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Parikh vectors
9KRI_1 1QFT_1 5LDZ_1 Letter Amino acid
33 4 16 P Proline
30 12 17 T Threonine
19 4 13 R Arginine
34 5 17 I Isoleucine
42 8 11 F Phenylalanine
32 11 15 Y Tyrosine
7 1 5 H Histidine
73 5 30 L Leucine
19 5 11 M Methionine
20 8 27 K Lycine
19 18 13 N Asparagine
22 4 7 C Cysteine
13 5 7 Q Glutamine
60 9 27 G Glycine
34 4 20 S Serine
24 4 0 W Tryptophan
60 15 31 V Valine
51 20 30 A Alanine
22 18 24 D Aspartic acid
21 15 17 E Glutamic acid 

9KRI_1|Chain A|Sodium- and chloride-dependent creatine transporter 1|Homo sapiens (9606)
>1QFT_1|Chains A, B|PROTEIN (FEMALE-SPECIFIC HISTAMINE BINDING PROTEIN 2)|Rhipicephalus appendiculatus (34631)
>5LDZ_1|Chains A, B, C, D, E, F|Fructose-1,6-bisphosphatase 1|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 \(w=f(c)\)
9KRI , Knot 247 635 0.83 40 276 591 
MAKKSAENGIYSVSGDEKKGPLIAPGPDGAPAKGDGPVGLGTPGGRLAVPPRETWTRQMDFIMSCVGFAVGLGNVWRFPYLCYKNGGGVFLIPYVLIALVGGIPIFFLEISLGQFMKAGSINVWNICPLFKGLGYASMVIVFYCNTYYIMVLAWGFYYLVKSFTTTLPWATCGHTWNTPDCVEIFRHEDCANASLANLTCDQLADRRSPVIEFWENKVLRLSGGLEVPGALNWEVTLCLLACWVLVYFCVWKGVKSTGKIVYFTATFPYVVLVVLLVRGVLLPGALDGIIYYLKPDWSKLGSPQVWIDAGTQIFFSYAIGLGALTALGSYNRFNNNCYKDAIILALINSGTSFFAGFVVFSILGFMAAEQGVHISKVAESGPGLAFIAYPRAVTLMPVAPLWAALFFFMLLLLGLDSQFVGVEGFITGLLDLLPASYYFRFQREISVALCCALCFVIDLSMVTDGGMYVFQLFDYYSASGTTLLWQAFWECVVVAWVYGADRFMDDIACMIGYRPCPWMKWCWSFFTPLVCMGIFIFNVVYYEPLVYNNTYVYPWWGEAMGWAFALSSMLCVPLHLLGCLLRAKGTMAERWQHLTQPIWGLHHLEYRAQDADVRGLTTLTPVSESSKVVVVESVM
1QFT , Knot 85 175 0.83 40 132 171 
NQPDWADEAANGAHQDAWKSLKADVENVYYMVKATYKNDPVWGNDFTCVGVMANDVNEDEKSIQAEFLFMNNADTNMQFATEKVTAVKMYGYNRENAFRYETEDGQVFTDVIAYSDDNCDVIYVPGTDGNEEGYELWTTDYDNIPANCLNKFNEYAVGRETRDVFTSACLEIAAA
5LDZ , Knot 141 338 0.81 38 200 323 
MADQAPFDTDVNTLTRFVMEEGRKARGTGELTQLLNSLCTAVKAISSAVRKAGIAHLYGIAGSTNVTGDQVKKLDVLSNDLVMNMLKSSFATCVLVSEEDKHAIIVEPEKRGKYVVCFDPLDGSSNIDCLVSVGTIFGIYRKKSTDEPSEKDALQPGRNLVAAGYALYGSATMLVLAMDCGVNCFMLDPAIGEFILVDKDVKIKKKGKIYSLNEGYARDFDPAVTEYIQRKKFPPDNSAPYGARYVGSMVADVHRTLVYGGIFLYPANKKSPNGKLRLLYECNPMAYVMEKAGGMATTGKEAVLDVIPTDIHQRAPVILGSPDDVLEFLKVYEKHSAQ

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 \(c\).

\(|P_{f(9KRI_1)}(2) \setminus P_{f(1QFT_1)}(2)|=168\), \(|P_{f(1QFT_1)}(2) \setminus P_{f(9KRI_1)}(2)|=24\). 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 1:11000100110010100001111111101111010111111011101111100010001011100111111110110110100001111111101111111111111101011011011010110101110111010111110000001111111100110010001111001001001001011000001010110100001100001110110001101011101111101010101110111101011011000101101010110111111110111111110111001010100110101110110011100111111101110000100000001111111001001111111101111111001101001100111111110101101111111111111111111110001111011101110111100010100010111001101110101100111011011000010100111011100111111011001100110111001011101010110111011111101100011100000101111011111111001101110111011010101100100100111110010001001010110010110000011110011
Pair \(Z_2\) Length of longest common subsequence
9KRI_1,1QFT_1 192 3
9KRI_1,5LDZ_1 170 4
1QFT_1,5LDZ_1 156 4

Newick tree

 
[
	9KRI_1:94.51,
	[
		5LDZ_1:78,1QFT_1:78
	]:16.51
]

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 \((0.85)(0.8)(\frac{810 }{\log_{20} 810}-\frac{175}{\log_{20}175})=177.\)
Status Protein1 Protein2 d d1/2
Query variables 9KRI_1 1QFT_1 221 140.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]