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Parikh vectors
5HAG_1 7ADE_1 3AXD_1 Letter Amino acid
6 21 10 N Asparagine
16 11 13 Q Glutamine
9 29 27 G Glycine
12 16 14 F Phenylalanine
15 23 18 D Aspartic acid
18 28 14 K Lycine
7 16 6 M Methionine
16 21 20 S Serine
6 1 8 W Tryptophan
19 25 15 V Valine
15 31 7 I Isoleucine
30 44 16 L Leucine
12 23 17 T Threonine
11 7 10 Y Tyrosine
19 31 16 A Alanine
10 30 10 R Arginine
8 1 0 C Cysteine
17 36 16 E Glutamic acid
6 8 3 H Histidine
21 17 9 P Proline 

5HAG_1|Chain A|Deubiquitinase and deneddylase Dub1|Chlamydia trachomatis (813)
>7ADE_1|Chains A[auth a], B[auth b], C[auth c], D[auth d], E[auth e], F[auth f]|Transcription termination factor Rho|Escherichia coli (562)
>3AXD_1|Chains A, B|Beta-glucanase|Fibrobacter succinogenes (59374)
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 \(w=f(c)\)
5HAG , Knot 126 273 0.86 40 176 265 
GPVKTQEDLLPLVPEQVFVEMYEDMARRQTIEALVPAWDSDIIFKCLCYFHTLYPGLIPLETFPPATIFNFKQKIISILEDKKAVLRGEPIKGPLPICCSKENYRRHLQRTTLLPVFMWYHPTPKTLSDTMQTMKQLAIKGSVGASHWLLVIVDIQARRLVYFDSLYNYVMPPENMKKELQSFAQQLDQVYPAYDSKKFSVKIAAKEVIQRGSGSSCGAWCCQFLHWYLKDPLTDALNDLPVDSVERHENLASFVQACEAAVQDLPELSWPEA
7ADE , Knot 177 419 0.85 40 219 405 
MNLTELKNTPVSELITLGENMGLENLARMRKQDIIFAILKQHAKSGEDIFGDGVLEILQDGFGFLRSADSSYLAGPDDIYVSPSQIRRFNLRTGDTISGKIRPPKEGERYFALLKVNEVNFDKPENARNKILFENLTPLHANSRLRMERGNGSTEDLTARVLDLASPIGRGQRGLIVAPPKAGKTMLLQNIAQSIAYNHPDCVLMVLLIDERPEEVTEMQRLVKGEVVASTFDEPASRHVQVAEMVIEKAKRLVEHKKDVIILLDSITRLARAYNTVVPASGKVLTGGVDANALHRPKRFFGAARNVEEGGSLTIIATALIDTGSKMDEVIYEEFKGTGNMELHLSRKIAEKRVFPAIDYNRSGTRKEELLTTQEELQKMWILRKIIHPMGEIDAMEFLINKLAMTKTNDDFFEMMKRS
3AXD , Knot 116 249 0.85 38 164 242 
MVSMKDFSGAELYTLEEYQYGKFEARMKMAAASGTVSSMFLYQNGSEIADGRPWVEVDIEVLGKSPGSFQSNIITGKAGAQKTSEKHHAVSPAADQAFHTYGLEWTPNYVRWTVDGQEVRKTEGGQVSNLTGTQGLRFNLWSSESAAWVGQFDESKLPLFQFINWVKVYKYTPGQGEGGSDFTLDWTDNFDTFDGSRWGKGDYTFDGNRVDLTDKNIYSRDGMLILALTRKGQESFNGQVPRDDEPAPL

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 \(c\).

\(|P_{f(5HAG_1)}(2) \setminus P_{f(7ADE_1)}(2)|=64\), \(|P_{f(7ADE_1)}(2) \setminus P_{f(5HAG_1)}(2)|=107\). 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 1:111000001111110011101000110000101111110001110010010010111111001111011010001101100001110101101111100000000001000011111110010100100010010011101011100111111010100110100100011110010001001100100101100000101011100110010100011100011010100110011001110010000011011010011100110101101
Pair \(Z_2\) Length of longest common subsequence
5HAG_1,7ADE_1 171 4
5HAG_1,3AXD_1 190 3
7ADE_1,3AXD_1 169 4

Newick tree

 
[
	5HAG_1:92.25,
	[
		7ADE_1:84.5,3AXD_1:84.5
	]:7.75
]

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 \((0.85)(0.8)(\frac{692 }{\log_{20} 692}-\frac{273}{\log_{20}273})=116.\)
Status Protein1 Protein2 d d1/2
Query variables 5HAG_1 7ADE_1 147 121.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]