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Parikh vectors
7RAF_1 7QYQ_1 8RSS_1 Letter Amino acid
4 5 7 W Tryptophan
10 21 6 D Aspartic acid
8 24 23 G Glycine
4 7 2 H Histidine
32 42 20 L Leucine
10 20 4 P Proline
10 4 7 N Asparagine
2 1 1 C Cysteine
20 16 5 Q Glutamine
14 6 7 K Lycine
21 12 26 V Valine
9 37 24 A Alanine
12 18 10 R Arginine
14 18 14 E Glutamic acid
9 5 14 Y Tyrosine
11 6 14 I Isoleucine
13 4 9 M Methionine
16 14 6 F Phenylalanine
21 17 10 S Serine
10 10 11 T Threonine 

7RAF_1|Chain A|Ancestral androgen receptor|Escherichia phage EcSzw-2 (2419741)
>7QYQ_1|Chains A, B, C, D|Dyp-type peroxidase family protein|Pseudomonas putida (303)
>8RSS_1|Chain A|Bacteriorhodopsin|Candidatus Actinomarina minuta (1389454)
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 \(w=f(c)\)
7RAF , Knot 115 250 0.84 40 178 243 
IPIFLSVLQSIEPEVVYAGYDNTQPDTSASLLTSLNELGERQLVRVVKWAKALPGFRNLHVDDQMTLIQYSWMGVMVFAMGWRSYKNVNSRMLYFAPDLVFNEQRMQKSTMYNLCVRMRHLSQEFVWLQVTQEEFLCMKALLLFSIIPVEGLKNQKYFDELRMNYIKELDRVISFQGKNPTSSSQRFYQLTKLLDSLQDLVRKLHQFTFDTFVQSQSLSVEFPEMMSEIISAQVPKILAGMVKPLLFHKQ
7QYQ , Knot 121 287 0.79 40 164 266 
MPFQQGLLATPVPAHARHLFFTLQSPEALPAALDALLPQVDGKQLLLGVGAPLAKALGREIPGLRPFPLLDAAVENPSTQHALWLWLRGDERGDLLLRTQALEQALAPALSLADSVDGFLHRGGHDLTGYEDGTENPTDEEAVQAAIAADGSSFAAFQLWKHDLQYFKSLPQADQDNIIGRRLSDNEELDDAPASAHVKRTAQESFEPEAFMVRRSVSWADQRGAGLAFVALGKSFEAFEVQLRRMSGLEDGIIDGLYRFSRPLTGGYYWCPPMSETGVDLSPLLRA
8RSS , Knot 101 220 0.82 40 156 214 
MEELTYRLFMVATVGMLAGTVFLLASSREVKPEHRRGVYISALVCGIAWYHYQKMGASWESGSYDTGLRYVDWVLTVPLMFVEVLAVTRKGAAYNEAVRNWGIAATVMIGAGYYGETSAAGSNEYWTGFVIAMATYVWLMRNLQAEGEGLKGDQAVAFENIKNLILVGWIIYPLGYIAPVVGDFDAIREVLYTIADIINKVGLGVLVLQMARVQSGEKVS

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 \(c\).

\(|P_{f(7RAF_1)}(2) \setminus P_{f(7QYQ_1)}(2)|=87\), \(|P_{f(7QYQ_1)}(2) \setminus P_{f(7RAF_1)}(2)|=73\). 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 1:1111101100101011011000001000101100100110001101101101111100101000101100011111111111000001000110111011100001000010010101001000111101000011010111110111101100000100101001001001101010010000001001001100100110010010100110000101011011001101011011111101111000
Pair \(Z_2\) Length of longest common subsequence
7RAF_1,7QYQ_1 160 4
7RAF_1,8RSS_1 178 4
7QYQ_1,8RSS_1 168 3

Newick tree

 
[
	8RSS_1:88.60,
	[
		7RAF_1:80,7QYQ_1:80
	]:8.60
]

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 \((0.85)(0.8)(\frac{537 }{\log_{20} 537}-\frac{250}{\log_{20}250})=81.7\)
Status Protein1 Protein2 d d1/2
Query variables 7RAF_1 7QYQ_1 100 95
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]