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Parikh vectors
4IGN_1 7CFZ_1 5VJL_1 Letter Amino acid
8 2 8 Y Tyrosine
10 0 19 N Asparagine
9 2 6 H Histidine
12 2 18 I Isoleucine
17 3 13 F Phenylalanine
22 5 16 P Proline
13 1 23 T Threonine
13 3 30 S Serine
25 8 20 V Valine
7 3 9 C Cysteine
10 5 13 Q Glutamine
23 3 15 E Glutamic acid
14 1 6 M Methionine
37 4 22 L Leucine
26 1 17 K Lycine
8 1 3 W Tryptophan
21 4 20 A Alanine
12 4 20 R Arginine
18 5 14 D Aspartic acid
27 8 33 G Glycine 

4IGN_1|Chains A, B, C, D, E, F|2-amino-3-carboxymuconate-6-semialdehyde decarboxylase|Homo sapiens (9606)
>7CFZ_1|Chains A, B|NADPH oxidase activator 1|Homo sapiens (9606)
>5VJL_1|Chain A|Hemagglutinin HA1|Influenza A virus (1332244)
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 \(w=f(c)\)
4IGN , Knot 146 332 0.85 40 200 324 
MKIDIHSHILPKEWPDLKKRFGYGGWVQLQHHSKGEAKLLKDGKVFAVVRENCWDPEVRIREMDQKGVTVQALSTVPVMFSYWAKPEDTLNLCQLLNNDLASTVVSYPRRFVGLGTLPMQAPELAVKEMERCVKELGFPGVQIGTHVNEWDLNAQELFPVYAAAERLKCSLFVHPWDMQMDGRMAKYWLPWLVGMPAETTIAICSMIMGGVFEKFPKLKVCFAHGGGAFPFTVGRISHGFSMRPDLCAQDNPMNPKKYLGSFYTDALVHDPLSLKLLTDVIGKDKVILGTDYPFPLGELEPGKLIESMEEFDEETKNKLKAGNALAFLGLER
7CFZ , Knot 38 65 0.81 38 57 63 
GRPVLYQVVAQHSYSAQGPEDLGFRQGDTVDVLCEVDQAWLEGHCDGRIGIFPKCFVVPAGPRMS
5VJL , Knot 141 325 0.83 40 209 318 
ADPGDKICLGHHAVSNGTKVNTLTERGVEVVNATETVERTNIPRICSKGKRTVDLGQCGLLGTITGPPQCDQFLEFSADLIIERREGSDVCYPGKFVNEEALRQILRESGGIDKEAMGFTYSGIRTNGATSACRRSGSSFYAEMKWLLSNTDNAAFPQMTKSYKNTRKSPALIVWGIHHSKSTAEQTTLYGSGNKLVTVGSSNYQQSFVPSPGARPQVNGLSSRIDFHWLMLNPNDTVTFSFNGAFIAPDRASFLRGKSMGIQSGVQVDANCEGDCYHSGGTIISNLPFQNIDSRAVGKCPRYVKQRSLLLATGMKNVPEIPKGR

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 \(c\).

\(|P_{f(4IGN_1)}(2) \setminus P_{f(7CFZ_1)}(2)|=165\), \(|P_{f(7CFZ_1)}(2) \setminus P_{f(4IGN_1)}(2)|=22\). 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 1:10101000111001101000110111101000001010110010111110000101010100100011010110011111001101000101001100011001100100111110111011011100100010011111101100100101010011110111001000111011010101011001111111111000111001111111001101010110111111101101001101010101000110100011010001110011010110011100011110001111101011011001001000000010110111111100
Pair \(Z_2\) Length of longest common subsequence
4IGN_1,7CFZ_1 187 3
4IGN_1,5VJL_1 175 4
7CFZ_1,5VJL_1 190 3

Newick tree

 
[
	7CFZ_1:96.39,
	[
		4IGN_1:87.5,5VJL_1:87.5
	]:8.89
]

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 \((0.85)(0.8)(\frac{397 }{\log_{20} 397}-\frac{65}{\log_{20}65})=103.\)
Status Protein1 Protein2 d d1/2
Query variables 4IGN_1 7CFZ_1 131 77
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]