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Parikh vectors
1EID_1 3IAN_1 2VWL_1 Letter Amino acid
4 7 5 M Methionine
15 23 11 S Serine
4 5 14 R Arginine
3 17 13 I Isoleucine
4 12 7 P Proline
12 22 14 A Alanine
10 29 7 N Asparagine
7 9 9 Q Glutamine
4 27 23 G Glycine
2 23 14 L Leucine
10 19 20 T Threonine
5 24 11 D Aspartic acid
8 0 9 C Cysteine
5 16 20 E Glutamic acid
4 9 5 H Histidine
10 26 19 K Lycine
2 15 12 F Phenylalanine
0 6 4 W Tryptophan
6 17 8 Y Tyrosine
9 15 16 V Valine 

1EID_1|Chain A|RIBONUCLEASE A|Bos taurus (9913)
>3IAN_1|Chain A|Chitinase|Lactococcus lactis subsp. lactis (1360)
>2VWL_1|Chain A|ACTIVATED FACTOR XA HEAVY CHAIN|HOMO SAPIENS (9606)
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 \(w=f(c)\)
1EID , Knot 64 124 0.83 38 107 121 
KETAAAKFERQHMDSSTSAASSSNYCNQMMKSRNLTKDRCKPVNTFVHESLADVQAVCSQKNVACKNGQTNCYQSYSTMSITDCRETGSSKYPNCAYKTTQANKHIIVACEGNPYVPVHGDASV
3IAN , Knot 137 321 0.82 38 187 308 
MSLDKVLVGYWHNWKSTGKDGYKGGSSADFNLSSTQEGYNVINVSFMKTPEGQTLPTFKPYNKTDTEFRAEISKLNAEGKSVLIALGGADAHIELKKSQESDFVNEIIRLVDTYGFDGLDIDLEQAAIEAADNQTVIPSALKKVKDHYRKDGKNFMITMAPEFPYLTSSGKYAPYINNLDSYYDFINPQYYNQGGDGFWDSDLNMWISQSNDEKKEDFLYGLTQRLVTGTDGFIKIPASKFVIGLPSNNDAAATGYVKDPNAVKNALNRLKASGNEIKGLMTWSVNWDAGTNSNGEKYNNTFVNTYAPMLFNNEGHHHHHH
2VWL , Knot 111 241 0.84 40 174 233 
IVGGQECKDGECPWQALLINEENEGFCGGTILSEFYILTAAHCLYQAKRFKVRVGDRNTEQEEGGEAVHEVEVVIKHNRFTKETYDFDIAVLRLKTPITFRMNVAPACLPERDWAESTLMTQKTGIVSGFGRTHEKGEQSTRLKMLEVPYVDRNSCKLSSSFIITQNMFCAGYDTKQEDACQGDSGGPHVTRFKDTYFVTGIVSWGEGCARKGKYGIYTKVTAFLKWIDRSMKTRGLPKAK

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 \(c\).

\(|P_{f(1EID_1)}(2) \setminus P_{f(3IAN_1)}(2)|=49\), \(|P_{f(3IAN_1)}(2) \setminus P_{f(1EID_1)}(2)|=129\). 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 1:0001110100001000001100000000110000100000011001100011010110000011000100000000001010000001000010010000010001111001010111010101
Pair \(Z_2\) Length of longest common subsequence
1EID_1,3IAN_1 178 4
1EID_1,2VWL_1 189 3
3IAN_1,2VWL_1 165 3

Newick tree

 
[
	1EID_1:94.68,
	[
		3IAN_1:82.5,2VWL_1:82.5
	]:12.18
]

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 \((0.85)(0.8)(\frac{445 }{\log_{20} 445}-\frac{124}{\log_{20}124})=96.2\)
Status Protein1 Protein2 d d1/2
Query variables 1EID_1 3IAN_1 120 83
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]