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Parikh vectors
4OWX_1 5IJH_1 2MHY_1 Letter Amino acid
0 13 1 H Histidine
0 20 4 L Leucine
0 4 2 P Proline
0 12 2 S Serine
0 16 2 A Alanine
0 9 3 I Isoleucine
0 19 2 K Lycine
0 14 0 F Phenylalanine
0 2 0 W Tryptophan
0 12 0 R Arginine
0 2 8 C Cysteine
0 14 2 Q Glutamine
12 14 3 T Threonine
0 8 3 Y Tyrosine
0 11 3 V Valine
0 8 3 N Asparagine
0 8 4 D Aspartic acid
0 22 10 E Glutamic acid
0 3 5 G Glycine
0 2 0 M Methionine 

4OWX_1|Chain A[auth L]|DNA (5'-D(P*TP*TP*TP*TP*TP*TP*TP*TP*T)-3')|Homo sapiens (9606)
>5IJH_1|Chains A, B|Xenotropic and polytropic retrovirus receptor 1|Homo sapiens (9606)
>2MHY_1|Chain A|Plethodontid modulating factor|Plethodon shermani (263671)
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 \(w=f(c)\)
4OWX , Knot 2 12 0.13 2 1 1 
TTTTTTTTTTTT
5IJH , Knot 98 213 0.82 40 143 200 
MKFAEHLSAHITPEWRKQYIQYEAFKDMLYSAQDQAPSVEVTDEDTVKRYFAKFEEKFFQTCEKELAKINTFYSEKLAEAQRRFATLQNELQSSLDAQKESTGVTTLRQRRKPVFHLSHEERVQHRNIKDLKLAFSEFYLSLILLQNYQNLNFTGFRKILKKHDKILETSRGADWRVAHVEVAPFYTCKKINQLISETEAVVTNELEHHHHHH
2MHY , Knot 32 57 0.75 32 46 53 
LQCNTLDGGTEECIPGIYNVCVHYKSEDEEYKSCGIQEECEDAEGATVLCCPEDLCN

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 \(c\).

\(|P_{f(4OWX_1)}(2) \setminus P_{f(5IJH_1)}(2)|=0\), \(|P_{f(5IJH_1)}(2) \setminus P_{f(4OWX_1)}(2)|=142\). 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 1:000000000000
Pair \(Z_2\) Length of longest common subsequence
4OWX_1,5IJH_1 142 2
4OWX_1,2MHY_1 47 1
5IJH_1,2MHY_1 153 2

Newick tree

 
[
	5IJH_1:84.13,
	[
		4OWX_1:23.5,2MHY_1:23.5
	]:60.63
]

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 \((0.85)(0.8)(\frac{225 }{\log_{20} 225}-\frac{12}{\log_{20}12})=74.7\)
Status Protein1 Protein2 d d1/2
Query variables 4OWX_1 5IJH_1 96 48.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]

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