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Parikh vectors
4GNQ_1 2FFG_1 8WOY_1 Letter Amino acid
44 3 42 A Alanine
26 2 30 D Aspartic acid
51 11 53 E Glutamic acid
51 7 55 L Leucine
44 3 26 P Proline
22 5 38 T Threonine
14 3 28 Y Tyrosine
27 5 30 N Asparagine
57 3 32 G Glycine
12 6 13 H Histidine
16 0 20 W Tryptophan
39 6 21 I Isoleucine
29 2 23 R Arginine
13 1 8 C Cysteine
18 7 31 Q Glutamine
40 5 33 K Lycine
19 4 21 M Methionine
28 6 27 F Phenylalanine
32 4 36 S Serine
32 4 29 V Valine 

4GNQ_1|Chain A|Phosphoenolpyruvate carboxykinase, cytosolic [GTP]|Rattus norvegicus (10116)
>2FFG_1|Chains A, B|ykuJ|Bacillus subtilis (1423)
>8WOY_1|Chain A|Angiotensin-converting enzyme|Oryctolagus cuniculus (9986)
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 \(w=f(c)\)
4GNQ , Knot 251 614 0.87 40 278 572 
MPPQLHNGLDFSAKVIQGSLDSLPQEVRKFVEGNAQLCQPEYIHICDGSEEEYGRLLAHMQEEGVIRKLKKYDNCWLALTDPRDVARIESKTVIITQEQRDTVPIPKSGQSQLGRWMSEEDFEKAFNARFPGCMKGRTMYVIPFSMGPLGSPLAKIGIELTDSPYVVASMRIMTRMGTSVLEALGDGEFIKCLHSVGCPLPLKKPLVNNWACNPELTLIAHLPDRREIISFGSGYGGNSLLGKKCFALRIASRLAKEEGWLAEHMLILGITNPEGKKKYLAAAFPSACGKTNLAMMNPTLPGWKVECVGDDIAWMKFDAQGNLRAINPENGFFGVAPGTSVKTNPNAIKTIQKNTIFTNVAETSDGGVYWEGIDEPLAPGVTITSWKNKEWRPQDEEPCAHPNSRFCTPASQCPIIDPAWESPEGVPIEGIIFGGRRPAGVPLVYEALSWQHGVFVGAAMRSEGGGIMHDPFAMRPFFGYNFGKYLAHWLSMAHRPAAKLPKIFHVNWFRKDKNGKFLWPGFGENSRVLEWMFGRIEGEDSAKLTPIGYVPKEDALNLKGLGDVNVEELFGISKEFWEKEVEEIDKYLEDQVNADLPYEIERELRALKQRISQM
2FFG , Knot 44 87 0.75 38 71 79 
MSQLMGIITRLQSLQETAEAANEPMQRYFEVNGEKICSVKYFEKNQTFELTVFQKGEKPNTYPFDNIDMVSIEIFELLQLEHHHHHH
8WOY , Knot 241 596 0.86 40 283 557 
STIEELAKTFLEKFNQEAEDLSYQSALASWDYNTNITEENVQKMNDAEAKWSAFYEEQSKLAKTYPSQEVQNLTVKRQLQALQQSGSSALSADKSKQLNTILSTMSTIYSTGKVCNQSNPQECFLLEPGLDEIMAKSTDYNERLWAWEGWRSVVGKQLRPLYEEYVVLKNEMARANNYEDYGDYWRADYEAEGADGYDYSRSQLIDDVERTFSEIKPLYEQLHAFVRTKLMDAYPSRISPTGCLPAHLLGDMWGRFWTNLYSLTVPFGQKPNIDVTDTMVNQGWDAERIFKEAEKFFVSVGLPSMTQGFWENSMLTEPGDGRKVVCHPTAWDLGKGDFRIKMCTKVTMDNFLTAHHEMGHIQYDMAYATQPFLLRNGANEGFHEAVGEIMSLSAATPEHLKSIGLLPYDFHEDNETEINFLLKQALTIVGTLPFTYMLEKWRWMVFKGEIPKEQWMQKWWEMKREIVGVVEPMPHDETYCDPAALFHVANDYSFIRYYTRTIYQFQFQEALCQAAQHEGPLHKCDISNSTEAGQKLLNMLRLGRSEPWTLALENVVGAKNMDVRPLLNYFEPLFTWLKEQNRNSFVGWSTEWTPYA

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 \(c\).

\(|P_{f(4GNQ_1)}(2) \setminus P_{f(2FFG_1)}(2)|=228\), \(|P_{f(2FFG_1)}(2) \setminus P_{f(4GNQ_1)}(2)|=21\). 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 1:11101001101010110101001100100110101010010010100100000101110100011100100000011110010011010000111000000011110010001101100001001101011101010010111101111101110111010001011101011001100110111010110010011011110011100110010101110110000110110101100111000111011001100011110011111100101000011111101010001111010111101001100111101010101011010011111111001000101100100001100110000111010110011111101001000010100001010100010011000111011100101111011111100111111100110100111111110001111100111101111001100110110110011101101101011000001011111110000110111101010001010111011000110101110101001111000110001001000100010101100100010110001001
Pair \(Z_2\) Length of longest common subsequence
4GNQ_1,2FFG_1 249 3
4GNQ_1,8WOY_1 147 3
2FFG_1,8WOY_1 230 3

Newick tree

 
[
	2FFG_1:13.71,
	[
		4GNQ_1:73.5,8WOY_1:73.5
	]:58.21
]

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 \((0.85)(0.8)(\frac{701 }{\log_{20} 701}-\frac{87}{\log_{20}87})=178.\)
Status Protein1 Protein2 d d1/2
Query variables 4GNQ_1 2FFG_1 233 131.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]