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Parikh vectors
5JYN_1 2QLF_1 9HLJ_1 Letter Amino acid
2 10 12 N Asparagine
0 8 5 C Cysteine
6 13 16 L Leucine
4 9 28 S Serine
0 11 13 T Threonine
2 0 2 W Tryptophan
0 4 4 M Methionine
1 5 7 P Proline
5 9 13 V Valine
1 10 11 A Alanine
0 14 20 D Aspartic acid
1 4 10 Q Glutamine
0 8 8 E Glutamic acid
0 3 1 H Histidine
2 4 8 Y Tyrosine
4 9 6 R Arginine
3 11 8 G Glycine
7 10 9 I Isoleucine
0 20 12 K Lycine
2 11 11 F Phenylalanine 

5JYN_1|Chains A, B, C|Envelope glycoprotein gp160|Human immunodeficiency virus 1 (11676)
>2QLF_1|Chains A, C|Caspase-7|Homo sapiens (9606)
>9HLJ_1|Chain A[auth D]|GV37-TCR alpha chain|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 \(w=f(c)\)
5JYN , Knot 23 40 0.70 26 34 38 
NWLWYIRIFIIIVGSLIGLRIVFAVLSLVNRVRQGYSPLS
2QLF , Knot 83 173 0.82 38 126 170 
AKPDRSSFVPSLFSKKKKNVTMRSIKTTRDRVPTYQYNMNFEKLGKCIIINNKNFDKVTGMGVRNGTDKDAEALFKCFRSLGFDVIVYNDCSCAKMQDLLKKASEEDHTNAACFACILLSHGEENVIYGKDGVTPIKDLTAHFRGDRCKTLLEKPKLFFIQACRGTELDDGIQ
9HLJ , Knot 95 204 0.82 40 141 198 
GEDVEQSLFLSVREGDSSVINCTYTDSSSTYLYWYKQEPGAGLQLLTYIFSNMDMKQDQRLTVLLNKKDKHLSLRIADTQTGDSAIYFCAEVEGNTPLVFGKGTRLSVIANIQNPDPAVYQLRDSKSSDKSVCLFTDFDSQTNVSQSKDSDVYITDKCVLDMRSMDFKSNSAVAWSNKSDFACANAFNNSIIPEDTFFPSPESS

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 \(c\).

\(|P_{f(5JYN_1)}(2) \setminus P_{f(2QLF_1)}(2)|=26\), \(|P_{f(2QLF_1)}(2) \setminus P_{f(5JYN_1)}(2)|=118\). 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 1:0111010111111101111011111101100100100110
Pair \(Z_2\) Length of longest common subsequence
5JYN_1,2QLF_1 144 3
5JYN_1,9HLJ_1 141 3
2QLF_1,9HLJ_1 163 3

Newick tree

 
[
	2QLF_1:78.91,
	[
		5JYN_1:70.5,9HLJ_1:70.5
	]:8.41
]

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 \((0.85)(0.8)(\frac{213 }{\log_{20} 213}-\frac{40}{\log_{20}40})=58.8\)
Status Protein1 Protein2 d d1/2
Query variables 5JYN_1 2QLF_1 75 46
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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