010001111001010111010101
Pair \(Z_2\) Length of longest common subsequence
2BLP_1,3EYF_1 162 4
2BLP_1,5OHX_1 207 3
3EYF_1,5OHX_1 195 3

Newick tree

 
[
	5OHX_1:10.26,
	[
		2BLP_1:81,3EYF_1:81
	]:25.26
]

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 \((0.85)(0.8)(\frac{340 }{\log_{20} 340}-\frac{124}{\log_{20}124})=66.4\)
Status Protein1 Protein2 d d1/2
Query variables 2BLP_1 3EYF_1 83 66
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]