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Parikh vectors
6VLA_1 5TYE_1 8WKD_1 Letter Amino acid
0 22 13 S Serine
2 27 7 R Arginine
2 3 11 N Asparagine
2 12 22 K Lycine
0 21 1 P Proline
0 5 1 W Tryptophan
0 30 6 Q Glutamine
0 27 19 E Glutamic acid
1 27 14 G Glycine
0 19 11 T Threonine
0 5 2 Y Tyrosine
0 28 18 A Alanine
0 13 8 D Aspartic acid
3 45 11 L Leucine
1 4 6 M Methionine
2 18 9 F Phenylalanine
2 22 19 V Valine
0 7 3 C Cysteine
0 15 0 H Histidine
2 6 11 I Isoleucine 

6VLA_1|Chain A|Intragenic antimicrobial peptide|Homo sapiens (9606)
>5TYE_1|Chain A|DNA-directed DNA/RNA polymerase mu|Homo sapiens (9606)
>8WKD_1|Chain A|DUF2121 domain-containing protein|Methanosarcina mazei (2209)
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 \(w=f(c)\)
6VLA , Knot 12 17 0.66 18 14 14 
LMGLFNRIIRKVVKLFN
5TYE , Knot 147 356 0.80 40 198 344 
GSAAAPLSPAWMPAYACQRPTPLTHHNTGLSEALEILAEAAGFEGSEGRLLTFCRAASVLKALPSPVTTLSQLQGLPHFGEHSSRVVQELLEHGVCEEVERVRRSERYQTMKLFTQIFGVGVKTADRWYREGLRTLDDLREQPQKLTQQQKAGLQHHQDLSTPVLRSDVDALQQVVEEAVGQALPGATVTLTGGFRRGKLQGHDVDFLITHPKEGQEAGLLPRVMCRLQDQGLILYHQHQHSCCESPTRLAQQSHMDAFERSFCIFRLPQPGSWKAVRVDLVVAPVSQFPFALLGWTGSKLFQRELRRFSRKEKGLWLNSHGLFDPEQKTFFQAASEEDIFRHLGLEYLPPEQRNA
8WKD , Knot 88 192 0.80 38 132 186 
ALVIAFIGKNGAVMAGDMREITFEGEKPDREKLEKELYSGSIVTDEEMQKKAEEFGVKITVADCKEKVSERNGVLVGEVSSAEGGVVKKRRLYASAGNFAIAELINTEMTLTSQGKGSNFIAFGNEFTKQVANKCFKDNWTKKSNLQDAVKILILCMETVARKTASVSKQFMIVQTASNADVLKVVEKDRNS

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 \(c\).

\(|P_{f(6VLA_1)}(2) \setminus P_{f(5TYE_1)}(2)|=8\), \(|P_{f(5TYE_1)}(2) \setminus P_{f(6VLA_1)}(2)|=192\). 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 1:11111001100110110
Pair \(Z_2\) Length of longest common subsequence
6VLA_1,5TYE_1 200 3
6VLA_1,8WKD_1 136 3
5TYE_1,8WKD_1 176 3

Newick tree

 
[
	5TYE_1:10.42,
	[
		6VLA_1:68,8WKD_1:68
	]:33.42
]

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 \((0.85)(0.8)(\frac{373 }{\log_{20} 373}-\frac{17}{\log_{20}17})=116.\)
Status Protein1 Protein2 d d1/2
Query variables 6VLA_1 5TYE_1 145 76
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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