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Parikh vectors
3ZOZ_1 1VOP_1 8KAK_1 Letter Amino acid
16 0 0 F Phenylalanine
24 0 0 S Serine
4 0 0 Y Tyrosine
23 0 0 N Asparagine
5 0 0 H Histidine
17 0 0 P Proline
39 0 0 V Valine
11 0 0 R Arginine
23 0 0 D Aspartic acid
27 0 0 E Glutamic acid
42 0 0 K Lycine
14 0 0 M Methionine
17 0 0 T Threonine
4 0 0 W Tryptophan
41 1 30 A Alanine
7 3 16 C Cysteine
6 0 0 Q Glutamine
40 7 25 G Glycine
19 0 0 I Isoleucine
38 0 0 L Leucine 

3ZOZ_1|Chain A|PHOSPHOGLYCERATE KINASE 1|HOMO SAPIENS (9606)
>1VOP_1|Chain A|23S RIBOSOMAL RNA|
>8KAK_1|Chains A, E|RNA (98-MER)|Streptococcus pyogenes serotype M1 (301447)
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 \(w=f(c)\)
3ZOZ , Knot 168 417 0.81 40 209 380 
MSLSNKLTLDKLDVKGKRVVMRVDFNVPMKNNQITNNQRIKAAVPSIKFCLDNGAKSVVLMSHLGRPDGVPMPDKYSLEPVAVELKSLLGKDVLFLKDCVGPEVEKACANPAAGSVILLENLRFHVEEEGKGKDASGNKVKAEPAKIEAFRASLSKLGDVYVNDAFGTAHRAHSSMVGVNLPQKAGGFLMKKELNYFAKALESPERPFLAILGGAKVADKIQLINNMLDKVNEMIIGGGMAFTFLKVLNNMEIGTSLFDEEGAKIVKDLMSKAEKNGVKITLPVDFVTADKFDENAKTGQATVASGIPAGWMGLDCGPESSKKYAEAVTRAKQIVWNGPVGVFEWEAFARGTKALMDEVVKATSRGCITIIGGGDTATCCAKWNTEDKVSHVSTGGGASLELLEGKVLPGVDALSNI
1VOP , Knot 8 13 0.52 8 9 10 
GACUGGGGCGGUC
8KAK , Knot 32 98 0.49 8 16 46 
GGUCCGUGUUCUAAUUGAAGGUUUUAGAGCUAGAAAUAGCAAGUUAAAAUAAGGCUAGUCCGUUAUCAACUUGAAAAAGUGGCACCGAGUCGGUGCUC

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 \(c\).

\(|P_{f(3ZOZ_1)}(2) \setminus P_{f(1VOP_1)}(2)|=204\), \(|P_{f(1VOP_1)}(2) \setminus P_{f(3ZOZ_1)}(2)|=4\). 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 1:101000101001010100111010101110000100000101111010101001100111100110101111100001011110100111001111000111010010101111011110010101000101001010010101101011010100110101001110100100011110110011111100010011011001001111111110110010110011001001111111110110110010110011000110110011001000110101110110100100010010101101111111110011000000101100100111011111101011101001110011010001010111110010001010000010010011110101101011111011001
Pair \(Z_2\) Length of longest common subsequence
3ZOZ_1,1VOP_1 208 3
3ZOZ_1,8KAK_1 207 3
1VOP_1,8KAK_1 7 4

Newick tree

 
[
	3ZOZ_1:11.78,
	[
		8KAK_1:3.5,1VOP_1:3.5
	]:11.28
]

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 \((0.85)(0.8)(\frac{430 }{\log_{20} 430}-\frac{13}{\log_{20}13})=134.\)
Status Protein1 Protein2 d d1/2
Query variables 3ZOZ_1 1VOP_1 165 85
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]