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Parikh vectors
2MRA_1 3RYD_1 2KPJ_1 Letter Amino acid
6 13 6 H Histidine
1 10 2 M Methionine
3 13 3 F Phenylalanine
1 3 1 W Tryptophan
12 20 6 A Alanine
10 17 5 D Aspartic acid
6 13 4 G Glycine
9 28 6 L Leucine
0 10 3 P Proline
7 14 5 T Threonine
3 15 6 N Asparagine
2 18 4 Q Glutamine
2 9 7 S Serine
6 10 4 V Valine
4 6 1 R Arginine
21 21 11 K Lycine
6 22 10 I Isoleucine
2 11 3 Y Tyrosine
2 5 1 C Cysteine
14 15 6 E Glutamic acid 

2MRA_1|Chain A|De novo designed protein OR459|artificial gene (32630)
>3RYD_1|Chains A, B[auth C]|Inositol monophosphatase family protein|Staphylococcus aureus (282459)
>2KPJ_1|Chain A|SOS-response transcriptional repressor, LexA|Eubacterium rectale ATCC 33656 (515619)
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 \(w=f(c)\)
2MRA , Knot 56 117 0.76 38 77 110 
MAGKELRVEIKIDCGNDDKETTYDLYFSKAEEAKELLKKVAEKAADKIKKQGCKRVKIRFEKKGLDDDARKKAKKWALEVANKIANELGAKQSTTTTDGDTFEVEVILELEHHHHHH
3RYD , Knot 123 273 0.84 40 183 261 
ELHHHHHHMTDKTLQQIDKLICSWLKQIDNVIPQLIMEMTTETKRHRFDLVTNVDKQIQQQFQQFLATYFPEHQLLAEEKSNAMITNEINHLWIMDPIDGTANLVKQQEDYCIILAYFYEGKPMLSYVYDYPHKKLYKAIRGEGAFCNGIKMEEPPSLKLEDAIISFNAQVMNLDTVQDLFDASFSYRLVGACGLDSMRVAKGQFGAHINTNPKPWDIAAQFLFAELLNLKMTTLDGKAIDHLKGAPFIISNKACHETVLKILNANGGYQKYR
2KPJ , Knot 50 94 0.80 40 76 87 
MVKDKQKAIFSENLNSYIAKSEKTQLEIAKSIGVSPQTFNTWCKGIAIPRMGKVQALADYFNINKSDLIEDKKLNIDTVPIESGYTLEHHHHHH

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 \(c\).

\(|P_{f(2MRA_1)}(2) \setminus P_{f(3RYD_1)}(2)|=28\), \(|P_{f(3RYD_1)}(2) \setminus P_{f(2MRA_1)}(2)|=134\). 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 1:111001010101001000000000101001001001100110011001000100010101000110001000100111011001100111000000001001010111010000000
Pair \(Z_2\) Length of longest common subsequence
2MRA_1,3RYD_1 162 6
2MRA_1,2KPJ_1 103 8
3RYD_1,2KPJ_1 171 6

Newick tree

 
[
	3RYD_1:91.45,
	[
		2MRA_1:51.5,2KPJ_1:51.5
	]:39.95
]

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 \((0.85)(0.8)(\frac{390 }{\log_{20} 390}-\frac{117}{\log_{20}117})=83.1\)
Status Protein1 Protein2 d d1/2
Query variables 2MRA_1 3RYD_1 108 74.5
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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