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Parikh vectors
5ZEU_1 1UBI_1 4INS_1 Letter Amino acid
522 6 1 G Glycine
0 9 2 L Leucine
0 1 0 M Methionine
0 2 0 F Phenylalanine
366 0 4 C Cysteine
0 6 2 Q Glutamine
0 7 2 I Isoleucine
0 7 0 K Lycine
0 4 1 V Valine
0 0 0 W Tryptophan
329 2 0 A Alanine
0 4 0 R Arginine
0 2 2 N Asparagine
0 3 2 S Serine
0 7 1 T Threonine
0 5 0 D Aspartic acid
0 6 2 E Glutamic acid
0 1 0 H Histidine
0 3 0 P Proline
0 1 2 Y Tyrosine 

5ZEU_1|Chain A[auth a]|16S rRNA|Mycobacterium smegmatis (strain ATCC 700084 / mc(2)155) (246196)
>1UBI_1|Chain A|UBIQUITIN|Homo sapiens (9606)
>4INS_1|Chains A, C|INSULIN (CHAIN A)|Sus scrofa (9823)
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 \(w=f(c)\)
5ZEU , Knot 276 1528 0.44 8 16 64 
UUUUUGUUUGGAGAGUUUGAUCCUGGCUCAGGACGAACGCUGGCGGCGUGCUUAACACAUGCAAGUCGAACGGAAAGGCCCUUUCGGGGGUACUCGAGUGGCGAACGGGUGAGUAACACGUGGGUGAUCUGCCCUGCACUUUGGGAUAAGCCUGGGAAACUGGGUCUAAUACCGAAUACACCCUGCUGGUCGCAUGGCCUGGUAGGGGAAAGCUUUUGCGGUGUGGGAUGGGCCCGCGGCCUAUCAGCUUGUUGGUGGGGUGAUGGCCUACCAAGGCGACGACGGGUAGCCGGCCUGAGAGGGUGACCGGCCACACUGGGACUGAGAUACGGCCCAGACUCCUACGGGAGGCAGCAGUGGGGAAUAUUGCACAAUGGGCGCAAGCCUGAUGCAGCGACGCCGCGUGAGGGAUGACGGCCUUCGGGUUGUAAACCUCUUUCAGCACAGACGAAGCGCAAGUGACGGUAUGUGCAGAAGAAGGACCGGCCAACUACGUGCCAGCAGCCGCGGUAAUACGUAGGGUCCGAGCGUUGUCCGGAAUUACUGGGCGUAAAGAGCUCGUAGGUGGUUUGUCGCGUUGUUCGUGAAAACUCACAGCUUAACUGUGGGCGUGCGGGCGAUACGGGCAGACUAGAGUACUGCAGGGGAGACUGGAAUUCCUGGUGUAGCGGUGGAAUGCGCAGAUAUCAGGAGGAACACCGGUGGCGAAGGCGGGUCUCUGGGCAGUAACUGACGCUGAGGAGCGAAAGCGUGGGGAGCGAACAGGAUUAGAUACCCUGGUAGUCCACGCCGUAAACGGUGGGUACUAGGUGUGGGUUUCCUUCCUUGGGAUCCGUGCCGUAGCUAACGCAUUAAGUACCCCGCCUGGGGAGUACGGCCGCAAGGCUAAAACUCAAAGGAAUUGACGGGGGCCCGCACAAGCGGCGGAGCAUGUGGAUUAAUUCGAUGCAACGCGAAGAACCUUACCUGGGUUUGACAUGCACAGGACGCCGGCAGAGAUGUCGGUUCCCUUGUGGCCUGUGUGCAGGUGGUGCAUGGCUGUCGUCAGCUCGUGUCGUGAGAUGUUGGGUUAAGUCCCGCAACGAGCGCAACCCUUGUCUCAUGUUGCCAGCACGUUAUGGUGGGGACUCGUGAGAGACUGCCGGGGUCAACUCGGAGGAAGGUGGGGAUGACGUCAAGUCAUCAUGCCCCUUAUGUCCAGGGCUUCACACAUGCUACAAUGGCCGGUACAAAGGGCUGCGAUGCCGUGAGGUGGAGCGAAUCCUUUCAAAGCCGGUCUCAGUUCGGAUCGGGGUCUGCAACUCGACCCCGUGAAGUCGGAGUCGCUAGUAAUCGCAGAUCAGCAACGCUGCGGUGAAUACGUUCCCGGGCCUUGUACACACCGCCCGUCACGUCAUGAAAGUCGGUAACACCCGAAGCCGGUGGCCUAACCCUUGUGGAGGGAGCCGUCGAAGGUGGGAUCGGCGAUUGGGACGAAGUCGUAACAAGGUAGCCGUACCGGAAGGUGCGGCUGGAUCACCUCCUUUCU
1UBI , Knot 40 76 0.76 36 61 74 
MQIFVKTLTGKTITLEVEPSDTIENVKAKIQDKEGIPPDQQRLIFAGKQLEDGRTLSDYNIQKESTLHLVLRLRGG
4INS , Knot 15 21 0.72 22 20 19 
GIVEQCCTSICSLYQLENYCN

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 \(c\).

\(|P_{f(5ZEU_1)}(2) \setminus P_{f(1UBI_1)}(2)|=14\), \(|P_{f(1UBI_1)}(2) \setminus P_{f(5ZEU_1)}(2)|=59\). 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 1:0000010001111110001100001100011110111010011011010100011010101011100111011111110000000111110100011101101110111011101101010111011000100001010000111101110001111110011100011010011101010000100110010101100011011111111100000101101011110111000101100010011000100110111101101100010011110110110111011001100011111110110011001010011110011110101100011100000101111110110110111111010010101101110101110001101011011010010101111110110110000011100101110000000011010111011110101110110110101010111111111100110011001010100110110010110110101011110001110100100011110010011101011111100010111011000100101001000101111100010110001100101110101011101101011101110011110100101111111100111100000110101101101111010101110100111111110100110110111110111000001110110110011010011111101111101011111101110111100111010000110110001010010111011011101001110101110000000000011110001010010110011010100111010000100011111101011001011110011110001111111001101111100010101110110111101010111001100011010110101111110000100011100011010101011110100110111110100110000000101100010101011101101010110010010011000101001011110100111001110000101101110101100000100001010010011010100101101111100010111111001001111001100011111111101111101101001110010010100000010100011110000101010100101101100110101111110010110100101111011110111000000011110011000011000111001111000101100011000010111100111100100110110010111001101101001011011101010000011100001010101001000100101001011111001101101000111100110110001100000101111111100100111110111100110110011110111100101101111011001010011111101011001110010000000000
Pair \(Z_2\) Length of longest common subsequence
5ZEU_1,1UBI_1 73 2
5ZEU_1,4INS_1 34 2
1UBI_1,4INS_1 71 3

Newick tree

 
[
	1UBI_1:40.39,
	[
		5ZEU_1:17,4INS_1:17
	]:23.39
]

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 \((0.85)(0.8)(\frac{1604 }{\log_{20} 1604}-\frac{76}{\log_{20}76})=406.\)
Status Protein1 Protein2 d d1/2
Query variables 5ZEU_1 1UBI_1 274 156.5
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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