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Parikh vectors
5OLE_1 9FZC_1 1AND_1 Letter Amino acid
8 0 12 C Cysteine
3 6 11 Q Glutamine
1 4 5 H Histidine
8 10 19 L Leucine
3 5 4 F Phenylalanine
7 14 8 T Threonine
11 6 3 R Arginine
10 7 17 S Serine
6 10 20 V Valine
12 25 23 G Glycine
14 12 18 N Asparagine
7 10 12 D Aspartic acid
6 7 13 I Isoleucine
6 7 9 K Lycine
6 5 4 W Tryptophan
3 13 8 Y Tyrosine
12 13 15 A Alanine
2 5 2 M Methionine
2 8 12 P Proline
2 6 8 E Glutamic acid 

5OLE_1|Chain A|Lysozyme C|Gallus gallus (9031)
>9FZC_1|Chains A, B|Outer membrane protein A|Escherichia coli K-12 (83333)
>1AND_1|Chain A|ANIONIC TRYPSIN|Rattus rattus (10117)
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 \(w=f(c)\)
5OLE , Knot 66 129 0.82 40 104 127 
KVFGRCELAAAMKRHGLDNYRGYSLGNWVCAAKFESNFNTQATNRNTDGSTDYGILQINSRWWCNDGRTPGSRNLCNIPCSALLSSDITASVNCAKKIVSDGNGMNAWVAWRNRCKGTDVQAWIRGCRL
9FZC , Knot 84 173 0.83 38 127 165 
MSAPKDNTWYTGAKLGWSQYHDTGFINNNGPTHENQLGAGAFGGYQVNPYVGFEMGYDWLGRMPYKGSVENGAYKAQGVQLTAKLGYPITDDLDIYTRLGGMVWRADTKSNVYGKNHDTGVSPVFAGGVEYAITPEIATRLEYQWTNNIGDAHTIGTRPDNGMLSLGVSYRFA
1AND , Knot 104 223 0.84 40 158 216 
IVGGYTCQENSVPYQVSLNSGYHFCGGSLINDQWVVSAAHCYKSRIQVRLGEHNINVLEGNEQFVNAAKIIKHPNFDHKTLNNDIMLIKLSSPVKLNARVATVALPSSCAPAGTQCLISGWGNTLSSGVNEPDLLQCLDAPLLPQADCEASYPGKITDNMVCVGFLEGGKDSCQGDSGGPVVCNGELQGIVSWGYGCALPDNPGVYTKVCNYVDWIQDTIAAN

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 \(c\).

\(|P_{f(5OLE_1)}(2) \setminus P_{f(9FZC_1)}(2)|=68\), \(|P_{f(9FZC_1)}(2) \setminus P_{f(5OLE_1)}(2)|=91\). 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 1:011100011111000110000100110110110100010001000000100001110100011000100110001001100111000101010010011001011011111000001001011101001
Pair \(Z_2\) Length of longest common subsequence
5OLE_1,9FZC_1 159 3
5OLE_1,1AND_1 158 3
9FZC_1,1AND_1 151 3

Newick tree

 
[
	5OLE_1:80.46,
	[
		1AND_1:75.5,9FZC_1:75.5
	]:4.96
]

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 \((0.85)(0.8)(\frac{302 }{\log_{20} 302}-\frac{129}{\log_{20}129})=53.6\)
Status Protein1 Protein2 d d1/2
Query variables 5OLE_1 9FZC_1 70 61
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]