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Parikh vectors
2MSF_1 5QQE_1 8UEV_1 Letter Amino acid
1 6 5 N Asparagine
0 5 3 Q Glutamine
0 2 0 H Histidine
1 18 27 L Leucine
0 3 6 M Methionine
1 16 3 V Valine
1 12 2 D Aspartic acid
0 9 6 E Glutamic acid
3 13 4 K Lycine
0 6 8 F Phenylalanine
0 2 4 W Tryptophan
1 13 12 A Alanine
5 11 3 G Glycine
0 10 5 I Isoleucine
1 11 6 P Proline
4 8 7 S Serine
0 13 9 T Threonine
1 8 3 Y Tyrosine
2 7 1 R Arginine
8 5 1 C Cysteine 

2MSF_1|Chain A|Peptide TsPep1|Tityus serrulatus (6887)
>5QQE_1|Chain A|Ras-related C3 botulinum toxin substrate 1|Homo sapiens (9606)
>8UEV_1|Chain A[auth 1A]|NADH-ubiquinone oxidoreductase chain 3|Sus scrofa (9823)
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 \(w=f(c)\)
2MSF , Knot 17 29 0.65 24 24 27 
KPKCGLCRYRCCSGGCSSGKCVNGACDCS
5QQE , Knot 84 178 0.81 40 140 171 
SMQAIKCVVVGDGAVGKTCLLISYTTNAFPGEYIPTVFDNYSANVMVDGKPVNLGLWDTAGQEDYDRLRPLSYPQTDVFLICFSLVSPASFENVRAKWYPEVRHHCPNTPIILVGTKLDLRDDKDTIEKLKEKKLTPITYPQGLAMAKEIGAVKYLECSALTQRGLKTVFDEAIRAVL
8UEV , Knot 54 115 0.74 38 80 110 
MNIMLTLLTNVTLASLLVLIAFWLPQLNAYSEKTSPYECGFDPMGSARLPFSMKFFLVAITFLLFDLEIALLLPLPWASQTNNLKTMLTMALFLLILLAASLAYEWTQKGLEWAE

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 \(c\).

\(|P_{f(2MSF_1)}(2) \setminus P_{f(5QQE_1)}(2)|=16\), \(|P_{f(5QQE_1)}(2) \setminus P_{f(2MSF_1)}(2)|=132\). 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 1:01001100000001100010010110000
Pair \(Z_2\) Length of longest common subsequence
2MSF_1,5QQE_1 148 3
2MSF_1,8UEV_1 100 2
5QQE_1,8UEV_1 144 3

Newick tree

 
[
	5QQE_1:79.20,
	[
		2MSF_1:50,8UEV_1:50
	]:29.20
]

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 \((0.85)(0.8)(\frac{207 }{\log_{20} 207}-\frac{29}{\log_{20}29})=61.5\)
Status Protein1 Protein2 d d1/2
Query variables 2MSF_1 5QQE_1 77 45
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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