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Parikh vectors
7ZHG_1 1EJM_1 1BCS_1 Letter Amino acid
0 14 7 K Lycine
0 34 21 S Serine
0 4 5 W Tryptophan
0 10 18 Y Tyrosine
0 2 3 M Methionine
0 3 13 F Phenylalanine
0 10 11 T Threonine
0 17 18 V Valine
0 2 12 R Arginine
29 16 12 N Asparagine
0 6 19 D Aspartic acid
410 12 4 C Cysteine
0 4 13 E Glutamic acid
554 25 23 G Glycine
0 8 15 P Proline
0 14 20 L Leucine
288 14 25 A Alanine
0 10 5 Q Glutamine
0 3 9 H Histidine
0 15 10 I Isoleucine 

7ZHG_1|Chain A[auth 2]|rRNA 16S|Pyrococcus abyssi GE5 (272844)
>1EJM_1|Chains A, C, E|BETA-TRYPSIN|Bos taurus (9913)
>1BCS_1|Chain A|SERINE CARBOXYPEPTIDASE II|Triticum aestivum (4565)
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 \(w=f(c)\)
7ZHG , Knot 271 1497 0.44 10 18 73 
AUUCNGGUUGAUCCUGCCGGAGGCCACUGCUAUGGGGGUCNGACUAAGCCAUGCGAGUCAAGGGGGCGUCCCUUCUGGGACGCCACCGGCGGACGGCUCAGUAACACGUCGGUAACCUACCCUCGGGAGGGGGAUAACCCCGGGAAACUGGGGCUAAUCCCCCAUAGGCCUGGGGUACUGGAAGGUCCCCAGGCCGAAAGGGAGCCGUAAGGCUCCGCCCGAGGAUGGGCCGGCGGCCGAUUAGGUAGUUGGUGGGGUAACGGCCCACCAAGCNGAAGAUCGGUACGGGCNGUGAGAGCGGGAGCCNGGAGAUGGACACUGAGACACGGGUCCAGGCCCUACGGGGCGCAGCAGGCGCGAAACCUCNGCAAUGCGGGAAACNGCGACGGGGGGACCCCCAGUGCCGUGCCUCUGGCACGGCUUUUCCGGAGUGUAAAAAGCUCCGGGAAUAAGGGCUGGGCAAGGCNGGUGGCAGCCGCCGCGGUAAUACCGGCGGCCNGAGUGGUGGCCACUAUUAUUGGGCCUAAAGCGGCNGUAGCCGGGCCCGUAAGUCCCUGGCGAAAUCCCACGGCUCAACNGUGGGGCUCGCUGGGGAUACUGCGGGCCUUGGGACNGGGAGAGGCNGGGGGUACCCCCGGGGUAGGGGUGAAAUCCUAUAAUCCCGGGGGGACCGCCAGUGGCGAAGGCGCCNGGCUGGAACGGGUCNGACGGUGAGGGCNGAAGGCCAGGGGAGCGAACNGGAUUAGAUACCCGGGUAGUCCUGGCUGUAAAGGAUGCGGGCUAGGUGUCGGGCGAGCUUCGAGCUCGCCCGGUGCNGUAGGGAAGCNGUUAAGCCNGCNGCCUGGGGAGUACGGCNGCAAGGCUGAAACUUAAAGGAAUUGGCGGGGGAGCACUACAAGGGGUGGAGCGUGCGGUUUAAUUGGAUUCAACGCCGGGAACCUCACNGGGGGCGACGGCAGGAUGAAGGCCAGGCUGAAGGUCUUGCCGGACGCGCCGAGAGGAGGUGCAUGGCCGCNGUCAGCUCGUACCGUGAGGCGUCCACUUAAGUGUGGUAACGAGCGAGACCCGCGCCCCCAGUUGCCAGUCCCUCCCGCUCGGGAGGGAGGCACUCUGGGGGGACUGCCGGCGAUAAGCNGGAGGAAGGGGCGGGCGACGGUAGGUCAGUAUGCCCNGAAACCCCCGGGCUACACGCGCGCUACAAUGGGCGGGACAAUGGGUGCNGACCCNGAAAGGGGGAGGUAAUCCCCUAAACCCGCCCUCAGUUCGGAUCGCGGGCUGCAACUCGCCCGCGUGAAGCUGGAAUCCCUAGUACCCGCGCGUCAUCAUCGCGCGGCGAAUACGUCCCUGCUCCUUGCACACACCGCCCGUCACUCCACCCGAGCGGGGCCUAGGUGAGGCCCGAUCUCCUUCGGGAGGUCGGGUCGAGCCUAGGCUCCGUGAGGGGGGAGAAGUCGUAACAAGGUAGCNGUAGGGGAACCUACGGCUCGAUCACCUCCU
1EJM , Knot 101 223 0.81 40 140 216 
IVGGYTCGANTVPYQVSLNSGYHFCGGSLINSQWVVSAAHCYKSGIQVRLGEDNINVVEGNEQFISASKSIVHPSYNSNTLNNDIMLIKLKSAASLNSRVASISLPTSCASAGTQCLISGWGNTKSSGTSYPDVLKCLKAPILSDSSCKSAYPGQITSNMFCAGYLEGGKDSCQGDSGGPVVCSGKLQGIVSWGSGCAQKNKPGVYTKVCNYVSWIKQTIASN
1BCS , Knot 119 263 0.84 40 182 252 
VEPSGHAADRIARLPGQPAVDFDMYSGYITVDEGAGRSLFYLLQEAPEDAQPAPLVLWLNGGPGCSSVAYGASEELGAFRVKPRGAGLVLNEYRWNKVANVLFLDSPAGVGFSYTNTSSDIYTSGDNRTAHDSYAFLAKWFERFPHYKYRDFYIAGESYAGHYVPELSQLVHRSKNPVINLKGFMVGNGLIDDYHDYVGTFEFWWNHGIVSDDTYRRLKEACLHDSFIHPSPACDAATDVATAEQGNIDMYSLYTPVCNITSS

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 \(c\).

\(|P_{f(7ZHG_1)}(2) \setminus P_{f(1EJM_1)}(2)|=10\), \(|P_{f(1EJM_1)}(2) \setminus P_{f(7ZHG_1)}(2)|=132\). 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 1:100001100110000100111110010010010111110001100111001010111001111111010000000011110100100110111011000110110101001101100010000011111111110110000111111001111001100000010111000111101001111110000011100111111111001011110000100011111011100110110011001110110011011110110110001001110011111001101011100101111101111100011111011101001111010111000111000010111101011011101011110000010110101111110010110111111100000110100101000001101011000000011110101111110000111110111110011101111001101101100100101101101001101100011101101100100100100111000111101100101100111000101110000011011110000101100011001011110001001111101001011100001111001111111100111110100000111101111101111000010110000111111100100110110111110100011001111011100011011011111001111100111111101110011100111010001110110000110010111111010111001110100111011100001110001000110100101111111001001110001001000111111010110010111100111100011111110011011111110100101111110111101010110001100111000110100111110000100111110110110111101111100111001111100001001110101001111111110101011001001001100010100101111010001000111010110110111011110001010000011001001100000000100011111111110100001111111001001101101110011111111111011101101101110011010100001111000001110010101010100101101110111101101110100110000111111111111011000000111000100000110001110010111001011000100010101111001111000001101000101010010010010101101110101000001000000101010100100010010000100011101111000111011110001100000000111111001110011100011100001011111111111110010110111101100101111111000101100011001000000
Pair \(Z_2\) Length of longest common subsequence
7ZHG_1,1EJM_1 142 3
7ZHG_1,1BCS_1 184 3
1EJM_1,1BCS_1 164 4

Newick tree

 
[
	1BCS_1:91.89,
	[
		7ZHG_1:71,1EJM_1:71
	]:20.89
]

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 \((0.85)(0.8)(\frac{1720 }{\log_{20} 1720}-\frac{223}{\log_{20}223})=386.\)
Status Protein1 Protein2 d d1/2
Query variables 7ZHG_1 1EJM_1 269 182
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]

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