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Parikh vectors
1GCI_1 3VCP_1 5SYT_1 Letter Amino acid
3 7 10 M Methionine
2 15 42 F Phenylalanine
17 23 24 T Threonine
22 12 14 N Asparagine
7 14 8 H Histidine
19 39 67 L Leucine
35 25 31 G Glycine
5 17 35 K Lycine
13 26 17 P Proline
40 38 25 A Alanine
5 25 16 D Aspartic acid
10 20 16 Q Glutamine
8 24 14 R Arginine
9 13 29 I Isoleucine
3 9 11 W Tryptophan
7 19 21 Y Tyrosine
25 31 27 V Valine
0 9 5 C Cysteine
5 22 34 E Glutamic acid
34 24 34 S Serine 

1GCI_1|Chain A|SUBTILISIN|Bacillus lentus (1467)
>3VCP_1|Chain A|Ring-hydroxylating dioxygenase|Sinorhizobium meliloti (266834)
>5SYT_1|Chain A|CAAX prenyl protease 1 homolog|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 \(w=f(c)\)
1GCI , Knot 111 269 0.77 38 132 242 
AQSVPWGISRVQAPAAHNRGLTGSGVKVAVLDTGISTHPDLNIRGGASFVPGEPSTQDGNGHGTHVAGTIAALNNSIGVLGVAPSAELYAVKVLGASGSGSVSSIAQGLEWAGNNGMHVANLSLGSPSPSATLEQAVNSATSRGVLVVAASGNSGAGSISYPARYANAMAVGATDQNNNRASFSQYGAGLDIVAPGVNVQSTYPGSTYASLNGTSMATPHVAGAAALVKQKNPSWSNVQIRNHLKNTATSLGSTNLYGSGLVNAEAATR
3VCP , Knot 175 412 0.85 40 243 394 
MTANPTSIHQRLDRRLSGFSLEQPFYTSPEVYALDLQHIFYKQWLYAVPVCQLAKAGSYTTLRVGAYEVVIVRSRDGEVRAFHNSCRHRGSLICKARQGQVAKLVCPYHQWTYELDGKLIWANDMGPDFDASKYGLKPVNLRNLDGLIYICLSDTPPDFQTFAQLARPYLEVHDLKDAKVAFTSTIIEKGNWKLVWENNRECYHCSSNHPALCRSFPLDPEVAGVQADGGVSKKLQAHFDRCEAAGTPAQFVLAGDGQYRLARMPLQEKALSYTMDGKAAVSRHLGRVAPPDAGTLLMFHYPSTWNHFLPDHSLTFRVMPISPTETEVTTTWLVHKDAVEGVDYDLKRLTEVWIATNDEDREIVETNQQGILSPAYVPGPYSPGQESGVMQFVDWYAASLERALAPRQVAAE
5SYT , Knot 189 480 0.81 40 239 438 
MGMWASLDALWEMPAEKRIFGAVLLFSWTVYLWETFLAQRQRRIYKTTTHVPPELGQIMDSETFEKSRLYQLDKSTFSFWSGLYSETEGTLILLFGGIPYLWRLSGRFCGYAGFGPEYEITQSLVFLLLATLFSALTGLPWSLYNTFVIEEKHGFNQQTLGFFMKDAIKKFVVTQCILLPVSSLLLYIIKIGGDYFFIYAWLFTLVVSLVLVTIYADYIAPLFDKFTPLPEGKLKEEIEVMAKSIDFPLTKVYVVEGSKRSSHSNAYFYGFFKNKRIVLFDTLLEEYSVLNKDIQEDSGMEPRNEEEGNSEEIKAKVKNKKQGCKNEEVLAVLGHELGHWKLGHTVKNIIISQMNSFLCFFLFAVLIGRKELFAAFGFYDSQPTLIGLLIIFQFIFSPYNEVLSFCLTVLSRRFEFQADAFAKKLGKAKDLYSALIKLNKDNLGFPVSDWLFSMWHYSHPPLLERLQALKTMKQSGLEVL

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 \(c\).

\(|P_{f(1GCI_1)}(2) \setminus P_{f(3VCP_1)}(2)|=41\), \(|P_{f(3VCP_1)}(2) \setminus P_{f(1GCI_1)}(2)|=152\). 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 1:10011111001011110001101011011110011000101010111011110100001010100111011110001111111101010110111101010100110110111001101101011010101010011001000111111101001110100110010111111000000010100011110111111010000110001010100110101111111100001010010100010001001100010101110101100
Pair \(Z_2\) Length of longest common subsequence
1GCI_1,3VCP_1 193 4
1GCI_1,5SYT_1 199 3
3VCP_1,5SYT_1 178 4

Newick tree

 
[
	1GCI_1:10.83,
	[
		3VCP_1:89,5SYT_1:89
	]:11.83
]

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 \((0.85)(0.8)(\frac{681 }{\log_{20} 681}-\frac{269}{\log_{20}269})=114.\)
Status Protein1 Protein2 d d1/2
Query variables 1GCI_1 3VCP_1 153 120
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]