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Parikh vectors
1YXP_1 8AVV_1 1DBZ_1 Letter Amino acid
7 114 17 A Alanine
0 44 16 Q Glutamine
0 29 27 I Isoleucine
0 56 17 T Threonine
0 62 11 R Arginine
0 48 21 D Aspartic acid
0 62 27 E Glutamic acid
0 24 11 F Phenylalanine
0 15 2 W Tryptophan
6 6 6 C Cysteine
0 142 32 L Leucine
0 8 22 K Lycine
0 16 4 M Methionine
0 50 35 S Serine
0 46 25 V Valine
0 15 16 N Asparagine
6 78 33 G Glycine
0 32 3 H Histidine
0 74 14 P Proline
0 17 18 Y Tyrosine 

1YXP_1|Chains A, B|5'-R(*CP*UP*(5BU)P*GP*CP*UP*GP*AP*AP*GP*UP*GP*CP*AP*CP*AP*CP*AP*GP*CP*AP*AP*G)-3'|
>8AVV_1|Chains A, B|Bacteriophytochrome,Response regulator|Deinococcus radiodurans R1 (243230)
>1DBZ_1|Chains A, B, C, D|FRUCTOSE-1,6-BISPHOSPHATASE|Pisum sativum (3888)
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 \(w=f(c)\)
1YXP , Knot 10 23 0.45 8 10 16 
CUUGCUGAAGUGCACACAGCAAG
8AVV , Knot 335 938 0.81 40 276 781 
MASMTGGQQMGRGSMSRDPLPFFPPLYLGGPEITTENCEREPIHIPGSIQPHGALLTADGHSGEVLQMSLNAATFLGQEPTVLRGQTLAALLPEQWPALQAALPPGCPDALQYRATLDWPAAGHLSLTVHRVGELLILEFEPTEAWDSTGPHALRNAMFALESAPNLRALAEVATQTVRELTGFDRVMLYKFAPDATGEVIAEARREGLHAFLGHRFPASDIPAQARALYTRHLLRLTADTRAAAVPLDPVLNPQTNAPTPLGGAVLRATSPMHMQYLRNMGVGSSLSVSVVVGGQLWGLIACHHQTPYVLPPDLRTTLEYLGRLLSLQVQVKEAADVAAFRQSLREHHARVALAAAHSLSPHDTLSDPALDLLGLMRAGGLILRFEGRWQTLGEVPPAPAVDALLAWLETQPGALVQTDALGQLWPAGADLAPSAAGLLAISVGEGWSECLVWLRPELRLEVAWGGATPDQAKDDLGPRHSFDTYLEEKRGYAEPWHPGEIEEAQDLRDTLTGALGERLSVIRDLNRALTQSNAEWRQYGFVISHHMQEPVRLISQFAELLTRQPRAQDGSPDSPQTERITGFLLRETSRLRSLTQDLHTYTALLSAPPPVRRPTPLGRVVDDVLQDLEPRIADTGASIEVAPELPVIAADAGLLRDLLLHLIGNALTFGGPEPRIAVRTERQGAGWSIAVSDQGAGIAPEYQERIFLLFQRLGSLDEALGNGLGLPLCRKIAELHGGTLTVESAPGEGSTFRCWLPDAGPLPGAADAASSAGGSAGSAGMPERASVPLRLLLVEDNAADIFLMEMALEYSSVHTELLVARDGLEALELLEQAKTGGPFPDLILLDLNMPRVDGFELLQALRADPHLAHLPAIVLTTSNDPSDVKRAYALQANSYLTKPSTLEDFLQLIERLTAYWFGTAAIPQTYQPQLEHHHHHH
1DBZ , Knot 146 357 0.80 40 190 330 
MAVKEATSETKKRSGYEIITLTSWLLQQEQKGIIDAELTIVLSSISMACKQIASLVQRANISNLTGTQGAVNIQGEDQKKLDVISNEVFSNCLRSSGRTGIIASEEEDVPVAVEESYSGNYIVVFDPLDGSSNLDAAVSTGSIFGIYSPNDESLPDFGDDSDDNTLGTEEQRCIVNVCQPGSNLLAAGYCMYSSSVIFVLTIGKGVFVFTLDPLYGEFVLTQENLQIPKSGKIYSFNEGNYKLWDENLKKYIDDLKEPGPSGKPYSARYIGSLVGDFHRTLLYGGIYGYPRDKKSKNGKLRLLYECAPMSFIVEQAGGKGSDGHQRVLDIQPTEIHQRVPLYIGSTEEVEKVEKYLA

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 \(c\).

\(|P_{f(1YXP_1)}(2) \setminus P_{f(8AVV_1)}(2)|=5\), \(|P_{f(8AVV_1)}(2) \setminus P_{f(1YXP_1)}(2)|=271\). 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 1:00010011110101010110111
Pair \(Z_2\) Length of longest common subsequence
1YXP_1,8AVV_1 276 3
1YXP_1,1DBZ_1 190 3
8AVV_1,1DBZ_1 176 4

Newick tree

 
[
	1YXP_1:12.00,
	[
		1DBZ_1:88,8AVV_1:88
	]:39.00
]

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 \((0.85)(0.8)(\frac{961 }{\log_{20} 961}-\frac{23}{\log_{20}23})=270.\)
Status Protein1 Protein2 d d1/2
Query variables 1YXP_1 8AVV_1 333 170.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]

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