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Parikh vectors
2DFC_1 2AHY_1 9IPE_1 Letter Amino acid
16 10 35 T Threonine
14 9 33 V Valine
7 4 28 A Alanine
6 4 28 R Arginine
20 3 45 N Asparagine
0 1 50 C Cysteine
8 11 18 F Phenylalanine
7 3 32 P Proline
10 3 24 Q Glutamine
4 2 40 E Glutamic acid
9 11 34 I Isoleucine
1 2 10 M Methionine
17 3 16 Y Tyrosine
4 4 28 D Aspartic acid
27 8 51 G Glycine
4 5 36 K Lycine
3 1 23 H Histidine
5 18 50 L Leucine
22 7 40 S Serine
6 1 6 W Tryptophan 

2DFC_1|Chain A|Endo-1,4-beta-xylanase 2|Hypocrea jecorina (51453)
>2AHY_1|Chains A, B|Potassium channel protein|Bacillus cereus (1396)
>9IPE_1|Chain A|Epidermal growth factor receptor|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 \(w=f(c)\)
2DFC , Knot 84 190 0.77 38 128 178 
QTIQPGTGYNNGYFYSYWNDGHGGVTYTNGPGGQFSVNWSNSGNFVGGKGWQPGTKNKVINFSGSYNPNGNSYLSVYGWSRNPLIEYYIVENFGTYNPSTGATKLGEVTSDGSVYDIYRTQRVNQPSIIGTATFYQYWSVRRNHRSSGSVNTANHFNAWAQQGLTLGTMDYQIVAVEGYFSSGSASITVS
2AHY , Knot 54 110 0.77 40 85 105 
MLSFLLTLKRMLRACLRAWKDKEFQVLFVLTILTLISGTIFYSTVEGLRPIDALYFSVVTLTTVGDGNFSPQTDFGKIFTILYIFIGIGLVFGFIHKLAVNVQLPSILSN
9IPE , Knot 240 627 0.82 40 274 566 
LEEKKVCQGTSNKLTQLGTFEDHFLSLQRMFNNCEVVLGNLEITYVQRNYDLSFLKTIQEVAGYVLIALNTVERIPLENLQIIRGNMYYENSYALAVLSNYDANKTGLKELPMRNLQEILHGAVRFSNNPALCNVESIQWRDIVSSDFLSNMSMDFQNHLGSCQKCDPSCPNGSCWGAGEENCQKLTKIICAQQCSGRCRGKSPSDCCHNQCAAGCTGPRESDCLVCRKFRDEATCKDTCPPLMLYNPTTYQMDVNPEGKYSFGATCVKKCPRNYVVTDHGSCVRACGADSYEMEEDGVRKCKKCEGPCRKVCNGIGIGEFKDSLSINATNIKHFKNCTSISGDLHILPVAFRGDSFTHTPPLDPQELDILKTVKEITGFLLIQAWPENRTDLHAFENLEIIRGRTKQHGQFSLAVVSLNITSLGLRSLKEISDGDVIISGNKNLCYANTINWKKLFGTSGQKTKIISNRGENSCKATGQVCHALCSPEGCWGPEPRDCVSCRNVSRGRECVDKCNLLEGEPREFVENSECIQCHPECLPQAMNITCTGRGPDNCIQCAHYIDGPHCVKTCPAGVMGENNTLVWKYADAGHVCHLCHPNCTYGCTGPGLEGCPTNGPKIPSHHHHHH

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 \(c\).

\(|P_{f(2DFC_1)}(2) \setminus P_{f(2AHY_1)}(2)|=93\), \(|P_{f(2AHY_1)}(2) \setminus P_{f(2DFC_1)}(2)|=50\). 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 1:0010110100010100010010111000011110101010001011110110110000110101000101000101011000111000110011000100110011010001010010000010010111010100010100000001010010010111001101101000111101010010101010
Pair \(Z_2\) Length of longest common subsequence
2DFC_1,2AHY_1 143 3
2DFC_1,9IPE_1 212 4
2AHY_1,9IPE_1 219 4

Newick tree

 
[
	9IPE_1:11.38,
	[
		2DFC_1:71.5,2AHY_1:71.5
	]:45.88
]

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 \((0.85)(0.8)(\frac{300 }{\log_{20} 300}-\frac{110}{\log_{20}110})=59.4\)
Status Protein1 Protein2 d d1/2
Query variables 2DFC_1 2AHY_1 73 57.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]