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Parikh vectors
8ONQ_1 3UQN_1 8QES_1 Letter Amino acid
1 35 27 A Alanine
1 29 19 L Leucine
1 1 10 F Phenylalanine
0 14 8 P Proline
1 4 4 W Tryptophan
1 21 13 E Glutamic acid
0 6 4 M Methionine
0 8 7 Q Glutamine
0 22 12 G Glycine
1 28 7 I Isoleucine
0 9 8 R Arginine
0 2 1 C Cysteine
0 5 7 H Histidine
0 18 18 K Lycine
0 11 14 S Serine
0 20 11 T Threonine
0 7 4 Y Tyrosine
0 20 20 V Valine
0 16 8 N Asparagine
0 15 18 D Aspartic acid 

8ONQ_1|Chains A, B|Peptide Ac-EFIAWL from hGLP-1|synthetic construct (32630)
>3UQN_1|Chains A, B|Dihydrodipicolinate synthase|Acinetobacter baumannii (575584)
>8QES_1|Chains A, B, C, D|Oxygen-insensitive NAD(P)H nitroreductase|Escherichia coli (562)
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 \(w=f(c)\)
8ONQ , Knot 7 7 0.64 14 6 5 
XEFIAWL
3UQN , Knot 125 291 0.81 40 172 275 
TIQGSIVAIVTPMLKDGGVDWKSLEKLVEWHIEQGTNSIVAVGTTGEASTLSMEEHTQVIKEIIRVANKRIPIIAGTGANSTREAIELTKAAKDLGADAALLVTPYYNKPTQEGLYQHYKAIAEAVELPLILYNVPGRTGVDLSNDTAVRLAEIPNIVGIKDATGDVPRGKALIDALNGKMAVYSGDDETAWELMLLGADGNISVTANIAPKAMSEVCAVAIAKDEQQAKTLNNKIANLHNILFCESNPIPVKWALHEMGLIDTGIRLPLTPLAEQYREPLRNALKDAGII
8QES , Knot 102 221 0.83 42 156 211 
DRWGSDIISVALKRHSTKAFDASKKLTPEQAEQIKTLLQYSPSSTNSQPWHFIVASTEEGKARVAKSAAGNYVFNERKMLDASHVVVFCAKTAMDDVWLKLVVDQEDADGRFATPEAKAANDKGRKFXADMHRKDLHDDAEWMAKQVYLNVGNFLLGVAALGLDAVPIEGFDAAILDAEFGLKEKGYTSLVVVPVGHHSVEDFNATLPKSRLPQNITLTEV

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 \(c\).

\(|P_{f(8ONQ_1)}(2) \setminus P_{f(3UQN_1)}(2)|=4\), \(|P_{f(3UQN_1)}(2) \setminus P_{f(8ONQ_1)}(2)|=170\). 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 1:0011111
Pair \(Z_2\) Length of longest common subsequence
8ONQ_1,3UQN_1 174 2
8ONQ_1,8QES_1 156 2
3UQN_1,8QES_1 156 3

Newick tree

 
[
	3UQN_1:84.10,
	[
		8ONQ_1:78,8QES_1:78
	]:6.10
]

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 \((0.85)(0.8)(\frac{298 }{\log_{20} 298}-\frac{7}{\log_{20}7})=99.2\)
Status Protein1 Protein2 d d1/2
Query variables 8ONQ_1 3UQN_1 123 63.5
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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