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Parikh vectors
3LZF_1 4GUB_1 7FRC_1 Letter Amino acid
9 10 3 Q Glutamine
16 22 6 E Glutamic acid
25 14 13 G Glycine
9 16 6 H Histidine
17 19 6 I Isoleucine
19 14 5 P Proline
21 13 15 A Alanine
12 26 2 R Arginine
9 2 3 C Cysteine
28 30 19 L Leucine
19 23 13 K Lycine
9 13 5 F Phenylalanine
33 14 11 S Serine
18 26 22 V Valine
22 15 16 N Asparagine
14 27 9 D Aspartic acid
2 10 2 M Methionine
27 11 6 T Threonine
7 5 0 W Tryptophan
15 23 7 Y Tyrosine 

3LZF_1|Chain A|Hemagglutinin, HA1 Subunit|Influenza A virus (59375)
>4GUB_1|Chain A|Casein kinase II subunit alpha|Homo sapiens (9606)
>7FRC_1|Chains A, B|Non-structural protein 3|Severe acute respiratory syndrome coronavirus 2 (2697049)
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 \(w=f(c)\)
3LZF , Knot 143 331 0.83 40 204 315 
ADPGDTICIGYHANNSTDTVDTVLEKNVTVTHSVNLLEDSHNGKLCKLKGIAPLQLGKCNIAGWLLGNPECDLLLTASSWSYIVETSNSENGTCYPGDFIDYEELREQLSSVSSFEKFEIFPKTSSWPNHETTKGVTAACSYAGASSFYRNLLWLTKKGSSYPKLSKSYVNNKGKEVLVLWGVHHPPTGTDQQSLYQNADAYVSVGSSKYNRRFTPEIAARPKVRDQAGRMNYYWTLLEPGDTITFEATGNLIAPWYAFALNRGSGSGIITSDAPVHDCNTKCQTPHGAINSSLPFQNIHPVTIGECPKYVRSTKLRMATGLRNIPSIQSR
4GUB , Knot 149 333 0.86 40 196 318 
MSGPVPSRARVYTDVNTHRPREYWDYESHVVEWGNQDDYQLVRKLGRGKYSEVFEAINITNNEKVVVKILKPVKKKKIKREIKILENLRGGPNIITLADIVKDPVSRTPALVFEHVNNTDFKQLYQTLTDYDIRFYMYEILKALDYCHSMGIMHRDVKPHNVMIDHEHRKLRLIDWGLAEFYHPGQEYNVRVASRYFKGPELLVDYQMYDYSLDMWSLGCMLASMIFRKEPFFHGHDNYDQLVRIAKVLGTEDLYDYIDKYNIELDPRFNDILGRHSRKRWERFVHSENQHLVSPEALDFLDKLLRYDHQSRLTAREAMEHPYFYTVVKDQAR
7FRC , Knot 80 169 0.81 38 119 162 
SMVNSFSGYLKLTDNVYIKNADIVEEAKKVKPTVVVNAANVYLKHGGGVAGALNKATNNAMQVESDDYIATNGPLKVGGSCVLSGHNLAKHCLHVVGPNVNKGEDIQLLKSAYENFNQHEVLLAPLLSAGIFGADPIHSLRVCVDTVRTNVYLAVFDKNLYDKLVSSFL

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 \(c\).

\(|P_{f(3LZF_1)}(2) \setminus P_{f(4GUB_1)}(2)|=102\), \(|P_{f(4GUB_1)}(2) \setminus P_{f(3LZF_1)}(2)|=94\). 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 1:1011001011001000000100110001010001011000001010010111110110001111111010001110100100110000000100011011000010001001001001011100001100000011011000111001000111100010001010000100010011111110011010000010001010101100000001010111010100011010001011011001010101011111011110010101110001110000000001011100011100101101100100100001011011001101000
Pair \(Z_2\) Length of longest common subsequence
3LZF_1,4GUB_1 196 3
3LZF_1,7FRC_1 179 4
4GUB_1,7FRC_1 171 4

Newick tree

 
[
	3LZF_1:96.46,
	[
		7FRC_1:85.5,4GUB_1:85.5
	]:10.96
]

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 \((0.85)(0.8)(\frac{664 }{\log_{20} 664}-\frac{331}{\log_{20}331})=91.9\)
Status Protein1 Protein2 d d1/2
Query variables 3LZF_1 4GUB_1 122 120.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]