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Parikh vectors
1OPB_1 8QML_1 7VMF_1 Letter Amino acid
3 19 5 A Alanine
9 13 3 N Asparagine
7 13 6 F Phenylalanine
11 26 7 G Glycine
1 6 3 H Histidine
15 25 8 K Lycine
3 11 1 M Methionine
14 21 7 T Threonine
3 11 1 Y Tyrosine
11 22 12 V Valine
11 20 5 D Aspartic acid
7 6 3 Q Glutamine
11 30 6 E Glutamic acid
4 22 4 I Isoleucine
9 38 8 L Leucine
4 5 1 W Tryptophan
6 29 0 R Arginine
3 5 1 C Cysteine
0 21 4 P Proline
2 14 11 S Serine 

1OPB_1|Chains A, B, C, D|CELLULAR RETINOL BINDING PROTEIN II|Rattus rattus (10117)
>8QML_1|Chain A|[FeFe] hydrogenase maturase subunit HydE|Thermotoga maritima MSB8 (243274)
>7VMF_1|Chains A, B, C, D, E|Histone deacetylase HDT2|Arabidopsis thaliana (3702)
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 \(w=f(c)\)
1OPB , Knot 67 134 0.81 38 103 130 
MTKDQNGTWEMESNENFEGYMKALDIDFATRKIAVRLTQTKIIVQDGDNFKTKTNSTFRNYDLDFTVGVEFDEHTKGLDGRNVKTLVTWEGNTLVCVQKGEKENRGWKQWVEGDKLYLELTCGDQVCRQVFKKK
8QML , Knot 157 358 0.86 42 210 338 
MWSHPQFEKASTGREILEKLERREFTREVLKEALSINDRGFNEALFKLADEIRRKYVGDEVHIRAIIEFSNVCRKNCLYCGLRRDNKNLKRYRMTPEEIVERARLAVQFGAKTIVLQSGEDPYXMPDVISDIVKEIKKMGVAVTLSLGEWPREYYEKWKEAGADRYLLRHETANPVLHRKLRPDTSFENRLNCLLTLKELGYETGAGSMVGLPGQTIDDLVDDLLFLKEHDFDMVGIGPFIPHPDTPLANEKKGDFTLTLKMVALTRILLPDSNIPATTAMGTIVPGGREITLRCGANVIMPNWTPSPYRQLYQLYPGKISVFEKDTASIPSVMKMIELLGRKPGRDWGGRKRVFETV
7VMF , Knot 51 96 0.80 38 74 93 
MEFWGVAVTPKNATKVTPEEDSLVHISQASLDCTVKSGESVVLSVTVGGAKLVIGTLSQDKFPQISFDLVFDKEFELSHSGTKANVHFIGYKSPNL

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 \(c\).

\(|P_{f(1OPB_1)}(2) \setminus P_{f(8QML_1)}(2)|=38\), \(|P_{f(8QML_1)}(2) \setminus P_{f(1OPB_1)}(2)|=145\). 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 1:10000010101000001010101101011000111010000111001001000000010000101011101000001101001001101010011010010000011001101001010100100100011000
Pair \(Z_2\) Length of longest common subsequence
1OPB_1,8QML_1 183 3
1OPB_1,7VMF_1 121 3
8QML_1,7VMF_1 180 5

Newick tree

 
[
	8QML_1:98.79,
	[
		1OPB_1:60.5,7VMF_1:60.5
	]:38.29
]

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 \((0.85)(0.8)(\frac{492 }{\log_{20} 492}-\frac{134}{\log_{20}134})=105.\)
Status Protein1 Protein2 d d1/2
Query variables 1OPB_1 8QML_1 140 94
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]