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Parikh vectors
7FQO_1 5NAF_1 3LTI_1 Letter Amino acid
23 20 20 K Lycine
14 13 14 F Phenylalanine
25 38 18 S Serine
13 28 12 T Threonine
13 29 15 A Alanine
29 29 31 L Leucine
14 13 4 Q Glutamine
11 7 7 M Methionine
6 13 1 W Tryptophan
17 25 17 V Valine
19 11 25 R Arginine
11 25 11 N Asparagine
11 21 5 H Histidine
18 20 24 I Isoleucine
30 15 27 E Glutamic acid
14 35 18 G Glycine
22 15 12 P Proline
9 5 9 Y Tyrosine
18 29 25 D Aspartic acid
4 11 1 C Cysteine 

7FQO_1|Chain A|Tyrosine-protein phosphatase non-receptor type 1|Homo sapiens (9606)
>5NAF_1|Chains A, B, C, D|F-box-like/WD repeat-containing protein TBL1XR1|Mus musculus (10090)
>3LTI_1|Chain A|DNA-directed RNA polymerase subunit beta|Escherichia coli (83333)
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 \(w=f(c)\)
7FQO , Knot 141 321 0.84 40 205 309 
MEMEKEFEQIDKSGSWAAIYQDIRHEASDFPSRVAKLPKNKNRNRYRDVSPFDHSRIKLHQEDNDYINASLIKMEEAQRSYILTQGPLPNTVGHFWEMVWEQKSRGVVMLNRVMEKGSLKCAQYWPQKEEKEMIFEDTNLKLTLISEDIKSYYTVRQLELENLTTQETREILHFHYTTWPDFGVPESPASFLNFLFKVRESGSLSPEHGPVVVHCSAGIGRSGTFCLADTCLLLMDKRKDPSSVDIKKVLLEMRKFRMGLIQTADQLRFSYLAVIEGAKFIMGDSSVQDQWKELSHEDLEPPPEHIPPPPRPPKRILEPHN
5NAF , Knot 162 402 0.80 40 211 366 
MGSSHHHHHHSSGLEVLFQGPEENGAHTIANNHTDMMEVDGDVEIPSNKAVVLRGHESEVFICAWNPVSDLLASGSGDSTARIWNLSENSTSGPTQLVLRHCIREGGQDVPSNKDVTSLDWNSEGTLLATGSYDGFARIWTKDGNLASTLGQHKGPIFALKWNKKGNFILSAGVDKTTIIWDAHTGEAKQQFPFHSAPALDVDWQSNNTFASCSTDMCIHVCKLGQDRPIKTFQGHTNEVNAIKWDPTGNLLASCSDDMTLKIWSMKQDNCVHDLQAHNKEIYTIKWSPTGPGTNNPNANLMLASASFDSTVRLWDVDRGICIHTLTKHQEPVYSVAFSPDGRYLASGSFDKCVHIWNTQTGALVHSYRGTGGIFEVCWNAAGDKVGASASDGSVCVLDLRK
3LTI , Knot 131 296 0.84 40 176 282 
GHPMSPGVFFDSDKGKTHSSGKVLYNARIIPYRGSWLDFEFDPKDNLFVRIDRRRKLPATIILRALNYTTEQILDLFFEKVIFEIRDNKLQMELVPERLRGETASFDIEANGKVYVEKGRRITARHIRQLEKDDVKLIEVPVEYIAGKVVAKDYIDESTGELICAANMELSLDLLAKLSQSGHKRIETLFTNDLDHGPYISETLRVDPTNDRLSALVEIYRMMRPGEPPTREAAESLFENLFFSEDRYDLSAVGRMKFNRSLLREEIEGSGILSKDDIIDVMKKLIDIRNGKGEVD

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 \(c\).

\(|P_{f(7FQO_1)}(2) \setminus P_{f(5NAF_1)}(2)|=83\), \(|P_{f(5NAF_1)}(2) \setminus P_{f(7FQO_1)}(2)|=89\). 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 1:101000100100010111100010001001100110110000000000101100001010000000101011010010000110011110011011011100000111110011001010010011000000111000010101100010000010010100100000001101000011011110011011011101000101010011111000111100101011000111100000100101001110100101111001001010011110110111100010001001000010111001111101100110100
Pair \(Z_2\) Length of longest common subsequence
7FQO_1,5NAF_1 172 3
7FQO_1,3LTI_1 145 4
5NAF_1,3LTI_1 163 4

Newick tree

 
[
	5NAF_1:87.21,
	[
		7FQO_1:72.5,3LTI_1:72.5
	]:14.71
]

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 \((0.85)(0.8)(\frac{723 }{\log_{20} 723}-\frac{321}{\log_{20}321})=110.\)
Status Protein1 Protein2 d d1/2
Query variables 7FQO_1 5NAF_1 142 128.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]