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Parikh vectors
1JKS_1 5MOC_1 6GBG_1 Letter Amino acid
15 14 10 G Glycine
22 10 5 I Isoleucine
18 12 9 V Valine
10 10 6 Y Tyrosine
13 13 3 R Arginine
17 14 3 D Aspartic acid
17 5 4 F Phenylalanine
29 29 7 E Glutamic acid
29 24 10 L Leucine
18 19 5 S Serine
3 2 1 W Tryptophan
17 24 6 A Alanine
12 7 8 N Asparagine
12 7 10 Q Glutamine
3 9 2 M Methionine
12 5 6 P Proline
15 10 9 T Threonine
1 2 0 C Cysteine
7 3 8 H Histidine
24 17 3 K Lycine 

1JKS_1|Chain A|DEATH-ASSOCIATED PROTEIN KINASE|Homo sapiens (9606)
>5MOC_1|Chain A|14-3-3 protein sigma|Homo sapiens (9606)
>6GBG_1|Chain A[auth D]|Carcinoembryonic antigen-related cell adhesion molecule 1|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 \(w=f(c)\)
1JKS , Knot 124 294 0.80 40 179 284 
TVFRQENVDDYYDTGEELGSGQFAVVKKCREKSTGLQYAAKFIKKRRTKSSRRGVSREDIEREVSILKEIQHPNVITLHEVYENKTDVILILELVAGGELFDFLAEKESLTEEEATEFLKQILNGVYYLHSLQIAHFDLKPENIMLLDRNVPKPRIKIIDFGLAHKIDFGNEFKNIFGTPEFVAPEIVNYEPLGLEADMWSIGVITYILLSGASPFLGDTKQETLANVSAVNYEFEDEYFSNTSALAKDFIRRLLVKDPKKRMTIQDSLQHPWIKPKDTQQALSSAWSHPQFEK
5MOC , Knot 108 236 0.83 40 164 229 
GAMGSMERASLIQKAKLAEQAERYEDMAAFMKGAVEKGEELSCEERNLLSVAYKNVVGGQRAAWRVLSSIEQKSNEEGSEEKGPEVREYREKVETELQGVCDTVLGLLDSHLIKEAGDAESRVFYLKMKGDYYRYLAEVATGDDKKRIIDSARSAYQEAMDISKKEMPPTNPIRLGLALNFSVFHYEIANSPEEAISLAKTTFDEAMADLHTLSEDSYKDSTLIMQLLRDNLTLWT
6GBG , Knot 60 115 0.82 38 98 110 
MQLTTESMPFNVAEGKEVLLLVHNLPQQLFGYSWYKGERVDGNRQIVGYAIGTQQATPGPANSGRETIYPNASLLIQNVTQNDTGFYTLQVIKSDLVNEEATGQFHVYPHHHHHH

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 \(c\).

\(|P_{f(1JKS_1)}(2) \setminus P_{f(5MOC_1)}(2)|=85\), \(|P_{f(5MOC_1)}(2) \setminus P_{f(1JKS_1)}(2)|=70\). 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 1:011000010000001001101011110000000011001101100000000001100001000101100100101101001000000111110111110110111000010000100110011011001001011010101001111000110101011011110010110010011101011110110001111010110111100111011011110000001101011000100001000011100110011100100010100010011101000001100110010100
Pair \(Z_2\) Length of longest common subsequence
1JKS_1,5MOC_1 155 4
1JKS_1,6GBG_1 169 4
5MOC_1,6GBG_1 166 3

Newick tree

 
[
	6GBG_1:85.73,
	[
		1JKS_1:77.5,5MOC_1:77.5
	]:8.23
]

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 \((0.85)(0.8)(\frac{530 }{\log_{20} 530}-\frac{236}{\log_{20}236})=84.1\)
Status Protein1 Protein2 d d1/2
Query variables 1JKS_1 5MOC_1 105 95
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]