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Parikh vectors
8CAI_1 9BVF_1 8UXQ_1 Letter Amino acid
0 34 9 R Arginine
0 26 10 N Asparagine
0 51 29 E Glutamic acid
487 18 9 G Glycine
0 51 11 L Leucine
0 7 4 M Methionine
388 17 11 A Alanine
0 32 7 F Phenylalanine
0 31 7 P Proline
0 5 4 W Tryptophan
0 30 9 V Valine
352 5 3 C Cysteine
0 21 3 Q Glutamine
0 17 2 H Histidine
0 22 7 T Threonine
0 17 5 Y Tyrosine
0 19 13 D Aspartic acid
0 36 7 I Isoleucine
0 40 29 K Lycine
0 35 16 S Serine 

8CAI_1|Chain A|16S rRNA|Escherichia coli BW25113 (679895)
>9BVF_1|Chain A|Son of sevenless homolog 2|Homo sapiens (9606)
>8UXQ_1|Chains A, B[auth C]|Chromobox protein homolog 5|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 \(w=f(c)\)
8CAI , Knot 282 1540 0.44 8 16 64 
AAAUUGAAGAGUUUGAUCAUGGCUCAGAUUGAACGCUGGCGGCAGGCCUAACACAUGCAAGUCGAACGGUAACAGGAAGAAGCUUGCUUCUUUGCUGACGAGUGGCGGACGGGUGAGUAAUGUCUGGGAAACUGCCUGAUGGAGGGGGAUAACUACUGGAAACGGUAGCUAAUACCGCAUAACGUCGCAAGACCAAAGAGGGGGACCUUCGGGCCUCUUGCCAUCGGAUGUGCCCAGAUGGGAUUAGCUAGUAGGUGGGGUAACGGCUCACCUAGGCGACGAUCCCUAGCUGGUCUGAGAGGAUGACCAGCCACACUGGAACUGAGACACGGUCCAGACUCCUACGGGAGGCAGCAGUGGGGAAUAUUGCACAAUGGGCGCAAGCCUGAUGCAGCCAUGCCGCGUGUAUGAAGAAGGCCUUCGGGUUGUAAAGUACUUUCAGCGGGGAGGAAGGGAGUAAAGUUAAUACCUUUGCUCAUUGACGUUACCCGCAGAAGAAGCACCGGCUAACUCCGUGCCAGCAGCCGCGGUAAUACGGAGGGUGCAAGCGUUAAUCGGAAUUACUGGGCGUAAAGCGCACGCAGGCGGUUUGUUAAGUCAGAUGUGAAAUCCCCGGGCUCAACCUGGGAACUGCAUCUGAUACUGGCAAGCUUGAGUCUCGUAGAGGGGGGUAGAAUUCCAGGUGUAGCGGUGAAAUGCGUAGAGAUCUGGAGGAAUACCGGUGGCGAAGGCGGCCCCCUGGACGAAGACUGACGCUCAGGUGCGAAAGCGUGGGGAGCAAACAGGAUUAGAUACCCUGGUAGUCCACGCCGUAAACGAUGUCGACUUGGAGGUUGUGCCCUUGAGGCGUGGCUUCCGGAGCUAACGCGUUAAGUCGACCGCCUGGGGAGUACGGCCGCAAGGUUAAAACUCAAAUGAAUUGACGGGGGCCCGCACAAGCGGUGGAGCAUGUGGUUUAAUUCGAUGCAACGCGAAGAACCUUACCUGGUCUUGACAUCCACGGAAGUUUUCAGAGAUGAGAAUGUGCCUUCGGGAACCGUGAGACAGGUGCUGCAUGGCUGUCGUCAGCUCGUGUUGUGAAAUGUUGGGUUAAGUCCCGCAACGAGCGCAACCCUUAUCCUUUGUUGCCAGCGGUCCGGCCGGGAACUCAAAGGAGACUGCCAGUGAUAAACUGGAGGAAGGUGGGGAUGACGUCAAGUCAUCAUGGCCCUUACGACCAGGGCUACACACGUGCUACAAUGGCGCAUACAAAGAGAAGCGACCUCGCGAGAGCAAGCGGACCUCAUAAAGUGCGUCGUAGUCCGGAUUGGAGUCUGCAACUCGACUCCAUGAAGUCGGAAUCGCUAGUAAUCGUGGAUCAGAAUGCCACGGUGAAUACGUUCCCGGGCCUUGUACACACCGCCCGUCACACCAUGGGAGUGGGUUGCAAAAGAAGUAGGUAGCUUAACCUUCGGGAGGGCGCUUACCACUUUGUGAUUCAUGACUGGGGUGAAGUCGUAACAAGGUAACCGUAGGGGAACCUGCGGUUGGAUCACCUCCU
9BVF , Knot 208 514 0.84 40 247 481 
MSYYHHHHHHLESTSLYKKAGFENLYFQEQPLRLPSPEVYRFVVKDSEENIVFEDNLQSRSGIPIIKGGTVVKLIERLTYHMYADPNFVRTFLTTYRSFCKPQELLSLLIERFEIPEPEPTDADKLAIEKGEQPISADLKRFRKEYVQPVQLRILNVFRHWVEHHFYDFERDLELLERLESFISSVRGKAMKKWVESIAKIIRRKKQAQANGVSHNITFESPPPPIEWHISKPGQFETFDLMTLHPIEIARQLTLLESDLYRKVQPSELVGSVWTKEDKEINSPNLLKMIRHTTNLTLWFEKCIVEAENFEERVAVLSRIIEILQVFQDLNNFNGVLEIVSAVNSVSVYRLDHTFEALQERKRKILDEAVELSQDHFKKYLVKLKSINPPCVPFFGIYLTNILKTEEGNNDFLKKKGKDLINFSKRRKVAEITGEIQQYQNQPYCLRIEPDMRRFFENLNPMGSASEKEFTDYLFNKSLEIEPRNCKQPPRFPRKSTFSLKSPGIRPNTGRHGS
8UXQ , Knot 87 195 0.78 40 127 189 
GSPEFGKKTKRTADSSSSEDEEEYVVEKVLDRRVVKGQVEYLLKWKGFSEEHNTWEPEKNLDCPELISEFMKKYKKMKEGENNKPREKSESNKRKSNFSNSADDIKSKKKREQSNDIARGFERGLEPEKIIGATDSCGDLMFLMKWKDTDEADLVLAKEANVKCPQIVIAFYEERLTWHAYPEDAENKEKETAKS

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 \(c\).

\(|P_{f(8CAI_1)}(2) \setminus P_{f(9BVF_1)}(2)|=14\), \(|P_{f(9BVF_1)}(2) \setminus P_{f(8CAI_1)}(2)|=245\). 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 1:1110011111100011001011000111001110100110110111000110101010111001110110110111111111000100000001001101110110111011101110110100011111100100011011111111101100100111110110110011010010101101001011110011111111111000001110000001001001110101000111011110011001101110111101101100010001110110110000011001100011111110110011001010011110011110101100011100000101111110110110111111010010101101110101110001101011001010010101010111111110000011100101111010000011011111111111111011110011010000010001001101001000101111111101001100110000101001101100101101101011111101011101001100111100100111010111101010101110110001001110011101011110000011100011000111110010100011010011011100011100001011111111101111000011101011011011110101011111000111111101001101101111101100000011101111100110100011101011111010111111011101111001110100001101100010100101110110100110001111100101000001111010110000011110011010100111001100100011111101011001011110011110001110111001101111100010101110110111101010110001100011010110101111110000100011000011010001011111000001111101111101010000011111001011110111010010101100100100110001010010111101001110011100001011011101011000001000000100100110110001100111110001111111100100110110111001111111110111110110100111001001011000001011001111001010101010010110110101010111111111011000010111110111011100001011110101001011000111001111000101100011000010111100111100100110110010111001111010010110111010100000111000010101010010001001010010111110111001011111111011101100011000001111111010001001000010110001011001111011110010110111101100101111111000101100111001000000
Pair \(Z_2\) Length of longest common subsequence
8CAI_1,9BVF_1 259 2
8CAI_1,8UXQ_1 139 2
9BVF_1,8UXQ_1 178 4

Newick tree

 
[
	9BVF_1:12.86,
	[
		8CAI_1:69.5,8UXQ_1:69.5
	]:52.36
]

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 \((0.85)(0.8)(\frac{2054 }{\log_{20} 2054}-\frac{514}{\log_{20}514})=380.\)
Status Protein1 Protein2 d d1/2
Query variables 8CAI_1 9BVF_1 281 243.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]

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