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Parikh vectors
1UCJ_1 3GMW_1 6ZKK_1 Letter Amino acid
2 8 5 N Asparagine
5 20 2 E Glutamic acid
5 14 6 V Valine
2 2 3 C Cysteine
5 15 5 I Isoleucine
0 11 0 K Lycine
3 5 4 F Phenylalanine
14 20 6 T Threonine
2 5 2 H Histidine
6 31 23 L Leucine
6 11 1 P Proline
7 15 10 S Serine
0 4 0 W Tryptophan
6 24 7 A Alanine
5 18 1 R Arginine
7 16 1 D Aspartic acid
5 8 2 Q Glutamine
8 21 4 G Glycine
0 9 12 M Methionine
8 4 4 Y Tyrosine 

1UCJ_1|Chains A, B|Guanyl-specific ribonuclease Sa|Streptomyces aureofaciens (1894)
>3GMW_1|Chains A, C|B-lactamase|Escherichia sp. Sflu5 (299586)
>6ZKK_11|Chain K|NADH-ubiquinone oxidoreductase chain 4L|Ovis aries (9940)
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 \(w=f(c)\)
1UCJ , Knot 49 96 0.77 34 80 93 
DVSGTVCLSALPPEATDTLNLIASDGPFPYSQDGVTFQNRESVLPTQSYGYYHEYTVITPGARTRGTRRIITGEATQEDYYTGDHYATFSLIDQTC
3GMW , Knot 115 261 0.81 40 175 254 
ETLVKVKDAEDQLGARVGYIELDLNSGKILESFRPEERFPMMSTFKVLLCGAVLSRVDAGQEQLGRRIHYSQNDLVEYSPVTEKHLTDGMTVRELCSAAITMSDNTAANLLLTTIGGPKELTAFLHNMGDHVTRLDRWEPELNEAIPNDERDTTMPVAMATTLRKLLTGELLTLASRQQLIDWMEADKVAGPLLRSALPAGWFIADKSGAGERGSRGIIAALGPDGKPSRIVVIYTTGSQATMDERNRQIAEIGASLIKHW
6ZKK , Knot 50 98 0.78 36 70 93 
MSLVYMNIMMAFTVSLTGLLMYRSHLMSSLLCLEGMMLSLFILATLMILNSHFTLASMMPIILLVFAACEAALGLSLLVMVSNTYGTDYVQNLNLLQC

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 \(c\).

\(|P_{f(1UCJ_1)}(2) \setminus P_{f(3GMW_1)}(2)|=28\), \(|P_{f(3GMW_1)}(2) \setminus P_{f(1UCJ_1)}(2)|=123\). 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 1:010101010111101000101110011110000110100000111000010000001101110001000110101000000010001010110000
Pair \(Z_2\) Length of longest common subsequence
1UCJ_1,3GMW_1 151 4
1UCJ_1,6ZKK_1 112 3
3GMW_1,6ZKK_1 177 4

Newick tree

 
[
	3GMW_1:89.31,
	[
		1UCJ_1:56,6ZKK_1:56
	]:33.31
]

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 \((0.85)(0.8)(\frac{357 }{\log_{20} 357}-\frac{96}{\log_{20}96})=80.8\)
Status Protein1 Protein2 d d1/2
Query variables 1UCJ_1 3GMW_1 100 67.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]

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