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Parikh vectors
9IJV_1 5YCG_1 4WYC_1 Letter Amino acid
21 2 26 R Arginine
8 6 12 Q Glutamine
38 16 44 G Glycine
9 11 24 H Histidine
23 8 19 I Isoleucine
4 3 15 M Methionine
11 7 12 F Phenylalanine
12 14 67 A Alanine
42 6 24 S Serine
12 2 10 Y Tyrosine
20 18 46 L Leucine
11 2 9 W Tryptophan
31 7 40 V Valine
23 4 3 N Asparagine
18 0 8 C Cysteine
19 13 15 E Glutamic acid
19 19 4 K Lycine
14 4 26 P Proline
27 6 28 T Threonine
25 9 25 D Aspartic acid

9IJV_1|Chains A, B, C, D|Neuraminidase|Influenza A virus (11320)
>5YCG_1|Chain A|Ancestral myoglobin aMbWp of Pakicetus relative|Physeter catodon (9755)
>4WYC_1|Chains A, B|Adenosylmethionine-8-amino-7-oxononanoate aminotransferase|Mycobacterium tuberculosis (83332)
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 \(w=f(c)\)
9IJV , Knot 166 387 0.85 40 217 367
EYRNWSKPQCGITGFAPFSKDNSIRLSAGGDIWVTREPYVSCDPDKCYQFALGQGTTLNNVHSNNTVRDRTPYRTLLMNELGVPFHLGTKQVCIAWSSSSCHDGKAWLHVCITGDDKNATASFIYNGRLVDSVVSWSKDILRTQESECVCINGTCTVVMTDGSASGKADTKILFIEEGKIVHTSTLSGSAQHVEECSCYPRYPGVRCVCRDNWKGSNRPIVDINIKDHSIVSSYVCSGLVGDTPRKNDSSSSSHCLDPNNEEGGHGVKGWAFDDGNDVWMGRTINETSRLGYETFKVIEGWSNPKSKLQTNRQVIVDRGDRSGYSGIFSVEGKSCINRCFYVELIRGRKEETEVLWTSNSIVVFCGTSGTYGTGSWPDGADLNLMPI
5YCG , Knot 77 157 0.82 38 121 154
GSHMGLSDGEWQLVLNVWGKVEADVAGHGQEVLIRLFTGHPETLEKFDKFKHLKTEDEMKASEDLKKHGNTVLTALGGILKKKGHHEAELKPLAQSHATKHKIPIKYLEFISDAIIHVLQSKHPGDFGADAQGAMNKALELFRNDIAAKYKELGFQG
4WYC , Knot 185 457 0.82 40 219 411
MGSSHHHHHHSSGLVPRGSHMAAATGGLTPEQIIAVDGAHLWHPYSSIGREAVSPVVAVAAHGAWLTLIRDGQPIEVLDAMSSWWTAIHGHGHPALDQALTTQLRVMNHVMFGGLTHEPAARLAKLLVDITPAGLDTVFFSDSGSVSVEVAAKMALQYWRGRGLPGKRRLMTWRGGYHGDTFLAMSICDPHGGMHSLWTDVLAAQVFAPQVPRDYDPAYSAAFEAQLAQHAGELAAVVVEPVVQGAGGMRFHDPRYLHDLRDICRRYEVLLIFDEIATGFGRTGALFAADHAGVSPDIMCVGKALTGGYLSLAATLCTADVAHTISAGAAGALMHGPTFMANPLACAVSVASVELLLGQDWRTRITELAAGLTAGLDTARALPAVTDVRVCGAIGVIECDRPVDLAVATPAALDRGVWLRPFRNLVYAMPPYICTPAEITQITSAMVEVARLVGSLP

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 \(c\).

\(|P_{f(9IJV_1)}(2) \setminus P_{f(5YCG_1)}(2)|=146\), \(|P_{f(5YCG_1)}(2) \setminus P_{f(9IJV_1)}(2)|=50\). 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 1:000010010011011111000001010111011100010100010000011110100100100000100001000111001111101100010111000000010111010101000010101100101100110100011000000010101000111001010101000111100101100001010100100000010011100100001010001110101000011000100111100100000000000101000011011011110010011110010000011000101101100100010000011100100010011101010001000101011010000001110000111101001001010110110101111
Pair \(Z_2\) Length of longest common subsequence
9IJV_1,5YCG_1 196 3
9IJV_1,4WYC_1 176 4
5YCG_1,4WYC_1 186 4

Newick tree

 
[
	5YCG_1:97.91,
	[
		9IJV_1:88,4WYC_1:88
	]:9.91
]

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 \((0.85)(0.8)(\frac{544 }{\log_{20} 544}-\frac{157}{\log_{20}157})=112.\)
Status Protein1 Protein2 d d1/2
Query variables 9IJV_1 5YCG_1 142 99
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]