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Parikh vectors
6ECE_1 8CPO_1 1LLO_1 Letter Amino acid
11 9 13 T Threonine
5 10 21 N Asparagine
34 11 32 G Glycine
18 10 2 H Histidine
8 5 2 M Methionine
11 16 25 S Serine
1 0 7 W Tryptophan
9 12 17 Y Tyrosine
18 3 7 R Arginine
2 1 6 C Cysteine
34 12 23 L Leucine
9 24 14 K Lycine
17 12 4 E Glutamic acid
14 14 20 I Isoleucine
21 2 14 P Proline
19 22 12 V Valine
28 12 20 A Alanine
24 20 14 D Aspartic acid
8 6 11 Q Glutamine
12 6 9 F Phenylalanine 

6ECE_1|Chains A, B|Vlm2|Streptomyces tsusimaensis (285482)
>8CPO_1|Chain A|PolB16 Intein Cys-less|metagenome (256318)
>1LLO_1|Chain A|Hevamine-A|Hevea brasiliensis (3981)
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 \(w=f(c)\)
6ECE , Knot 126 303 0.79 40 173 277 
MHHHHHHHHENLYFQGGSSTAGDPTAKLVRLNPRGGDGPGIVFAPPAGGTVLGYIELARHLKGFGEIHGVEAPGLGAGETPVYPSFEEMVQFCSDSAAGVAGDGVYIGGHALGGHIAFYLATMLLDRGIRPKGLIILDTPPRLGDIPVADADLTEEETKVFILAMGIGGMLDQDRDALKDLPYEEAKQLLLDRAKNDPRVSAFLSEDYLDRFLRLQMHQLMYSRDVVLPQRKLDIPIHVFRTKNHAPEVARLFSAWENYAAGEVTFVDIPGDHATMLRAPHVSEVAQLLDRHCGLPSDDGPRG
8CPO , Knot 98 207 0.84 38 139 201 
MKTEFSGDTDAVHGKTHVFIRSIKNGSHMQEAKIDIKSLYDSLAKKYDVQHKNSYEVIYPKGYEIKVLGNKYVKLVAMSRHKTQKHLVKIVVKSEKTIDSLDPIRQKSLLKKQDEVVVTTDHIAMVYNDDHFFENVNAKNLKVGNYVSVYDEASDKEVIGEIASIEDLGMTDDYVYDAEVDDDSHAFYASNILVHASQFCNGTKLGG
1LLO , Knot 119 273 0.81 40 170 261 
GGIAIYWGQNGNEGTLTQTCSTRKYSYVNIAFLNKFGNGQTPQINLAGHCNPAAGGCTIVSNGIRSCQIQGIKVMLSLGGGIGSYTLASQADAKNVADYLWNNFLGGKSSSRPLGDAVLDGIDFDIEHGSTLYWDDLARYLSAYSKQGKKVYLTAAPQCPFPDRYLGTALNTGLFDYVWVQFYNNPPCQYSSGNINNIINSWNRWTTSINAGKIFLGLPAAPEAAGSGYVPPDVLISRILPEIKKSPKYGGVMLWSKFYDDKNGYSSSILDSV

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 \(c\).

\(|P_{f(6ECE_1)}(2) \setminus P_{f(8CPO_1)}(2)|=94\), \(|P_{f(8CPO_1)}(2) \setminus P_{f(6ECE_1)}(2)|=60\). 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 1:100000000001010110001101010110101011011111111111101110101100101110101101111111001101010011010000111111011011101111011101101110011010111110011011011110101000000111111111111000001100110001001110010001010111000010011010100110000111100010111011000001101101101100011101011011100101101101001101100001110001101
Pair \(Z_2\) Length of longest common subsequence
6ECE_1,8CPO_1 154 3
6ECE_1,1LLO_1 185 3
8CPO_1,1LLO_1 161 3

Newick tree

 
[
	1LLO_1:89.71,
	[
		6ECE_1:77,8CPO_1:77
	]:12.71
]

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 \((0.85)(0.8)(\frac{510 }{\log_{20} 510}-\frac{207}{\log_{20}207})=87.5\)
Status Protein1 Protein2 d d1/2
Query variables 6ECE_1 8CPO_1 104 88.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]