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Parikh vectors
6JSB_1 7PQY_1 4CCM_1 Letter Amino acid
10 14 21 F Phenylalanine
7 10 31 P Proline
8 15 24 S Serine
3 2 7 W Tryptophan
5 15 19 Y Tyrosine
1 8 5 C Cysteine
17 7 34 E Glutamic acid
6 7 10 I Isoleucine
15 13 63 L Leucine
26 12 21 T Threonine
13 12 40 A Alanine
10 17 13 N Asparagine
10 7 13 H Histidine
18 16 31 V Valine
5 9 38 R Arginine
4 6 25 Q Glutamine
8 10 7 K Lycine
14 9 27 D Aspartic acid
21 16 27 G Glycine
2 0 11 M Methionine 

6JSB_1|Chains A, B|Hypothetical membrane protein|Corynebacterium glutamicum (strain ATCC 13032 / DSM 20300 / JCM 1318 / LMG 3730 / NCIMB 10025) (196627)
>7PQY_1|Chains A, B[auth E]|Spike protein S1|Severe acute respiratory syndrome coronavirus 2 (2697049)
>4CCM_1|Chains A, B|BIFUNCTIONAL LYSINE-SPECIFIC DEMETHYLASE AND HISTIDYL-HYDROXYLASE NO66|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 \(w=f(c)\)
6JSB , Knot 92 203 0.80 40 135 194 
MKCRVVTTTGTADWSVRESFNNYLEGPIANGAAYKYHGGIEVRDGVETTGTKSAREFTWPVLGSEEGAVKLGGGVHWTGHNHYSGDDESQAPDNFILDLDFSNPTVKFDGNEGTLLVDFKSREFVDTKTVADFLTGTQAELATITFDEPIDLTQENVTVTGQTKLTATGVDVMGTFYPEGEALAPITLNLTNEVVLEHHHHHH
7PQY , Knot 94 205 0.81 38 142 195 
ETGHHHHHHTNLCPFGEVFNATRFASVYAWNRKRISNCVADYSVLYNSASFSTFKCYGVSPTKLNDLCFTNVYADSFVIRGDEVRQIAPGQTGKIADYNYKLPDDFTGCVIAWNSNNLDSKVGGNYNYLYRLFRKSNLKPFERDISTEIYQAGSTPCNGVEGFNCYFPLQSYGFQPTNGVGYQPYRVVVLSFELLHAPATVCGKK
4CCM , Knot 189 467 0.83 40 226 437 
MSPLRRVLAELNRIPSSRRRAARLFEWLIAPMPPDHFYRRLWEREAVLVRRQDHTYYQGLFSTADLDSMLRNEEVQFGQHLDAARYINGRRETLNPPGRALPAAAWSLYQAGCSLRLLCPQAFSTTVWQFLAVLQEQFGSMAGSNVYLTPPNSQGFAPHYDDIEAFVLQLEGRKLWRVYRPRVPTEELALTSSPNFSQDDLGEPVLQTVLEPGDLLYFPRGFIHQAECQDGVHSLHLTLSTYQRNTWGDFLEAILPLAVQAAMEENVEFRRGLPRDFMDYMGAQHSDSKDPRRTAFMEKVRVLVARLGHFAPVDAVADQRAKDFIHDSLPPVLTDRERALSVYGLPIRWEAGEPVNVGAQLTTETEVHMLQDGIARLVGEGGHLFLYYTVENSRVYHLEEPKCLEIYPQQADAMELLLGSYPEFVRVGDLPCDSVEDQLSLATTLYDKGLLLTKMPLALNAENLYFQ

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 \(c\).

\(|P_{f(6JSB_1)}(2) \setminus P_{f(7PQY_1)}(2)|=75\), \(|P_{f(7PQY_1)}(2) \setminus P_{f(6JSB_1)}(2)|=82\). 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 1:10001100010101010001000101111011100001110100110001000100101111100011101111101010000010000011001110101001010101001011101000011000011011010010110101001101000010101000101011011101010101111101010001110000000
Pair \(Z_2\) Length of longest common subsequence
6JSB_1,7PQY_1 157 6
6JSB_1,4CCM_1 187 3
7PQY_1,4CCM_1 180 4

Newick tree

 
[
	4CCM_1:95.78,
	[
		6JSB_1:78.5,7PQY_1:78.5
	]:17.28
]

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 \((0.85)(0.8)(\frac{408 }{\log_{20} 408}-\frac{203}{\log_{20}203})=60.4\)
Status Protein1 Protein2 d d1/2
Query variables 6JSB_1 7PQY_1 77 75
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]