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Parikh vectors
3TCP_1 2DYP_1 7WPY_1 Letter Amino acid
9 17 11 Q Glutamine
22 26 13 E Glutamic acid
13 9 16 H Histidine
19 14 18 V Valine
20 17 22 D Aspartic acid
6 5 3 C Cysteine
40 20 27 L Leucine
14 14 19 P Proline
4 9 2 W Tryptophan
13 14 8 Y Tyrosine
17 23 15 R Arginine
16 5 18 I Isoleucine
23 14 16 S Serine
9 6 17 N Asparagine
21 19 25 G Glycine
16 8 12 M Methionine
9 7 7 F Phenylalanine
12 17 17 T Threonine
12 25 24 A Alanine
18 8 15 K Lycine 

3TCP_1|Chains A, B|Tyrosine-protein kinase Mer|Homo sapiens (9606)
>2DYP_1|Chain A|HLA class I histocompatibility antigen, alpha chain G|Homo sapiens (9606)
>7WPY_1|Chains A, B, C, D|Dioxygenase andA|Aspergillus stellatus (1549217)
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 \(w=f(c)\)
3TCP , Knot 136 313 0.83 40 196 303 
MGSSHHHHHHSSGLVPRGSEELQNKLEDVVIDRNLLILGKILGEGEFGSVMEGNLKQEDGTSLKVAVKTMKLDNSSQREIEEFLSEAACMKDFSHPNVIRLLGVCIEMSSQGIPKPMVILPFMKYGDLHTYLLYSRLETGPKHIPLQTLLKFMVDIALGMEYLSNRNFLHRDLAARNCMLRDDMTVCVADFGLSKKIYSGDYYRQGRIAKMPVKWIAIESLADRVYTSKSDVWAFGVTMWEIATRGMTPYPGVQNHEMYDYLLHGHRLKQPEDCLDELYEIMYSCWRTDPLDRPTFSVLRLQLEKLLESLPDV
2DYP , Knot 123 277 0.83 40 189 266 
MGSHSMRYFSAAVSRPGRGEPRFIAMGYVDDTQFVRFDSDSASPRMEPRAPWVEQEGPEYWEEETRNTKAHAQTDRMNLQTLRGYYNQSEASSHTLQWMIGCDLGSDGRLLRGYEQYAYDGKDYLALNEDLRSWTAADTAAQISKRKCEAANVAEQRRAYLEGTCVEWLHRYLENGKEMLQRADPPKTHVTHHPVFDYEATLRCWALGFYPAEIILTWQRDGEDQTQDVELVETRPAGDGTFQKWAAVVVPSGEEQRYTCHVQHEGLPEPLMLRWKQ
7WPY , Knot 132 305 0.82 40 192 294 
MGSSHHHHHHSSGLVPRGSMPPIRRVNASQGSDAAYQILQEDGCVIVEQVICPNIIAKISDDVNRVMDKATIGAKKGEQTHIINMHNRTIHMGDLVLTSKTYRDELLNLPFAHEVLEKVFKKDSGDYWLNAGVILNMLPGAEAQRPHRDDYLYPVSQHMDPATSPDLMINITFPLNEFRHDNGGTLLLPKSHTGPNADFYANAEDLPAAEMQVGDALIFTGKCVHGGGANRSDKPRIGLALAAQPGYLTPRESNVNVPRDIVETMTPLAQRMIGWGTVRTKDTYGLNMLQDKDFHEALGLKSKTA

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 \(c\).

\(|P_{f(3TCP_1)}(2) \setminus P_{f(2DYP_1)}(2)|=99\), \(|P_{f(2DYP_1)}(2) \setminus P_{f(3TCP_1)}(2)|=92\). 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 1:1100000000001111010001000100111000111110111010110110101000010010111001010000000100110011010010010110111101010001110111111110010100011000100110011100110111011111001000011000111000110001010110111000100100000101101110111100110010000001111110110110011010111000010001101001001000100100110001000110010101101010011001101
Pair \(Z_2\) Length of longest common subsequence
3TCP_1,2DYP_1 191 4
3TCP_1,7WPY_1 170 19
2DYP_1,7WPY_1 165 3

Newick tree

 
[
	3TCP_1:92.88,
	[
		7WPY_1:82.5,2DYP_1:82.5
	]:10.38
]

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 \((0.85)(0.8)(\frac{590 }{\log_{20} 590}-\frac{277}{\log_{20}277})=88.0\)
Status Protein1 Protein2 d d1/2
Query variables 3TCP_1 2DYP_1 112 105.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]