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Parikh vectors
4JWI_1 2CMK_1 3RVC_1 Letter Amino acid
13 12 9 G Glycine
20 28 15 L Leucine
16 8 10 T Threonine
0 1 2 W Tryptophan
0 2 1 Y Tyrosine
6 6 1 H Histidine
18 7 7 K Lycine
4 7 4 F Phenylalanine
12 20 10 V Valine
7 19 11 R Arginine
16 15 7 D Aspartic acid
0 1 1 C Cysteine
6 4 8 M Methionine
12 12 11 S Serine
25 34 12 A Alanine
9 5 7 N Asparagine
16 9 4 Q Glutamine
19 17 12 E Glutamic acid
14 11 11 I Isoleucine
6 9 9 P Proline 

4JWI_1|Chains A, B|Chaperone protein DnaK|Escherichia coli (83333)
>2CMK_1|Chain A|PROTEIN (CYTIDINE MONOPHOSPHATE KINASE)|Escherichia coli (562)
>3RVC_1|Chain A|Non-structural protein 1|Influenza A virus (211044)
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 \(w=f(c)\)
4JWI , Knot 95 219 0.78 34 145 212 
VLLLDVTPLSLGIETMGGVMTTLIAKNTTIPTKHSQVFSTAEDNQSAVTIHVLQGERKRAADNKSLGQFNLDGINPAPRGMPQIEVTFDIDADGILHVSAKDKNSGKEQKITIKASSGLNEDEIQKMVRDAEANAEADRKFEELVQTRNQGDHLLHSTRKQVEEAGDKLPADDKTAIESALTALETALKGEDKAAIEAKMQELAQVSQKLMEIAQQQHA
2CMK , Knot 100 227 0.79 40 140 215 
MTAIAPVITIDGPSGAGKGTLCKAMAEALQWHLLDSGAIYRVLALAALHHHVDVASEDALVPLASHLDVRFVSTNGNLEVILEGEDVSGEIRTQEVANAASQVAAFPRVREALLRRQRAFRELPGLIADGRDMGTVVFPDAPVKIFLDASSEERAHRRMLQLQEKGFSVNFERLLAEIKERDDRDRNRAVAPLVPAADALVLDSTTLSIEQVIEKALQYARQKLALA
3RVC , Knot 75 152 0.82 40 120 147 
MTMASVPASRYLTDMTLEEMSRDWSMLIPKQKVAGPLCIRMDQAIMDKNIILKANFSVIFDRLETLILLRAFTEEGAIVGEISPLPSLPGHTAEDVKNAVGVLIGGLEANDNTVRVSETLQRFAWRSSNENGRPPLTPKQKREMAGTIRSEV

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 \(c\).

\(|P_{f(4JWI_1)}(2) \setminus P_{f(2CMK_1)}(2)|=69\), \(|P_{f(2CMK_1)}(2) \setminus P_{f(4JWI_1)}(2)|=64\). 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 1:111101011011100111110011100001100000110010000011010110100001100001101010110111011101010101010111010100000100001010100110000100110010101010001001100000100110000001001100111000011001101100110100011101010011010001101100001
Pair \(Z_2\) Length of longest common subsequence
4JWI_1,2CMK_1 133 3
4JWI_1,3RVC_1 157 4
2CMK_1,3RVC_1 126 3

Newick tree

 
[
	4JWI_1:75.71,
	[
		2CMK_1:63,3RVC_1:63
	]:12.71
]

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 \((0.85)(0.8)(\frac{446 }{\log_{20} 446}-\frac{219}{\log_{20}219})=66.1\)
Status Protein1 Protein2 d d1/2
Query variables 4JWI_1 2CMK_1 82 81
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]