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Parikh vectors
4XHP_1 7ZHT_1 5MVJ_1 Letter Amino acid
20 23 10 Q Glutamine
90 40 7 E Glutamic acid
38 26 7 F Phenylalanine
22 31 4 R Arginine
48 22 5 N Asparagine
0 11 4 C Cysteine
74 37 16 G Glycine
6 13 2 H Histidine
68 46 12 L Leucine
26 20 12 Y Tyrosine
52 43 14 A Alanine
67 39 8 D Aspartic acid
66 40 5 I Isoleucine
78 38 13 K Lycine
30 8 4 M Methionine
58 32 32 S Serine
6 6 5 W Tryptophan
25 25 15 P Proline
40 26 24 T Threonine
52 36 22 V Valine 

4XHP_1|Chain A|ParM hybrid fusion protein|Bacillus thuringiensis serovar kurstaki str. YBT-1520 (570416)
>7ZHT_1|Chains A, B, C, D|Glucose-6-phosphate 1-dehydrogenase|Leishmania donovani (5661)
>5MVJ_1|Chains A, C, E, G[auth H]|antibody Fab heavy chain|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 \(w=f(c)\)
4XHP , Knot 183 866 0.47 38 217 410 
MVEQMLSKNMLLGGFDTGNIKAKISFLNEKGNIESFAIPTVIAEAPPAKIDLKSAPSKKNDYVNEKDEDIELLHVRIISNSLDGDARSRAWYVGAYAKDQEDRQEPTVDEMGKTEDKFSQKNKKLHLIPLFTSMAVAAARIGKEEVSVPFSGGMPIEDYKLRGEEQILEMLYGEHTVEFLDGTYEGKKIKITINDGTMNVEGVSSVLAILFDIVNGEIVEVEGMDAEIGESYAINDLGAGTSDNAFFEDGELNKKLSTNTDLGTNKYIDEILKNIKERFMENEILKSDDTDEIESPFKTREDFIQRLVMPEVEKMIEDDTYKPTFSVKWGDVKENVTDIVMDGMLKYAEDQKASLMKFWFKTNADKNIVVGGGVLFGYAGLRDLKEQDGFILPKNIQESAYFTSRSYLIANLLEQLNKEGVEAGSGSGSGSGSGSGSGSGSGSMVEQMLSKNMLLGGFDTGNIKAKISFLNEKGNIESFAIPTVIAEAPPAKIDLKSAPSKKNDYVNEKDEDIELLHVRIISNSLDGDARSRAWYVGAYAKDQEDRQEPTVDEMGKTEDKFSQKNKKLHLIPLFTSMAVAAARIGKEEVSVPFSGGMPIEDYKLRGEEQILEMLYGEHTVEFLDGTYEGKKIKITINDGTMNVEGVSSVLAILFDIVNGEIVEVEGMDAEIGESYAINDLGAGTSDNAFFEDGELNKKLSTNTDLGTNKYIDEILKNIKERFMENEILKSFMTDEIESPFKTREDFIQRLVMPEVEKMIEDDTYKPTFSVKWGPVKENVTDIVMDGMLKYAEDQKASLDKFWDKTNADKNIVVGGGVLFGYAGLRDLKEQDGFILPKNIQESAYFTSRSYLIANLLEQLNKEGVEA
7ZHT , Knot 230 562 0.86 40 279 537 
MSEEQSHADQDAYVADVDGILDVLRAQVLERKPDDIFQFISKSALSLQKDRGAESCDRINCKVKDEQKSRALTIIVFGASGDLAKKKTFPALFDLYCGGLLPPEVNIIGYARTKVDDVEKWKHETLMKYFSNLSERGCHAEDFLKHISYFCGAYDSVDDFKRLDAVIREKENAFKGPEKGGNRLFYLALPPSVFASVCESIHKGAMPQEVGGWVRVIIEKPFGRDTKSSAELSQALEPFFDESQLYRIDHYLGKEMVQNIITTRFANRIFSAVWNASNIACVQITFKETIGTEGRGGYFDNIGIIRDVMQNHLTQILALLAMEKPRSLDAECIRDEKVSVLKCIEPITKENCVLGQYTASADGSIPGYLEDVTVPEGSTCPTFAVMRLNINNDRWAGVPFILKAGKAVEQKYVAIRIQFRDEVHPYGEATQRNELVIRAQPSEAMYVKITTKVPGLSGDLRQTHQTELDLTYHTRYDVRLPDAYESLINDALLGNSTNFVRKDELDVAWRIFTPLLHQIDSGEIKPIPYQAGTRGPKEADEFIANNGFKHQKGYHWLPSNKL
5MVJ , Knot 99 221 0.80 40 146 204 
QVQLQQSGPELVKPGTSVKMPCKASGYIFTDYVISWVKQRTGQGLEWIGEIFPRSGSTYYNEKFKGKATLTADKSSNTAYMQLSSVTSEDSAVYFCARDYYGTSFAMDYWGQGTSVTVSSAKTTPPSVYPLAPGSAAQTNSMVTLGCLVKGYFPEPVTVTWNSGSLSSGVHTFPAVLQSDLYTLSSSVTVPSSTWPSETVTCNVAHPASSTKVDKKIVPRD

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 \(c\).

\(|P_{f(4XHP_1)}(2) \setminus P_{f(7ZHT_1)}(2)|=40\), \(|P_{f(7ZHT_1)}(2) \setminus P_{f(4XHP_1)}(2)|=102\). 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 1:11001100011111100101010101100010100111101110111101010011000000100000010110101100010101000110111010000000010100110000010000001011111001111110110001011101111100001010001101101000101101000100101010010101011001111110110101101011010110001100111100001110010100010000011000010011001000110001100000001001100000110011110100110000001010101101000100111011100100001011011100010001111111111011100100001111100100010100000111011001000110110101010101010101010110011000111111001010101011000101001111011101111010100110000001000000101101011000101010001101110100000000101001100000100000010111110011111101100010111011111000010100011011010001011010001001010100101010110011111101101011010110101100011001111000011100101000100000110000100110010001100011001100010011000001100111101001100000010101011110001001110111001000010100110000100011111111110111001000011111001000101000001110110010001101
Pair \(Z_2\) Length of longest common subsequence
4XHP_1,7ZHT_1 142 3
4XHP_1,5MVJ_1 177 3
7ZHT_1,5MVJ_1 207 4

Newick tree

 
[
	5MVJ_1:10.35,
	[
		4XHP_1:71,7ZHT_1:71
	]:32.35
]

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 \((0.85)(0.8)(\frac{1428 }{\log_{20} 1428}-\frac{562}{\log_{20}562})=219.\)
Status Protein1 Protein2 d d1/2
Query variables 4XHP_1 7ZHT_1 195 169.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]