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Parikh vectors
2MPJ_1 3TNE_1 5IFZ_1 Letter Amino acid
0 23 10 L Leucine
0 15 8 K Lycine
1 3 0 W Tryptophan
0 17 4 Y Tyrosine
0 5 8 E Glutamic acid
1 1 9 H Histidine
1 0 4 M Methionine
1 12 4 F Phenylalanine
1 24 30 A Alanine
1 20 9 N Asparagine
1 25 12 D Aspartic acid
3 4 3 C Cysteine
5 34 14 G Glycine
0 21 11 I Isoleucine
2 8 1 P Proline
3 42 11 S Serine
2 6 10 R Arginine
1 36 8 T Threonine
1 24 13 V Valine
1 19 3 Q Glutamine 

2MPJ_1|Chain A|RECQL4-helicase-like protein|Xenopus laevis (8355)
>3TNE_1|Chains A, B|Secreted aspartic protease|Candida parapsilosis (5480)
>5IFZ_1|Chains A, B|Ribose 5-phosphate isomerase|Brucella melitensis bv. 1 str. 16M (224914)
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 \(w=f(c)\)
2MPJ , Knot 19 25 0.81 30 24 23 
NRSGDTCFRCGGMGHWASQCPGSVP
3TNE , Knot 141 339 0.80 38 176 320 
DSISLSLINEGPSYASKVSVGSNKQQQTVIIDTGSSDFWVVDSNAQCGKGVDCKSSGTFTPSSSSSYKNLGAAFTIRYGDGSTSQGTWGKDTVTINGVSITGQQIADVTQTSVDQGILGIGYTSNEAVYDTSGRQTTPNYDNVPVTLKKQGKIRTNAYSLYLNSPSAETGTIIFGGVDNAKYSGKLVAEQVTSSQALTISLASVNLKGSSFSFGDGALLDSGTTLTYFPSDFAAQLADKAGARLVQVARDQYLYFIDCNTDTSGTTVFNFGNGAKITVPNTEYVYQNGDGTCLWGIQPSDDTILGDNFLRHAYLLYNLDANTISIAQVKYTTDSSISAV
5IFZ , Knot 78 172 0.77 38 117 162 
MAHHHHHHMRHQGSSELRSKIMKVAVAGDSAGEGLAKVLADHLKDRFEVSEISRTDAGADAFYANLSDRVASAVLDGTYDRAILVCGTGIGVCIAANKVPGIRAALTHDTYSAERAALSNNAQIITMGARVIGAEVAKTIADAFLAQTFDENGRSAGNVNAINEVDAKYNKC

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 \(c\).

\(|P_{f(2MPJ_1)}(2) \setminus P_{f(3TNE_1)}(2)|=12\), \(|P_{f(3TNE_1)}(2) \setminus P_{f(2MPJ_1)}(2)|=164\). 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 1:0001000100111101100011011
Pair \(Z_2\) Length of longest common subsequence
2MPJ_1,3TNE_1 176 2
2MPJ_1,5IFZ_1 123 2
3TNE_1,5IFZ_1 139 3

Newick tree

 
[
	3TNE_1:84.39,
	[
		2MPJ_1:61.5,5IFZ_1:61.5
	]:22.89
]

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 \((0.85)(0.8)(\frac{364 }{\log_{20} 364}-\frac{25}{\log_{20}25})=109.\)
Status Protein1 Protein2 d d1/2
Query variables 2MPJ_1 3TNE_1 130 69.5
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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