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Parikh vectors
9IHU_1 6DVE_1 7KOZ_1 Letter Amino acid
8 16 13 H Histidine
28 21 11 I Isoleucine
14 1 13 W Tryptophan
23 23 15 A Alanine
25 25 10 D Aspartic acid
60 31 18 E Glutamic acid
17 28 24 G Glycine
3 1 5 C Cysteine
20 20 14 P Proline
18 24 12 T Threonine
21 18 13 R Arginine
22 10 12 N Asparagine
50 36 30 L Leucine
29 34 17 V Valine
18 26 26 S Serine
29 9 8 Y Tyrosine
9 9 17 Q Glutamine
47 15 10 K Lycine
7 4 3 M Methionine
29 8 14 F Phenylalanine 

9IHU_1|Chains A, B|Glycoside hydrolase family 57 N-terminal domain-containing protein|Aquifex aeolicus (strain VF5) (224324)
>6DVE_1|Chains A, B|DNA-directed RNA polymerase subunit alpha|Mycobacterium tuberculosis (strain ATCC 25618 / H37Rv) (83332)
>7KOZ_1|Chain A|T-cell surface glycoprotein CD1a|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 \(w=f(c)\)
9IHU , Knot 188 477 0.81 40 223 432 
MKKLFLVFWWHMHQPLYREPYTGEYLLPWTFFHAVKDYYDMPAYLKDFEIKLNFNLTPVLIDQIQEYAQGKAKDVFLEAIRKDPDDLEKEEVEKLIEFTKLNYEKPIYRFERIRELMNKEKLNREELLDLQTLNLLAWCGRTLRKDLKDLLNKGRNYTQEEKEYVLNKYFEIIKKTLSIYREIKEEGKGSVSTSPYYHPLIPILLNPNCVYETTPNVKIPDFAVSFREDASKHVELAKEKYFEIFGEHPVYMWPPEASVSNEALELYYEKGINMLATDEVILKNSVERASPYLRYYFRELISVFFRDKTLSDLIGFSYHAWNAEDAVRDFIGRLKKIHESVDFQPVVFVVLDGENCWEYYEENGIPFLEKLYSTLEKEEWIETLTLEEAMRKEDVKTEVIESVKAGTWFDGNFLKWIGNKEKNEYWKILIEAKKKAKNDYILVAEGSDWFWWQGEEKAPFVEVFDKLFRSFVRRAQE
6DVE , Knot 152 359 0.83 40 192 344 
MGHHHHHHHHHHMLISQRPTLSEDVLTDNRSQFVIEPLEPGFGYTLGNSLRRTLLSSIPGAAVTSIRIDGVLHEFTTVPGVKEDVTEIILNLKSLVVSSEEDEPVTMYLRKQGPGEVTAGDIVPPAGVTVHNPGMHIATLNDKGKLEVELVVERGRGYVPAVQNRASGAEIGRIPVDSIYSPVLKVTYKVDATRVEQRTDFDKLILDVETKNSISPRDALASAGKTLVELFGLARELNVEAEGIEIGPSPAEADHIASFALPIDDLDLTVRSYNCLKREGVHTVGELVARTESDLLDIRNFGQKSIDEVKIKLHQLGLSLKDSPPSFDPSEVAGYDVATGTWSTEGAYDEQDYAETEQL
7KOZ , Knot 131 285 0.86 40 185 277 
ATGLKEPLSFHVIWIASFYNHSWKQNLVSGWLSDLQTHTWDSNSSTIVFLWPWSRGNFSNEEWKELETLFRIRTIRSFEGIRRYAHELQFEYPFEIQVTGGCELHSGKVSGSFLQLAYQGSDFVSFQNNSWLPYPVAGNMAKHFCKVLNQNQHENDITHNLLSDTCPRFILGLLDAGKAHLQRQVKPEAWLSHGPSPGPGHLQLVCHVSGFYPKPVWVMWMRGEQEQQGTQRGDILPSADGTWYLRATLEVAAGEAADLSCRVKHSSLEGQDIVLYWEGSLVPRG

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 \(c\).

\(|P_{f(9IHU_1)}(2) \setminus P_{f(6DVE_1)}(2)|=95\), \(|P_{f(6DVE_1)}(2) \setminus P_{f(9IHU_1)}(2)|=64\). 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 1:100111111101001100010010011110110110000011101001010101010111100100010101001110110001001000010011010010000110010010011000010000110100101111010010001001100100000000001100010110001010001000101010001000111111101001000010101101110100010001011000010111001101111010100011010000110111000111000100101010001001101110000100111100011010011001110100100010101111111010001000000111110010001000011001010011000010001100101101101011011100000001011101000100001111010011110100011110110011001100100
Pair \(Z_2\) Length of longest common subsequence
9IHU_1,6DVE_1 159 4
9IHU_1,7KOZ_1 200 4
6DVE_1,7KOZ_1 177 4

Newick tree

 
[
	7KOZ_1:98.90,
	[
		9IHU_1:79.5,6DVE_1:79.5
	]:19.40
]

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 \((0.85)(0.8)(\frac{836 }{\log_{20} 836}-\frac{359}{\log_{20}359})=128.\)
Status Protein1 Protein2 d d1/2
Query variables 9IHU_1 6DVE_1 159 139
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]