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Parikh vectors
8HZE_1 9DKB_1 4OWX_1 Letter Amino acid
0 13 0 E Glutamic acid
8 47 0 G Glycine
0 85 0 L Leucine
0 4 0 K Lycine
0 24 0 F Phenylalanine
0 10 0 N Asparagine
0 23 0 Q Glutamine
0 30 0 P Proline
0 13 0 W Tryptophan
0 8 0 Y Tyrosine
0 47 0 V Valine
13 61 0 A Alanine
0 13 0 D Aspartic acid
0 16 0 I Isoleucine
0 14 0 M Methionine
0 31 0 S Serine
0 33 12 T Threonine
0 28 0 R Arginine
0 5 0 H Histidine
8 13 0 C Cysteine 

8HZE_1|Chains A, B|RNA (36-MER)|synthetic construct (32630)
>9DKB_1|Chain A|URAT1|Homo sapiens (9606)
>4OWX_1|Chain A[auth L]|DNA (5'-D(P*TP*TP*TP*TP*TP*TP*TP*TP*T)-3')|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 \(w=f(c)\)
8HZE , Knot 15 35 0.50 8 16 28 
GAAGAUUGUAAACAUGCCGAAAGGCAGACACUUCC
9DKB , Knot 203 518 0.81 40 220 466 
MAFSELLDQVGGLGRFQVLQTVALVVPIMWLCTQSMLENFSAAVPSHRCWVPLLDNSTAQASVPGALGPEALLAVSIPPGPNQGPHQCRRFRQPQWQLLDPNATATNWSEAATEPCVDGWVYDRSTFTSTIVAKWDLVCDSQALKPMAQSIYLAGILVGAAVCGPASDRFGRRLVLTWSYLQMAVSGTAAAFAPTFPVYCLFRFLVAFAVAGVMMNTGTLVMEWTSAQARPLVMTLNSLGFSFGHVLMAAVAYGVRDWALLQLVVSVPFFLCFVYSCWLAESARWLLITGRLDRGLRELQRVAAINGKRAVGDTLTPQVLLSAMQEELSVGQAPASLGTLLRTPGLRLRTCISTLCWFAFGFTFFGLALDLQALGSNIFLLQVLIGVVDIPAKIGTLLLLSRLGRRPTQAASLVLAGLCILANTLVPHEMGALRSALAVLGLGGLGAAFTCITIYSGELFPTVLRMTAVGLGQMAARGGAILGPLVRLLGVHGPWLPLLVYGTVPVLSGLAALLLPET
4OWX , Knot 2 12 0.13 2 1 1 
TTTTTTTTTTTT

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 \(c\).

\(|P_{f(8HZE_1)}(2) \setminus P_{f(9DKB_1)}(2)|=11\), \(|P_{f(9DKB_1)}(2) \setminus P_{f(8HZE_1)}(2)|=215\). 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 1:11111001011101010011111101110100000
Pair \(Z_2\) Length of longest common subsequence
8HZE_1,9DKB_1 226 3
8HZE_1,4OWX_1 17 0
9DKB_1,4OWX_1 221 1

Newick tree

 
[
	9DKB_1:12.95,
	[
		8HZE_1:8.5,4OWX_1:8.5
	]:12.45
]

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 \((0.85)(0.8)(\frac{553 }{\log_{20} 553}-\frac{35}{\log_{20}35})=158.\)
Status Protein1 Protein2 d d1/2
Query variables 8HZE_1 9DKB_1 200 106.5
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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