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Parikh vectors
8CTG_1 2PRX_1 5BQG_1 Letter Amino acid
11 21 18 A Alanine
2 6 6 H Histidine
4 7 8 S Serine
4 3 3 N Asparagine
6 16 7 G Glycine
4 8 6 I Isoleucine
13 12 22 L Leucine
6 5 9 F Phenylalanine
5 10 13 V Valine
10 4 4 C Cysteine
9 11 4 E Glutamic acid
6 7 4 K Lycine
12 10 11 P Proline
2 1 2 W Tryptophan
6 5 6 Y Tyrosine
6 6 11 R Arginine
8 7 4 D Aspartic acid
8 8 4 Q Glutamine
5 6 6 M Methionine
5 7 6 T Threonine 

8CTG_1|Chain A|Frizzled-8|Mus musculus (10090)
>2PRX_1|Chains A, B|Thioesterase superfamily protein|Shewanella loihica (323850)
>5BQG_1|Chain A|Prostaglandin E synthase|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 \(w=f(c)\)
8CTG , Knot 67 132 0.82 40 109 129 
ASAKELACQEITVPLCKGIGYNYTYMPNQFNHDTQDEAGLEVHQFWPLVEIQCSPDLKFFLCSMYTPICLEDYKKPLPPCRSVCERAKAGCAPLMRQYGFAWPDRMRCDRLPEQGNPDTLCMDAAALEVLFQ
2PRX , Knot 78 160 0.82 40 123 157 
GMQGIAFQDAYPDDLSHCYGCGRNNEQGHQLKSYWRGEQTIAHFMPKPFHTAIPGFVYGGLIASLIDCHGTGSASAAAQRALEQAGEQLDEPPRFVTAALNIDYLAPTPMGVELELVGEIKEVKPRKVVVEIALSADGKLCARGHMVAVKMPETMAATSA
5BQG , Knot 71 154 0.77 40 117 146 
MALPAHSLVMSSPALPAFLLCSTLLVIKMYVVAIITGQVRLRKKAFANPEDALRHGGPQYCRSDPDVERCLRAHRNDMETIYPFLFLGFVYSFLGPNPFVAWMHFLVFLVGRVAHTVAYLGKLRAPIRSVTYTLAQLPCASMALQILWEAARHL

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 \(c\).

\(|P_{f(8CTG_1)}(2) \setminus P_{f(2PRX_1)}(2)|=71\), \(|P_{f(2PRX_1)}(2) \setminus P_{f(8CTG_1)}(2)|=85\). 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 1:101001100010111001110000011001000000011101001111101000101011100100110100000111100010001011011110001111100100001100101001010111101110
Pair \(Z_2\) Length of longest common subsequence
8CTG_1,2PRX_1 156 3
8CTG_1,5BQG_1 138 3
2PRX_1,5BQG_1 148 4

Newick tree

 
[
	2PRX_1:78.22,
	[
		8CTG_1:69,5BQG_1:69
	]:9.22
]

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 \((0.85)(0.8)(\frac{292 }{\log_{20} 292}-\frac{132}{\log_{20}132})=49.7\)
Status Protein1 Protein2 d d1/2
Query variables 8CTG_1 2PRX_1 63 57
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]