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Parikh vectors
5OEM_1 5OAL_1 1MMF_1 Letter Amino acid
19 12 21 P Proline
7 16 31 T Threonine
2 4 3 W Tryptophan
7 19 35 D Aspartic acid
17 11 42 E Glutamic acid
24 21 40 L Leucine
7 8 14 K Lycine
11 13 18 F Phenylalanine
11 23 43 V Valine
7 9 4 H Histidine
5 8 37 I Isoleucine
15 23 37 S Serine
6 10 23 N Asparagine
4 5 5 C Cysteine
13 12 23 Q Glutamine
4 12 15 Y Tyrosine
11 14 61 A Alanine
7 12 37 R Arginine
10 11 41 G Glycine
2 6 25 M Methionine 

5OEM_1|Chain A|Interferon regulatory factor 9|Mus musculus (10090)
>5OAL_1|Chains A, B, C, D, E, F, G, H, I, J, K, L, M, N, O|Soluble acetylcholine receptor|Aplysia californica (6500)
>1MMF_1|Chains A, D[auth L]|glycerol dehydrase alpha subunit|Klebsiella pneumoniae (573)
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 \(w=f(c)\)
5OEM , Knot 89 189 0.82 40 133 182 
EDPVFLEHQLPLNSDYSLLLTFIYGGRVVGKTQVHSLDCRLVAEASDSESSMEQVEFPKPDPLEPTQHLLNQLDRGVLVASNSRGLFVQRLCPIPISWNAPEAPPGPGPHLLPSNKCVELFKTTYFCRDLAQYFQGQGPPPKFQATLHFWEESPGSSHSQENLITVQMEQAFARHLLEKIPEEEKAALF
5OAL , Knot 112 249 0.82 40 175 238 
MLVSVYLALLVACVGQAHSQANLMRLKSDLFNRSPMYPGPTKDDPLTVTLGFFLQDIVKVDSSTNEVDLVYYERQRWKLNSLMWDPNEYGNITDFRTSAADIWTPDITAASSTRPVQVLSPQIAVVTHDGSVMFSPAQRLSFMCDPTGVDSEEGVTCAVKFESWVYSGFEIDLKTDTDQVDLSSYYASSKYEILSATQTRQVQHYSCCPEPYIDVNLVVKFRERRAGNGFFRNLFDENLYFQGHHHHHH
1MMF , Knot 219 555 0.83 40 244 514 
MKRSKRFAVLAQRPVNQDGLIGEWPEEGLIAMDSPFDPVSSVKVDNGLIVELDGKRRDQFDMIDRFIADYAINVERTEQAMRLEAVEIARMLVDIHVSREEIIAITTAITPAKAVEVMAQMNVVEMMMALQKMRARRTPSNQCHVTNLKDNPVQIAADAAEAGIRGFSEQETTVGIARYAPFNALALLVGSQCGRPGVLTQCSVEEATELELGMRGLTSYAETVSVYGTEAVFTDGDDTPWSKAFLASAYASRGLKMRYTSGTGSEALMGYSESKSMLYLESRCIFITKGAGVQGLQNGAVSCIGMTGAVPSGIRAVLAENLIASMLDLEVASANDQTFSHSDIRRTARTLMQMLPGTDFIFSGYSAVPNYDNMFAGSNFDAEDFDDYNILQRDLMVDGGLRPVTEAETIAIRQKAARAIQAVFRELGLPPIADEEVEAATYAHGSNEMPPRNVVEDLSAVEEMMKRNITGLDIVGALSRSGFEDIASNILNMLRQRVTGDYLQTSAILDRQFEVVSAVNDINDYQGPGTGYRISAERWAEIKNIPGVVQPDTIE

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 \(c\).

\(|P_{f(5OEM_1)}(2) \setminus P_{f(5OAL_1)}(2)|=71\), \(|P_{f(5OAL_1)}(2) \setminus P_{f(5OEM_1)}(2)|=113\). 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 1:001111000111000001110110110111000100100011101000000100101101011010001100100111110000111100101111010110111111101110000101100001000110010101111010101011000110000000110101001110011001100001111
Pair \(Z_2\) Length of longest common subsequence
5OEM_1,5OAL_1 184 4
5OEM_1,1MMF_1 201 4
5OAL_1,1MMF_1 173 5

Newick tree

 
[
	5OEM_1:99.40,
	[
		5OAL_1:86.5,1MMF_1:86.5
	]:12.90
]

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 \((0.85)(0.8)(\frac{438 }{\log_{20} 438}-\frac{189}{\log_{20}189})=73.2\)
Status Protein1 Protein2 d d1/2
Query variables 5OEM_1 5OAL_1 93 81
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]