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Parikh vectors
2MCN_1 4MSR_1 1GRA_1 Letter Amino acid
6 0 21 D Aspartic acid
10 0 29 E Glutamic acid
1 0 16 H Histidine
4 0 34 L Leucine
2 0 14 F Phenylalanine
1 0 42 A Alanine
5 0 17 R Arginine
1 0 24 P Proline
1 0 3 W Tryptophan
0 5 10 C Cysteine
1 0 11 Q Glutamine
4 0 34 K Lycine
0 0 31 S Serine
5 0 29 I Isoleucine
1 0 15 M Methionine
2 0 31 T Threonine
3 0 13 Y Tyrosine
6 0 44 V Valine
3 0 17 N Asparagine
5 5 43 G Glycine 

2MCN_1|Chain A|CD2-associated protein|Mus musculus (10090)
>4MSR_1|Chains A, B|RNA 10mer duplex with six 2'-5'-linkages|null
>1GRA_1|Chain A|GLUTATHIONE REDUCTASE|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 \(w=f(c)\)
2MCN , Knot 33 61 0.74 36 52 59 
VDYIVEYDYDAVHDDELTIRVGEIIRNVKKLQEEGWLEGELNGRRGMFPDNFVKEIKRETE
4MSR , Knot 5 10 0.38 4 4 6 
CCGGCGCCGG
1GRA , Knot 198 478 0.85 40 246 462 
ACRQEPQPQGPPPAAGAVASYDYLVIGGGSGGLASARRAAELGARAAVVESHKLGGTCVNVGCVPKKVMWNTAVHSEFMHDHADYGFPSCEGKFNWRVIKEKRDAYVSRLNAIYQNNLTKSHIEIIRGHAAFTSDPKPTIEVSGKKYTAPHILIATGGMPSTPHESQIPGASLGITSDGFFQLEELPGRSVIVGAGYIAVEMAGILSALGSKTSLMIRHDKVLRSFDSMISTNCTEELENAGVEVLKFSQVKEVKKTLSGLEVSMVTAVPGRLPVMTMIPDVDCLLWAIGRVPNTKDLSLNKLGIQTDDKGHIIVDEFQNTNVKGIYAVGDVCGKALLTPVAIAAGRKLAHRLFEYKEDSKLDYNNIPTVVFSHPPIGTVGLTEDEAIHKYGIENVKTYSTSFTPMYHAVTKRKTKCVMKMVCANKEEKVVGIHMQGLGCDEMLQGFAVAVKMGATKADFDNTVAIHPTSSEELVTLR

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 \(c\).

\(|P_{f(2MCN_1)}(2) \setminus P_{f(4MSR_1)}(2)|=52\), \(|P_{f(4MSR_1)}(2) \setminus P_{f(2MCN_1)}(2)|=4\). 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 1:1001100000110000101011011001001000111010101001111001100100000
Pair \(Z_2\) Length of longest common subsequence
2MCN_1,4MSR_1 56 1
2MCN_1,1GRA_1 214 4
4MSR_1,1GRA_1 244 2

Newick tree

 
[
	1GRA_1:13.50,
	[
		2MCN_1:28,4MSR_1:28
	]:10.50
]

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 \((0.85)(0.8)(\frac{71 }{\log_{20} 71}-\frac{10}{\log_{20}10})=25.0\)
Status Protein1 Protein2 d d1/2
Query variables 2MCN_1 4MSR_1 32 18.5
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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