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Parikh vectors
1DSD_1 2JVK_1 7QAM_1 Letter Amino acid
0 6 0 N Asparagine
2 7 6 G Glycine
0 4 1 P Proline
1 8 10 A Alanine
0 5 2 Q Glutamine
0 14 4 L Leucine
0 4 2 F Phenylalanine
0 4 0 Y Tyrosine
0 5 2 R Arginine
0 7 0 H Histidine
0 11 1 K Lycine
0 7 0 M Methionine
0 2 2 S Serine
3 4 0 T Threonine
0 11 1 D Aspartic acid
2 0 1 C Cysteine
0 11 0 E Glutamic acid
0 15 0 I Isoleucine
0 0 0 W Tryptophan
0 7 3 V Valine 

1DSD_1|Chains A, B|DNA (5'-D(*GP*AP*TP*GP*CP*TP*TP*C)-3')|
>2JVK_1|Chain A|Sporulation initiation phosphotransferase F|Bacillus subtilis (1423)
>7QAM_1|Chain A|Serine/threonine-protein phosphatase PGAM5, 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 \(w=f(c)\)
1DSD , Knot 6 8 0.52 8 7 6 
GATGCTTC
2JVK , Knot 65 132 0.80 36 103 125 
MMNEKILIVDDQYGIRILLNEVFNKEGYQTFQAANGLQALDIVTKERPDLVLLDMKIPGMDGIEIAKRMKVIDENIRVIIMTAYGELDMIQESKELGALTHFAKPFDIDEIRDAVKKYLPLKSNLEHHHHHH
7QAM , Knot 19 35 0.64 24 27 32 
AFRQALQLAACGLAGGSAAVLFSAVAVGKPRAGGD

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 \(c\).

\(|P_{f(1DSD_1)}(2) \setminus P_{f(2JVK_1)}(2)|=6\), \(|P_{f(2JVK_1)}(2) \setminus P_{f(1DSD_1)}(2)|=102\). 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 1:11010000
Pair \(Z_2\) Length of longest common subsequence
1DSD_1,2JVK_1 108 2
1DSD_1,7QAM_1 34 1
2JVK_1,7QAM_1 114 3

Newick tree

 
[
	2JVK_1:63.35,
	[
		1DSD_1:17,7QAM_1:17
	]:46.35
]

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 \((0.85)(0.8)(\frac{140 }{\log_{20} 140}-\frac{8}{\log_{20}8})=49.8\)
Status Protein1 Protein2 d d1/2
Query variables 1DSD_1 2JVK_1 62 33
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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