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Parikh vectors
5RXN_1 6PFR_1 9MUS_1 Letter Amino acid
4 21 22 K Lycine
3 16 16 T Threonine
5 17 17 V Valine
10 19 19 D Aspartic acid
4 2 3 C Cysteine
6 16 17 E Glutamic acid
0 15 15 H Histidine
1 22 20 L Leucine
1 1 2 W Tryptophan
0 7 7 R Arginine
0 7 6 Q Glutamine
6 23 27 G Glycine
1 7 6 M Methionine
5 11 10 P Proline
0 8 10 A Alanine
2 12 12 I Isoleucine
1 13 12 N Asparagine
2 12 13 F Phenylalanine
0 12 10 S Serine
3 11 9 Y Tyrosine 

5RXN_1|Chain A|RUBREDOXIN|Clostridium pasteurianum (1501)
>6PFR_1|Chain A|Green fluorescent protein|Aequorea victoria (6100)
>9MUS_1|Chains A, B|Fluorescent thiol-disulfide redox biosensor|Aequorea victoria (6100)
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 \(w=f(c)\)
5RXN , Knot 29 54 0.71 30 44 51 
MKKYTCTVCGYIYDPEDGDPDDGVNPGTDFKDIPDDWVCPLCGVGKDEFEEVEE
6PFR , Knot 111 252 0.81 40 163 237 
MRGSHHHHHHTDPMVSKGEELFTGVVPILVELDGDVNGHKFSVSGEGEGDATYGKLTLKFICTTGKLPVPWPTLVTTLAYGVLCFSRYPDHMKQHDFFKSAMPEGYVQERTIFFKDDGNYKTRAEVKFEGDTLVNRIELKGIDFKEDGNILGHKLEYNYNSHNVYIMADKQKNGIKSNFKIRHNIEDGSVQLADHYQQNTPIGDGPVLLPDNHYLSTQSKLSKDPNEKRDHMVLLEFVTAAGITLGMDELYK
9MUS , Knot 114 253 0.83 40 164 237 
MKHHHHHHHGGASMVSKGEELFTGVVPILVELDGDVNGHKFSVSGEGEGDATYGKLTLKFISTTGKLPVPWPTLVTTLSWGVQCFARYPDHMKRHDFFKSAMPEGYVQERTIFFKDDGNYKTRAEVKFEGDTLVNRIELKGIDFKEDGNILGHKLEYNGGCEDNVYITADKQKNGIKANFKIRHNIEDGGVQLADHYQQNTPIGDGPVLLPDNHYLSTCSKLSKDPNEKRDHMVFLEFVTAAGITLGMDELYK

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 \(c\).

\(|P_{f(5RXN_1)}(2) \setminus P_{f(6PFR_1)}(2)|=19\), \(|P_{f(6PFR_1)}(2) \setminus P_{f(5RXN_1)}(2)|=138\). 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 1:100000010101001001010011011001001100110110111000100100
Pair \(Z_2\) Length of longest common subsequence
5RXN_1,6PFR_1 157 3
5RXN_1,9MUS_1 162 3
6PFR_1,9MUS_1 39 64

Newick tree

 
[
	5RXN_1:91.40,
	[
		6PFR_1:19.5,9MUS_1:19.5
	]:71.90
]

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 \((0.85)(0.8)(\frac{306 }{\log_{20} 306}-\frac{54}{\log_{20}54})=81.3\)
Status Protein1 Protein2 d d1/2
Query variables 5RXN_1 6PFR_1 100 59.5
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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