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Parikh vectors
4TXT_1 5KKA_1 1DWW_1 Letter Amino acid
19 8 23 R Arginine
43 12 19 D Aspartic acid
34 20 30 E Glutamic acid
57 36 26 G Glycine
47 17 19 S Serine
30 18 27 T Threonine
48 4 17 Y Tyrosine
64 35 26 A Alanine
20 14 21 Q Glutamine
32 0 13 W Tryptophan
29 34 21 V Valine
36 13 23 P Proline
3 3 9 C Cysteine
18 2 16 H Histidine
21 17 26 I Isoleucine
40 33 38 L Leucine
47 21 18 K Lycine
14 4 12 M Methionine
26 10 19 F Phenylalanine
28 11 17 N Asparagine 

4TXT_1|Chain A|Glycoside hydrolase family 48|Caldicellulosiruptor bescii DSM 6725 (521460)
>5KKA_1|Chains A, B|Malate dehydrogenase|Escherichia coli (strain K12) (83333)
>1DWW_1|Chains A, B|NITRIC OXIDE SYNTHASE|MUS MUSCULUS (10090)
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 \(w=f(c)\)
4TXT , Knot 262 656 0.86 40 286 609 
PTPSSTPSVLGEYGQRFMWLWNKIHDPANGYFNQDGIPYHSVETLICEAPDYGHLTTSEAFSYYVWLEAVYGKLTGDWSKFKTAWDTLEKYMIPSAEDQPMRSYDPNKPATYAGEWETPDKYPSPLEFNVPVGKDPLHNELVSTYGSTLMYGMHWLMDVDNWYGYGKRGDGVSRASFINTFQRGPEESVWETVPHPSWEEFKWGGPNGFLDLFIKDQNYSKQWRYTDAPDADARAIQATYWAKVWAKEQGKFNEISSYVAKAAKMGDYLRYAMFDKYFKPLGCQDKNAAGGTGYDSAHYLLSWYYAWGGALDGAWSWKIGSSHVHFGYQNPMAAWALANDSDMKPKSPNGASDWAKSLKRQIEFYRWLQSAEGAIAGGATNSWNGRYEKYPAGTATFYGMAYEPNPVYHDPGSNTWFGFQAWSMQRVAEYYYVTGDKDAGALLEKWVSWVKSVVKLNSDGTFAIPSTLDWSGQPDTWNGAYTGNSNLHVKVVDYGTDLGITASLANALLYYSAGTKKYGVFDEGAKNLAKELLDRMWKLYRDEKGLSAPEKRADYKRFFEQEVYIPAGWIGKMPNGDVIKSGVKFIDIRSKYKQDPDWPKLEAAYKSGQAPEFRYHRFWAQCDIAIANATYEILFGNQSSSVDKLAAALEHHHHHH
5KKA , Knot 132 312 0.81 38 168 290 
MKVAVLGAAGGIGQALALLLKTQLPSGSELSLYDIAPVTPGVAVDLSHIPTAVKIKGFSGEDATPALEGADVVLISAGVARKPGMDRSDLFNVNAGIVKNLVQQVAKTCPKACIGIITNPVNTTVAIAAEVLKKAGVYDKNKLFGVTTLDIIRSNTFVAELKGKQPGEVEVPVIGGHSGVTILPLLSQVPGVSFTEQEVADLTKRIQNAGTEVVEAKAGGGSATLSMGQAAARFGLSLVRALQGEQGVVECAYVEGDGQYARFFSQPLLLGKNGVEERKSIGTLSAFEQNALEGMLDTLKKDIALGEEFVNK
1DWW , Knot 180 420 0.86 40 251 408 
QYVRIKNWGSGEILHDTLHHKATSDFTCKSKSCLGSIMNPKSLTRGPRDKPTPLEELLPHAIEFINQYYGSFKEAKIEEHLARLEAVTKEIETTGTYQLTLDELIFATKMAWRNAPRCIGRIQWSNLQVFDARNCSTAQEMFQHICRHILYATNNGNIRSAITVFPQRSDGKHDFRLWNSQLIRYAGYQMPDGTIRGDAATLEFTQLCIDLGWKPRYGRFDVLPLVLQADGQDPEVFEIPPDLVLEVTMEHPKYEWFQELGLKWYALPAVANMLLEVGGLEFPACPFNGWYMGTEIGVRDFCDTQRYNILEEVGRRMGLETHTLASLWKDRAVTEINVAVLHSFQKQNVTIMDHHTASESFMKHMQNEYRARGGCPADWIWLVPPVSGSITPVFHQEMLNYVLSPFYYYQIEPWKTHIWQ

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 \(c\).

\(|P_{f(4TXT_1)}(2) \setminus P_{f(5KKA_1)}(2)|=146\), \(|P_{f(5KKA_1)}(2) \setminus P_{f(4TXT_1)}(2)|=28\). 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 1:10100010111001001111100100110101000111000100110011001010000110001110110101010100100110010001110100011000010011001101001000101101011110011000110001001101101110100101010010110010110010011000110011010100101111011101110000000010000110101011010011011100010100100011011011001001110001011100000111101000100110100111111011101011000101100011111111000010100101100110010001010011001011111110001010000011101010111001011000110001111011010011000010100011111001101100110100010111100101010100101100100010101100100111010110111000110000111001100110011001101000001101100010000110001011111110110101100110110100000001011010110001011010000111000111101000111100000100111110000000
Pair \(Z_2\) Length of longest common subsequence
4TXT_1,5KKA_1 174 4
4TXT_1,1DWW_1 161 4
5KKA_1,1DWW_1 181 4

Newick tree

 
[
	5KKA_1:91.35,
	[
		4TXT_1:80.5,1DWW_1:80.5
	]:10.85
]

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 \((0.85)(0.8)(\frac{968 }{\log_{20} 968}-\frac{312}{\log_{20}312})=176.\)
Status Protein1 Protein2 d d1/2
Query variables 4TXT_1 5KKA_1 225 162
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]