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Parikh vectors
5VOG_1 9DCH_1 4WOV_1 Letter Amino acid
15 2 17 A Alanine
15 0 13 D Aspartic acid
9 0 19 H Histidine
5 0 9 F Phenylalanine
7 0 19 P Proline
5 0 8 N Asparagine
3 0 15 Q Glutamine
12 0 38 L Leucine
8 0 6 Y Tyrosine
2 0 6 C Cysteine
15 0 25 E Glutamic acid
15 0 8 I Isoleucine
5 0 8 M Methionine
7 0 16 T Threonine
5 0 4 W Tryptophan
8 0 24 R Arginine
13 5 27 G Glycine
14 0 8 K Lycine
3 0 25 S Serine
17 0 22 V Valine 

5VOG_1|Chain A|Putative phosphoribosyltransferase|Neisseria gonorrhoeae (485)
>9DCH_1|Chain A[auth G]|TERRAmut RNA|Homo sapiens (9606)
>4WOV_1|Chains A, B|Non-receptor tyrosine-protein kinase TYK2|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 \(w=f(c)\)
5VOG , Knot 85 183 0.80 40 132 173 
MAHHHHHHMKQKIWYTYDDIHRVIKALAEKIRNAGVKYDAMIAIGGGGFIPARMLRCFLEIPIYAVTTAYYDSDNEGQVTEEVKKVQWLDPVPEVLRGKNVLVVDEVDDSRVTMEFCLKELLKEDFDTVGVAVLHEKIKAKAGKIPEGIPYFSGITVEDWWINYPWDALDIDEHNRLAEAGRG
9DCH , Knot 6 10 0.46 6 4 5 
UGAGUGUGAG
4WOV , Knot 134 317 0.81 40 188 303 
MGSSHHHHHHSSGETVRFQGHMNLSQLSFHRVDQKEITQLSHLGQGTRTNVYEGRLRVEGSGDPEEGKMDDEDPLVPGRDRGQELRVVLKVLDPSHHDIALAFYETASLMSQVSHTHLAFVHGVCVRGPENIMVTEYVEHGPLDVWLRRERGHVPMAWKMVVAQQLASALSYLENKNLVHGNVCGRNILLARLGLAEGTSPFIKLSDPGVGLGALSREERVERIPWLAPECLPGGANSLSTAMDKWGFGATLLEICFDGEAPLQSRSPSEKEHFYQRQHRLPEPSCPQLATLTSQCLTYEPTQRPSFRTILRDLTRL

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 \(c\).

\(|P_{f(5VOG_1)}(2) \setminus P_{f(9DCH_1)}(2)|=131\), \(|P_{f(9DCH_1)}(2) \setminus P_{f(5VOG_1)}(2)|=3\). 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 1:110000001000110000010011011100100111000111111111111101100110111011001000000010100010010110111011010011110010000101010100110001001111110001010110110111010110100111001101101000001101101
Pair \(Z_2\) Length of longest common subsequence
5VOG_1,9DCH_1 134 2
5VOG_1,4WOV_1 182 6
9DCH_1,4WOV_1 190 2

Newick tree

 
[
	4WOV_1:10.20,
	[
		5VOG_1:67,9DCH_1:67
	]:33.20
]

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 \((0.85)(0.8)(\frac{193 }{\log_{20} 193}-\frac{10}{\log_{20}10})=65.8\)
Status Protein1 Protein2 d d1/2
Query variables 5VOG_1 9DCH_1 83 43
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]