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Parikh vectors
1VTR_1 1MHJ_1 6TBL_1 Letter Amino acid
0 2 2 Y Tyrosine
0 2 1 N Asparagine
0 0 4 D Aspartic acid
0 0 2 H Histidine
0 0 8 P Proline
0 0 1 W Tryptophan
0 0 7 R Arginine
0 2 5 E Glutamic acid
0 2 27 L Leucine
0 0 3 M Methionine
0 0 2 F Phenylalanine
2 0 10 A Alanine
0 2 5 Q Glutamine
0 1 6 G Glycine
0 0 6 K Lycine
2 1 6 T Threonine
0 4 3 C Cysteine
0 2 3 I Isoleucine
0 2 15 S Serine
0 1 9 V Valine 

1VTR_1|Chain A|DNA (5'-D(*AP*TP*AP*T)-3')|
>1MHJ_1|Chain A|INSULIN|Homo sapiens (9606)
>6TBL_1|Chains A, B|MMS19 nucleotide excision repair protein homolog|Mus musculus (10090)
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 \(w=f(c)\)
1VTR , Knot 3 4 0.34 4 2 2 
ATAT
1MHJ , Knot 15 21 0.72 22 20 19 
GIVEQCCTSICSLYQLENYCN
6TBL , Knot 57 125 0.73 40 90 118 
GGGRELPTLLSLLLEALSCPDSVVQLSTLSCLQPLLLEAPQIMSLHVDTLVTKFLNLSSSYSMAVRIAALQCMHALTRLPTSVLLPYKSQVIRALAKPLDDKKRLVRKEAVSARGEWFLLGSPGS

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 \(c\).

\(|P_{f(1VTR_1)}(2) \setminus P_{f(1MHJ_1)}(2)|=2\), \(|P_{f(1MHJ_1)}(2) \setminus P_{f(1VTR_1)}(2)|=20\). 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 1:1010
Pair \(Z_2\) Length of longest common subsequence
1VTR_1,1MHJ_1 22 1
1VTR_1,6TBL_1 92 1
1MHJ_1,6TBL_1 100 2

Newick tree

 
[
	6TBL_1:55.10,
	[
		1VTR_1:11,1MHJ_1:11
	]:44.10
]

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 \((0.85)(0.8)(\frac{25 }{\log_{20} 25}-\frac{4}{\log_{20}4})=9.94\)
Status Protein1 Protein2 d d1/2
Query variables 1VTR_1 1MHJ_1 14 8
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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