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Parikh vectors
5KXX_1 5LPN_1 9DAK_1 Letter Amino acid
6 7 38 V Valine
11 9 41 R Arginine
8 2 10 C Cysteine
6 17 37 I Isoleucine
2 2 42 P Proline
6 2 26 W Tryptophan
3 11 43 F Phenylalanine
10 7 61 S Serine
7 14 44 T Threonine
7 13 50 D Aspartic acid
2 11 64 E Glutamic acid
1 4 25 H Histidine
6 18 61 K Lycine
2 5 27 M Methionine
3 4 37 Y Tyrosine
14 8 44 N Asparagine
12 14 69 G Glycine
8 15 84 L Leucine
12 10 53 A Alanine
3 4 35 Q Glutamine 

5KXX_1|Chain A|Lysozyme C|Gallus gallus (9031)
>5LPN_1|Chains A, C|Ras-related protein Rab-10|Homo sapiens (9606)
>9DAK_1|Chain A|ACE2|Pipistrellus nathusii (59473)
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 \(w=f(c)\)
5KXX , Knot 66 129 0.82 40 104 127 
KVFGRCELAAAMKRHGLDNYRGYSLGNWVCAAKFESNFNTQATNRNTDGSTDYGILQINSRWWCNDGRTPGSRNLCNIPCSALLSSDITASVNCAKKIVSDGNGMNAWVAWRNRCKGTDVQAWIRGCRL
5LPN , Knot 85 177 0.82 40 130 174 
GHMAKKTYDLLFKLLLIGDSGVGKTCVLFRFSDDAFNTTFISTIGIDFKIKTVELQGKKIKLQIWDTAGQERFHTITTSYYRGAMGIMLVYDITNGKSFENISKWLRNIDEHANEDVERMLLGNKCDMDDKRVVPKGKGEQIAREHGIRFFETSAKANINIEKAFLTLAEDILRKTP
9DAK , Knot 330 891 0.83 40 324 786 
MGGEGLRASPRRRPLLPLQPRGCPRGDGCLRGGRGRAGFGFWRVTGGSRASANHVHAFFFFLQLLGNVLVIVLSHHFGKEFATMPMGSLQPLATLYLLGMLVASVLAQSLTTEEKAREFLDKFNSEAENWSHESALASWDYNTNINDKNAQKMNEADSKWSAFYKEHSKLAQGFPLQEIQNSTIKLQLQILQQNGSSVLTAEKSKRLSTILTTMSTIYSTGKVCNPNNPQQCFTLSGLEDIMEKSKDYHQRLWIWEGWRSEVGKQLRPLYEEYVALKNEMARGNNYKDYGDYWRGDYETEGGDGYNYSRNHLIEDVDRIFLEIKPLYEQLHAYVRAKLMNAYPSRISPTGCLPAHLLGDMWGRFWTNLYNLTVPFEKKQNIDVTDTMKKQSWDAEKIFKEAEKFYLSVGLHNMTPEFWNNSMLTEPSDGRQVVCHPTAWDLGKNDFRIKMCTKVTMDDFLTAHHEMGHIQYDMAYAKQPYLLRNGANEGFHEAVGEIMSLSAATPKHLKDLGLLAQNYPEDYETEINFLLKQALNIVGTLPFTYMLEKWRWMVFEGKIPKEQWMEKWWEMKREIVGVVEPLPHDETYCDPASLFHVANDYSFIRYFTRTILEFQFQEALCQIANHTGPLHKCDISNSTEAGKQLKNMLELGKSKPWTFALEQIARTKEMDAKPLLNYFKPLFSWLKELNGNSVGWSADWSPYSEQSIKVRISLKSALGEKAYEWNDNEMYLFRSSVAYAMRVYFLKVKNETIPFRAEDVWVSDEKIRVSFKFFVTSPTNVSDIIPRSEVEDAIRMSRGRINDAFRLDDKTLEFLGIQPTLGPPYQPPVTGTGGGSWSHPQFEKGGGSGGGSGGSAWSHPQFEKGGGGSDYKDHDGDYKDHDIDYKDDDDKL

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 \(c\).

\(|P_{f(5KXX_1)}(2) \setminus P_{f(5LPN_1)}(2)|=69\), \(|P_{f(5LPN_1)}(2) \setminus P_{f(5KXX_1)}(2)|=95\). 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 1:011100011111000110000100110110110100010001000000100001110100011000100110001001100111000101010010011001011011111000001001011101001
Pair \(Z_2\) Length of longest common subsequence
5KXX_1,5LPN_1 164 3
5KXX_1,9DAK_1 256 3
5LPN_1,9DAK_1 220 5

Newick tree

 
[
	9DAK_1:12.41,
	[
		5KXX_1:82,5LPN_1:82
	]:47.41
]

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 \((0.85)(0.8)(\frac{306 }{\log_{20} 306}-\frac{129}{\log_{20}129})=54.8\)
Status Protein1 Protein2 d d1/2
Query variables 5KXX_1 5LPN_1 72 61
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]