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Parikh vectors
6LGA_1 8FID_1 5VPA_1 Letter Amino acid
34 8 0 Y Tyrosine
33 24 3 V Valine
38 24 17 E Glutamic acid
30 24 1 I Isoleucine
20 7 7 K Lycine
14 10 0 M Methionine
28 35 11 R Arginine
30 9 2 N Asparagine
47 31 2 D Aspartic acid
19 11 3 Q Glutamine
43 36 2 S Serine
29 23 2 T Threonine
2 2 1 C Cysteine
40 25 0 G Glycine
51 50 8 L Leucine
35 26 0 P Proline
34 49 7 A Alanine
26 21 1 H Histidine
31 22 1 F Phenylalanine
14 4 0 W Tryptophan 

6LGA_1|Chains A, B|Sucrose hydrolase|Bombyx mori (7091)
>8FID_1|Chain A|Cytochrome P450|Erwinia tracheiphila (65700)
>5VPA_1|Chain A|Protein fosB|Homo sapiens (9606)
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 \(w=f(c)\)
6LGA , Knot 241 598 0.86 40 284 568 
MGSSHHHHHHSSGLVPRGSHMSNQNAPTPPPTEVIQLDWWKNCVLYQIYPRSFKDSDGDGIGDLKGIISELKHFVDAGVDAIWMSPIFESPMVDFGYDISNFYDIHYEYGTMEDFEELLDKAHELGLKVLLDFVPNHASNESEYFIKSEAREPGYENFFIWADPLPNPENPGVRLPPSNWVSQFGGSAWEWSEKRQQYYLHQFAIQQVDFDFRNPAVKQEMFNIMKFWLDKGADGFRLDALPYLIEADPADHEGRYPDDPLSGLTQFESHQLGYTIPLYTKDLIELYDVVYEWREFLDEYNKNHGGDTRVVFSEGYANVSMTMLYYGNEDGAIGAHFPFNFDFITDLSSKSNARDFVYIILRWLTYMPYGGIPNWVFGNHDNNRMPTRFRHDMVDGLNIINMLLPGVAVTYQGEEIGMRDGYVSWEDTVDIEACNRGDPDTYHLYSRDPARTPYHWDNSTSAGFSTSTNTWLPVAEDYQEINLAKQKETARSHFKNYQALTKLRKQATLSHGEYDIRALSDRTFYLVRSLPTHDTYVLLFNVSERRDTVDLGRVPHLTLPATVYVSSIHSARLAGHEITSSQLSLEAGEALVLKAQPI
8FID , Knot 178 441 0.82 40 215 399 
MAHHHHHHGSTQEHSTPSATDYRDAFAALAAPENNSNPYEYMRWLREHDPVHRSASGIFLLSRHADIYWALKATGDTFRGPSAGDLARLFPRAAISRSLNLLASTLAMKEPPTHTRLRRLISRDFTVRQIDSLRPSIVQVTQTRLGNIASSLEKGHVVNLHEVFSVVLPMLIFAELFGMTQADIFELAASIRTILEGLNPHASDEQLAASDAASLEVERYFEGLIEKKRREPQSDLVSVLVSTHDDDDDKLSDTELISMLWGILLGGFATTAAMIDHAILDMLAFPEQRRWLEGDAASVKAFIEEVLRYDAPAMFSSIPRIAQYDITIGDVTIPEGADVRVLLASGNRDPAAFPDPDRFDPSRFHGTAPGMSTYGHVLLSFGHGIHFCLGAQLARVQLAECLPVINARFPHLVLAGDPVREPSAFLRTFRSLPVRLDTSGR
5VPA , Knot 31 68 0.64 30 43 61 
SEEEEKRRVRRERNKLAAAKCRNRRRELTDRLQAETDQLEEEKAELESEIAELQKEKERLEFVLVAHK

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 \(c\).

\(|P_{f(6LGA_1)}(2) \setminus P_{f(8FID_1)}(2)|=106\), \(|P_{f(8FID_1)}(2) \setminus P_{f(6LGA_1)}(2)|=37\). 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 1:1100000000001111010010000110111001101011000110010100100001011101011100100110111011110111001110110010010010000101001001100100111011101110010000001100010011000111110111010011101110011001110110100000000100111001010100111000110110111001101101011101101011000100100110110010000110011100001101001100100110000000110001110010101010110010001111101110101100100000100110111011001101111011110000001100100011011011011111111000100111001010100010101000101000010000110010010000011100000011111000001011000001000100001100100010100100010110000101100110000011110100000010110110101110101001001011100100001010110111101011
Pair \(Z_2\) Length of longest common subsequence
6LGA_1,8FID_1 143 6
6LGA_1,5VPA_1 255 3
8FID_1,5VPA_1 206 3

Newick tree

 
[
	5VPA_1:12.30,
	[
		6LGA_1:71.5,8FID_1:71.5
	]:55.80
]

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 \((0.85)(0.8)(\frac{1039 }{\log_{20} 1039}-\frac{441}{\log_{20}441})=157.\)
Status Protein1 Protein2 d d1/2
Query variables 6LGA_1 8FID_1 204 171.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]