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Parikh vectors
1OSM_1 5KCF_1 6XBL_1 Letter Amino acid
0 4 22 C Cysteine
20 9 19 Q Glutamine
16 14 23 K Lycine
3 3 15 W Tryptophan
27 15 59 A Alanine
35 13 28 D Aspartic acid
9 15 28 E Glutamic acid
9 13 30 I Isoleucine
19 7 34 F Phenylalanine
21 20 44 S Serine
23 11 39 T Threonine
27 9 22 N Asparagine
27 48 67 L Leucine
2 9 40 P Proline
28 4 19 Y Tyrosine
10 12 38 R Arginine
2 13 12 H Histidine
2 15 13 M Methionine
19 13 47 V Valine
43 10 53 G Glycine 

1OSM_1|Chains A, B, C|OMPK36|Klebsiella pneumoniae (573)
>5KCF_1|Chains A, B|Estrogen receptor|Homo sapiens (9606)
>6XBL_1|Chain A[auth R]|Smoothened homolog|Homo sapiens (9606)
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 \(w=f(c)\)
1OSM , Knot 145 342 0.82 38 165 318 
AEIYNKDGNKLDLYGKIDGLHYFSDDKDVDGDQTYMRLGVKGETQINDQLTGYGQWEYNVQANNTESSSDQAWTRLAFAGLKFGDAGSFDYGRNYGVVYDVTSWTDVLPEFGGDTYGSDNFLQSRANGVATYRNSDFFGLVDGLNFALQYQGKNGSVSGEGATNNGRGALKQNGDGFGTSVTYDIFDGISAGFAYANSKRTDDQNQLLLGEGDHAETYTGGLKYDANNIYLATQYTQTYNATRAGSLGFANKAQNFEVAAQYQFDFGLRPSVAYLQSKGKDLNGYGDQDILKYVDVGATYYFNKNMSTYVDYKINLLDDNSFTRSAGISTDDVVALGLVYQF
5KCF , Knot 117 257 0.84 40 165 246 
IKRSKKNSLALSLTADQMVSALLDAEPPILYSEYDPTRPFSEASMMGLLTNLADRELVHMINWAKRVPGFVDLTLHDQVHLLECAWLEILMIGLVWRSMEHPGKLLFAPNLLLDRNQGKCVEGMVEIFDMLLATSSRFRMMNLQGEEFVCLKSIILLNSGVYTFLSSTLKSLEEKDHIHRVLDKITDTLIHLMAKAGLTLQQQHQRLAQLLLILSHIRHMSNKGMEHLYSMKCKNVVPLSDLLLEMLDAHRLHAPTS
6XBL , Knot 251 652 0.83 40 290 603 
MAAARPARGPELPLLGLLLLLLLGDPGRGAASSGNATGPGPRSAGGSARRSAAVTGPPPPLSHCGRAAPCEPLRYNVCLGSVLPYGATSTLLAGDSDSQEEAHGKLVLWSGLRNAPRCWAVIQPLLCAVYMPKCENDRVELPSRTLCQATRGPCAIVERERGWPDFLRCTPDRFPEGCTNEVQNIKFNSSGQCEVPLVRTDNPKSWYEDVEGCGIQCQNPLFTEAEHQDMHSYIAAFGAVTGLCTLFTLATFVADWRNSNRYPAVILFYVNACFFVGSIGWLAQFMDGARREIVCRADGTMRLGEPTSNETLSCVIIFVIVYYALMAGVVWFVVLTYAWHTSFKALGTTYQPLSGKTSYFHLLTWSLPFVLTVAILAVAQVDGDSVSGICFVGYKNYRYRAGFVLAPIGLVLIVGGYFLIRGVMTLFSIKSNHPGLLSEKAASKINETMLRLGIFGFLAFGFVLITFSCHFYDFFNQAEWERSFRDYVLCQANVTIGLPTKQPIPDCEIKNRPSLLVEKINLFAMFGTGIAMSTWVWTKATLLIWRRTWCRLTGQSDDEPKRIKKSKMIAKAFSKRHELLQNPGQELSFSMHTVSHDGPVAGLAFDLNEPSADVSSAWAQHVTKMVARRGAILPQDISVTPVATDYKDDDDK

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 \(c\).

\(|P_{f(1OSM_1)}(2) \setminus P_{f(5KCF_1)}(2)|=88\), \(|P_{f(5KCF_1)}(2) \setminus P_{f(1OSM_1)}(2)|=88\). 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 1:101000010010101010110010000010100001011101000100010101010001010000000001100111111011011010010001110010010011101110001000110001011100000011111011011100010010101011000101110001011100100011011011110100000000001111010010000111000100101100000000100110111100100101110001011101011010001001010100011001011100010001000100010110000100011100001111111001
Pair \(Z_2\) Length of longest common subsequence
1OSM_1,5KCF_1 176 4
1OSM_1,6XBL_1 195 4
5KCF_1,6XBL_1 203 4

Newick tree

 
[
	6XBL_1:10.07,
	[
		1OSM_1:88,5KCF_1:88
	]:15.07
]

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 \((0.85)(0.8)(\frac{599 }{\log_{20} 599}-\frac{257}{\log_{20}257})=96.4\)
Status Protein1 Protein2 d d1/2
Query variables 1OSM_1 5KCF_1 123 109
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]