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Parikh vectors
5EFT_1 5ZUH_1 8QSA_1 Letter Amino acid
0 2 14 G Glycine
2 4 24 L Leucine
0 1 2 W Tryptophan
0 0 2 C Cysteine
0 2 9 M Methionine
1 6 5 F Phenylalanine
0 8 5 P Proline
0 2 3 H Histidine
0 4 13 R Arginine
0 1 7 N Asparagine
0 10 29 E Glutamic acid
4 5 19 S Serine
0 10 10 T Threonine
1 11 24 A Alanine
0 5 7 Q Glutamine
1 1 10 I Isoleucine
2 10 17 K Lycine
0 4 10 Y Tyrosine
2 9 12 V Valine
3 4 14 D Aspartic acid 

5EFT_1|Chains A, C, E, G|p9-1|Rice black-streaked dwarf virus 2 (566568)
>5ZUH_1|Chain A|RNA binding protein La-like protein|Trypanosoma brucei (5691)
>8QSA_1|Chain A|14-3-3 protein sigma|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 \(w=f(c)\)
5EFT , Knot 12 16 0.69 16 15 14 
ADSVIKSDFSSLKLDV
5ZUH , Knot 52 99 0.80 38 81 95 
TDHQTVYVKPVPPTATLEQLTEFFSKHGTVQAVWRRYFAGKKDAPPESRTKPSVFVVFNSSEEAEAFQKAPPMYDDVQLTAEMKTTYLERKAEEIAAKK
8QSA , Knot 108 236 0.83 40 164 229 
GAMGSMERASLIQKAKLAEQAERYEDMAAFMKGAVEKGEELSCEERNLLSVAYKNVVGGQRAAWRVLSSIEQKSNEEGSEEKGPEVREYREKVETELQGVCDTVLGLLDSHLIKEAGDAESRVFYLKMKGDYYRYLAEVATGDDKKRIIDSARSAYQEAMDISKKEMPPTNPIRLGLALNFSVFHYEIANSPEEAISLAKTTFDEAMADLHTLSEDSYKDSTLIMQLLRDNLTLWT

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 \(c\).

\(|P_{f(5EFT_1)}(2) \setminus P_{f(5ZUH_1)}(2)|=11\), \(|P_{f(5ZUH_1)}(2) \setminus P_{f(5EFT_1)}(2)|=77\). 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 1:1001100010010101
Pair \(Z_2\) Length of longest common subsequence
5EFT_1,5ZUH_1 88 2
5EFT_1,8QSA_1 157 2
5ZUH_1,8QSA_1 155 3

Newick tree

 
[
	8QSA_1:86.41,
	[
		5EFT_1:44,5ZUH_1:44
	]:42.41
]

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 \((0.85)(0.8)(\frac{115 }{\log_{20} 115}-\frac{16}{\log_{20}16})=37.6\)
Status Protein1 Protein2 d d1/2
Query variables 5EFT_1 5ZUH_1 48 27.5
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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