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Parikh vectors
2VNL_1 6KFK_1 9DQJ_1 Letter Amino acid
12 15 5 K Lycine
11 41 17 A Alanine
0 0 11 C Cysteine
5 27 54 L Leucine
3 16 18 F Phenylalanine
8 34 37 S Serine
7 11 8 Y Tyrosine
10 9 11 E Glutamic acid
23 17 9 I Isoleucine
1 2 8 W Tryptophan
8 39 16 G Glycine
1 8 13 M Methionine
10 20 4 D Aspartic acid
9 24 10 Q Glutamine
3 2 6 H Histidine
7 12 14 P Proline
6 44 23 T Threonine
13 29 27 V Valine
3 6 20 R Arginine
11 46 11 N Asparagine 

2VNL_1|Chain A|BIFUNCTIONAL TAIL PROTEIN, PIIGCN4|ENTEROBACTERIA PHAGE P22 (10754)
>6KFK_1|Chain A|Flagellar hook protein FlgE|Salmonella enterica subsp. enterica serovar Typhimurium (90371)
>9DQJ_1|Chain A|Mas-related G-protein coupled receptor member D|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 \(w=f(c)\)
2VNL , Knot 71 151 0.78 38 106 142 
TDITANVVVSNPRPIFTESRSFKAVANGKIYIGQIDTDPVNPANQIPVYIENEDGSHVQITQPLIINAAGKIVYNGQLVKIVTVQGHSMAIYDANGSQVDYIANVLKWDPDQYSIEADKKFKQIEDKIEEILSKIYHIENEIARIKKLIGE
6KFK , Knot 161 402 0.80 38 189 367 
SFSQAVSGLNAAATNLDVIGNNIANSATYGFKSGTASFADMFAGSKVGLGVKVAGITQDFTDGTTTNTGRGLDVAISQNGFFRLVDSNGSVFYSRNGQFKLDENRNLVNMQGMQLTGYPATGTPPTIQQGANPAPITIPNTLMAAKSTTTASMQINLNSTDPVPSKTPFSVSDADSYNKKGTVTVYDSQGNAHDMNVYFVKTKDNEWAVYTHDSSDPAATAPTTASTTLKFNENGILESGGTVNITTGTINGATAATFSLSFLNSMQQNTGANNIVATNQNGYKPGDLVSYQINNDGTVVGNYSNEQEQVLGQIVLANFANNEGLASQGDNVWAATQASGVALLGTAGSGNFGKLTNGALEASNVDLSKELVNMIVAQRNYQSNAQTIKTQDQILNTLVNLR
9DQJ , Knot 134 322 0.80 40 177 306 
GPNQTLNSSGTVESALNYSRGSTVHTAYLVLSSLAMFTCLCGMAGNSMVIWLLGFRMHRNPFCIYILNLAAADLLFLFSMASTLSLETQPLVNTTDKVHELMKRLMYFAYTVGLSLLTAISTQRCLSVLFPIWFKCHRPRHLSAWVCGLLWTLCLLMNGLTSSFCSKFLKFNEDRCFRVDMVQAALIMGVLTPVMTLSSLTLFVWVRRSSQQWRRQPTRLFVVVLASVLVFLICSLPLSIYWFVLYWLSLPPEMQVLCFSLSRLSSSVSSSANPVIYFLVGSRRSHRLPTRSLGTVLQQALREEPELEGGETPTVGTNEMGA

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 \(c\).

\(|P_{f(2VNL_1)}(2) \setminus P_{f(6KFK_1)}(2)|=38\), \(|P_{f(6KFK_1)}(2) \setminus P_{f(2VNL_1)}(2)|=121\). 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 1:0010101110010111000001011101010110100011011001110100001001010011110111011001011011010100111001010010011011010100001010001001000100110010010001101001110
Pair \(Z_2\) Length of longest common subsequence
2VNL_1,6KFK_1 159 4
2VNL_1,9DQJ_1 179 4
6KFK_1,9DQJ_1 164 5

Newick tree

 
[
	9DQJ_1:87.84,
	[
		2VNL_1:79.5,6KFK_1:79.5
	]:8.34
]

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 \((0.85)(0.8)(\frac{553 }{\log_{20} 553}-\frac{151}{\log_{20}151})=117.\)
Status Protein1 Protein2 d d1/2
Query variables 2VNL_1 6KFK_1 141 97
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]