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Parikh vectors
7YPK_1 7CRI_1 8OFH_1 Letter Amino acid
0 4 19 E Glutamic acid
0 18 13 G Glycine
0 31 18 L Leucine
0 12 4 M Methionine
0 14 14 S Serine
0 23 10 V Valine
0 9 6 N Asparagine
0 10 4 D Aspartic acid
0 13 3 Q Glutamine
0 2 5 H Histidine
0 14 4 Y Tyrosine
0 5 10 R Arginine
0 24 18 I Isoleucine
0 8 12 K Lycine
0 8 1 W Tryptophan
0 29 14 A Alanine
0 2 0 C Cysteine
0 16 7 F Phenylalanine
0 5 7 P Proline
0 17 12 T Threonine 

7YPK_1|Chain A[auth S]|alpha-S1-casein|Bos taurus (9913)
>7CRI_1|Chain A|Chloride pumping rhodopsin|Nonlabens marinus S1-08 (1454201)
>8OFH_1|Chains A, B, C, D, E, F|Diadenylate cyclase|Streptococcus pneumoniae R6 (171101)
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 \(w=f(c)\)
7YPK , Knot 2 19 0.10 2 1 1 
XXXXXXXXXXXXXXXXXXX
7CRI , Knot 120 264 0.84 40 161 255 
KNIESLFDYSAGQFEFIDHLLTMGVGVHFAALIFFLVVSQFVAPKYRIATALSCIVMVSAGLILNSQAVMWTDAYAYVDGSYQLQDLTFSNGYRYVNWMATIPCLLLQLLIVLNLKGKELFSTATWLILAAWGMIITGYVGQLYEVDDIAQLMIWGAVSTAFFVVMNWIVGTKIFKNRATMLGGTDSTITKVFWLMMFAWTLYPIAYLVPAFMNNADGVVLRQLLFTIADISSKVIYGLMITYIAIQQSAAAGYVPAQQALGRI
8OFH , Knot 85 181 0.81 38 130 173 
GSHMASMNAPISAEEQMIRAFVKSVEYMSPRKIGALVAIQRVRTLQEYISTGIPLDAKISAELLINIFIPNTPLHDGAVIIKEERIAVTSAYLPLTKNTGISKEFGTRHRAAIGLSEVSDALTFVVSEETGGISITYNGRFKHNLTLDEFETELREILLPKEEVGLSFKERLLGGWKHEKK

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 \(c\).

\(|P_{f(7YPK_1)}(2) \setminus P_{f(7CRI_1)}(2)|=1\), \(|P_{f(7CRI_1)}(2) \setminus P_{f(7YPK_1)}(2)|=161\). 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 1:0000000000000000000
Pair \(Z_2\) Length of longest common subsequence
7YPK_1,7CRI_1 162 0
7YPK_1,8OFH_1 131 0
7CRI_1,8OFH_1 157 3

Newick tree

 
[
	7CRI_1:83.97,
	[
		7YPK_1:65.5,8OFH_1:65.5
	]:18.47
]

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 \((0.85)(0.8)(\frac{283 }{\log_{20} 283}-\frac{19}{\log_{20}19})=88.9\)
Status Protein1 Protein2 d d1/2
Query variables 7YPK_1 7CRI_1 119 60
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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