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Parikh vectors
3TEA_1 7PNK_1 8UXS_1 Letter Amino acid
12 12 19 R Arginine
13 30 30 G Glycine
2 1 14 K Lycine
5 14 18 P Proline
13 16 19 T Threonine
3 14 13 Y Tyrosine
19 31 16 A Alanine
3 10 14 W Tryptophan
6 11 6 M Methionine
2 5 9 C Cysteine
2 6 8 Q Glutamine
9 17 19 E Glutamic acid
3 28 14 I Isoleucine
4 25 18 F Phenylalanine
8 7 19 D Aspartic acid
8 10 16 H Histidine
8 28 25 L Leucine
11 24 26 S Serine
17 24 21 V Valine
3 23 17 N Asparagine 

3TEA_1|Chains A, B|4-hydroxybenzoyl-CoA thioesterase|Arthrobacter sp. (1667)
>7PNK_1|Chains A, Y[auth a]|Photosystem II protein D1|Dunaliella salina (3046)
>8UXS_1|Chains A, B|Kelch domain-containing protein 2|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 \(w=f(c)\)
3TEA , Knot 75 151 0.83 40 115 148 
MHRTSNGSHATGGNLPDVASHYPVAYEQTLDGTVGFVIDEMTPERATASVEVTDTLRERWGLVHGGAYCALAAMLATEATVAVVHEKGMMAVGQSNHTSFFRPVKEGHVRAEAVRIHAGSTTWFWDVSLRDDAGRLCAVSSMSIAVRPRRD
7PNK , Knot 144 336 0.83 40 192 318 
ENTSLWARFCEWITSTENRLYIGWFGVIMIPTLLTAISVYIIAFIAAPPVDIDGIREPVSGSLLYGNNIITGAVVPTSNAIGLHFYPIWEAASLDEWLYNGGPYQLVVCHFFLGVCCYMGREWELSYRLGMRPWIAVAYSAPVAAATAVFIIYPIGQGSFSDGMPLGISGTFNFMIVFQAEHNILMHPFHMFGVAGVFGGSLFSAMHGSLVTSSLIRETTENESANAGYKFGQEEETYNIVAAHGYFGRLIFQYASFNNSRSLHFFLAVWPVVCIWLTALGISTMAFNLNGFNFNQSVVDSNGRVLNTWADIINRANLGMEVMHERNAHNFPLDLA
8UXS , Knot 149 341 0.85 40 219 322 
ESMELACPAERSGHVAVSDGRHMFVWGGYKSNQVRGLYDFYLPREELWIYNMETGRWKKINTEGDVPPSMSGSCAVCVDRVLYLFGGHHSRGNTNKFYMLDSRSTDRVLQWERIDCQGIPPSSKDKLGVWVYKNKLIFFGGYGYLPEDKVLGTFEFDETSFWNSSHPRGWNDHVHILDTETFTWSQPITTGKAPSPRAAHACATVGNRGFVFGGRYRDARMNDLHYLNLDTWEWNELIPQGICPVGRSWHSLTPVSSDHLFLFGGFTTDKQPLSDAWTYCISKNEWIQFNHPYTEKPRLWHTACASDEGEVIVFGGCANNLLVHHRAAHSNEILIFSVQPK

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 \(c\).

\(|P_{f(3TEA_1)}(2) \setminus P_{f(7PNK_1)}(2)|=53\), \(|P_{f(7PNK_1)}(2) \setminus P_{f(3TEA_1)}(2)|=130\). 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 1:1000001001011011011000111000010101111100101001010101000100011110111001111111001011110001111110000001101100101010110101100011101010001101011001011101000
Pair \(Z_2\) Length of longest common subsequence
3TEA_1,7PNK_1 183 3
3TEA_1,8UXS_1 186 3
7PNK_1,8UXS_1 177 4

Newick tree

 
[
	3TEA_1:93.47,
	[
		7PNK_1:88.5,8UXS_1:88.5
	]:4.97
]

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 \((0.85)(0.8)(\frac{487 }{\log_{20} 487}-\frac{151}{\log_{20}151})=99.0\)
Status Protein1 Protein2 d d1/2
Query variables 3TEA_1 7PNK_1 123 88.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]