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Parikh vectors
1JUH_1 5WNU_1 1IQE_1 Letter Amino acid
3 0 5 M Methionine
26 311 13 A Alanine
2 428 9 C Cysteine
17 0 9 Q Glutamine
30 545 22 G Glycine
38 0 11 S Serine
8 0 15 R Arginine
12 0 19 E Glutamic acid
12 0 13 I Isoleucine
5 0 17 K Lycine
14 0 7 N Asparagine
10 0 5 H Histidine
21 0 6 P Proline
39 0 20 T Threonine
21 0 8 Y Tyrosine
29 0 16 V Valine
21 0 11 D Aspartic acid
18 0 13 L Leucine
18 0 12 F Phenylalanine
6 0 4 W Tryptophan 

1JUH_1|Chains A, B, C, D|quercetin 2,3-dioxygenase|Aspergillus japonicus (34381)
>5WNU_1|Chain A|16S Ribosomal RNA rRNA|Thermus thermophilus (strain HB8 / ATCC 27634 / DSM 579) (300852)
>1IQE_1|Chain A|coagulation Factor Xa|Homo sapiens (9606)
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 \(w=f(c)\)
1JUH , Knot 146 350 0.81 40 195 328 
DTSSLIVEDAPDHVRPYVIRHYSHARAVTVDTQLYRFYVTGPSSGYAFTLMGTNAPHSDALGVLPHIHQKHYENFYCNKGSFQLWAQSGNETQQTRVLSSGDYGSVPRNVTHTFQIQDPDTEMTGVIVPGGFEDLFYYLGTNATDTTHTPYIPSSSDSSSTTGPDSSTISTLQSFDVYAELSFTPRTDTVNGTAPANTVWHTGANALASTAGDPYFIANGWGPKYLNSQYGYQIVAPFVTATQAQDTNYTLSTISMSTTPSTVTVPTWSFPGACAFQVQEGRVVVQIGDYAATELGSGDVAFIPGGVEFKYYSEAYFSKVLFVSSGSDGLDQNLVNGGEEWSSVSFPADW
5WNU , Knot 274 1522 0.44 8 16 63 
UUUGUUGGAGAGUUUGAUCCUGGCUCAGGGUGAACGCUGGCGGCGUGCCUAAGACAUGCAAGUCGUGCGGGCCGCGGGGUUUUACUCCGUGGUCAGCGGCGGACGGGUGAGUAACGCGUGGGUGACCUACCCGGAAGAGGGGGACAACCCGGGGAAACUCGGGCUAAUCCCCCAUGUGGACCCGCCCCUUGGGGUGUGUCCAAAGGGCUUUGCCCGCUUCCGGAUGGGCCCGCGUCCCAUCAGCUAGUUGGUGGGGUAAUGGCCCACCAAGGCGACGACGGGUAGCCGGUCUGAGAGGAUGGCCGGCCACAGGGGCACUGAGACACGGGCCCCACUCCUACGGGAGGCAGCAGUUAGGAAUCUUCCGCAAUGGGCGCAAGCCUGACGGAGCGACGCCGCUUGGAGGAAGAAGCCCUUCGGGGUGUAAACUCCUGAACCCGGGACGAAACCCCCGACGAGGGGACUGACGGUACCGGGGUAAUAGCGCCGGCCAACUCCGUGCCAGCAGCCGCGGUAAUACGGAGGGCGCGAGCGUUACCCGGAUUCACUGGGCGUAAAGGGCGUGUAGGCGGCCUGGGGCGUCCCAUGUGAAAGACCACGGCUCAACCGUGGGGGAGCGUGGGAUACGCUCAGGCUAGACGGUGGGAGAGGGUGGUGGAAUUCCCGGAGUAGCGGUGAAAUGCGCAGAUACCGGGAGGAACGCCGAUGGCGAAGGCAGCCACCUGGUCCACCCGUGACGCUGAGGCGCGAAAGCGUGGGGAGCAAACCGGAUUAGAUACCCGGGUAGUCCACGCCCUAAACGAUGCGCGCUAGGUCUCUGGGUCUCCUGGGGGCCGAAGCUAACGCGUUAAGCGCGCCGCCUGGGGAGUACGGCCGCAAGGCUGAAACUCAAAGGAAUUGACGGGGGCCCGCACAAGCGGUGGAGCAUGUGGUUUAAUUCGAAGCAACGCGAAGAACCUUACCAGGCCUUGACAUGCUAGGGAACCCGGGUGAAAGCCUGGGGUGCCCCGCGAGGGGAGCCCUAGCACAGGUGCUGCAUGGCCGUCGUCAGCUCGUGCCGUGAGGUGUUGGGUUAAGUCCCGCAACGAGCGCAACCCCCGCCGUUAGUUGCCAGCGGUUCGGCCGGGCACUCUAACGGGACUGCCCGCGAAAGCGGGAGGAAGGAGGGGACGACGUCUGGUCAGCAUGGCCCUUACGGCCUGGGCGACACACGUGCUACAAUGCCCACUACAAAGCGAUGCCACCCGGCAACGGGGAGCUAAUCGCAAAAAGGUGGGCCCAGUUCGGAUUGGGGUCUGCAACCCGACCCCAUGAAGCCGGAAUCGCUAGUAAUCGCGGAUCAGCCAUGCCGCGGUGAAUACGUUCCCGGGCCUUGUACACACCGCCCGUCACGCCAUGGGAGCGGGCUCUACCCGAAGUCGCCGGGAGCCUACGGGCAGGCGCCGAGGGUAGGGCCCGUGACUGGGGCGAAGUCGUAACAAGGUAGCUGUACCGGAAGGUGCGGCUGGAUCCACUCCUUUCU
1IQE , Knot 109 235 0.84 40 171 227 
IVGGQECKDGECPWQALLINEENEGFCGGTILSEFYILTAAHCLYQAKRFKVRVGDRNTEQEEGGEAVHEVEVVIKHNRFTKETYDFDIAVLRLKTPITFRMNVAPACLPERDWAESTLMTQKTGIVSGFGRTHEKGRQSTRLKMLEVPYVDRNSCKLSSSFIITQNMFCAGYDTKQEDACQGDSGGPHVTRFKDTYFVTGIVSWGEGCARKGKYGIYTKVTAFLKWIDRSMKTR

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 \(c\).

\(|P_{f(1JUH_1)}(2) \setminus P_{f(5WNU_1)}(2)|=189\), \(|P_{f(5WNU_1)}(2) \setminus P_{f(1JUH_1)}(2)|=10\). 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 1:00001110011001010110000010110100010010101100101101110011000111111010000000100001010111001000000011001001011001000101001000101111111100110011001000000101100000000011000010010010101010101000010101110011001101110011010111011110010000100111111010010000001001010001001011010111101101001011101100110011010111111110100000101001111001001100011011001001011101
Pair \(Z_2\) Length of longest common subsequence
1JUH_1,5WNU_1 199 4
1JUH_1,1IQE_1 168 3
5WNU_1,1IQE_1 173 3

Newick tree

 
[
	5WNU_1:96.10,
	[
		1JUH_1:84,1IQE_1:84
	]:12.10
]

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 \((0.85)(0.8)(\frac{1872 }{\log_{20} 1872}-\frac{350}{\log_{20}350})=384.\)
Status Protein1 Protein2 d d1/2
Query variables 1JUH_1 5WNU_1 272 207.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]