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Parikh vectors
5JCV_1 7XJM_1 4TLW_1 Letter Amino acid
11 18 32 N Asparagine
14 28 45 L Leucine
1 0 6 C Cysteine
6 9 38 Q Glutamine
13 23 34 E Glutamic acid
8 20 31 G Glycine
17 28 25 K Lycine
16 12 36 T Threonine
6 26 37 A Alanine
18 22 37 D Aspartic acid
15 18 26 Y Tyrosine
9 14 40 V Valine
4 12 11 M Methionine
4 5 25 H Histidine
12 22 27 I Isoleucine
3 21 33 P Proline
10 21 45 S Serine
1 9 17 W Tryptophan
6 5 30 R Arginine
9 13 36 F Phenylalanine 

5JCV_1|Chain A|Lmo2181 protein|Listeria monocytogenes serovar 1/2a (strain ATCC BAA-679 / EGD-e) (169963)
>7XJM_1|Chains A, B, C, D|Putrescine-binding periplasmic protein|Vibrio cholerae O1 biovar El Tor str. N16961 (243277)
>4TLW_1|Chain A|ADP-ribosylating toxin CARDS|Mycoplasma pneumoniae (272634)
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 \(w=f(c)\)
5JCV , Knot 84 183 0.79 40 121 173 
SNALRDLQKLNKDMVGWLTIIDTEIDYPILQSKDNDYYLHHNYKNEKARAGSIFKDYRNTNEFLDKNTIIYGHNMKDGSMFADLRKYLDKDFLVAHPTFSYESGLTNYEVEIFAVYETTTDFYYIETEFPETTDFEDYLQKVKQQSVYTSNVKVSGKDRIITLSTCDTEKDYEKGRMVIQGKL
7XJM , Knot 142 326 0.84 38 201 316 
AMAKDQELYFYNWSEYIPSEVLEDFTKETGIKVIYSTYESNESMYAKLKTQGAGYDLVVPSTYFVSKMRKEGMLQEIDHSKLSHFKDLDPNYLNKPFDPGNKFSIPYIWGATGIGINTDMLDKKSLKNWGDLWDAKWAGQLMLMDDAREVFHIALSKLGYSPNTTNPKEIKAAYRELKKLMPNVLVFNSDFPANPYLAGEVSLGMLWNGSAYMARQEGAPIQIIWPEKGTIFWMDSISIPAGAKNIEAAHKMIDFLLRPENAAKIALEIGYPTPVKTAHDLLPKEFANDPSIYPPQSVIDNGEWQDEVGEASVLYDEYFQKLKVND
4TLW , Knot 247 611 0.86 40 285 579 
MDHHHHHHHHASENLYFQGHMPNPVRFVYRVDLRSPEEIFEHGFSTLGDVRNFFEHILSTNFGRSYFISTSETPTAAIRFFGSWLREYVPEHPRRAYLYEIRADQHFYNARATGENLLDLMRQRQVVFDSGDREMAQMGIRALRTSFAYQREWFTDGPIAAANVRSAWLVDAVPVEPGHAHHPAGRVVETTRINEPEMHNPHYQELQTQANDQPWLPTPGIATPVHLSIPQAASVADVSEGTSASLSFACPDWSPPSSNGENPLDKCIAEKIDNYNLQSLPQYASSVKELEDTPVYLRGIKTQKTFMLQADPQNNNVFLVEVNPKQKSSFPQTIFFWDVYQRICLKDLTGAQISLSLTAFTTQYAGQLKVHLSVSAVNAVNQKWKMTPQDIAITQFRVSSELLGQTENGLFWNTKSGGSQHDLYVCPLKNPPSDLEELQIIVDECTTHAQFVTMRAASTFFVDVQLGWYWRGYYYTPQLSGWSYQMKTPDGQIFYDLKTSKIFFVQDNQNVFFLHNKLNKQTGYSWDWVEWLKHDMNEDKDENFKWYFSRDDLTIPSVEGLNFRHIRCYADNQQLKVIISGSRWGGWYSTYDKVESNVEDKILVKDGFDRF

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 \(c\).

\(|P_{f(5JCV_1)}(2) \setminus P_{f(7XJM_1)}(2)|=47\), \(|P_{f(7XJM_1)}(2) \setminus P_{f(5JCV_1)}(2)|=127\). 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 1:001100100100011111011000100111000000001000000001011011000000001100001101001001011101000100011110101000011000010111100000010010001100001000100100001000010101000110100000000000101110101
Pair \(Z_2\) Length of longest common subsequence
5JCV_1,7XJM_1 174 3
5JCV_1,4TLW_1 210 4
7XJM_1,4TLW_1 172 4

Newick tree

 
[
	5JCV_1:99.65,
	[
		7XJM_1:86,4TLW_1:86
	]:13.65
]

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 \((0.85)(0.8)(\frac{509 }{\log_{20} 509}-\frac{183}{\log_{20}183})=94.8\)
Status Protein1 Protein2 d d1/2
Query variables 5JCV_1 7XJM_1 121 94
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]