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Parikh vectors
6AWY_1 1TOI_1 5PAB_1 Letter Amino acid
1 22 15 R Arginine
6 23 16 S Serine
0 5 7 W Tryptophan
7 12 7 Y Tyrosine
17 30 25 G Glycine
2 6 9 H Histidine
12 18 11 I Isoleucine
10 40 28 L Leucine
9 21 12 D Aspartic acid
0 5 9 C Cysteine
1 16 12 Q Glutamine
16 27 10 E Glutamic acid
5 20 7 F Phenylalanine
13 20 16 T Threonine
11 45 13 A Alanine
18 17 8 K Lycine
1 8 4 M Methionine
9 15 14 P Proline
6 21 7 N Asparagine
12 25 24 V Valine 

6AWY_1|Chains A, B|Ara h 8 allergen|Arachis hypogaea (3818)
>1TOI_1|Chain A|Aspartate aminotransferase|Escherichia coli (562)
>5PAB_1|Chain A[auth H]|Coagulation factor VII heavy chain|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 \(w=f(c)\)
6AWY , Knot 73 156 0.78 36 106 148 
GVFTFEDEITSTVPPAKLYNAMKDADSITPKIIDDVKSVEIVEGNGGPGTIKKLTIVEDGETKFILHKVESIDEANYAYNYSVVGGVALPPTAEKITFETKLVEGPNGGSIGKLTLKYHTKGDAKPDEEELKKGKAKGEGLFRAIEGYVLANPTQY
1TOI , Knot 166 396 0.83 40 217 370 
MFENITAAPADPILGLADLFRADERPGKINLGIGLYYDETGKIPVLTSVKKAEQYLLENETTKLYLGIDGIPEFGRCTQELLFGKGSALINDKRARTAQTPGGSGALRVAADFLAKNTSVKRVWVSNPSWPNHKSVFNSAGLEVREYAYYDAENHTLDFDALINSLNEAQAGDVVLFHGCCHNPTGIDPTLEQWQTLAQLSVEKGWLPLFDFAYQGFARGLEEDAEGLRAFAAMHKELIVASSYSKNFGLYNERVGACTLVAADSETVDRAFSQMKAAIRDNYSSPPAHGASVVATILSNDALRAIWEQELTDMRQRIQRMRQLFVNTLQEKGANRDFSFIIKQNGMFSFSGLTKEQVLRLREEFGVYAVASGRVNVAGMTPDNMAPLCEAIVAVL
5PAB , Knot 114 254 0.82 40 184 245 
IVGGKVCPKGECPWQVLLLVNGAQLCGGTLINTIWVVSAAHCFDKIKNWRNLIAVLGEHDLSEHDGDEQSRRVAQVIIPSTYVPGTTNHDIALLRLHQPVVLTDHVVPLCLPERTFSERTLAFVRFSLVSGWGQLLDRGATALELMVLNVPRLMTQDCLQQSRKVGDSPNITEYMFCAGYSDGSKDSCKGDSGGPHATHYRGTWYLTGIVSWGQGCATVGHFGVYTRVSQYIEWLQKLMRSEPRPGVLLRAPFP

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 \(c\).

\(|P_{f(6AWY_1)}(2) \setminus P_{f(1TOI_1)}(2)|=25\), \(|P_{f(1TOI_1)}(2) \setminus P_{f(6AWY_1)}(2)|=136\). 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 1:111010001000111101001100100101011001001011010111101001011001000111001001001001000011111111101001010001101101101101010000010101000010010101011101101011101000
Pair \(Z_2\) Length of longest common subsequence
6AWY_1,1TOI_1 161 4
6AWY_1,5PAB_1 192 3
1TOI_1,5PAB_1 179 4

Newick tree

 
[
	5PAB_1:96.56,
	[
		6AWY_1:80.5,1TOI_1:80.5
	]:16.06
]

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 \((0.85)(0.8)(\frac{552 }{\log_{20} 552}-\frac{156}{\log_{20}156})=115.\)
Status Protein1 Protein2 d d1/2
Query variables 6AWY_1 1TOI_1 145 98.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]