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Parikh vectors
1KZI_1 4BRQ_1 9KGP_1 Letter Amino acid
10 28 17 S Serine
21 21 34 D Aspartic acid
16 29 23 Q Glutamine
13 10 24 F Phenylalanine
16 25 24 I Isoleucine
14 19 26 P Proline
14 16 22 T Threonine
7 7 5 W Tryptophan
9 22 12 Y Tyrosine
13 4 20 R Arginine
11 28 19 N Asparagine
13 18 35 E Glutamic acid
13 19 23 V Valine
13 19 29 A Alanine
12 14 35 K Lycine
28 35 50 L Leucine
7 9 14 M Methionine
5 6 3 C Cysteine
18 21 27 G Glycine
11 18 13 H Histidine 

1KZI_1|Chains A, B|Thymidylate synthase|Escherichia coli (562)
>4BRQ_1|Chains A, B|ECTONUCLEOSIDE TRIPHOSPHATE DIPHOSPHOHYDROLASE I|LEGIONELLA PNEUMOPHILA (446)
>9KGP_1|Chains A, B|Bifunctional cytochrome P450/NADPH--P450 reductase|Priestia megaterium NBRC 15308 = ATCC 14581 (1348623)
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 \(w=f(c)\)
1KZI , Knot 124 264 0.87 40 186 256 
MKQYLELMQKVLDEGTQKNDRTGTGTLSIFGHQMRFNLQDGFPLVTTKRCHLRSIIHELLWFLQGDTNIAYLHENNVTIWDEWADENGDLGPVYGKQWRAWPTPDGRHIDQITTVLNQLKNDPDSRRIIVSAWNVGELDKMALAPCHAFFQFYVADGKLSCQLYQRSCDVFLGLPFNIASYALLVHMMAQQCDLEVGDFVWTGGDTHLYSNHMDQTHLQLSREPRPLPKLIIKRKPESIFDYRFEDFEIEGYDPHPGIKAPVAI
4BRQ , Knot 160 368 0.85 40 222 353 
MDTNPCEKHSCIAVIDAGSTGSRLHIYSYDTDDTNTPIHIEEIWNKKIKPGFASIQPNSVTIDAYLTMLLADAPIHNIPVYFYATAGMRLLPQSQQKKYYDELDYWFRQQSQWQLVEAKTITGNDEALFDWLAVNYKLDTLKSVQNKSVGVMDMGGASVQIVFPMPKNAEISKHNQVELNIYGQNINLYVHSFLGLGQTEMSHQFLNSPSCFANDYPLPDGESGQGNAPSCKEEVTSLMNSVHKVNQQIQPLLALNPVNEWYSIGGISNLASSQLFHFENSELTNQSLLQQGDNQICHQQWDILNGQYPDDEYLYQYCLLSSYYYALMVDGYGINPNQTIHYIPPEQNLDWTIGVVLHRALEHHHHHH
9KGP , Knot 191 455 0.85 40 242 437 
TIKEMPQPKTFGELKNLPLLNTDKPVQALMKIADELGEIFKFEAPGRVTRYLSSQRLIKEACDESRFDKNLSQGLKFVRDFAGDGLVTSWTHEKNWKKAHNILLPSFSQQAMKGYHAMMVDIAVQLIQKWERLNADEHIEVPEDMTRLTLDTIGLCGFNYRFNSFYRDQPHPFITSMVRALDEIMNKLQRANPDDPAYDENKRQFQDDIKVMNDLVDKIIADRKASGEQSDDLLTHMLNGKDPETGEPLDDENIRYQIITFLGGGHETTSGLLSFALYFLVKNPHVLQKAAEEAARILVDPVPSYKQVKQLKYVGMVLNEALRLWPTGPAFSLYAKEDTVLGGEYPLEKGDELMVLIPQLHRDKTIWGDDVEEFRPERFENPSAIPQHAFKPFGNGQRACIGQQFALHEATLVLGMMLKHFDFEDHTNYELDIKETLTLKPEGFVVKAKSKQIPL

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 \(c\).

\(|P_{f(1KZI_1)}(2) \setminus P_{f(4BRQ_1)}(2)|=79\), \(|P_{f(4BRQ_1)}(2) \setminus P_{f(1KZI_1)}(2)|=115\). 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 1:100010110011001000000010101011100101010011111000000100110011111010001101000010110011000101111010010111010100100100110010001000011101101101001111100111010110101000100000011111110110011110111000010110111011000100001000010100010111011100010011000100101010010111011111
Pair \(Z_2\) Length of longest common subsequence
1KZI_1,4BRQ_1 194 4
1KZI_1,9KGP_1 172 3
4BRQ_1,9KGP_1 166 3

Newick tree

 
[
	1KZI_1:94.37,
	[
		9KGP_1:83,4BRQ_1:83
	]:11.37
]

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 \((0.85)(0.8)(\frac{632 }{\log_{20} 632}-\frac{264}{\log_{20}264})=103.\)
Status Protein1 Protein2 d d1/2
Query variables 1KZI_1 4BRQ_1 132 114.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]