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Parikh vectors
3MBI_1 8CEP_1 5AIQ_1 Letter Amino acid
16 0 2 Y Tyrosine
1 353 1 C Cysteine
18 0 15 E Glutamic acid
5 0 3 M Methionine
26 0 5 S Serine
13 0 7 T Threonine
13 0 7 N Asparagine
25 0 8 D Aspartic acid
16 486 7 G Glycine
8 0 1 H Histidine
29 0 12 I Isoleucine
20 0 24 L Leucine
22 0 8 K Lycine
7 0 3 F Phenylalanine
18 388 13 A Alanine
6 0 9 Q Glutamine
0 0 1 W Tryptophan
23 0 5 V Valine
14 0 11 R Arginine
7 0 4 P Proline 

3MBI_1|Chains A, B, C, D|Ribose-phosphate pyrophosphokinase|Thermoplasma volcanium (50339)
>8CEP_1|Chain A|16S rRNA|Escherichia coli BW25113 (679895)
>5AIQ_1|Chains A, B, C, D|TRANSCRIPTIONAL REGULATOR, MARR FAMILY|NEISSERIA MENINGITIDIS SEROGROUP B (491)
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 \(w=f(c)\)
3MBI , Knot 121 287 0.79 38 175 272 
SGMKIIALRSSLKLAARIAEELKTEPVMPDERRFPDGELYLRYDEDLTGHNIFIIGNTHSDAEVMEMILTLSAIQDYRTKSVNIIAPYYGYARQHQRYKNGEPISSQILTEIYSSYSNSIATVDIHDEKTLSYSKVKFSDLHANDAIVRYYKNVDVDYVVSPDDGGLARVADISAKLGKKHFFIEKKRIDDRTVEMKVPNVDVNGKKLLIVDDIISTGGTIAKSSGLLREKGASKIYVSAVHGLFVNGSENKILQNADEIHVTDTVESKFSDISVYQEVCNYIRDID
8CEP , Knot 282 1540 0.44 8 16 64 
AAAUUGAAGAGUUUGAUCAUGGCUCAGAUUGAACGCUGGCGGCAGGCCUAACACAUGCAAGUCGAACGGUAACAGGAAGAAGCUUGCUUCUUUGCUGACGAGUGGCGGACGGGUGAGUAAUGUCUGGGAAACUGCCUGAUGGAGGGGGAUAACUACUGGAAACGGUAGCUAAUACCGCAUAACGUCGCAAGACCAAAGAGGGGGACCUUCGGGCCUCUUGCCAUCGGAUGUGCCCAGAUGGGAUUAGCUAGUAGGUGGGGUAACGGCUCACCUAGGCGACGAUCCCUAGCUGGUCUGAGAGGAUGACCAGCCACACUGGAACUGAGACACGGUCCAGACUCCUACGGGAGGCAGCAGUGGGGAAUAUUGCACAAUGGGCGCAAGCCUGAUGCAGCCAUGCCGCGUGUAUGAAGAAGCCCUUCGGGUUGUAAAGUACUUUCAGCGGGGAGGAAGGGAGUAAAGUUAAUACCUUUGCUCAUUGACGUUACCCGCAGAAGAAGCACCGGCUAACUCCGUGCCAGCAGCCGCGGUAAUACGGAGGGUGCAAGCGUUAAUCGGAAUUACUGGGCGUAAAGCGCACGCAGGCGGUUUGUUAAGUCAGAUGUGAAAUCCCCGGGCUCAACCUGGGAACUGCAUCUGAUACUGGCAAGCUUGAGUCUCGUAGAGGGGGGUAGAAUUCCAGGUGUAGCGGUGAAAUGCGUAGAGAUCUGGAGGAAUACCGGUGGCGAAGGCGGCCCCCUGGACGAAGACUGACGCUCAGGUGCGAAAGCGUGGGGAGCAAACAGGAUUAGAUACCCUGGUAGUCCACGCCGUAAACGAUGUCGACUUGGAGGUUGUGCCCUUGAGGCGUGGCUUCCGGAGCUAACGCGUUAAGUCGACCGCCUGGGGAGUACGGCCGCAAGGUUAAAACUCAAAUGAAUUGACGGGGGCCCGCACAAGCGGUGGAGCAUGUGGUUUAAUUCGAUGCAACGCGAAGAACCUUACCUGGUCUUGACAUCCACGGAAGUUUUCAGAGAUGAGAAUGUGCCUUCGGGAACCGUGAGACAGGUGCUGCAUGGCUGUCGUCAGCUCGUGUUGUGAAAUGUUGGGUUAAGUCCCGCAACGAGCGCAACCCUUAUCCUUUGUUGCCAGCGGUCCGGCCGGGAACUCAAAGGAGACUGCCAGUGAUAAACUGGAGGAAGGUGGGGAUGACGUCAAGUCAUCAUGGCCCUUACGACCAGGGCUACACACGUGCUACAAUGGCGCAUACAAAGAGAAGCGACCUCGCGAGAGCAAGCGGACCUCAUAAAGUGCGUCGUAGUCCGGAUUGGAGUCUGCAACUCGACUCCAUGAAGUCGGAAUCGCUAGUAAUCGUGGAUCAGAAUGCCACGGUGAAUACGUUCCCGGGCCUUGUACACACCGCCCGUCACACCAUGGGAGUGGGUUGCAAAAGAAGUAGGUAGCUUAACCUUCGGGAGGGCGCUUACCACUUUGUGAUUCAUGACUGGGGUGAAGUCGUAACAAGGUAACCGUAGGGGAACCUGCGGUUGGAUCACCUCCU
5AIQ , Knot 72 146 0.82 40 105 140 
MPTQSKHASINIGLIQAREALMTQFRPILNQANITDQQWRIIRLLAENGTLDFQDLANQACILRPSLTGILTRLEKAGLVVRLKPSNDQRRVFLKLTAEGEKLYEEIGEEVDERYDAIEEVLGREKMLLLKDLLAELAKIEDALNS

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 \(c\).

\(|P_{f(3MBI_1)}(2) \setminus P_{f(8CEP_1)}(2)|=172\), \(|P_{f(8CEP_1)}(2) \setminus P_{f(3MBI_1)}(2)|=13\). 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 1:01101111000101110110010001111000011010101000001010011111000001011011101011000000010111100101000000001011000110010000000110101000001000010100101001110000010100110100111101101010110001110000100001010110101010011110011001101100011100011001010110111101000011001001010001000100101000100010010
Pair \(Z_2\) Length of longest common subsequence
3MBI_1,8CEP_1 185 2
3MBI_1,5AIQ_1 154 3
8CEP_1,5AIQ_1 117 2

Newick tree

 
[
	3MBI_1:92.28,
	[
		5AIQ_1:58.5,8CEP_1:58.5
	]:33.78
]

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 \((0.85)(0.8)(\frac{1827 }{\log_{20} 1827}-\frac{287}{\log_{20}287})=392.\)
Status Protein1 Protein2 d d1/2
Query variables 3MBI_1 8CEP_1 282 200
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]