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Parikh vectors
2ENV_1 2BXX_1 7DSD_1 Letter Amino acid
5 8 19 F Phenylalanine
1 6 17 T Threonine
3 5 13 Y Tyrosine
2 4 18 V Valine
4 10 25 A Alanine
3 2 8 H Histidine
9 10 29 K Lycine
3 1 12 M Methionine
1 13 14 P Proline
8 8 15 R Arginine
1 12 14 D Aspartic acid
5 2 24 E Glutamic acid
3 4 23 I Isoleucine
10 8 26 S Serine
0 6 7 W Tryptophan
3 7 11 N Asparagine
9 1 11 C Cysteine
3 7 15 Q Glutamine
12 16 15 G Glycine
3 4 30 L Leucine 

2ENV_1|Chain A|Peroxisome proliferator-activated receptor delta|Homo sapiens (9606)
>2BXX_1|Chains A, B|NUCLEOCAPSID PROTEIN|AVIAN INFECTIOUS BRONCHITIS VIRUS (11120)
>7DSD_1|Chains A, B, C, D, E, F, G|Calcium homeostasis modulator 1|Danio rerio (7955)
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 \(w=f(c)\)
2ENV , Knot 47 88 0.79 38 69 83 
GSSGSSGMECRVCGDKASGFHYGVHACEGCKGFFRRTIRMKLEYEKCERSCKIQKKNRNKCQYCRFQKCLALGMSHNAIRFGSGPSSG
2BXX , Knot 69 134 0.84 40 108 131 
HMSSGNASWFQAIKAKKLNTPPPKFEGSGVPDNENIKPSQQHGYWRRQARFKPGKGGRCPVPDAWYFYYTGTGPAADLNWGDTQDGIVWVAAKGADTKSRSNQGTRDPDKFDQYPLRFSDGGPDGNFRWDFIPL
7DSD , Knot 153 346 0.86 40 219 336 
MDKFRIMVQFLQANQESFMNGICGIMALASAQMYSSFEFTCPCLPDYNYAYGIGILIVPPIWFFLLGYVMNNNISVLTEEWKRPVGKRSKDPAVLRYMFSSMTQRALIAPAVWIAVTLMDGKSFLCAFSPTADLSEFVNESYQSLSQKELLKIQAKIPCKDIFEEHEIISREAATRYIRCLSQACGWTFLMVITLVAFLVRAIRPCFTQAAFLKTKYWSHYIDTERKLFDETCKEHAKSFAKVCIQQYFESISGEIVSQLPQSPAKKGKGNKDEDGEKQKSDEERLLGIRKEGDMNKVLWNWHTCKPPLLLSKRTEEMNGHAHLDTHSLTDERHTKKKAVVYYSKV

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 \(c\).

\(|P_{f(2ENV_1)}(2) \setminus P_{f(2BXX_1)}(2)|=46\), \(|P_{f(2BXX_1)}(2) \setminus P_{f(2ENV_1)}(2)|=85\). 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 1:1001001100010100101100110100100111000101010000000000100000000000010001111100011011011001
Pair \(Z_2\) Length of longest common subsequence
2ENV_1,2BXX_1 131 3
2ENV_1,7DSD_1 218 3
2BXX_1,7DSD_1 207 3

Newick tree

 
[
	7DSD_1:11.75,
	[
		2ENV_1:65.5,2BXX_1:65.5
	]:51.25
]

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 \((0.85)(0.8)(\frac{222 }{\log_{20} 222}-\frac{88}{\log_{20}88})=43.6\)
Status Protein1 Protein2 d d1/2
Query variables 2ENV_1 2BXX_1 58 46.5
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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