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Parikh vectors
5CIX_1 3ZZJ_1 8ELD_1 Letter Amino acid
9 18 14 K Lycine
11 15 15 P Proline
21 21 18 S Serine
23 46 33 A Alanine
9 27 19 E Glutamic acid
3 6 7 H Histidine
1 5 5 W Tryptophan
13 11 11 Y Tyrosine
17 26 26 V Valine
28 38 23 L Leucine
3 8 9 M Methionine
19 22 24 T Threonine
11 18 12 Q Glutamine
17 16 18 I Isoleucine
8 20 12 F Phenylalanine
13 21 18 R Arginine
10 20 27 D Aspartic acid
0 5 3 C Cysteine
16 23 19 N Asparagine
14 30 18 G Glycine 

5CIX_1|Chain A|Ribosome inactivating protein|Momordica balsamina (3672)
>3ZZJ_1|Chain A|ASPARTATE AMINOTRANSFERASE|ESCHERICHIA COLI (562)
>8ELD_1|Chains A, B, C, D, E, F, G|Gp37|Shigella phage Buco (2530183)
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 \(w=f(c)\)
5CIX , Knot 110 246 0.82 38 157 234 
DVSFRLSGADPSSYGMFIKDLRNALPHTEKVYNIPLLLPSVSGAGRYLLMHLFNYDGNTITVAVDVTNVYIMGYLALTTSYFFNEPAADLASQYVFRSARRKITLPYSGNYERLQIAAGKPREKIPIGLPALDTAISTLLHYDSTAAAGALLVLIQTTAEAARFKYIEQQIQERAYRDEVPSSATISLENSWSGLSKQIQLAQGNNGVFRTPTVLVDSKGNRVQITNVTSNVVTSNIQLLLNTKNI
3ZZJ , Knot 169 396 0.85 40 219 372 
MFENITAAPADPILGLADLFRADERPGKINLGQGVYKDETGKTPVLTSVKKAEQYLLENETTKNYLGIDGIPEFGRCTQELLFGKGSALINDKRARTAQTPGGTGALRVAADFLAKNTSVKRVWVSNPSWPNHKSVFNSAGLEVREYAYYDAENHTLDFDALINSLNEAQAGDVVLFHGCCHNPTGIDPTLEQWQTLAQLSVEKGWLPLFDFAQQGFARGLEEDAEGLRAFAAMHKELIVASSYSKNFGLYNERVGACTLVAADSETVDRAFSQMKAAIYANYSNPPAHGASVVATILSNDALRAIWEQELTDMRQRIQRMRQLFVNTLQEKGANRDFSFIIKQNGMFSFSGLTKEQVLRLREEFGVYAVASGRVNVAGMTPDNMAPLCEAIVAVL
8ELD , Knot 145 331 0.84 40 209 320 
MANTDYAGNLTRPHWGGAASDVDIHLEVYQNEVDTRFQYQAMFLGLSSQRSVADRSNTYRIDRLNTSSVKGRTSGVALEPTPVRNDKMLIVVDTVLYIRNPIDYQDDWTAPDFLTEMGQNNGSEFAEVFDQAHLIQLIKGRSWVAPAHLKPAFSDGIEIEATIDSDVTTQAGMEANAIAINQAHKAGIDELIKRKVPLNDMITLVSTEIYSLLLEHPKLFNKDWGDANANGYKERRAVLMNGIPVVECTEFPDAGTHPLGSAYTVTADDAKCRMVTFSKSRTLVTVEAKPFTSRIWDDEQNFANVLDCYAMYQVGERRPDTAAVVKFNEAP

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 \(c\).

\(|P_{f(5CIX_1)}(2) \setminus P_{f(3ZZJ_1)}(2)|=44\), \(|P_{f(3ZZJ_1)}(2) \setminus P_{f(5CIX_1)}(2)|=106\). 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 1:010101011010001111001001110000100111111010111001110110001001011101001011101110000110011101100011001000101100100001011110100011111111001100110000011111111110001011010010001000100001100101010001011000101101001110010111000100101001000110001011100001
Pair \(Z_2\) Length of longest common subsequence
5CIX_1,3ZZJ_1 150 3
5CIX_1,8ELD_1 162 3
3ZZJ_1,8ELD_1 154 3

Newick tree

 
[
	8ELD_1:80.32,
	[
		5CIX_1:75,3ZZJ_1:75
	]:5.32
]

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 \((0.85)(0.8)(\frac{642 }{\log_{20} 642}-\frac{246}{\log_{20}246})=111.\)
Status Protein1 Protein2 d d1/2
Query variables 5CIX_1 3ZZJ_1 143 114
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]