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Parikh vectors
1FRG_1 1XHO_1 1KIJ_1 Letter Amino acid
10 13 39 A Alanine
15 12 25 K Lycine
7 5 25 R Arginine
8 10 14 I Isoleucine
15 12 39 L Leucine
3 8 4 M Methionine
10 6 13 N Asparagine
4 2 0 C Cysteine
9 2 10 Q Glutamine
4 8 11 H Histidine
9 2 14 F Phenylalanine
4 2 1 W Tryptophan
9 2 15 Y Tyrosine
11 10 36 V Valine
12 11 14 D Aspartic acid
9 8 39 E Glutamic acid
13 9 43 G Glycine
11 4 17 P Proline
31 11 13 S Serine
23 11 18 T Threonine 

1FRG_1|Chain A[auth L]|IGG2A 26/9 FAB (LIGHT CHAIN)|Mus musculus (10090)
>1XHO_1|Chains A, B, C|Chorismate mutase|Clostridium thermocellum (1515)
>1KIJ_1|Chains A, B|DNA GYRASE SUBUNIT B|Thermus thermophilus (274)
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 \(w=f(c)\)
1FRG , Knot 103 217 0.85 40 155 210 
DIVMTQSPSSLTVTAGEKVTMSCKSSQSLFNSGKRKNFLTWYHQKPGQPPKLLIYWASTRESGVPDRFSGSGSGTDFTLTITSVQAEDLAIYYCQNDYSHPLTFGAGTKLELKRADAAPTVSIFPPSSEQLTSGGASVVCFLNNFYPKDINVKWKIDGSERQNGVLNSWTDQDSKDSTYSMSSTLTLTKDEYERHNSYTCEATHKTSTSPIVKSFNR
1XHO , Knot 69 148 0.77 40 111 143 
MGSSHHHHHHSSGLVPRGSQSTSLYKKAGLMVWAIRGATTVSDNTADEIVAETQKLLKEMAEKNGLEEDDIISIIFTVTKDLDAAFPAIAARNMGWTSTALMCMNEIDVPGSLEKCIRVMMHVNTDKDKKDIKHVYLNGAKVLRPDLT
1KIJ , Knot 163 390 0.83 38 193 364 
MSYDASAIRVLKGLEGVRHRPAMYIGGTGVEGYHHLFKEILDNAVDEALAGYATEILVRLNEDGSLTVEDNGRGIPVDLMPEEGKPAVEVIYNTLHSGGKFEQGAYKVSGGLHGVGASVVNALSEWTVVEVFREGKHHRIAFSRGEVTEPLRVVGEAPRGKTGTRVTFKPDPEIFGNLRFDPSKIRARLREVAYLVAGLKLVFQDRQHGKEEVFLDKGGVASFAKALAEGEDLLYEKPFLIRGTHGEVEVEVGFLHTQGYNAEILTYANMIPTRDGGTHLTAFKSAYSRALNQYAKKAGLNKEKGPQPTGDDLLEGLYAVVSVKLPNPQFEGQTKGKLLNPEAGTAVGQVVYERLLEILEENPRIAKAVYEKALRAAQAREAARKARELV

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 \(c\).

\(|P_{f(1FRG_1)}(2) \setminus P_{f(1XHO_1)}(2)|=102\), \(|P_{f(1XHO_1)}(2) \setminus P_{f(1FRG_1)}(2)|=58\). 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 1:0111000100101011001010000000110010000110100001101101110110000011100101010100101010010100111000000000110111100101001011101011110000100111011011001010010101010100000111001000000000001000101000000000000001000000011100100
Pair \(Z_2\) Length of longest common subsequence
1FRG_1,1XHO_1 160 3
1FRG_1,1KIJ_1 178 5
1XHO_1,1KIJ_1 168 5

Newick tree

 
[
	1KIJ_1:88.60,
	[
		1FRG_1:80,1XHO_1:80
	]:8.60
]

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 \((0.85)(0.8)(\frac{365 }{\log_{20} 365}-\frac{148}{\log_{20}148})=65.6\)
Status Protein1 Protein2 d d1/2
Query variables 1FRG_1 1XHO_1 85 69
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]