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Parikh vectors
3CTG_1 3FUE_1 5UJT_1 Letter Amino acid
7 38 6 A Alanine
10 30 9 G Glycine
9 16 7 H Histidine
6 21 10 N Asparagine
2 11 3 C Cysteine
5 35 11 P Proline
1 23 6 R Arginine
6 26 5 Q Glutamine
13 69 17 L Leucine
3 11 0 M Methionine
0 13 4 W Tryptophan
4 34 12 D Aspartic acid
10 41 15 E Glutamic acid
6 31 8 I Isoleucine
12 40 8 K Lycine
3 27 12 F Phenylalanine
12 45 17 S Serine
6 40 11 T Threonine
3 22 7 Y Tyrosine
11 38 17 V Valine 

3CTG_1|Chain A|Glutaredoxin-2|Saccharomyces cerevisiae (4932)
>3FUE_1|Chain A|Leukotriene A-4 hydrolase|Homo sapiens (9606)
>5UJT_1|Chains A, D, G|MHC class II antigen|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 \(w=f(c)\)
3CTG , Knot 62 129 0.77 38 90 122 
MGSSHHHHHHSSGLVPRGSHMVSQETVAHVKDLIGQKEVFVAAKTYCPYCKATLSTLFQELNVPKSKALVLELDEMSNGSEIQDALEEISGQKTVPNVYINGKHIGGNSDLETLKKNGKLAEILKPVFQ
3FUE , Knot 247 611 0.86 40 286 571 
MPEIVDTCSLASPASVCRTKHLHLRCSVDFTRRTLTGTAALTVQSQEDNLRSLVLDTKDLTIEKVVINGQEVKYALGERQSYKGSPMEISLPIALSKNQEIVIEISFETSPKSSALQWLTPEQTSGKEHPYLFSQCQAIHCRAILPCQDTPSVKLTYTAEVSVPKELVALMSAIRDGETPDPEDPSRKIYKFIQKVPIPCYLIALVVGALESRQIGPRTLVWSEKEQVEKSAYEFSETESMLKIAEDLGGPYVWGQYDLLVLPPSFPYGGMENPCLTFVTPTLLAGDKSLSNVIAHEISHSWTGNLVTNKTWDHFWLNEGHTVYLERHICGRLFGEKFRHFNALGGWGELQNSVKTFGETHPFTKLVVDLTDIDPDVAYSSVPYEKGFALLFYLEQLLGGPEIFLGFLKAYVEKFSYKSITTDDWKDFLYSYFKDKVDVLNQVDWNAWLYSPGLPPIKPNYDMTLTNACIALSQRWITAKEDDLNSFNATDLKDLSSHQLNEFLAQTLQRAPLPLGHIKRMQEVYNFNAINNSEIRFRWLRLCIQSKWEDAIPLALKMATEQGRMKFTRPLFKDLAAFDKSHDQAVRTYQEHKASMHPVTAMLVGKDLKVD
5UJT , Knot 87 185 0.81 38 140 178 
GEDIVADHVASYGVNLYQSYGPSGQYSHEFDGDEEFYVDLERKETVWQLPLFRRFRRFDPQFALTNIAVLKHNLNCVIKRSNSTAATNEVPEVTVFSKSPVTLGQPNTLICLVDNIFPPVVNITWLSNGHSVTEGVSETSFLSKSDHSFFKISYLTFLPSDDEIYDCKVEHWGLDEPLLKHWEPE

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 \(c\).

\(|P_{f(3CTG_1)}(2) \setminus P_{f(3FUE_1)}(2)|=18\), \(|P_{f(3FUE_1)}(2) \setminus P_{f(3CTG_1)}(2)|=214\). 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 1:110000000000111101001100001101001110001111100001000101001100101100011110100100100100110010100011010101001110001001000101101101110
Pair \(Z_2\) Length of longest common subsequence
3CTG_1,3FUE_1 232 3
3CTG_1,5UJT_1 142 3
3FUE_1,5UJT_1 192 3

Newick tree

 
[
	3FUE_1:11.90,
	[
		3CTG_1:71,5UJT_1:71
	]:44.90
]

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 \((0.85)(0.8)(\frac{740 }{\log_{20} 740}-\frac{129}{\log_{20}129})=174.\)
Status Protein1 Protein2 d d1/2
Query variables 3CTG_1 3FUE_1 223 133
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]