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Parikh vectors
3MCF_1 8AZP_1 7NBL_1 Letter Amino acid
3 3 3 C Cysteine
12 19 0 L Leucine
6 11 0 S Serine
3 6 1 T Threonine
8 15 2 A Alanine
6 6 0 P Proline
3 0 0 W Tryptophan
5 7 0 Y Tyrosine
1 16 0 N Asparagine
3 3 0 Q Glutamine
19 9 0 E Glutamic acid
3 6 0 I Isoleucine
10 15 0 K Lycine
2 3 0 M Methionine
14 22 0 V Valine
10 2 0 R Arginine
6 10 0 D Aspartic acid
10 15 2 G Glycine
9 6 0 H Histidine
3 5 0 F Phenylalanine 

3MCF_1|Chains A, B|Diphosphoinositol polyphosphate phosphohydrolase 3-alpha|Homo sapiens (9606)
>8AZP_1|Chains A, B|Papain-like protease nsp3|Severe acute respiratory syndrome coronavirus 2 (2697049)
>7NBL_1|Chain A|DNA (5'-D(*CP*TP*AP*(FFC)P*AP*CP*GP*G)-3')|synthetic construct (32630)
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 \(w=f(c)\)
3MCF , Knot 66 136 0.79 40 102 130 
MKKRAACLCFRSEREDEVLLVSSSRYPDRWIVPGGGMEPEEEPGGAAVREVYEEAGVKGKLGRLLGVFEQNQDPEHRTYVYVLTVTELLEDWEDSVSIGRKREWFKVEDAIKVLQCHKPVHAEYLEKLKAHHHHHH
8AZP , Knot 87 179 0.84 38 125 172 
GPMDGEVNSFSGYLKLTDNVYIKNADIVEEAKKVKPTVVVNAANVYLKHGGGVAGALNKATNNAMQVESDDYIATNGPLKVGGSCVLSGHNLAKHCLHVVGPNVNKGEDIQLLKSAYENFNQHEVLLAPLLSAGIFGADPIHSLRVCVDTVRTNVYLAVFDKNLYDKLVSSFLEMKSEK
7NBL , Knot 6 8 0.52 8 6 6 
CTACACGG

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 \(c\).

\(|P_{f(3MCF_1)}(2) \setminus P_{f(8AZP_1)}(2)|=61\), \(|P_{f(8AZP_1)}(2) \setminus P_{f(3MCF_1)}(2)|=84\). 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 1:1000110101000000011110000010011111111010001111110010001110101101111100000100000101101001100100010110000110100110110000110100100101000000
Pair \(Z_2\) Length of longest common subsequence
3MCF_1,8AZP_1 145 4
3MCF_1,7NBL_1 104 2
8AZP_1,7NBL_1 129 2

Newick tree

 
[
	8AZP_1:73.32,
	[
		3MCF_1:52,7NBL_1:52
	]:21.32
]

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 \((0.85)(0.8)(\frac{315 }{\log_{20} 315}-\frac{136}{\log_{20}136})=55.1\)
Status Protein1 Protein2 d d1/2
Query variables 3MCF_1 8AZP_1 71 60.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]