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Parikh vectors
4FPH_1 7SZB_1 7FQK_1 Letter Amino acid
58 28 39 L Leucine
77 17 27 S Serine
49 9 22 I Isoleucine
42 10 27 D Aspartic acid
26 7 14 Q Glutamine
56 17 24 G Glycine
11 5 33 H Histidine
53 14 24 K Lycine
11 6 16 M Methionine
21 13 24 P Proline
32 28 23 A Alanine
48 5 20 N Asparagine
35 21 26 E Glutamic acid
29 11 9 F Phenylalanine
43 14 24 T Threonine
8 0 5 W Tryptophan
37 6 26 Y Tyrosine
38 17 36 V Valine
22 25 17 R Arginine
1 2 8 C Cysteine 

4FPH_1|Chain A|Sialidase B|Streptococcus pneumoniae (1313)
>7SZB_1|Chains A, C|EKC/KEOPS complex subunit TP53RK|Homo sapiens (9606)
>7FQK_1|Chains A, B, C, D|Legumain|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 \(w=f(c)\)
4FPH , Knot 269 697 0.84 40 267 628 
MNKRGLYSKLGISVVGISLLMGVPTLIHANELNYGQLSISPIFQGGSYQLNNKSIDISSLLLDKLSGESQTVVMKFKADKPNSLQALFGLSNSKAGFKNNYFSIFMRDSGEIGVEIRDAQKGINYLFSRPASLWGKHKGQAVENTLVFVSDSKDKTYTMYVNGIEVFSETVDTFLPISNINGIDKATLGAVNREGKEHYLAKGSIDEISLFNKAISDQEVSTIPLSNPFQLIFQSGDSTQANYFRIPTLYTLSSGRVLSSIDARYGGTHDSKSKINIATSYSDDNGKTWSEPIFAMKFNDYEEQLVYWPRDNKLKNSQISGSASFIDSSIVEDKKSGKTILLADVMPAGIGNNNANKADSGFKEINGHYYLKLKKNGDNDFRYTVRENGVVYNETTNKPTNYTINDKYEVLEGGKSLTVEQYSVDFDSGSLRERHNGKQVPMNVFYKDSLFKVTPTNYIAMTTSQNRGESWEQFKLLPPFLGEKHNGTYLCPGQGLALKSSNRLIFATYTSGELTYLISDDSGQTWKKSSASIPFKNATAEAQMVELRDGVIRTFFRTTTGKIAYMTSRDSGETWSKVSYIDGIQQTSYGTQVSAIKYSQLIDGKEAVILSTPNSRSGRKGGQLVVGLVNKEDDSIDWKYHYDIDLPSYGYAYSAITELPNHHIGVLFEKYDSWSRNELHLSNVVQYIDLEINDLTK
7SZB , Knot 108 255 0.78 38 162 242 
GPMAAARATTPADGEEPAPEAEALAAARERSSRFLSGLELVKQGAEARVFRGRFQGRAAVIKHRFPKGYRHPALEARLGRRRTVQEARALLRCRRAGISAPVVFFVDYASNCLYMEEIEGSVTVRDYIQSTMETEKTPQGLSNLAKTIGQVLARMHDEDLIHGDLTTSNMLLKPPLEQLNIVLIDFGLSFISALPEDKGVDLYVLEKAFLSTHPNTETVFEAFLKSYSTSSKKARPVLKKLDEVRLRGRKRSMVG
7FQK , Knot 185 444 0.84 40 250 423 
MKLCILLAVVAFVGLSLGVPIDDPEDGGKHWVVIVAGSNGWYNYRHQADACHAYQIIHRNGIPDEQIVVMMYDDIAYSEDNPTPGIVINRPNGTDVYQGVPKDYTGEDVTPQNFLAVLRGDAEAVKGIGSGKVLKSGPQDHVFIYFTNHGSTGILVFPNEDLHVKDLNETIHYMYKHKMYRKMVFYIEACESGSMMNHLPDNINVYATTAANPRESSYACYYDEKRSTYLGDWYSVNWMEDSDVEDLTKETLHKQYHLVKSHTNTSHVMQYGQKTISTMKVMQFQGMKRKASSPVPLPPVTHLDLTPSPDVPLTIMKRKLMNTNDLEESRQLTEEIQRHLDARHLIEKSVRKIVSLLAASEAEVEQLLSERAPLTGHSCYPEALLHFRTHCFNWHSPTYEYALRHLYVLVNLCEKPYPLHRIKLSMDHVCLGHYVDHHHHHHHH

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 \(c\).

\(|P_{f(4FPH_1)}(2) \setminus P_{f(7SZB_1)}(2)|=134\), \(|P_{f(7SZB_1)}(2) \setminus P_{f(4FPH_1)}(2)|=29\). 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 1:1000110001110111101111110110100100101010111011000100001010011100101000011101010010010111110000111000010111000101110100100110011001101110001011000111100000000010101101100010011110010110010111100010000110101001011001100001001110011011100100001001011010010010110010100110000000101100000001001001111101000000110110000100001010101100011000001001111011111110001001001100101000101000100010001000111000000010000100000110110010100001010010100000100111011000011010100011100000010010010111111100001001011011110000011110000101001100001001000010111001010101101001110011000010110100000100100100101100000100101100001101001111001000010011011111100000010100000101100101001100110001111100000100001010011001010100100
Pair \(Z_2\) Length of longest common subsequence
4FPH_1,7SZB_1 163 4
4FPH_1,7FQK_1 139 4
7SZB_1,7FQK_1 164 4

Newick tree

 
[
	7SZB_1:85.44,
	[
		4FPH_1:69.5,7FQK_1:69.5
	]:15.94
]

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 \((0.85)(0.8)(\frac{952 }{\log_{20} 952}-\frac{255}{\log_{20}255})=189.\)
Status Protein1 Protein2 d d1/2
Query variables 4FPH_1 7SZB_1 237 160.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]