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Parikh vectors
1HEV_1 3CDL_1 9IQW_1 Letter Amino acid
0 6 6 M Methionine
1 16 6 R Arginine
4 8 12 N Asparagine
5 8 7 G Glycine
2 21 12 L Leucine
1 9 6 T Threonine
1 3 6 Y Tyrosine
0 14 8 V Valine
3 7 11 D Aspartic acid
4 11 1 Q Glutamine
0 8 10 I Isoleucine
2 10 7 P Proline
1 23 8 A Alanine
2 7 9 K Lycine
4 14 5 S Serine
2 6 2 W Tryptophan
8 1 2 C Cysteine
2 17 6 E Glutamic acid
1 3 5 H Histidine
0 11 5 F Phenylalanine 

1HEV_1|Chain A|HEVEIN|Hevea brasiliensis (3981)
>3CDL_1|Chains A, B|Transcriptional regulator AefR|Pseudomonas syringae pv. tomato (223283)
>9IQW_1|Chains A, B, C, D[auth E]|Capsule biosynthesis protein|Phage #D (77920)
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 \(w=f(c)\)
1HEV , Knot 26 43 0.75 32 38 41 
EQCGRQAGGKLCPNNLCCSQWGWCGSTDEYCSPDHNCQSNCKD
3CDL , Knot 95 203 0.83 40 142 198 
SNAMRLTDQKRESIVQAAIAEFGDRGFEITSMDRIAARAEVSKRTVYNHFPSKEELFAEMLQRLWNCAPPQSEVVYRPLVSLREQLLELLWGKMRNLTDSSFLDLARVVVGATIHSPERAQVWLARINEREETFSAWIRAAQKDGRLKPVDPGFAATQMHALLKSFAFWPQVTFNAALLTPQEQSNVVESALNMFLGWYEIPG
9IQW , Knot 70 134 0.85 40 111 132 
VPRGSHMHRVENMLNLCFDVDDCITEWNNNRDYVNFKPDVEMVSAINALYDAGHTITLYTARGMKSVGPGRIAIDILPSLIQNLANIGLKYHNLLTHKPVYDWIIDDKAMRPDEFKALMNKGEFETFKSYKPNL

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 \(c\).

\(|P_{f(1HEV_1)}(2) \setminus P_{f(3CDL_1)}(2)|=25\), \(|P_{f(3CDL_1)}(2) \setminus P_{f(1HEV_1)}(2)|=129\). 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 1:0001001110101001000011101000000010000000000
Pair \(Z_2\) Length of longest common subsequence
1HEV_1,3CDL_1 154 3
1HEV_1,9IQW_1 129 3
3CDL_1,9IQW_1 167 3

Newick tree

 
[
	3CDL_1:84.93,
	[
		1HEV_1:64.5,9IQW_1:64.5
	]:20.43
]

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 \((0.85)(0.8)(\frac{246 }{\log_{20} 246}-\frac{43}{\log_{20}43})=67.7\)
Status Protein1 Protein2 d d1/2
Query variables 1HEV_1 3CDL_1 85 51
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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