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Parikh vectors
1GUK_1 8FLI_1 5YFZ_1 Letter Amino acid
12 0 20 F Phenylalanine
12 0 16 R Arginine
10 0 22 Q Glutamine
12 279 28 G Glycine
27 0 38 L Leucine
8 0 20 T Threonine
0 229 10 C Cysteine
19 0 26 E Glutamic acid
13 0 30 I Isoleucine
18 0 37 K Lycine
10 0 7 M Methionine
9 0 28 S Serine
10 0 13 Y Tyrosine
13 0 32 V Valine
17 227 33 A Alanine
11 0 24 D Aspartic acid
2 0 10 H Histidine
13 0 17 P Proline
2 0 2 W Tryptophan
4 0 29 N Asparagine 

1GUK_1|Chains A, B|GLUTATHIONE S-TRANSFERASE A4-4|Mus musculus (10090)
>8FLI_1|Chain A|Group II Intron|Thermosynechococcus vestitus (146786)
>5YFZ_1|Chains A, B, C|Serine hydroxymethyltransferase|Plasmodium vivax (5855)
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 \(w=f(c)\)
1GUK , Knot 101 222 0.82 38 145 213 
MAAKPKLYYFNGRGRMESIRWLLAAAGVEFEEEFLETREQYEKMQKDGHLLFGQVPLVEIDGMMLTQTRAILSYLAAKYNLYGKDLKERVRIDMYADGTQDLMMMIAVAPFKTPKEKEESYDLILSRAKTRYFPVFEKILKDHGEAFLVGNQLSWADIQLLEAILMVEELSAPVLSDFPLLQAFKTRISNIPTIKKFLQPGSQRKPPPDGPYVEVVRIVLKF
8FLI , Knot 172 893 0.43 8 16 64 
CCCAGGGUUGGCCGAGCGUUGCGACGCGAAAGCUAGCCAGAUGAUUGUCCCACUAGCCCAACAAGCUAGAACGGGACCGGUUGUUCCCCCAACCGUAGCCUAGGGAGGCAUGCGUGACUGGUAACGGUCAGGUGUGAAGCCCUCCCGACAAUGUAGCCCGAACCGCAAGGUUGAAGCUGAAUCCGUGAGGAGGAAGCAACUUCACCAGUGUCAGGUGAUAGGGAACUAGGCUUGAGGGUAUGGUGAGCACAUGCGAAGUGAUGUCAGAAGCCUCGUCACAGACCAACAGGCCAAAGACACUGAUAGGCCUGAGCCAAAACGGCAAAUGGAUAGGCUACAUCGCUCGCUCGUCGGUGUACGGGGACGUCAAUCCAUCGGGGCACAGUCACCACCUAACCCCUCGUGUCAUCUGGUUGGAACGCGGUAAGCCCGUAUCCUCGCCUUGAACACUCAAGGCAGGCAAACCUUCGGGAAUGCUGAUGGGGGUGCGGGUAUGGGAUGCAGGAGAAAGCGAAUGCCGGUCUGUAAUGGACCGGAUAGGGGUUGAGGAGACAAUCCAACAUCACCCCGCCCGAAAGGGAGCAGACUUCCUGCUGGUCUCUCUUUGCGAGAUAGCCUGUAGAACCUCUUGAAUGGAGACAAGGCAAAUGGCAGUGGAACAAACCACUGGUGCGGUCACCAACCAAACGGAAACAAGCUGGCACAGCAUAGACUGGGCCAAAGCCAACCGUGAGGUAAAGAGGCUGCAAGUGCGUAUCGCAAAGGCGUUCGCGCCGGUUCCUCUUGAAAGAGGGGCUUUGAGAGGCCUGAGCCGGAUGUGGGGAAACUCACAAGUCCGGUUCUUAGGGGGCGGGGAUGGCAUUCGUGCCUCCCUGCUACCCGGCGAUGAGGCA
5YFZ , Knot 184 442 0.84 40 236 419 
MFNNEPLEQIDKELHDILADEEKRQRETINLIASENLTNGAVRECLGNRVSNKYSEGYPKKRYYGGNDFIDKIEELCQKRALEAFNVSDEEWGVNVQPLSGSAANVQALYALVGVKGKIMGMHLCSGGHLTHGFFDEKKKVSITSDMFESKLYKCNSQGYVDLDAVREMALSFKPKVIICGYTSYPRDIDYQQFRQICDEVNAYLFADISHISSFVACNILNNPFLHADVVTTTTHKILRGPRSALIFFNKKRNPGIEQKINSAVFPSFQGGPHNNKIAAVACQLKEVHSPAFKEYTQQVLLNSKALAKALISKQIDLVTNGTDNHLIVVDLRKFSITGSKLQETCNAINVSLNKNTIPSDVDCVSPSGVRIGTPAMTTRGAKEKDMEFIADVLARAIKITVDLQEQYGKKLVDFKKGLPGNAQLQQLKQEVVTWAGALPFP

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 \(c\).

\(|P_{f(1GUK_1)}(2) \setminus P_{f(8FLI_1)}(2)|=143\), \(|P_{f(8FLI_1)}(2) \setminus P_{f(1GUK_1)}(2)|=14\). 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 1:111010100101010100101111111101000110000000010001011110111101011110000111001110001010010001010101010001111111111001000000001110010000111100110001011111001011010110111110010111100111101100010011010011011000011101101011011101
Pair \(Z_2\) Length of longest common subsequence
1GUK_1,8FLI_1 157 4
1GUK_1,5YFZ_1 169 3
8FLI_1,5YFZ_1 240 3

Newick tree

 
[
	5YFZ_1:11.93,
	[
		1GUK_1:78.5,8FLI_1:78.5
	]:32.43
]

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 \((0.85)(0.8)(\frac{1115 }{\log_{20} 1115}-\frac{222}{\log_{20}222})=240.\)
Status Protein1 Protein2 d d1/2
Query variables 1GUK_1 8FLI_1 172 134.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]