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Parikh vectors
6CFH_1 8DVI_1 1QLK_1 Letter Amino acid
2 23 4 G Glycine
0 31 3 I Isoleucine
3 15 5 M Methionine
0 13 0 P Proline
0 17 1 R Arginine
0 4 2 C Cysteine
0 39 17 E Glutamic acid
0 38 8 K Lycine
2 36 5 S Serine
0 19 1 Y Tyrosine
1 42 4 A Alanine
0 26 7 D Aspartic acid
1 21 3 Q Glutamine
1 41 8 L Leucine
0 24 3 T Threonine
1 9 0 W Tryptophan
0 31 7 V Valine
0 26 2 N Asparagine
0 7 5 H Histidine
0 13 7 F Phenylalanine 

6CFH_1|Chains A, B|TAR DNA-binding protein 43|Homo sapiens (9606)
>8DVI_1|Chains A, B, C, D, E, F|DnaB-like replicative helicase|Escherichia phage T4 (2681598)
>1QLK_1|Chains A, B|S-100 PROTEIN|Rattus norvegicus (10116)
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 \(w=f(c)\)
6CFH , Knot 8 11 0.58 14 9 9 
SWGMMGMLASQ
8DVI , Knot 194 475 0.84 40 241 447 
MVEIILSHLIFDQAYFSKVWPYMDSEYFESGPAKNTFKLIKSHVNEYHSVPSINALNVALENSSFTETEYSGVKTLISKLADSPEDHSWLVKETEKYVQQRAMFNATSKIIEIQTNAELPPEKRNKKMPDVGAIPDIMRQALSISFDSYVGHDWMDDYEARWLSYMNKARKVPFKLRILNKITKGGAETGTLNVLMAGVNVGKSLGLCSLAADYLQLGHNVLYISMEMAEEVCAKRIDANMLDVSLDDIDDGHISYAEYKGKMEKWREKSTLGRLIVKQYPTGGADANTFRSLLNELKLKKNFVPTIIIVDYLGICKSCRIRVYSENSYTTVKAIAEELRALAVETETVLWTAAQVGKQAWDSSDVNMSDIAESAGLPATADFMLAVIETEELAAAEQQLIKQIKSRYGDKNKWNKFLMGVQKGNQKWVEIEQDSTPTEVNEVAGSQQIQAEQNRYQRNESTRAQLDALANELKF
1QLK , Knot 50 92 0.82 36 76 89 
MSELEKAMVALIDVFHQYSGREGDKHKLKKSELKELINNELSHFLEEIKEQEVVDKVMETLDEDGDGECDFQEFMAFVSMVTTACHEFFEHE

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 \(c\).

\(|P_{f(6CFH_1)}(2) \setminus P_{f(8DVI_1)}(2)|=4\), \(|P_{f(8DVI_1)}(2) \setminus P_{f(6CFH_1)}(2)|=236\). 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 1:01111111100
Pair \(Z_2\) Length of longest common subsequence
6CFH_1,8DVI_1 240 3
6CFH_1,1QLK_1 85 1
8DVI_1,1QLK_1 213 4

Newick tree

 
[
	8DVI_1:12.68,
	[
		6CFH_1:42.5,1QLK_1:42.5
	]:86.18
]

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 \((0.85)(0.8)(\frac{486 }{\log_{20} 486}-\frac{11}{\log_{20}11})=150.\)
Status Protein1 Protein2 d d1/2
Query variables 6CFH_1 8DVI_1 190 97.5
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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