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Parikh vectors
3WGB_1 4JZT_1 4ICC_1 Letter Amino acid
11 7 20 I Isoleucine
5 20 32 K Lycine
0 3 6 W Tryptophan
8 8 11 Y Tyrosine
23 6 11 R Arginine
33 13 15 G Glycine
10 10 15 F Phenylalanine
29 12 23 V Valine
43 7 24 A Alanine
4 2 4 C Cysteine
18 14 16 S Serine
12 4 13 T Threonine
17 18 25 E Glutamic acid
11 4 4 M Methionine
16 5 12 Q Glutamine
14 10 9 H Histidine
47 15 30 L Leucine
13 8 18 P Proline
6 4 11 N Asparagine
21 8 17 D Aspartic acid 

3WGB_1|Chains A, B, C, D|L-allo-threonine aldolase|Aeromonas jandaei (650)
>4JZT_1|Chains A, B[auth D]|dGTP pyrophosphohydrolase|Bacillus subtilis subsp. subtilis (224308)
>4ICC_1|Chain A[auth X]|Aldo-keto reductase family 1 member B10|Homo sapiens (9606)
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 \(w=f(c)\)
3WGB , Knot 143 341 0.81 38 181 316 
GSHMRYIDLRSDTVTQPTDAMRQCMLHAEVGDDVYGEDPGVNALEAYGADLLGKEAALFVPSGTMSNLLAVMSHCQRGEGAVLGSAAHIYRYEAQGSAVLGSVALQPVPMQADGSLALADVRAAIAPDDVHFTPTRLVCLENTHNGKVLPLPYLREMRELVDEHGLQLHLDGARLFNAVVASGHTVRELVAPFDSVSICLSKGLGAPVGSLLVGSHAFIARARRLRKMVGGGMRQAGILAQAGLFALQQHVVRLADDHRRARQLAEGLAALPGIRLDLAQVQTNMVFLQLTSGESAPLLAFMKARGILFSGYGELRLVTHLQIHDDDIEEVIDAFTEYLGA
4JZT , Knot 85 178 0.82 40 133 170 
MGSSHHHHHHSSGLVPRGSHMYEFKDYYQNTVQLSFDDQPFSDSPKHVWVICRFGGKWLLTEHEDRGYEFPGGKVEPMECAEEAALRAVKEETGARVKSLKYLGQYKVLGKEKVIVKNIYFADIEKLEKQADYFETKGPVLFHELPENLSRNKKFSFIMKDSVLPISLKKLKESGWIE
4ICC , Knot 139 316 0.84 40 207 301 
MATFVELSTKAKMPIVGLGTWKSPLGKVKEAVKVAIDAGYRHIDCAYVYQNEHEVGEAIQEKIQEKAVKREDLFIVSKLWPTFFERPLVRKAFEKTLKDLKLSYLDVYLIHWPQGFKSGDDLFPRDDKGNAIGGKATFLDAWEAMEELVDEGLVKALGVSNFSHFQIEKLLNKPGLKYKPVTNQVECHPYLTQEKLIQYCHSKGITVTAYSPLGSPDRPWAKPEDPSLLEDPKIKEIAAKHKKTAAQVLIRFHIQRNVIVIPKSVTPARIVENIQVFDFKLSDEEMATILSFNRNWRACNLLQSSHLEDYPFNAEY

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 \(c\).

\(|P_{f(3WGB_1)}(2) \setminus P_{f(4JZT_1)}(2)|=113\), \(|P_{f(4JZT_1)}(2) \setminus P_{f(3WGB_1)}(2)|=65\). 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 1:10010010100001001001100011010110010100111011010110111001111110101001111100000101111101101000010101111011101111010101111010111110010101001101000001011111010010011000110101011011011110100100111110010101001111111011110011110100100111111001111101111110001101100000100110111111110101101000111101001001111111010111101010101100101000010011011000111
Pair \(Z_2\) Length of longest common subsequence
3WGB_1,4JZT_1 178 4
3WGB_1,4ICC_1 176 5
4JZT_1,4ICC_1 174 4

Newick tree

 
[
	3WGB_1:88.99,
	[
		4ICC_1:87,4JZT_1:87
	]:1.99
]

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 \((0.85)(0.8)(\frac{519 }{\log_{20} 519}-\frac{178}{\log_{20}178})=99.1\)
Status Protein1 Protein2 d d1/2
Query variables 3WGB_1 4JZT_1 122 93.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]