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Parikh vectors
4ASO_1 6GIJ_1 1WJU_1 Letter Amino acid
8 1 5 N Asparagine
5 0 2 D Aspartic acid
5 0 2 F Phenylalanine
7 1 5 V Valine
8 1 7 I Isoleucine
5 3 10 K Lycine
6 0 7 T Threonine
10 1 4 A Alanine
5 0 6 R Arginine
3 0 0 C Cysteine
4 0 5 Q Glutamine
7 6 12 L Leucine
4 1 9 S Serine
5 0 2 Y Tyrosine
1 1 3 P Proline
0 0 0 W Tryptophan
9 0 8 E Glutamic acid
5 0 10 G Glycine
1 0 2 H Histidine
6 0 1 M Methionine 

4ASO_1|Chains A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P|TUBR FROM BACILLUS THURINGIENSIS PBTOXIS|BACILLUS THURINGIENSIS (1428)
>6GIJ_1|Chain A|temporinB_KKG6A|synthetic construct (32630)
>1WJU_1|Chain A|NEDD8 ultimate buster-1|Homo sapiens (9606)
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 \(w=f(c)\)
4ASO , Knot 55 104 0.81 38 89 100 
MNRDHFYTLNIAEIAERIGNDDCAYQVLMAFINENGEAQMLNKTAVAEMIQLSKPTVFATVNSFYCAGYIDETRVGRSKIYTLSDLGVEIVECFKQKAMEMRNL
6GIJ , Knot 10 15 0.60 16 12 13 
KKLLPIVANLLKSLL
1WJU , Knot 53 100 0.81 36 78 94 
GSSGSSGDNYRTTGIATIEVFLPPRLKKDRKNLLETRLHITGRELRSKIAETFGLQENYIKIVINKKQLQLGKTLEEQGVAHNVKAMVLELKQSSGPSSG

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 \(c\).

\(|P_{f(4ASO_1)}(2) \setminus P_{f(6GIJ_1)}(2)|=86\), \(|P_{f(6GIJ_1)}(2) \setminus P_{f(4ASO_1)}(2)|=9\). 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 1:10000100101101100110000100111111000101011000111011010010111010010011010000110001001001110110010001101001
Pair \(Z_2\) Length of longest common subsequence
4ASO_1,6GIJ_1 95 2
4ASO_1,1WJU_1 115 4
6GIJ_1,1WJU_1 76 3

Newick tree

 
[
	4ASO_1:56.80,
	[
		6GIJ_1:38,1WJU_1:38
	]:18.80
]

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 \((0.85)(0.8)(\frac{119 }{\log_{20} 119}-\frac{15}{\log_{20}15})=39.4\)
Status Protein1 Protein2 d d1/2
Query variables 4ASO_1 6GIJ_1 51 29
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]