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Parikh vectors
4JOA_1 5VNN_1 2GPV_1 Letter Amino acid
16 36 17 D Aspartic acid
15 29 20 K Lycine
23 55 13 S Serine
13 25 8 Y Tyrosine
18 48 20 A Alanine
11 16 1 C Cysteine
26 39 18 E Glutamic acid
33 75 34 L Leucine
13 26 11 M Methionine
17 42 16 V Valine
19 43 5 R Arginine
12 25 5 N Asparagine
27 50 7 G Glycine
22 42 13 I Isoleucine
15 44 8 T Threonine
4 10 2 W Tryptophan
13 54 10 Q Glutamine
6 17 7 H Histidine
14 38 5 F Phenylalanine
27 50 10 P Proline 

4JOA_1|Chain A|ALK tyrosine kinase receptor|Homo sapiens (9606)
>5VNN_1|Chain A|Protein transport protein Sec23A|Homo sapiens (9606)
>2GPV_1|Chains A, B, C, D, E, F|Estrogen-related receptor gamma|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 \(w=f(c)\)
4JOA , Knot 147 344 0.83 40 208 330 
GAMGSELQSPEYKLSKLRTSTIMTDYNPNYCFAGKTSSISDLKEVPRKNITLIRGLGHGAFGEVYEGQVSGMPNDPSPLQVAVKTLPEVCSEQDELDFLMEALIISKFNHQNIVRCIGVSLQSLPRFILLELMAGGDLKSFLRETRPRPSQPSSLAMLDLLHVARDIACGCQYLEENHFIHRDIAARNCLLTCPGPGRVAKIGDFGMARDIYRAGYYRKGGCAMLPVKWMPPEAFMEGIFTSKTDTWSFGVLLWEIFSLGYMPYPSKSNQEVLEFVTSGGRMDPPKNCPGPVYRIMTQCWQHQPEDRPNFAIILERIEYCTQDPDVINTALPIEYGPLVEEEEK
5VNN , Knot 296 764 0.85 40 315 707 
MTTYLEFIQQNEERDGVRFSWNVWPSSRLEATRMVVPVAALFTPLKERPDLPPIQYEPVLCSRTTCRAVLNPLCQVDYRAKLWACNFCYQRNQFPPSYAGISELNQPAELLPQFSSIEYVVLRGPQMPLIFLYVVDTCMEDEDLQALKESMQMSLSLLPPTALVGLITFGRMVQVHELGCEGISKSYVFRGTKDLSAKQLQEMLGLSKVPLTQATRGPQVQQPPPSNRFLQPVQKIDMNLTDLLGELQRDPWPVPQGKRPLRSSGVALSIAVGLLECTFPNTGARIMMFIGGPATQGPGMVVGDELKTPIRSWHDIDKDNAKYVKKGTKHFEALANRAATTGHVIDIYACALDQTGLLEMKCCPNLTGGYMVMGDSFNTSLFKQTFQRVFTKDMHGQFKMGFGGTLEIKTSREIKISGAIGPCVSLNSKGPCVSENEIGTGGTCQWKICGLSPTTTLAIYFEVVNQHNAPIPQGGRGAIQFVTQYQHSSGQRRIRVTTIARNWADAQTQIQNIAASFDQEAAAILMARLAIYRAETEEGPDVLRWLDRQLIRLCQKFGEYHKDDPSSFRFSETFSLYPQFMFHLRRSSFLQVFNNSPDESSYYRHHFMRQDLTQSLIMIQPILYAYSFSGPPEPVLLDSSSILADRILLMDTFFQILIYHGETIAQWRKSGYQDMPEYENFRHLLQAPVDDAQEILHSRFPMPRYIDTEHGGSQARFLLSKVNPSQTHNNMYAWGQESGAPILTDDVSLQVFMDHLKKLAVSSA
2GPV , Knot 105 230 0.82 40 153 223 
PAKKPYNKIVSHLLVAEPEKIYAMPDPTVPDSDIKALTTLCDLADRELVVIIGWAKHIPGFSTLSLADQMSLLQSAWMEILILGVVYRSLSFEDELVYADDYIMDEDQSKLAGLLDLNNAILQLVKKYKSMKLEKEEFVTLKAIALANSDSMHIEDVEAVQKLQDVLHEALQDYEAGQHMEDPRRAGKMLMTLPLLRQTSTKAVQHFYNIKLEGKVPMHKLFLEMLEAKV

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 \(c\).

\(|P_{f(4JOA_1)}(2) \setminus P_{f(5VNN_1)}(2)|=40\), \(|P_{f(5VNN_1)}(2) \setminus P_{f(4JOA_1)}(2)|=147\). 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 1:11110010010001001000011000010001110000100100110001011011101111010010101110010110111001101000000101110111100100001100111010011011110111110100110000101001001111011011001101000100001100011100011001111011011011110010011000011011111011110111011100000010111111011011011010000001101100110101100011110011000100010001011111001000000101100111100111100000
Pair \(Z_2\) Length of longest common subsequence
4JOA_1,5VNN_1 187 4
4JOA_1,2GPV_1 173 4
5VNN_1,2GPV_1 206 4

Newick tree

 
[
	5VNN_1:10.01,
	[
		4JOA_1:86.5,2GPV_1:86.5
	]:15.51
]

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 \((0.85)(0.8)(\frac{1108 }{\log_{20} 1108}-\frac{344}{\log_{20}344})=201.\)
Status Protein1 Protein2 d d1/2
Query variables 4JOA_1 5VNN_1 258 185
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]