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Parikh vectors
9AXC_1 8YOZ_1 5SMA_1 Letter Amino acid
8 4 23 C Cysteine
33 8 29 G Glycine
17 0 20 H Histidine
34 1 25 I Isoleucine
55 11 40 L Leucine
44 8 30 K Lycine
29 7 22 R Arginine
24 0 31 T Threonine
9 1 10 W Tryptophan
29 2 34 A Alanine
34 4 34 D Aspartic acid
20 3 14 M Methionine
22 4 30 F Phenylalanine
37 0 29 S Serine
24 3 25 Y Tyrosine
18 1 27 N Asparagine
25 2 16 Q Glutamine
35 13 15 E Glutamic acid
32 4 26 P Proline
34 6 43 V Valine 

9AXC_1|Chains A, C|GST26/CRAF chimera|Homo sapiens (9606)
>8YOZ_1|Chains A, B|Transcription factor zinc-finger domain-containing protein|Thermus thermophilus HB8 (300852)
>5SMA_1|Chain A[auth D]|Proofreading exoribonuclease nsp14|Severe acute respiratory syndrome coronavirus 2 (2697049)
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 \(w=f(c)\)
9AXC , Knot 231 563 0.86 40 286 539 
MSPILGYWKIKGLVQPTRLLLEYLEEKYEEHLYERDEGDKWRNKKFELGLEFPNLPYYIDGDVKLTQSMAIIRYIADKHNMLGGCPKERAEISMLEGAVLDIRYGVSRIAYSKDFETLKVDFLSKLPEMLKMFEDRLCHKTYLNGDHVTHPDFMLYDALDVVLYMDPMCLDAFPKLVCFKKRIEAIPQIDKYLKSSKYIAWPLQGWQATFGGGDHPPKSDSQPKTPVPAQRERAPVSGTQEKNKIRPRGQRDSSDDWEIEASEVMLSTRIGSGSFGTVYKGKWHGDVAVKILKVVDPTPEQFQAFRNEVAVLRKTRHVNILLFMGYMTKDNLAIVTQWCEGSSLYKHLHVQETKFQMFQLIDIARQTAQGMDYLHAKNIIHRDMKSNNIFLHEGLTVKIGDFGLATVKSRWSGSQQVEQPTGSVLWMAPEVIRMQDNNPFSFQSDVYSYGIVLYELMTGELPYSHINNRDQIIFMVGRGYASPDLSKLYKNCPKAMKRLVADCVKKVKEERPLFPQILSSIELLQHSLPKINRSASEPSLHRAAHTEDINACTLTTSPRLPVF
8YOZ , Knot 41 82 0.73 34 61 75 
MPLLLCPNCQVGMREVERRGVLIDVCPQCGGVWLDKGELEKLLAEAEEVERRYEEELEGFYRKEGKPYKRKKGFMKLFDLFD
5SMA , Knot 217 523 0.86 40 267 499 
SMLFKDCSKVITGLHPTQAPTHLSVDTKFKTEGLCVDIPGIPKDMTYRRLISMMGFKMNYQVNGYPNMFITREEAIRHVRAWIGFDVEGCHATREAVGTNLPLQLGFSTGVNLVAVPTGYVDTPNNTDFSRVSAKPPPGDQFKHLIPLMYKGLPWNVVRIKIVQMLSDTLKNLSDRVVFVLWAHGFELTSMKYFVKIGPERTCCLCDRRATCFSTASDTYACWHHSIGFDYVYNPFMIDVQQWGFTGNLQSNHDLYCQVHGNAHVASCDAIMTRCLAVHECFVKRVDWTIEYPIIGDELKINAACRKVQHMVVKAALLADKFPVLHDIGNPKAIKCVPQADVEWKFYDAQPCSDKAYKIEELFYSYATHSDKFTDGVCLFWNCNVDRYPANSIVCRFDTRVLSNLNLPGCDGGSLYVNKHAFHTPAFDKSAFVNLKQLPFFYYSDSPCESHGKQVVSDIDYVPLKSATCITRCNLGGAVCRHHANEYRLYLDAYNMMISAGFSLWVYKQFDTYNLWNTFTRLQ

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 \(c\).

\(|P_{f(9AXC_1)}(2) \setminus P_{f(8YOZ_1)}(2)|=238\), \(|P_{f(8YOZ_1)}(2) \setminus P_{f(9AXC_1)}(2)|=13\). 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 1:10111101010111010011100100000001000001001000010111011011001010101000111100110000111101000101011011110100110011000010010101100110110110001000001010010010111001101110101101011101101000101110100010000011111011010111100110000010011110000111010000001010100000001010100111000110101101001010101110110110101001011000111100000101111110100001111001001001000101000010110110110001011001010011000100001110011010110111101000101000100101011111101101000011010001000111100110101100010000011111101010101001000010110011100100100001111011001011000110100010010100110000101001000101111
Pair \(Z_2\) Length of longest common subsequence
9AXC_1,8YOZ_1 251 3
9AXC_1,5SMA_1 159 4
8YOZ_1,5SMA_1 244 3

Newick tree

 
[
	8YOZ_1:13.33,
	[
		9AXC_1:79.5,5SMA_1:79.5
	]:55.83
]

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 \((0.85)(0.8)(\frac{645 }{\log_{20} 645}-\frac{82}{\log_{20}82})=165.\)
Status Protein1 Protein2 d d1/2
Query variables 9AXC_1 8YOZ_1 212 119.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]