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Parikh vectors
1PDB_1 7KGN_1 7JQP_1 Letter Amino acid
9 11 33 F Phenylalanine
3 14 2 W Tryptophan
6 21 21 Y Tyrosine
5 71 12 A Alanine
9 34 22 D Aspartic acid
7 36 8 Q Glutamine
16 17 33 E Glutamic acid
3 8 11 H Histidine
1 1 2 C Cysteine
7 28 18 T Threonine
14 44 25 V Valine
8 36 22 R Arginine
10 31 28 N Asparagine
19 54 33 L Leucine
17 18 30 K Lycine
12 48 9 P Proline
13 34 18 G Glycine
9 25 24 I Isoleucine
6 19 9 M Methionine
12 33 14 S Serine 

1PDB_1|Chain A|Dihydrofolate reductase|Homo sapiens (9606)
>7KGN_1|Chain A|L,D-transpeptidase|Salmonella enterica (28901)
>7JQP_1|Chains A, B, C, D, E|Encapsidation protein|Lactococcus phage asccphi28 (503388)
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 \(w=f(c)\)
1PDB , Knot 87 186 0.81 40 138 177 
VGSLNCIVAVSQNMGIGKNGDLPWPPLRNEFRYFQRMTTTSSVEGKQNLVIMGKKTWFSIPEKNRPLKGRINLVLSRELKEPPQGAHFLSRSLDDALKLTEQPELANKVDMVWIVGGSSVYKEAMNHPGHLKLFVTRIMQDFESDTFFPEIDLEKYKLLPEYPGVLSDVQEEKGIKYKFEVYEKND
7KGN , Knot 232 583 0.84 40 253 541 
DEPEMIPGDSAVAATDLAGPQKQSAATAIMAGIQPLPEGVSAEKVRADLQSQLPSGYTPVYMSQLTLLYAARDMKPMWDNRDAVKAFQQQLAEVAIAGFQPQFTAWVALLTDPAVNGMARDVVLSDAMMGYLHFIANIPVKGQRWLYSNKPYALATPPVSVINQWQIALEEGQLPMFVASLAPQHPQYAPMHDALLKLVADSRPWPQLTNTATLRPGQWSNDVPALREILQRTGMLDGGPKIALPGDNTADSAVVSPSAVVDETSVAHDEPTARRSKPAPAARAYDRELVEAVKRFQAWQGLGADGVIGPATRNWLNMTPAQRAGVLALNIQRLRLLPAELSTGIMVNIPAYSLVYYQNGNQVLASRVIVGRPDRKTPMMSSALNNVVVNPPWNVPPTLARKDILPKVWNDPGYLERHGYTVMRGWNSKEAIDPWQVDWATITPSNLPFRFQQAPGAHNSLGRYKFNMPSSDAIYLHDTPNHTLFQRDARALSSGCVRVNKASELANMLLQDAGWNDARISGALKQGDTRYVNIRQNIPVNLYYLTAFVGADGRMQYRTDIYNYDLTARSSAQIVPKVEQLIR
7JQP , Knot 155 374 0.81 40 207 352 
MITKSNEFWTPKRLLETDDRIFLVVGGRGVGKTFNVTGEALDDLFFNNVSMVYLRRLGVEIDELEKNNFITEEMLRVYFGNRFSDFNADESKQIMRFSIDGAIHEIKAIRNKIFFDDRCIVYFIALSRAGHVKSNNYPDVKYLVFDEVIIDRSIMPNARYIRNEFTVLLNLIETIKRKREDFYLFMLSNVGENFNPIFAGLGYYLTHEDIKKGFVKREDYCVQFVENKQEELNMTDPFVRLGAKNRDFSNSKTNAFENIRTPYFKHYGKKPKLLVKYDRQYLGIAERKIPSGLEYYYQVYKTLDGLENITVFNNNFDTLMEDEVFLEETQLKKKFKTYFELFQQNMVYHESPETFLEWSKFVYALKLEHHHHHH

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 \(c\).

\(|P_{f(1PDB_1)}(2) \setminus P_{f(7KGN_1)}(2)|=35\), \(|P_{f(7KGN_1)}(2) \setminus P_{f(1PDB_1)}(2)|=150\). 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 1:110100111100011110010111111000100100100000101000111110001101100001101010111000100110110110001001101000101100101111111001000110011010111001100100001110101000011100111100100001100010100000
Pair \(Z_2\) Length of longest common subsequence
1PDB_1,7KGN_1 185 4
1PDB_1,7JQP_1 179 4
7KGN_1,7JQP_1 178 4

Newick tree

 
[
	1PDB_1:91.67,
	[
		7JQP_1:89,7KGN_1:89
	]:2.67
]

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 \((0.85)(0.8)(\frac{769 }{\log_{20} 769}-\frac{186}{\log_{20}186})=163.\)
Status Protein1 Protein2 d d1/2
Query variables 1PDB_1 7KGN_1 205 133
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]