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Parikh vectors
3HDP_1 7OJB_1 5SXT_1 Letter Amino acid
7 0 14 Y Tyrosine
4 16 28 N Asparagine
1 0 2 C Cysteine
3 13 28 Q Glutamine
9 19 66 L Leucine
5 9 32 F Phenylalanine
4 17 41 T Threonine
0 0 23 W Tryptophan
7 18 49 D Aspartic acid
4 2 17 H Histidine
14 16 31 K Lycine
4 7 37 P Proline
9 18 35 S Serine
9 37 67 G Glycine
12 30 42 V Valine
7 38 100 A Alanine
5 8 43 R Arginine
13 23 41 E Glutamic acid
14 25 18 I Isoleucine
2 13 14 M Methionine 

3HDP_1|Chain A|Glyoxalase-I|Clostridium acetobutylicum (1488)
>7OJB_1|Chain A|Cell division protein FtsZ|Staphylococcus aureus (1280)
>5SXT_1|Chains A, B|Catalase-peroxidase|Burkholderia pseudomallei (strain 1710b) (320372)
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 \(w=f(c)\)
3HDP , Knot 65 133 0.79 38 99 129 
GSHMSLKVHHIGYAVKNIDSALKKFKRLGYVEESEVVRDEVRKVYIQFVINGGYRVELVAPDGEDSPINKTIKKGSTPYHICYEVEDIQKSIEEMSQIGYTLFKKAEIAPAIDNRKVAFLFSTDIGLIELLEK
7OJB , Knot 133 309 0.82 34 174 292 
GSHMATLKVIGVGGGGNNAVNRMIDHGMNNVEFIAINTDGQALNLSKAESKIQIGEKLTRGLGAGANPEIGKKAAEESREQIEDAIQGADMVFVTSGMGGGTGTGAAPVVAKIAKEMGALTVGVVTRPFSFEGRKKQTQAAAGVEAMKAAVDTLIVIPNDRLLDIVDKSTPMMEAFKEADNVLRQGVQGISDLIAVSGEVNLDFADVKTIMSNQGSALMGIGVSSGENRAVEAAKKAISSPLLETSIVGAQGVLMNITGGESLSLFEAQEAADIVQDAADEDVNMIFGTVINPELQDEIVVTVIATGFD
5SXT , Knot 278 728 0.84 40 274 656 
MSNEAKCPFHQAAGNGTSNRDWWPNQLDLSILHRHSSLSDPMGKDFNYAQAFEKLDLAAVKRDLHALMTTSQDWWPADFGHYGGLFIRMAWHSAGTYRTADGRGGAGEGQQRFAPLNSWPDNANLDKARRLLWPIKQKYGRAISWADLLILTGNVALESMGFKTFGFAGGRADTWEPEDVYWGSEKIWLELSGGPNSRYSGDRQLENPLAAVQMGLIYVNPEGPDGNPDPVAAARDIRDTFARMAMNDEETVALIAGGHTFGKTHGAGPASNVGAEPEAAGIEAQGLGWKSAYRTGKGADAITTGLEVTWTTTPTQWSHNFFENLFGYEWELTKSPAGAHQWVAKGADAVIPDAFDPSKKHRPTMLTTDLSLRFDPAYEKISRRFHENPEQFADAFARAWFKLTHRDMGPRARYLGPEVPAEVLLWQDPIPAVDHPLIDAADAAELKAKVLASGLTVSQLVSTAWAAASTFRGSDKRGGANGARIRLAPQKDWEANQPEQLAAVLETLEAIRTAFNGAQRGGKQVSLADLIVLAGCAGVEQAAKNAGHAVTVPFAPGRADASQEQTDVESMAVLEPVADGFRNYLKGKYRVPAEVLLVDKAQLLTLSAPEMTVLLGGLRVLGANVGQSRHGVFTAREQALTNDFFVNLLDMGTEWKPTAADADVFEGRDRATGELKWTGTRVDLVFGSHSQLRALAEVYGSADAQEKFVRDFVAVWNKVMNLDRFDLA

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 \(c\).

\(|P_{f(3HDP_1)}(2) \setminus P_{f(7OJB_1)}(2)|=35\), \(|P_{f(7OJB_1)}(2) \setminus P_{f(3HDP_1)}(2)|=110\). 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 1:1001010100110110010011001001101000011000100101011101100101111010001100010010010010001001000100100110011001011111000011111000111101100
Pair \(Z_2\) Length of longest common subsequence
3HDP_1,7OJB_1 145 4
3HDP_1,5SXT_1 219 3
7OJB_1,5SXT_1 152 4

Newick tree

 
[
	5SXT_1:10.45,
	[
		3HDP_1:72.5,7OJB_1:72.5
	]:27.95
]

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 \((0.85)(0.8)(\frac{442 }{\log_{20} 442}-\frac{133}{\log_{20}133})=92.4\)
Status Protein1 Protein2 d d1/2
Query variables 3HDP_1 7OJB_1 112 78.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]