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Parikh vectors
8UFM_1 5BPP_1 2MAI_1 Letter Amino acid
5 27 1 F Phenylalanine
31 40 0 T Threonine
1 13 0 W Tryptophan
10 34 0 D Aspartic acid
5 26 1 Q Glutamine
17 31 0 I Isoleucine
17 50 0 S Serine
13 22 0 Y Tyrosine
20 38 1 A Alanine
7 24 4 R Arginine
15 33 2 G Glycine
19 41 0 E Glutamic acid
3 23 0 H Histidine
22 40 0 K Lycine
5 12 0 M Methionine
10 36 0 P Proline
25 39 1 V Valine
13 21 2 N Asparagine
5 11 0 C Cysteine
26 70 3 L Leucine 

8UFM_1|Chain A|Papain-like protease nsp3|Severe acute respiratory syndrome coronavirus 2 (2697049)
>5BPP_1|Chain A|Leukotriene A-4 hydrolase|Homo sapiens (9606)
>2MAI_1|Chain A|Lassomycin|Lentzea kentuckyensis (360086)
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 \(w=f(c)\)
8UFM , Knot 117 269 0.81 40 161 250 
SNAKIKACVEEVTTTLEETKFLTENLLLYIDINGNLHPDSATLVSDIDITFLKKDAPYIVGDVVQEGVLTAVVIPTKKAGGTTEMLAKALRKVPTDNYITTYPGQGCNGYTVEEAKTVLKKCKSAFYILPSIISNEKQEILGTVSWNLREMLAHAEETRKLMPVCVETKAIVSTIQRKYKGIKIQEGVVDYGARFYFYTSKTTVASLINTLNDLNETLVTMPLGYVTHGLNLEEAARCMRSLKVPATVSVSSPDAVTAYNGYLTSSSKT
5BPP , Knot 253 631 0.86 40 290 587 
MGSSHHHHHHSSGLVPRGSHMPEIVDTCSLASPASVCRTKHLHLRCSVDFTRRTLTGTAALTVQSQEDNLRSLVLDTKDLTIEKVVINGQEVKYALGERQSYKGSPMEISLPIALSKNQEIVIEISFETSPKSSALQWLTPEQTSGKEHPYLFSQCQAIHCRAILPCQDTPSVKLTYTAEVSVPKELVALMSAIRDGETPDPEDPSRKIYKFIQKVPIPCYLIALVVGALESRQIGPRTLVWSEKEQVEKSAYEFSETESMLKIAEDLGGPYVWGQYDLLVLPPSFPYGGMENPCLTFVTPTLLAGDKSLSNVIAHEISHSWTGNLVTNKTWDHFWLNEGHTVYLERHICGRLFGEKFRHFNALGGWGELQNSVKTFGETHPFTKLVVDLTDIDPDVAYSSVPYEKGFALLFYLEQLLGGPEIFLGFLKAYVEKFSYKSITTDDWKDFLYSYFKDKVDVLNQVDWNAWLYSPGLPPIKPNYDMTLTNACIALSQRWITAKEDDLNSFNATDLKDLSSHQLNEFLAQTLQRAPLPLGHIKRMQEVYNFNAINNSEIRFRWLRLCIQSKWEDAIPLALKMATEQGRMKFTRPLFKDLAAFDKSHDQAVRTYQEHKASMHPVTAMLVGKDLKVD
2MAI , Knot 12 16 0.69 18 14 14 
GLRRLFANQLVGRRNX

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 \(c\).

\(|P_{f(8UFM_1)}(2) \setminus P_{f(5BPP_1)}(2)|=29\), \(|P_{f(5BPP_1)}(2) \setminus P_{f(8UFM_1)}(2)|=158\). 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 1:00101010100100010000110001110101010101001011001010110001101110110011101111100011100011101100110000100011010010010010011000001101110110000001110101010011101000001111010001110010000011010011100110101000000110110010010001101111010011010011001001011101010010110100101000000
Pair \(Z_2\) Length of longest common subsequence
8UFM_1,5BPP_1 187 3
8UFM_1,2MAI_1 167 2
5BPP_1,2MAI_1 282 3

Newick tree

 
[
	5BPP_1:12.45,
	[
		8UFM_1:83.5,2MAI_1:83.5
	]:45.95
]

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 \((0.85)(0.8)(\frac{900 }{\log_{20} 900}-\frac{269}{\log_{20}269})=171.\)
Status Protein1 Protein2 d d1/2
Query variables 8UFM_1 5BPP_1 223 155
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]