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Parikh vectors
5UZL_1 2ZII_1 6XMF_1 Letter Amino acid
2 14 66 A Alanine
1 17 63 K Lycine
3 6 48 P Proline
4 17 35 V Valine
1 13 28 T Threonine
2 8 20 Y Tyrosine
1 16 32 N Asparagine
0 3 13 C Cysteine
1 6 54 Q Glutamine
1 30 64 E Glutamic acid
0 14 36 G Glycine
0 9 7 M Methionine
5 18 81 R Arginine
2 17 28 D Aspartic acid
1 41 63 L Leucine
5 21 33 S Serine
0 3 14 W Tryptophan
2 2 16 H Histidine
1 22 24 I Isoleucine
1 11 42 F Phenylalanine 

5UZL_1|Chain A|O-acyltransferase|Brassica napus (3708)
>2ZII_1|Chains A, B, C, D|Vacuolar protein sorting-associated protein 74|Saccharomyces cerevisiae (4932)
>6XMF_1|Chain A|CRISPR-Cas|metagenome (256318)
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 \(w=f(c)\)
5UZL , Knot 23 33 0.81 32 31 31 
NVDVRYTYRPSVPAHRRVRESPLSSDAIFKQSH
2ZII , Knot 126 288 0.82 40 170 273 
GSDNINIPTLTLMEEVLLMGLRDREGYLSFWNDSISYALRGCIIIELALRGKIRILDDSARKRFDLSERLIEVIDSSKTGEVLLDETLQLMKNDEPLSISNWIDLLSGETWNLLKINYQLKQVRERLAKGLVDKGVLRTEMKNFFLFDMATHPIADASCKEAIKRRVLSVLVSRNMELSYNEYFPETTSFKIIRTLALICGSYGANVLENVLTTLEYEKRDKAISRAEEIMAQFSQYPFDLEKETELGVSVNLNKEVKEEIENNPGHDLQLEVIAGVFEVFSRMDMLL
6XMF , Knot 286 767 0.82 40 287 699 
MAQASSTPAVSPRPRPRYREERTLVRKLLPRPGQSKQEFRENVKKLRKAFLQFNADVSGVCQWAIQFRPRYGKPAEPTETFWKFFLEPETSLPPNDSRSPEFRRLQAFEAAAGINGAAALDDPAFTNELRDSILAVASRPKTKEAQRLFSRLKDYQPAHRMILAKVAAEWIESRYRRAHQNWERNYEEWKKEKQEWEQNHPELTPEIREAFNQIFQQLEVKEKRVRICPAARLLQNKDNCQYAGKNKHSVLCNQFNEFKKNHLQGKAIKFFYKDAEKYLRCGLQSLKPNVQGPFREDWNKYLRYMNLKEETLRGKNGGRLPHCKNLGQECEFNPHTALCKQYQQQLSSRPDLVQHDELYRKWRREYWREPRKPVFRYPSVKRHSIAKIFGENYFQADFKNSVVGLRLDSMPAGQYLEFAFAPWPRNYRPQPGETEISSVHLHFVGTRPRIGFRFRVPHKRSRFDCTQEELDELRSRTFPRKAQDQKFLEAARKRLLETFPGNAEQELRLLAVDLGTDSARAAFFIGKTFQQAFPLKIVKIEKLYEQWPNQKQAGDRRDASSKQPRPGLSRDHVGRHLQKMRAQASEIAQKRQELTGTPAPETTTDQAAKKATLQPFDLRGLTVHTARMIRDWARLNARQIIQLAEENQVDLIVLESLRGFRPPGYENLDQEKKRRVAFFAHGRIRRKVTEKAVERGMRVVTVPYLASSKVCAECRKKQKDNKQWEKNKKRGLFKCEGCGSQAQVDENAARVLGRVFWGEIELPTAIP

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 \(c\).

\(|P_{f(5UZL_1)}(2) \setminus P_{f(2ZII_1)}(2)|=16\), \(|P_{f(2ZII_1)}(2) \setminus P_{f(5UZL_1)}(2)|=155\). 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 1:010100000101110001000110001110000
Pair \(Z_2\) Length of longest common subsequence
5UZL_1,2ZII_1 171 3
5UZL_1,6XMF_1 264 4
2ZII_1,6XMF_1 177 4

Newick tree

 
[
	6XMF_1:12.00,
	[
		5UZL_1:85.5,2ZII_1:85.5
	]:34.50
]

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 \((0.85)(0.8)(\frac{321 }{\log_{20} 321}-\frac{33}{\log_{20}33})=94.0\)
Status Protein1 Protein2 d d1/2
Query variables 5UZL_1 2ZII_1 118 65
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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