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Parikh vectors
6IMG_1 3HDC_1 1MMU_1 Letter Amino acid
1 7 21 E Glutamic acid
0 6 18 H Histidine
0 11 25 K Lycine
0 7 15 F Phenylalanine
0 5 10 Y Tyrosine
0 12 16 A Alanine
1 12 8 R Arginine
1 11 31 G Glycine
2 5 24 I Isoleucine
0 5 24 T Threonine
0 6 16 N Asparagine
0 5 16 Q Glutamine
3 11 10 P Proline
1 11 24 S Serine
0 14 21 V Valine
0 8 29 D Aspartic acid
4 2 1 C Cysteine
1 14 32 L Leucine
0 3 3 M Methionine
2 3 3 W Tryptophan 

6IMG_1|Chain A|(ACE)-GLY-CYS-PRO-CYS-ILE-TRP-PRO-GLU-LEU-CYS-PRO-TRP-ILE-ARG-SER-CYS-(NH2)|Enterobacteria phage M13 (1977402)
>3HDC_1|Chain A|Thioredoxin family protein|Geobacter metallireducens GS-15 (269799)
>1MMU_1|Chains A, B|Aldose 1-epimerase|Lactococcus lactis (1358)
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 \(w=f(c)\)
6IMG , Knot 12 18 0.64 20 16 16 
XGCPCIWPELCPWIRSCX
3HDC , Knot 74 158 0.79 40 113 148 
MSLAPGKAESDAPLVRTGALAPNFKLPTLSGENKSLAQYRGKIVLVNFWASWCPYCRDEMPSMDRLVKSFPKGDLVVLAVNVEKRFPEKYRRAPVSFNFLSDATGQVQQRYGANRLPDTFIVDRKGIIRQRVTGGIEWDAPKVVSYLKSLEGHHHHHH
1MMU , Knot 149 347 0.83 40 199 329 
MSIKIRDFGLGSDLISLTNKAGVTISFTNLGARIVDWQKDGKHLILGFDSAKEYLEKDAYPGATVGPTAGRIKDGLVKISGKDYILNQNEGPQTLHGGEESIHTKLWTYEVTDLGAEVQVKFSLVSNDGTNGYPGKIEMSVTHSFDDDNKWKIHYEAISDKDTVFNPTGHVYFNLNGDASESVENHGLRLAASRFVPLKDQTEIVRGDIVDIKNTDLDFRQEKQLSNAFNSNMEQVQLVKGIDHPFLLDQLGLDKEQARLTLDDTSISVFTDQPSIVIFTANFGDLGTLYHEKKQVHHGGITFECQVSPGSEQIPELGDISLKAGEKYQATTIYSLHTKLEHHHHHH

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 \(c\).

\(|P_{f(6IMG_1)}(2) \setminus P_{f(3HDC_1)}(2)|=13\), \(|P_{f(3HDC_1)}(2) \setminus P_{f(6IMG_1)}(2)|=110\). 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 1:010101110101110000
Pair \(Z_2\) Length of longest common subsequence
6IMG_1,3HDC_1 123 2
6IMG_1,1MMU_1 209 3
3HDC_1,1MMU_1 184 6

Newick tree

 
[
	1MMU_1:10.99,
	[
		6IMG_1:61.5,3HDC_1:61.5
	]:46.49
]

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 \((0.85)(0.8)(\frac{176 }{\log_{20} 176}-\frac{18}{\log_{20}18})=56.6\)
Status Protein1 Protein2 d d1/2
Query variables 6IMG_1 3HDC_1 70 38.5
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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