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Parikh vectors
8JSY_1 9HCH_1 8FFA_1 Letter Amino acid
6 3 5 Y Tyrosine
10 5 5 F Phenylalanine
14 14 7 S Serine
6 1 2 W Tryptophan
11 10 18 A Alanine
16 10 18 L Leucine
0 2 0 C Cysteine
4 5 7 Q Glutamine
13 3 15 E Glutamic acid
21 13 9 G Glycine
13 9 9 I Isoleucine
19 9 4 K Lycine
4 13 10 R Arginine
12 4 7 D Aspartic acid
9 8 6 P Proline
13 14 9 V Valine
5 4 6 M Methionine
14 7 11 T Threonine
6 7 4 N Asparagine
7 4 4 H Histidine 

8JSY_1|Chains A, B, C, D, E, F, G, H, I, J|Dihydrofolate reductase family protein|Leptospira interrogans serovar Pomona (44276)
>9HCH_10|Chain J[auth L]|Large ribosomal subunit protein uL14m|Mus musculus (10090)
>8FFA_1|Chains A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P|Probable dna-binding stress protein|Pseudomonas aeruginosa PAO1 (208964)
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 \(w=f(c)\)
8JSY , Knot 91 203 0.79 38 142 196 
MRKITVLSFITLDGVMQAPGGPEEDTSGGFKYGGWTAPYEDEVSGKIMEKQMKPADYLLGRKTFEIFASYWPEHADFWPGINDGTKYVMSKTVKKSDWKNSVFLESLADIKKLKNSEGSDIQVWGSGELIQLLFKNDLVDELWLKIFPVTLNTGKRLFGDGTIPAAFTLIESSVTPSGVIIANYKRAGEVKTGTVGAHHHHHH
9HCH , Knot 69 145 0.79 40 110 142 
MAALTGLWGSFAHVSRAFSQRCFSTSGSLSAVQKMTRVRVVDNSALGSTPYHRPPRCIHVYNKSGVGKVGDQILLAIRGQKKKALIVGHRMPGSRMTPKFDSNNVVLIEDNGNPVGTRIKIPIPTSLRRREGEYSKVLAIAQNFV
8FFA , Knot 76 156 0.82 38 120 152 
MEINIGIGEQDRAAIAEGLSRLLADTYTLYLKTHNFHWNVTGPMFNTLHLMFEGQYTELAVAVDDIAERIRALGFPAPGTYAAYARLSSIKEEEGVPEAEEMIRQLVQGQEAVVRTARSIFPLLDKVSDEPTADLLTQRMQVHEKTAWMLRSLLAS

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 \(c\).

\(|P_{f(8JSY_1)}(2) \setminus P_{f(9HCH_1)}(2)|=100\), \(|P_{f(9HCH_1)}(2) \setminus P_{f(8JSY_1)}(2)|=68\). 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 1:10010110110101110111110000011100111011000010101100010110011100010111001100101111100100011000100001000111001101001000010010111010110111000110011101111010010011101011111011000101011111000011010010111000000
Pair \(Z_2\) Length of longest common subsequence
8JSY_1,9HCH_1 168 3
8JSY_1,8FFA_1 156 3
9HCH_1,8FFA_1 158 3

Newick tree

 
[
	9HCH_1:82.68,
	[
		8JSY_1:78,8FFA_1:78
	]:4.68
]

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 \((0.85)(0.8)(\frac{348 }{\log_{20} 348}-\frac{145}{\log_{20}145})=61.7\)
Status Protein1 Protein2 d d1/2
Query variables 8JSY_1 9HCH_1 81 68.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]