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Parikh vectors
3GJP_1 3BEX_1 6MSP_1 Letter Amino acid
3 9 9 R Arginine
2 10 5 N Asparagine
1 15 2 D Aspartic acid
5 17 5 K Lycine
0 14 4 T Threonine
0 3 1 W Tryptophan
2 26 6 G Glycine
1 7 2 M Methionine
0 9 0 P Proline
5 17 9 L Leucine
0 14 3 F Phenylalanine
2 12 5 S Serine
1 7 1 Y Tyrosine
1 11 0 A Alanine
0 2 0 C Cysteine
1 4 3 Q Glutamine
6 15 19 E Glutamic acid
1 7 10 H Histidine
2 18 6 I Isoleucine
2 32 7 V Valine 

3GJP_1|Chains A, B, C|General control protein GCN4|Saccharomyces cerevisiae (4932)
>3BEX_1|Chains A, B, C, D, E, F|Type III pantothenate kinase|Thermotoga maritima (2336)
>6MSP_1|Chain A|De novo Designed Protein Foldit3|synthetic construct (32630)
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 \(w=f(c)\)
3GJP , Knot 24 35 0.81 30 33 33 
GSRMKQLEDKVEELLSKNYHLENEIARIKKLVGER
3BEX , Knot 112 249 0.82 40 165 239 
MDPMYLLVDVGNTHSVFSITEDGKTFRRWRLSTGVFQTEDELFSHLHPLLGDAMREIKGIGVASVVPTQNTVIERFSQKYFHISPIWVKAKNGCVKWNVKNPSEVGADRVANVVAFVKEYGKNGIIIDMGTATTVDLVVNGSYEGGAILPGFFMMVHSLFRGTAKLPLVEVKPADFVVGKDTEENIRLGVVNGSVYALEGIIGRIKEVYGDLPVVLTGGQSKIVKDMIKHEIFDEDLTIKGVYHFCFGD
6MSP , Knot 49 97 0.77 34 73 88 
MGHHHHHHENLYFQSHMTDELLERLRQLFEELHERGTEIVVEVHINGERDEIRVRNISKEELKKLLERIREKIEREGSSEVEVNVHSGGQTWTFNEK

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 \(c\).

\(|P_{f(3GJP_1)}(2) \setminus P_{f(3BEX_1)}(2)|=13\), \(|P_{f(3BEX_1)}(2) \setminus P_{f(3GJP_1)}(2)|=145\). 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 1:10010010001001100000100011010011100
Pair \(Z_2\) Length of longest common subsequence
3GJP_1,3BEX_1 158 3
3GJP_1,6MSP_1 74 3
3BEX_1,6MSP_1 142 3

Newick tree

 
[
	3BEX_1:84.05,
	[
		3GJP_1:37,6MSP_1:37
	]:47.05
]

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 \((0.85)(0.8)(\frac{284 }{\log_{20} 284}-\frac{35}{\log_{20}35})=82.3\)
Status Protein1 Protein2 d d1/2
Query variables 3GJP_1 3BEX_1 102 57.5
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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