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Parikh vectors
1AWV_1 8PVP_1 3CIG_1 Letter Amino acid
3 10 31 Q Glutamine
23 15 27 G Glycine
4 11 4 M Methionine
6 14 30 P Proline
8 17 72 S Serine
2 23 18 Y Tyrosine
14 23 37 K Lycine
15 13 38 F Phenylalanine
9 13 19 A Alanine
6 26 23 R Arginine
7 27 27 D Aspartic acid
4 2 10 C Cysteine
12 22 33 E Glutamic acid
4 16 25 H Histidine
9 15 73 N Asparagine
10 19 34 I Isoleucine
7 30 128 L Leucine
1 5 9 W Tryptophan
11 11 40 T Threonine
9 26 19 V Valine 

1AWV_1|Chains A, B, C, D, E, F|CYCLOPHILIN A|Homo sapiens (9606)
>8PVP_1|Chains A, B|Casein kinase II subunit alpha|Homo sapiens (9606)
>3CIG_1|Chain A|Toll-like receptor 3|Mus musculus (10090)
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 \(w=f(c)\)
1AWV , Knot 81 164 0.84 40 128 160 
VNPTVFFDIAVDGEPLGRVSFELFADKVPKTAENFRALSTGEKGFGYKGSCFHRIIPGFMCQGGDFTRHNGTGGKSIYGEKFEDENFILKHTGPGILSMANAGPNTNGSQFFICTAKTEWLDGKHVVFGKVKEGMNIVEAMERFGSRNGKTSKKITIADCGQLE
8PVP , Knot 151 338 0.86 40 199 323 
SMSGPVPSRARVYTDVNTHRPREYWDYESHVVEWGNQDDYQLVRKLGRGKYSEVFEAINITNNEKVVVKILKPVKKKKIKREIKILENLRGGPNIITLADIVKDPVSRTPALVFEHVNNTDFKQLYQTLTDYDIRFYMYEILKALDYCHSMGIMHRDVKPHNVMIDHEHRKLRLIDWGLAEFYHPGQEYNVRVASRYFKGPELLVDYQMYDYSLDMWSLGCMLASMIFRKEPFFHGHDNYDQLVRIAKVLGTEDLYDYIDKYNIELDPRFNDILGRHSRKRWERFVHSENQHLVSPEALDFLDKLLRYDHQSRLTAREAMEHPYFYTVVKDQARMGSS
3CIG , Knot 256 697 0.80 40 249 579 
QCTVRYNVADCSHLKLTHIPDDLPSNITVLNLTHNQLRRLPPTNFTRYSQLAILDAGFNSISKLEPELCQILPLLKVLNLQHNELSQISDQTFVFCTNLTELDLMSNSIHKIKSNPFKNQKNLIKLDLSHNGLSSTKLGTGVQLENLQELLLAKNKILALRSEELEFLGNSSLRKLDLSSNPLKEFSPGCFQTIGKLFALLLNNAQLNPHLTEKLCWELSNTSIQNLSLANNQLLATSESTFSGLKWTNLTQLDLSYNNLHDVGNGSFSYLPSLRYLSLEYNNIQRLSPRSFYGLSNLRYLSLKRAFTKQSVSLASHPNIDDFSFQWLKYLEYLNMDDNNIPSTKSNTFTGLVSLKYLSLSKTFTSLQTLTNETFVSLAHSPLLTLNLTKNHISKIANGTFSWLGQLRILDLGLNEIEQKLSGQEWRGLRNIFEIYLSYNKYLQLSTSSFALVPSLQRLMLRRVALKNVDISPSPFRPLRNLTILDLSNNNIANINEDLLEGLENLEILDFQHNNLARLWKRANPGGPVNFLKGLSHLHILNLESNGLDEIPVGVFKNLFELKSINLGLNNLNKLEPFIFDDQTSLRSLNLQKNLITSVEKDVFGPPFQNLNSLDMRFNPFDCTCESISWFVNWINQTHTNISELSTHYLCNTPHHYYGFPLKLFDTSSCKDSAPFENLYFQGHHHHHHWSHPQFEK

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 \(c\).

\(|P_{f(1AWV_1)}(2) \setminus P_{f(8PVP_1)}(2)|=58\), \(|P_{f(8PVP_1)}(2) \setminus P_{f(1AWV_1)}(2)|=129\). 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 1:10101110111010111010101110011001001011001001110010010011111100110100001011001010010000111000111110110111000100111001000110100111101001101101100110001000001011001010
Pair \(Z_2\) Length of longest common subsequence
1AWV_1,8PVP_1 187 4
1AWV_1,3CIG_1 195 3
8PVP_1,3CIG_1 186 4

Newick tree

 
[
	1AWV_1:96.34,
	[
		8PVP_1:93,3CIG_1:93
	]:3.34
]

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 \((0.85)(0.8)(\frac{502 }{\log_{20} 502}-\frac{164}{\log_{20}164})=98.9\)
Status Protein1 Protein2 d d1/2
Query variables 1AWV_1 8PVP_1 128 94
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]