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Parikh vectors
5CMS_1 4OPY_1 4TTL_1 Letter Amino acid
6 4 1 Y Tyrosine
8 16 1 R Arginine
3 2 4 C Cysteine
8 15 1 I Isoleucine
5 11 0 K Lycine
3 5 0 F Phenylalanine
6 7 1 N Asparagine
10 16 2 D Aspartic acid
2 9 0 M Methionine
10 20 0 T Threonine
1 4 0 W Tryptophan
10 8 0 Q Glutamine
9 20 1 E Glutamic acid
12 6 1 H Histidine
19 14 0 V Valine
25 25 2 A Alanine
16 23 5 G Glycine
16 30 0 L Leucine
8 12 2 P Proline
6 16 1 S Serine 

5CMS_1|Chains A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R|O-acetyl-ADP-ribose deacetylase|Escherichia coli (562)
>4OPY_1|Chain A|Extended spectrum beta-lactamase TEM-63|Escherichia coli (562)
>4TTL_1|Chain A|Alpha-conotoxin Vc1A|Conus victoriae (319920)
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 \(w=f(c)\)
5CMS , Knot 84 183 0.79 40 126 171 
HHHHHHMKTRIHVVQGDITKLAVDVIVNAANPSLMGGGGVDGAIHRAAGPALLDACLKVRQQQGDCPTGHAVITLAGDLPAKAVVHTVGPVWRGGEQNEDQLLQDAYLNSLRLVAANSYTSVAFPAISTGVAGYPRAAAAEIAVKTVSEFITRHALPEQVYFVCYDEENAHLYERLLTQQGDE
4OPY , Knot 116 263 0.82 40 172 253 
HPETLVKVKDAEDQLGGRVGYIELDLASGKILESFRPEERFPMMSTFKVLLCGAVLSRVDAGQEQLGRRIHYSQNDLVEYSPVTEKHLTDGMTVGELCSAAITMSDNTAANLLLTTIGGPKELTAFLHNMGDHVTRLDSWEPELNEAIPNDERDTTTPVAMATTLRKLLTGELLTAASRQQLIDWMEADKVAGPLLRSALPAGWFIADKSGAGERGSRGIIAALGPDGKPSRIVVIYMTGSQATMDERNRQIAEIGASLIKHW
4TTL , Knot 15 22 0.70 24 20 20 
GCCSDPRCNYDHPEICGGAAGG

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 \(c\).

\(|P_{f(5CMS_1)}(2) \setminus P_{f(4OPY_1)}(2)|=59\), \(|P_{f(4OPY_1)}(2) \setminus P_{f(5CMS_1)}(2)|=105\). 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 1:000000100010110101001110111011010111111101110011111110101010000100101011101110111011100111110110000001100101001011110000011111100111101011110111001001100011100101100000010100011000100
Pair \(Z_2\) Length of longest common subsequence
5CMS_1,4OPY_1 164 4
5CMS_1,4TTL_1 130 3
4OPY_1,4TTL_1 170 3

Newick tree

 
[
	4OPY_1:88.83,
	[
		5CMS_1:65,4TTL_1:65
	]:23.83
]

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 \((0.85)(0.8)(\frac{446 }{\log_{20} 446}-\frac{183}{\log_{20}183})=77.3\)
Status Protein1 Protein2 d d1/2
Query variables 5CMS_1 4OPY_1 98 81.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]