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Parikh vectors
2EBH_1 3FGR_1 7VSX_1 Letter Amino acid
15 3 4 H Histidine
33 10 7 R Arginine
39 12 8 N Asparagine
3 2 1 C Cysteine
47 16 20 G Glycine
25 4 2 M Methionine
37 13 6 P Proline
54 16 9 E Glutamic acid
48 4 18 I Isoleucine
78 30 16 L Leucine
40 6 9 F Phenylalanine
10 6 3 W Tryptophan
53 13 3 A Alanine
39 10 12 D Aspartic acid
34 8 8 Q Glutamine
26 9 6 Y Tyrosine
33 9 16 V Valine
38 6 7 K Lycine
59 13 6 S Serine
35 12 10 T Threonine 

2EBH_1|Chain A[auth X]|Dermonecrotic toxin|Pasteurella multocida (747)
>3FGR_1|Chain A|Putative phospholipase B-like 2 28 kDa form|Mus musculus (10090)
>7VSX_1|Chain A|QLnK|Oplophorus gracilirostris (727944)
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 \(w=f(c)\)
2EBH , Knot 290 746 0.85 40 298 701 
MGHHHHHHDYDIPTTENLYFQGAHMGIQRVSNAKLLGGSPYSPFRIGLEGVWTPEVLKARASVIGKPIGESYKRILAKLQRIHNSNILDERQGLMHELMELIDLYEESQPSSERLNAFRELRTQLEKALYLPEMEALKKQILQIPNKGSGAARFLLRTAMNEMAGKTSESTADLIRFALQDTVISAPFRGYAGAIPEAIDFPVKYVIEDISVFDKIQTNYWELPAYESWNEGSNSALLPGLLRESQSKGMLSKCRIIENSLYIGHSYEEMFYSISPYSNQVGGPYELYPFTFFSMLQEVQGDLGFEQAFATRNFFNTLVSDRLSLMENTMLLTESFDYTPWDAIYGDINYDEQFAAMSINERIEKCMNTYRGVAFQNSSKSIDFFLNNLTTFIDNGLTEIAISDLPYDIVQQEISQFLQGSNEWKTLDAMLFNLDKGDINGAFRKLLQSAKDNNIKFRAIGHSDNSVPPFNNPYKSLYYKGNIIAEAIEKLDREGQKFVVFADSSLLNSTPGTGRPMPGLVQYLKIPATVVDSDGAWQFLPDVASSRVPIEVTELENWQVLTPPQGKILGLKQFKLTAGFPTEQSRLPLLENSVSEDLREELMQKIDAIKNDVKMNSLVCMEAGSSDSVSPKVAARLKDMGLEAGMGASITWWRREGGMEFSHQMHTTASFKFAGKEFAVDASHLQFVHDQLDTTILILPVDDWALEIAQRNRAINPFVEYVSKTGNMLALFMPPLFTKPRLTRAL
3FGR , Knot 93 202 0.81 40 137 192 
LPTLGPGWQRQNPDPPVSRTRSLLLDAASGQLRLEDGFHPDAVAWANLTNAIRETGWAYLDLSTNGRYNDSLQAYAAGVVEASVSEELIYMHWMNTVVNYCGPFEYEVGYCEKLKNFLEANLEWMQREMELNPDSPYWHQVRLTLLQLKGLEDSYEGRLTFPTGRFTIKPLGFLLLQISGDLEDLEPALNKTNTKPSLGSGS
7VSX , Knot 77 171 0.77 40 124 160 
EFFTLEDFVGDWRQTAGYNQDQVLEQGGLSSLFQNLGVSVTPIQRIVLSGENGLKIDIHVIIPYEGLSGDQMGQIEKIFKVVYPVDDHHFKVILHYGTLVIDGVTPNMIDYFGRPYEGIAVFDGKKITVTGTLWNGNKIIDERLINPDGSLLFRVTINGVTGWRLCERILA

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 \(c\).

\(|P_{f(2EBH_1)}(2) \setminus P_{f(3FGR_1)}(2)|=180\), \(|P_{f(3FGR_1)}(2) \setminus P_{f(2EBH_1)}(2)|=19\). 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 1:11000000000110000101011011100100101111010011011101110101101010111011100000111010010000110000111001101101000001000010110010001001101101011000110110010111011100110011100000010110111000110111010111110110111001100101100100001011100010010001111111000000111000011000101100000110010100001111001011011011001010111001110001100110001011000111000100011011010100000111101000100010000111100000010111001001100110011100110011000100110100010010111101001010111001100100001010111000001111001000100010111011001000100111110001100011010111111001011101100011101110110001110100100101101101011110010101111000001111000100010001100101100010100110101100001010111010011101111101011000111010001000101011100111010010110001000111111001110110000110111001000101111111111001010011
Pair \(Z_2\) Length of longest common subsequence
2EBH_1,3FGR_1 199 4
2EBH_1,7VSX_1 202 4
3FGR_1,7VSX_1 167 3

Newick tree

 
[
	2EBH_1:10.24,
	[
		3FGR_1:83.5,7VSX_1:83.5
	]:21.74
]

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 \((0.85)(0.8)(\frac{948 }{\log_{20} 948}-\frac{202}{\log_{20}202})=204.\)
Status Protein1 Protein2 d d1/2
Query variables 2EBH_1 3FGR_1 264 165.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]