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Parikh vectors
7DSG_1 5TGN_1 1VIY_1 Letter Amino acid
5 3 10 S Serine
4 2 4 Y Tyrosine
1 3 10 N Asparagine
2 0 0 C Cysteine
8 5 21 L Leucine
4 0 6 K Lycine
8 8 13 D Aspartic acid
4 11 14 I Isoleucine
2 2 2 W Tryptophan
10 8 19 V Valine
4 4 15 Q Glutamine
9 3 13 H Histidine
4 5 9 P Proline
5 6 9 T Threonine
2 2 3 M Methionine
0 4 4 F Phenylalanine
13 19 30 A Alanine
6 12 13 R Arginine
12 5 12 E Glutamic acid
5 9 11 G Glycine 

7DSG_1|Chain A|Brucella Abortus PhiA|Brucella abortus biovar 1 (strain 9-941) (262698)
>5TGN_1|Chains A, B, C, D|Uncharacterized protein|Sphaerobacter thermophilus (479434)
>1VIY_1|Chains A, B, C|Dephospho-CoA kinase|Escherichia coli (562)
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 \(w=f(c)\)
7DSG , Knot 53 108 0.76 38 85 103 
MDEEQPPIERVDYICERSVVVPVTYIRSNGAPAAAVLEVEGKMVALQWHGDLKKYVAIDEQDSYRWADRGGQATLSHLEADHTAKEVTLLSACRADTAEELEHHHHHH
5TGN , Knot 57 111 0.80 36 88 108 
SNAMAAIDLAREYISRVNGRDGSGAAALFAQDGEIIAPVGRVYRGWDAIAAFIEAAPPATTAQIAERTMGTHRVVLHGVVQTPRFAPAQIEWIFDVDGDRIRRLTINHLRD
1VIY , Knot 98 218 0.80 38 141 203 
MSLRYIVALTGGIGSGKSTVANAFADLGINVIDADIIARQVVEPGAPALHAIADHFGANMIAADGTLQRRALRERIFANPEEKNWLNALLHPLIQQETQHQIQQATSPYVLWVVPLLVENSLYKKANRVLVVDVSPETQLKRTMQRDDVTREHVEQILAAQATREARLAVADDVIDNNGAPDAIASDVARLHAHYLQLASQFVSQEKPEGGSHHHHHH

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 \(c\).

\(|P_{f(7DSG_1)}(2) \setminus P_{f(5TGN_1)}(2)|=54\), \(|P_{f(5TGN_1)}(2) \setminus P_{f(7DSG_1)}(2)|=57\). 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 1:100001110010010000111110010001111111101010111101010100011100000001100110101001010001001011010010010010000000
Pair \(Z_2\) Length of longest common subsequence
7DSG_1,5TGN_1 111 3
7DSG_1,1VIY_1 146 6
5TGN_1,1VIY_1 141 4

Newick tree

 
[
	1VIY_1:76.41,
	[
		7DSG_1:55.5,5TGN_1:55.5
	]:20.91
]

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 \((0.85)(0.8)(\frac{219 }{\log_{20} 219}-\frac{108}{\log_{20}108})=35.7\)
Status Protein1 Protein2 d d1/2
Query variables 7DSG_1 5TGN_1 45 43
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]