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Parikh vectors
3BXQ_1 6JBU_1 7BWX_1 Letter Amino acid
4 3 0 C Cysteine
2 16 11 I Isoleucine
2 17 20 L Leucine
0 15 5 K Lycine
0 14 3 P Proline
1 17 27 G Glycine
0 6 2 W Tryptophan
2 22 4 Y Tyrosine
2 11 1 E Glutamic acid
0 12 8 H Histidine
0 2 1 M Methionine
0 19 34 A Alanine
0 12 2 R Arginine
2 17 12 N Asparagine
0 26 11 D Aspartic acid
2 7 8 Q Glutamine
0 9 9 F Phenylalanine
1 17 19 V Valine
2 17 19 S Serine
1 17 26 T Threonine 

3BXQ_1|Chains A, C|insulin A chain|null (9606)
>6JBU_1|Chain A|Adhesion G protein-coupled receptor L3|Mus musculus (10090)
>7BWX_1|Chains A, B, C, D, E, F|Ice-binding protein isoform1a|Antarctomyces psychrotrophicus (89416)
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 \(w=f(c)\)
3BXQ , Knot 15 21 0.72 22 20 19 
GIVEQCCTSICSLYQLENYCN
6JBU , Knot 122 276 0.82 40 188 263 
APGDPRECPGLLKGVYQSEHLFESDHQSGAWCKDPLQASDKIYYMPWTPYRTDTLTEYSSKDDFIAGRPTTTYKLPHRVDGTGFVVYDGALFFNKERTRNIVKFDLRTRIKSGEAIIANANYHDTSPYRWGGKSDIDLAVDENGLWVIYATEQNNGKIVISQLNPYTLRIEGTWDTAYDKRSASNAFMICGILYVVKSVYEDDDNEATGNKIDYIYNTDQSKDSLVDVPFPNSYQYIAAVDYNPRDNLLYVWNNYHVVKYSLDFGPAAAHHHHHHH
7BWX , Knot 91 222 0.73 38 119 198 
HHHHHHTGLDLGAASSFGALAPQGVANAGATVINGDMGTTGTSITGFPPGLITGQLHINDDTSTQAFADSRTAFVAGQALIATVDQAGTATLGGNTFVAGVYKYDSAVGLDGVLTLDGAGDASSVWVFQLATTLVTYASSSIILTNGAKANNVFWIVGSSATLGTYSHLEGNVIANALIAAQTGATINGALLAGSAVTLDSNTVTVQNSASKLVRSAKFFKV

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 \(c\).

\(|P_{f(3BXQ_1)}(2) \setminus P_{f(6JBU_1)}(2)|=9\), \(|P_{f(6JBU_1)}(2) \setminus P_{f(3BXQ_1)}(2)|=177\). 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 1:111000000100100100000
Pair \(Z_2\) Length of longest common subsequence
3BXQ_1,6JBU_1 186 2
3BXQ_1,7BWX_1 129 3
6JBU_1,7BWX_1 161 6

Newick tree

 
[
	6JBU_1:93.27,
	[
		3BXQ_1:64.5,7BWX_1:64.5
	]:28.77
]

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 \((0.85)(0.8)(\frac{297 }{\log_{20} 297}-\frac{21}{\log_{20}21})=92.2\)
Status Protein1 Protein2 d d1/2
Query variables 3BXQ_1 6JBU_1 115 62
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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