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Parikh vectors
3AUQ_1 5YMS_1 6MYU_1 Letter Amino acid
17 10 41 E Glutamic acid
8 8 24 Y Tyrosine
6 1 17 C Cysteine
14 5 24 Q Glutamine
18 10 30 I Isoleucine
14 8 25 K Lycine
8 6 22 F Phenylalanine
25 15 36 S Serine
13 5 39 R Arginine
7 18 19 N Asparagine
15 13 41 V Valine
10 17 43 T Threonine
12 9 61 G Glycine
14 7 25 P Proline
10 3 21 H Histidine
34 10 50 L Leucine
9 3 12 M Methionine
1 4 5 W Tryptophan
12 5 54 A Alanine
16 4 32 D Aspartic acid 

3AUQ_1|Chain A|Vitamin D3 receptor|Homo sapiens (9606)
>5YMS_1|Chains A, B, C, D, E, F, G, H|Outer capsid protein VP4|Human rotavirus A (10941)
>6MYU_1|Chain A|Succinate dehydrogenase [ubiquinone] flavoprotein subunit, mitochondrial|Gallus gallus (9031)
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 \(w=f(c)\)
3AUQ , Knot 117 263 0.82 40 174 250 
GSHMDSLRPKLSEEQQRIIAILLDAHHKTYDPTYSDFCQFRPPVRVNDGGGSVTLELSQLSMLPHLADLVSYSIQKVIGFAKMIPGFRDLTSEDQIVLLKSSAIEVIMLRSNESFTMDDMSWTCGNQDYKYRVSDVTKAGHSLELIEPLIKFQVGLKKLNLHEEEHVLLMAICIVSPDRPGVQDAALIEAIQDRLSNTLQTYIRCRHPPPGSHLLYAKMIQKLADLRSLNEEHSKQYRCLSFQPECSMKLTPLVLEVFGNEIS
5YMS , Knot 77 161 0.81 40 121 154 
MVLDGPYQPVAFKPPNDYWILVNSNSNGVVLEGTNNTDVWVAIISIEPNVNSESRQYSLFGVNKQITVVNTSNKWKFMEMFRNNSNAEFQHKRTLTSSTKLVGILKHGGRLWTYHGETPNATTDYSTTSNLNEISVTTYAEFYIIPRSQESKCTEYINTGL
6MYU , Knot 249 621 0.86 40 280 584 
STKVSDSISTQYPVVDHEFDAVVVGAGGAGLRAAFGLSEAGFNTACVTKLFPTRSHTVAAQGGINAALGNMEDDNWRWHFYDTVKGSDWLGDQDAIHYMTEQAPAAVIELENYGMPFSRTEEGKIYQRAFGGQSLQFGKGGQAHRCCCVADRTGHSLLHTLYGRSLRYDTSYFVEYFALDLLMENGECRGVIALCIEDGTIHRFRAKNTVIATGGYGRTYFSCTSAHTSTGDGTAMVTRAGLPCQDLEFVQFHPTGIYGAGCLITEGCRGEGGILINSQGERFMERYAPVAKDLASRDVVSRSMTIEIREGRGCGPEKDHVYLQLHHLPPQQLATRLPGISETAMIFAGVDVTKEPIPVLPTVHYNMGGIPTNYKGQVITHVNGEDKVVPGLYACGEAASASVHGANRLGANSLLDLVVFGRACALTIAETCKPGEPVPSIKPNAGEESVANLDKLRFADGTIRTSEARLNMQKTMQSHAAVFRTGSILQEGCEKLSQIYRDLAHLKTFDRGIVWNTDLVETLELQNLMLCALQTIYGAEARKESRGAHAREDYKLRIDEFDYSKPLQGQQKRPFEEHWRKHTLSYVDVKSGKVTLKYRPVIDRTLNEEDCSSVPPAIRSY

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 \(c\).

\(|P_{f(3AUQ_1)}(2) \setminus P_{f(5YMS_1)}(2)|=114\), \(|P_{f(5YMS_1)}(2) \setminus P_{f(3AUQ_1)}(2)|=61\). 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 1:10010010101000000111111010000001000010010111010011101010100101110110110001001111101111100100000111100011011110000010100101001000000010010011001011011101011100101000001111110110100111001111011000100010001000011110011010110011010010000000000101010001010111101110010
Pair \(Z_2\) Length of longest common subsequence
3AUQ_1,5YMS_1 175 4
3AUQ_1,6MYU_1 184 4
5YMS_1,6MYU_1 221 3

Newick tree

 
[
	6MYU_1:10.97,
	[
		3AUQ_1:87.5,5YMS_1:87.5
	]:18.47
]

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 \((0.85)(0.8)(\frac{424 }{\log_{20} 424}-\frac{161}{\log_{20}161})=78.2\)
Status Protein1 Protein2 d d1/2
Query variables 3AUQ_1 5YMS_1 102 81.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]