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Parikh vectors
1QVL_1 8CWR_1 5HEM_1 Letter Amino acid
0 4 8 M Methionine
2 3 3 W Tryptophan
0 1 7 Y Tyrosine
0 4 5 D Aspartic acid
3 10 4 R Arginine
0 2 7 N Asparagine
0 4 2 C Cysteine
0 6 10 Q Glutamine
0 2 3 H Histidine
0 8 10 L Leucine
1 2 5 F Phenylalanine
0 5 7 A Alanine
0 1 7 T Threonine
0 3 9 V Valine
0 7 8 P Proline
0 2 12 K Lycine
0 5 3 S Serine
0 9 6 E Glutamic acid
0 2 7 I Isoleucine
0 10 3 G Glycine 

1QVL_1|Chain A|c-RW|null
>8CWR_1|Chain A|Redox- and pH-responsive transcriptional regulator WhiB3|Mycobacterium tuberculosis (83332)
>5HEM_1|Chains A, B|Bromodomain-containing protein 2|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 \(w=f(c)\)
1QVL , Knot 4 6 0.39 6 5 4 
RRWWRF
8CWR , Knot 49 90 0.81 40 77 88 
MPQPEQLPGPNADIWNWQLQGLCRGMDSSMFFHPDGERGRARTQREQRAKEMCRRCPVIEACRSHALEVGEPYGVWGGLSESERDLLLKG
5HEM , Knot 65 126 0.83 40 107 124 
SMKPGRVTNQLQYLHKVVMKALWKHQFAWPFRQPVDAVKLGLPDYHKIIKQPMDMGTIKRRLENNYYWAASECMQDFNTMFTNCYIYNKPTDYIVLMAQTLEKIFLQKVASMPQEEQELVVTIPKN

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 \(c\).

\(|P_{f(1QVL_1)}(2) \setminus P_{f(8CWR_1)}(2)|=4\), \(|P_{f(8CWR_1)}(2) \setminus P_{f(1QVL_1)}(2)|=76\). 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 1:001101
Pair \(Z_2\) Length of longest common subsequence
1QVL_1,8CWR_1 80 2
1QVL_1,5HEM_1 110 2
8CWR_1,5HEM_1 146 3

Newick tree

 
[
	5HEM_1:70.96,
	[
		1QVL_1:40,8CWR_1:40
	]:30.96
]

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 \((0.85)(0.8)(\frac{96 }{\log_{20} 96}-\frac{6}{\log_{20}6})=36.0\)
Status Protein1 Protein2 d d1/2
Query variables 1QVL_1 8CWR_1 49 25.5
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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