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Parikh vectors
8VMZ_1 2EBQ_1 4PFQ_1 Letter Amino acid
0 1 13 R Arginine
4 5 0 C Cysteine
4 5 15 G Glycine
0 4 7 K Lycine
6 3 13 A Alanine
0 1 6 Q Glutamine
0 2 12 E Glutamic acid
0 2 12 I Isoleucine
0 3 23 L Leucine
0 3 9 P Proline
0 1 5 N Asparagine
0 0 6 M Methionine
0 4 16 S Serine
7 6 5 T Threonine
0 1 2 W Tryptophan
0 4 23 V Valine
0 2 20 D Aspartic acid
0 0 3 F Phenylalanine
0 0 9 Y Tyrosine
0 0 8 H Histidine 

8VMZ_1|Chains A[auth C], B[auth D]|DNA (5'-D(*TP*TP*GP*AP*CP*TP*CP*TP*CP*TP*TP*AP*AP*GP*AP*GP*AP*GP*TP*CP*A)-3')|synthetic construct (32630)
>2EBQ_1|Chain A|Nuclear pore complex protein Nup153|Homo sapiens (9606)
>4PFQ_1|Chains A, B, C, D, E, F, G, H|Hypoxanthine phosphoribosyltransferase|Brachybacterium faecium (446465)
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 \(w=f(c)\)
8VMZ , Knot 10 21 0.48 8 11 15 
TTGACTCTCTTAAGAGAGTCA
2EBQ , Knot 28 47 0.76 32 38 43 
GSSGSSGVIGTWDCDTCLVQNKPEAIKCVACETPKPGTCVKRALTLT
4PFQ , Knot 90 207 0.77 38 130 196 
MHHHHHHSSGVDLGTENLYFQSMVPQTHPHPDVDRVLLDEQQIRDRLAELGEQIAADYAEEPPVLVGVLRGAVMVMADLARQIDLKVEMDWMAVSSYGSGTKSSGVVRILKDLSGDITDRNVLIVEDIIDSGLTLKWLLSNLRSRGPKSVEVAALLRKPDAARVDIDVKYIGFDIPSEFVIGYGLDYAENYRNLPYVGVLSRSVYED

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 \(c\).

\(|P_{f(8VMZ_1)}(2) \setminus P_{f(2EBQ_1)}(2)|=8\), \(|P_{f(2EBQ_1)}(2) \setminus P_{f(8VMZ_1)}(2)|=35\). 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 1:001100000001111111001
Pair \(Z_2\) Length of longest common subsequence
8VMZ_1,2EBQ_1 43 3
8VMZ_1,4PFQ_1 135 2
2EBQ_1,4PFQ_1 138 4

Newick tree

 
[
	4PFQ_1:77.82,
	[
		8VMZ_1:21.5,2EBQ_1:21.5
	]:56.32
]

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 \((0.85)(0.8)(\frac{68 }{\log_{20} 68}-\frac{21}{\log_{20}21})=18.7\)
Status Protein1 Protein2 d d1/2
Query variables 8VMZ_1 2EBQ_1 24 15.5
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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