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Parikh vectors
2MIE_1 8QIA_1 4HQR_1 Letter Amino acid
0 8 20 D Aspartic acid
0 3 10 C Cysteine
1 14 17 G Glycine
2 6 16 S Serine
0 9 20 K Lycine
0 5 15 F Phenylalanine
0 4 9 Y Tyrosine
0 10 14 T Threonine
0 9 16 A Alanine
0 7 14 R Arginine
1 7 16 E Glutamic acid
0 10 21 L Leucine
0 2 6 M Methionine
3 7 11 P Proline
2 3 12 N Asparagine
0 3 9 Q Glutamine
1 14 13 H Histidine
1 4 16 I Isoleucine
0 1 2 W Tryptophan
1 12 15 V Valine 

2MIE_1|Chain A|CLAVATA3/ESR (CLE)-related protein 44|Arabidopsis thaliana (3702)
>8QIA_1|Chain A|Phototropin|Chlamydomonas reinhardtii (3055)
>4HQR_1|Chains A, B|Caspase-7|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 \(w=f(c)\)
2MIE , Knot 9 12 0.62 16 11 10 
HEVPSGPNPISN
8QIA , Knot 64 138 0.76 40 106 129 
HHHHHHHHHHSSGHIEGRHMAGLRHTFVVADATLPDCPLVYASEGFYAMTGYGPDEVLGHNCRFLQGEGTDPKEVQKIRDAIKKGEACSVRLLNYRKDGTPFWNLLTVTPIKTPDGRVSKFVGVQVDVTSKTEGKALA
4HQR , Knot 121 272 0.83 40 179 262 
MSIKTTRDRVPTYQYNMNFEKLGKCIIINNKNFDKVTGMGVRNGTDKDAEALFKCFRSLGFDVIVYNDCSCAKMQDLLKKASEEDHTNAACFACILLSHGEENVIYGKDGVTPIKDLTAHFRGDRCKTLLEKPKLFFIQACRGTELDDGIQAASGPINDLVPRGSTDANPRYKIPVEADFLFAYSTVPGYYSWRSPGRGSWFVQALCSILEEHGKDLEIMQILTRVNDRVARHFESQSDDPHFHEKKQIPCVVSMLTKELYFSQLEHHHHHH

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 \(c\).

\(|P_{f(2MIE_1)}(2) \setminus P_{f(8QIA_1)}(2)|=7\), \(|P_{f(8QIA_1)}(2) \setminus P_{f(2MIE_1)}(2)|=102\). 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 1:001101101100
Pair \(Z_2\) Length of longest common subsequence
2MIE_1,8QIA_1 109 2
2MIE_1,4HQR_1 178 3
8QIA_1,4HQR_1 173 6

Newick tree

 
[
	4HQR_1:96.32,
	[
		2MIE_1:54.5,8QIA_1:54.5
	]:41.82
]

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 \((0.85)(0.8)(\frac{150 }{\log_{20} 150}-\frac{12}{\log_{20}12})=51.1\)
Status Protein1 Protein2 d d1/2
Query variables 2MIE_1 8QIA_1 61 33
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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