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Parikh vectors
3CPE_1 7TOP_1 7FKS_1 Letter Amino acid
23 3 10 F Phenylalanine
45 7 18 S Serine
34 15 14 V Valine
23 2 14 N Asparagine
48 8 23 I Isoleucine
16 2 8 M Methionine
14 4 5 W Tryptophan
32 22 18 A Alanine
24 11 11 R Arginine
36 5 15 E Glutamic acid
36 7 7 G Glycine
21 4 10 P Proline
42 17 19 K Lycine
30 7 21 T Threonine
23 1 7 Y Tyrosine
46 4 14 D Aspartic acid
9 1 1 C Cysteine
20 7 9 Q Glutamine
26 0 5 H Histidine
44 11 29 L Leucine 

3CPE_1|Chain A|DNA packaging protein Gp17|Bacteriophage T4 (10665)
>7TOP_16|Chain P[auth AL14]|60S ribosomal protein L14-A|Saccharomyces cerevisiae (4932)
>7FKS_1|Chain A|Pre-mRNA-splicing factor 8|Saccharomyces cerevisiae S288C (559292)
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 \(w=f(c)\)
3CPE , Knot 242 592 0.87 40 287 562 
MGSSHHHHHHSSGLVPRGSHMLEDPMEQPINVLNDFHPLNEAGKILIKHPSLAERKDEDGIHWIKSQWDGKWYPEKFSDYLRLHKIVKIPNNSDKPELFQTYKDKNNKRSRYMGLPNLKRANIKTQWTREMVEEWKKCRDDIVYFAETYCAITHIDYGVIKVQLRDYQRDMLKIMSSKRMTVCNLSRQLGKTTVVAIFLAHFVCFNKDKAVGILAHKGSMSAEVLDRTKQAIELLPDFLQPGIVEWNKGSIELDNGSSIGAYASSPDAVRGNSFAMIYIEDCAFIPNFHDSWLAIQPVISSGRRSKIIITTTPNGLNHFYDIWTAAVEGKSGFEPYTAIWNSVKERLYNDEDIFDDGWQWSIQTINGSSLAQFRQEHTAAFEGTSGTLISGMKLAVMDFIEVTPDDHGFHQFKKPEPDRKYIATLDCSEGRGQDYHALHIIDVTDDVWEQVGVLHSNTISHLILPDIVMRYLVEYNECPVYIELNSTGVSVAKSLYMDLEYEGVICDSYTDLGMKQTKRTKAVGCSTLKDLIEKDKLIIHHRATIQEFRTFSEKGVSWAAEEGYHDDLVMSLVIFGWLSTQSKFIDYADKDD
7TOP , Knot 67 138 0.79 38 97 131 
MSTDSIVKASNWRLVEVGRVVLIKKGQSAGKLAAIVEIIDQKKVLIDGPKAGVPRQAINLGQVVLTPLTFALPRGARTATVSKKWAAAAVCEKWAASSWAKKIAQRERRAALTDFERFQVMVLRKQKRYTVKKALAKA
7FKS , Knot 117 258 0.84 40 175 251 
GAMNSSNYAELFNNDIKLFVDDTNVYRVTVHKTFEGNVATKAINGCIFTLNPKTGHLFLKIIHTSVWAGQKRLSQLAKWKTAEEVSALVRSLPKEEQPKQIIVTRKAMLDPLEVHMLDFPNIAIRPTELRLPFSAAMSIDKLSDVVMKATEPQMVLFNIYDDWLDRISSYTAFSRLTLLLRALKTNEESAKMILLSDPTITIKSYHLWPSFTDEQWITIESQMRDLILTEYGRKYNVNISALTQTEIKDIILGQNIKA

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 \(c\).

\(|P_{f(3CPE_1)}(2) \setminus P_{f(7TOP_1)}(2)|=202\), \(|P_{f(7TOP_1)}(2) \setminus P_{f(3CPE_1)}(2)|=12\). 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 1:1100000000001111010011001100110110010110011011100101100000011011000101010100100010100110110000010110000000000000111101001010001000110010000001101100001100100111010100000011011000010100100011000111111101101000011111100101010110000011011101101111010010101001001110100101101001111010001111010001111011100100001110001011001001101110100110100111001000100000110011010100101001101000001110100101101101111011010100011001001010000110100001010000110110100011001111000010011110111001100000110101000110110010101000111000000111000000011100010011000011100010100100100011011100100001110111111100000110010000
Pair \(Z_2\) Length of longest common subsequence
3CPE_1,7TOP_1 214 3
3CPE_1,7FKS_1 178 3
7TOP_1,7FKS_1 162 3

Newick tree

 
[
	3CPE_1:10.56,
	[
		7FKS_1:81,7TOP_1:81
	]:22.56
]

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 \((0.85)(0.8)(\frac{730 }{\log_{20} 730}-\frac{138}{\log_{20}138})=168.\)
Status Protein1 Protein2 d d1/2
Query variables 3CPE_1 7TOP_1 213 129
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]