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Parikh vectors
3ORC_1 7TKD_1 4QPF_1 Letter Amino acid
0 1 18 R Arginine
0 2 12 N Asparagine
1 10 23 G Glycine
0 9 21 I Isoleucine
0 12 41 L Leucine
0 6 15 F Phenylalanine
0 5 17 S Serine
3 10 30 A Alanine
1 1 6 C Cysteine
0 2 22 K Lycine
0 3 8 M Methionine
3 3 12 T Threonine
0 0 4 W Tryptophan
0 1 23 Y Tyrosine
0 1 26 Q Glutamine
0 2 12 P Proline
0 6 23 V Valine
0 1 23 D Aspartic acid
0 1 28 E Glutamic acid
0 0 11 H Histidine 

3ORC_1|Chains A[auth R], B[auth S]|DNA (5'-D(*TP*AP*TP*CP*GP*AP*TP*A)-3')|
>7TKD_1|Chains A[auth 0], B[auth 1], C[auth 2], D[auth 3], E[auth 4], F[auth 5], G[auth 6], H[auth 7], I[auth 8], J[auth 9]|ATP synthase subunit 9, mitochondrial|Saccharomyces cerevisiae (4932)
>4QPF_1|Chain A|Farnesyl pyrophosphate synthase|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 \(w=f(c)\)
3ORC , Knot 5 8 0.43 8 5 6 
TATCGATA
7TKD , Knot 40 76 0.76 36 57 72 
MQLVLAAKYIGAGISTIGLLGAGIGIAIVFAALINGVSRNPSIKDTVFPMAILGFALSEATGLFCLMVSFLLLFGV
4QPF , Knot 158 375 0.83 40 213 352 
MGSSHHHHHHSSGRENLYFQGHMNGDQNSDVYAQEKQDFVQHFSQIVRVLTEDEMGHPEIGDAIARLKEVLEYNAIGGKYNRGLTVVVAFRELVEPRKQDADSLQRAWTVGWCVELLQAFFLVADDIMDSSLTRRGQICWYQKPGVGLDAINDANLLEACIYRLLKLYCREQPYYLNLIELFLQSSYQTEIGQTLDLLTAPQGNVDLVRFTEKRYKSIVKYKTAFYSFYLPIAAAMYMAGIDGEKEHANAKKILLEMGEFFQIQDDYLDLFGDPSVTGKIGTDIQDNKCSWLVVQCLQRATPEQYQILKENYGQKEAEKVARVKALYEELDLPAVFLQYEEDSYSHIMALIEQYAAPLPPAVFLGLARKIYKRRK

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 \(c\).

\(|P_{f(3ORC_1)}(2) \setminus P_{f(7TKD_1)}(2)|=3\), \(|P_{f(7TKD_1)}(2) \setminus P_{f(3ORC_1)}(2)|=55\). 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 1:01001101
Pair \(Z_2\) Length of longest common subsequence
3ORC_1,7TKD_1 58 2
3ORC_1,4QPF_1 214 2
7TKD_1,4QPF_1 202 3

Newick tree

 
[
	4QPF_1:11.96,
	[
		3ORC_1:29,7TKD_1:29
	]:89.96
]

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 \((0.85)(0.8)(\frac{84 }{\log_{20} 84}-\frac{8}{\log_{20}8})=30.7\)
Status Protein1 Protein2 d d1/2
Query variables 3ORC_1 7TKD_1 39 21
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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