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Parikh vectors
4UZC_1 6VCL_1 2VYV_1 Letter Amino acid
7 5 8 Y Tyrosine
16 14 9 P Proline
3 1 3 W Tryptophan
8 20 12 R Arginine
3 11 18 N Asparagine
3 9 6 H Histidine
5 11 20 I Isoleucine
9 21 20 L Leucine
1 9 8 M Methionine
5 6 3 C Cysteine
12 10 5 Q Glutamine
1 18 15 E Glutamic acid
11 26 30 G Glycine
13 7 27 K Lycine
7 14 11 F Phenylalanine
8 9 15 S Serine
3 11 27 T Threonine
9 26 35 A Alanine
7 16 25 D Aspartic acid
8 31 34 V Valine 

4UZC_1|Chains A, B, C, D|ORF 73|HUMAN HERPESVIRUS 8 (37296)
>6VCL_1|Chain A|Diaminopimelate epimerase|Escherichia coli (strain K12) (83333)
>2VYV_1|Chains A, B, C|GLYCERALDEHYDE-3-PHOSPHATE DEHYDROGENASE|ESCHERICHIA COLI BL21(DE3) (469008)
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 \(w=f(c)\)
4UZC , Knot 69 139 0.81 40 111 137 
GSRYQQPPVPYRQIDDCPAKARPQHIFYRRFLGKDGRRDPKCQWKFAVIFWGNDPYGLKKLSQAFQFGGVKAGPVSCLPHPGPDQSPITYCVYVYCQNKDTSKKVQMARLAWEASHPLAGNLQSSIVKFKKPLPLTQPG
6VCL , Knot 125 275 0.85 40 178 268 
SMQFSKMHGLGNDFMVVDAVTQNVFFSPELIRRLADAHLGVGFDQLLVVEPPYDPELDFHYRIFNADGSEVAQCGNGARCFARFVRLKGLTNKRDIRVSTANGRMVLTVTDDDLVRVNMGEPNFEPSAVPFRANKAEKTYIMRAAEQTILCGVVSMGNPHCVIQVDDVDTAAVETLGPVLESHERFPERANIGFMQVVKREHIRLRVYERGAGETQACGSGACAAVAVGIQQGLLAEEVRVELPGGRLDIAWKGPGHPLYMTGPAVHVADGFIHL
2VYV , Knot 144 331 0.84 40 191 317 
MTIKVGINGFGRIGRIVFRAAQKRSDIEIVAINDLLDADYMAYMLKYDSTHGRFDGTVEVKDGHLIVNGKKIRVTAERDPANLKWDEVGVDVVAEATGLFLTDETARKHITAGAKKVVMTGPSKDNTPMFVKGANFDKYAGQDIVSNASCTTNCLAPLAKVINDNFGIIEGLMTTVHATTATQKTVDGPSHKDWRGGRGASQNIIPSSTGAAKAVGKVLPELNGKLTGMAFRVPTPNVSVVDLTVRLEKAATYEQIKAAVKAAAEGEMKGVLGYTEDDVVSTDFNGEVCTSVFDAKAGIALNDNFVKLVSWYDNETGYSNKVLDLIAHISK

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 \(c\).

\(|P_{f(4UZC_1)}(2) \setminus P_{f(6VCL_1)}(2)|=55\), \(|P_{f(6VCL_1)}(2) \setminus P_{f(4UZC_1)}(2)|=122\). 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 1:1000001111000100011010100110001110010001000101111111001011001001101111011110011011100011000101000000000010110111010011110100011010011110011
Pair \(Z_2\) Length of longest common subsequence
4UZC_1,6VCL_1 177 3
4UZC_1,2VYV_1 192 3
6VCL_1,2VYV_1 155 4

Newick tree

 
[
	4UZC_1:96.76,
	[
		6VCL_1:77.5,2VYV_1:77.5
	]:19.26
]

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 \((0.85)(0.8)(\frac{414 }{\log_{20} 414}-\frac{139}{\log_{20}139})=82.5\)
Status Protein1 Protein2 d d1/2
Query variables 4UZC_1 6VCL_1 108 79.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]