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Parikh vectors
1QSX_1 2EEZ_1 5GXX_1 Letter Amino acid
2 1 8 C Cysteine
0 8 21 Q Glutamine
0 22 26 P Proline
0 13 46 D Aspartic acid
0 12 26 I Isoleucine
0 6 23 F Phenylalanine
0 9 28 S Serine
0 9 11 M Methionine
0 1 17 W Tryptophan
0 9 47 Y Tyrosine
0 18 25 R Arginine
2 47 50 G Glycine
0 8 22 H Histidine
0 17 27 K Lycine
4 26 31 T Threonine
0 43 28 V Valine
4 40 64 A Alanine
0 7 28 N Asparagine
0 27 36 E Glutamic acid
0 46 46 L Leucine 

1QSX_1|Chains A, B|5'-D(CP*TP*TP*TP*TP*GP*CP*AP*AP*AP*AP*G)-3'|null
>2EEZ_1|Chains A, B[auth C], C[auth D], D[auth E], E[auth F], F[auth G]|Alanine dehydrogenase|Thermus thermophilus (300852)
>5GXX_1|Chains A, B|Glucanase|Clostridium thermocellum (1515)
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 \(w=f(c)\)
1QSX , Knot 5 12 0.34 8 7 8 
CTTTTGCAAAAG
2EEZ , Knot 151 369 0.80 40 185 344 
MVIGVPKEIKTLENRVALTPGGVESLVRRGHTVLVERGAGEGSGLSDAEYARAGAELVGREEAWGAEMVVKVKEPLPEEYGFLREGLILFTYLHLAADRGLTEAMLRSGVTGIAYETVQLPDGTLPLLVPMSEVAGRMAPQVGAQFLEKPKGGRGVLLGGVPGVAPASVVILGGGTVGTNAAKIALGMGAQVTILDVNHKRLQYLDDVFGGRVITLTATEANIKKSVQHADLLIGAVLVPGAKAPKLVTRDMLSLMKEGAVIVDVAVDQGGCVETIRPTTHAEPTYVVDGVVHYGVANMPGAVPRTSTFALTNQTLPYVLKLAEKGLDALLEDAALLKGLNTHKGRLTHPGVAEAFGLPYTPPEEALRG
5GXX , Knot 242 610 0.84 40 275 573 
MAGSYNYAEALQKAIYFYECQQAGPLPEWNRVEWRGDATMNDEVLGGWYDAGDHVKFNLPMAYSAAMLGWALYEYGDDIEASGQRLHLERNLAFALDYLVACDRGDSVVYQIGDGAADHKWWGSAEVIEKEMTRPYFVGKGSAVVGQMAAALAVGSIVLKNDTYLRYAKKYFELADATRSDSTYTAANGFYSSHSGFWDELLWASTWLYLATGDRNYLDKAESYTPKLNRQNQTTDIEYQWAHCWDDCHYGAMILLARATGKEEYHKFAQMHLDWWTPQGYNGKRVAYTPGGLAHLDTWGPLRYATTEAFLAFVYADSINDPALKQKYYNFAKSQIDYALGSNPDNRSYVVGFGNNPPQRPHHRTAHGTWLDKRDIPEKHRHVLYGALVGGPGRDDSYEDNIEDYVKNEVACDYNAGFVGALCRLTAEYGGTPLANFPPPEQRDDEFFVEAAINQASDHFTEIKALLNNRSSWPARLIKDLSYNYYMDLTEVFEAGYSVDDIKVTIGYCESGMDVEISPITHLYDNIYYIKISYIDGTNICPIGQEQYAAELQFRIAAPQGTKFWDPTNDFSYQGLTRELAKTKYMPVFDGATKIFGEVPGGLEHHHHHH

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 \(c\).

\(|P_{f(1QSX_1)}(2) \setminus P_{f(2EEZ_1)}(2)|=2\), \(|P_{f(2EEZ_1)}(2) \setminus P_{f(1QSX_1)}(2)|=180\). 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 1:000001011111
Pair \(Z_2\) Length of longest common subsequence
1QSX_1,2EEZ_1 182 2
1QSX_1,5GXX_1 274 3
2EEZ_1,5GXX_1 174 4

Newick tree

 
[
	1QSX_1:12.54,
	[
		2EEZ_1:87,5GXX_1:87
	]:37.54
]

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 \((0.85)(0.8)(\frac{381 }{\log_{20} 381}-\frac{12}{\log_{20}12})=120.\)
Status Protein1 Protein2 d d1/2
Query variables 1QSX_1 2EEZ_1 150 77
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]

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