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Parikh vectors
1JKK_1 4FVN_1 6COP_1 Letter Amino acid
17 13 1 A Alanine
17 19 1 D Aspartic acid
24 24 3 K Lycine
3 2 1 M Methionine
17 12 3 F Phenylalanine
18 18 1 S Serine
15 12 0 T Threonine
13 7 2 R Arginine
12 10 0 N Asparagine
15 22 0 G Glycine
10 8 0 Y Tyrosine
29 26 2 L Leucine
12 17 0 P Proline
12 11 1 Q Glutamine
29 13 1 E Glutamic acid
22 9 1 I Isoleucine
18 17 0 V Valine
1 1 0 C Cysteine
7 12 0 H Histidine
3 7 0 W Tryptophan 

1JKK_1|Chain A|DEATH-ASSOCIATED PROTEIN KINASE|Homo sapiens (9606)
>4FVN_1|Chain A|Carbonic anhydrase 2|Homo sapiens (9606)
>6COP_1|Chain A|Competence-stimulating peptide type 1|Streptococcus pneumoniae (1313)
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 \(w=f(c)\)
1JKK , Knot 124 294 0.80 40 179 284 
TVFRQENVDDYYDTGEELGSGQFAVVKKCREKSTGLQYAAKFIKKRRTKSSRRGVSREDIEREVSILKEIQHPNVITLHEVYENKTDVILILELVAGGELFDFLAEKESLTEEEATEFLKQILNGVYYLHSLQIAHFDLKPENIMLLDRNVPKPRIKIIDFGLAHKIDFGNEFKNIFGTPEFVAPEIVNYEPLGLEADMWSIGVITYILLSGASPFLGDTKQETLANVSAVNYEFEDEYFSNTSALAKDFIRRLLVKDPKKRMTIQDSLQHPWIKPKDTQQALSSAWSHPQFEK
4FVN , Knot 112 260 0.79 40 176 249 
MSHHWGYGKHNGPEHWHKDFPIAKGERQSPVDIDTHTAKYDPSLKPLSVSYDQATSLRILNNGHAFNVEFDDSQDKAVLKGGPLDGTYRLIQFHFHWGSLDGQGSEHTVDKKKYAAELHLVHWNTKYGDFGKAVQQPDGLAVLGIFLKVGSAKPGLQKVVDVLDSIKTKGKSADFTNFDPRGLLPESLDYWTYPGSLTTPPLLECVTWIVLKEPISVSSEQVLKFRKLNFNGEGEPEELMVDNWRPAQPLKNRQIKASFK
6COP , Knot 13 17 0.72 22 16 15 
EMALSKFFRDFILQRKK

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 \(c\).

\(|P_{f(1JKK_1)}(2) \setminus P_{f(4FVN_1)}(2)|=84\), \(|P_{f(4FVN_1)}(2) \setminus P_{f(1JKK_1)}(2)|=81\). 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
1JKK_1,4FVN_1 165 3
1JKK_1,6COP_1 177 3
4FVN_1,6COP_1 176 2

Newick tree

 
[
	6COP_1:90.08,
	[
		1JKK_1:82.5,4FVN_1:82.5
	]:7.58
]

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 \((0.85)(0.8)(\frac{554 }{\log_{20} 554}-\frac{260}{\log_{20}260})=83.4\)
Status Protein1 Protein2 d d1/2
Query variables 1JKK_1 4FVN_1 105 101
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]