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Parikh vectors
3MOE_1 8CUF_1 8AUE_1 Letter Amino acid
26 0 18 D Aspartic acid
13 0 4 C Cysteine
52 0 20 E Glutamic acid
57 0 38 G Glycine
33 1 25 S Serine
23 0 19 T Threonine
27 0 18 N Asparagine
14 0 19 Y Tyrosine
33 0 28 V Valine
42 1 15 K Lycine
19 0 8 M Methionine
47 1 36 A Alanine
29 0 26 R Arginine
18 0 17 Q Glutamine
13 0 17 H Histidine
39 0 25 I Isoleucine
51 3 32 L Leucine
28 1 13 F Phenylalanine
44 0 21 P Proline
16 0 3 W Tryptophan 

3MOE_1|Chain A|Phosphoenolpyruvate carboxykinase, cytosolic [GTP]|Rattus norvegicus (10116)
>8CUF_1|Chains A, B|Synthetic epi-Novo29 (2R,3S)|Eleftheria (1597779)
>8AUE_1|Chains A, B|12-oxophytodienoate reductase 3|Solanum lycopersicum (4081)
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 \(w=f(c)\)
3MOE , Knot 255 624 0.87 40 281 580 
GSMPPQLHNGLDFSAKVIQGSLDSLPQEVRKFVEGNAQLCQPEYIHICDGSEEEYGRLLAHMQEEGVIRKLKKYDNCWLALTDPRDVARIESKTVIITQEQRDTVPIPKSGQSQLGRWMSEEDFEKAFNARFPGCMKGRTMYVIPFSMGPLGSPLAKIGIELTDSPYVVASMRIMTRMGTSVLEALGDGEFIKCLHSVGCPLPLKKPLVNNWACNPELTLIAHLPDRREIISFGSGYGGNSLLGKKCFALRIASRLAKEEGWLAEHMLILGITNPEGKKKYLAAAFPSACGKTNLAMMNPTLPGWKVECVGDDIAWMKFDAQGNLRAINPENGFFGVAPGTSVKTNPNAIKTIQKNTIFTNVAETSDGGVYWEGIDEPLAPGVTITSWKNKEWRPQDEEPCAHPNSRFCTPASQCPIIDPAWESPEGVPIEGIIFGGRRPAGVPLVYEALSWQHGVFVGAAMRSEATAGAEHKGKVIMHDPFAMRPFFGYNFGKYLAHWLSMAHRPAAKLPKIFHVNWFRKDKNGKFLWPGFGENSRVLEWMFGRIEGEDSAKLTPIGYVPKEDALNLKGLGDVNVEELFGISKEFWEKEVEEIDKYLEDQVNADLPYEIERELRALKQRISQM
8CUF , Knot 7 8 0.60 12 7 6 
FLKSXALL
8AUE , Knot 168 402 0.83 40 223 376 
HHHHHHMASSAQDGNNPLFSPYKMGKFNLSHRVVLAPMTRCRALNNIPQAALGEYYEQRATAGGFLITEGTMISPTSAGFPHVPGIFTKEQVREWKKIVDVVHAKGAVIFCQLWHVGRASHEVYQPAGAAPISSTEKPISNRWRILMPDGTHGIYPKPRAIGTYEISQVVEDYRRSALNAIEAGFDGIEIHGAHGYLIDQFLKDGINDRTDEYGGSLANRCKFITQVVQAVVSAIGADRVGVRVSPAIDHLDAMDSNPLSLGLAVVERLNKIQLHSGSKLAYLHVTQPRYVAYGQTEAGRLGSEEEEARLMRTLRNAYQGTFICSGGYTRELGIEAVAQGDADLVSYGRLFISNPDLVMRIKLNAPLNKYNRKTFYTQDPVVGYTDYPFLQGNGSNGPLSRL

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 \(c\).

\(|P_{f(3MOE_1)}(2) \setminus P_{f(8CUF_1)}(2)|=276\), \(|P_{f(8CUF_1)}(2) \setminus P_{f(3MOE_1)}(2)|=2\). 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 1:101110100110101011010100110010011010101001001010010000010111010001110010000001111001001101000011100000001111001000110110000100110101110101001011110111110111011101000101110101100110011011101011001001101111001110011001010111011000011011010110011100011101100110001111001111110010100001111110101000111101011110100110011110101010101101001111111100100010110010000110011000011101011001111110100100001010000101010001001100011101110010111101111110011111110011010011111111000101110001011100111101111001100110110110011101101101011000001011111110000110111101010001010111011000110101110101001111000110001001000100010101100100010110001001
Pair \(Z_2\) Length of longest common subsequence
3MOE_1,8CUF_1 278 2
3MOE_1,8AUE_1 164 5
8CUF_1,8AUE_1 224 3

Newick tree

 
[
	8CUF_1:13.84,
	[
		3MOE_1:82,8AUE_1:82
	]:55.84
]

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 \((0.85)(0.8)(\frac{632 }{\log_{20} 632}-\frac{8}{\log_{20}8})=191.\)
Status Protein1 Protein2 d d1/2
Query variables 3MOE_1 8CUF_1 249 126
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]