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Parikh vectors
4ZPZ_1 5BWO_1 2QRR_1 Letter Amino acid
2 0 6 N Asparagine
6 1 4 Q Glutamine
9 0 10 L Leucine
7 1 3 K Lycine
2 0 3 F Phenylalanine
4 1 5 R Arginine
6 0 8 E Glutamic acid
3 0 4 P Proline
7 2 2 T Threonine
1 1 8 A Alanine
6 2 7 G Glycine
7 0 5 I Isoleucine
3 1 9 S Serine
0 1 0 W Tryptophan
5 0 7 D Aspartic acid
1 0 0 H Histidine
1 0 4 M Methionine
1 0 6 Y Tyrosine
4 0 10 V Valine
1 0 0 C Cysteine 

4ZPZ_1|Chains A, B|Polyubiquitin-B|Homo sapiens (9606)
>5BWO_1|Chain A[auth B]|Palmitoyl H3K9 Peptide|synthetic construct (32630)
>2QRR_1|Chains A, B|Methionine import ATP-binding protein metN|Vibrio parahaemolyticus (223926)
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 \(w=f(c)\)
4ZPZ , Knot 40 76 0.76 38 61 74 
MQIFVKTLTGKTITLEVEPSDTIENVKAKIQDKEGIPPDQQRLIFCGKQLEDGRTLSDYNIQKESTLHLVLRLRGG
5BWO , Knot 8 10 0.61 16 9 8 
QTARKSTGGW
2QRR , Knot 55 101 0.83 34 92 99 
SNALSIPEDYQARLQPNRVEGSYPLVRMEFTGATVDAPLMSQISRKYNIDVSILSSDLDYAGGVKFGMMVAELFGNEQDDSAAIEYLRENNVKVEVLGYVL

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 \(c\).

\(|P_{f(4ZPZ_1)}(2) \setminus P_{f(5BWO_1)}(2)|=58\), \(|P_{f(5BWO_1)}(2) \setminus P_{f(4ZPZ_1)}(2)|=6\). 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 1:1011100101001010101000100101010000111100001110100100100100001000001011101011
Pair \(Z_2\) Length of longest common subsequence
4ZPZ_1,5BWO_1 64 2
4ZPZ_1,2QRR_1 109 3
5BWO_1,2QRR_1 93 2

Newick tree

 
[
	2QRR_1:55.50,
	[
		4ZPZ_1:32,5BWO_1:32
	]:23.50
]

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 \((0.85)(0.8)(\frac{86 }{\log_{20} 86}-\frac{10}{\log_{20}10})=30.4\)
Status Protein1 Protein2 d d1/2
Query variables 4ZPZ_1 5BWO_1 37 20.5
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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