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Parikh vectors
7SIN_1 8ILK_1 2PRC_1 Letter Amino acid
50 11 24 V Valine
62 7 35 A Alanine
38 12 9 D Aspartic acid
74 8 6 I Isoleucine
41 18 16 K Lycine
8 6 9 M Methionine
81 13 28 L Leucine
29 12 15 Y Tyrosine
18 3 5 W Tryptophan
40 10 15 N Asparagine
32 5 9 C Cysteine
34 15 27 P Proline
69 15 16 S Serine
44 15 31 T Threonine
70 9 9 F Phenylalanine
35 4 24 R Arginine
22 11 13 Q Glutamine
61 15 19 E Glutamic acid
52 21 19 G Glycine
18 14 7 H Histidine

7SIN_1|Chains A, B|Isoform 1 of Extracellular calcium-sensing receptor|Homo sapiens (9606)
>8ILK_1|Chains A, B|Green FLUORESCENT PROTEIN|synthetic construct (32630)
>2PRC_1|Chain A[auth C]|PHOTOSYNTHETIC REACTION CENTER|Blastochloris viridis (1079)
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 \(w=f(c)\)
7SIN , Knot 334 878 0.86 40 309 795
MAFYSCCWVLLALTWHTSAYGPDQRAQKKGDIILGGLFPIHFGVAAKDQDLKSRPESVECIRYNFRGFRWLQAMIFAIEEINSSPALLPNLTLGYRIFDTCNTVSKALEATLSFVAQNKIDSLNLDEFCNCSEHIPSTIAVVGATGSGVSTAVANLLGLFYIPQVSYASSSRLLSNKNQFKSFLRTIPNDEHQATAMADIIEYFRWNWVGTIAADDDYGRPGIEKFREEAEERDICIDFSELISQYSDEEEIQHVVEVIQNSTAKVIVVFSSGPDLEPLIKEIVRRNITGKIWLASEAWASSSLIAMPQYFHVVGGTIGFALKAGQIPGFREFLKKVHPRKSVHNGFAKEFWEETFNCHLQEGAKGPLPVDTFLRGHEESGDRFSNSSTAFRPLCTGDENISSVETPYIDYTHLRISYNVYLAVYSIAHALQDIYTCLPGRGLFTNGSCADIKKVEAWQVLKHLRHLNFTNNMGEQVTFDECGDLVGNYSIINWHLSPEDGSIVFKEVGYYNVYAKKGERLFINEEKILWSGFSREVPFSNCSRDCLAGTRKGIIEGEPTCCFECVECPDGEYSDETDASACNKCPDDFWSNENHTSCIAKEIEFLSWTEPFGIALTLFAVLGIFLTAFVLGVFIKFRNTPIVKATNRELSYLLLFSLLCCFSSSLFFIGEPQDWTCRLRQPAFGISFVLCISCILVKTNRVLLVFEAKIPTSFHRKWWGLNLQFLLVFLCTFMQIVICVIWLYTAPPSSYRNQELEDEIIFITCHEGSLMALGFLIGYTCLLAAICFFFAFKSRKLPENFNEAKFITFSMLIFFIVWISFIPAYASTYGKFVSAVEVIAILAASFGLLACIFFNKIYIILFKPSRNTIEDYKDDDDK
8ILK , Knot 101 224 0.81 40 162 219
GPGYQDPMASTPFKFQLKGTINGKSFTVEGEGEGNSHEGSHKGKYVCTSGKLPMSWAALGTSFGYGMKYYTKYPSGLKNWFHEVMPEGFTYDRHIQYKGDGSIHAKHQHFMKNGTYHNIVEFTGQDFKENSPVLTGDMNVSLPNEVQHIPRDDGVECPVTLLYPLLSDKSKCVEAHQNTICKPLHNQPAPDVPYHWIRKQYTQSKDDTEERDHICQSETLEAHL
2PRC , Knot 140 336 0.80 40 197 318
CFEPPPATTTQTGFRGLSMGEVLHPATVKAKKERDAQYPPALAAVKAEGPPVSQVYKNVKVLGNLTEAEFLRTMTAITEWVSPQEGCTYCHDENNLASEAKYPYVVARRMLEMTRAINTNWTQHVAQTGVTCYTCHRGTPLPPYVRYLEPTLPLNNRETPTHVERVETRSGYVVRLAKYTAYSALNYDPFTMFLANDKRQVRVVPQTALPLVGVSRGKERRPLSDAYATFALMMSISDSLGTNCTFCHNAQTFESWGKKSTPQRAIAWWGIRMVRDLNMNYLAPLNASLPASRLGRQGEAPQADCRTCHQGVTKPLFGASRLKDYPELGPIKAAAK

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 \(c\).

\(|P_{f(7SIN_1)}(2) \setminus P_{f(8ILK_1)}(2)|=178\), \(|P_{f(8ILK_1)}(2) \setminus P_{f(7SIN_1)}(2)|=31\). 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 1:11100001111110100010110001000101111111110111110000100010010010001011011011111100100011111010110011000001001101010111000100101001000000110011111101011001110111110110100100001100000100110011000001011101100101011101110000101110010001000010101001100000000100110110000101111100110101110011000101011110011100011111001011110111110110111100110010100010011100110001000100110111110011010000100100000110110010001001001010000101000101110011011001000111011100100101001011011001001010001100101000101110001101010100101110011000101001001110000111011000111000000011100011101010001001001010000000101000010011000000001100101101001111110111111111011111111010001110100001001111011001000111110100100010011111011101001110000111110101100100011110101111110011011101111001110000000100011110000101111111110001111101111100001100100101101011111111101111010001011011011111110111110111001011110100001000000000
Pair \(Z_2\) Length of longest common subsequence
7SIN_1,8ILK_1 209 4
7SIN_1,2PRC_1 170 5
8ILK_1,2PRC_1 193 3

Newick tree

 
[
	8ILK_1:10.26,
	[
		7SIN_1:85,2PRC_1:85
	]:20.26
]

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 \((0.85)(0.8)(\frac{1102 }{\log_{20} 1102}-\frac{224}{\log_{20}224})=236.\)
Status Protein1 Protein2 d d1/2
Query variables 7SIN_1 8ILK_1 301 187.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]