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Parikh vectors
5HVD_1 5EJE_1 3GNU_1 Letter Amino acid
8 17 15 D Aspartic acid
2 3 2 C Cysteine
11 6 4 M Methionine
18 10 9 T Threonine
5 7 10 Y Tyrosine
26 19 14 V Valine
10 4 10 N Asparagine
15 8 14 Q Glutamine
12 18 5 E Glutamic acid
16 19 14 G Glycine
9 18 13 K Lycine
19 19 25 A Alanine
16 13 5 R Arginine
22 14 8 I Isoleucine
36 16 11 L Leucine
17 5 8 F Phenylalanine
17 5 20 S Serine
4 0 6 W Tryptophan
5 3 5 H Histidine
9 10 15 P Proline 

5HVD_1|Chain A|Ion transport protein|Magnetococcus marinus (strain ATCC BAA-1437 / JCM 17883 / MC-1) (156889)
>5EJE_1|Chains A, B|Adenylate kinase|Escherichia coli (331111)
>3GNU_1|Chain A[auth P]|25 kDa protein elicitor|Pythium aphanidermatum (65070)
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 \(w=f(c)\)
5HVD , Knot 128 277 0.86 40 181 272 
GSHMSRKIRDLIESKRFQNVITAIIVLNGAVLGLLTDTTLSASSQNLLERVDQLCLTIFIVEISLKIYAYGVRGFFRSGWNLFDFVIVAIALMPAQGSLSVLRTFRIFRVMRLVSVIPTMRRVVQGMLLALPGVGSVAALLTVVFYIAAVMATNLYGATFPEWFGDLSKSLYTLFQVMTLESWSMGIVRPVMNVHPNAWVFFIPFIMLTTFTVLNLFIGICVDAMAITKEQEEEAKTGHHQEPISQTLLHLGDRLDRIEKQLAQNNELLQRQQPQKK
5EJE , Knot 94 214 0.78 38 147 207 
MRIILLGAPGAGKGTQAQFIMEKYGIPQISTGDMLRAAVKSGSELGKQAKDIMDACKLVTDELVIALVKERIAQEDCRNGFLLDGFPRTIPQADAMKEAGINVDYVLEFDVPDELIVDRIVGRRVHAPSGRVYHVKFNPPKVEGKDDVTGEELTTRKDDQEECVRKRLVEYHQMTAPLIGYYSKEAEAGNTKYAKVDGTKPVAEVRADLEKILG
3GNU , Knot 98 213 0.82 40 151 207 
AVINHDAVPVWPQPEPADATQALAVRFKPQLDVVNGCQPYPAVDPQGNTSGGLKPSGSQAAACRDMSKAQVYSRSGTYNGYYDIMYSWYMPKDSPSIGIGHRHDWENVVVWLDNAASANIVALSASAHSGYKKSFPADKSYLDGITAKISYKSTWPLDHELGFTTSAGKQQPLIQWEQMTQAARDALESTDFGNANVPFKSNFQDKLVKAFFQ

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 \(c\).

\(|P_{f(5HVD_1)}(2) \setminus P_{f(5EJE_1)}(2)|=103\), \(|P_{f(5EJE_1)}(2) \setminus P_{f(5HVD_1)}(2)|=69\). 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 1:1001000100110000100110111110111111100001010000110010010101111010101010110111001101101111111111101010110010110110110111010011011111111110111110111011111100101101101110100010011011010010111101110101011111111111001011011111010111100000001001000011000110110010010001100001100001000
Pair \(Z_2\) Length of longest common subsequence
5HVD_1,5EJE_1 172 3
5HVD_1,3GNU_1 186 4
5EJE_1,3GNU_1 166 3

Newick tree

 
[
	5HVD_1:91.65,
	[
		5EJE_1:83,3GNU_1:83
	]:8.65
]

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 \((0.85)(0.8)(\frac{491 }{\log_{20} 491}-\frac{214}{\log_{20}214})=80.1\)
Status Protein1 Protein2 d d1/2
Query variables 5HVD_1 5EJE_1 104 90.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]