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Parikh vectors
3HPO_1 5TEX_1 7RRP_1 Letter Amino acid
25 20 7 G Glycine
31 18 6 I Isoleucine
65 31 22 L Leucine
23 15 3 P Proline
5 7 1 W Tryptophan
18 7 9 Y Tyrosine
45 22 6 V Valine
31 19 13 D Aspartic acid
57 22 15 E Glutamic acid
12 11 10 H Histidine
39 7 13 K Lycine
21 14 6 F Phenylalanine
23 19 9 S Serine
60 9 12 A Alanine
15 3 4 M Methionine
19 6 11 N Asparagine
2 5 3 C Cysteine
27 11 10 Q Glutamine
24 7 5 T Threonine
38 21 7 R Arginine 

3HPO_1|Chain A|DNA polymerase I, large fragment|Geobacillus stearothermophilus (1422)
>5TEX_1|Chain A|Serine/threonine-protein kinase pim-1|Homo sapiens (9606)
>7RRP_1|Chains A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T, U, V, W, X|Ferritin 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 \(w=f(c)\)
3HPO , Knot 227 580 0.83 40 252 535 
AKMAFTLADRVTEEMLADKAALVVEVVEENYHAAPIVGIAVVNEHGRFFLRPETALADPQFVAWLGDETKKKSMFDSKRAAVALKWKGIELCGVSFDLLLAAYLLDPAQGVDDVAAAAKMKQYEAVRPDEAVYGKGAKRAVPDEPVLAEHLVRKAAAIWELERPFLDELRRNEQDRLLVELEQPLSSILAEMEFAGVKVDTKRLEQMGKELAEQLGTVEQRIYELAGQEFNINSPKQLGVILFEKLQLPVLKKTKTGYSTSADVLEKLAPYHEIVENILHYRQLGKLQSTYIEGLLKVVRPDTKKVHTIFNQALTQTGRLSSTEPNLQNIPIRLEEGRKIRQAFVPSESDWLIFAADYSQIELRVLAHIAEDDNLMEAFRRDLDIHTKTAMDIFQVSEDEVTPNMRRQAKAVNFGIVSGISDYGLAQNLNISRKEAAEFIERYFESFPGVKRYMENIVQEAKQKGYVTTLLHRRRYLPDITSRNFNVRSFAERMAMNTPIQGSAADIIKKAMIDLNARLKEERLQAHLLLQVHDELILEAPKEEMERLCRLVPEVMEQAVTLRVPLKVDYHYGSTWYDAK
5TEX , Knot 121 274 0.82 40 176 257 
PLESQYQVGPLLGSGGFGSVYSGIRVSDNLPVAIKHVEKDRISDWGELPNGTRVPMEVVLLKKVSSGFSGVIRLLDWFERPDSFVLILERPEPVQDLFDFITERGALQEELARSFFWQVLEAVRHCHNCGVLHRDIKDENILIDLNRGELKLIDFGSGALLKDTVYTDFDGTRVYSPPEWIRYHRYHGRSAAVWSLGILLYDMVCGDIPFEHDEEIIRGQVFFRQRVSSECQHLIRWCLALRPSDRPTFEEIQNHPWMQDVLLPQETAEIHLHS
7RRP , Knot 82 172 0.81 40 130 167 
TSQVRQNYHQDSEAAINRQINLELYASYVYLSMSYYFDRDDVALKNFAKYFLHQSHEEREHAEKLMKLQNQRGGRIFLQDIKKPDCDDWESGLNAMECALHLEKNVNQSLLELHKLATDKNDPHLCDFIETHYLNEQVKAIKELGDHVTNLRKMGAPESGLAEYLFDKHTLG

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 \(c\).

\(|P_{f(3HPO_1)}(2) \setminus P_{f(5TEX_1)}(2)|=129\), \(|P_{f(5TEX_1)}(2) \setminus P_{f(3HPO_1)}(2)|=53\). 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 1:1011101100100011100111110110000011111111110001011101001110101111110000000110000111110101101011010111110110110110011111010000110100110101100111001111001100111110100111001000000011101001100111010111101000010011001100110100010011100101001001111110010111100000100001011001110001100110000110100001011101101000010011001100010100001010011101001001001111000011111100001010111011000011011000101000011011010000101010001011011110110001110010100001101100010011110001001100100010100110000011010000101001100111001101011011001110101010000101011101000111011000100100111011001101011101000010010010
Pair \(Z_2\) Length of longest common subsequence
3HPO_1,5TEX_1 182 4
3HPO_1,7RRP_1 186 4
5TEX_1,7RRP_1 178 3

Newick tree

 
[
	3HPO_1:92.98,
	[
		5TEX_1:89,7RRP_1:89
	]:3.98
]

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 \((0.85)(0.8)(\frac{854 }{\log_{20} 854}-\frac{274}{\log_{20}274})=158.\)
Status Protein1 Protein2 d d1/2
Query variables 3HPO_1 5TEX_1 194 142.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]