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Parikh vectors
9QQA_1 8SZG_1 4YZZ_1 Letter Amino acid
0 30 16 D Aspartic acid
0 12 17 Q Glutamine
0 8 4 H Histidine
419 30 47 A Alanine
2 17 19 N Asparagine
0 12 22 K Lycine
0 7 13 M Methionine
0 12 14 F Phenylalanine
0 9 10 W Tryptophan
0 18 21 V Valine
495 14 4 C Cysteine
550 40 37 G Glycine
0 28 14 T Threonine
0 8 16 Y Tyrosine
0 23 16 R Arginine
0 31 27 L Leucine
0 5 18 P Proline
0 37 18 S Serine
0 9 19 E Glutamic acid
0 19 15 I Isoleucine 

9QQA_1|Chain A[auth A2]|18S rRNA|Oryctolagus cuniculus (9986)
>8SZG_1|Chain A[auth D]|Guanine nucleotide-binding protein G(I)/G(S)/G(T) subunit beta-1|Homo sapiens (9606)
>4YZZ_1|Chains A, B[auth D]|TRAP TRANSPORTER SOLUTE BINDING PROTEIN|BORDETELLA BRONCHISEPTICA (257310)
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 \(w=f(c)\)
9QQA , Knot 344 1870 0.46 12 20 71 
UACCUGGUUGAUCCUGCCAGUAGCAUAUGCUUGUCUCAAAGAUUAAGCCAUGCAUGUCUAAGUACGCACGGCCGGUACAGUGAAACUGCGAAUGGCUCAUUAAAUCAGUUAUGGUUCCUUUGGUCGCUCGCUCCUCUCCUACUUGGAUAACUGUGGUAAUUCUAGAGCUAAUACAUGCCGACGGGCGCUGACCCCCUUCGCGGGGGGGAUGCGUGCAUUUAUCAGAUCAAAACCAACCCGGUCAGCCUCCCCCCGGCCCCGGCCGGGGGGGUGGGCGCCGGCGGCUUUGGUGACUCUAGAUAACCUCGGGCCGAUCGCACGCCCUCCGUGGCGGCGACGACCCAUUCGAACGUCUGCCCUAUCAACUUUCGAUGGUAGUCGCCGUGCCUACCAUGGUGACCACGGGUGACGGGGAAUCAGGGUUCGAUUCCGGAGAGGGAGCCUGAGAAACGGCUACCACAUCCAAGGAAGGCAGCAGGCGCGCAAAUUACCCACUCCCGACCCGGGGAGGUAGUGACGAAAAAUAACAAUACAGGACUCUUUCGAGGCCCUGUAAUUGGAAUGAGUCCACUUUAAAUCCUUUAACGAGGAUCCAUUGGAGGGCAAGUCUGGUGCCAGCAGCCGCGGUAAUUCCAGCUCCAAUAGCGUAUAUUAAAGUUGCUGCAGUUAAAAAGCUCGUAGUUGGAUCUUGGGAGCGGGCGGGCGGUCCGCCGCGAGGCGAGCCACCGCCCGUCCCCGCCCCUUGCCUCUCGGCGCCCCCUCGAUGCUCUUAGCUGAGUGUCCCGCGGGGCCCGAAGCGUUUACUUUGAAAAAAUUAGAGUGUUCAAAGCAGGCCCGAGCCGCCUGGAUACCGCAGCUAGGAAUAAUGGAAUAGGACCGCGGUUCUAUUUUGUUGGUUUUCGGAACUGAGGCCAUGAUUAAGAGGGACGGCCGGGGGCAUUCGUAUUGCGCCGCUAGAGGUGAAAUUCUUGGACCGGCGCAAGACGGACCAGAGCGAAAGCAUUUGCCAAGAAUGUUUUCAUUAAUCAAGAACGAAAGUCGGAGGUUCGAAGACGAUCAGAUACCGUCGUAGUUCCGACCAUAAACGAUGCCGACUGGCGAUGCGGCGGCGUUAUUCCCAUGACCCGCCGGGCAGCUUCCGGGAAACCAAAGUCUUUGGGUUCCGGGGGGAGUAUGGUUGCAAAGCUGAAACUUAAAGGAAUUGACGGAAGGGCACCACCAGGAGUGGAGCCUGCGGCUUAAUUUGACXCAACACGGGAAACCUCACCCGGCCCGGACACGGACAGGAUUGACAGAUUGAUAGCUCUUUCUCGAUUCCGUGGGUGGUGGUGCAUGGCNGUUCUUAGUUGGUGGAGCGAUUUGUCUGGUUAAUUCCGAUAACGAACGAGACUCUGGCAUGCUAACUAGUUACGCGACCCCCGAGCGGUCGGCGUCCCCCAACUUCUUAGAGGGACAAGUGGCGUUCAGCCACCCGAGAUUGAGCAAUAACAGGUCUGUGAUGCCCUUAGAUGUCCGGGGCUGCACGCGCGCUACACUGACUGGCUCAGCGUGUGCCUACCCUACGCCGGCAGGCGCGGGUAACCCGUUGAACCCCAUUCGUGAUGGGGAUCGGGGAUUGCAAUUAUUCCCCAUGAACGAGGAAUUCCCAGUAAGUGCGGGUCAUAAGCUUGCGUUGAUUAAGUCCCUGCCCUUUGUACACACCGCCCGUCGCUACUACCGAUUGGAUGGUUUAGUGAGGCCCUCGGAUCGGCCCCGCCGGGGUCGGCCCACGGCCCUGGCGGAGCGCUGAGAAGACGGUCGAACUUGACUAUCUAGAGGAAGUAAAAGUCGUAACAAGGUUUCNGUAGGUGAACCUGCGGAAGGAUCAUUA
8SZG , Knot 153 369 0.81 40 197 334 
GSSGSELDQLRQEAEQLKNQIRDARKACADATLSQITNNIDPVGRIQMRTRRTLRGHLAKIYAMHWGTDSRLLVSASQDGKLIIWDSYTTNKVHAIPLRSSWVMTCAYAPSGNYVACGGLDNICSIYNLKTREGNVRVSRELAGHTGYLSCCRFLDDNQIVTSSGDTTCALWDIETGQQTTTFTGHTGDVMSLSLAPDTRLFVSGACDASAKLWDVREGMCRQTFTGHESDINAICFFPNGNAFATGSDDATCRLFDLRADQELMTYSHDNIICGITSVSFSKSGRLLLAGYDDFNCNVWDALKADRAGVLAGHDNRVSCLGVTDDGMAVATGSWDSFLKIWNGSSGGGGSGGGGSSGVSGWRLFKKIS
4YZZ , Knot 152 367 0.81 40 213 347 
MQRRRFLAQAAGAAGAGLAAVGMPAMAQANPTVRWRMSTSWPKSLDTIYGSADELCKRVGQLTDGKFEIRAFPGGELVPSAQNMDAVSNGTVECNHVLSTMYIGKNTALTFDTGLSFGLNARQHNAWIHYGGGLQQLRELYKKYNIVNHVCGNVGVQMGGWYRKEIKSTADLNGLNMRIGGIGGMVLSKLGVVPQQIPPGDIYPALEKGTIDAAEWIGPYDDEKLGFNKVAPYYYSPGWFEGSASITSMVNDKAWEALPPAYQAAFEAACGEQSMRMLANYDARNPLALRKLIAGGAKVSFFPKEVMDAVYKASQQLWTELSEKNPDFKAIYPGWKKFQEDEAGWFRVAENALDNYTFAAVARAQAK

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 \(c\).

\(|P_{f(9QQA_1)}(2) \setminus P_{f(8SZG_1)}(2)|=11\), \(|P_{f(8SZG_1)}(2) \setminus P_{f(9QQA_1)}(2)|=188\). 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 1:0100011001100001001101101010100010000111110011100101010100011101010101100110101101111001011101100010011100110010110000000110010001000000000010001110110010110110000111100110101010011011101001100000000101111111101010101000100111001111001100011001100000000011000011001111111011101001101100001101100001110110000111001100101010000001011011011011000100011101000100001001100000110110110010010100010010110110010111011011111100111100011000011111111110001111110110010010100011111111011011101010111001000100000110001111111011011011111101101101011110000000111100001011001111011100010000111000000110111110001001111110111000110100110110010110110000110000110110101010011110010010110011111100010110011100001111101110111011000100101111011100100100010000010000001000000110100000001101000001100111010000101111000111101000100001111111001111010001111011100011100100011101001011001111101101111011110010110000100001001100000111100111100101100111111110110011111010001010010100100111110111100000111001101011110111001111011111010001001111101000001001100111110111110011111000111110110011101001001011000011001011101101001100110110101101101001000001011000100111011000001111110011110000011100001111111101011001011110011110001111111001101111111010010011111011110001011000110001100011010111111000010001100011101011101111001101110011011000000000110000101110110110101011001000001100110111101100010001100110000110110111011110000110101001100110010101100000111011001101000000110000001111111011101101000110010001111001110110110111000101101000001110100011110010101010100101001100110001101010100010000101001101110101110110001001110000100010110111110011111001011001000000101110111111000001101110101110010111000101001100111000001000000101010100100010010010010011001110110001101111000001110011000010011110011000101100001101111010011111110110011100011001000111111110111110010110111100000101110111000101111111001001
Pair \(Z_2\) Length of longest common subsequence
9QQA_1,8SZG_1 199 4
9QQA_1,4YZZ_1 217 6
8SZG_1,4YZZ_1 178 3

Newick tree

 
[
	9QQA_1:10.66,
	[
		8SZG_1:89,4YZZ_1:89
	]:19.66
]

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 \((0.85)(0.8)(\frac{2239 }{\log_{20} 2239}-\frac{369}{\log_{20}369})=464.\)
Status Protein1 Protein2 d d1/2
Query variables 9QQA_1 8SZG_1 342 245
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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