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Parikh vectors
1IMS_1 6BWZ_1 4PRE_1 Letter Amino acid
0 0 12 V Valine
1 0 23 A Alanine
0 0 6 N Asparagine
0 0 9 H Histidine
0 0 3 M Methionine
0 3 13 S Serine
0 0 27 R Arginine
0 0 18 D Aspartic acid
2 1 19 G Glycine
0 0 7 K Lycine
0 0 8 I Isoleucine
0 0 16 P Proline
0 2 14 Y Tyrosine
1 0 22 T Threonine
0 0 9 W Tryptophan
2 0 4 C Cysteine
0 0 17 Q Glutamine
0 0 23 E Glutamic acid
0 0 19 L Leucine
0 0 7 F Phenylalanine 

1IMS_1|Chain A|DNA (5'-D(*CP*GP*AP*TP*CP*G)-3')|
>6BWZ_1|Chain A|SYSGYS peptide  from low-complexity domain of FUS|Homo sapiens (9606)
>4PRE_1|Chain A|MHC class I antigen|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 \(w=f(c)\)
1IMS , Knot 5 6 0.49 8 4 4 
CGATCG
6BWZ , Knot 4 6 0.39 6 4 4 
SYSGYS
4PRE , Knot 121 276 0.82 40 182 265 
GSHSMRYFYTAMSRPGRGEPRFIAVGYVDDTQFVRFDSDAASPRTEPRAPWIEQEGPEYWDRNTQIFKTNTQTYRESLRNLRGYYNQSEAGSHIIQRMYGCDLGPDGRLLRGHDQSAYDGKDYIALNEDLSSWTAADTAAQITQRKWEAARVAEQRRAYLEGLCVEWLRRYLENGKETLQRADPPKTHVTHHPVSDHEATLRCWALGFYPAEITLTWQRDGEDQTQDTELVETRPAGDRTFQKWAAVVVPSGEEQRYTCHVQHEGLPKPLTLRWEP

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 \(c\).

\(|P_{f(1IMS_1)}(2) \setminus P_{f(6BWZ_1)}(2)|=4\), \(|P_{f(6BWZ_1)}(2) \setminus P_{f(1IMS_1)}(2)|=4\). 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 1:011001
Pair \(Z_2\) Length of longest common subsequence
1IMS_1,6BWZ_1 8 1
1IMS_1,4PRE_1 182 2
6BWZ_1,4PRE_1 182 2

Newick tree

 
[
	4PRE_1:10.05,
	[
		1IMS_1:4,6BWZ_1:4
	]:10.05
]

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 \((0.85)(0.8)(\frac{12 }{\log_{20} 12}-\frac{6}{\log_{20}6})=3.01\)
Status Protein1 Protein2 d d1/2
Query variables 1IMS_1 6BWZ_1 3 3
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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