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Parikh vectors
2MOM_1 1HJP_1 7JMR_1 Letter Amino acid
0 0 8 W Tryptophan
0 11 11 R Arginine
1 4 5 N Asparagine
0 1 6 C Cysteine
1 8 17 S Serine
3 3 5 H Histidine
3 12 9 I Isoleucine
1 11 10 P Proline
2 4 10 Y Tyrosine
7 24 20 A Alanine
3 7 17 D Aspartic acid
1 8 9 Q Glutamine
1 19 16 E Glutamic acid
4 16 20 G Glycine
1 12 17 K Lycine
0 6 5 M Methionine
0 9 17 T Threonine
7 23 17 L Leucine
3 8 12 F Phenylalanine
4 17 12 V Valine 

2MOM_1|Chains A, B, C|Lysosome-associated membrane glycoprotein 2|Homo sapiens (9606)
>1HJP_1|Chain A|RUVA|Escherichia coli (562)
>7JMR_1|Chain A|Pea pathogenicity protein 2|Madurella mycetomatis (100816)
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 \(w=f(c)\)
2MOM , Knot 26 42 0.77 30 35 40 
SADDDNFLVPIAVGAALAGVLILVLLAYFIGLKHHHAGYEQF
1HJP , Knot 92 203 0.80 38 137 193 
MIGRLRGIIIEKQPPLVLIEVGGVGYEVHMPMTCFYELPEAGQEAIVFTHFVVREDAQLLYGFNNKQERTLFKELIKTNGVGPKLALAILSGMSAQQFVNAVEREEVGALVKLPGIGKKTAERLIVEMKDRFKGLHGDLFTPAADLVLTSPASPATDDAEQEAVAALVALGYKPQEASRMVSKIARPDASSETLIREALRAAL
7JMR , Knot 112 243 0.84 40 174 235 
MTSPLDPHSKCYSHINGSAEELLDRLAVSELCKGWPVYRDASEWKNYRSLFTEDATVWTTWSGPRPVDEFITISKAGKEQGVFIMHRECGTLVELSPQQGRAIGKMKATITQRFSFPAASGAATNGTTGTTSDAIEFDVDCDCRFIFFCEKDTASGAWKAKYVKLFYEKDKVVSVDGHQAPKFTKDELAKYPQGYRYLGAAQARLGYDIDLQLPTSSGQLWDRMYGEMENWLGGNKVDLFWEH

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 \(c\).

\(|P_{f(2MOM_1)}(2) \setminus P_{f(1HJP_1)}(2)|=12\), \(|P_{f(1HJP_1)}(2) \setminus P_{f(2MOM_1)}(2)|=114\). 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 1:010000111111111111111111111011110000110001
Pair \(Z_2\) Length of longest common subsequence
2MOM_1,1HJP_1 126 3
2MOM_1,7JMR_1 181 3
1HJP_1,7JMR_1 163 3

Newick tree

 
[
	7JMR_1:92.54,
	[
		2MOM_1:63,1HJP_1:63
	]:29.54
]

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 \((0.85)(0.8)(\frac{245 }{\log_{20} 245}-\frac{42}{\log_{20}42})=67.8\)
Status Protein1 Protein2 d d1/2
Query variables 2MOM_1 1HJP_1 82 49
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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