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Parikh vectors
2GGA_1 1HOZ_1 6SWI_1 Letter Amino acid
53 21 3 G Glycine
16 16 6 K Lycine
25 20 9 E Glutamic acid
6 6 6 Q Glutamine
46 29 12 L Leucine
19 13 4 M Methionine
1 7 0 W Tryptophan
3 8 6 Y Tyrosine
28 13 10 R Arginine
14 23 2 N Asparagine
30 24 3 D Aspartic acid
11 14 5 F Phenylalanine
25 18 4 P Proline
29 14 12 S Serine
37 19 7 T Threonine
44 28 7 A Alanine
5 9 10 H Histidine
23 13 9 I Isoleucine
38 33 6 V Valine
2 11 0 C Cysteine 

2GGA_1|Chain A|3-phosphoshikimate 1-carboxyvinyltransferase|Agrobacterium sp. (268951)
>1HOZ_1|Chains A, B|INOSINE-ADENOSINE-GUANOSINE-PREFERRING NUCLEOSIDE HYDROLASE|Trypanosoma vivax (5699)
>6SWI_1|Chain A|Two-component response regulator|Geobacillus stearothermophilus (1422)
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 \(w=f(c)\)
2GGA , Knot 183 455 0.82 40 200 419 
MSHGASSRPATARKSSGLSGTVRIPGDKSISHRSFMFGGLASGETRITGLLEGEDVINTGKAMQAMGARIRKEGDTWIIDGVGNGGLLAPEAPLDFGNAATGCRLTMGLVGVYDFDSTFIGDASLTKRPMGRVLNPLREMGVQVKSEDGDRLPVTLRGPKTPTPITYRVPMASAQVKSAVLLAGLNTPGITTVIEPIMTRDHTEKMLQGFGANLTVETDADGVRTIRLEGRGKLTGQVIDVPGDPSSTAFPLVAALLVPGSDVTILNVLMNPTRTGLILTLQEMGADIEVINPRLAGGEDVADLRVRSSTLKGVTVPEDRAPSMIDEYPILAVAAAFAEGATVMNGLEELRVKESDRLSAVANGLKLNGVDCDEGETSLVVRGRPDGKGLGNASGAAVATHLDHRIAMSFLVMGLVSENPVTVDDATMIATSFPEFMDLMAGLGAKIELSDTKAA
1HOZ , Knot 149 339 0.85 40 200 324 
MRGSPHHHHHHGSAKNVVLDHDGNLDDFVAMVLLASNTEKVRLIGALCTDADCFVENGFNVTGKIMCLMHNNMNLPLFPIGKSAATAVNPFPKEWRCLAKNMDDMPILNIPENVELWDKIKAENEKYEGQQLLADLVMNSEEKVTICVTGPLSNVAWCIDKYGEKFTSKVEECVIMGGAVDVRGNVFLPSTDGTAEWNIYWDPASAKTVFGCPGLRRIMFSLDSTNTVPVRSPYVQRFGEQTNFLLSILVGTMWAMCTHCELLRDGDGYYAWDALTAAYVVDQKVANVDPVPIDVVVDKQPNEGATVRTDAENYPLTFVARNPEAEFFLDMLLRSARAC
6SWI , Knot 60 121 0.79 36 92 114 
SRLSPIVRRARQYIQKHFSQSDLTLESVAEFLNVSPVYLSRMIKQELGISFIHLLTKIRMEKAAELLLSTELPIHEIAERVGYDTQHYFSTAFKKAMGVSPNQYRRTRMISEYTDHHHHHH

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 \(c\).

\(|P_{f(2GGA_1)}(2) \setminus P_{f(1HOZ_1)}(2)|=80\), \(|P_{f(1HOZ_1)}(2) \setminus P_{f(2GGA_1)}(2)|=80\). 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 1:10011000110100001101010111000100001111111010001011101001100101101111010001001110111011111101110110110100101111110010001110101000111011011001110100001001110101100101100011110101001111111001110011011100000001101111010100010110010101010101011011101000111111111111001011011101000111101001110101101011110011010100001011011000110110001111111111011011011001010000010111011010110000100011101010101110101111100100011101111111000110100101110011011011111110101000011
Pair \(Z_2\) Length of longest common subsequence
2GGA_1,1HOZ_1 160 5
2GGA_1,6SWI_1 176 4
1HOZ_1,6SWI_1 210 6

Newick tree

 
[
	6SWI_1:10.87,
	[
		2GGA_1:80,1HOZ_1:80
	]:21.87
]

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 \((0.85)(0.8)(\frac{794 }{\log_{20} 794}-\frac{339}{\log_{20}339})=123.\)
Status Protein1 Protein2 d d1/2
Query variables 2GGA_1 1HOZ_1 155 135.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]