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Parikh vectors
8CLR_1 4ZYX_1 3HCP_1 Letter Amino acid
0 8 22 R Arginine
0 9 13 N Asparagine
0 9 14 D Aspartic acid
0 11 11 H Histidine
0 2 5 W Tryptophan
5 15 20 G Glycine
0 3 9 M Methionine
0 21 19 V Valine
0 1 14 Y Tyrosine
0 8 17 Q Glutamine
0 16 22 I Isoleucine
0 19 41 L Leucine
0 7 11 F Phenylalanine
0 10 20 T Threonine
0 14 25 S Serine
1 21 16 A Alanine
5 3 9 C Cysteine
0 14 26 E Glutamic acid
0 12 21 K Lycine
0 7 24 P Proline 

8CLR_1|Chain A|RNA hairpin with CUUG tetraloop|synthetic RNA (2086595)
>4ZYX_1|Chain A|Trifunctional purine biosynthetic protein adenosine-3|Homo sapiens (9606)
>3HCP_1|Chains A, B|Ferrochelatase, mitochondrial|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 \(w=f(c)\)
8CLR , Knot 7 14 0.44 8 8 10 
GGCAGCUUGCUGCC
4ZYX , Knot 93 210 0.79 40 139 199 
MARVAVLISGTGSNLQALIDSTREPNSSAQIDIVISNKAAVAGLDKAERAGIPTRVINHKLYKNRVEFDSAIDLVLEEFSIDIVCLAGFMRILSGPFVQKWNGKMLNIHPSLLPSFKGSNAHEQALETGVTVTGCTVHFVAEDVDAGQIILQEAVPVKRGDTVATLSERVKLAEHKIFPAALQLVASGTVQLGENGKICWVKEEHHHHHH
3HCP , Knot 152 359 0.83 40 219 343 
RKPKTGILMLNMGGPETLGDVHDFLLRLFLDRDLMTLPIQNKLAPAIAKRRTPKIQEQYRRIGGGSPIKIWTSKQGEGMVKLLDELSPNTAPHKYYIGFRYVHPLTEEAIEEMERDGLERAIAFTQYPQYSCSTTGSSLNAIYRYYNQVGRKPTMKWSTIDRWPTHHLLIQCFADHILKELDHFPLEKRSEVVILFSAHSLPMSVVNRGDPYPQEVSATVQKVMERLEYCNPYRLVWQSKVGPMPWLGPQTDESIKGLCERGRKNILLVPIAFTSDHIETLYELDIEYSQVLAKECGVENIRRAESLNGNPLFSKALADLVHSHIQSNELCSKQLTLSCPLCVNPVCRETKSFFTSQQL

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 \(c\).

\(|P_{f(8CLR_1)}(2) \setminus P_{f(4ZYX_1)}(2)|=6\), \(|P_{f(4ZYX_1)}(2) \setminus P_{f(8CLR_1)}(2)|=137\). 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 1:11011000100100
Pair \(Z_2\) Length of longest common subsequence
8CLR_1,4ZYX_1 143 2
8CLR_1,3HCP_1 225 2
4ZYX_1,3HCP_1 188 3

Newick tree

 
[
	3HCP_1:11.35,
	[
		8CLR_1:71.5,4ZYX_1:71.5
	]:40.85
]

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 \((0.85)(0.8)(\frac{224 }{\log_{20} 224}-\frac{14}{\log_{20}14})=73.5\)
Status Protein1 Protein2 d d1/2
Query variables 8CLR_1 4ZYX_1 91 48
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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