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Parikh vectors
5AUY_1 1FLH_1 4ZJT_1 Letter Amino acid
1 6 4 C Cysteine
15 34 5 G Glycine
16 15 6 F Phenylalanine
11 17 11 P Proline
2 5 5 W Tryptophan
18 28 22 V Valine
16 17 9 A Alanine
12 16 9 N Asparagine
17 21 21 D Aspartic acid
23 0 9 K Lycine
29 14 14 E Glutamic acid
22 25 14 I Isoleucine
30 20 13 L Leucine
4 5 1 M Methionine
16 41 27 S Serine
15 26 14 T Threonine
10 16 8 Y Tyrosine
13 3 14 R Arginine
11 16 9 Q Glutamine
12 1 3 H Histidine 

5AUY_1|Chain A|Death-associated protein kinase 1|Homo sapiens (9606)
>1FLH_1|Chain A|UROPEPSIN|Homo sapiens (9606)
>4ZJT_1|Chains A, B, C, D, E, F, G, H, I, J|Acetylcholine-binding protein|Lymnaea stagnalis (6523)
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 \(w=f(c)\)
5AUY , Knot 123 293 0.79 40 178 280 
MTVFRQENVDDYYDTGEELGSGQFAVVKKCREKSTGLQYAAKFIKKRRTKSSRRGVSREDIEREVSILKEIQHPNVITLHEVYENKTDVILILELVAGGELFDFLAEKESLTEEEATEFLKQILNGVYYLHSLQIAHFDLKPENIMLLDRNVPKPRIKIIDFGLAHKIDFGNEFKNIFGTPEFVAPEIVNYEPLGLEADMWSIGVITYILLSGASPFLGDTKQETLANVSAVNYEFEDEYFSNTSALAKDFIRRLLVKDPKKRMTIQDSLQHPWIKPKDTQQALSLEHHHHHH
1FLH , Knot 137 326 0.81 38 178 304 
VDEQPLENYLDMEYFGTIGIGTPAQDFTVVFDTGSSNLWVPSVYCSSLACTNHNRFNPEDSSTYQSTSETVSITYGTGSMTGILGYDTVQVGGISDTNQIFGLSETEPGSFLYYAPFDGILGLAYPSISSSGATPVFDNIWNQGLVSQDLFSVYLSADDQSGSVVIFGGIDSSYYTGSLNWVPVTVEGYWQITVDSITMNGEAIACAEGCQAIVDTGTSLLTGPTSPIANIQSDIGASENSDGDMVVSCSAISSLPDIVFTINGVQYPVPPSAYILQSEGSCISGFQGMNVPTESGELWILGDVFIRQYFTVFDRANNQVGLAPVA
4ZJT , Knot 99 218 0.81 40 145 210 
DYKDDDDKLDRADILYNIRQTSRPDVIPTQRDRPVAVSVSLKFINILEVNEITNEVDVVFWQQTTWSDRTLAWNSSHSPDQVSVPISSLWVPDLAAYNAISKPEVLTPQLARVVSDGEVLYMPSIRQRFSCDVSGVDTESGATCRIKIGSWTHHSREISVDPTTENSDDSEYFSQYSRFEILDVTQKKNSVTYSCCPEAYEDVEVSLNFRKKGRSEIL

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 \(c\).

\(|P_{f(5AUY_1)}(2) \setminus P_{f(1FLH_1)}(2)|=78\), \(|P_{f(1FLH_1)}(2) \setminus P_{f(5AUY_1)}(2)|=78\). 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 1:10110000100000010011010111100000000110011011000000000011000010001011001001011010010000001111101111101101110000100001001100110110010010110101010011110001101010110111100101100100111010111101100011110101101111001110110111100000011010110001000010000111001100111001000101000100111010000011010000000
Pair \(Z_2\) Length of longest common subsequence
5AUY_1,1FLH_1 156 3
5AUY_1,4ZJT_1 163 4
1FLH_1,4ZJT_1 167 4

Newick tree

 
[
	4ZJT_1:83.95,
	[
		5AUY_1:78,1FLH_1:78
	]:5.95
]

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 \((0.85)(0.8)(\frac{619 }{\log_{20} 619}-\frac{293}{\log_{20}293})=91.0\)
Status Protein1 Protein2 d d1/2
Query variables 5AUY_1 1FLH_1 114 107
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]