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Parikh vectors
1IFV_1 8FEO_1 2RMN_1 Letter Amino acid
10 0 10 D Aspartic acid
0 3 8 C Cysteine
3 0 14 Q Glutamine
18 3 14 G Glycine
5 0 6 H Histidine
10 0 10 L Leucine
8 0 18 T Threonine
0 0 1 W Tryptophan
13 4 14 A Alanine
13 0 13 E Glutamic acid
12 0 14 I Isoleucine
5 0 19 P Proline
5 0 7 Y Tyrosine
2 0 15 R Arginine
4 0 9 N Asparagine
15 0 11 K Lycine
7 0 8 F Phenylalanine
0 0 5 M Methionine
12 0 20 S Serine
13 0 17 V Valine 

1IFV_1|Chains A, B|PROTEIN LLR18B|Lupinus luteus (3873)
>8FEO_1|Chain A[auth AAA]|RNA 16mer|synthetic construct (32630)
>2RMN_1|Chain A|Tumor protein 63|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 \(w=f(c)\)
1IFV , Knot 75 155 0.81 34 110 146 
GVFAFEDEHPSAVAQAKLFKALTKDSDDIIPKVIEQIQSVEIVEGNGGPGTVKKITASHGGHTSYVLHKIDAIDEASFEYNYSIVGGTGLDESLEKITFESKLLSGPDGGSIGKIKVKFHTKGDVLSDAVREEAKARGTGLFKAVEGYVLANPNY
8FEO , Knot 8 16 0.46 8 8 11 
AGAGUAGAUCUUCUCU
2RMN , Knot 106 233 0.82 40 171 228 
GSSTFDALSPSPAIPSNTDYPGPHSFDVSFQQSSTAKSATWTYSTELKKLYCQIAKTCPIQIKVMTPPPQGAVIRAMPVYKKAEHVTEVVKRCPNHELSREFNEGQIAPPSHLIRVEGNSHAQYVEDPITGRQSVLVPYEPPQVGTEFTTVLYNFMCNSSCVGGMNRRPILIIVTLETRDGQVLGRRCFEARICACPGRDRKADEDSIRKQQVSDSTKNGDAFRQNTHGIQMT

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 \(c\).

\(|P_{f(1IFV_1)}(2) \setminus P_{f(8FEO_1)}(2)|=110\), \(|P_{f(8FEO_1)}(2) \setminus P_{f(1IFV_1)}(2)|=8\). 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 1:11111000010111010110110000001110110010010110101111010010100110000110010110010100000111101100010010100011011011011010101000101100110001010101110110101110100
Pair \(Z_2\) Length of longest common subsequence
1IFV_1,8FEO_1 118 1
1IFV_1,2RMN_1 157 4
8FEO_1,2RMN_1 177 2

Newick tree

 
[
	2RMN_1:90.38,
	[
		1IFV_1:59,8FEO_1:59
	]:31.38
]

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 \((0.85)(0.8)(\frac{171 }{\log_{20} 171}-\frac{16}{\log_{20}16})=55.9\)
Status Protein1 Protein2 d d1/2
Query variables 1IFV_1 8FEO_1 74 40.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]