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Parikh vectors
4GRS_1 4HJJ_1 6ZZK_1 Letter Amino acid
18 14 15 D Aspartic acid
11 6 9 Q Glutamine
30 12 12 E Glutamic acid
25 14 19 I Isoleucine
11 11 8 F Phenylalanine
19 15 13 S Serine
8 9 12 N Asparagine
4 0 0 C Cysteine
14 6 12 P Proline
14 6 21 T Threonine
33 6 21 V Valine
11 4 3 Y Tyrosine
23 8 22 A Alanine
18 8 11 R Arginine
25 6 15 G Glycine
6 1 6 H Histidine
32 11 24 L Leucine
22 14 14 K Lycine
7 6 4 M Methionine
2 0 0 W Tryptophan 

4GRS_1|Chains A, B, C, D|Phospho-2-dehydro-3-deoxyheptonate aldolase, 2-dehydro-3-deoxyphosphoheptonate aldolase|Thermotoga maritima (243274)
>4HJJ_1|Chain A|Interleukin-18|Homo sapiens (9606)
>6ZZK_1|Chains A, B|Dihydrolipoyllysine-residue acetyltransferase component of pyruvate dehydrogenase complex|Corynebacterium glutamicum ATCC 13032 (196627)
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 \(w=f(c)\)
4GRS , Knot 143 333 0.83 40 197 316 
MIVVLKPGSTEEDIRKVVKLAESYNLKCHISKGQERTVIGIIGDDRYVVADKFESLDCVESVVRVLKPYKLVSREFHPEDTVIDLGDVKIGNGYFTIIAGPCSIESRDQIMKVAEFLAEVGIKVLRGGAFKPRTSPYSFQGYGEKALRWMREAADEYGLVTVTEVMDTRHVELVAKYSDILQIGARNSQNFELLKEVGKVENPVLLKRGMGNTIQELLYSAEYIMAQGNENVILCERGIRTFETATRFTLDISAVPVVKELSHLPIIVDPSHPAGRRSLVIPLAKAAYAIGADGIMVEVHPEPEKALSDSQQQLTFDDFLQLLKELEALGWKG
4HJJ , Knot 75 157 0.80 36 119 153 
YFGKLESKLSVIRNLNDQVLFIDQGNRPLFEDMTDSDARDNAPRTIFIISMYKDSQPRGMAVTISVKAEKISTLSAENKIISFKEMNPPDNIKDTKSDIIFFQRSVPGHDNKMQFESSSYEGYFLAAEKERDLFKLILKKEDELGDRSIMFTVQNEA
6ZZK , Knot 106 241 0.80 36 162 230 
GSLRGTTQKVNRIREITAMKTVEALQISAQLTQLHEVDMTRVAELRKKNKPAFIEKHGVNLTYLPFFVKAVVEALVSHPNVNASFNAKTKEMTYHSSVNLSIAVDTPAGLLTPVIHDAQDLSIPEIAKAIVDLADRSRNNKLKPNDLSGGTFTITNIGSEGALSDTPILVPPQAGILGTGAIVKRPVVITEDGIDSIAIRQMVFLPLTYDHQVVDGADAGRFLTTIKDRLETANFEGDLQL

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 \(c\).

\(|P_{f(4GRS_1)}(2) \setminus P_{f(4HJJ_1)}(2)|=120\), \(|P_{f(4HJJ_1)}(2) \setminus P_{f(4GRS_1)}(2)|=42\). 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 1:111110110000010011011000010001001000011111100001110010010010011011010011000101000110110101101010111110010000011011011101110110111101000100101010011011001100011101001100001011100001101110000010110011010011110011100100110010011101000111000110010010010101011111001001111101001110001111110110111101111010101001100000010100110110010111101
Pair \(Z_2\) Length of longest common subsequence
4GRS_1,4HJJ_1 162 3
4GRS_1,6ZZK_1 161 4
4HJJ_1,6ZZK_1 157 3

Newick tree

 
[
	4GRS_1:81.48,
	[
		6ZZK_1:78.5,4HJJ_1:78.5
	]:2.98
]

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 \((0.85)(0.8)(\frac{490 }{\log_{20} 490}-\frac{157}{\log_{20}157})=97.8\)
Status Protein1 Protein2 d d1/2
Query variables 4GRS_1 4HJJ_1 123 89
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]