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Parikh vectors
7JFL_1 6OVB_1 2NRU_1 Letter Amino acid
9 23 21 D Aspartic acid
7 19 11 Q Glutamine
24 37 18 G Glycine
4 4 10 M Methionine
4 28 15 N Asparagine
16 29 27 E Glutamic acid
5 6 23 K Lycine
15 55 21 S Serine
4 28 9 Y Tyrosine
9 38 21 A Alanine
14 44 10 R Arginine
5 10 9 H Histidine
4 46 19 I Isoleucine
9 8 1 W Tryptophan
5 4 5 C Cysteine
24 51 28 L Leucine
8 35 12 F Phenylalanine
15 33 9 P Proline
11 47 17 T Threonine
21 32 21 V Valine 

7JFL_1|Chains A, C[auth B]|Interferon regulatory factor 3|Homo sapiens (9606)
>6OVB_1|Chain A|Active core crystal toxin protein 1D|Bacillus thuringiensis (1428)
>2NRU_1|Chains A, B, C, D|Interleukin-1 receptor-associated kinase 4|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 \(w=f(c)\)
7JFL , Knot 97 213 0.81 40 147 205 
SEFENPLKRLLVPGEEWEFEVTAFYRGRQVFQQTISCPEGLRLVGSEVGDRTLPGWPVTLPDPGMSLTDRGVMSYVRHVLSCLGGGLALWRAGQWLWAQRLGHCHTYWAVSEELLPNSGHGPDGEVPKDKEGGVFDLGPFIVDLITFTEGSGRSPRYALWFCVGESWPQDQPWTKRLVMVKVVPTCLRALVEMARVGGASSLENTVDLHISNS
6OVB , Knot 230 577 0.84 40 249 534 
RLETGNTVADISLGLINFLYSNFVPGGGFIVGLLELIWGFIGPSQWDIFLAQIEQLISQRIEEFARNQAISRLEGLSNLYKCYVRAFSDWEKDPTNPALRECMRIQFNDMNSALITAIPLFRVQNYEVALLSVYVQAANLHLSILRDVSVFGERWGYDTATINNRYSDLTSLIHVYTNHCVDTYNQGLRRLEGRFLSDWIVYNRFRRQLTISVLDIVAFFPNYDIRTYPIQTATQLTREVYLDLPFINENLSPAAVYPTFSAAESAIIRSPHLVDFLNSFTIYTDSLARSAYWGGHLVNSFRTGTTTNLIRSPLYGREGNTERPVTITASPSVPIFRTLSYPTGLDNSNPVAGIEGVEFQNTISRSIYRKSGPIDSFSELPPQDASVSPAIGYSHRLCHATFLERISGPRIAGTVFSWTHRSASPTNEVSPSRITQIPWVKAHTLASGASVIKGPGFTGGDILTRNSMGELGTLRVTFTGRLPQSYYIRFRYASVANRSGTFRYSQPPSYGISFPKTMDAGEPLTSRSFAHTTLFTPITFSRAQEEFDLYIQSGVYIDRIEFIPVTATFEAEYDLER
2NRU , Knot 135 307 0.84 40 193 294 
ENKSLEVSDTRFHSFSFYELKNVTNNFDERPISVGGNKMGEGGFGVVYKGYVNNTTVAVKKLAAMVDITTEELKQQFDQEIKVMAKCQHENLVELLGFSSDGDDLCLVYVYMPNGSLLDRLSCLDGTPPLSWHMRCKIAQGAANGINFLHENHHIHRDIKSANILLDEAFTAKISDFGLARASEKFAQTVMTSRIVGTTAYMAPEALRGEITPKSDIYSFGVVLLEIITGLPAVDEHREPQLLLDIKEEIEDEEKTIEDYIDKKMNDADSTSVEAMYSVASQCLHEKKNKRPDIKKVQQLLQEMTAS

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 \(c\).

\(|P_{f(7JFL_1)}(2) \setminus P_{f(6OVB_1)}(2)|=40\), \(|P_{f(6OVB_1)}(2) \setminus P_{f(7JFL_1)}(2)|=142\). 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 1:001001100111110010101011001001100010010110111001100011111101101110100011100100110011111111011011110011000001110001110010110101100001111011111101101001010010011110110011000110001111011100101110110111100100010101000
Pair \(Z_2\) Length of longest common subsequence
7JFL_1,6OVB_1 182 3
7JFL_1,2NRU_1 184 4
6OVB_1,2NRU_1 174 4

Newick tree

 
[
	7JFL_1:92.95,
	[
		6OVB_1:87,2NRU_1:87
	]:5.95
]

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 \((0.85)(0.8)(\frac{790 }{\log_{20} 790}-\frac{213}{\log_{20}213})=160.\)
Status Protein1 Protein2 d d1/2
Query variables 7JFL_1 6OVB_1 202 137
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]