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Parikh vectors
1OUK_1 7VGI_1 5KZZ_1 Letter Amino acid
13 20 2 F Phenylalanine
17 28 10 P Proline
25 20 9 A Alanine
19 19 6 R Arginine
4 7 10 C Cysteine
13 10 1 H Histidine
19 21 8 K Lycine
12 9 2 M Methionine
5 6 3 W Tryptophan
24 14 8 E Glutamic acid
16 26 8 G Glycine
44 29 13 L Leucine
22 16 7 V Valine
14 29 3 N Asparagine
15 5 3 Q Glutamine
22 30 4 I Isoleucine
21 23 9 S Serine
19 22 9 T Threonine
15 16 4 Y Tyrosine
27 20 8 D Aspartic acid 

1OUK_1|Chain A|Mitogen-activated protein kinase 14|Homo sapiens (9606)
>7VGI_1|Chain A|Cell cycle control protein 50A|Homo sapiens (9606)
>5KZZ_1|Chain A|Smoothened|Xenopus laevis (8355)
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 \(w=f(c)\)
1OUK , Knot 162 366 0.87 40 225 352 
GSHMLEMSQERPTFYRQELNKTIWEVPERYQNLSPVGSGAYGSVCAAFDTKTGLRVAVKKLSRPFQSIIHAKRTYRELRLLKHMKHENVIGLLDVFTPARSLEEFNDVYLVTHLMGADLNNIVKCQKLTDDHVQFLIYQILRGLKYIHSADIIHRDLKPSNLAVNEDCELKILDFGLARHTDDEMTGYVATRWYRAPEIMLNWMHYNQTVDIWSVGCIMAELLTGRTLFPGTDHIDQLKLILRLVGTPGAELLKKISSESARNYIQSLTQMPKMNFANVFIGANPLAVDLLEKMLVLDSDKRITAAQALAHAYFAQYHDPDDEPVADPYDQSFESRDLLIDEWKSLTYDEVISFVPPPLDQEEMES
7VGI , Knot 162 370 0.86 40 226 354 
MASMAMNYNAKDEVDGGPPCAPGGTAKTRRPDNTAFKQQRLPAWQPILTAGTVLPIFFIIGLIFIPIGIGIFVTSNNIREIEIDYTGTEPSSPCNKCLSPDVTPCFCTINFTLEKSFEGNVFMYYGLSNFYQNHRRYVKSRDDSQLNGDSSALLNPSKECEPYRRNEDKPIAPCGAIANSMFNDTLELFLIGNDSYPIPIALKKKGIAWWTDKNVKFRNPPGGDNLEERFKGTTKPVNWLKPVYMLDSDPDNNGFINEDFIVWMRTAALPTFRKLYRLIERKSDLHPTLPAGRYSLNVTYNYPVHYFDGRKRMILSTISWMGGKNPFLGIAYIAVGSISFLLGVVLLVINHKYRNSSNTADITIHHHHHH
5KZZ , Knot 65 127 0.82 40 107 125 
AMADIGSDKCKKTTTCEPLKYNICLGSVLPYALTSTVLAEDSSSQDEVHDKLSLWSGLRNAPRCWDAIRPLLCAVYMPKCEGGKVELPSQGLCQTTRVPCAIVARERGWPDFLKCTTDYFPEGCPNE

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 \(c\).

\(|P_{f(1OUK_1)}(2) \setminus P_{f(7VGI_1)}(2)|=84\), \(|P_{f(7VGI_1)}(2) \setminus P_{f(1OUK_1)}(2)|=85\). 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 1:100110100001010000100011011000001011101101010111000011011100100110011010000001011001000011111011011001001001011001111010011000010000101110011011001001011000101001110000010110111100000010101100100110111011000001011011011101101001111000100101110111011101100100001000100100110101101111101111011001111000001011011101011000010001110100001000011100100100001101111110000100
Pair \(Z_2\) Length of longest common subsequence
1OUK_1,7VGI_1 169 4
1OUK_1,5KZZ_1 192 3
7VGI_1,5KZZ_1 197 3

Newick tree

 
[
	5KZZ_1:10.15,
	[
		1OUK_1:84.5,7VGI_1:84.5
	]:16.65
]

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 \((0.85)(0.8)(\frac{736 }{\log_{20} 736}-\frac{366}{\log_{20}366})=100.\)
Status Protein1 Protein2 d d1/2
Query variables 1OUK_1 7VGI_1 129 128.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]